SHOCKS (Do you want to identify them?) Preliminary draft. Please do not quote it yet A more complete draft will be available soon at http://www.eco.uc3m.es/jgonzalo.

Jes´ us Gonzalo ∗ Economics U. Carlos III de Madrid [email protected]

Oscar Mart´ınez Economics U. Rovira i Virgili [email protected]

01 March 2008 Abstract The purpose of this paper is to develop econometric tools to test what type of shocks are permanent and which ones are transitory in an univariate time series econometric model. These shocks are characterized according to their own properties (size, sign, etc) or according to the characteristics of the economy (expansion - recession). In this way the identification, opposite to the standard literature where at every time t there is always a permanent shock, is achieved by allowing for periods with no permanent shocks. In order to perform this analysis, we introduce a new set of threshold models: autoregressive threshold integrated moving average models (TARIMA). These are integrated models with a threshold non-linearity in the moving average part. The threshold structure allows for the existence of non-invertible as well as invertible regimes. The former regime corresponds to transitory shocks, while the latter to permanent ones. The paper fully develops the properties of the model (stationarity-ergodicity, invertibility and its impulse response function) and the asymptotic theory of LS (consistency and asymptotic normality). From these results we present a general two steps testing strategy for permanent and transitory shocks. The paper ends with an application to GNP where we analyse the long run effects of large versus small shocks, positive versus negative shocks, and expansion versus recession shocks.

Keywords: Asymmetries, Identification of Shocks, Moving Averaged Models, Nonlinearities, Permanent Shocks, Persistence, Threshold Models, Transitory Shocks. JEL: C22, C51 ∗

Corresponding Address: http://www.eco.uc3m.es/jgonzalo. Financial support DGCYT Grant (SEC010890) is gratefully acknowledged.

1

Introduction

What are the shocks that drive the economy? Where does the persistence present in most of the economic time series come from? Are there always permanent shocks at every t? Is the persistence of output shocks symmetric or asymmetric? Do negative oil price shocks have different effect on gasoline price than positive ones? What fraction of output variation is due to supply or demand shocks? All the attempts to answer these questions pass through the critical decision of shock identification plus being able to test if a given shock is permanent or transitory. Traditionally this has been done by decomposing the analyzed variable into unobserved permanent and transitory components (see Watson, 1986). Some examples are: (1) Beveridge and Nelson (1981) decomposition (B−N hereafter), where the permanent component is a random walk and there is perfect correlation between permanent and transitory shocks due to the fact that there is only one shock, the permanent one; (2) Unobserved component models with uncorrelated components (U C − 0) (see Harvey, 1985; Clark, 1987), where the permanent component is also assumed to be a random walk but its innovation is uncorrelated with the one in the transitory term. Note that not all the ARIMA models admit an U C − 0 decomposition. This problem can be solved eliminating the random walk constraint on the permanent component. In this case we have an extra identification problem that is overcome by imposing an ad-hoc smooth condition on the permanent term (see the “canonical” decomposition in Pierce, 1979). All these decompositions present two basic problems that are not solved in the literature: (i) at every time period t there is always a permanent shock, and (ii) none of the assumptions on permanent and transitory components are testable, and therefore we can not test whether a type of shocks is permanent or transitory. The approach proposed in this paper tries to solve the identification and testing issue without encountering these problems. Our proposal is based on two pillars. First, as in the above mentioned identification attempts we accept that there are permanent and transitory shocks, and secondly, shocks behave differently in the long-run according to their own characteristics (sign, size, etc.) or some characteristics of the economy (expansion-recession). The first premise, as is well known (see Quah (1992)) is not enough to identify the permanent and transitory shocks of a univariate time series. This paper shows that with our second assumption we can not only identify the permanent and transitory shocks of a single economic variable, but we can also test whether in fact these shocks are transitory or permanent. In order to implement our proposal we use a class of threshold models, the Autoregressive Threshold Integrated Moving Average models (TARIMA), (a particular case of these models was pre-

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viously introduced in the literature by Gonzalo and Mart´ınez, 2006, where a new non-linear permanent-transitory decomposition is obtained). These are models with a unit root in the autoregressive part to capture persistence, and a threshold design in the moving average side to allow for asymmetries. The threshold variable can represent any characteristic of the shocks (large or small, positive or negative, etc.) or of the economy (expansion or recession, inflation or deflation, etc.) we are interested on. This threshold design is able to capture the possibility that any of these characteristics may have different long run effects. By allowing the existence of a unit root in some of the moving average regime, we permit that the characteristics triggering those regimes have only a transitory effect. In principle, TARIMA models can deal with very general types of asymmetries. In this paper we divide them in two types: asymmetries caused by observable variables (expansionrecession, etc) and asymmetries caused by non-observable variables, in particular those generated by the characteristic (size, sign) of the model’ shocks. The identification of permanent and transitory shocks with our procedure, allows to construct an orthogonal nonlinear permanent-transitory decomposition of the original variable. We also obtain a non-linear B − N decomposition that while sharing part of the spirit of both, the standard linear B − N and the U C − 0 decompositions, its behavior lies in between. This is proposed and study for a particular case in Gonzalo and Martinez (2006). A complete analysis of a threshold permanent-transitory decomposition is developed in Gonzalo and Martinez (2007). Threshold moving average (TMA) models have already been considered in Wecker (1981) and in De Gooijer (1998), and very recently in Guay and Scaillet (2003) and Ling and Tong (2005). The former two works are centered on presenting the new TMA model and on analyzing some of the moment properties in detail. They both assume normality, the threshold parameter r to be known and equal to zero, and they do not present asymptotic results. These can be found in the latter two. Guay and Scaillet use indirect inference and focus on the shock sign case in an stationary model. Ling and Tong develop a test for threshold effect in a stationary self-exciting TMA model. Our paper is more general in the sense of admitting a reacher family of threshold variables, we admit non-invertible moving average regime and we present asymptotic results for LS estimation as well as for inference. The paper finishes with an application of the TARIMA model to GNP where three types of asymmetries are considered, asymmetries caused by shock sign, shock size and recession-expansion. It is concluded that the asymmetry are statistical significant in the last two and permanent and transitory shocks are found in the shock size case. The rest of the paper is structured as follows. Section 2 introduces the TARIMA model

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and analyzes in detail its properties (stationarity-ergodicity, invertibility and its impulse response function). Section 3 presents the consistence of the LS estimation. In section 4, the asymptotic theory results needed for testing the hypothesis of interests are developed. Section 5 shows the finite sample performance of the test. Section 6 presents the empirical application of the TARIMA models to GNP. Finally, Section 7 draws some concluding remarks. The appendix contains technical derivations and proofs of the results in the main text.

2

TARIMA Models

In its general form, the threshold autoregressive and threshold integrated moving average, TARIMA(n, m, p, 1, q), model is: n+1 X

[Φp,l (L)(1 − L)yt + µl ] 1 (zt ∈ Al ) =

l=1

m+1 X

[Θq,j (L)εt ] 1 (zt ∈ Bj ) ,

t = 0, ±1, ±2, . . .

j=1

(1) where Φp,l (L) is a p − th order lag polynomial with all its roots outside the unit circle, Θq,j (L) is a q − th order lag polynomial, 1(.) denotes the indicator function and zt ∈ 0. G.2 zt and xt are strictly stationary, ergodic and φ − mixing of size −a with a > 1. G.3 Φ0p (L) has all its roots outside the unit circle, and µ0 ∈ [−µ, µ] with µ < ∞. I.0 θi = 1 for some i = 1, 2. I.1 E [ εt / =t−1 ] = 0 and E [ εt (1 − θt )/ =t−1 ] = 0. hQ . i q−1 2 AX.0 ∃q < ∞ s.t. E i=0 θt−i =t−q = λ1 < 1. . n ³Q ´ o q−1 4 AX.1 θ0 ∈ Θθ = [−1 + δ, 1 + δ] × [−1 + δ, 1 − δ] × [r, r] s.t. E =t−q ≤ λ∗1 < 1 , i=0 |θ t−i | with q < ∞, δ > 0, and θt = (1 + δ)1(zt ≤ r) + (1 − δ)1(zt > r). ³ . ´ 2γ AX.2 supz0 ∈[r,r] E |εt−1 | z 0 − ν ≤ zt−1 ≤ z 0 + ν ≤ σ 2γ ε < ∞, with γ > 2, and, supz0 ∈[r,r] E ( 1 (z 0 − ν < zt < z 0 + ν)/ =t−1 ) ≤ νM, with M < ∞, and ν > 0. ³ ´ ¡ ¢ 2 AX.3 inf z0 ∈[r,r] E |εt−1 | 1 (z 0 − ν < zt−1 < z 0 + ν) ≥ min νσ 2ε , σ 2ε with σ ε2 > 0 £ ¤ 0 M (r) = E 1(zt−1 < r)Xt−1 Xt−1 >0 £ ¤ 0 M (r) = E 1(zt−1 > r)Xt−1 Xt−1 >0 1

This could be concluded by the Wold’s representation, but it is worth to note that in this case the shocks are the same in both series

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0 with Xt−1 = [xt−1 , ..., xt−p , 1, εt−1 ], and ν > 0.

ASZ.0 [a]

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If |θ1 | > |θ2 |, λ1 = [E (|θ (εt−1 ) |) + rhf ∗ ] < 1,

[b] If |θ2 | > |θ1 |, λ1 = |θ2 | + rhf ∗ < 1 where, h = |θ1 − θ2 |, and f ∗ = 2 maxε fε (ε) ASZ.1 For θ = (θ1 , θ2 , r) and h = |θ1 − θ2 | and f ∗ = 2 maxε fε (ε), [a] If |θ1 | > |θ2 |: θ0 ∈ Θθ = {θ ∈ (−1, 1] × (−1, 1) × s.t. λ1 (θ) = λ2 (θ) + 2rhf ∗ ≤ λ∗1 < 1}, where λ2 (θ) = |θ1 | p (r) + |θ2 | (1 − p (r)), p (r) ≥ supk P (|εt + k| < r) , r > 0 and r < M . [b] If |θ2 | > |θ1 |: n 2 θ0 ∈ Θθ = θ ∈ (−1, 1) × [r, r]

s.t.

o λ1 (θ) = |θ2 | + 2rhf ∗ ≤ λ∗1 < 1 , where r > 0

and r < M . ASG.0 λ1 = maxi |θi | + 12 |r|hf ∗ < 1 where h and f ∗ were defined on ASZ.0. n o 2 ASG.1 θ0 ∈ Θθ = θ ∈ (−1, 1) × [r, r] s.t. λ1 (θ) = maxi |θi | + |r|hf ∗ ≤ λ∗1 < 1 , where h, r, r and f ∗ were defined on ASZ.0. The general assumptions G.0 to G.2 are usual in threshold models. G.3 define the parametric space for the part of the model no related with the threshold. I.0 and I.1 are sufficient conditions to obtain transitory and permanent shocks in terms of the Impulse Response Function. I.0 implies that a unit root in one of the MA regimes is necessary. A sufficient condition for I.1 is that the prediction of εt on each regime is 0. εt has to be a martingale difference sequence in each regime. AX.0, ASZ.0 ([a] or [b]) and ASG.0 are used in the invertibility of the TARIMA-X, TARIMA-Shock-Size and TARIMA-ShockSign model respectively. AX.1-AX.3 are used in the proof of consistency and asymptotic normality for the TARIMA-X. AX.1 describes the parametric space. It is worth to note that only the MA regime of zt ≤ r can be non-invertible. However it is only a notational issue, given that it does no rule out the other possibility, since {zt ≤ r} ≡ {−zt ≥ −r}. So the only implications is that we have to know in which regime can be the transitory shock. AX.2 is a conditional moment bound on εt and zt , and AX.3 is a full rank condition in the sense of condition 1.8 of Hansen (2000). Finally ASZ.1 ([a] or [b]) and ASG.1 describes the parametric space for TARIMA-Shock-Size and TARIMA-Shock-Sign respectively. It is easy to prove that under G.0 to G.3, all parameters of these spaces satisfy the corresponding AX.2 and AX.3 assumptions. After describing the TARIMA models and the sufficient assumptions, this section focus on the main properties of TARIMA models, i.e. stationarity, invertibility and persistence (through the Impulse Response Function). 2

Maybe can be changed to kθ(b et )k2 < 1 when εt is normal, using that ε2 f (ε) decreasing with the variance

of ε

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2.1

Stationarity and Invertibility

In this section, we study the stationarity, ergodicity and invertibility of TARIMA models. We concentrate only on causal solutions of (3), taking as the analysed variable xt = (1 − L)yt . The assumptions which guarantee the stationarity and ergodicity of xt are in the following lemma. Lemma 1. Let yt be an TARIMA process as (3) and g(.) any measurable function on p, with p = max{p, j + 1} and Ip−1 the identity matrix of order {p − 1} × {p − 1}. b) when zt 6= g(xt−j ) for j = 0, 1, 2, ..; if the sequence {εt , zt } is strictly stationary and ergodic, then {xt } is strictly stationary and ergodic. Proof: See appendix. About invertibility, the non-linearity of TARIMA models disables the use of the classical definition of invertibility. Then, we use the general invertibility definition introduced by Granger and Andersen (1978) and later improved by Hallin (1980). Hallin proved that under nonlinearity with constant coefficients both definitions are equivalent. Clearly, the TARIMA models are within this class. The definition of Granger and Andersen is the following: Definition 1. (Granger and Andersen) The process xt = g(xt−1 , εt−1 , ..., xt−p , εt−p ) + εt is invertible if ¡ ¢ lim E vt2 = 0

t→∞

with vt = εt − εbt = εt − (xt − g(xt−1 , εbt−1 , ..., xt−p , εbt−p ))

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This definition guarantees the possibility to recover the shocks from the past of the series. Working with MA models, linear or non-linear, the forecast depends on the past innovations that are not observed. So we have to recover them from the observable series. In that sense, the invertibility is a necessary condition for prediction. Moreover we are interested in identify them, in some cases based on some of their properties (size, sign...), so recover them seems a necessary condition. Note that this definition of invertibility implies the convergence of vt to 0 in L2 whereas the Ling, Tong an Li (2007) definition implies a.s. convergence. The results about the invertibility property for TARIMA models are summarized in the following lemma: Lemma 2. Let yt be an TARIMA process as (3), ¡ ¢ a) if zt is observable, (TARIMA-X), under E ε20 < ∞ and AX.0 {yt } is invertible. b) if zt = |εt |, (TARIMA-Shock-Size), under G.0 and ASZ.0 a) or b), {yt } is invertible. c) if zt = εt , (TARIMA-Shock-Sign), under G.0 and ASG.0, {yt } is invertible. Proof: See appendix. Given that the definition of invertibility is not the same, the condition AX.0 can not be compared directly with the conditions of Ling, Tong and Li (2007) for invertibility. However, the condition AX.0 can be substituted by,

AX.0’ E (log(θt )) < 0 and

³Q

T 2 t=0 θt

´

v02 is an uniformly integrable sequence, where

v0 = ε0 − εb0 .

The second part of AX.0’,

³Q

T 2 t=0 θt

´

v02 is an uniformly integrable sequence, guarantees

that the convergence in probability implies L2 convergence. For the TARIMA-X model, the condition for invertibility is quite similar to the stationarity condition of TAR models. However, for TARIMA-Shock, if we study the assumption ASZ.0 or ASG.0 we can distinguish two different parts. In the first one, we have E (|θ(εt−1 )|) or maxi θi , that must be less than 1. This checks the case of overdifferentiability, specially when we allow a unit root in the MA. But non-invertibility is not only a problem of overdifferentiability, but a problem of discontinuity3 too. The TARIMA-X is linear in shocks, but TARIMA-Shock no. The second part, rhf ∗ , checks the discontinuity degree, measured as a product of the gap, rh, and its upper limit probability, f ∗ . ¯ ¯ ¯ t ¯ When the function is continuous, the ¯ ∂ε∂εt−1 ¯ determines the overdifferentiability, having to be less than 1. If there is linearity, it is |θ| < 1. When the model is not continuous on εt−1 , the previous condition is not enough. 3

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In the case of TARIMA-X models, we can obtain transitory shocks in any of both regimes, allowing the series to be invertible. In the TARIMA-Shock-Size models, Lemma 2 b) allows that θ1 = 1; this is the case of small shock transitory, and big shock permanent. Unfortunately, this assumption excludes the opposite case, big transitory shock. Finally, for TARIMA-Shock-Sign models, as we see in the next section, with constant mean it is not possible to obtain transitory shocks. Then, we simplify the conditions not allowing a unit root in any of both regimes. It is word to note that as in size case, when r is different from 0, it is necessary to bound the discontinuity to guarantee invertibility. It is easy to see that for r = 0 the process is continuous on εt−1 .

2.2

Impulse Response Function

One of the distinguishing feature of TARIMA process is that the effect of shocks can be transitory or permanent with positive probability. The only previous attempts are the Stochastic Permanent Break (STOPBREAK) model of Engle and Smith (1999) and the unobserved component model of Kuan, Huang and Tsay (2005). However, in the former case the probability of a transitory shock is zero. In the latter, we are not able to identify the issue of transitory or permanent effect and there is no asymptotic result for estimation and inference. To define the permanent effect of a shock we use the Impulse Response Function (IR). This function measures the effect of a perturbation at time t in the sample path of the ∞

series, {yt+k }k=0 . If this effect on yt+k does not vanish when k → ∞ we say that the shock is permanent. With the linear models the traditional IR function does not have any conceptual problem in the definitions of permanent and transitory shocks. However with nonlinear models three main aspects of the time series come up to determine the relationship of the IR function with the persistence definitions. These aspects are the history of the series at time t − 1, the shock εt and future shocks. To capture all these three new aspects we use the Generalized Impulse Response Function (GI) introduced and defined in Koop, Pesaran and Potter (1996) and in Potter (2000) as:

GI (k, εt , wt−1 ) = E [ yt+k | εt , wt−1 ] − E [ yt+k | wt−1 ]

f or k = 0, 1, 2, ..

