Modelling good and bad volatility - Core

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ond moments of positive returns (good volatility) to evolve differently from the ..... is a nonrandom point in R, ρ(Yn) is the spectral radius of the matrix Yn and [Yn]ij.
Modelling good and bad volatility Matteo M. Pelagatti∗ Department of Statistics, Universit` a degli Studi di Milano-Bicocca, Milan, Italy

Abstract The returns of many financial assets show significant skewness, but in the literature this issue is only marginally dealt with. Our conjecture is that this distributional asymmetry may be due to two different dynamics in positive and negative returns. In this paper we propose a process that allows the simultaneous modelling of skewed conditional returns and different dynamics in their conditional second moments. The main stochastic properties of the model are analyzed and necessary and sufficient conditions for weak and strict stationarity are derived. An application to the daily returns on the principal index of the London Stock Exchange supports our model when compared to other frequently used GARCH-type models, which are nested into ours. Keywords: Volatility, Skewness, GARCH, Asymmetric Dynamics, Stationarity. JEL codes: C22, C53, G10.

1

Introduction

It is quite common for financial assets returns to show conditional heteroskedasticity, leptokurtosis and skewness. The first two properties are usually dealt with GARCH or Stochastic Volatility type models possibly with fat-tailed distributions. However, skewness has received much less attention in literature. In fact, skewness is usually obtained as a byproduct of models for leverage effects such as the Exponential GARCH (Nelson 1991) and the GJR-GARCH (Glosten et al. 1993). In most cases, though, the standardized residuals of these models still show significant skewness. Table 1 reports the sample skewness and its significance for the daily percentage returns of the FTSE100 index and for the standardized residuals with respect to asymmetric GARCH models with GED (conditional) distribution: the significance of the skewness coefficient appears even enhanced in the standardized residuals. Similarly, the absolute values of the minima are circa twice as large as the maxima, evidencing a significantly fatter negative tail. On some stock returns like ENEL1 , the skewness coefficient gets even larger after passing the EGARCH and GJR-GARCH filters. Some authors have modelled skewness by using GARCH-type processes with conditional skewed distributions (Harris et al. 2004, Lambert and Laurent 2001, Lambert and Laurent 2002, Lanne and Saikkonnen 2004, Miettinen 2005), and in the OxMetrics package G@rch ∗ Full address: Department of Statistics, Universit` a degli Studi di Milano-Bicocca, Via Bicocca degli Arcimboldi, 8, I-20126 Milano. E-mail: [email protected] 1 ENEL is Italy’s largest power company (the privatized former public monopolist) and Europe’s third largest listed utility.

1

Returns (%) St.Resid EGARCH St.Resid GJR-GARCH

Mean 0.013 -0.011 -0.012

Max 3.299 5.830 6.144

Min -5.658 -13.047 -13.326

St.Dev. 0.439 1.002 1.004

Skew -0.734∗ -0.530∗∗ -0.554∗∗

Ex.Kurt. 10.426 5.285 5.666



and ∗∗ denote significance at, respectively, 5% and 1% levels, and are based on the bootstrap test of Lisi (2005).

Table 1: Daily returns on FTSE100 from 2nd Jan 1984 to 11th Oct 2007. 4 by Laurent and Peters2 the skewed Student’s t distribution may be used for modelling conditional returns. Some other authors have built GARCH type models for the conditional skewness (Hansen 1994, Harvey and Siddique 1999). In this paper we pursue a different approach. We argue that the dynamics of negative returns may differ from that of positive returns and propose a class of processes that may be used to model these different behaviours3 . More precisely, we allow the conditional second moments of positive returns (good volatility) to evolve differently from the conditional second moments of negative returns (bad volatility). The plan of the rest of the article is as follows. In Section 2 we define the process in terms of its conditional distribution. The main stochastic properties of the process, such as the conditional moments and the conditions for the existence of weakly and strictly stationary solutions are analyzed in Section 3. Section 4 contains an application of the process to the main index of the London Stock Exchange (FTSE100), which demonstrates that the dynamics of real data may be captured by our model better than by other frequently used GARCH type processes. Finally, we give some concluding remarks in Section 5. Two appendices at the very end of the article report the density and the moments of the half generalized error distribution used throughout the paper and some eigenvalues and matrix norms necessary for assessing the stationarity of the process.

