Macroeconomic Dynamics, 13, 2009, 149–166. Printed in the United States of America. doi:10.1017/S1365100508070478

ARTICLES

THE NEW KEYNESIAN PHILLIPS CURVE IN A TIME-VARYING COEFFICIENT ENVIRONMENT: SOME EUROPEAN EVIDENCE GEORGE HONDROYIANNIS Bank of Greece and Harokopio University

P.A.V.B. SWAMY U.S. Bureau of Labor Statistics

GEORGE S. TAVLAS Bank of Greece

We examine whether the importance of lagged inflation in the New Keynesian Phillips Curve (NKPC) may be a result of specification biases. NKPCs are estimated for four countries: France, Germany, Italy, and the United Kingdom. Using time-varying coefficient (TVC) estimation, a procedure that can deal with possible specification biases, we find support for the NKPC model that excludes lagged inflation. Our results indicate a Phillips-curve relationship for the countries considered that does not contain an inertial element. Keywords: New Keynesian Phillips Curve, Specification Biases, Time-Varying Coefficients

1. INTRODUCTION The New Keynesian Phillips Curve (NKPC) has been widely used in empirical work dealing with the relationship between inflation and real economic activity. As discussed in the next section, two types of NKPCs have figured prominently in empirical studies: (1) a “pure” NKPC, under which inflation expectations are entirely forward-looking, and (2) a “hybrid” NKPC, which includes lagged inflation We thank Harris Dellas, Stephen Hall, Peter von zur Muehlen, and Arnold Zellner for helpful comments. The questions and comments of an Associate Editor and two referees were especially stimulating. The views expressed are those of the authors and should not be interpreted as those of their respective institutions. Address correspondence to: George Tavlas, Economics Research Department, Bank of Greece, 21, El. Venizelos Ave, 102 50 Athens, Greece; e-mail: [email protected] c 2009 Cambridge University Press

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as an explanatory variable, in addition to expectations of next period’s inflation (as under the pure NKPC). In this note, we apply a time-varying-coefficient (TVC) procedure that corrects for specification errors as a result of incorrect functional forms, omitted variables, and measurement errors to both versions of the NKPC for four countries—France, Germany, Italy, and the United Kingdom (UK). The purpose of the note is to investigate whether the role of lagged inflation under the hybrid NKPC might be the result of specification errors. Our findings are generally favorable to pure NKPC specification. What follows consists of three sections. Section 2 discusses fixed coefficient and TVC representations of the NKPC and the conditions under which TVC estimation produces what we call bias-free coefficients. Section 3 presents empirical results. As a benchmark set of results, we follow the approach used in much of the literature, applying GMM estimation with no corrections for specification biases to the hybrid NKPC. We find that the coefficient on lagged inflation is positive and significant. Then, we apply TVC estimation to the hybrid of NKPC. We find that lagged inflation is no longer significant. Finally, we apply TVC estimation to the pure NKPC and find that the coefficient on inflation expectations is near unity. Although this experiment is not a controlled one (because GMM and TVC estimate expected inflation differently), our results provide some support for the view that the role of lagged inflation found in studies that use GMM estimation might reflect specification biases. Section 4 provides conclusions. An Appendix compares the TVC procedure used here with other TVC procedures and with GMM estimation.

2. REPRESENTATIONS OF THE NKPC 2.1. Pure and hybrid variants with fixed coefficients The NKPC can be derived from Calvo’s (1983) contract-pricing model, under which each firm keeps its price fixed until it receives a random signal that allows it to change its price. In this framework, each firm faces a constant probability (1−θ ) of adjusting its price in any given period, independent of the history of previous price adjustments. Under certain conditions, aggregation of all firms produces the following pure NKPC equation in log-linearized form: p˙ t = βEt p˙ t+1 + λ1 st + η0t ,

(1)

where p˙ t is the inflation rate, Et p˙ t+1 is the expected inflation in period t + 1 as it is formulated in period t, st is the (log of) average real marginal cost in per cent deviation from its steady state level, and η0t is an error term. The coefficient, β, is a stochastic discount factor for profits that is on average between 0 and 1, and λ1 =

(1 − θ )(1 − βθ ) θ

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is a parameter that is positive; p˙ t increases when real marginal cost, which is a measure of excess demand, increases (since there is a tendency for inflation to increase).1 In empirical applications, because marginal cost is unobserved, real unit labor cost (rulct ) is often used as its proxy. Two types of problems have confronted empirical researchers in the empirical implementation of equation (1). First, some authors have added a lagged inflation term to the equation, yielding the hybrid variant, and have found that inflation is predominantly backward looking [e.g., Fuhrer (1997); Rudebusch (2002); Rudd and Whelan (2005)]. An implication of this finding is that there is an inertial element in the inflation process regardless of the credibility of the monetary authority. Second, considerable uncertainty surrounds the value of λ1 , the coefficient for the driving process of the NKPC [Kurmann (2005, p. 1131)], with some authors [e.g., Linde (2005)] finding a coefficient near zero. If λ1 equals zero, there is no (short-run or long-run) Phillips curve. The hybrid NKPC is based on a modified Calvo model, under which a proportion of firms, ρ, sets prices using a backward-looking rule.2 This modification yields a hybrid model as follows: p˙ t = ωf Et p˙ t+1 + λ2 st + ωb p˙ t−1 + η1t ,

(2)

where p˙ t−1 is the lagged inflation and η1t is an error term. The reduced form parameter λ2 is defined as λ2 = (1 − ρ)(1 − θ )(1 − βθ )φ −1 with φ = θ + ρ[1 − θ (1 − β)]. The two reduced form parameters, ωf and ωb , can be interpreted as the weights on “backward-looking” and “forward-looking” components of inflation and are defined as ωf = βθ φ −1 and ωb = ρφ −1 , respectively.3 To estimate the hybrid NKPC using GMM, the following procedure is typically adopted. Assuming rational expectations and that the η˜ 1t = p˙ t − ωf p˙ t+1 − λ2 rulct − ωb p˙ t−1 are identically and independently distributed (i.i.d.), Et p˙ t+1 is replaced by the actual inflation, p˙ t+1 , and orthogonality conditions are imposed between η˜ 1t and a set of instruments that are correlated with p˙ t+1 , rulct , and p˙ t−1 but not with η˜ 1t . In the exactly identified case, where there are exactly as many moment equations as there are parameters to be estimated [Greene (2003, pp. 536– 538)], GMM estimation procedure is considered appealing because it only requires identifying the relevant instruments and does not require strong assumptions on the underlying model. The orthogonality conditions can be stated as Et {(p˙ t − λ2 rulct − ωf p˙ t+1 − ωb p˙ t−1 )zt } = 0,

(3)

where zt is a vector of instruments dated t and earlier. 2.2. TVC Representations The TVC approach [Swamy and Tavlas (1995, 2001, 2005, 2007)] takes as its point of departure the idea that there is a true, changing economy, but that any of its econometric representations is almost certainly a misspecified version of

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the truth. This misspecification may take the form of omitted variables, measurement errors, and incorrect functional form (broadly, the dynamic modeling ideas of marginalization, conditioning, and model specification). These problems are expected to produce biased coefficient estimates. Hence, the TVC estimation technique tries to correct for these biases by using a set of “driving” variables.4 The idea underlying the technique is that causal relationships cannot be recovered unless the coefficients are corrected for specification biases. Unlike other TVC approaches, the approach used here aims to simultaneously remove biases stemming from incorrect functional forms, omitted variables, and measurement errors. Consider the functional form problem.5 TVC begins with the following presumptions. (1) Any specific functional form [e.g., the log-linear approximations in equations (1) and (2)] is likely to be incorrect. (2) A finite class of functional forms may not include the true (unknown) functional form as a special case. (3) Therefore, an infinite class of functional forms is needed to encompass the true (but unknown) functional form. The TVC procedure generates an infinite class by allowing all the coefficients of all the correctly-measured determinants of the dependent variable to vary freely (so that each coefficient can have infinitely many time paths).6,7 The solution to the functional form problem still leaves us with biases as a result of omitted-variables and measurement-errors. If omitted variables and measurement errors are present, the coefficients on the included explanatory variables can be unstable even if a true, stable linear relationship exists between the dependent variable and its determinants. This situation can arise because the coefficients on the included explanatory variables have not been corrected for omitted variables and measurement errors, the effects of which can themselves be time-varying. Alternatively, if the relationship between the dependent variable and its determinants is truly nonlinear, the coefficients of the relationship’s linearin-variables representation will be time-varying even without omitted variables and measurement errors.8 If there are variables omitted from, and measurement errors in, the variables included in a linear-in-variables representation of a truly nonlinear relationship, additional components representing omitted-variable and measurement-error biases appear in each coefficient of the linear-in-variables representation. Consequently, the TVC technique aims to identify the biases stemming from omitted variables and measurement errors, and to remove them. To do so, TVC estimation uses “coefficient drivers.” These can be thought of as having to do with missing variables and measurement errors, at least in part, because they help explain the movement of the coefficients of the included variables caused by omitted variables and measurement errors. If there is a true linear process with constant coefficients, the included coefficient drivers can remove the time-variation in the estimated coefficients caused by time-varying omitted-variables and timevarying measurement-errors. The TVC procedure involves experimentation in the selection of drivers to yield stable bias-free coefficients, if they exist, or timevarying, bias-free coefficients, if there is a true, stable, and nonlinear relationship.