(5)

where wt is the sample history until time t. According to this definition, a shock is persistent if the effect of including in the information set on the conditional expectation of yt+k given the past does not vanish when k → ∞.4 In the TARIMA-X case, although we are in an 4 So the GIRF measures the effect on the conditional expectation, not on the sample path. However it can be proved that if there is a noninvertible MA regime, the shocks of that regime are not in the future sample

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univariate analysis, when zt is not a function of yt , the persistence properties depends on two variables, εt and zt . As zt is observable in t, we propose the following generalization of the GI for vt = (εt , zt ): GI(k, vt , wt−1 ) = E [ yt+k | vt , wt−1 ] − E [ yt+k | wt−1 ]

f or k = 0, 1, 2, ..

This allows a correct interpretation of the GI function in a more general context. In any case, when zt is =t−1 -measurable it is not necessary this generalization. In the following lemma we establish the relation between a unit root in some of the MA regimes and the existence of transitory and permanent shocks. Lemma 3. Let yt be an TARIMA process as (3) which satisfies G.3. Then, Φp (L)−1 = Ψ(L) =

∞ X

ψj Lj ,

j=0

where ψ0 = 1 and ψ∞ = 0; under I.0 and I.1, the GI function of yt is given by ψk−1 εt i h GI (k, vt , wt−1 ) = (1 − θ) Pk−2 ψ + ψ k−1 εt j=0 j

if

θt = 1

if

θt = θ 6= 1.

Clearly, when θt = 1, which implies a non-invertible MA regime, the shock εt is transitory, its effect goes to zero when k → ∞ given that limk→∞ ψk−1 = 0. On the other hand, when θt 6= 1, the shock εt will be permanent. It is worthy to note that it is zt , although the threshold variable is zt−1 , what determines the persistent or transitory effect of εt . For example, in the particular case of zt = |εt |, it is the size of the own shock which determines its persistent or transitory effect. The case of TARIMA-Shock-Sign does not hold assumption I.1. It can be proved that E [ εt (1 − θt )/ =t−1 ] ≡ µE 6= 0, so the expected effect of εt , conditioned on t − 1, is different from zero. Then, this implies the effect of knowing εt on the conditional expectation will be always different from zero and permanent. In this case does no matter the existence of a noninvertible MA regime. To correct this effect, it will be enough to allow a threshold in the mean. This case can be studied easily following the proofs of this paper, although for space reason we do no include here. So for sign case, we can study here asymmetric responses, but not in terms of transitory and permanent responses. path of the series.

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2.3

3

Permanent-Transitory Decomposition

Estimation

In this section it is proved the main results about estimation distinguishing between the cases of observable and unobservable threshold variable. We propose the Conditional Least Squares (CLS) method introduced by Klimko and Nelson (1978) to estimate the parameters of TARIMA models, establishing the conditions under which these estimators are consistent and their rate of convergence. For simplicity, without lack of generality, we concentrate in the simple case of TARIMA(n,m,p,d,q), TARIMA(0,1,1,1,1) model:

(1 − ρL) ∆yt = (1 − ρL) xt = µ + εt − θt−1 εt−1

εt − θ1 εt−1 =µ+ ε −θ ε t

2 t−1

if zt−1 ≤ r

(6)

if zt−1 > r

with ∆ = (1 − L). The most general case, p > 1, follows straightforward from the proofs of this paper. A general parameter in Θ is denoted by φ = (φ1 , φ2 , φ3 , φ4 , φ5 ) = ¢ ¢ ¡ ¡ (µ, ρ, θ1 , θ2 , r) = (β, r) and the true parameter φ0 = φ01 , φ02 , φ03 , φ04 , φ05 = µ0 , ρ0 , θ10 , θ20 , r0 = ¡ 0 0¢ β , r . In general we denote slope parameters to the vector of parameters β, which can be the continuous parameters. The objective function is the sum of the square errors, that is, QT (φ) =

T X

e2t (φ)

t=1

with

et (φ) = (1 − ρL) xt − µ + θt−1 et−1 (φ) , θ1 if zt−1 ≤ r θt−1 = θ if z > r. 2

e0 = 0

t−1

³ ´ b b For a given r, the Least Square (LS) solution is β(r). Second, we minimize QT β(r), r , ³ ´ ³ ´ obtaining rb. The CLS estimator of φ0 is φbT = φb1 , φb2 , φb3 , φb4 , φb5 = µ b, ρb, θb1 , θb2 , rb = ³ ´ b r), rb is the global minimizer of QT (φ), that is, β(b φbT = arg min QT (φ) φ∈Θ

where Θ = {φ

s.t

θ ∈ Θθ , ρ ∈ (−1, 1), µ ∈ [−µ, µ]}.

In general, for Threshold Autoregressive (TAR) models, the rate of convergence for the CLSE of the parameters are T 1/2 except for r, the threshold parameter, which is

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T . That result is due to the kind of discontinuity of the objective function in r via the indicator function. In the case of TARIMA-X models, the same results of TAR models is obtained given that QT (φ) is continuous in all the parameter, except r, through the indicator function. For the TARIMA-Shock case, however, the threshold variable is not observable, so it must be estimated. Thus, all the parameters to estimate enter in the indicator function and present the same discontinuity of r. In this case, the objective function is discontinuous in all the parameters through the indicator function, so all of them are T −consistent. However, there is an exception to this result in the Shock case, the TARIMA-Shock-Sign with r = 0 known. In that case, it can be proved that the objective function is continuous in all the parameters to estimate. From the previous comments, in the TARIMA-X case, the rate of convergence of βbi is T 1/2 for i = 1, ..., 4 and T for rb, that is, the threshold parameter. This result is established in the following Theorem. Theorem 1. [Consistency for TARIMA-X] Under G.0 a),G.1, G.2, G.3 and AX.1¡ ¢ ¡ ¢ as AX.3, φbT −→ φ0 , βbi,T = βi0 + Op T −1/2 for i = 1, ..., 4 and rbT = r0 + Op T −1 . Proof: See appendix. This Theorem establishes the existence on Θ, defined through AX.1 and GX.3, of an unique asymptotic minimum in QT (φ) reached at φ0 , the consistence of the CLSE and the rate of convergence5 . The assumptions G.1 and AX.3 guarantee the identification of this minimum. In assumption G.2, it is assumed that xt and zt are φ − mixing. This assumption is made for simplicity of the proof. However, it can be relaxed to xt and zt be L2 − N ear Epoch Dependence of size −∞ on φ − mixing processes. For a complete description of this kind of processes see Billingsley(1968), McLeish (1975), Bierens (1983), and Davidson (1994) between others. For the case of TARIMA-Shock-Size, the results about the consistence and rate of convergence are in the following Theorem. Theorem 2. [Consistency for TARIMA-Shock-Size] Under G.0-G.3, and ASZ.1, ¢ ¡ as φbT −→ φ0 , and φbi,T = φ0i + Op T −1 for i = 1, ..., 5. Proof: See appendix. As in the previous case, this Theorem establishes the existence on Θ, defined through ASZ.1 and G.3, of a unique asymptotic minimum in the true values of the parameters and the convergence of the estimators to this minimum at a T rate. This Theorem prove the intuition developed previously where all the parameters share the same properties of r. 5

In the paper we define Op (T −α ) in the following way, XT is Op (T −α ) if for ∀ε > 0 ∃∆ε and an integer Tε ≥ 1 such that P [|T α XT | > ∆ε ] < 1 − ε for ∀T ≥ Tε .

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Finally, for the TARIMA-Shock-Sign case, these results are established in the following Theorem. Theorem 3. [Consistency for TARIMA-Shock-Sign] Under G.0-G.3, and ASG.1, as φbT −→ φ0 , and

¢ ¡ a) If it is known that r0 = 0, βbi,T = βi0 + Op T −1/2 for i = 1, ..., 4 ¡ ¢ b) If it is known that |r0 | > 0, φbi,T = φ0i + Op T −1 for i = 1, ..., 5 . Proof: See appendix. In the Sign case, we have two different results depending on the value of r0 . When r0 = 0, we are in the continuous threshold case, (see for example Chan and Tsay (1998) for the continuous SETAR case). As in that case, we need to assume the continuity of the objective function. Here, this implies to assume that r0 = 0, which is the only possible value to obtain the continuity. That is the pure sign asymmetric case. Under this assumption the estimators converge at a T 1/2 rate, given that in this case, the objective function is continuous in all the parameters. For the discontinuous case, |r0 | > 0, we obtain the same results that in the Size case.

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Inference

In this section we present the testing strategy for the main hypothesis in the three models and develop the necessary asymptotic results to make inference. Attending the properties of the models, there is a main property to test, the existence of asymmetries in terms of permanent and transitory shocks. We propose to test it in two step. In a first step, we develop the threshold effect test based on statistic W1 , defined later, which corresponds with the existence of asymmetries in the IRF. The null in this test, as usual, is the symmetry of the IRF. In the second step, if we reject the null of symmetry in the first step, so the existence of a threshold effect can be assumed, we test the existence of transitory shocks in a particular MA regime with positive probability. For that, we test the existence of a unit root in that regime through the statistic W2 , defined later. If we assume I.1, under the null of a unit root, the shocks in that regime are transitory, as we have proved by Lemma 3. As it is well known from the threshold literature, the threshold effect test is deeply affected by knowing r0 . If r0 is unknown, we are in the case of inference when a parameter is not identified under the null hypothesis. Following the literature we propose a supremum type test, see for example Andrews (1993), Andrews and Ploberger (1994), Davies (1987) and Hansen (1996) between others. In general, the result for r0 known case can be obtained from the unknown r0 case. On the other side, given the T −consistence of rb when a

15

threshold effect exists, the asymptotic distribution of the slope parameters is not affected by unknowing r0 . In this testing strategy we distinguish between the observable and non observable threshold variable case, focusing the results in the unknown r0 case which is the habitual case. Only for the TARIMA-Shock-Sign, the r0 = 0 known case is considered as a particular case. In the observable case the same results for TAR models of Chan (1993) are obtained, that is, the asymptotic distribution of the slope parameter estimators, βb = (b µ, ρb1 , ..., ρbp , θb1 , θb2 ) is Normal and independent of the threshold parameter estimator. In the shock case, the objective function is not continuous on the slope parameters, so we develop a two step estimation to obtain statistics with asymptotic Normal distributions independents of the threshold parameter. In the rest of the section, the theory for threshold effect and unit root in the MA regime tests is developed for each model, TARIMA-X and Shock, distinguishing the case of TARIMA-Shock-Sign when r0 = 0 is known as a particular case.

4.1

Hypothesis testing in the Observable case

As we say above, in a first step we test the existence of a threshold effect. In order to do that we rewrite the model of equation (6) as

xt = µ + ρxt−1 − θεt−1 − αεt−1 1(zt−1 ≤ r) + εt ,

(7)

where the redefined slope parameters β = (µ, ρ, θ, α) are estimated by CLS in a similar way described in the previous section. The main reason to change the specification of equation (6) to (7) is that the asymptotic theory is more easy given that on equation (7) only one regressor depends on r. Apart of this, nothing changes and it is easy to see that θ = θ1 and α = (θ2 − θ1 ). Through this new equation, the null hypothesis of symmetry is: H0 : α(r) = 0

∀r ∈ [r, r],

versus the alternative, Ha : α(r) 6= 0 for some r ∈ [r, r]. So, the proposed statistic for this test is based on

W1,T (r) =

Tα b2 (r) , Vb (b α (r))

16

where: b −1 (r, r) R2 Vb (b α (r)) = R40 K b (r, r) = σ K bε2 T −1

T X

R40 = (0, 0, 0, 1),

with

³ ´ ³ ´ b r κ0 β, br , κt β, t

t=1 ¯ ¡ ¢0 ∂et (φ) ¯¯ κ(φ? ) = κ1t (φ? ), κ2t (φ? ), κ3t (φ? ), κ4t (φ? ) = , ∂β ¯φ?

κ1t (φ) = −

j t Y X

θ,

κ3t (φ) =

j t Y X

j=0 k=1

κ2t (φ) = −

j t Y X

θet−j−1 (φ),

j=0 k=1

θxt−j−1

and κ4t (φ) =

j=0 k=1

j t Y X

θet−j−1 (φ)1 (zt−j−1 ≤ r) ,

j=0 k=1

with φ = (β, r). Given that under the null r is not identified, we propose to test H0 with ΓT = supr∈[r,r] W1,T (r), as it is suggested by Davies (1977, 1987) between others. To study the asymptotic behavior of this statistic under the null it is necessary to define the following processes. First, let s (r) be a gaussian process with mean zero and ¡ ¡ ¢ ¡ ¢¢ covariance function K (r1 , r2 ) = σε2 E κt β 0 , r1 κ0t β 0 , r2 . Second, the gaussian process S (r) = R40 K −1 (r, r) s (r), with mean zero and covariance function given by K (r1 , r2 ) = R40 K −1 (r1 , r1 ) K (r1 , r2 ) K −1 (r2 , r2 ) R4 . The following Theorem shows the asymptotic distribution of W1,T (r) =

Tα b2 (r) , b (b V α(r))

where

⇒ denote weak convergence with respect to the Skorohod metric. Theorem 4. Under H0 and assumptions G.0, G.2, G.3, and A.1-A.2, 0

(1) W1,T (r) ⇒ S (r) K

−1

(r, r) S (r), d

0

(2) ΓT = supr∈[r,r] W1,T (r) −→ supr∈[r,r] S (r) K

−1

(r, r) S (r).

Proof: See Appendix. The Theorem 4 (1) gives the asymptotic distribution of test statistic W1,T (r), which 0

is given by S (r) K

−1

(r, r) S (r). The result (2) follows straight forward given that the

supr∈[r,r] is a continuous functional on [r, r] to R. Since the null distribution of ΓT = supr∈[r,r] W1,T (r) depends, in general, upon the covariance function K (r1 , r2 ), critical values cannot be tabulated except in special cases. So to be able to carry out this test T

we propose a bootstrap approximation. Algorithm 1 generates bootstrap samples {yt? }t=1 from model (7) and calculate the bootstrap approximation of the null distribution of ΓT = supr∈[r,r] W1,T (r). Algorithm 1. (Model-based Bootstrap procedure) 1. l = 1. T

2. Generate {ε?t }t=1 resampling from

n oT εbt − εb

t=1

17

with εb = T −1

PT

bt . t=1 ε

T b ? + ε? . 3. Generate {yt? }t=1 from (1 − ρbL)(1 − L)yt? = µ b − θε t t−1 T f ? (r). 4. From model (7), with bootstrap sample {yt? }t=1 , obtain Γ?,l = supr∈[r,r] W 0

5. l = l + 1. Go to step 2 while l ≤ B. 6. Estimate the p-value, pv , from bootstrap approximation, B ¢ 1 X ¡ ?,l 1 Γ >Γ . pv = T l=1

The proof of the validity of this procedure is not within the scope of the paper, but it follows that of the bootstrap procedure proposed by Caner-Hansen (2001). When r0 is known we can use the statistic W1,T (r0 ). As it is easy to prove, for each r ∈ [r, r] the asymptotic distribution of W1,T (r) has a marginal chi-square distribution. So under the null, W1,T (r0 ) converges to a χ21 . The second step of the strategy, if we reject the null of no threshold effect, is to test the existence of a unit root in a MA regime in model (6). In this case, the null hypothesis is

H0 : θi = 1, versus the alternative, Ha : θi 6= 1 for i = 1, 2. For that we propose a Wald statistic

W2,T (b r) =

T (θbi − 1)2 . b −1 (b r)Ri σ bε2 R0 H i

where φb is the CLS estimator of φ on equation (7), so β = (µ, ρ, θ1 , θ2 ), and Ri s.t. Ri0 β = θi . To obtain the asymptotic distribution of the W2,T (b r) statistic, we use the following Theorem. Theorem 5. Under assumptions G.0-G.3 and A.1-A.3, ´ ³ ´ ³→ − d a) T 1/2 βb − β −→ N 0 , σε2 H −1 p

b r) −→ σε2 H b) σ bε2 H(b

18

with σ bε2

=T

−1

T X

b e2t (φ),

t=1

H = lim T −1 T →∞

T X

£ ¤ E ht (φ0 )h0t (φ0 ) ,

t=1

¡

¯? ∂et (φ) ¯¯ , ∂β ¯φ Ã j ! t X Y h3t (φ) = θt−k et−j−1 (φ)1(zt−j−1 ≤ r),

ht (φ? ) = h1t (φ? ), h2t (φ? ), h3t (φ? ), h4t (φ? ) h1t (φ) =

t X j=0

h2t (φ)

=

t X

Ã

Ã

j=0

and

b = T −1 H

j Y

! θt−k

k=1 j Y

,

=

j=0

! θt−k

¢0

xt−j−1 ,

h4t (φ)

k=1

=

j=0

k=1 T X

t X

Ã

j Y

! θt−k

et−j−1 (φ)1(zt−j−1 > r),

k=1

³ ´ ³ ´ ht φb h0t φb .

t=1

Proof: See Appendix. From the prove of this theorem it is easy to see that when r0 is known the asymptotic distribution is the same. So, as in TAR models, see Chan (1993) for example, the asymptotic distribution of βb is independent of the rb. In fact, when r0 is known the result of Theorem 5 holds without assuming G.1. So this Theorem in that case can be used to test the existence of a threshold effect. Under the null, from Theorem 5, W2,T (b r) converges to a χ21 , so we reject the null hypothesis at the level α when W2,T (b r) > χ21,α with χ21,α s.t. P [χ21 > χ21,α ] = α. If there is no evidence against the null, the shocks in the regime i are transitory.

4.2

Hypothesis testing in the TARIMA-Shock case

As in the observable case we start with the threshold effect test assuming that r0 is unknown, so excluding the TARIMA-Shock-Sign case with r0 = 0 known. The test follows the same steps of the observable case. First, similar to equation (7) consider:

xt = µ + ρxt−1 − θεt−1 − αεt−1 1(g(εt−1 ) < r) + εt ,

(8)

where g(εt ) = |εt | in the Size case, and g(εt ) = εt in the Sign case. The parameters β = (µ, ρ, θ, α) are estimated by CLS and the null hypothesis of symmetry is:

H0 : α(r) = 0

19

∀r ∈ [r, r],

versus the alternative, Ha : α(r) 6= 0 for some r. So, the proposed statistic for this test is based again on

W1,T (r) =

Tα b2 (r) . Vb (b α (r))

Under the null, the asymptotic distribution is given by the processes S(r) and the covariance function K(r1 , r2 ) defined in the observable case, although in this case:

κ1t (φ) = −

j t Y X

θ,

κ3t (φ) =

j=0 k=1

κ2t (φ) = −

j t Y X

j t Y X

θet−j−1 (φ),

j=0 k=1

θxt−j−1

and

κ4t (φ) =

j=0 k=1

j t Y X

θet−j−1 (φ)1 (g(εt−j−1 ) ≤ r) .

j=0 k=1

The following Theorem resume this result, where ⇒ denote weak convergence with respect to the Skorohod metric. Theorem 6. Under H0 and assumptions G.0, G.2, G.3, and ASZ.1 (for Size case) and ASG.1 (for Sign case), 0

(1) W1,T (r) ⇒ S (r) K

−1

(r, r) S (r), d

0

(2) ΓT = supr∈[r,r] W1,T (r) −→ supr∈[r,r] S (r) K

−1

(r, r) S (r).