2

The Model

Let rt be the time series of returns. Define the series of positive and negative returns as rt− := rt · I(rt < 0)

rt+ := rt · I(rt ≥ 0),

with I(·) indicator function. The returns are modelled as rt |{rt < 0, Ft−1 } ∼ D− (h− t )

rt |{rt > 0, Ft−1 } ∼ D+ (h+ t ),

where Ft is the σ-field generated by rt , rt−1 , . . ., D+ a family of distributions with positive − support and second non-central moment h+ t and D a family of distributions with negative − support and second non-central moment ht . The two distributions may, of course, depend on other parameters, such as location and tail parameters. 2 As far as we know, this is the only mainstream software package that allows for skewed conditional distributions in GARCH modelling. We used this package for estimating the EGARCH and GJR-GARCH models of Table 1. 3 While presenting this work at the 5th OxMetrics User Conference, S´ ebastien Laurent acquainted me with the article by El Babsiri and Zakoian (2001), whose basic idea of treating positive and negative returns asymmetrically is the same as the one we pursue in this paper. However, the model we propose is different from theirs and has the advantage of nesting other frequently used models such as the GARCH and the GJR-GARCH, both with possibly fat-tailed and skewed conditional distributions. This allows the use of maximum-likelihood based statistics for testing against these subset models.

2

It is useful for later developments to use conditional distributions with unitary second moments, such as the Half-GED, whose main properties are reported in Appendix A. By doing so, we can write q q h− rt |Ft−1 = ξt I(ξt ≥ 0) h+ + ξ (ξ < 0) (2.1) tI t t t with ξt i.i.d. process with 2 2 E[ξt |ξt ≥ 0] = E[ξt |ξt < 0] = 1,

p := E[I(ξt ≥ 0)],

q := E[I(ξt < 0)] = 1 − p.

Thus, p and q are the probabilities of, respectively, positive and negative returns, and will be assumed constant throughout the paper. The conditional second moments are let evolve according to the following bivariate difference equation # "  +   + + 2 ht−1 ht rt−1 + B (2.2) = ω + A − 2 h− h− r t t−1 t−1 with ω (2 × 1) non-negative vector, A and B (2 × 2) matrices with non-negative elements. The non-negativity conditions are sufficient and necessary for the non-negativity of the conditional second moments. The matrix B will be taken diagonal, so that each element of ht depends only on the same element in ht−1 . This will avoid collinearity problems, since we expect the two components of ht to be highly correlated in real world applications.

3

Stochastic Properties of the Model

Form equations (2.1) and (2.2) we have the following representation of the square of positive and negative returns: rt2 |Ft−1 = Xt ht (3.3) ht = ω + (AXt−1 + B)ht−1 with

rt+ rt−

 rt :=



h+ t h− t

 ht :=

 Xt :=

(3.4)

 2  ξt I(ξt ≥ 0) 0 . 0 ξt2 I(ξt < 0)

Notice that Xt is an i.i.d. sequence of nonnegative random matrices.

3.1

Conditional moments

It is easy to see that the return process rt is not a martingale difference (MD) sequence. In fact, if we model rt+ and rt− using two independent distributions, the conditional expectation may be easily obtained as µt|t−1 := E[rt |Ft−1 ]     q q − F + ξ (ξ < 0) h F = E ξt I(ξt ≥ 0) h+ E I t−1 t t t−1 t t q q − h− = µ+ · p · h+ t +µ ·q· t , where µ+ := E(ξt |ξt ≥ 0) and µ− := E(ξt |ξt < 0). 3

(3.5)

Using the Half-GED described in Appendix A, we have  −1/2 µ+ = Γ(2/ν + ) Γ(1/ν + )Γ(3/ν + )  −1/2 . µ− = −Γ(2/ν − ) Γ(1/ν − )Γ(3/ν − ) If the MD property is considered an essential feature, it may be obtained by subtracting the conditional mean from the process rt : r˜t =: rt − µt|t−1 . We do not use the process r˜t in this paper directly, since a financial economist would anyway change the conditional expectation of the process for embedding risk-preferences or some form of absence of arbitrage. The k-step ahead prediction (conditional expectation) of rt is nontrivial since it involves √ conditional expectations of the process ht . Probably the easiest way to get approximations is by means of the following Taylor expansion. Let y be a nonnegative random variable with expectation µ, then, if the needed moments exist, we have 1 −5/2 1 −3/2 √ 1/2 2 3 E[ y] ≈ µ − µ E[(y − µ) ] + µ E[(y − µ) ]. 8 16