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If the process is successful, we observe a set of biased coefficients, which should exhibit considerable time variation and a set of bias-corrected coefficients. The latter should reveal the coefficients of the underlying true relationship of interest. These bias-corrected coefficients can be such that the hypothesis that they are equal to zero cannot be rejected (at a given level of significance). If so, they imply the absence of a relationship between the explanatory variable or variables in question and the dependent variable. Under the TVC approach, the coefficient of each explanatory variable can be viewed as the sum of three terms: (1) a component measuring the direct effect of the explanatory variable on the dependent variable without specification biases, that is, the bias-free component; (2) the omitted-variables-bias component; and (3) the measurement-error-bias component. Also, the intercept term is the sum of components representing, in addition to the intercept of the true underlying process, measurement errors in the dependent variable and the portions of the excluded variables remaining after the effects of the included explanatory variables have been removed. Thus, the intercept term in a TVC environment is not comparable to the constant term under linear regression estimation. The intercept can be time-varying, and it captures the effects of portions of omitted variables and measurement errors in the dependent variable. In fact, under the TVC procedure, the intercept is retained even if it is insignificant because it absorbs the effects of parts of excluded explanatory variables. In the tables that follow, we refer to the intercept as “remainder effects.” Further discussion of these arguments is provided in the Appendix. A TVC version of the hybrid NKPC is p˙ t = γ0t + γ1t x1t + γ2t x2t + γ3t x3t ,

(4)

where x1t is a proxy for Et p˙ t+1 , x2t = rulct that is, x2t is a proxy for st , and x3t = p˙ t−1 . In a TVC context, the explanatory variables in model (4) are called “the included explanatory variables.” Model (4) without the last term is a TVC version of model (1).9 We now assume that γj t = πj 0 +

p−1

πj d zdt + εj t

(j = 0, 1, 2, 3),

(5)

d=1

where the π s are fixed parameters, zdt = 1 for all d and t, the zs are the coefficient drivers. It is assumed that the mean of εj t is zero and the εj t are serially and contemporaneously correlated. Because of the correlations between the explanatory variables of model (4) and their coefficients, it is assumed that these explanatory variables are conditionally independent of their coefficients given the coefficient drivers. The drivers help to (1) identify the components of the coefficients of the TVC model in equation (4) and (2) decompose the coefficients into their respective components described in the preceding paragraph. The TVC technique involves

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the simultaneous estimation of all the time-vary coefficients [in equation (4)] and the fixed parameters [in equation (5)] of the model. 2.3. TVC and structural changes During the time period considered in this paper, a combination of supply shocks that are hard to measure, changes in monetary-policy regimes, and structural changes in labor markets that may have altered the natural unemployment rate, make it likely that NKPCs involve nonlinear structures (or equivalently, structures that are linear-in-variables with changing coefficients) in each of the four economies under investigation. How does TVC estimation deal with structural changes? Consider the case in which a dummy variable is used to capture a change in structure. Unlike fixedcoefficient estimation, under which the dummy variable is added to a regression, in TVC estimation the dummy variable first appears as a coefficient driver, as in equation (5), and the resulting expression is substituted into equation (4), so that the coefficient driver can affect all the coefficients of (4) (it also affects the variances and covariances of the errors). As we have stressed, drivers are selected with the aim of removing biases stemming from time-varying omitted variables and measurement errors so that, if successfully done, the remaining parts of coefficients are bias-free. Thus, the coefficient drivers are chosen to capture the policy changes impacting the economy and to isolate the effects of those changes so that the underlying bias-free coefficients are revealed.10 3. DATA AND EMPIRICAL RESULTS The NKPC estimates reported here are based on quarterly data for France, Germany, Italy, and the United Kingdom. The inflation rate (p˙ t ), used as the dependent variable, is the annualized quarterly percentage change in the implicit GDP deflator. Real unit labor cost (rulc), a forcing variable, is estimated using the deviation of the log of the labor income share from its average value; the labor income share is the ratio of total compensation of employees in the economy to nominal GDP. Three coefficient drivers are used: change in the inflation rate in period t − 1 (as measured by the GDP deflator), change in the three-month t-bill rate in period t − 1, and change in wage inflation in period t − 1. Wage inflation is the annualized quarterly percentage change in hourly earnings in manufacturing. The CPI inflation rate (the annualized percentage change in the consumer price index) was used as an instrument in the GMM estimation, as well as to estimate expected prices for use in TVC estimation. (As discussed later, several of the other variables described earlier also were used as instruments.) An output gap measure and the yield on 10-year government bonds (referred to as “bond rate”) were also used to estimate expected inflation for use in TVC models.11 We first report GMM results for the hybrid model. The estimation periods for the four countries were as follows: France 1971:2–2005:2; Germany 1971:1–2005:2;

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TABLE 1. GMM estimation of hybrid NKPC

p˙ t+1 rulct p˙ t−1 J-test

France

Germany

Italy

UK

— 0.729∗∗∗ [43.60] 0.081∗∗∗ [7.20] 0.223∗∗∗ [13.40] 0.04

— 0.899∗∗∗ [20.50] 0.014 [0.76] 0.068∗ [1.64] 0.08

— 0.764∗∗∗ [21.94] 0.011∗ [1.74] 0.284∗∗∗ [9.09] 0.07

— 0.895∗∗∗ [9.78] 0.050∗ [1.67] 0.149∗∗∗ [1.83] 0.06

Notes: Figures in brackets are t-statistics. ∗∗∗ , ∗∗ , ∗ indicate significance at 1%, 5%, and 10% level, respectively.

Italy 1971:2–2005:2 and the UK 1970:4–2005:1.12 In estimating equation (2) with GMM, we followed the usual practice of using actual inflation (as measured by the change in the GDP deflator) in period t + 1 to measure inflation expectations. In applying GMM, the vector zt of instrumental variables in equation (3) included four lags of inflation, real unit labor costs, four lags of CPI inflation, four lags of wage inflation, and four lags of the t-bill rate.13 The standard errors of the estimated parameters were modified using a Barlett or quadratic Kernel with Newey-West bandwidth. In addition, prewhitening was used [Greene (2003), pp. 140 and 200]. GMM results are reported in Table 1. All the coefficients have the appropriate sign and most are statistically significant. The exception is the rulc variable for Germany, which is not significant. For each country, the coefficient on lagged inflation is significant, although in the German equation it is significant only at the 10 per cent level.14 For the other countries, the coefficient on lagged inflation is significant at the 1% level. Next, we estimate the hybrid model using TVC technology. To put TVC estimation of equation (4) on a comparable basis with the GMM estimation of model (2), under TVC estimation we employed a proxy for Et p˙ t+1 that was related to the instruments used under GMM. Specifically, the estimated values of inflation were generated using ordinary least squares (OLS) under which, initially, the explanatory variables used to estimate inflation were essentially the same as those used as instruments in the GMM estimation and, consisted of the information set available at time t.15 Because our purpose was to estimate expected inflation for period t + 1 as it is formulated in period t, the information set should be the one available in period t. Therefore, in the OLS regression, information set at time t − 1 was employed to estimate inflation for period t. In other words, in the OLS regression, the dependent variable, the inflation rate (as measured by the percentage change in the implicit GDP deflator), was dated t and all the explanatory variables were dated t − 2 or earlier, except the output gap and the t-bill rate, which were dated t − 1 or earlier. The measure of the output gap (output gap) was computed as the deviation of actual output from the potential output.

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Potential output was computed with a Hodrick-Prescott filter. As discussed later, in the case of Germany, two dummy variables were employed. Any variables with statistically insignificant estimated coefficients were dropped, and the regression was reestimated. The estimates (pˆ˙ t+1 ) of inflation in period t + 1 given by the following regressions were used as proxies for Et p˙ t+1 : f f f f France: pˆ˙ t = αˆ 0 + αˆ 1 p˙ t−5 + αˆ 2 (wage inflation)t−5 + αˆ 3 (CPI inflation)t−2 f

f

+ αˆ 4 (CPI inflation)t−3 + αˆ 5 (t-bill rate)t−4 ,

(6)

f

where αˆ j , j = 0, 1, . . . , 5, are the OLS estimates computed from the French quarterly data for 1971:2–2005:2; g g g g Germany: pˆ˙ t = αˆ 0 + αˆ 1 p˙ t−4 + αˆ 2 (outputgap)t−1 + αˆ 3 (outputgap)t−4 g

g

g

+ αˆ 4 (wage inflation)t−2 + αˆ 5 (wage inflation)t−3 + αˆ 6 (CPI inflation)t−3 g

g

g

+ αˆ 7 (t-bill rate)t−1 + αˆ 8 (Dummy87) + αˆ 9 (Dummy90),

(7)

g

where αˆ j , j = 0, 1, . . . , 9, are the OLS estimates computed from the German quarterly data for 1971:1–2005:2, Dummy87 takes the value 1 for the period 1987:1–1992:2 and zero for other periods, and Dummy90 takes the value 1 for the period 1990:4–1991:4 and zero for other periods;16 Italy: pˆ˙ t = αˆ 0i + αˆ 1i p˙ t−4 + αˆ 2i (outputgap)t−1 + αˆ 3i (outputgap)t−2 + αˆ 4i (wage inflation)t−4 + αˆ 5i (CPI inflation)t−2 + αˆ 6i (t-bill rate)t−1 + αˆ 7i (t-bill rate)t−2 ,

(8)

where αˆ ji , j = 0, 1, . . . , 7, are the OLS estimates computed from the Italian quarterly data for 1971:2–2005:2; United Kingdom: pˆ˙ t = αˆ 0u + αˆ 1u p˙ t−3 + αˆ 2u (outputgap)t−2 + αˆ 3u (outputgap)t−5 + αˆ 4u (wage inflation)t−2 + αˆ 5u (CPI inflation)t−2 + αˆ 6u (t-bill rate)t−1 + αˆ 7u (bondrate)t−1 ,

(9)

where αˆ ju , j = 0, 1, . . . , 7, are the OLS estimates computed from the U.K. quarterly data for 1970:4–2005:1. Clearly, the inflation process in each of the four countries considered differs, each with its particular idiosyncratic elements. In such circumstances, a superior procedure to the one employed earlier would have been to use a direct measure of inflation expectations. Such a measure, however, was not available over the time period and with the data frequency that we employed.17 Consequently, we settled on this procedure, whereby we started with a set of explanatory variables that also were used as instruments under GMM estimation.18

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TABLE 2. TVC estimation of hybrid NKPC: Average bias-free effects

Remainder effect pˆ˙ t+1 from (6), (7), (8), or (9) rulct p˙ t−1

France

Germany

Italy

UK

0.351 [0.98] 0.704∗∗ [1.94] −0.163∗ [−1.80] 0.079 [0.42]

1.304∗∗∗ [6.34] 0.237∗∗ [1.93] 0.310∗∗∗ [6.01] 0.060 [0.510]

1.993∗∗ [2.85] 0.645∗∗∗ [5.66] 0.180∗∗∗ [4.41] 0.096 [0.95]

2.353∗∗∗ [5.67] 0.701∗∗∗ [8.44] 0.510∗∗∗ [5.30] 0.058 [0.57]

Notes: Figures in brackets are t-statistics. ∗∗∗ , ∗∗ , ∗ indicate significance at 1%, 5%, and 10% level, respectively. TVC models are estimated using three coefficient drivers: change in the t-bill rate in period t − 1, change in inflation rate in period t − 1 and change in wage inflation in period t-1. The bias-free effect is estimated using the change in inflation in the previous period for France and the change in t-bill rate in the previous period for all the other countries.