Proof: See Appendix. Again, since the null distribution of ΓT = supr∈[r,r] W1,T (r) depends, in general, upon the covariance function K (r1 , r2 ), critical values cannot be tabulated except in special cases. So to be able to carry out this test we propose a bootstrap approximation based on the same Algorithm 1 of the observable case, taking into account that in step 4 the model follows equation (8). When r0 is known, the marginal distribution of W1,T (r) for a given r is a χ2 , so we can test the threshold effect as in the observable case. If the symmetry is rejected, we can test the existence of transitory shocks in a given regime. For that we study the asymptotic distribution of the estimators under G.1. At first, we exclude the Shock-Sign case when r0 = 0, which is studied as a particular case in the next section, so the Shock case is characterized by the same discontinuity of parameter r in all the parameters. That prevents us from obtaining the asymptotic normality of CLS estimator for β parameters. Besides their asymptotic distribution is very difficult to obtain and we recognize that it is beyond the scope of this paper. Therefore we propose an alternative way of obtaining an statistic and its asymptotic distribution to test the existence of a unit root in a MA regime. In a first step, the TARIMA-Shock model is

20

estimated by CLS as previously described in Section 3. In a second step, we consider εt and r0 as known and equal to the CLS estimates of the first step. Basically, this can be b be done because in the first step everything is T-consistent. In more detail, let ebt = et (φ) the errors of the CLS estimation of model (6). The second step estimators are the ordinary least square (OLS) estimators of the model: 0 b (b b (b b + ut (9) xt = µ + ρxt−1 + θ1 et−1 (φ)1 zt−1 > rb) + θ2 et−1 (φ)1 zt−1 < rb) + ut = Zt−1 (φ)β

¯ ¯ ¯ b¯ b in the Sign case. In matrix way, with zbt = ¯et (φ) ¯ in the Size case and zbt = et (φ) b + U, X = Z(φ)β ³ ´−1 b φ) b b with Z 0 (φ) = (Z1 (φ), ..., ZT −1 (φ)). Then, the OLS estimator of β is βeOLS = Z 0 (φ)Z( Z 0 (φ)X. The asymptotic distribution of this OLS estimator is showed in the following Theorem. Theorem 7. Under assumptions G.0-G.3 and ASZ.1, for Shock-Size, or ASG.1, for Sign case with |r0 | > 0, ³ ´ d a) T 1/2 βbOLS − β 0 → N (0, Ωz ) p b → b z (φ) b) Ω Ωz , i−1 h h ¢i−1 PT −1 PT −1 ¡ b t (φ) b b =σ b z (φ) . and Ω bε2 T −1 t=0 Zt0 (φ)Z with Ωz = σε2 limT →∞ T −1 t=0 E Zt0 (φ0 )Zt (φ0 )

Proof: See Appendix. Then, we can test the transitory small shock hypothesis:

H0 :θ1 = 1 Ha :θ1 6= 1, with θ1 in model (9) and g(εt ) = |εt |. The test statistic is, W2,T (r0 ) =

T (βb3 − 1)2 , b rb)R3 b z (β, R0 Ω 3

which under the null, from Theorem 7, converges to a χ21 . The same statistic with rb = r0 can be used given that the Theorem 7 is equally valid when rb is substituted by r0 . However, as we commented above the result is based on the T −consistence of all the estimators, so in both cases the assumption G.1 is needed.

21

Hypothesis testing in the TARIMA-Shock-Sign case with r0 = 0

4.2.1 known

There are two main reasons to deal with this case as a particular case. First, although it is a case of TARIMA-Shock model, the objective function is continuous in the parameters β, as in the TARIMA-X. Second, we have to impose that r0 = 0 is known. As in the SETAR model case, the knowledge of the continuity of the model has to be imposed. Only the recent paper of Gonzalo and Wolf (2005) allows to test the continuity of the SETAR model. This issue is under current investigation by the authors. To carry the testing strategy out for this case we use the following theorem which give the asymptotic distribution of the CLS estimators of β and can be used for the threshold effect test, given that G.1 is not assumed. Theorem 8. For TARIMA-Shock-Sign model, under G.0, G.2, G.3, ASG.1 and r0 = 0 ´ ³− ´ ³ → d (1) βb − β −→ N 0 , σε2 H −1 p

b −→ σε2 H (2) σ bε2 H with,

σ bε2 = T −1

T X

b e2t (φ),

t=1

H = lim T −1

T X

T →∞

£ ¡ ¢ ¡ ¢¤ E ht φ0 h0t φ0 ,

t=1

b = lim T −1 H

T X

T →∞

³ ´ ³ ´ b r0 h0 β, b r0 , ht β, t

t=1

¢0 ¡ ht (φ? ) = h1t (φ? ), h2t (φ? ), h3t (φ? ), h4t (φ? ) , h1t (φ) =

j t Y X

θt−k ,

h3t (φ) =

j=0 k=1

h2t (φ) =

j t Y X

j t Y X

θt−k et−j−1 (φ)1 (et−j−1 (φ) ≤ 0) ,

j=0 k=1

θt−k xt−j−1

and

h4t (φ) =

j=0 k=1

j t Y X

θt−k et−j−1 (φ)1 (et−j−1 (φ) > 0) .

j=0 k=1

Proof: See Appendix. The threshold effect test is based on the following null, H0 : θ1 = θ2 ⇔ R0 β = 0 for R0 = (0, 0, 1, −1), versus the alternative: Ha : θ1 6= θ2 ⇔ R0 β 6= 0,

22

and the proposed statistic is: W1 (r0 ) =

b2 T (R0 β) , b −1 R σ bε2 R0 H

(10)

which under the null, and using the Theorem 8, will converge to a χ21 . This statistic has been proposed by other authors, see for example Wecker (1981) and Elwood (1998), but nobody to the best of our knowledge has proved the asymptotic distribution. The second step, in the strategy testing, is not possible. As we see in section 2.2, in the TARIMA-Shock-Sign case the shocks coming from a noninvertible MA are not transitory given that assumption I.1 does not hold.

5

Finite Sample Performance

This section examines the performance of the proposed statistics in finite samples through Monte Carlo experiments. We focus on the threshold effect test when r0 is unknown which need a bootstrap approximation for the asymptotic distribution. However, given its widely use, the results for the TARIMA-Shock-Sign case when r0 = 0 is known are equally provided. In all the experiments we consider εt iid N (0, 1), a sample size T = 200 and significant levels α = 0.05 and 0.10. Due to the large computational requirements of the simulation design, the number of Bootstrap replications was set at B = 200 and M = 400 for Monte Carlo replications. We start the experiments with the TARIMA-X model. In order to examine the size and power of the proposed threshold effect test we consider the following simple TARIMA-X model: (1 − L)yt = xt = εt − θ1 εt−1 1(|xt−1 | ≤ r) − θ2 εt−1 1(|xt−1 | > r),

(11)

where we set µ = 0 and Φp (L) = 1 to reduce the computational burden. Under the null, θ1 = θ2 = θ. We report in Table 1 the actual size for θ = −0.5, 0 and 0.5.

Table 1: Finite Sample Size for Bootstrap Threshold Effect Test Observable α − level Sup W1,T Mean W1,T Exp W1,T

θ = −0.5 0.05 0.10 0.0450 0.0850 0.0525 0.1000 0.0450 0.0875

θ=0 0.05 0.10 0.0700 0.1350 0.0525 0.0950 0.0700 0.1375

θ = 0.5 0.05 0.10 0.0450 0.0975 0.0525 0.0875 0.0550 0.0875

Note: The DGP is the model (11) with θ1 = θ2 = θ. The test and the estimation is based on model (7) and Algorithm 1

The second experiment consider the TARIMA-Shock. We focus on the size case, con-

23

sidering the model:

(1 − L)yt = xt = εt − θ1 εt−1 1(|εt−1 | ≤ r) − θ2 εt−1 1(|εt−1 | > r).

(12)

As in previous case, to reduce the computational burden we set µ = 0 and Φp (L) = 1. The null is θ1 = θ2 = θ, an IMA(1,1) again, and the results about the size are listed in Table 2.

Table 2: Finite Sample Size for Bootstrap Threshold Effect Test Shock-Size α − level Sup W1,T Mean W1,T Exp W1,T

θ = −0.5 0.05 0.10 0.0425 0.0775 0.0400 0.0800 0.0450 0.0700

θ=0 0.05 0.10 0.0525 0.0975 0.0350 0.0775 0.0500 0.0950

θ = 0.5 0.05 0.10 0.0450 0.0800 0.0350 0.0775 0.0425 0.0800

Note: The DGP is the model (12) with θ1 = θ2 = θ. The test and the estimation is based on model (8) and Algorithm 1

6

Empirical Application

In this section we apply the TARIMA model to test the existence of asymmetries in the persistence of the shocks to US GNP. We start with the sign of the shock. The previous works of Beaudry and Koop (1993) and Pesaran and Potter (1997) conclude the existence of asymmetries in the persistence of the shock based on the sign of the shock. Both papers have opposite conclusions about which shocks are more persistent6 . On the other hand, the work of Elwood (1998) reject the existence of asymmetries in the persistence of the shocks produced by its sign. Finally we analyse the existence of asymmetries in the persistence of the shocks based on its own size. We consider, as in the paper of Pesaran and Potter, yt to be the 100log(RGN P ) such that xt = (1 − L)yt is the growth rate. The data corresponds to the Quarterly Seasonally Adjusted Real Gross National Product of the United States in Millions of chained 2000 Dollars for the period 1954:Q1 to 2006:Q1. As the series is the same for the three asymmetric cases studied here, the real GNP, in the Table 3 shows the estimation results of the Linear model, which correspond to the model under the null of symmetry. 6

In these cases all the shocks are permanent so the more persistent term has to be understood as having a bigger effect on the limit.

24

Table 3: Estimation results for ARIMA models φ1

φ2

0.481 (0.180) −0.405 (0.258) 0.283 (0.066)

0.286 (0.080)

ARIMA(0,1,1) ARIMA(1,1,1) ARIMA(2,1,1) ARIMA(1,1,0)

µ 0.823 (0.075) 0.417 (0.155) 0.902 (0.218) 0.579 (0.081)

θ −0.224 (0.068) 0.215 (0.204) 0.668 (0.265)

AIC/SC 2.603 2.636 2.566 2.614 2.570 2.635 2.564 2.597

Note: In brackets are the standard deviation. ARIMA model:(1 − φ1 L − φ2 L2 )(1 − L)yt = µ + (1 − θL)εt

6.1 6.1.1

Asymmetries in the GNP rate Asymmetries in the sign of the shock.TARIMA-Shock-Sign Case.

Now, given the previous results, we deal with the question about wether that asymmetric persistence behavior on the sign of the shock can be produced by the sign of the own shock. This question has been considered previously by Elwood (1998). For that he proposed a pure TAR and TMA model with the sign of the shock as threshold variable. He uses a Maximum Likelihood estimation method and a Quasi-likelihood ratio test, although no asymptotic theory was proved. So, as well as redefining the data, we answer the question also using the LS estimation method and the Wald type test proposed in the previous sections. The results are showed in Table 4. As in the case of Elwood we fail to reject the linearity when the source of the asymmetry is produced by the pure sign of the shock εt > 0.

Table 4: Estimation results for ARTIMA-Shock-Sign models φ1

φ2

ARTIMA-S-Sign (0,1,1,1) ARTIMA-S-Sign (1,1,1,1) ARTIMA-S-Sign (2,1,1,1)

0.469 (0.095) −0.387 (0.284)

0.283 (0.083)

µ 0.853 (0.094) 0.433 (0.093) 0.901 (0.227)

θ2 −0.161 (0.104) 0.218 (0.165) −0.623 (0.305)

θ1 −0.281 (0.112) 0.183 (0.092) −0.681 (0.296)

statistic 0.447 0.038 0.130

Note: In brackets are the standard deviation. ARTIMA-S-Sign model:(1 − φ1 L − φ2 L2 )(1 − L)yt = µ + 1 − θL)εt 1(εt ≤ 0) + 1 − θ2 L)εt 1(εt > 0)

25

AIC/SC 2.608 2.657 2.570 2.635 2.574 2.656

6.2

Asymmetries persistence on the size of the shock

Finally, as in Gonzalo and Martinez (2006), we test the existence of asymmetries in the persistence of the shocks due to its size and the validity of the identification of transitory shocks based on that criterion. However, in this case we apply the Wald type test proposed in the current paper to the extended real GNP data. The testing results for the null of symmetry are shown in Table 5.

Table 5: Testing results for ARTIMA-Shock-Size models ARTIMA-S-Size

φ1

φ2

(0,1,1,1) 0.525 (0.077) 0.301 (0.298)

(1,1,1,1) (2,1,1,1)

0.090 (0.117)

µ 0.831 (0.035) 0.404 (0.072) 0.492 (0.175)

θ2 −0.225 (0.061) 0.260 (0.093) 0.042 (0.304)

θ1 0.971 (0.567) 0.872 (0.353) 0.655 (0.452)

r

p-value

0.249

0.288

0.407

0.046

0.394

0.392

Note: In brackets are the standard deviation. ARTIMA-S-Size model:(1 − φ1 L − φ2 L2 )(1 − L)yt = µ + (1 − θ1 L)εt 1(|εt | ≤ r) + (1 − θ2 L)εt 1(|εt | > r).

The p − value has been obtained using the proposed Bootstrap method for B=500 and ri ∈ [b ε(0.25T ) , εb(0.75T ) ] for i = 1, ..., 50.

For the TARIMA-Shock-Size(1,1,1,1) we reject the null of linearity at a significant level of 5%, as in the previous work of Gonzalo and Martinez. In the Table 6 we can see the estimation results for the two step estimation method and the estimation under the null of θ2 = 1. Using the Theorem 7 we can test that θ2 = 1. The p − value of this test is 0.2919. Assuming that the shocks have a symmetric distribution and both assumptions, I.0 and I.1, holds we can not reject that the small shocks are transitory.

Table 6: Estimation results for ARTIMA-Shock-Size models ARTIMA-Shock-Size(1,1,1,1) First Step Second Step Restricted θ2 = 1

φ1 0.525 (0.077) 0.705 (0.191) 0.461 (0.049)

µ 0.404 (0.072) 0.205 (0.176) 0.455 (0.050)

θ2 0.260 (0.093) 0.436 (0.206) 0.216 (0.052)

θ1 0.872 (0.353) 1.476 (0.451) 1.000

r 0.407 0.407 0.407

AIC/SC 2.558 2.639 2.543 2.607 2.550 2.599

Note: In brackets are the standard deviation. ARTIMA-S-Size model:(1 − φ1 L − φ2 L2 )(1 − L)yt = µ + (1 − θL)εt 1(|εt | ≤ r) + (1 − θ2 L)εt 1(|εt | > r).

The number of parameters for the AIC and BC criterions is k = 3.

Using the estimation results of the estimation when θ2 = 1 is imposed, we obtain the new Permanent-Transitory (P-T) decompositions introduced in Gonzalo and Martinez

26

AIC/SC 2.586 2.651 2.558 2.639 2.569 2.667

(2006). In the Figure 2 we can see the logarithm of the real GNP times 100 with the NBER reference points. The shady bars cover the periods from peaks to troughs.

Figure 2: Real GNP and NBER reference points. 1954:04-2006:01 57III

69IV

100*Log(RGNP)

934

61I

80I 75I

90III 82IV

01IV

850

766

60II

73IV

58II

70IV

81III

01I

80III

91I

Time

Figure 3: Orthogonal P-T Decomposition and NBER reference points 57III

Permanent Component

934

69IV 61I

80I 75I

90III 82IV

01IV

850

60II

73IV

81III

01I

766 58II

70IV

80III

91I

Time

57III

69IV

80I

90III

0.3929

Transitory Component

61I

75I

82IV

01IV

0.0

60II

73IV

81III

01I

-0.6089 58II

70IV

80III Time

27

91I

The Figure 3 shows the orthogonal P-T Decomposition, which is given by: (1 − φL)(1 − L)ytP = µ + (1 − θ2 L)εt 1(|εt | > r) (1 − φL)ytT = (1 − θ1 L)εt 1(|εt | ≤ r). Figures 4 and 6 show the Beveridge and Nelson Decomposition for the TARIMA and ARIMA(1,1,1) models respectively, which is given by the following expressions: µ 1 − θ2 + εt 1(|εt | > r) 1−φ 1−φ θ2 − φ θ1 − φ (1 − φL)ytT = εt 1(|εt | > r) + εt 1(|εt | ≤ r), 1−φ 1−φ (1 − L)ytP =

for the TARIMA model, and 1−θ µ + εt 1−φ 1−φ θ−φ (1 − φL)ytT = εt , 1−φ (1 − L)ytP =

for the ARIMA(1,1,1) model. The permanent component is defined by a random walk. In the linear case, the permanent and transitory shocks are perfect correlated (they are the same). However in the nonlinear BN P-T decomposition, they are never perfectly correlated and the correlation depends on the value of θ1 , θ2 and φ. From the three previous figures we can conclude than in the orthogonal decomposition the permanent component is really smoother. For the Beveridge and Nelson decompositions, nonlinear and linear, the transitory component is contra-cyclical. For the TARIMA model, a deeper knowledge of the business cycle can be obtained focusing on the graph of permanent and transitory shocks. These are in Figure 7. Clearly the recession period are characterized by negative permanent shocks and the recovery from the crisis for positive permanent shocks. The transitory shocks has a main role in the expansion periods. Equally it can be observed the effect of the Great Moderation, a scaling down of the size in permanent shock (the bigger ones)7 . Finally, the Table 7 shows some results about the variance of the permanent and transitory shocks and components. We can see how in the TARIMA model the variance of the permanent shocks is less than half in the linear case. Spite of this, the increment of permanent component in B-N decompositions are the same in all the cases by definition. To finish with the application we estimate the GIRF for the estimated TARIMA-Shock7 It can be of interest to consider the threshold variable r depending on the conditional variance or almost an structural change in that parameter.