(3.6)

Of course, the above approximation may be stopped at any desired order. The equations for forecasting the volatilities are easily obtained by the law of iterated expectations: E[ht+1 |Ft ] = ht+1 "k−2 # X i (AX + B) ω + (AX + B)k−1 ht+1 , E[ht+k |Ft ] =

k = 2, 3, . . .

i=0

As for the conditional variance, since ht is Ft−1 -measurable, we have E[rt2 |Ft−1 ] = ht , and after some easy but tedious algebra we obtain m2t|t−1 := E[˜ rt2 |Ft−1 ] q     − + −2 +2 + − h+ = p 1 − pµ h− ht − 2pqµ µ t ht + q 1 − qµ t .

(3.7)

The third conditional central moment, necessary for computing the conditional skewness, is given by   3/2 + 2 +3 m3t|t−1 := p µ+ − 2pµ + p µ (h+ + t ) 3   q 2 − 2pqµ− 1 − 2pµ+ h+ h− t t + (3.8) q  2 − − 2pqµ+ 1 − 2qµ− h+ t ht +   − 2 −3 3/2 + q µ− − 2qµ + q µ (h− , t ) 3 − where µ+ 3 and µ3 are the third moments of the distributions of ξt |ξ ≥ 0 and ξt |ξ < 0, respectively (refer to Appendix A for the moments of the HGED).

4

Dividing equation (3.8) by equation (3.7) raised to the power of 3/2 , we obtain the formula for the conditional skewness: m3t|t−1 γ1 t|t−1 := . 3/2 m2 t|t−1

3.2

Weak stationarity and second moments

By taking expectations of both sides of equation (3.4), we get ¯ t = ω + (AX ¯ t−1 , ¯ + B)h h ¯ t := E[ht ] and where we have set h  ¯ := E(Xt ) = p X 0

 0 . q

¯ t to converge to an equilibrium, It is well-known that a necessary and sufficient condition for h ¯ say h, which does not depend on initial conditions is ¯ + B) < 1, ρ(AX

(3.9)

¯ +B are reported in Appendix where ρ(·) denotes the spectral radius. The eigenvalues of AX B. When this condition holds the long run volatilities are given by ¯ = (I − AX ¯ − B)−1 ω, h ¯ − B) is invertible, and provided that (I − AX 2 ¯ ¯ h. E[rt ] = X

(3.10)

The same results may be obtained from the VARMA representation of the squared positive and negative returns process. Let’s define the process ¯ t. ηt := rt2 − Xh

(3.11)

It easy to show that ηt is a MD sequence: 2 ¯ E[ht |Ft−1 ] = Xh ¯ t − Xh ¯ t = 0. E[ηt |Ft−1 ] = E[rt |Ft−1 ] − X

From equation (3.11) we have ¯ −1 r 2 − X ¯ −1 ηt , ht = X t and substituting in equation (2.2), we obtain the VARMA representation of rt2 , ¯ + (XA ¯ + XB ¯ X ¯ −1 )r 2 + ηt + XB ¯ X ¯ −1 ηt−1 , rt2 = Xω t−1 which is well known to have causal stationary solutions iff ¯ + XB ¯ X ¯ −1 ) < 1. ρ(XA

5

(3.12)

Since the spectrum of a matrix is invariant under similarity transformations, conditions (3.9) and (3.12) are equivalent, in fact: ¯ −1 (XA ¯ + XB ¯ X ¯ −1 )X ¯ = AX ¯ + B. X The expectation of rt2 may be derived thorough the VARMA representation as well: 2 ¯ − XB ¯ X ¯ −1 )−1 Xω. ¯ E(rt ) = (I2 − XA

(3.13)

The equivalence of this expressions with (3.10) is easily proved applying the result (L¨ utkepohl 1996, p.29) (I + XY )−1 X = X(I + Y X)−1 that holds if I + XY is non-singular and X and Y are square matrices. We summarize these results in the following proposition. For notational brevity’s sake put Gt = AXt + B. Proposition 1. The process rt is weakly stationary if and only if ρ(E[G0 ]) < 1. Under this condition, we have ¯ := E(ht ) = (I − E[G0 ])−1 ω, h 2 ¯ ¯ h, E(rt ) = X p + − E(rt ) = [µ · p µ · q] E ht . An approximation to the unconditional expectation of rt may be obtained using the Taylor expansion (3.6).