Table 2 presents TVC estimates of the hybrid model. We used πˆ j 0 + πˆ j 1 z1t−1 , where πˆ j 0 and πˆ j 1 are the iteratively rescaled generalized least squares (IRSGLS) estimates of πj 0 and πj 1 in (5) respectively,as an estimator of the bias-free component of γj t .19 This means that we used 3d=2 πˆ j d zdt−1 + εˆ j t , where πˆ j d is the IRSGLS estimate of πj d , as an estimator of the sum of omitted-variables and measurement-error bias components of γj t . (These z’s are defined at the bottom of Table 2.) We report the means, πˆ j 0 + πˆ j 1 (1/T ) Tt=1 z1t−1 , where T is the number of quarters in the period 1971:2–2005:2, 1971:1–2005:2, 1971:2–2005:2, 1970:4– 2005:1 for France, Germany, Italy, the United Kingdom, respectively. The bias-free effects of lagged inflation are insignificant, whereas the bias-free effects of estimated future inflation are significant for all the four countries. With the exception of the equation for France, the bias-free effects of rulct are of the correct sign and significant.20 The hypothesis that the coefficient on estimated future inflation is equal to unity at the 5% level can be rejected for each of the four countries. Next, we dropped lagged inflation and estimated the pure NKPC model. These results are reported in Table 3. (The coefficient drivers used to obtain these results are defined at the bottom of Table 3.) The coefficients on real unit labor costs are significant and in a range (0.11 to 0.67) that is consistent with estimates reported in the literature. The coefficients of the terms representing expected inflation are in the range of 0.89 to 0.95 and all are significant. For all the four countries, the hypothesis that the average bias-free effect of the estimated future inflation equals unity cannot be rejected at the 5% level. As noted earlier, TVC estimation provides total (biased) coefficients that can be expected to exhibit considerable time variation and bias-corrected coefficients that should reveal the underlying true coefficients of interest. To illustrate, Figures 1–4 provide total and bias-free coefficients of the lagged inflation term in the hybrid NKPC for the four countries considered. The bias-free coefficients all hover near zero, with stable behavioral patterns, whereas the total coefficients are highly volatile.

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TABLE 3. TVC Estimation of “pure” NKPC: Average bias-free effects

Remainder effect pˆ˙ t+1 from (6), (7), (8), or (9) rulct p˙ t−1

France

Germany

Italy

UK

1.334∗∗∗ [3.05] 0.913∗∗∗ [3.64] 0.679∗∗∗ [5.82] —

0.546∗ [1.78] 0.890∗∗∗ [6.47] 0.295∗∗ [2.46] —

0.506 [0.85] 0.944∗∗∗ [10.25] 0.111∗∗∗ [3.47] —

1.383∗∗∗ [3.14] 0.946∗∗∗ [9.39] 0.258∗∗ [2.56] —

Notes: Figures in brackets are t-statistics. ∗∗∗ , ∗∗ , ∗ indicate significance at 1%, 5% and 10% level, respectively. TVC models are estimated using three coefficient drivers: change in the t-bill rate in period t − 1, change in inflation rate in period t − 1 and change in wage inflation in period t − 1. The bias-free effect is estimated using the change in inflation in the previous period for France and the change in t-bill rate in the previous period for all the other countries.

Apart from the absence of both (a) inflation inertia (as a result of backwardlooking inflation expectations) and (b) a long-run Phillips-curve trade-off, what are the economic implications of our results? The coefficients of the NKPCs can be used to determine (1 − θ ), the probability of adjusting prices in any given quarter, from the formula for λ1 = (1 − θ )(1 − βθ )(1/θ ) given below equation (1). Our results imply the following probabilities of adjusting prices in any given quarter: For France, 0.54 is the fraction of firms each quarter, implying that on average firms adjust prices every two quarters. For Germany, 0.39 is the fraction of firms that adjusts prices each quarter so that price adjustment takes place every 2.5 quarters. For Italy, the fraction of firms that adjusts each quarter is 0.27 so that price adjustment takes place every four quarters. For the United Kingdom, 0.38

FIGURE 1. France: Estimates of total and bias-free effects of lag inflation.

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FIGURE 2. Germany: Estimates of total and bias-free effects of lag inflation.

is the fraction of firms that adjusts prices each quarter so that price adjustment takes place every 2.5 quarters. Our findings, based on the percentage change in the GDP deflator, suggest somewhat faster price adjustment among the (three) euro-area economies considered (that is, excluding the United Kingdom) than the survey results reported by the euro-area’s Inflation Persistence Network (IPN). For the euro-area as a whole, the IPN survey found that consumer prices were adjusted every four to five quarters and producer prices were adjusted every 3.5

FIGURE 3. Italy: Estimates of total and bias-free effects of lag inflation.

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FIGURE 4. UK: Estimates of total and bias-free effects of lag inflation.

quarters [Altissimo, Ehrmann, and Smets (2006)]. Although our results are broadly comparable with those of the IPN, the more frequent price adjustments implied by our results may reflect such factors as the wider country-coverage and the particular (i.e., survey) methodology of the IPN. 4. CONCLUSION TVC estimation provides a methodology for determining whether the role of lagged inflation in some estimates of the hybrid NKPC may reflect specification errors. Our results are generally favorable to the pure NKPC, and imply the existence of a short-run Phillips-curve trade-off in the countries considered and the absence of an inertial element in the Phillips relationship. NOTES 1. In models with some form of strategic complementarity, λ1 also may depend on the elasticity of substitution among differentiated goods and the elasticity of marginal cost of firms’ output [Sbordone (2007)]. 2. See, for example, Christiano, Eichenbaum, and Evans (2005), who assumed that all firms adjust their prices each period, but some are not able to reoptimize so they index their price adjustment to lagged inflation. 3. The errors in the specification of equations (1) and (2), such as incorrect functional form, omitted variables and measurement errors in the included variables do not always take the form of additive error terms. Therefore, the error terms in equations (1) and (2) may represent factors other than missing variables. Without knowledge of what these factors are and how they are reflected in the error term, it is not possible to assess whether the assumptions about the errors are correct [Pratt and Schlaifer (1984, p. 11)]. The issue of correctly interpreting the additive errors is taken up in the Appendix. In the Appendix, it should be noted that the additive error term of (1) or (2) appears as a term in its intercept. 4. As noted later, these variables are called “coefficient drivers.”

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5. Our aim is to provide an intuitive explanation of the TVC procedure used here. A more formal discussion is contained in the Appendix. 6. The Appendix explains how this procedure is implemented in light of the fact that data for only a subset of the determinants of p˙ t are available. 7. The procedure can be contrasted with other time-varying coefficient procedures, such as the Kalman filter, which assumes a true functional form under which the coefficients are time-varying (in the case of the Kalman filter, the model coefficients are updated starting with some initial values, which are possibly inconsistent with the processes generating the coefficients). β 8. For example, a linear-in-variables representation of the nonlinear relationship, yt = β1 + β2 xt 3 , β3 −1 and xt is correlated with γ1t of yt with its determinant, xt , is yt = β1 + γ1t xt , where γ1t = β2 xt unless β3 = 1. 9. The conditions under which the TVCs of (1) are unique are given in Swamy and Tavlas (2007, Proposition 3, p. 300). 10. For example, one coefficient driver that we used is the change in each country’s t-bill rate. Clearly, the t-bill rate changes whenever policy is changed. 11. The t-bill rate is in line 60.c of the International Financial Statistics (IFS). Wage in manufacturing is in line 65.c of the IFS. The government bond yield is in line 61 of the IFS. The remaining data are from the Datastream OECD Economic Outlook. 12. Differences among the estimation periods reflect differences in data availability. 13. The instruments and the particular lags are based on those used in the literature; see Gali and Gertler (1999) and Jondeau and Bihan (2005). 14. These results may change depending on the set of instruments used. 15. The only difference is that we used the bond rate as a proxy for long-term interest rate and a measure of the output gap instead of the real unit labor costs. The reason for using the output gap was to avoid possible collinearity with the forcing variable (rulc) in the NKPC estimation. 16. These dummies correspond to the “new European Monetary System (EMS)” and Germany’s re-unification, respectively. Under the “new EMS,” which lasted from the beginning of 1987 until the exchange rates crisis of September 1992, the Deutsche Mark took over the role of anchor currency and there were no exchange-rate requirements within the system. 17. OECD forecasts of inflation based on the GDP deflator are available on a semiannual frequency beginning with the mid-1970s. Use of semiannual data requires repeating forecasts for each two-quarter interval. Since 1989, Consensus Economics has published a quarterly special edition of Consensus Forecasts. In this special edition, averages of consumer price growth forecasts (but not of the GDP deflator) are provided on a quarterly basis for individual quarters. Use of these data would involve a sharp reduction of our sample period. 18. As noted in the case of the German equation, two dummy variables were used to estimate expected inflation. The inclusion of the dummies was made to improve the fit of the German equation. The adjusted R-squares for the German equation with and without dummies were 0.62 and 0.54, respectively. For the other countries, the adjusted R-squares were: 0.85 for France; 0.69 for Italy; and 0.68 for the United Kingdom. 19. Formulas for computing the standard errors of IRSGLS estimates are provided in Chang, Hallahan, and Swamy (1992) and Swamy, Yaghi, Mehta, and Chang (2007). These standard errors measure the impact of sampling fluctuations in πˆ j d . 20. In the case of the French equation, the sign of rulct is negative.

REFERENCES Altissimo, Fillipo, Michael Ehrmann, and Frank Smets (2006) Inflation Persistence and Price-Setting Behavior in the Euro Area: A Summary of the Inflation Persistence Network Evidence. Working paper research, No. 95, Bank of Belgium. Ascari, Guido (2004) Staggered prices and trend inflation: Some nuisances. Review of Economic Dynamics 7, 642–667.