28

Figure 4: Nonlinear Beveridge and Nelson P-T Decomposition and NBER reference points 57III

Permanent Component

934

69IV 61I

80I 75I

90III 82IV

01IV

850

60II

73IV

81III

01I

766 58II

70IV

80III

91I

Time

57III

69IV

80I

90III

1.8459

Transitory Component

61I

75I

82IV

01IV

0.0

60II

73IV

81III

01I

-1.2898 58II

70IV

80III

91I

Time

Figure 5: Linear Beveridge and Nelson P-T Decomposition and NBER reference points 57III

Permanent Component

934

69IV 61I

80I 75I

90III 82IV

01IV

850

60II

73IV

81III

01I

766 58II

70IV

80III

91I

Time

57III

69IV

80I

90III

2.0285

Transitory Component

61I

75I

82IV

01IV

0.0

60II

73IV

81III

01I

-1.4066 58II

70IV

80III Time

29

91I

Figure 6: Permanent and Transitory shocks and NBER reference points 57III

69IV

80I

61I

90III

75I

82IV

01IV

Permanent Shocks

2

0

60II

73IV

81III

01I

-3 58II

70IV

80III

91I

Time

57III

69IV

80I

90III

0.3830

Transitory Shocks

61I

75I

82IV

01IV

0.0

60II

73IV

81III

01I

-0.4044 58II

70IV

80III

91I

Time

Figure 7: Median of the

GIRF (k,εt ,=t−1 ) GIRF (0,εt ,=t−1 ) (b)

10

20

1.0 0.6

30

0

10

20 Horizon

(c)

(d)

30

0

10

20

0.5 0.0 -0.5

Positive Shock Negative Shock

Positive Shock Negative Shock

-1.0

% Change in Level of RGNP

-0.5

1.0

0.5 1.0 1.5

Horizon

-1.5

% Change in Level of RGNP

0

Big Shock Small Shock

0.2

% Change in Level of RGNP

1.4 1.3 1.2 1.1

Positive Shock Negative Shock

1.0

% Change in Level of RGNP

1.4

(a)

30

0

Horizon

10

20 Horizon

(a)Unconditional. (b)Unconditional. (c)Big Shocks. (d)Small Shocks.

30

30

Figure 8: The GIRF (22, εt , =t−1 ) (b) 0 -1 -2 -4

-3

% Change in Level of RGNP

-1 -2 -3 -4 -5

% Change in Level of RGNP

0

(a)

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

-3.0

-2.5

-2.0

-1.5

Shock

Shock

(c)

(d)

-1.0

-0.5

0.0

0.0

0.5

1.0

1.5

2.0

1 0 -1 -2

% Change in Level of RGNP

0 -5 -10 -15

% Change in Level of RGNP

2

-3.0

2.5

0.0

0.5

Shock

1.0

1.5

2.0

2.5

Shock

(a)Negative Shocks. (b)Negative Shocks, without extremes. (c)Positives Shocks. (d)Positive Shocks, without extremes. Table 7: Variance results for P-T decompositions

Ortogonal P-T Non linear BN P-T Linear BN P-T, ARIMA(1,1,1) Linear BN P-T, ARIMA(1,1,0)

V (εt ) 0.7261 0.7261 0.7362 0.7425

V (εPt ) 0.7064 0.7064 1.6835 1.4444

V (εTt ) 0.0207 0.0207 0.1931 0.1157

V (∆ytP ) 0.8099 1.5219 1.7123 1.4750

V (ytT ) 0.0247 0.2221 0.2528 0.1255

Note: In the linear BN decomposition the variance of the permanent shock and the increment of perµ manent component has to be the same. The difference is due to the approximation to 1−φ .

Size, in the same way described in the TARIMA-X case. The result is shown in Figures 7 y 8. The GIRF for small shocks converge to zero so the small shocks are transitory. Moreover the size asymmetry is the only one that seen significative, given that there is no asymmetry in the sign. In Figure 8 it is shown the problems to estimate the GIRF for very small values of εt .

31

7

Conclusion

In time series econometric models shock issues play a key role, particularly identification of permanent and transitory shocks. In the literature this has been done in a very ad hoc way. In this paper we link and also explain the persistence of the shocks to some characteristic of them or of the economy. This is done by proposing a new type of threshold model. This model allows to test wether a given characteristic causes the different degrees of persistence. In this way in our GNP application we conclude that large shocks are persistent while small ones are transitory. Also we find that (in mean) shocks in recessions are slightly more persistent than shocks in expansions. Extensions and applications to a multivariate framework are under current research by the authors.

32

8

Appendix

8.1

Proofs

Proof of Lemma 1. From equation (3): εt − θ1 εt−1 if zt−1 ≤ r Φp (L)(1 − L)yt = Φp (L)xt = ut = µ + ε −θ ε if zt−1 > r, t 2 t−1 When zt = g(xt−j ), the strictly stationarity and ergodicity of xt can be proved in the same way of Theorem 4.1 of Ling, Tong and Li (2006), defining in this case: Φp (L)Sn (t) = µ +

εt

if n < max{p, j}

[−θ + ψ1(g (S 2 n−j−1 (t − j − 1)) ≤ r)] εt−1 + εt

if n ≥ max{p, j}

with ψ = (θ2 − θ1 ) and LSn (t) = Sn−1 (t − 1). If zt is different from g(xt−j ), from Theorem 1.3.3 of Taniguchi and Kakizawa (2000), it is enough to prove the strictly stationarity and ergodicity of ut given that G.3 holds. In this sense, the strictly stationarity and ergodicity of {εt , zt } it is enough for the strictly stationarity and ergodicity of ut , from Theorem 1.3.3 again, and then for xt . Proof of Lemma 2. For the case a), as zt is observable, we have Φp (L) xt = µ + εt − θt−1 (zt−1 ) εt−1 , εbt = Φp (L) xt − µ + θt−1 (zt−1 ) εbt−1 , vt = εbt − εt = θt−1 (zt−1 ) (b εt−1 − εt−1 ) . Then it is easy to see that vt2 =

t Y

2 θt−j (zt−j ) v02 .

j=1

The condition for invertibility of yt is lim E

t→∞

t Y

2 θt−j (zt−j ) v02 = 0.

j=1

33

If AX.0 holds, using the Iterated Law of Expectation (ILE), lim E

t→∞

t Y

2 θt−j (zt−j ) v02 = lim E E t→∞

j=1

≤

[ t−1 ] lim λ1 q t→∞

q Y

, 2 θt−j (zt−j )

=t−q−1

j=1

t Y

2 θt−j (zt−j ) v02

j=q+1

¡ ¢ 2q (1 + δ) E v02 = 0,

¡ ¢ ¡ ¢ where [x] is the integer part of x and given that E v02 < ∞. This is implied by E ε20 < ∞. For the case b), TARIMA-Shock-Size, from equation (3),

εbt = Φp (L) xt + µ + θ1 1 (|b εt−1 | ≤ r) εbt−1 + θ2 1 (|b εt−1 | > r) εbt−1 . Define the events

A1,t−1 = {|εt−1 | ≤ r, |b εt−1 | ≤ r}

A2,t−1 = {|εt−1 | > r, |b εt−1 | > r}

A3,t−1 = {|εt−1 | > r, |b εt−1 | ≤ r}

A4,t−1 = {|εt−1 | ≤ r, |b εt−1 | > r} .

(13)

Then, if |θ1 | > |θ2 | , we have E |vt | = E |b εt − εt | ≤ hrE [1 (A3,t−1 ) + 1 (A4,t−1 )]+E {[|θ1 | 1 (|εt−1 | ≤ r) + |θ2 | 1 (|εt−1 | > r)] |vt−1 |} . First we calculate E [1 (A3,t−1 ) + 1 (A4,t−1 )]. Using that vt−1 is =t−2 measurable and taking f ∗ = maxe 2f (e),

E [1 (A3,t−1 ) + 1 (A4,t−1 )] ≤ E {E [ 1 (−r ≤ εt−1 ≤ −r − vt−1 ) + 1 (r − vt−1 ≤ εt−1 ≤ r)| Ft−2 ]} + E {E [ 1 (−r − vt−1 ≤ εt−1 ≤ −r) + 1 (r ≤ εt−1 ≤ r − vt−1 )| Ft−2 ]} ¯ ( "Z #) Z r Z −r Z r−vt−1 ¯ −r−vt−1 ¯ =E E f (ε) ∂ε + f (ε) ∂ε + f (ε) ∂ε + f (ε) ∂ε¯ Ft−2 ¯ −r r−vt−1 −r−vt−1 r ≤ f ∗ E (|vt−1 |) , so, E |vt | ≤ [hrf ∗ + E(|θ (εt−1 )|)] E |vt−1 | = λt1 E |v0 | . ¡ 2 ¢ For invertibility we need the convergence of E vt−1 , then £ ¤ 2 vt2 ≤ h2 r2 [1 (A3,t−1 ) + 1 (A4,t−1 )] + θ12 1 (|εt−1 | ≤ r) + θ22 1 (|εt−1 | > r) vt−1 + + 2hr [1 (A4,t−1 ) + 1 (A3,t−1 )] [|θ1 | 1 (|εt−1 | > r) + |θ2 | 1 (|εt−1 | ≤ r)] |vt−1 | .

34

Taking λ1 = [hrf ∗ + E(|θ(εt−1 )|)] and K = h2 r2 f ∗ + 4hr, ¡ ¢ ¡ 2 ¢ ¡ 2 ¢ t−1 E vt2 ≤ KE |vt−1 | + E(|θ(εt−1 )|)E vt−1 ≤ Kλt−1 1 E |v0 | + λ1 E vt−1 ≤ tKλ1 E |v0 | . ¡ ¢ Then, by ASZ.0 a), λ1 < 1 and we obtain that limt→∞ E vt2 = 0. If |θ2 | > |θ1 |, using ASZ.0 b), we obtain |vt | ≤ hr [1 (A3,t−1 ) |b εt−1 | + 1 (A4,t−1 ) |εt−1 |] + |θ2 | , E |vt | ≤ λt1 E |v0 | . The rest of the proof is similar to the other case. Finally, for case c), in a similar way of case b),

vt = (b εt − εt ) = θ1 vt−1 1 (εt−1 ≤ r, εbt−1 ≤ r) + θ2 vt−1 1 (εt−1 > r, εbt−1 > r) + (θ2 εbt−1 − θ1 εt−1 ) 1 (εt−1 ≤ r, εbt−1 > r) + (θ1 εbt−1 − θ2 εt−1 ) 1 (εt−1 > r, εbt−1 ≤ r) . Using that

|θ2 εbt−1 − θ1 εt−1 | 1 (εt−1 ≤ r, εbt−1 > r) ≤ [maxi |θi | |vt−1 | + h|r|] 1 (εt−1 ≤ r, εbt−1 > r) |θ1 εbt−1 − θ2 εt−1 | 1 (εt−1 > r, εbt−1 ≤ r) ≤ [maxi |θi | |vt−1 | + h|r|] 1 (εt−1 > r, εbt−1 ≤ r) , it can be obtained, using the assumption ASG.0,that

|vt | ≤ max |θi | |vt−1 | + h|r| [1 (r ≤ εt−1 ≤ r − vt−1 ) + 1 (r − vt−1 ≤ εt−1 ≤ r)] i

1 kvt k1 ≤ max |θi | kvt−1 k1 + h|r| f ∗ kvt−1 k1 ≤ λ1 kvt−1 k1 . i 2 Now, following the same steps of the previous cases, we obtain the desired result, ¯ ¯ limt→∞ E ¯vt2 ¯ = 0. Proof of Lemma 3. Given that Φp (L) has all its roots outside the unit circle, Ψ (L) with

35

ψ0 = 1 and ψ∞ = 0 always exits. First we write (3) in the following way, (1 − L) yt+k = Ψ (1) µ + Ψ (L) (εt+k − θt+k−1 εt+k−1 ) yt+k = (t + k)Ψ (1) µ + y 0 + εt+k +

t+k−1 X

εt+k−s (1 − θt+k−s )

s=1

= (t + k)Ψ (1) µ + y 0 + εt+k + + εt (1 − θt )

k−1 X

ψj + ψk +

with y 0 = y0 +

t−1 X

s−1 X j=0

εt+k−s (1 − θt+k−s )

s=1

j=0

k−1 X

s−1 X

εt−s (1 − θt−s )

s=1

s−1+k X

ψj + ψs

ψj + ψs +

j=0

ψj + ψs+k ,

j=0

h Pt+k−1+s i ε ψ − θ ψ + (1 − θ ) ψj . Using I.1 and the −s t+k+s −s s+1 −s s=0 j=s+2

P∞

Law of Iterated Expectation (LIE), E ( yt+k | vt , wt−1 ) − E ( yt+k | wt−1 ) = (1 − θt )

k−1 X

ψj + ψk εt .

j=0

Then, using I.0, ψk ε t h i GI (k, vt , wt−1 ) = (1 − θ) Pk−1 ψ + ψ ε k t j=0 j

9

if

θt = 1

if

θt 6= 1.

References

Andrews, D.W.K., 1993. Tests for Parameter Instability and Structural Change with Unknown Change Point, Econometrica 61, 821-856. Andrews, D.W.K., and Ploberger, W. 1994. Optimal Tests When a Nuisance Parameter Is Present Only Under the Alternative. Econometrica, 62, 1383-1414. Beaudry, P., Koop, G., 1993. Do recessions permanently change output?. Journal of Monetary Economics 31, 149-164. Beveridge, S., Nelson, C.R., 1981. A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘Business Cycle’. Journal of Monetary Economics 7, 151-174. Bierens, H.J., 1983. Uniform consistency of kernel estimators of a regression function under generalized conditions. Journal of the American Statistical Association 77, 699-707. Billingsley, P. 1968. Convergence of Probability Measures. John Wiley & Sons, New York.

36

Caner, M., Hansen, B.E., 2001. Threshold autoregression with a unit root. Econometrica, 69, 1555-1596. Chan, K.S., 1993. Consistency and limiting distribution of the least squares estimator of a Threshold Autoregressive model. The Annals of Statistics 21, 520-533. Chan K.S., Tsay, R.S., 1998. Limiting properties of the least squares estimator of a continuous threshold autoregressive model. Biometrika 85, 413-426. Clark, P.K., 1987. The cyclical component of U.S. economic activity. Quarterly Journal of Economics 102, 797-814. Davies, R.B., 1977. Hypothesis Testing When a Nuisance Parameter Is Present Only Under the Alternative. Biometrika, 64, 247-254. Davies, R.B., 1987. Hypothesis Testing When a Nuisance Parameter Is Present Only Under the Alternative. Biometrika, 74, 33-43. Davidson, J., 1994. Stochastic Limit Theory. Oxford University Press. De Gooijer, J.G., 1998. On threshold moving-average models. Journal of Time Series Analysis 19, 1-18. Elwood, S., 1998. Is the persistence of shocks to output asymmetric?. Journal of Monetary Economics 41, 411-426. Engle, R.F., Smith, A.D., 1999. Stochastic Permanent Breaks. Review of Economics and Statistics 84, 553-574. Gonzalo, J., Wolf, M. 2005. Subsampling inference in threshold autoregressive models. Journal of econometrics, 127, 201-224. Gonzalo, J., Martinez, O., 2006. Large shocks vs. small shocks. Or does size matter? May be so. Journal of Econometrics, 127(1-2), pages 311-347. Gonzalo, J., Martinez, O., 2007. Threshold permanent-transitory decomposition. Mimeo. Granger, C., Andersen, A.P., 1978. On the invertibility of time series models. Stochastic Processes and Their Applications, 8, 87-92. Guay, A., Scaillet, O., 2003. Indirect inference, nuisance parameter and threshold moving average models. Journal of Business and Economic Statistics 21, 122-132. Hallin, M., 1980. Invertibility and generalized invertibility of time-series models. Journal of the Royal Statistical Society Series B 42, 210-212. Hansen, B.E., 1996. Inference When a Nuisance Parameter Is Not Identified Under the Null Hypothesis. Econometrica, 64, 413-430. Hansen, B.E., 2000. Sample Splitting and Threshold Estimation. Econometrica, 68, 575-603. Harvey, A.C., 1985. Trends and cycles in macroeconomics time series. Journal of Business and Economic Statistics 3, 216-27.

37

Klimko, L.A., Nelson, P.I., 1978. On conditional least squares estimation for stochastic processes. Annals of Statistics 6, 629-642. Koop, G., Pesaran, M.H., Potter, S.M., 1996. Impulse response analysis in nonlinear multivariate models. Journal of Econometrics 74, 119-147. Kuan, C.M., Huang, Y.L. and Tsay, R.S., 2005. An unobserved-component model with switching permanent and transitory innovations. Journal of Business and Economic Statistics, 23, 443454. Ling, S., Tong, H. and Li, D., 2006. Ergodicity and Invertibility of Threshold MA Models.Bernoulli (forthcoming). Ling, S., Tong, H., 2005. Testing for a linear MA model against threshold MA models. Annals of Statistics, 33, no. 6, 25292552. McLeish, D.L., 1975. A maximal inequality and dependent strong laws. Annals of Probability, 3, 829-839. Pesaran, M.H., Potter, S.M., 1997. A floor and ceiling model of US Output. Journal of Economic Dynamics and Control, 21, 661-695. Pierce, D.A., 1979. Signal extraction error in nonstationary time series. The Annals of Statistics 7, 1303-1320. Potter, S.M., 2000. Nonlinear Impulse Response Functions. Journal of Economic Dynamics and Control 24, 1425-46. Quah, D., 1992. The relative importance of permanent and transitory components: identification and some theoretical bounds. Econometrica 60, 107-118. Taniguchi, M., and Kakizawa, Y., 2000. Asymptotic Theory of Statistical Inference for Time Series. New York, Springer-Verlag. Watson, M.W., 1986. Univariate detrending methods with stochastic trends. Journal of Monetary Economics 18, 49-75. Wecker, W.E., 1981. Asymmetric time series. Journal of the American Statistical Association 76, 16-21.