3.3

Strict stationarity and ergodicity

In order to generalize the results of Nelson (1990) we have to rely on the literature on strong law of large numbers and weak convergence for products of i.i.d. random matrices originated by the articles of Bellman (1954), Furstenberg and Kesten (1960) and Kesten and Spitzer (1984). In fact, by iterating equation (3.4) for the times 1, 2, . . . , t we can write  ht = I + Gt−1 + Gt−1 Gt−2 + Gt−1 Gt−2 Gt−3 + . . . + Gt−1 · · · G1 ω + Gt−1 · · · G1 G0 h0 , (3.14) where Gt = AXt + B and h0 is strictly positive and finite with probability one and independent of G0 , G1 , . . .. For notational convenience put Ht = Gt−1 Gt−2 · · · G0 for t ∈ N. In order to make this paper as much self-contained as possible, we summarize theorems 1 and 2 by Furstenberg and Kesten (1960), theorem 5 by Kingman (1973) and theorem 2 by Hennion (1997) in the follwing Theorem 1. In the rest of the paper we use the following notation: let x be a q-vector and X a q × q

6

matrix, kxk1 =

q X

|xi |

i=1 q

kXk1 = sup{kXxk1 : x ∈ R , kxk1 = 1} = max j

v(X) = inf{kXxk1 : x ≥ 0, kxk1 = 1} = min j

q X

|xij |

i=1

q X

|xij |,

i=1

and the relational operators (, =) applied to vectors and matrices are to be intended elementwise (i.e. X > Y is true iff every element of X is greater than the corresponding element of Y .). Theorem 1 (Furstenberg, Kesten, Kingman and Hennion). Let X1 , X2 , X3 , . . . form a stationary and ergodic stochastic process with values in the set of k × k nonnegative real matrices. If 4 0 0 E | log kX1 k1 | + E | log v(X1 )| < +∞, and for the products Yn = Xn · · · X1 Pr{Yn > 0} > 0

for some n,

holds, then a) b) c) d)

lim n−1 E[log kYn k1 ] = λ

n→∞

lim n−1 log kYn k1 = λ

a.s.,

lim n−1 log ρ(Yn ) = λ

a.s.,

n→∞

n→∞

lim n

−1

n→∞

log[Yn ]ij = λ

a.s.,

where λ, usually referred to as greatest Lyapunov exponent or greatest characteristic exponent, is a nonrandom point in R, ρ(Yn ) is the spectral radius of the matrix Yn and [Yn ]ij is the (i, j)-th element of Yn . Proof. See Hennion (1997) sections 5-7. Now, paralleling Nelson (1990), we are in the condition to prove the following propositions. Let λ be the greatest Lyapunov exponent relative to the matrix product Ht and suppose that the assumptions of Theorem 1 hold for the sequence of matrices Gi . Proposition 2. Let ω = 0, as t → ∞: a) if λ > 0, then ht → +∞ a.s.; 2

b) if λ = 0, then ht is tight in R+ and has a limit distribution which depends on h0 and for some h0 is concentrated on strictly positive vectors; c) if λ < 0, then ht → 0 a.s.; 4 The

easier to verify condition maxi,j E | log[X1 ]ij | < +∞ implies the condition stated in the proposition.

7

Proof. Let (Ω, F, µ) be the probability space on which Gi , for i = 1, 2, . . ., are defined. By Theorem 1 point d) and the definition of almost sure convergence, there is a F-measurable M , such that for n > M and λ 6= 0 −

1 |λ| |λ| < log [Ht ]ij − λ < 2 t 2

from which e−

t|λ| 2 +tλ

< [Ht ]ij < e

t|λ| 2 +tλ

a.s.

a.s.