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Bakhshi, Hasan, Hashmat Khan, Pablo Burriel-Llombart, and Barbara Rudolf (2007) The New Keynesian Phillips curve under trend inflation and strategic complementarity. Journal of Macroeconomics 29, 37–59. Calvo, Guillermo (1983) Staggered prices in a utility-maximizing framework. Journal of Monetary Economics 12, 383–398. Chang, I-Lok, Charles Hallahan, and P.A. V.B. Swamy (1992) Efficient computation of stochastic coefficient models. In Hans M. Amman, David A. Belsley, and Louis F. Pau (eds.), Computational Economics and Econometrics, pp. 43–53. London: Kluwer Academic Publishers. Christiano, Lawrence, Martin Eichenbaum, and Charles Evans (2005) Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy 113, 1–45. Cogley, Timothy and Argia M. Sbordone (2006) Trend Inflation and Inflation Persistence in the New Keynesian Phillips Curve. Federal Reserve Bank of New York Staff Reports, No. 270. Fuhrer, Jeff C. (1997) The (un)importance of forward–looking behavior in price setting. Journal of Money, Credit and Banking 29, 338–350. Gali, Jordi and Mark Gertler (1999) Inflation dynamics: A structural econometric approach. Journal of Monetary Economics 44, 195–222. Granger, Clive W.J. and Paul Newbold (1974). Spurious regressions in econometrics. Journal of Econometrics 2, 111–a20. Greene, William H. (2003) Econometric Analysis. 5th edition, Upper Saddle River, NJ: Prentice Hall. Jondeau, Eric and Herv´e Le Bihan (2005) Testing for the New Keynesian Phillips curve: Additional international evidence. Economic Modelling 22, 521–550. Kozicki, Sharon and Peter A. Tinsley (2003) Alternative sources of the lag dynamics of inflation. In Price Adjustment and Monetary Policy: Proceedings of a Conference Held at the Bank of Canada, pp. 3–47, November 2002, Bank of Canada, Ottawa. Kurmann, Andr´e (2005) Quantifying the uncertainty about the fit of a New Keynesian pricing model. Journal of Monetary Economics 52, 1119–1134. Linde, Jesper (2005) Estimating New-Keynesian Phillips curves: A full information maximum likelihood approach. Journal of Monetary Economics 52, 1135–1149. McCallum, Bennett T. and Edward Nelson (1999) Nominal income targeting in an open-economy optimizing model. Journal of Monetary Economics 43, 553–578. McCallum, Bennett T. and Edward Nelson (2000) Monetary policy for an open economy: An alternative approach with optimizing agents and sticky prices. Oxford Review of Economic Policy 16, 74–91. Pratt, John W. and Robert Schlaifer (1984) On the nature and discovery of structure (with discussion). Journal of the American Statistical Association 79, 9–21, 29–33. Pratt, John W. and Robert Schlaifer (1988) On the interpretation and observation of laws. Journal of Econometrics, Annals 39, 23–52. Rudebusch, Glenn D. (2002) Assessing nominal income rules for monetary policy with model and data uncertainty. Economic Journal 112, 402–432. Rudd, Jeremy and Karl Whelan (2005) New tests of the New Keynesian Phillips curve. Journal of Monetary Economics 52, 1167–1181. Sbordone, Argia (2007) Inflation Persistence: Alternative Interpretations and Policy Implications. Federal Reserve Bank of New York Staff Reports, No. 286. Swamy, P.A. V.B. and George S. Tavlas (1995) Random coefficient models: Theory and applications. Journal of Economic Surveys 9, 165–182. Swamy, P.A. V.B. and George S. Tavlas (2001) Random coefficient models. In Badi H. Baltagi (ed.), A Companion to Theoretical Econometrics, pp. 410–428. Malden, MA: Blackwell Publishers. Swamy, P.A.V.B. and George S. Tavlas (2005) Theoretical conditions under which monetary policies are effective and practical obstacles to their verification. Economic Theory 25, 999–1005. Swamy, P.A. V.B. and George S. Tavlas (2007) The new Keynesian Phillips curve and inflation expectations: Re-specification and interpretation. Economic Theory 31, 293–306. Swamy, P.A. V.B., George S. Tavlas and Jatinder S. Mehta (2007) Methods of distinguishing between spurious regressions and causation. Journal of Statistical Theory and Applications 1, 83–96.

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Swamy. P.A. V.B., Wisam Yaghi, Jatinder S. Mehta, and I-Lok Chang (2007) Empirical best linear unbiased prediction in misspecified and improved panel data models with an application to gasoline demand. Computational Statistics & Data Analysis 51, 3381–3392.

APPENDIX Models (1) and (2) provide log-linear approximations, around a zero-steady-state inflation rate, to Calvo’s nonlinear model of inflation dynamics. The assumption of zero-steady-state inflation has been questioned by a number of authors, including Ascari (2004), Bakhshi, Khan, Burriel-Llombart, and Rudolf (2007), Cogley and Sbordone (2006), Kozicki and Tinsley (2003), and Sbordone (2007). To avoid this assumption, Cogley and Sbordone (2006) rewrite model (2) as p˙ t − p¯˙ t = α1t Et (p˙ t+1 − p¯˙ t+1 ) + α2t (st − s¯t ) + α3t (p˙ t−1 − p¯˙ t−1 ) + α4t

∞

j −1 ϕ1t Et (p˙ t+j − p¯˙ t+j ) + η2t ,

(A.1)

j =2

where a bar over a variable denotes its value in a steady state characterized by shifting trend inflation, denoted by p¯˙ t , Et is as explained in equation (1), and η2t is an added error term. Two restrictions have been imposed on equation (A.1): (i) each steady-state value and the corresponding non–steady state value have the same coefficient value. (ii) The error term, η2t , captures the deviation of the functional form of (A.1) from the true functional form of NKPC and other possible misspecifications in (A.1). These restrictions might assign a wrong functional form to (A.1). With respect to the functional form of (A.1), TVC estimation recognizes that any finite class of the functional forms of NKPC may not encompass the true functional form. Only an infinite class of the functional forms of NKPC will encompass the true functional from. To obtain such a class, consider p˙ t = p¯˙ t + α1t Et p˙ t+1 + α2t st + α3t p˙ t−1 + α4t

∞

j −1

ϕ1t Et p˙ t+j + α5t Et p¯˙ t+1

j =2

+ α6t s¯t + α7t p¯˙ t−1 + α8t

∞ j =2

j −1

ϕ1t Et p¯˙ t+j +

mt

αkt xkt∗ ,

(A.2)

k=9

where the steady-state values are allowed to have their own coefficients that might be different from the coefficients of the corresponding non-steady-state values, the variables represented by η2t are explicitly written as xkt∗ , k = 9, . . . , mt , and mt is allowed to depend on time because the number of the determinants of p˙ t might change over time. Model (A.2) has no excluded variables because every possible determinant of p˙ t is included in (A.2). The variable, xkt∗ with any one of the values of k ≥ 9 and ≤ mt , in the last term of (A.2) represents imported raw materials that are treated as inputs to the home country’s productive process. This is the variable that should be included in equations (1) and (2) to make these equations open-economy specifications [McCallum and Nelson (1999, 2000)]. An infinite class of functional forms that encompasses the true functional form of NKPC is obtained by allowing all the coefficients of (A.2) to vary freely. Note that this result holds

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only when no determinant of p˙ t is excluded from (A.2). The member of this class with the true functional form is given by the coefficients of (A.2) with the correct time profiles because the coefficients of a linear-in-variables representation of a nonlinear equation with fixed parameters will be time-varying, as shown in footnote 8. We simply denote the coefficients of (A.2) with the correct time profiles by αkt∗ , k = 1, . . . , mt , because we do not know the shapes of these time profiles. The NKPC is truly linear if the αkt∗ and p¯˙ t are constant and is truly nonlinear otherwise. The coefficients of (A.2) with the correct time profiles cannot be obtained by increasing the order of the Taylor approximation to Calvo’s optimal pricing equation that is different from (A.2). For the simplicity of exposition, we change the symbols used to denote the explanatory variables of (A.2) other than p¯˙ t and xkt∗ with k ≥ 9 to xkt∗ , k = 1, . . . , 8, respectively. The variables, xkt∗ with k ≥ 4, are called “the excluded variables” because they are excluded from model (2). The steady-state variables in (A.2) are unobservable and their proxies are measured with error. Rather than use some proxies in place of the steady-state variables with the neglect of measurement errors in those proxies, we will do much better to treat those variables as omitted variables. The hybrid NKPC in (2) can be at odds with a rational behavior of households if an incorrect assumption is made about η1t . If we assume that η0t and η1t represent “the” explanatory variables excluded from models (1) and (2), respectively, and assume, in addition, that the explanatory variables included in (1) and (2) are independent of these error terms, respectively, then the assumption will be meaningless. For a proof of this proposition, see Pratt and Schlaifer (1988, p. 34), who showed that the “condition [that the explanatory variables of (1) or (2) be independent of ‘the’ excluded variables themselves] is meaningless unless the definite article is deleted and can then be satisfied only for certain ‘sufficient sets’ of excluded variables . . . ” To find such sufficient sets, we assume that xkt∗ = µ∗0kt + µ∗1kt x1t∗ + µ∗2kt x2t∗ + µ∗3kt x3t∗

(k = 4, . . . , mt ).

(A.3)

Inserting (A.3) into (A.2) gives p˙ t = p¯˙ t + + α3t∗ +

mt k=4 mt

αkt∗ µ∗0kt

+

αkt∗ µ∗3kt x3t∗ .

α1t∗

+

mt k=4

αkt∗ µ∗1kt

x1t∗

+

α2t∗

+

mt

αkt∗ µ∗2kt

x2t∗

k=4

(A.4)

k=4

The expected inflation used in (1) should be consistent with (A.4) because the meaningful assumption in (A.3) leads to (A.4). It is not possible to generate the values of x1t∗ = Et p˙ t+1 from (A.4) because (A.4) has too many unknowns. For this reason, we use models (6)–(9) to generate the proxies for x1t∗ . This procedure and our timing for the variables introduced in the dynamics of the expected inflation in (6)–(9) are valid, provided we take into account the measurement errors contained in these proxies. Let x1t , x2t , x3t denote the available observations on the proxies for x1t∗ , x2t∗ , x3t∗ , respectively. We treat the observed measurements, xkt = xkt∗ + vkt , k = 1, 2, 3, as the sums of (unobserved) “true” values and (unknown) measurement errors. The symbols, x ∗ , with an asterisk denote “true” values and the symbols, x, without an asterisk denote observable variables. The variables, xj t , j = 1, 2, 3, are called “the included explanatory variables” because they are included in