38

Jes´ us Gonzalo ∗ Economics U. Carlos III de Madrid [email protected]

Oscar Mart´ınez Economics U. Rovira i Virgili [email protected]

01 March 2008 Abstract The purpose of this paper is to develop econometric tools to test what type of shocks are permanent and which ones are transitory in an univariate time series econometric model. These shocks are characterized according to their own properties (size, sign, etc) or according to the characteristics of the economy (expansion - recession). In this way the identification, opposite to the standard literature where at every time t there is always a permanent shock, is achieved by allowing for periods with no permanent shocks. In order to perform this analysis, we introduce a new set of threshold models: autoregressive threshold integrated moving average models (TARIMA). These are integrated models with a threshold non-linearity in the moving average part. The threshold structure allows for the existence of non-invertible as well as invertible regimes. The former regime corresponds to transitory shocks, while the latter to permanent ones. The paper fully develops the properties of the model (stationarity-ergodicity, invertibility and its impulse response function) and the asymptotic theory of LS (consistency and asymptotic normality). From these results we present a general two steps testing strategy for permanent and transitory shocks. The paper ends with an application to GNP where we analyse the long run effects of large versus small shocks, positive versus negative shocks, and expansion versus recession shocks.

Keywords: Asymmetries, Identification of Shocks, Moving Averaged Models, Nonlinearities, Permanent Shocks, Persistence, Threshold Models, Transitory Shocks. JEL: C22, C51 ∗

Corresponding Address: http://www.eco.uc3m.es/jgonzalo. Financial support DGCYT Grant (SEC010890) is gratefully acknowledged.

1

Introduction

What are the shocks that drive the economy? Where does the persistence present in most of the economic time series come from? Are there always permanent shocks at every t? Is the persistence of output shocks symmetric or asymmetric? Do negative oil price shocks have different effect on gasoline price than positive ones? What fraction of output variation is due to supply or demand shocks? All the attempts to answer these questions pass through the critical decision of shock identification plus being able to test if a given shock is permanent or transitory. Traditionally this has been done by decomposing the analyzed variable into unobserved permanent and transitory components (see Watson, 1986). Some examples are: (1) Beveridge and Nelson (1981) decomposition (B−N hereafter), where the permanent component is a random walk and there is perfect correlation between permanent and transitory shocks due to the fact that there is only one shock, the permanent one; (2) Unobserved component models with uncorrelated components (U C − 0) (see Harvey, 1985; Clark, 1987), where the permanent component is also assumed to be a random walk but its innovation is uncorrelated with the one in the transitory term. Note that not all the ARIMA models admit an U C − 0 decomposition. This problem can be solved eliminating the random walk constraint on the permanent component. In this case we have an extra identification problem that is overcome by imposing an ad-hoc smooth condition on the permanent term (see the “canonical” decomposition in Pierce, 1979). All these decompositions present two basic problems that are not solved in the literature: (i) at every time period t there is always a permanent shock, and (ii) none of the assumptions on permanent and transitory components are testable, and therefore we can not test whether a type of shocks is permanent or transitory. The approach proposed in this paper tries to solve the identification and testing issue without encountering these problems. Our proposal is based on two pillars. First, as in the above mentioned identification attempts we accept that there are permanent and transitory shocks, and secondly, shocks behave differently in the long-run according to their own characteristics (sign, size, etc.) or some characteristics of the economy (expansion-recession). The first premise, as is well known (see Quah (1992)) is not enough to identify the permanent and transitory shocks of a univariate time series. This paper shows that with our second assumption we can not only identify the permanent and transitory shocks of a single economic variable, but we can also test whether in fact these shocks are transitory or permanent. In order to implement our proposal we use a class of threshold models, the Autoregressive Threshold Integrated Moving Average models (TARIMA), (a particular case of these models was pre-

2

viously introduced in the literature by Gonzalo and Mart´ınez, 2006, where a new non-linear permanent-transitory decomposition is obtained). These are models with a unit root in the autoregressive part to capture persistence, and a threshold design in the moving average side to allow for asymmetries. The threshold variable can represent any characteristic of the shocks (large or small, positive or negative, etc.) or of the economy (expansion or recession, inflation or deflation, etc.) we are interested on. This threshold design is able to capture the possibility that any of these characteristics may have different long run effects. By allowing the existence of a unit root in some of the moving average regime, we permit that the characteristics triggering those regimes have only a transitory effect. In principle, TARIMA models can deal with very general types of asymmetries. In this paper we divide them in two types: asymmetries caused by observable variables (expansionrecession, etc) and asymmetries caused by non-observable variables, in particular those generated by the characteristic (size, sign) of the model’ shocks. The identification of permanent and transitory shocks with our procedure, allows to construct an orthogonal nonlinear permanent-transitory decomposition of the original variable. We also obtain a non-linear B − N decomposition that while sharing part of the spirit of both, the standard linear B − N and the U C − 0 decompositions, its behavior lies in between. This is proposed and study for a particular case in Gonzalo and Martinez (2006). A complete analysis of a threshold permanent-transitory decomposition is developed in Gonzalo and Martinez (2007). Threshold moving average (TMA) models have already been considered in Wecker (1981) and in De Gooijer (1998), and very recently in Guay and Scaillet (2003) and Ling and Tong (2005). The former two works are centered on presenting the new TMA model and on analyzing some of the moment properties in detail. They both assume normality, the threshold parameter r to be known and equal to zero, and they do not present asymptotic results. These can be found in the latter two. Guay and Scaillet use indirect inference and focus on the shock sign case in an stationary model. Ling and Tong develop a test for threshold effect in a stationary self-exciting TMA model. Our paper is more general in the sense of admitting a reacher family of threshold variables, we admit non-invertible moving average regime and we present asymptotic results for LS estimation as well as for inference. The paper finishes with an application of the TARIMA model to GNP where three types of asymmetries are considered, asymmetries caused by shock sign, shock size and recession-expansion. It is concluded that the asymmetry are statistical significant in the last two and permanent and transitory shocks are found in the shock size case. The rest of the paper is structured as follows. Section 2 introduces the TARIMA model

3

and analyzes in detail its properties (stationarity-ergodicity, invertibility and its impulse response function). Section 3 presents the consistence of the LS estimation. In section 4, the asymptotic theory results needed for testing the hypothesis of interests are developed. Section 5 shows the finite sample performance of the test. Section 6 presents the empirical application of the TARIMA models to GNP. Finally, Section 7 draws some concluding remarks. The appendix contains technical derivations and proofs of the results in the main text.

2

TARIMA Models

In its general form, the threshold autoregressive and threshold integrated moving average, TARIMA(n, m, p, 1, q), model is: n+1 X

[Φp,l (L)(1 − L)yt + µl ] 1 (zt ∈ Al ) =

l=1

m+1 X

[Θq,j (L)εt ] 1 (zt ∈ Bj ) ,

t = 0, ±1, ±2, . . .

j=1

(1) where Φp,l (L) is a p − th order lag polynomial with all its roots outside the unit circle, Θq,j (L) is a q − th order lag polynomial, 1(.) denotes the indicator function and zt ∈ 0. G.2 zt and xt are strictly stationary, ergodic and φ − mixing of size −a with a > 1. G.3 Φ0p (L) has all its roots outside the unit circle, and µ0 ∈ [−µ, µ] with µ < ∞. I.0 θi = 1 for some i = 1, 2. I.1 E [ εt / =t−1 ] = 0 and E [ εt (1 − θt )/ =t−1 ] = 0. hQ . i q−1 2 AX.0 ∃q < ∞ s.t. E i=0 θt−i =t−q = λ1 < 1. . n ³Q ´ o q−1 4 AX.1 θ0 ∈ Θθ = [−1 + δ, 1 + δ] × [−1 + δ, 1 − δ] × [r, r] s.t. E =t−q ≤ λ∗1 < 1 , i=0 |θ t−i | with q < ∞, δ > 0, and θt = (1 + δ)1(zt ≤ r) + (1 − δ)1(zt > r). ³ . ´ 2γ AX.2 supz0 ∈[r,r] E |εt−1 | z 0 − ν ≤ zt−1 ≤ z 0 + ν ≤ σ 2γ ε < ∞, with γ > 2, and, supz0 ∈[r,r] E ( 1 (z 0 − ν < zt < z 0 + ν)/ =t−1 ) ≤ νM, with M < ∞, and ν > 0. ³ ´ ¡ ¢ 2 AX.3 inf z0 ∈[r,r] E |εt−1 | 1 (z 0 − ν < zt−1 < z 0 + ν) ≥ min νσ 2ε , σ 2ε with σ ε2 > 0 £ ¤ 0 M (r) = E 1(zt−1 < r)Xt−1 Xt−1 >0 £ ¤ 0 M (r) = E 1(zt−1 > r)Xt−1 Xt−1 >0 1

This could be concluded by the Wold’s representation, but it is worth to note that in this case the shocks are the same in both series

7

0 with Xt−1 = [xt−1 , ..., xt−p , 1, εt−1 ], and ν > 0.

ASZ.0 [a]

2

If |θ1 | > |θ2 |, λ1 = [E (|θ (εt−1 ) |) + rhf ∗ ] < 1,

[b] If |θ2 | > |θ1 |, λ1 = |θ2 | + rhf ∗ < 1 where, h = |θ1 − θ2 |, and f ∗ = 2 maxε fε (ε) ASZ.1 For θ = (θ1 , θ2 , r) and h = |θ1 − θ2 | and f ∗ = 2 maxε fε (ε), [a] If |θ1 | > |θ2 |: θ0 ∈ Θθ = {θ ∈ (−1, 1] × (−1, 1) × s.t. λ1 (θ) = λ2 (θ) + 2rhf ∗ ≤ λ∗1 < 1}, where λ2 (θ) = |θ1 | p (r) + |θ2 | (1 − p (r)), p (r) ≥ supk P (|εt + k| < r) , r > 0 and r < M . [b] If |θ2 | > |θ1 |: n 2 θ0 ∈ Θθ = θ ∈ (−1, 1) × [r, r]

s.t.

o λ1 (θ) = |θ2 | + 2rhf ∗ ≤ λ∗1 < 1 , where r > 0

and r < M . ASG.0 λ1 = maxi |θi | + 12 |r|hf ∗ < 1 where h and f ∗ were defined on ASZ.0. n o 2 ASG.1 θ0 ∈ Θθ = θ ∈ (−1, 1) × [r, r] s.t. λ1 (θ) = maxi |θi | + |r|hf ∗ ≤ λ∗1 < 1 , where h, r, r and f ∗ were defined on ASZ.0. The general assumptions G.0 to G.2 are usual in threshold models. G.3 define the parametric space for the part of the model no related with the threshold. I.0 and I.1 are sufficient conditions to obtain transitory and permanent shocks in terms of the Impulse Response Function. I.0 implies that a unit root in one of the MA regimes is necessary. A sufficient condition for I.1 is that the prediction of εt on each regime is 0. εt has to be a martingale difference sequence in each regime. AX.0, ASZ.0 ([a] or [b]) and ASG.0 are used in the invertibility of the TARIMA-X, TARIMA-Shock-Size and TARIMA-ShockSign model respectively. AX.1-AX.3 are used in the proof of consistency and asymptotic normality for the TARIMA-X. AX.1 describes the parametric space. It is worth to note that only the MA regime of zt ≤ r can be non-invertible. However it is only a notational issue, given that it does no rule out the other possibility, since {zt ≤ r} ≡ {−zt ≥ −r}. So the only implications is that we have to know in which regime can be the transitory shock. AX.2 is a conditional moment bound on εt and zt , and AX.3 is a full rank condition in the sense of condition 1.8 of Hansen (2000). Finally ASZ.1 ([a] or [b]) and ASG.1 describes the parametric space for TARIMA-Shock-Size and TARIMA-Shock-Sign respectively. It is easy to prove that under G.0 to G.3, all parameters of these spaces satisfy the corresponding AX.2 and AX.3 assumptions. After describing the TARIMA models and the sufficient assumptions, this section focus on the main properties of TARIMA models, i.e. stationarity, invertibility and persistence (through the Impulse Response Function). 2

Maybe can be changed to kθ(b et )k2 < 1 when εt is normal, using that ε2 f (ε) decreasing with the variance

of ε

8

2.1

Stationarity and Invertibility

In this section, we study the stationarity, ergodicity and invertibility of TARIMA models. We concentrate only on causal solutions of (3), taking as the analysed variable xt = (1 − L)yt . The assumptions which guarantee the stationarity and ergodicity of xt are in the following lemma. Lemma 1. Let yt be an TARIMA process as (3) and g(.) any measurable function on p, with p = max{p, j + 1} and Ip−1 the identity matrix of order {p − 1} × {p − 1}. b) when zt 6= g(xt−j ) for j = 0, 1, 2, ..; if the sequence {εt , zt } is strictly stationary and ergodic, then {xt } is strictly stationary and ergodic. Proof: See appendix. About invertibility, the non-linearity of TARIMA models disables the use of the classical definition of invertibility. Then, we use the general invertibility definition introduced by Granger and Andersen (1978) and later improved by Hallin (1980). Hallin proved that under nonlinearity with constant coefficients both definitions are equivalent. Clearly, the TARIMA models are within this class. The definition of Granger and Andersen is the following: Definition 1. (Granger and Andersen) The process xt = g(xt−1 , εt−1 , ..., xt−p , εt−p ) + εt is invertible if ¡ ¢ lim E vt2 = 0

t→∞

with vt = εt − εbt = εt − (xt − g(xt−1 , εbt−1 , ..., xt−p , εbt−p ))

9

This definition guarantees the possibility to recover the shocks from the past of the series. Working with MA models, linear or non-linear, the forecast depends on the past innovations that are not observed. So we have to recover them from the observable series. In that sense, the invertibility is a necessary condition for prediction. Moreover we are interested in identify them, in some cases based on some of their properties (size, sign...), so recover them seems a necessary condition. Note that this definition of invertibility implies the convergence of vt to 0 in L2 whereas the Ling, Tong an Li (2007) definition implies a.s. convergence. The results about the invertibility property for TARIMA models are summarized in the following lemma: Lemma 2. Let yt be an TARIMA process as (3), ¡ ¢ a) if zt is observable, (TARIMA-X), under E ε20 < ∞ and AX.0 {yt } is invertible. b) if zt = |εt |, (TARIMA-Shock-Size), under G.0 and ASZ.0 a) or b), {yt } is invertible. c) if zt = εt , (TARIMA-Shock-Sign), under G.0 and ASG.0, {yt } is invertible. Proof: See appendix. Given that the definition of invertibility is not the same, the condition AX.0 can not be compared directly with the conditions of Ling, Tong and Li (2007) for invertibility. However, the condition AX.0 can be substituted by,

AX.0’ E (log(θt )) < 0 and

³Q

T 2 t=0 θt

´

v02 is an uniformly integrable sequence, where

v0 = ε0 − εb0 .

The second part of AX.0’,

³Q

T 2 t=0 θt

´

v02 is an uniformly integrable sequence, guarantees

that the convergence in probability implies L2 convergence. For the TARIMA-X model, the condition for invertibility is quite similar to the stationarity condition of TAR models. However, for TARIMA-Shock, if we study the assumption ASZ.0 or ASG.0 we can distinguish two different parts. In the first one, we have E (|θ(εt−1 )|) or maxi θi , that must be less than 1. This checks the case of overdifferentiability, specially when we allow a unit root in the MA. But non-invertibility is not only a problem of overdifferentiability, but a problem of discontinuity3 too. The TARIMA-X is linear in shocks, but TARIMA-Shock no. The second part, rhf ∗ , checks the discontinuity degree, measured as a product of the gap, rh, and its upper limit probability, f ∗ . ¯ ¯ ¯ t ¯ When the function is continuous, the ¯ ∂ε∂εt−1 ¯ determines the overdifferentiability, having to be less than 1. If there is linearity, it is |θ| < 1. When the model is not continuous on εt−1 , the previous condition is not enough. 3

10

In the case of TARIMA-X models, we can obtain transitory shocks in any of both regimes, allowing the series to be invertible. In the TARIMA-Shock-Size models, Lemma 2 b) allows that θ1 = 1; this is the case of small shock transitory, and big shock permanent. Unfortunately, this assumption excludes the opposite case, big transitory shock. Finally, for TARIMA-Shock-Sign models, as we see in the next section, with constant mean it is not possible to obtain transitory shocks. Then, we simplify the conditions not allowing a unit root in any of both regimes. It is word to note that as in size case, when r is different from 0, it is necessary to bound the discontinuity to guarantee invertibility. It is easy to see that for r = 0 the process is continuous on εt−1 .

2.2

Impulse Response Function

One of the distinguishing feature of TARIMA process is that the effect of shocks can be transitory or permanent with positive probability. The only previous attempts are the Stochastic Permanent Break (STOPBREAK) model of Engle and Smith (1999) and the unobserved component model of Kuan, Huang and Tsay (2005). However, in the former case the probability of a transitory shock is zero. In the latter, we are not able to identify the issue of transitory or permanent effect and there is no asymptotic result for estimation and inference. To define the permanent effect of a shock we use the Impulse Response Function (IR). This function measures the effect of a perturbation at time t in the sample path of the ∞

series, {yt+k }k=0 . If this effect on yt+k does not vanish when k → ∞ we say that the shock is permanent. With the linear models the traditional IR function does not have any conceptual problem in the definitions of permanent and transitory shocks. However with nonlinear models three main aspects of the time series come up to determine the relationship of the IR function with the persistence definitions. These aspects are the history of the series at time t − 1, the shock εt and future shocks. To capture all these three new aspects we use the Generalized Impulse Response Function (GI) introduced and defined in Koop, Pesaran and Potter (1996) and in Potter (2000) as:

GI (k, εt , wt−1 ) = E [ yt+k | εt , wt−1 ] − E [ yt+k | wt−1 ]

f or k = 0, 1, 2, ..

(5)

where wt is the sample history until time t. According to this definition, a shock is persistent if the effect of including in the information set on the conditional expectation of yt+k given the past does not vanish when k → ∞.4 In the TARIMA-X case, although we are in an 4 So the GIRF measures the effect on the conditional expectation, not on the sample path. However it can be proved that if there is a noninvertible MA regime, the shocks of that regime are not in the future sample

11

univariate analysis, when zt is not a function of yt , the persistence properties depends on two variables, εt and zt . As zt is observable in t, we propose the following generalization of the GI for vt = (εt , zt ): GI(k, vt , wt−1 ) = E [ yt+k | vt , wt−1 ] − E [ yt+k | wt−1 ]

f or k = 0, 1, 2, ..