Now, for λ > 0 the (almost sure) limit of the above expression is +∞, while for λ < 0 the limit is 0. When λ = 0, we have to relay on the results of Kesten and Spitzer (1984). Since G0 is non-negative and, when βii > 0 for i = {1, 2}, Pr{G0 has a zero row or column} = 0, then assumptions H1 and H2 of Kesten and Spitzer (1984) are fulfilled and by their Theorem 1 CI : {Ht is tight with limit distribution concentrated on positive matrices} ⇓ H3 : {E[G0 ]ij < ∞ and log ρ(E[G0 ]) = 0} and CIV : {λ = 0 a.s.}. By Theorem 2 of Kesten and Spitzer (1984), under assumptions H1 , H2 and H3 CIV ⇔ CI . Therefore, if H1 , H2 and E[G0 ]ij < ∞ hold, the condition λ = 0 is necessary and sufficient for Ht to be tight and converge in probability to a nonzero random matrix. This implies that for arbitrary positive h0 < ∞ the random vector ht = Ht h0 is tight and its limit2 ing distribution depends on h0 , and, apart from possible subspaces of R+ where ht may converge to 0 almost surely, ht converges weakly to a distribution with positive support. Proposition 3. Let ω > 0, as t → ∞: a) if λ ≥ 0, then ht → +∞ a.s.; b) if λ < 0, then ht is asymptotically strictly stationary and ergodic. Proof. Part a) is a trivial consequence of Proposition 3a). As for part b), we have already proven that the second addend of equation (3.14) converges almost surly to zero when λ < 0. In the proof of Proposition 3 we have shown that the rate of almost sure convergence to zero of all the elements of Ht is O(etλ/2 ), which assures the convergence of the partial sums of matrix products in (3.14). Now consider the process ˜ t := (I + Gt−1 + Gt−1 Gt−2 + Gt−1 Gt−2 Gt−3 + . . .)ω h ˜ t is asymptotically stationary and ergodic if By Stout (1974, Th. 3.5.8) h Rt∞ := I + Gt−1 + Gt−1 Gt−2 + Gt−1 Gt−2 Gt−3 + . . . is measurable. Now, take the process Rtk := I + Gt−1 + Gt−1 Gt−2 + Gt−1 Gt−2 Gt−3 + . . . + Gt−1 Gt−2 Gt−3 · · · Gt−k .

8

Rtk involves only sums and products of measurable functions and therefore is measurable for any finite k. Since Rtk is increasing in k we have supk Rtk = Rt∞ , which is measurable. ˜ t = R∞ ω is stationary and ergodic. Thus, h t ˜ t as t increases. Consider the Now we have to show that ht converges almost surely to h difference ˜ t =Gt−1 Gt−2 · · · G0 h0 ht − h  − Gt−1 Gt−2 · · · G0 + Gt−1 Gt−2 · · · G−1  + Gt−1 Gt−2 · · · G−2 + . . . ω. We have proven in Proposition 2 that the first addend on the right hand side converges almost surely to zero as t increases if λ < 0. Under the same condition, also the second addend converges almost surely to zero, since each element of the infinite sum converges almost surely to zero at exponential rate as t increases. This proves the asymptotic stationarity and ergodicity of the process ht when λ < 0. Remarks i) Lyapunov exponents are usually very hard to compute, but under the conditions of Theorem 1 it is easy to see that ρ(E(G0 )) < 1 implies λ < 0 (i.e. weak stationarity is sufficient for strict stationarity), in fact by equation (1.4) in Kesten and Spitzer (1984) λ = lim

t→∞

1 log kHn k1 < log ρ(E[G0 ]). t

ii) Exploiting the submultiplicativity of any operator norm k · k, another sufficient condition for strict stationarity could be derived using the inequality t

1 1X log kHt k ≤ kGi−1 k, t t i=1 that for t → ∞ becomes λ ≤ E[log kG0 k]. If the right hand side is smaller then zero, the strict stationarity condition holds. But since by a well known result of matrix algebra ρ(E[G0 ]) ≤ k E[G0 ]k, the bound of the previous remark is tighter.