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t ∗ ∗ − vkt for xkt∗ in (A.4) gives equation (4) with γ0t = (p¯˙ t + m (2). Substituting xkt k=4 αkt µ0kt ) mt ∗ ∗ ∗ and γj t = (αj t + k=4 αkt µj kt )(1 − (vj t /xj t )), j = 1, 2, 3. The intercept, γ0t , contains an additional term if the available data on p˙ t contain measurement errors. The following interpretations of the terms of γ s may be given: t ∗ ∗ ˙ t of the portions of excluded 1. The term, m k=4 αkt µ0kt , of γ0t is the joint effect on p variables remaining after the effects of the “true” values of the included explanatory variables have been removed. This interpretation shows that in (4), γ0t = 0 even when p¯˙ t = 0. This disproves the claim that having a nonzero γ0t in (4) and a nonzero NKPC. The hybrid NKPC π00 in (5) amounts to having in (4) something that is not a t ∗ ∗ in (2) is misspecified because η1t is not equal to (p¯˙ t + m k=4 αkt µ0kt ) and ωf , λ2 , ωb are not equal to γ1t , γ2t , γ3t , respectively. The coefficient, ωf , does not become a function of the measurement error, p˙ t+1 − Et p˙ t+1 , when Et p˙ t+1 is replaced by p˙ t+1 , as under GMM estimation of (2). The correct interpretation of γ0t suggested by (A.4) results in the inclusion of an intercept in (4), (5), and Tables 2 and 3, and it is the misinterpretation of η0t and η1t that led to the elimination of a constant term from (1), (2), andTable 1. t ∗ ∗ 2. The term, m k=4 αkt µj kt , of γj t with j > 0 measures a omitted-variables bias as a result of excluded variables. This term is nonzero when the j th included variable acts partly as a stand-in variable for some of the excluded variables. It can be interpreted as the “indirect” effect because xj t affects some of omitted variables, which affect p˙ t . The omitted-variables bias can be time-varying even when the NKPC is truly linear ∗ because some of the µt j kt ∗can∗be time-varying. 3. The term, −(αj∗t + m k=4 αkt µj kt )(vj t /xj t ), of γj t with j > 0 measures a measurementerror bias as a result of mismeasuring xj t . It can be time-varying even when the NKPC is truly linear because the (vj t /xj t ) are time-varying and some of the µ∗j kt and mt can be time-varying. Typically, the output gap used in (7)–(9) is known to be measured with large error, and probably also the wage inflation used in (6)–(9). Consequently, the measurement error, v1t , in our proxy, x1t , for expected inflation can be large in magnitude. This does not imply a large absolute value t for ∗the∗ measurement-error bias component of the coefficient γ1t of x1t if |α1t∗ + m k=4 αkt µ1kt | is small and |x1t | is large relative to |v1t |. Similarly, the conditions under which the magnitude of the measurement-error bias component of the coefficient, γ2t , of rulct is small can be derived. 4. The term, αj∗t , is the direct effect of xj t on p˙ t and it is devoid of omitted-variable and measurement-error biases. This direct effect is constant if the NKPC is truly linear and is time-varying otherwise. If it is zero for all t, then the nonzero correlation between p˙ t and xj t is spurious [Swamy, Tavlas, and Mehta (2007)]. This definition is different from that of Granger and Newbold (1974) and applies to both linear and nonlinear models, whereas Granger and Newbold’s definition applies only to linear models. [Swamy and Tavlas (2007, pp. 299 and 301) proved that the nonzero values of α3t∗ are spurious.] The estimates in Table 2 support this analytical result. It should be noted that the direct effect, α2t∗ , on p˙ t of the (log of) average real marginal cost, st , (the driving process) does not become zero when γ0t is not set equal to zero. It may appear to be zero if it is confused with γ2t . 5. When the interpretations of γ s given here are adopted, model (4) coincides with a NKPC based on an open-economy specification without the assumption of a zerosteady-state inflation rate. Therefore, it is incorrect to say that in (4), we chose a

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NKPC based on a closed-economy specification with a zero-steady-state inflation rate. 6. The coefficients of model (4) may not follow random walks because they are the sums of three terms, each of which may follow a different stochastic process. Without realizing this, if we assume incorrectly that the coefficients of (4) follow random walks, then the components of γ s are not identifiable and the following comment by Pratt and Schlaifer (1988) applies: “If a statistician with a Bayesian computer program treats the likelihood function of c as if it were the likelihood function of b, as he must if he supplies no proper prior distribution of b given c, then what his printout will contain will be neither beans nor corn but succotash” (p. 49). They further point out that rather than assess a judgmental distribution of “proxy effects” for excluded variables based on mere guesses about their signs and magnitudes, a Bayesian will do much better to search like a non-Bayesian for concomitants that absorb them. Using equation (5), we are able to identify the components of γ s and do exactly as Pratt and Schlaifer suggest. The coefficient drivers included in (5) are our concomitants. They are used to decompose the coefficients of (4) into their respective components. There is no such thing as measurement errors in coefficient drivers. The only property of coefficient drivers that is important here is the closeness of the correspondence of their time variations to time variations in the components of γ s. We judge that time variations in the coefficient drivers included in (5) correspond closely to time variations in the components of the coefficients of (4).

ARTICLES

THE NEW KEYNESIAN PHILLIPS CURVE IN A TIME-VARYING COEFFICIENT ENVIRONMENT: SOME EUROPEAN EVIDENCE GEORGE HONDROYIANNIS Bank of Greece and Harokopio University

P.A.V.B. SWAMY U.S. Bureau of Labor Statistics

GEORGE S. TAVLAS Bank of Greece

We examine whether the importance of lagged inflation in the New Keynesian Phillips Curve (NKPC) may be a result of specification biases. NKPCs are estimated for four countries: France, Germany, Italy, and the United Kingdom. Using time-varying coefficient (TVC) estimation, a procedure that can deal with possible specification biases, we find support for the NKPC model that excludes lagged inflation. Our results indicate a Phillips-curve relationship for the countries considered that does not contain an inertial element. Keywords: New Keynesian Phillips Curve, Specification Biases, Time-Varying Coefficients

1. INTRODUCTION The New Keynesian Phillips Curve (NKPC) has been widely used in empirical work dealing with the relationship between inflation and real economic activity. As discussed in the next section, two types of NKPCs have figured prominently in empirical studies: (1) a “pure” NKPC, under which inflation expectations are entirely forward-looking, and (2) a “hybrid” NKPC, which includes lagged inflation We thank Harris Dellas, Stephen Hall, Peter von zur Muehlen, and Arnold Zellner for helpful comments. The questions and comments of an Associate Editor and two referees were especially stimulating. The views expressed are those of the authors and should not be interpreted as those of their respective institutions. Address correspondence to: George Tavlas, Economics Research Department, Bank of Greece, 21, El. Venizelos Ave, 102 50 Athens, Greece; e-mail: [email protected] c 2009 Cambridge University Press

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as an explanatory variable, in addition to expectations of next period’s inflation (as under the pure NKPC). In this note, we apply a time-varying-coefficient (TVC) procedure that corrects for specification errors as a result of incorrect functional forms, omitted variables, and measurement errors to both versions of the NKPC for four countries—France, Germany, Italy, and the United Kingdom (UK). The purpose of the note is to investigate whether the role of lagged inflation under the hybrid NKPC might be the result of specification errors. Our findings are generally favorable to pure NKPC specification. What follows consists of three sections. Section 2 discusses fixed coefficient and TVC representations of the NKPC and the conditions under which TVC estimation produces what we call bias-free coefficients. Section 3 presents empirical results. As a benchmark set of results, we follow the approach used in much of the literature, applying GMM estimation with no corrections for specification biases to the hybrid NKPC. We find that the coefficient on lagged inflation is positive and significant. Then, we apply TVC estimation to the hybrid of NKPC. We find that lagged inflation is no longer significant. Finally, we apply TVC estimation to the pure NKPC and find that the coefficient on inflation expectations is near unity. Although this experiment is not a controlled one (because GMM and TVC estimate expected inflation differently), our results provide some support for the view that the role of lagged inflation found in studies that use GMM estimation might reflect specification biases. Section 4 provides conclusions. An Appendix compares the TVC procedure used here with other TVC procedures and with GMM estimation.

2. REPRESENTATIONS OF THE NKPC 2.1. Pure and hybrid variants with fixed coefficients The NKPC can be derived from Calvo’s (1983) contract-pricing model, under which each firm keeps its price fixed until it receives a random signal that allows it to change its price. In this framework, each firm faces a constant probability (1−θ ) of adjusting its price in any given period, independent of the history of previous price adjustments. Under certain conditions, aggregation of all firms produces the following pure NKPC equation in log-linearized form: p˙ t = βEt p˙ t+1 + λ1 st + η0t ,

(1)

where p˙ t is the inflation rate, Et p˙ t+1 is the expected inflation in period t + 1 as it is formulated in period t, st is the (log of) average real marginal cost in per cent deviation from its steady state level, and η0t is an error term. The coefficient, β, is a stochastic discount factor for profits that is on average between 0 and 1, and λ1 =

(1 − θ )(1 − βθ ) θ

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is a parameter that is positive; p˙ t increases when real marginal cost, which is a measure of excess demand, increases (since there is a tendency for inflation to increase).1 In empirical applications, because marginal cost is unobserved, real unit labor cost (rulct ) is often used as its proxy. Two types of problems have confronted empirical researchers in the empirical implementation of equation (1). First, some authors have added a lagged inflation term to the equation, yielding the hybrid variant, and have found that inflation is predominantly backward looking [e.g., Fuhrer (1997); Rudebusch (2002); Rudd and Whelan (2005)]. An implication of this finding is that there is an inertial element in the inflation process regardless of the credibility of the monetary authority. Second, considerable uncertainty surrounds the value of λ1 , the coefficient for the driving process of the NKPC [Kurmann (2005, p. 1131)], with some authors [e.g., Linde (2005)] finding a coefficient near zero. If λ1 equals zero, there is no (short-run or long-run) Phillips curve. The hybrid NKPC is based on a modified Calvo model, under which a proportion of firms, ρ, sets prices using a backward-looking rule.2 This modification yields a hybrid model as follows: p˙ t = ωf Et p˙ t+1 + λ2 st + ωb p˙ t−1 + η1t ,

(2)

where p˙ t−1 is the lagged inflation and η1t is an error term. The reduced form parameter λ2 is defined as λ2 = (1 − ρ)(1 − θ )(1 − βθ )φ −1 with φ = θ + ρ[1 − θ (1 − β)]. The two reduced form parameters, ωf and ωb , can be interpreted as the weights on “backward-looking” and “forward-looking” components of inflation and are defined as ωf = βθ φ −1 and ωb = ρφ −1 , respectively.3 To estimate the hybrid NKPC using GMM, the following procedure is typically adopted. Assuming rational expectations and that the η˜ 1t = p˙ t − ωf p˙ t+1 − λ2 rulct − ωb p˙ t−1 are identically and independently distributed (i.i.d.), Et p˙ t+1 is replaced by the actual inflation, p˙ t+1 , and orthogonality conditions are imposed between η˜ 1t and a set of instruments that are correlated with p˙ t+1 , rulct , and p˙ t−1 but not with η˜ 1t . In the exactly identified case, where there are exactly as many moment equations as there are parameters to be estimated [Greene (2003, pp. 536– 538)], GMM estimation procedure is considered appealing because it only requires identifying the relevant instruments and does not require strong assumptions on the underlying model. The orthogonality conditions can be stated as Et {(p˙ t − λ2 rulct − ωf p˙ t+1 − ωb p˙ t−1 )zt } = 0,