This allows a correct interpretation of the GI function in a more general context. In any case, when zt is =t−1 -measurable it is not necessary this generalization. In the following lemma we establish the relation between a unit root in some of the MA regimes and the existence of transitory and permanent shocks. Lemma 3. Let yt be an TARIMA process as (3) which satisfies G.3. Then, Φp (L)−1 = Ψ(L) =

∞ X

ψj Lj ,

j=0

where ψ0 = 1 and ψ∞ = 0; under I.0 and I.1, the GI function of yt is given by ψk−1 εt i h GI (k, vt , wt−1 ) = (1 − θ) Pk−2 ψ + ψ k−1 εt j=0 j

if

θt = 1

if

θt = θ 6= 1.

Clearly, when θt = 1, which implies a non-invertible MA regime, the shock εt is transitory, its effect goes to zero when k → ∞ given that limk→∞ ψk−1 = 0. On the other hand, when θt 6= 1, the shock εt will be permanent. It is worthy to note that it is zt , although the threshold variable is zt−1 , what determines the persistent or transitory effect of εt . For example, in the particular case of zt = |εt |, it is the size of the own shock which determines its persistent or transitory effect. The case of TARIMA-Shock-Sign does not hold assumption I.1. It can be proved that E [ εt (1 − θt )/ =t−1 ] ≡ µE 6= 0, so the expected effect of εt , conditioned on t − 1, is different from zero. Then, this implies the effect of knowing εt on the conditional expectation will be always different from zero and permanent. In this case does no matter the existence of a noninvertible MA regime. To correct this effect, it will be enough to allow a threshold in the mean. This case can be studied easily following the proofs of this paper, although for space reason we do no include here. So for sign case, we can study here asymmetric responses, but not in terms of transitory and permanent responses. path of the series.

12

2.3

3

Permanent-Transitory Decomposition

Estimation

In this section it is proved the main results about estimation distinguishing between the cases of observable and unobservable threshold variable. We propose the Conditional Least Squares (CLS) method introduced by Klimko and Nelson (1978) to estimate the parameters of TARIMA models, establishing the conditions under which these estimators are consistent and their rate of convergence. For simplicity, without lack of generality, we concentrate in the simple case of TARIMA(n,m,p,d,q), TARIMA(0,1,1,1,1) model:

(1 − ρL) ∆yt = (1 − ρL) xt = µ + εt − θt−1 εt−1

εt − θ1 εt−1 =µ+ ε −θ ε t

2 t−1

if zt−1 ≤ r

(6)

if zt−1 > r

with ∆ = (1 − L). The most general case, p > 1, follows straightforward from the proofs of this paper. A general parameter in Θ is denoted by φ = (φ1 , φ2 , φ3 , φ4 , φ5 ) = ¢ ¢ ¡ ¡ (µ, ρ, θ1 , θ2 , r) = (β, r) and the true parameter φ0 = φ01 , φ02 , φ03 , φ04 , φ05 = µ0 , ρ0 , θ10 , θ20 , r0 = ¡ 0 0¢ β , r . In general we denote slope parameters to the vector of parameters β, which can be the continuous parameters. The objective function is the sum of the square errors, that is, QT (φ) =

T X

e2t (φ)

t=1

with

et (φ) = (1 − ρL) xt − µ + θt−1 et−1 (φ) , θ1 if zt−1 ≤ r θt−1 = θ if z > r. 2

e0 = 0

t−1

³ ´ b b For a given r, the Least Square (LS) solution is β(r). Second, we minimize QT β(r), r , ³ ´ ³ ´ obtaining rb. The CLS estimator of φ0 is φbT = φb1 , φb2 , φb3 , φb4 , φb5 = µ b, ρb, θb1 , θb2 , rb = ³ ´ b r), rb is the global minimizer of QT (φ), that is, β(b φbT = arg min QT (φ) φ∈Θ

where Θ = {φ

s.t

θ ∈ Θθ , ρ ∈ (−1, 1), µ ∈ [−µ, µ]}.

In general, for Threshold Autoregressive (TAR) models, the rate of convergence for the CLSE of the parameters are T 1/2 except for r, the threshold parameter, which is

13

T . That result is due to the kind of discontinuity of the objective function in r via the indicator function. In the case of TARIMA-X models, the same results of TAR models is obtained given that QT (φ) is continuous in all the parameter, except r, through the indicator function. For the TARIMA-Shock case, however, the threshold variable is not observable, so it must be estimated. Thus, all the parameters to estimate enter in the indicator function and present the same discontinuity of r. In this case, the objective function is discontinuous in all the parameters through the indicator function, so all of them are T −consistent. However, there is an exception to this result in the Shock case, the TARIMA-Shock-Sign with r = 0 known. In that case, it can be proved that the objective function is continuous in all the parameters to estimate. From the previous comments, in the TARIMA-X case, the rate of convergence of βbi is T 1/2 for i = 1, ..., 4 and T for rb, that is, the threshold parameter. This result is established in the following Theorem. Theorem 1. [Consistency for TARIMA-X] Under G.0 a),G.1, G.2, G.3 and AX.1¡ ¢ ¡ ¢ as AX.3, φbT −→ φ0 , βbi,T = βi0 + Op T −1/2 for i = 1, ..., 4 and rbT = r0 + Op T −1 . Proof: See appendix. This Theorem establishes the existence on Θ, defined through AX.1 and GX.3, of an unique asymptotic minimum in QT (φ) reached at φ0 , the consistence of the CLSE and the rate of convergence5 . The assumptions G.1 and AX.3 guarantee the identification of this minimum. In assumption G.2, it is assumed that xt and zt are φ − mixing. This assumption is made for simplicity of the proof. However, it can be relaxed to xt and zt be L2 − N ear Epoch Dependence of size −∞ on φ − mixing processes. For a complete description of this kind of processes see Billingsley(1968), McLeish (1975), Bierens (1983), and Davidson (1994) between others. For the case of TARIMA-Shock-Size, the results about the consistence and rate of convergence are in the following Theorem. Theorem 2. [Consistency for TARIMA-Shock-Size] Under G.0-G.3, and ASZ.1, ¢ ¡ as φbT −→ φ0 , and φbi,T = φ0i + Op T −1 for i = 1, ..., 5. Proof: See appendix. As in the previous case, this Theorem establishes the existence on Θ, defined through ASZ.1 and G.3, of a unique asymptotic minimum in the true values of the parameters and the convergence of the estimators to this minimum at a T rate. This Theorem prove the intuition developed previously where all the parameters share the same properties of r. 5

In the paper we define Op (T −α ) in the following way, XT is Op (T −α ) if for ∀ε > 0 ∃∆ε and an integer Tε ≥ 1 such that P [|T α XT | > ∆ε ] < 1 − ε for ∀T ≥ Tε .

14

Finally, for the TARIMA-Shock-Sign case, these results are established in the following Theorem. Theorem 3. [Consistency for TARIMA-Shock-Sign] Under G.0-G.3, and ASG.1, as φbT −→ φ0 , and

¢ ¡ a) If it is known that r0 = 0, βbi,T = βi0 + Op T −1/2 for i = 1, ..., 4 ¡ ¢ b) If it is known that |r0 | > 0, φbi,T = φ0i + Op T −1 for i = 1, ..., 5 . Proof: See appendix. In the Sign case, we have two different results depending on the value of r0 . When r0 = 0, we are in the continuous threshold case, (see for example Chan and Tsay (1998) for the continuous SETAR case). As in that case, we need to assume the continuity of the objective function. Here, this implies to assume that r0 = 0, which is the only possible value to obtain the continuity. That is the pure sign asymmetric case. Under this assumption the estimators converge at a T 1/2 rate, given that in this case, the objective function is continuous in all the parameters. For the discontinuous case, |r0 | > 0, we obtain the same results that in the Size case.

4

Inference

In this section we present the testing strategy for the main hypothesis in the three models and develop the necessary asymptotic results to make inference. Attending the properties of the models, there is a main property to test, the existence of asymmetries in terms of permanent and transitory shocks. We propose to test it in two step. In a first step, we develop the threshold effect test based on statistic W1 , defined later, which corresponds with the existence of asymmetries in the IRF. The null in this test, as usual, is the symmetry of the IRF. In the second step, if we reject the null of symmetry in the first step, so the existence of a threshold effect can be assumed, we test the existence of transitory shocks in a particular MA regime with positive probability. For that, we test the existence of a unit root in that regime through the statistic W2 , defined later. If we assume I.1, under the null of a unit root, the shocks in that regime are transitory, as we have proved by Lemma 3. As it is well known from the threshold literature, the threshold effect test is deeply affected by knowing r0 . If r0 is unknown, we are in the case of inference when a parameter is not identified under the null hypothesis. Following the literature we propose a supremum type test, see for example Andrews (1993), Andrews and Ploberger (1994), Davies (1987) and Hansen (1996) between others. In general, the result for r0 known case can be obtained from the unknown r0 case. On the other side, given the T −consistence of rb when a

15

threshold effect exists, the asymptotic distribution of the slope parameters is not affected by unknowing r0 . In this testing strategy we distinguish between the observable and non observable threshold variable case, focusing the results in the unknown r0 case which is the habitual case. Only for the TARIMA-Shock-Sign, the r0 = 0 known case is considered as a particular case. In the observable case the same results for TAR models of Chan (1993) are obtained, that is, the asymptotic distribution of the slope parameter estimators, βb = (b µ, ρb1 , ..., ρbp , θb1 , θb2 ) is Normal and independent of the threshold parameter estimator. In the shock case, the objective function is not continuous on the slope parameters, so we develop a two step estimation to obtain statistics with asymptotic Normal distributions independents of the threshold parameter. In the rest of the section, the theory for threshold effect and unit root in the MA regime tests is developed for each model, TARIMA-X and Shock, distinguishing the case of TARIMA-Shock-Sign when r0 = 0 is known as a particular case.

4.1

Hypothesis testing in the Observable case

As we say above, in a first step we test the existence of a threshold effect. In order to do that we rewrite the model of equation (6) as

xt = µ + ρxt−1 − θεt−1 − αεt−1 1(zt−1 ≤ r) + εt ,

(7)

where the redefined slope parameters β = (µ, ρ, θ, α) are estimated by CLS in a similar way described in the previous section. The main reason to change the specification of equation (6) to (7) is that the asymptotic theory is more easy given that on equation (7) only one regressor depends on r. Apart of this, nothing changes and it is easy to see that θ = θ1 and α = (θ2 − θ1 ). Through this new equation, the null hypothesis of symmetry is: H0 : α(r) = 0

∀r ∈ [r, r],

versus the alternative, Ha : α(r) 6= 0 for some r ∈ [r, r]. So, the proposed statistic for this test is based on

W1,T (r) =

Tα b2 (r) , Vb (b α (r))

16

where: b −1 (r, r) R2 Vb (b α (r)) = R40 K b (r, r) = σ K bε2 T −1

T X

R40 = (0, 0, 0, 1),

with

³ ´ ³ ´ b r κ0 β, br , κt β, t

t=1 ¯ ¡ ¢0 ∂et (φ) ¯¯ κ(φ? ) = κ1t (φ? ), κ2t (φ? ), κ3t (φ? ), κ4t (φ? ) = , ∂β ¯φ?

κ1t (φ) = −

j t Y X

θ,

κ3t (φ) =

j t Y X

j=0 k=1

κ2t (φ) = −

j t Y X

θet−j−1 (φ),

j=0 k=1

θxt−j−1

and κ4t (φ) =

j=0 k=1

j t Y X

θet−j−1 (φ)1 (zt−j−1 ≤ r) ,

j=0 k=1

with φ = (β, r). Given that under the null r is not identified, we propose to test H0 with ΓT = supr∈[r,r] W1,T (r), as it is suggested by Davies (1977, 1987) between others. To study the asymptotic behavior of this statistic under the null it is necessary to define the following processes. First, let s (r) be a gaussian process with mean zero and ¡ ¡ ¢ ¡ ¢¢ covariance function K (r1 , r2 ) = σε2 E κt β 0 , r1 κ0t β 0 , r2 . Second, the gaussian process S (r) = R40 K −1 (r, r) s (r), with mean zero and covariance function given by K (r1 , r2 ) = R40 K −1 (r1 , r1 ) K (r1 , r2 ) K −1 (r2 , r2 ) R4 . The following Theorem shows the asymptotic distribution of W1,T (r) =

Tα b2 (r) , b (b V α(r))

where

⇒ denote weak convergence with respect to the Skorohod metric. Theorem 4. Under H0 and assumptions G.0, G.2, G.3, and A.1-A.2, 0

(1) W1,T (r) ⇒ S (r) K

−1

(r, r) S (r), d

0

(2) ΓT = supr∈[r,r] W1,T (r) −→ supr∈[r,r] S (r) K

−1

(r, r) S (r).

Proof: See Appendix. The Theorem 4 (1) gives the asymptotic distribution of test statistic W1,T (r), which 0

is given by S (r) K

−1

(r, r) S (r). The result (2) follows straight forward given that the

supr∈[r,r] is a continuous functional on [r, r] to R. Since the null distribution of ΓT = supr∈[r,r] W1,T (r) depends, in general, upon the covariance function K (r1 , r2 ), critical values cannot be tabulated except in special cases. So to be able to carry out this test T

we propose a bootstrap approximation. Algorithm 1 generates bootstrap samples {yt? }t=1 from model (7) and calculate the bootstrap approximation of the null distribution of ΓT = supr∈[r,r] W1,T (r). Algorithm 1. (Model-based Bootstrap procedure) 1. l = 1. T

2. Generate {ε?t }t=1 resampling from

n oT εbt − εb

t=1

17

with εb = T −1

PT

bt . t=1 ε

T b ? + ε? . 3. Generate {yt? }t=1 from (1 − ρbL)(1 − L)yt? = µ b − θε t t−1 T f ? (r). 4. From model (7), with bootstrap sample {yt? }t=1 , obtain Γ?,l = supr∈[r,r] W 0

5. l = l + 1. Go to step 2 while l ≤ B. 6. Estimate the p-value, pv , from bootstrap approximation, B ¢ 1 X ¡ ?,l 1 Γ >Γ . pv = T l=1

The proof of the validity of this procedure is not within the scope of the paper, but it follows that of the bootstrap procedure proposed by Caner-Hansen (2001). When r0 is known we can use the statistic W1,T (r0 ). As it is easy to prove, for each r ∈ [r, r] the asymptotic distribution of W1,T (r) has a marginal chi-square distribution. So under the null, W1,T (r0 ) converges to a χ21 . The second step of the strategy, if we reject the null of no threshold effect, is to test the existence of a unit root in a MA regime in model (6). In this case, the null hypothesis is

H0 : θi = 1, versus the alternative, Ha : θi 6= 1 for i = 1, 2. For that we propose a Wald statistic

W2,T (b r) =

T (θbi − 1)2 . b −1 (b r)Ri σ bε2 R0 H i

where φb is the CLS estimator of φ on equation (7), so β = (µ, ρ, θ1 , θ2 ), and Ri s.t. Ri0 β = θi . To obtain the asymptotic distribution of the W2,T (b r) statistic, we use the following Theorem. Theorem 5. Under assumptions G.0-G.3 and A.1-A.3, ´ ³ ´ ³→ − d a) T 1/2 βb − β −→ N 0 , σε2 H −1 p

b r) −→ σε2 H b) σ bε2 H(b

18

with σ bε2

=T

−1

T X

b e2t (φ),

t=1

H = lim T −1 T →∞

T X

£ ¤ E ht (φ0 )h0t (φ0 ) ,

t=1

¡

¯? ∂et (φ) ¯¯ , ∂β ¯φ Ã j ! t X Y h3t (φ) = θt−k et−j−1 (φ)1(zt−j−1 ≤ r),

ht (φ? ) = h1t (φ? ), h2t (φ? ), h3t (φ? ), h4t (φ? ) h1t (φ) =

t X j=0

h2t (φ)

=

t X

Ã

Ã

j=0

and

b = T −1 H

j Y

! θt−k

k=1 j Y

,

=

j=0

! θt−k

¢0

xt−j−1 ,

h4t (φ)

k=1

=

j=0

k=1 T X

t X

Ã

j Y

! θt−k

et−j−1 (φ)1(zt−j−1 > r),

k=1

³ ´ ³ ´ ht φb h0t φb .

t=1

Proof: See Appendix. From the prove of this theorem it is easy to see that when r0 is known the asymptotic distribution is the same. So, as in TAR models, see Chan (1993) for example, the asymptotic distribution of βb is independent of the rb. In fact, when r0 is known the result of Theorem 5 holds without assuming G.1. So this Theorem in that case can be used to test the existence of a threshold effect. Under the null, from Theorem 5, W2,T (b r) converges to a χ21 , so we reject the null hypothesis at the level α when W2,T (b r) > χ21,α with χ21,α s.t. P [χ21 > χ21,α ] = α. If there is no evidence against the null, the shocks in the regime i are transitory.

4.2

Hypothesis testing in the TARIMA-Shock case

As in the observable case we start with the threshold effect test assuming that r0 is unknown, so excluding the TARIMA-Shock-Sign case with r0 = 0 known. The test follows the same steps of the observable case. First, similar to equation (7) consider:

xt = µ + ρxt−1 − θεt−1 − αεt−1 1(g(εt−1 ) < r) + εt ,

(8)

where g(εt ) = |εt | in the Size case, and g(εt ) = εt in the Sign case. The parameters β = (µ, ρ, θ, α) are estimated by CLS and the null hypothesis of symmetry is:

H0 : α(r) = 0

19

∀r ∈ [r, r],

versus the alternative, Ha : α(r) 6= 0 for some r. So, the proposed statistic for this test is based again on

W1,T (r) =

Tα b2 (r) . Vb (b α (r))

Under the null, the asymptotic distribution is given by the processes S(r) and the covariance function K(r1 , r2 ) defined in the observable case, although in this case:

κ1t (φ) = −

j t Y X

θ,

κ3t (φ) =

j=0 k=1

κ2t (φ) = −

j t Y X

j t Y X

θet−j−1 (φ),

j=0 k=1

θxt−j−1

and

κ4t (φ) =

j=0 k=1

j t Y X

θet−j−1 (φ)1 (g(εt−j−1 ) ≤ r) .

j=0 k=1

The following Theorem resume this result, where ⇒ denote weak convergence with respect to the Skorohod metric. Theorem 6. Under H0 and assumptions G.0, G.2, G.3, and ASZ.1 (for Size case) and ASG.1 (for Sign case), 0

(1) W1,T (r) ⇒ S (r) K

−1

(r, r) S (r), d

0

(2) ΓT = supr∈[r,r] W1,T (r) −→ supr∈[r,r] S (r) K

−1

(r, r) S (r).