4

Application

The model, with p = q = 0.5, has been applied to the daily returns on the FTSE100 index, and the the estimated good and bad volatilities are depicted in log-scale in Figure 1. As expected the positive and negative volatilities are highly correlated, but negative volatility seems more reactive to shocks than positive volatility. By observing Table 2 it emerges that negative returns tend to raise volatilities much more than positive shocks. This feature, known as leverage effect, is well documented in the literature and can be captured using EGARCH and or GJR-GARCH processes. What these processes are not able to model is the case in which negative volatility is not driven

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50 1984 10

Good

Bad

Black Monday ’87

1986

1990

1988

1st Gulf War

1 0 0 200 7 1992

400

600

800

1000

1994

1200

1400

1600

1996

1800

2000

1998

2 1 0 10

2200

2400

2600 9/11

2000

2800

2002

3000

3200

2nd Gulf War

3400

3600

3800

4000

2006

2004

1 0

4200

4400

4600

4800

5000

5200

5400

5600

5800

6000

6200

Figure 1: Positive and negative volatilities (log scale) estimated on FTSE100 returns in the period 2nd Jan 1984 – 11th Oct 2007.

Const + 2 rt−1 − 2 rt−1 h+ t−1 h− t−1 Tail ρ(E[G0 ])

Estimate 0.016 0.057 0.090 0.908 0.000 1.479 0.987

Restrictions Skewed. GARCH Skewed. GJR-GARCH

h+ t St.Error 0.030 0.011 0.012 0.012 0.040

t-ratio 0.531 4.986 7.721 73.654 36.909

Wald Stat. 108.76 20.26

DoF 5 4

Estimate 0.023 0.014 0.156 0.000 0.904 1.967

h− t St.Error 0.037 0.009 0.023 0.015 0.049

t-ratio 0.603 1.486 6.683 58.338 40.133

p-value 0.000 0.000

Table 2: ML estimates of the model for daily FTSE100 returns from 2nd Jan 1984 to 11th Oct 2007. by positive shocks, which however have significant effects on positive volatility. This fact is a clear feature of our data, as it appears from the second line of Table 2. Figure 2 depicts the news impact curves, both for positive and negative volatilities. The curves are computed as 2 2 ¯+ h+ t = ω1 + a11 rt−1 I(rt−1 ≥ 0) + a12 rt−1 I(rt−1 < 0) + b11 h ¯− h− = ω2 + a21 r 2 I(rt−1 ≥ 0) + a22 r 2 I(rt−1 < 0) + b22 h t

t−1

t−1

¯ + and h ¯ − are the unconditional second moments and rt−1 ranges between the where h minimum and maximum observed on the FTSE100 daily percentage returns series. 10

h+

25.0

h−

22.5 20.0 17.5

ht

15.0 12.5 10.0 7.5 5.0 2.5 −12

−10

−8

−6

−4

−2

rt−1

0

2

4

6

8

Figure 2: News impact curve for positive and negative volatilities. Since our model nests the simple GARCH and the GJR-GARCH both skewed and symmetric, it is possible to carry out one of the classic ML-based tests (Wald, LR, LM). We report Wald type statistics in the lower panel of Table 2 and both hypotheses are to be rejected at any usual level.

5

Conclusion

We have proposed a novel approach to attack the issue of skewness in financial asset returns. The relevant literature dealing with this problem is still scarce and mostly relies on GARCH type processes with conditional skewed distributions (Harris et al. 2004, Lambert and Laurent 2001, Lambert and Laurent 2002, Lanne and Saikkonnen 2004, Miettinen 2005) or with time-varying conditional skewness (Hansen 1994, Harvey and Siddique 1999). Our approach consists in the asymmetrical modelling of second moments for positive and negative returns. The model we have developed nests the widespread GARCH and GJRGARCH processes as special cases and this allows the use of ML-based tests for assessing the relevance of our model when compared to these simpler ones. The application of our model to the main index of the London Stock Exchange reveals that the second moments of positive and negative returns do show different dynamics. In fact, the restrictions corresponding to the skewed GARCH and GJR-GARCH models are strongly rejected by the data. Practitioners should get benefits from the application of our model, both when evaluating the risk of holding an asset or by exploiting the asymmetric dynamics of positive and negative volatilities for implementing option strategies5 . 5 This was suggested to me by professor David Hendry, whom I greatly aknowledge, at the 5th OxMetrics User Conference, London, 20th-21st September 2007.