(3)

where zt is a vector of instruments dated t and earlier. 2.2. TVC Representations The TVC approach [Swamy and Tavlas (1995, 2001, 2005, 2007)] takes as its point of departure the idea that there is a true, changing economy, but that any of its econometric representations is almost certainly a misspecified version of

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the truth. This misspecification may take the form of omitted variables, measurement errors, and incorrect functional form (broadly, the dynamic modeling ideas of marginalization, conditioning, and model specification). These problems are expected to produce biased coefficient estimates. Hence, the TVC estimation technique tries to correct for these biases by using a set of “driving” variables.4 The idea underlying the technique is that causal relationships cannot be recovered unless the coefficients are corrected for specification biases. Unlike other TVC approaches, the approach used here aims to simultaneously remove biases stemming from incorrect functional forms, omitted variables, and measurement errors. Consider the functional form problem.5 TVC begins with the following presumptions. (1) Any specific functional form [e.g., the log-linear approximations in equations (1) and (2)] is likely to be incorrect. (2) A finite class of functional forms may not include the true (unknown) functional form as a special case. (3) Therefore, an infinite class of functional forms is needed to encompass the true (but unknown) functional form. The TVC procedure generates an infinite class by allowing all the coefficients of all the correctly-measured determinants of the dependent variable to vary freely (so that each coefficient can have infinitely many time paths).6,7 The solution to the functional form problem still leaves us with biases as a result of omitted-variables and measurement-errors. If omitted variables and measurement errors are present, the coefficients on the included explanatory variables can be unstable even if a true, stable linear relationship exists between the dependent variable and its determinants. This situation can arise because the coefficients on the included explanatory variables have not been corrected for omitted variables and measurement errors, the effects of which can themselves be time-varying. Alternatively, if the relationship between the dependent variable and its determinants is truly nonlinear, the coefficients of the relationship’s linearin-variables representation will be time-varying even without omitted variables and measurement errors.8 If there are variables omitted from, and measurement errors in, the variables included in a linear-in-variables representation of a truly nonlinear relationship, additional components representing omitted-variable and measurement-error biases appear in each coefficient of the linear-in-variables representation. Consequently, the TVC technique aims to identify the biases stemming from omitted variables and measurement errors, and to remove them. To do so, TVC estimation uses “coefficient drivers.” These can be thought of as having to do with missing variables and measurement errors, at least in part, because they help explain the movement of the coefficients of the included variables caused by omitted variables and measurement errors. If there is a true linear process with constant coefficients, the included coefficient drivers can remove the time-variation in the estimated coefficients caused by time-varying omitted-variables and timevarying measurement-errors. The TVC procedure involves experimentation in the selection of drivers to yield stable bias-free coefficients, if they exist, or timevarying, bias-free coefficients, if there is a true, stable, and nonlinear relationship.

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If the process is successful, we observe a set of biased coefficients, which should exhibit considerable time variation and a set of bias-corrected coefficients. The latter should reveal the coefficients of the underlying true relationship of interest. These bias-corrected coefficients can be such that the hypothesis that they are equal to zero cannot be rejected (at a given level of significance). If so, they imply the absence of a relationship between the explanatory variable or variables in question and the dependent variable. Under the TVC approach, the coefficient of each explanatory variable can be viewed as the sum of three terms: (1) a component measuring the direct effect of the explanatory variable on the dependent variable without specification biases, that is, the bias-free component; (2) the omitted-variables-bias component; and (3) the measurement-error-bias component. Also, the intercept term is the sum of components representing, in addition to the intercept of the true underlying process, measurement errors in the dependent variable and the portions of the excluded variables remaining after the effects of the included explanatory variables have been removed. Thus, the intercept term in a TVC environment is not comparable to the constant term under linear regression estimation. The intercept can be time-varying, and it captures the effects of portions of omitted variables and measurement errors in the dependent variable. In fact, under the TVC procedure, the intercept is retained even if it is insignificant because it absorbs the effects of parts of excluded explanatory variables. In the tables that follow, we refer to the intercept as “remainder effects.” Further discussion of these arguments is provided in the Appendix. A TVC version of the hybrid NKPC is p˙ t = γ0t + γ1t x1t + γ2t x2t + γ3t x3t ,

(4)

where x1t is a proxy for Et p˙ t+1 , x2t = rulct that is, x2t is a proxy for st , and x3t = p˙ t−1 . In a TVC context, the explanatory variables in model (4) are called “the included explanatory variables.” Model (4) without the last term is a TVC version of model (1).9 We now assume that γj t = πj 0 +

p−1

πj d zdt + εj t

(j = 0, 1, 2, 3),

(5)

d=1

where the π s are fixed parameters, zdt = 1 for all d and t, the zs are the coefficient drivers. It is assumed that the mean of εj t is zero and the εj t are serially and contemporaneously correlated. Because of the correlations between the explanatory variables of model (4) and their coefficients, it is assumed that these explanatory variables are conditionally independent of their coefficients given the coefficient drivers. The drivers help to (1) identify the components of the coefficients of the TVC model in equation (4) and (2) decompose the coefficients into their respective components described in the preceding paragraph. The TVC technique involves

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the simultaneous estimation of all the time-vary coefficients [in equation (4)] and the fixed parameters [in equation (5)] of the model. 2.3. TVC and structural changes During the time period considered in this paper, a combination of supply shocks that are hard to measure, changes in monetary-policy regimes, and structural changes in labor markets that may have altered the natural unemployment rate, make it likely that NKPCs involve nonlinear structures (or equivalently, structures that are linear-in-variables with changing coefficients) in each of the four economies under investigation. How does TVC estimation deal with structural changes? Consider the case in which a dummy variable is used to capture a change in structure. Unlike fixedcoefficient estimation, under which the dummy variable is added to a regression, in TVC estimation the dummy variable first appears as a coefficient driver, as in equation (5), and the resulting expression is substituted into equation (4), so that the coefficient driver can affect all the coefficients of (4) (it also affects the variances and covariances of the errors). As we have stressed, drivers are selected with the aim of removing biases stemming from time-varying omitted variables and measurement errors so that, if successfully done, the remaining parts of coefficients are bias-free. Thus, the coefficient drivers are chosen to capture the policy changes impacting the economy and to isolate the effects of those changes so that the underlying bias-free coefficients are revealed.10 3. DATA AND EMPIRICAL RESULTS The NKPC estimates reported here are based on quarterly data for France, Germany, Italy, and the United Kingdom. The inflation rate (p˙ t ), used as the dependent variable, is the annualized quarterly percentage change in the implicit GDP deflator. Real unit labor cost (rulc), a forcing variable, is estimated using the deviation of the log of the labor income share from its average value; the labor income share is the ratio of total compensation of employees in the economy to nominal GDP. Three coefficient drivers are used: change in the inflation rate in period t − 1 (as measured by the GDP deflator), change in the three-month t-bill rate in period t − 1, and change in wage inflation in period t − 1. Wage inflation is the annualized quarterly percentage change in hourly earnings in manufacturing. The CPI inflation rate (the annualized percentage change in the consumer price index) was used as an instrument in the GMM estimation, as well as to estimate expected prices for use in TVC estimation. (As discussed later, several of the other variables described earlier also were used as instruments.) An output gap measure and the yield on 10-year government bonds (referred to as “bond rate”) were also used to estimate expected inflation for use in TVC models.11 We first report GMM results for the hybrid model. The estimation periods for the four countries were as follows: France 1971:2–2005:2; Germany 1971:1–2005:2;

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TABLE 1. GMM estimation of hybrid NKPC

p˙ t+1 rulct p˙ t−1 J-test

France

Germany

Italy

UK

— 0.729∗∗∗ [43.60] 0.081∗∗∗ [7.20] 0.223∗∗∗ [13.40] 0.04

— 0.899∗∗∗ [20.50] 0.014 [0.76] 0.068∗ [1.64] 0.08

— 0.764∗∗∗ [21.94] 0.011∗ [1.74] 0.284∗∗∗ [9.09] 0.07

— 0.895∗∗∗ [9.78] 0.050∗ [1.67] 0.149∗∗∗ [1.83] 0.06

Notes: Figures in brackets are t-statistics. ∗∗∗ , ∗∗ , ∗ indicate significance at 1%, 5%, and 10% level, respectively.

Italy 1971:2–2005:2 and the UK 1970:4–2005:1.12 In estimating equation (2) with GMM, we followed the usual practice of using actual inflation (as measured by the change in the GDP deflator) in period t + 1 to measure inflation expectations. In applying GMM, the vector zt of instrumental variables in equation (3) included four lags of inflation, real unit labor costs, four lags of CPI inflation, four lags of wage inflation, and four lags of the t-bill rate.13 The standard errors of the estimated parameters were modified using a Barlett or quadratic Kernel with Newey-West bandwidth. In addition, prewhitening was used [Greene (2003), pp. 140 and 200]. GMM results are reported in Table 1. All the coefficients have the appropriate sign and most are statistically significant. The exception is the rulc variable for Germany, which is not significant. For each country, the coefficient on lagged inflation is significant, although in the German equation it is significant only at the 10 per cent level.14 For the other countries, the coefficient on lagged inflation is significant at the 1% level. Next, we estimate the hybrid model using TVC technology. To put TVC estimation of equation (4) on a comparable basis with the GMM estimation of model (2), under TVC estimation we employed a proxy for Et p˙ t+1 that was related to the instruments used under GMM. Specifically, the estimated values of inflation were generated using ordinary least squares (OLS) under which, initially, the explanatory variables used to estimate inflation were essentially the same as those used as instruments in the GMM estimation and, consisted of the information set available at time t.15 Because our purpose was to estimate expected inflation for period t + 1 as it is formulated in period t, the information set should be the one available in period t. Therefore, in the OLS regression, information set at time t − 1 was employed to estimate inflation for period t. In other words, in the OLS regression, the dependent variable, the inflation rate (as measured by the percentage change in the implicit GDP deflator), was dated t and all the explanatory variables were dated t − 2 or earlier, except the output gap and the t-bill rate, which were dated t − 1 or earlier. The measure of the output gap (output gap) was computed as the deviation of actual output from the potential output.