Proof: See Appendix. Again, since the null distribution of ΓT = supr∈[r,r] W1,T (r) depends, in general, upon the covariance function K (r1 , r2 ), critical values cannot be tabulated except in special cases. So to be able to carry out this test we propose a bootstrap approximation based on the same Algorithm 1 of the observable case, taking into account that in step 4 the model follows equation (8). When r0 is known, the marginal distribution of W1,T (r) for a given r is a χ2 , so we can test the threshold effect as in the observable case. If the symmetry is rejected, we can test the existence of transitory shocks in a given regime. For that we study the asymptotic distribution of the estimators under G.1. At first, we exclude the Shock-Sign case when r0 = 0, which is studied as a particular case in the next section, so the Shock case is characterized by the same discontinuity of parameter r in all the parameters. That prevents us from obtaining the asymptotic normality of CLS estimator for β parameters. Besides their asymptotic distribution is very difficult to obtain and we recognize that it is beyond the scope of this paper. Therefore we propose an alternative way of obtaining an statistic and its asymptotic distribution to test the existence of a unit root in a MA regime. In a first step, the TARIMA-Shock model is

20

estimated by CLS as previously described in Section 3. In a second step, we consider εt and r0 as known and equal to the CLS estimates of the first step. Basically, this can be b be done because in the first step everything is T-consistent. In more detail, let ebt = et (φ) the errors of the CLS estimation of model (6). The second step estimators are the ordinary least square (OLS) estimators of the model: 0 b (b b (b b + ut (9) xt = µ + ρxt−1 + θ1 et−1 (φ)1 zt−1 > rb) + θ2 et−1 (φ)1 zt−1 < rb) + ut = Zt−1 (φ)β

¯ ¯ ¯ b¯ b in the Sign case. In matrix way, with zbt = ¯et (φ) ¯ in the Size case and zbt = et (φ) b + U, X = Z(φ)β ³ ´−1 b φ) b b with Z 0 (φ) = (Z1 (φ), ..., ZT −1 (φ)). Then, the OLS estimator of β is βeOLS = Z 0 (φ)Z( Z 0 (φ)X. The asymptotic distribution of this OLS estimator is showed in the following Theorem. Theorem 7. Under assumptions G.0-G.3 and ASZ.1, for Shock-Size, or ASG.1, for Sign case with |r0 | > 0, ³ ´ d a) T 1/2 βbOLS − β 0 → N (0, Ωz ) p b → b z (φ) b) Ω Ωz , i−1 h h ¢i−1 PT −1 PT −1 ¡ b t (φ) b b =σ b z (φ) . and Ω bε2 T −1 t=0 Zt0 (φ)Z with Ωz = σε2 limT →∞ T −1 t=0 E Zt0 (φ0 )Zt (φ0 )

Proof: See Appendix. Then, we can test the transitory small shock hypothesis:

H0 :θ1 = 1 Ha :θ1 6= 1, with θ1 in model (9) and g(εt ) = |εt |. The test statistic is, W2,T (r0 ) =

T (βb3 − 1)2 , b rb)R3 b z (β, R0 Ω 3

which under the null, from Theorem 7, converges to a χ21 . The same statistic with rb = r0 can be used given that the Theorem 7 is equally valid when rb is substituted by r0 . However, as we commented above the result is based on the T −consistence of all the estimators, so in both cases the assumption G.1 is needed.

21

Hypothesis testing in the TARIMA-Shock-Sign case with r0 = 0

4.2.1 known

There are two main reasons to deal with this case as a particular case. First, although it is a case of TARIMA-Shock model, the objective function is continuous in the parameters β, as in the TARIMA-X. Second, we have to impose that r0 = 0 is known. As in the SETAR model case, the knowledge of the continuity of the model has to be imposed. Only the recent paper of Gonzalo and Wolf (2005) allows to test the continuity of the SETAR model. This issue is under current investigation by the authors. To carry the testing strategy out for this case we use the following theorem which give the asymptotic distribution of the CLS estimators of β and can be used for the threshold effect test, given that G.1 is not assumed. Theorem 8. For TARIMA-Shock-Sign model, under G.0, G.2, G.3, ASG.1 and r0 = 0 ´ ³− ´ ³ → d (1) βb − β −→ N 0 , σε2 H −1 p

b −→ σε2 H (2) σ bε2 H with,

σ bε2 = T −1

T X

b e2t (φ),

t=1

H = lim T −1

T X

T →∞

£ ¡ ¢ ¡ ¢¤ E ht φ0 h0t φ0 ,

t=1

b = lim T −1 H

T X

T →∞

³ ´ ³ ´ b r0 h0 β, b r0 , ht β, t

t=1

¢0 ¡ ht (φ? ) = h1t (φ? ), h2t (φ? ), h3t (φ? ), h4t (φ? ) , h1t (φ) =

j t Y X

θt−k ,

h3t (φ) =

j=0 k=1

h2t (φ) =

j t Y X

j t Y X

θt−k et−j−1 (φ)1 (et−j−1 (φ) ≤ 0) ,

j=0 k=1

θt−k xt−j−1

and

h4t (φ) =

j=0 k=1

j t Y X

θt−k et−j−1 (φ)1 (et−j−1 (φ) > 0) .

j=0 k=1

Proof: See Appendix. The threshold effect test is based on the following null, H0 : θ1 = θ2 ⇔ R0 β = 0 for R0 = (0, 0, 1, −1), versus the alternative: Ha : θ1 6= θ2 ⇔ R0 β 6= 0,

22

and the proposed statistic is: W1 (r0 ) =

b2 T (R0 β) , b −1 R σ bε2 R0 H

(10)

which under the null, and using the Theorem 8, will converge to a χ21 . This statistic has been proposed by other authors, see for example Wecker (1981) and Elwood (1998), but nobody to the best of our knowledge has proved the asymptotic distribution. The second step, in the strategy testing, is not possible. As we see in section 2.2, in the TARIMA-Shock-Sign case the shocks coming from a noninvertible MA are not transitory given that assumption I.1 does not hold.

5

Finite Sample Performance

This section examines the performance of the proposed statistics in finite samples through Monte Carlo experiments. We focus on the threshold effect test when r0 is unknown which need a bootstrap approximation for the asymptotic distribution. However, given its widely use, the results for the TARIMA-Shock-Sign case when r0 = 0 is known are equally provided. In all the experiments we consider εt iid N (0, 1), a sample size T = 200 and significant levels α = 0.05 and 0.10. Due to the large computational requirements of the simulation design, the number of Bootstrap replications was set at B = 200 and M = 400 for Monte Carlo replications. We start the experiments with the TARIMA-X model. In order to examine the size and power of the proposed threshold effect test we consider the following simple TARIMA-X model: (1 − L)yt = xt = εt − θ1 εt−1 1(|xt−1 | ≤ r) − θ2 εt−1 1(|xt−1 | > r),

(11)

where we set µ = 0 and Φp (L) = 1 to reduce the computational burden. Under the null, θ1 = θ2 = θ. We report in Table 1 the actual size for θ = −0.5, 0 and 0.5.

Table 1: Finite Sample Size for Bootstrap Threshold Effect Test Observable α − level Sup W1,T Mean W1,T Exp W1,T

θ = −0.5 0.05 0.10 0.0450 0.0850 0.0525 0.1000 0.0450 0.0875

θ=0 0.05 0.10 0.0700 0.1350 0.0525 0.0950 0.0700 0.1375

θ = 0.5 0.05 0.10 0.0450 0.0975 0.0525 0.0875 0.0550 0.0875

Note: The DGP is the model (11) with θ1 = θ2 = θ. The test and the estimation is based on model (7) and Algorithm 1

The second experiment consider the TARIMA-Shock. We focus on the size case, con-

23

sidering the model:

(1 − L)yt = xt = εt − θ1 εt−1 1(|εt−1 | ≤ r) − θ2 εt−1 1(|εt−1 | > r).

(12)

As in previous case, to reduce the computational burden we set µ = 0 and Φp (L) = 1. The null is θ1 = θ2 = θ, an IMA(1,1) again, and the results about the size are listed in Table 2.

Table 2: Finite Sample Size for Bootstrap Threshold Effect Test Shock-Size α − level Sup W1,T Mean W1,T Exp W1,T

θ = −0.5 0.05 0.10 0.0425 0.0775 0.0400 0.0800 0.0450 0.0700

θ=0 0.05 0.10 0.0525 0.0975 0.0350 0.0775 0.0500 0.0950

θ = 0.5 0.05 0.10 0.0450 0.0800 0.0350 0.0775 0.0425 0.0800

Note: The DGP is the model (12) with θ1 = θ2 = θ. The test and the estimation is based on model (8) and Algorithm 1

6

Empirical Application

In this section we apply the TARIMA model to test the existence of asymmetries in the persistence of the shocks to US GNP. We start with the sign of the shock. The previous works of Beaudry and Koop (1993) and Pesaran and Potter (1997) conclude the existence of asymmetries in the persistence of the shock based on the sign of the shock. Both papers have opposite conclusions about which shocks are more persistent6 . On the other hand, the work of Elwood (1998) reject the existence of asymmetries in the persistence of the shocks produced by its sign. Finally we analyse the existence of asymmetries in the persistence of the shocks based on its own size. We consider, as in the paper of Pesaran and Potter, yt to be the 100log(RGN P ) such that xt = (1 − L)yt is the growth rate. The data corresponds to the Quarterly Seasonally Adjusted Real Gross National Product of the United States in Millions of chained 2000 Dollars for the period 1954:Q1 to 2006:Q1. As the series is the same for the three asymmetric cases studied here, the real GNP, in the Table 3 shows the estimation results of the Linear model, which correspond to the model under the null of symmetry. 6

In these cases all the shocks are permanent so the more persistent term has to be understood as having a bigger effect on the limit.

24

Table 3: Estimation results for ARIMA models φ1

φ2

0.481 (0.180) −0.405 (0.258) 0.283 (0.066)

0.286 (0.080)

ARIMA(0,1,1) ARIMA(1,1,1) ARIMA(2,1,1) ARIMA(1,1,0)

µ 0.823 (0.075) 0.417 (0.155) 0.902 (0.218) 0.579 (0.081)

θ −0.224 (0.068) 0.215 (0.204) 0.668 (0.265)

AIC/SC 2.603 2.636 2.566 2.614 2.570 2.635 2.564 2.597

Note: In brackets are the standard deviation. ARIMA model:(1 − φ1 L − φ2 L2 )(1 − L)yt = µ + (1 − θL)εt

6.1 6.1.1

Asymmetries in the GNP rate Asymmetries in the sign of the shock.TARIMA-Shock-Sign Case.

Now, given the previous results, we deal with the question about wether that asymmetric persistence behavior on the sign of the shock can be produced by the sign of the own shock. This question has been considered previously by Elwood (1998). For that he proposed a pure TAR and TMA model with the sign of the shock as threshold variable. He uses a Maximum Likelihood estimation method and a Quasi-likelihood ratio test, although no asymptotic theory was proved. So, as well as redefining the data, we answer the question also using the LS estimation method and the Wald type test proposed in the previous sections. The results are showed in Table 4. As in the case of Elwood we fail to reject the linearity when the source of the asymmetry is produced by the pure sign of the shock εt > 0.

Table 4: Estimation results for ARTIMA-Shock-Sign models φ1

φ2

ARTIMA-S-Sign (0,1,1,1) ARTIMA-S-Sign (1,1,1,1) ARTIMA-S-Sign (2,1,1,1)

0.469 (0.095) −0.387 (0.284)

0.283 (0.083)

µ 0.853 (0.094) 0.433 (0.093) 0.901 (0.227)

θ2 −0.161 (0.104) 0.218 (0.165) −0.623 (0.305)

θ1 −0.281 (0.112) 0.183 (0.092) −0.681 (0.296)

statistic 0.447 0.038 0.130

Note: In brackets are the standard deviation. ARTIMA-S-Sign model:(1 − φ1 L − φ2 L2 )(1 − L)yt = µ + 1 − θL)εt 1(εt ≤ 0) + 1 − θ2 L)εt 1(εt > 0)

25

AIC/SC 2.608 2.657 2.570 2.635 2.574 2.656

6.2

Asymmetries persistence on the size of the shock

Finally, as in Gonzalo and Martinez (2006), we test the existence of asymmetries in the persistence of the shocks due to its size and the validity of the identification of transitory shocks based on that criterion. However, in this case we apply the Wald type test proposed in the current paper to the extended real GNP data. The testing results for the null of symmetry are shown in Table 5.

Table 5: Testing results for ARTIMA-Shock-Size models ARTIMA-S-Size

φ1

φ2

(0,1,1,1) 0.525 (0.077) 0.301 (0.298)

(1,1,1,1) (2,1,1,1)

0.090 (0.117)

µ 0.831 (0.035) 0.404 (0.072) 0.492 (0.175)

θ2 −0.225 (0.061) 0.260 (0.093) 0.042 (0.304)

θ1 0.971 (0.567) 0.872 (0.353) 0.655 (0.452)

r

p-value

0.249

0.288

0.407

0.046

0.394

0.392

Note: In brackets are the standard deviation. ARTIMA-S-Size model:(1 − φ1 L − φ2 L2 )(1 − L)yt = µ + (1 − θ1 L)εt 1(|εt | ≤ r) + (1 − θ2 L)εt 1(|εt | > r).

The p − value has been obtained using the proposed Bootstrap method for B=500 and ri ∈ [b ε(0.25T ) , εb(0.75T ) ] for i = 1, ..., 50.

For the TARIMA-Shock-Size(1,1,1,1) we reject the null of linearity at a significant level of 5%, as in the previous work of Gonzalo and Martinez. In the Table 6 we can see the estimation results for the two step estimation method and the estimation under the null of θ2 = 1. Using the Theorem 7 we can test that θ2 = 1. The p − value of this test is 0.2919. Assuming that the shocks have a symmetric distribution and both assumptions, I.0 and I.1, holds we can not reject that the small shocks are transitory.

Table 6: Estimation results for ARTIMA-Shock-Size models ARTIMA-Shock-Size(1,1,1,1) First Step Second Step Restricted θ2 = 1

φ1 0.525 (0.077) 0.705 (0.191) 0.461 (0.049)

µ 0.404 (0.072) 0.205 (0.176) 0.455 (0.050)

θ2 0.260 (0.093) 0.436 (0.206) 0.216 (0.052)

θ1 0.872 (0.353) 1.476 (0.451) 1.000

r 0.407 0.407 0.407

AIC/SC 2.558 2.639 2.543 2.607 2.550 2.599

Note: In brackets are the standard deviation. ARTIMA-S-Size model:(1 − φ1 L − φ2 L2 )(1 − L)yt = µ + (1 − θL)εt 1(|εt | ≤ r) + (1 − θ2 L)εt 1(|εt | > r).

The number of parameters for the AIC and BC criterions is k = 3.

Using the estimation results of the estimation when θ2 = 1 is imposed, we obtain the new Permanent-Transitory (P-T) decompositions introduced in Gonzalo and Martinez

26

AIC/SC 2.586 2.651 2.558 2.639 2.569 2.667

(2006). In the Figure 2 we can see the logarithm of the real GNP times 100 with the NBER reference points. The shady bars cover the periods from peaks to troughs.

Figure 2: Real GNP and NBER reference points. 1954:04-2006:01 57III

69IV

100*Log(RGNP)

934

61I

80I 75I

90III 82IV

01IV

850

766

60II

73IV

58II

70IV

81III

01I

80III

91I

Time

Figure 3: Orthogonal P-T Decomposition and NBER reference points 57III

Permanent Component

934

69IV 61I

80I 75I

90III 82IV

01IV

850

60II

73IV

81III

01I

766 58II

70IV

80III

91I

Time

57III

69IV

80I

90III

0.3929

Transitory Component

61I

75I

82IV

01IV

0.0

60II

73IV

81III

01I

-0.6089 58II

70IV

80III Time

27

91I

The Figure 3 shows the orthogonal P-T Decomposition, which is given by: (1 − φL)(1 − L)ytP = µ + (1 − θ2 L)εt 1(|εt | > r) (1 − φL)ytT = (1 − θ1 L)εt 1(|εt | ≤ r). Figures 4 and 6 show the Beveridge and Nelson Decomposition for the TARIMA and ARIMA(1,1,1) models respectively, which is given by the following expressions: µ 1 − θ2 + εt 1(|εt | > r) 1−φ 1−φ θ2 − φ θ1 − φ (1 − φL)ytT = εt 1(|εt | > r) + εt 1(|εt | ≤ r), 1−φ 1−φ (1 − L)ytP =

for the TARIMA model, and 1−θ µ + εt 1−φ 1−φ θ−φ (1 − φL)ytT = εt , 1−φ (1 − L)ytP =

for the ARIMA(1,1,1) model. The permanent component is defined by a random walk. In the linear case, the permanent and transitory shocks are perfect correlated (they are the same). However in the nonlinear BN P-T decomposition, they are never perfectly correlated and the correlation depends on the value of θ1 , θ2 and φ. From the three previous figures we can conclude than in the orthogonal decomposition the permanent component is really smoother. For the Beveridge and Nelson decompositions, nonlinear and linear, the transitory component is contra-cyclical. For the TARIMA model, a deeper knowledge of the business cycle can be obtained focusing on the graph of permanent and transitory shocks. These are in Figure 7. Clearly the recession period are characterized by negative permanent shocks and the recovery from the crisis for positive permanent shocks. The transitory shocks has a main role in the expansion periods. Equally it can be observed the effect of the Great Moderation, a scaling down of the size in permanent shock (the bigger ones)7 . Finally, the Table 7 shows some results about the variance of the permanent and transitory shocks and components. We can see how in the TARIMA model the variance of the permanent shocks is less than half in the linear case. Spite of this, the increment of permanent component in B-N decompositions are the same in all the cases by definition. To finish with the application we estimate the GIRF for the estimated TARIMA-Shock7 It can be of interest to consider the threshold variable r depending on the conditional variance or almost an structural change in that parameter.