11

References Bauwens, L., and S. Laurent (2002): “A new class of multivariate skew densities, with application to GARCH models”, Working Paper. Bellman, R. (1954): “Limit theorems for non-commutative operations I”, Duke Mathematical Journal, 21, 491-500. El Babsiri, M., and J.M. Zakoian (2001) “Contemporaneous asymmetry in GARCH processes”, Journal of Econometrics, 101, 257-294. Furstenberg, H. (1960): “ Product of Random Matrices”, Annals of Mathematical Statistics, 31, 457-469. Glosten, L., R. Jagannathan, and D. Runkle (1993): “On the relation between the expected value and the colatility of the nominal excess return on stocks”, Journal of Finance, 48(5), 1779-1801. Hansen, B.E. (1994): “Autoregressive conditional density estimation ”, International Economic Review, 35(3), 705-730. Harris, R., C. K¨ u¸cu ¨k¨ ozmen, and F. Yilmaz (2004): “Skewness in the conditional distribution of daily equity returns”, Applied Financial Economics, 14, 195–202. Harvey, C.R., and A. Siddique (1999): “Autoregressive conditional skewness”, Journal of Financial and Quantitative Analysis, 34(4), 465-487. Hennion, H. (1997): “Limit theorems for products of positive random matrices”, The Annals of Probability, 24(4), 1545–1587. Kesten, K., and F. Spitzer (1984): “Convergence in distribution of products of random matrices”, Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete, 67, 363386. Kingman, J.F.C. (1973): Subadditive ergodic theory. Annals of Probability, 1, 883-899. Lambert, P., and S. Laurent (2001): “Modelling financial time series using GARCH-type models and a skewed Student density”, Mimeo, Universit´e del Li`ege. Lambert, P., and S. Laurent (2002): “Modelling skewness dynamics in series of financial data using skewed location-scale distributions”, Working Paper. Lanne, M., and P. Saikkonnen (2004): “A skewed GARCH-in-mean model: an application to U.S. stock returns”, Conference Paper 469, Econometric Society 2004 North American Summer Meetings. Lisi, F. (2005) “Testing asymmetry in financial time series”, Working Paper n.15/2005, Dipartimento di Scienze Statistiche, Universit`a di Padova L¨ utkepohl, H. (1996): Handbook of Matrices. Chichester: John Wiley & Sons. Mannion, D. (1993) “Products of 2 × 2 random matrices”, The Annals of Applied Probabilities, 3(4), 1189-1218.

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Miettinen, J. (2005): “GARCH Modelling with Skewed Conditional Distributions”, Working Paper, RUESG Workshop on Financial Econometrics. Mukherjea, A. (1987): “Convergence in distribution of products of random matrices: a semigroup approach”, Transactions of the American Mathematical Society, 303(1), 395– 411. Nelson, D.B. (1990): “Stationarity and persistence in the GARCH(1,1) model”, Econometric Theory, 6, 318–334. Nelson, D.B. (1991): “Conditional heteroskedasticity in asset returns: a new approach”, Econometrica, 59, 347-370. Stout, W.F. (1974): Almost Sure Convergence. London: Academic Press.

Appendices A

Half Generalized Error Distribution

The half generalized error distribution (HGED) has positive support and in R+ is proportional to the well known GED. Its density is ν  ν exp − 21 ωx f (x) = 1 ω2 ν Γ( ν1 ) with − ν1

ω=2



Γ( ν1 ) Γ( ν3 )

 12 ,

Moments: µk = E(X k ) = Γ

x ∈ R+ ∪ {0},

ν ∈ R+ .

 1 + k   1  k2 −1  3 − k2 Γ ; Γ ν ν ν

particularly  2    1   3 − 12 Γ Γ µ = E(X) = Γ ν ν ν and µ2 = E(X 2 ) = 1.

B

Relevant norms and eigenvalues

¯ + B: Eigenvalues of AX  1n ρ1,2 = b11 + b22 + a11 p + a22 q ± (b11 + b22 + a11 p + a22 q)2 2 1/2 o − 4(b11 b22 + a11 b22 p + a22 b11 q − a12 a21 pq + a11 a22 pq) . Norms of G0t :     kG0t k1 = (a11 + a12 )ξt I(ξt ≥ 0) + b11 ∨ (a21 + a22 )ξt I(ξt < 0) + b22 ,     v(G0t ) = (a11 + a12 )ξt I(ξt ≥ 0) + b11 ∧ (a21 + a22 )ξt I(ξt < 0) + b22 .

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