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Potential output was computed with a Hodrick-Prescott filter. As discussed later, in the case of Germany, two dummy variables were employed. Any variables with statistically insignificant estimated coefficients were dropped, and the regression was reestimated. The estimates (pˆ˙ t+1 ) of inflation in period t + 1 given by the following regressions were used as proxies for Et p˙ t+1 : f f f f France: pˆ˙ t = αˆ 0 + αˆ 1 p˙ t−5 + αˆ 2 (wage inflation)t−5 + αˆ 3 (CPI inflation)t−2 f

f

+ αˆ 4 (CPI inflation)t−3 + αˆ 5 (t-bill rate)t−4 ,

(6)

f

where αˆ j , j = 0, 1, . . . , 5, are the OLS estimates computed from the French quarterly data for 1971:2–2005:2; g g g g Germany: pˆ˙ t = αˆ 0 + αˆ 1 p˙ t−4 + αˆ 2 (outputgap)t−1 + αˆ 3 (outputgap)t−4 g

g

g

+ αˆ 4 (wage inflation)t−2 + αˆ 5 (wage inflation)t−3 + αˆ 6 (CPI inflation)t−3 g

g

g

+ αˆ 7 (t-bill rate)t−1 + αˆ 8 (Dummy87) + αˆ 9 (Dummy90),

(7)

g

where αˆ j , j = 0, 1, . . . , 9, are the OLS estimates computed from the German quarterly data for 1971:1–2005:2, Dummy87 takes the value 1 for the period 1987:1–1992:2 and zero for other periods, and Dummy90 takes the value 1 for the period 1990:4–1991:4 and zero for other periods;16 Italy: pˆ˙ t = αˆ 0i + αˆ 1i p˙ t−4 + αˆ 2i (outputgap)t−1 + αˆ 3i (outputgap)t−2 + αˆ 4i (wage inflation)t−4 + αˆ 5i (CPI inflation)t−2 + αˆ 6i (t-bill rate)t−1 + αˆ 7i (t-bill rate)t−2 ,

(8)

where αˆ ji , j = 0, 1, . . . , 7, are the OLS estimates computed from the Italian quarterly data for 1971:2–2005:2; United Kingdom: pˆ˙ t = αˆ 0u + αˆ 1u p˙ t−3 + αˆ 2u (outputgap)t−2 + αˆ 3u (outputgap)t−5 + αˆ 4u (wage inflation)t−2 + αˆ 5u (CPI inflation)t−2 + αˆ 6u (t-bill rate)t−1 + αˆ 7u (bondrate)t−1 ,

(9)

where αˆ ju , j = 0, 1, . . . , 7, are the OLS estimates computed from the U.K. quarterly data for 1970:4–2005:1. Clearly, the inflation process in each of the four countries considered differs, each with its particular idiosyncratic elements. In such circumstances, a superior procedure to the one employed earlier would have been to use a direct measure of inflation expectations. Such a measure, however, was not available over the time period and with the data frequency that we employed.17 Consequently, we settled on this procedure, whereby we started with a set of explanatory variables that also were used as instruments under GMM estimation.18

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TABLE 2. TVC estimation of hybrid NKPC: Average bias-free effects

Remainder effect pˆ˙ t+1 from (6), (7), (8), or (9) rulct p˙ t−1

France

Germany

Italy

UK

0.351 [0.98] 0.704∗∗ [1.94] −0.163∗ [−1.80] 0.079 [0.42]

1.304∗∗∗ [6.34] 0.237∗∗ [1.93] 0.310∗∗∗ [6.01] 0.060 [0.510]

1.993∗∗ [2.85] 0.645∗∗∗ [5.66] 0.180∗∗∗ [4.41] 0.096 [0.95]

2.353∗∗∗ [5.67] 0.701∗∗∗ [8.44] 0.510∗∗∗ [5.30] 0.058 [0.57]

Notes: Figures in brackets are t-statistics. ∗∗∗ , ∗∗ , ∗ indicate significance at 1%, 5%, and 10% level, respectively. TVC models are estimated using three coefficient drivers: change in the t-bill rate in period t − 1, change in inflation rate in period t − 1 and change in wage inflation in period t-1. The bias-free effect is estimated using the change in inflation in the previous period for France and the change in t-bill rate in the previous period for all the other countries.

Table 2 presents TVC estimates of the hybrid model. We used πˆ j 0 + πˆ j 1 z1t−1 , where πˆ j 0 and πˆ j 1 are the iteratively rescaled generalized least squares (IRSGLS) estimates of πj 0 and πj 1 in (5) respectively,as an estimator of the bias-free component of γj t .19 This means that we used 3d=2 πˆ j d zdt−1 + εˆ j t , where πˆ j d is the IRSGLS estimate of πj d , as an estimator of the sum of omitted-variables and measurement-error bias components of γj t . (These z’s are defined at the bottom of Table 2.) We report the means, πˆ j 0 + πˆ j 1 (1/T ) Tt=1 z1t−1 , where T is the number of quarters in the period 1971:2–2005:2, 1971:1–2005:2, 1971:2–2005:2, 1970:4– 2005:1 for France, Germany, Italy, the United Kingdom, respectively. The bias-free effects of lagged inflation are insignificant, whereas the bias-free effects of estimated future inflation are significant for all the four countries. With the exception of the equation for France, the bias-free effects of rulct are of the correct sign and significant.20 The hypothesis that the coefficient on estimated future inflation is equal to unity at the 5% level can be rejected for each of the four countries. Next, we dropped lagged inflation and estimated the pure NKPC model. These results are reported in Table 3. (The coefficient drivers used to obtain these results are defined at the bottom of Table 3.) The coefficients on real unit labor costs are significant and in a range (0.11 to 0.67) that is consistent with estimates reported in the literature. The coefficients of the terms representing expected inflation are in the range of 0.89 to 0.95 and all are significant. For all the four countries, the hypothesis that the average bias-free effect of the estimated future inflation equals unity cannot be rejected at the 5% level. As noted earlier, TVC estimation provides total (biased) coefficients that can be expected to exhibit considerable time variation and bias-corrected coefficients that should reveal the underlying true coefficients of interest. To illustrate, Figures 1–4 provide total and bias-free coefficients of the lagged inflation term in the hybrid NKPC for the four countries considered. The bias-free coefficients all hover near zero, with stable behavioral patterns, whereas the total coefficients are highly volatile.

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TABLE 3. TVC Estimation of “pure” NKPC: Average bias-free effects

Remainder effect pˆ˙ t+1 from (6), (7), (8), or (9) rulct p˙ t−1

France

Germany

Italy

UK

1.334∗∗∗ [3.05] 0.913∗∗∗ [3.64] 0.679∗∗∗ [5.82] —

0.546∗ [1.78] 0.890∗∗∗ [6.47] 0.295∗∗ [2.46] —

0.506 [0.85] 0.944∗∗∗ [10.25] 0.111∗∗∗ [3.47] —

1.383∗∗∗ [3.14] 0.946∗∗∗ [9.39] 0.258∗∗ [2.56] —

Notes: Figures in brackets are t-statistics. ∗∗∗ , ∗∗ , ∗ indicate significance at 1%, 5% and 10% level, respectively. TVC models are estimated using three coefficient drivers: change in the t-bill rate in period t − 1, change in inflation rate in period t − 1 and change in wage inflation in period t − 1. The bias-free effect is estimated using the change in inflation in the previous period for France and the change in t-bill rate in the previous period for all the other countries.

Apart from the absence of both (a) inflation inertia (as a result of backwardlooking inflation expectations) and (b) a long-run Phillips-curve trade-off, what are the economic implications of our results? The coefficients of the NKPCs can be used to determine (1 − θ ), the probability of adjusting prices in any given quarter, from the formula for λ1 = (1 − θ )(1 − βθ )(1/θ ) given below equation (1). Our results imply the following probabilities of adjusting prices in any given quarter: For France, 0.54 is the fraction of firms each quarter, implying that on average firms adjust prices every two quarters. For Germany, 0.39 is the fraction of firms that adjusts prices each quarter so that price adjustment takes place every 2.5 quarters. For Italy, the fraction of firms that adjusts each quarter is 0.27 so that price adjustment takes place every four quarters. For the United Kingdom, 0.38

FIGURE 1. France: Estimates of total and bias-free effects of lag inflation.

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FIGURE 2. Germany: Estimates of total and bias-free effects of lag inflation.

is the fraction of firms that adjusts prices each quarter so that price adjustment takes place every 2.5 quarters. Our findings, based on the percentage change in the GDP deflator, suggest somewhat faster price adjustment among the (three) euro-area economies considered (that is, excluding the United Kingdom) than the survey results reported by the euro-area’s Inflation Persistence Network (IPN). For the euro-area as a whole, the IPN survey found that consumer prices were adjusted every four to five quarters and producer prices were adjusted every 3.5

FIGURE 3. Italy: Estimates of total and bias-free effects of lag inflation.

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FIGURE 4. UK: Estimates of total and bias-free effects of lag inflation.

quarters [Altissimo, Ehrmann, and Smets (2006)]. Although our results are broadly comparable with those of the IPN, the more frequent price adjustments implied by our results may reflect such factors as the wider country-coverage and the particular (i.e., survey) methodology of the IPN. 4. CONCLUSION TVC estimation provides a methodology for determining whether the role of lagged inflation in some estimates of the hybrid NKPC may reflect specification errors. Our results are generally favorable to the pure NKPC, and imply the existence of a short-run Phillips-curve trade-off in the countries considered and the absence of an inertial element in the Phillips relationship. NOTES 1. In models with some form of strategic complementarity, λ1 also may depend on the elasticity of substitution among differentiated goods and the elasticity of marginal cost of firms’ output [Sbordone (2007)]. 2. See, for example, Christiano, Eichenbaum, and Evans (2005), who assumed that all firms adjust their prices each period, but some are not able to reoptimize so they index their price adjustment to lagged inflation. 3. The errors in the specification of equations (1) and (2), such as incorrect functional form, omitted variables and measurement errors in the included variables do not always take the form of additive error terms. Therefore, the error terms in equations (1) and (2) may represent factors other than missing variables. Without knowledge of what these factors are and how they are reflected in the error term, it is not possible to assess whether the assumptions about the errors are correct [Pratt and Schlaifer (1984, p. 11)]. The issue of correctly interpreting the additive errors is taken up in the Appendix. In the Appendix, it should be noted that the additive error term of (1) or (2) appears as a term in its intercept. 4. As noted later, these variables are called “coefficient drivers.”