28

Figure 4: Nonlinear Beveridge and Nelson P-T Decomposition and NBER reference points 57III

Permanent Component

934

69IV 61I

80I 75I

90III 82IV

01IV

850

60II

73IV

81III

01I

766 58II

70IV

80III

91I

Time

57III

69IV

80I

90III

1.8459

Transitory Component

61I

75I

82IV

01IV

0.0

60II

73IV

81III

01I

-1.2898 58II

70IV

80III

91I

Time

Figure 5: Linear Beveridge and Nelson P-T Decomposition and NBER reference points 57III

Permanent Component

934

69IV 61I

80I 75I

90III 82IV

01IV

850

60II

73IV

81III

01I

766 58II

70IV

80III

91I

Time

57III

69IV

80I

90III

2.0285

Transitory Component

61I

75I

82IV

01IV

0.0

60II

73IV

81III

01I

-1.4066 58II

70IV

80III Time

29

91I

Figure 6: Permanent and Transitory shocks and NBER reference points 57III

69IV

80I

61I

90III

75I

82IV

01IV

Permanent Shocks

2

0

60II

73IV

81III

01I

-3 58II

70IV

80III

91I

Time

57III

69IV

80I

90III

0.3830

Transitory Shocks

61I

75I

82IV

01IV

0.0

60II

73IV

81III

01I

-0.4044 58II

70IV

80III

91I

Time

Figure 7: Median of the

GIRF (k,εt ,=t−1 ) GIRF (0,εt ,=t−1 ) (b)

10

20

1.0 0.6

30

0

10

20 Horizon

(c)

(d)

30

0

10

20

0.5 0.0 -0.5

Positive Shock Negative Shock

Positive Shock Negative Shock

-1.0

% Change in Level of RGNP

-0.5

1.0

0.5 1.0 1.5

Horizon

-1.5

% Change in Level of RGNP

0

Big Shock Small Shock

0.2

% Change in Level of RGNP

1.4 1.3 1.2 1.1

Positive Shock Negative Shock

1.0

% Change in Level of RGNP

1.4

(a)

30

0

Horizon

10

20 Horizon

(a)Unconditional. (b)Unconditional. (c)Big Shocks. (d)Small Shocks.

30

30

Figure 8: The GIRF (22, εt , =t−1 ) (b) 0 -1 -2 -4

-3

% Change in Level of RGNP

-1 -2 -3 -4 -5

% Change in Level of RGNP

0

(a)

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

-3.0

-2.5

-2.0

-1.5

Shock

Shock

(c)

(d)

-1.0

-0.5

0.0

0.0

0.5

1.0

1.5

2.0

1 0 -1 -2

% Change in Level of RGNP

0 -5 -10 -15

% Change in Level of RGNP

2

-3.0

2.5

0.0

0.5

Shock

1.0

1.5

2.0

2.5

Shock

(a)Negative Shocks. (b)Negative Shocks, without extremes. (c)Positives Shocks. (d)Positive Shocks, without extremes. Table 7: Variance results for P-T decompositions

Ortogonal P-T Non linear BN P-T Linear BN P-T, ARIMA(1,1,1) Linear BN P-T, ARIMA(1,1,0)

V (εt ) 0.7261 0.7261 0.7362 0.7425

V (εPt ) 0.7064 0.7064 1.6835 1.4444

V (εTt ) 0.0207 0.0207 0.1931 0.1157

V (∆ytP ) 0.8099 1.5219 1.7123 1.4750

V (ytT ) 0.0247 0.2221 0.2528 0.1255

Note: In the linear BN decomposition the variance of the permanent shock and the increment of perµ manent component has to be the same. The difference is due to the approximation to 1−φ .

Size, in the same way described in the TARIMA-X case. The result is shown in Figures 7 y 8. The GIRF for small shocks converge to zero so the small shocks are transitory. Moreover the size asymmetry is the only one that seen significative, given that there is no asymmetry in the sign. In Figure 8 it is shown the problems to estimate the GIRF for very small values of εt .

31

7

Conclusion

In time series econometric models shock issues play a key role, particularly identification of permanent and transitory shocks. In the literature this has been done in a very ad hoc way. In this paper we link and also explain the persistence of the shocks to some characteristic of them or of the economy. This is done by proposing a new type of threshold model. This model allows to test wether a given characteristic causes the different degrees of persistence. In this way in our GNP application we conclude that large shocks are persistent while small ones are transitory. Also we find that (in mean) shocks in recessions are slightly more persistent than shocks in expansions. Extensions and applications to a multivariate framework are under current research by the authors.

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8

Appendix

8.1

Proofs

Proof of Lemma 1. From equation (3): εt − θ1 εt−1 if zt−1 ≤ r Φp (L)(1 − L)yt = Φp (L)xt = ut = µ + ε −θ ε if zt−1 > r, t 2 t−1 When zt = g(xt−j ), the strictly stationarity and ergodicity of xt can be proved in the same way of Theorem 4.1 of Ling, Tong and Li (2006), defining in this case: Φp (L)Sn (t) = µ +

εt

if n < max{p, j}

[−θ + ψ1(g (S 2 n−j−1 (t − j − 1)) ≤ r)] εt−1 + εt

if n ≥ max{p, j}

with ψ = (θ2 − θ1 ) and LSn (t) = Sn−1 (t − 1). If zt is different from g(xt−j ), from Theorem 1.3.3 of Taniguchi and Kakizawa (2000), it is enough to prove the strictly stationarity and ergodicity of ut given that G.3 holds. In this sense, the strictly stationarity and ergodicity of {εt , zt } it is enough for the strictly stationarity and ergodicity of ut , from Theorem 1.3.3 again, and then for xt . Proof of Lemma 2. For the case a), as zt is observable, we have Φp (L) xt = µ + εt − θt−1 (zt−1 ) εt−1 , εbt = Φp (L) xt − µ + θt−1 (zt−1 ) εbt−1 , vt = εbt − εt = θt−1 (zt−1 ) (b εt−1 − εt−1 ) . Then it is easy to see that vt2 =

t Y

2 θt−j (zt−j ) v02 .

j=1

The condition for invertibility of yt is lim E

t→∞

t Y

2 θt−j (zt−j ) v02 = 0.

j=1

33

If AX.0 holds, using the Iterated Law of Expectation (ILE), lim E

t→∞

t Y

2 θt−j (zt−j ) v02 = lim E E t→∞

j=1

≤

[ t−1 ] lim λ1 q t→∞

q Y

, 2 θt−j (zt−j )

=t−q−1

j=1

t Y

2 θt−j (zt−j ) v02

j=q+1

¡ ¢ 2q (1 + δ) E v02 = 0,

¡ ¢ ¡ ¢ where [x] is the integer part of x and given that E v02 < ∞. This is implied by E ε20 < ∞. For the case b), TARIMA-Shock-Size, from equation (3),

εbt = Φp (L) xt + µ + θ1 1 (|b εt−1 | ≤ r) εbt−1 + θ2 1 (|b εt−1 | > r) εbt−1 . Define the events

A1,t−1 = {|εt−1 | ≤ r, |b εt−1 | ≤ r}

A2,t−1 = {|εt−1 | > r, |b εt−1 | > r}

A3,t−1 = {|εt−1 | > r, |b εt−1 | ≤ r}

A4,t−1 = {|εt−1 | ≤ r, |b εt−1 | > r} .

(13)

Then, if |θ1 | > |θ2 | , we have E |vt | = E |b εt − εt | ≤ hrE [1 (A3,t−1 ) + 1 (A4,t−1 )]+E {[|θ1 | 1 (|εt−1 | ≤ r) + |θ2 | 1 (|εt−1 | > r)] |vt−1 |} . First we calculate E [1 (A3,t−1 ) + 1 (A4,t−1 )]. Using that vt−1 is =t−2 measurable and taking f ∗ = maxe 2f (e),

E [1 (A3,t−1 ) + 1 (A4,t−1 )] ≤ E {E [ 1 (−r ≤ εt−1 ≤ −r − vt−1 ) + 1 (r − vt−1 ≤ εt−1 ≤ r)| Ft−2 ]} + E {E [ 1 (−r − vt−1 ≤ εt−1 ≤ −r) + 1 (r ≤ εt−1 ≤ r − vt−1 )| Ft−2 ]} ¯ ( "Z #) Z r Z −r Z r−vt−1 ¯ −r−vt−1 ¯ =E E f (ε) ∂ε + f (ε) ∂ε + f (ε) ∂ε + f (ε) ∂ε¯ Ft−2 ¯ −r r−vt−1 −r−vt−1 r ≤ f ∗ E (|vt−1 |) , so, E |vt | ≤ [hrf ∗ + E(|θ (εt−1 )|)] E |vt−1 | = λt1 E |v0 | . ¡ 2 ¢ For invertibility we need the convergence of E vt−1 , then £ ¤ 2 vt2 ≤ h2 r2 [1 (A3,t−1 ) + 1 (A4,t−1 )] + θ12 1 (|εt−1 | ≤ r) + θ22 1 (|εt−1 | > r) vt−1 + + 2hr [1 (A4,t−1 ) + 1 (A3,t−1 )] [|θ1 | 1 (|εt−1 | > r) + |θ2 | 1 (|εt−1 | ≤ r)] |vt−1 | .

34

Taking λ1 = [hrf ∗ + E(|θ(εt−1 )|)] and K = h2 r2 f ∗ + 4hr, ¡ ¢ ¡ 2 ¢ ¡ 2 ¢ t−1 E vt2 ≤ KE |vt−1 | + E(|θ(εt−1 )|)E vt−1 ≤ Kλt−1 1 E |v0 | + λ1 E vt−1 ≤ tKλ1 E |v0 | . ¡ ¢ Then, by ASZ.0 a), λ1 < 1 and we obtain that limt→∞ E vt2 = 0. If |θ2 | > |θ1 |, using ASZ.0 b), we obtain |vt | ≤ hr [1 (A3,t−1 ) |b εt−1 | + 1 (A4,t−1 ) |εt−1 |] + |θ2 | , E |vt | ≤ λt1 E |v0 | . The rest of the proof is similar to the other case. Finally, for case c), in a similar way of case b),

vt = (b εt − εt ) = θ1 vt−1 1 (εt−1 ≤ r, εbt−1 ≤ r) + θ2 vt−1 1 (εt−1 > r, εbt−1 > r) + (θ2 εbt−1 − θ1 εt−1 ) 1 (εt−1 ≤ r, εbt−1 > r) + (θ1 εbt−1 − θ2 εt−1 ) 1 (εt−1 > r, εbt−1 ≤ r) . Using that

|θ2 εbt−1 − θ1 εt−1 | 1 (εt−1 ≤ r, εbt−1 > r) ≤ [maxi |θi | |vt−1 | + h|r|] 1 (εt−1 ≤ r, εbt−1 > r) |θ1 εbt−1 − θ2 εt−1 | 1 (εt−1 > r, εbt−1 ≤ r) ≤ [maxi |θi | |vt−1 | + h|r|] 1 (εt−1 > r, εbt−1 ≤ r) , it can be obtained, using the assumption ASG.0,that

|vt | ≤ max |θi | |vt−1 | + h|r| [1 (r ≤ εt−1 ≤ r − vt−1 ) + 1 (r − vt−1 ≤ εt−1 ≤ r)] i

1 kvt k1 ≤ max |θi | kvt−1 k1 + h|r| f ∗ kvt−1 k1 ≤ λ1 kvt−1 k1 . i 2 Now, following the same steps of the previous cases, we obtain the desired result, ¯ ¯ limt→∞ E ¯vt2 ¯ = 0. Proof of Lemma 3. Given that Φp (L) has all its roots outside the unit circle, Ψ (L) with

35

ψ0 = 1 and ψ∞ = 0 always exits. First we write (3) in the following way, (1 − L) yt+k = Ψ (1) µ + Ψ (L) (εt+k − θt+k−1 εt+k−1 ) yt+k = (t + k)Ψ (1) µ + y 0 + εt+k +

t+k−1 X

εt+k−s (1 − θt+k−s )

s=1

= (t + k)Ψ (1) µ + y 0 + εt+k + + εt (1 − θt )

k−1 X

ψj + ψk +

with y 0 = y0 +

t−1 X

s−1 X j=0

εt+k−s (1 − θt+k−s )

s=1

j=0

k−1 X

s−1 X

εt−s (1 − θt−s )

s=1

s−1+k X

ψj + ψs

ψj + ψs +

j=0

ψj + ψs+k ,

j=0

h Pt+k−1+s i ε ψ − θ ψ + (1 − θ ) ψj . Using I.1 and the −s t+k+s −s s+1 −s s=0 j=s+2

P∞

Law of Iterated Expectation (LIE), E ( yt+k | vt , wt−1 ) − E ( yt+k | wt−1 ) = (1 − θt )

k−1 X

ψj + ψk εt .

j=0

Then, using I.0, ψk ε t h i GI (k, vt , wt−1 ) = (1 − θ) Pk−1 ψ + ψ ε k t j=0 j

9

if

θt = 1

if

θt 6= 1.

References

Andrews, D.W.K., 1993. Tests for Parameter Instability and Structural Change with Unknown Change Point, Econometrica 61, 821-856. Andrews, D.W.K., and Ploberger, W. 1994. Optimal Tests When a Nuisance Parameter Is Present Only Under the Alternative. Econometrica, 62, 1383-1414. Beaudry, P., Koop, G., 1993. Do recessions permanently change output?. Journal of Monetary Economics 31, 149-164. Beveridge, S., Nelson, C.R., 1981. A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘Business Cycle’. Journal of Monetary Economics 7, 151-174. Bierens, H.J., 1983. Uniform consistency of kernel estimators of a regression function under generalized conditions. Journal of the American Statistical Association 77, 699-707. Billingsley, P. 1968. Convergence of Probability Measures. John Wiley & Sons, New York.

36

Caner, M., Hansen, B.E., 2001. Threshold autoregression with a unit root. Econometrica, 69, 1555-1596. Chan, K.S., 1993. Consistency and limiting distribution of the least squares estimator of a Threshold Autoregressive model. The Annals of Statistics 21, 520-533. Chan K.S., Tsay, R.S., 1998. Limiting properties of the least squares estimator of a continuous threshold autoregressive model. Biometrika 85, 413-426. Clark, P.K., 1987. The cyclical component of U.S. economic activity. Quarterly Journal of Economics 102, 797-814. Davies, R.B., 1977. Hypothesis Testing When a Nuisance Parameter Is Present Only Under the Alternative. Biometrika, 64, 247-254. Davies, R.B., 1987. Hypothesis Testing When a Nuisance Parameter Is Present Only Under the Alternative. Biometrika, 74, 33-43. Davidson, J., 1994. Stochastic Limit Theory. Oxford University Press. De Gooijer, J.G., 1998. On threshold moving-average models. Journal of Time Series Analysis 19, 1-18. Elwood, S., 1998. Is the persistence of shocks to output asymmetric?. Journal of Monetary Economics 41, 411-426. Engle, R.F., Smith, A.D., 1999. Stochastic Permanent Breaks. Review of Economics and Statistics 84, 553-574. Gonzalo, J., Wolf, M. 2005. Subsampling inference in threshold autoregressive models. Journal of econometrics, 127, 201-224. Gonzalo, J., Martinez, O., 2006. Large shocks vs. small shocks. Or does size matter? May be so. Journal of Econometrics, 127(1-2), pages 311-347. Gonzalo, J., Martinez, O., 2007. Threshold permanent-transitory decomposition. Mimeo. Granger, C., Andersen, A.P., 1978. On the invertibility of time series models. Stochastic Processes and Their Applications, 8, 87-92. Guay, A., Scaillet, O., 2003. Indirect inference, nuisance parameter and threshold moving average models. Journal of Business and Economic Statistics 21, 122-132. Hallin, M., 1980. Invertibility and generalized invertibility of time-series models. Journal of the Royal Statistical Society Series B 42, 210-212. Hansen, B.E., 1996. Inference When a Nuisance Parameter Is Not Identified Under the Null Hypothesis. Econometrica, 64, 413-430. Hansen, B.E., 2000. Sample Splitting and Threshold Estimation. Econometrica, 68, 575-603. Harvey, A.C., 1985. Trends and cycles in macroeconomics time series. Journal of Business and Economic Statistics 3, 216-27.

37

Klimko, L.A., Nelson, P.I., 1978. On conditional least squares estimation for stochastic processes. Annals of Statistics 6, 629-642. Koop, G., Pesaran, M.H., Potter, S.M., 1996. Impulse response analysis in nonlinear multivariate models. Journal of Econometrics 74, 119-147. Kuan, C.M., Huang, Y.L. and Tsay, R.S., 2005. An unobserved-component model with switching permanent and transitory innovations. Journal of Business and Economic Statistics, 23, 443454. Ling, S., Tong, H. and Li, D., 2006. Ergodicity and Invertibility of Threshold MA Models.Bernoulli (forthcoming). Ling, S., Tong, H., 2005. Testing for a linear MA model against threshold MA models. Annals of Statistics, 33, no. 6, 25292552. McLeish, D.L., 1975. A maximal inequality and dependent strong laws. Annals of Probability, 3, 829-839. Pesaran, M.H., Potter, S.M., 1997. A floor and ceiling model of US Output. Journal of Economic Dynamics and Control, 21, 661-695. Pierce, D.A., 1979. Signal extraction error in nonstationary time series. The Annals of Statistics 7, 1303-1320. Potter, S.M., 2000. Nonlinear Impulse Response Functions. Journal of Economic Dynamics and Control 24, 1425-46. Quah, D., 1992. The relative importance of permanent and transitory components: identification and some theoretical bounds. Econometrica 60, 107-118. Taniguchi, M., and Kakizawa, Y., 2000. Asymptotic Theory of Statistical Inference for Time Series. New York, Springer-Verlag. Watson, M.W., 1986. Univariate detrending methods with stochastic trends. Journal of Monetary Economics 18, 49-75. Wecker, W.E., 1981. Asymmetric time series. Journal of the American Statistical Association 76, 16-21.

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