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5. Our aim is to provide an intuitive explanation of the TVC procedure used here. A more formal discussion is contained in the Appendix. 6. The Appendix explains how this procedure is implemented in light of the fact that data for only a subset of the determinants of p˙ t are available. 7. The procedure can be contrasted with other time-varying coefficient procedures, such as the Kalman filter, which assumes a true functional form under which the coefficients are time-varying (in the case of the Kalman filter, the model coefficients are updated starting with some initial values, which are possibly inconsistent with the processes generating the coefficients). β 8. For example, a linear-in-variables representation of the nonlinear relationship, yt = β1 + β2 xt 3 , β3 −1 and xt is correlated with γ1t of yt with its determinant, xt , is yt = β1 + γ1t xt , where γ1t = β2 xt unless β3 = 1. 9. The conditions under which the TVCs of (1) are unique are given in Swamy and Tavlas (2007, Proposition 3, p. 300). 10. For example, one coefficient driver that we used is the change in each country’s t-bill rate. Clearly, the t-bill rate changes whenever policy is changed. 11. The t-bill rate is in line 60.c of the International Financial Statistics (IFS). Wage in manufacturing is in line 65.c of the IFS. The government bond yield is in line 61 of the IFS. The remaining data are from the Datastream OECD Economic Outlook. 12. Differences among the estimation periods reflect differences in data availability. 13. The instruments and the particular lags are based on those used in the literature; see Gali and Gertler (1999) and Jondeau and Bihan (2005). 14. These results may change depending on the set of instruments used. 15. The only difference is that we used the bond rate as a proxy for long-term interest rate and a measure of the output gap instead of the real unit labor costs. The reason for using the output gap was to avoid possible collinearity with the forcing variable (rulc) in the NKPC estimation. 16. These dummies correspond to the “new European Monetary System (EMS)” and Germany’s re-unification, respectively. Under the “new EMS,” which lasted from the beginning of 1987 until the exchange rates crisis of September 1992, the Deutsche Mark took over the role of anchor currency and there were no exchange-rate requirements within the system. 17. OECD forecasts of inflation based on the GDP deflator are available on a semiannual frequency beginning with the mid-1970s. Use of semiannual data requires repeating forecasts for each two-quarter interval. Since 1989, Consensus Economics has published a quarterly special edition of Consensus Forecasts. In this special edition, averages of consumer price growth forecasts (but not of the GDP deflator) are provided on a quarterly basis for individual quarters. Use of these data would involve a sharp reduction of our sample period. 18. As noted in the case of the German equation, two dummy variables were used to estimate expected inflation. The inclusion of the dummies was made to improve the fit of the German equation. The adjusted R-squares for the German equation with and without dummies were 0.62 and 0.54, respectively. For the other countries, the adjusted R-squares were: 0.85 for France; 0.69 for Italy; and 0.68 for the United Kingdom. 19. Formulas for computing the standard errors of IRSGLS estimates are provided in Chang, Hallahan, and Swamy (1992) and Swamy, Yaghi, Mehta, and Chang (2007). These standard errors measure the impact of sampling fluctuations in πˆ j d . 20. In the case of the French equation, the sign of rulct is negative.

REFERENCES Altissimo, Fillipo, Michael Ehrmann, and Frank Smets (2006) Inflation Persistence and Price-Setting Behavior in the Euro Area: A Summary of the Inflation Persistence Network Evidence. Working paper research, No. 95, Bank of Belgium. Ascari, Guido (2004) Staggered prices and trend inflation: Some nuisances. Review of Economic Dynamics 7, 642–667.

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Bakhshi, Hasan, Hashmat Khan, Pablo Burriel-Llombart, and Barbara Rudolf (2007) The New Keynesian Phillips curve under trend inflation and strategic complementarity. Journal of Macroeconomics 29, 37–59. Calvo, Guillermo (1983) Staggered prices in a utility-maximizing framework. Journal of Monetary Economics 12, 383–398. Chang, I-Lok, Charles Hallahan, and P.A. V.B. Swamy (1992) Efficient computation of stochastic coefficient models. In Hans M. Amman, David A. Belsley, and Louis F. Pau (eds.), Computational Economics and Econometrics, pp. 43–53. London: Kluwer Academic Publishers. Christiano, Lawrence, Martin Eichenbaum, and Charles Evans (2005) Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy 113, 1–45. Cogley, Timothy and Argia M. Sbordone (2006) Trend Inflation and Inflation Persistence in the New Keynesian Phillips Curve. Federal Reserve Bank of New York Staff Reports, No. 270. Fuhrer, Jeff C. (1997) The (un)importance of forward–looking behavior in price setting. Journal of Money, Credit and Banking 29, 338–350. Gali, Jordi and Mark Gertler (1999) Inflation dynamics: A structural econometric approach. Journal of Monetary Economics 44, 195–222. Granger, Clive W.J. and Paul Newbold (1974). Spurious regressions in econometrics. Journal of Econometrics 2, 111–a20. Greene, William H. (2003) Econometric Analysis. 5th edition, Upper Saddle River, NJ: Prentice Hall. Jondeau, Eric and Herv´e Le Bihan (2005) Testing for the New Keynesian Phillips curve: Additional international evidence. Economic Modelling 22, 521–550. Kozicki, Sharon and Peter A. Tinsley (2003) Alternative sources of the lag dynamics of inflation. In Price Adjustment and Monetary Policy: Proceedings of a Conference Held at the Bank of Canada, pp. 3–47, November 2002, Bank of Canada, Ottawa. Kurmann, Andr´e (2005) Quantifying the uncertainty about the fit of a New Keynesian pricing model. Journal of Monetary Economics 52, 1119–1134. Linde, Jesper (2005) Estimating New-Keynesian Phillips curves: A full information maximum likelihood approach. Journal of Monetary Economics 52, 1135–1149. McCallum, Bennett T. and Edward Nelson (1999) Nominal income targeting in an open-economy optimizing model. Journal of Monetary Economics 43, 553–578. McCallum, Bennett T. and Edward Nelson (2000) Monetary policy for an open economy: An alternative approach with optimizing agents and sticky prices. Oxford Review of Economic Policy 16, 74–91. Pratt, John W. and Robert Schlaifer (1984) On the nature and discovery of structure (with discussion). Journal of the American Statistical Association 79, 9–21, 29–33. Pratt, John W. and Robert Schlaifer (1988) On the interpretation and observation of laws. Journal of Econometrics, Annals 39, 23–52. Rudebusch, Glenn D. (2002) Assessing nominal income rules for monetary policy with model and data uncertainty. Economic Journal 112, 402–432. Rudd, Jeremy and Karl Whelan (2005) New tests of the New Keynesian Phillips curve. Journal of Monetary Economics 52, 1167–1181. Sbordone, Argia (2007) Inflation Persistence: Alternative Interpretations and Policy Implications. Federal Reserve Bank of New York Staff Reports, No. 286. Swamy, P.A. V.B. and George S. Tavlas (1995) Random coefficient models: Theory and applications. Journal of Economic Surveys 9, 165–182. Swamy, P.A. V.B. and George S. Tavlas (2001) Random coefficient models. In Badi H. Baltagi (ed.), A Companion to Theoretical Econometrics, pp. 410–428. Malden, MA: Blackwell Publishers. Swamy, P.A.V.B. and George S. Tavlas (2005) Theoretical conditions under which monetary policies are effective and practical obstacles to their verification. Economic Theory 25, 999–1005. Swamy, P.A. V.B. and George S. Tavlas (2007) The new Keynesian Phillips curve and inflation expectations: Re-specification and interpretation. Economic Theory 31, 293–306. Swamy, P.A. V.B., George S. Tavlas and Jatinder S. Mehta (2007) Methods of distinguishing between spurious regressions and causation. Journal of Statistical Theory and Applications 1, 83–96.

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Swamy. P.A. V.B., Wisam Yaghi, Jatinder S. Mehta, and I-Lok Chang (2007) Empirical best linear unbiased prediction in misspecified and improved panel data models with an application to gasoline demand. Computational Statistics & Data Analysis 51, 3381–3392.

APPENDIX Models (1) and (2) provide log-linear approximations, around a zero-steady-state inflation rate, to Calvo’s nonlinear model of inflation dynamics. The assumption of zero-steady-state inflation has been questioned by a number of authors, including Ascari (2004), Bakhshi, Khan, Burriel-Llombart, and Rudolf (2007), Cogley and Sbordone (2006), Kozicki and Tinsley (2003), and Sbordone (2007). To avoid this assumption, Cogley and Sbordone (2006) rewrite model (2) as p˙ t − p¯˙ t = α1t Et (p˙ t+1 − p¯˙ t+1 ) + α2t (st − s¯t ) + α3t (p˙ t−1 − p¯˙ t−1 ) + α4t

∞

j −1 ϕ1t Et (p˙ t+j − p¯˙ t+j ) + η2t ,

(A.1)

j =2

where a bar over a variable denotes its value in a steady state characterized by shifting trend inflation, denoted by p¯˙ t , Et is as explained in equation (1), and η2t is an added error term. Two restrictions have been imposed on equation (A.1): (i) each steady-state value and the corresponding non–steady state value have the same coefficient value. (ii) The error term, η2t , captures the deviation of the functional form of (A.1) from the true functional form of NKPC and other possible misspecifications in (A.1). These restrictions might assign a wrong functional form to (A.1). With respect to the functional form of (A.1), TVC estimation recognizes that any finite class of the functional forms of NKPC may not encompass the true functional form. Only an infinite class of the functional forms of NKPC will encompass the true functional from. To obtain such a class, consider p˙ t = p¯˙ t + α1t Et p˙ t+1 + α2t st + α3t p˙ t−1 + α4t

∞

j −1

ϕ1t Et p˙ t+j + α5t Et p¯˙ t+1

j =2

+ α6t s¯t + α7t p¯˙ t−1 + α8t

∞ j =2

j −1

ϕ1t Et p¯˙ t+j +

mt

αkt xkt∗ ,

(A.2)

k=9

where the steady-state values are allowed to have their own coefficients that might be different from the coefficients of the corresponding non-steady-state values, the variables represented by η2t are explicitly written as xkt∗ , k = 9, . . . , mt , and mt is allowed to depend on time because the number of the determinants of p˙ t might change over time. Model (A.2) has no excluded variables because every possible determinant of p˙ t is included in (A.2). The variable, xkt∗ with any one of the values of k ≥ 9 and ≤ mt , in the last term of (A.2) represents imported raw materials that are treated as inputs to the home country’s productive process. This is the variable that should be included in equations (1) and (2) to make these equations open-economy specifications [McCallum and Nelson (1999, 2000)]. An infinite class of functional forms that encompasses the true functional form of NKPC is obtained by allowing all the coefficients of (A.2) to vary freely. Note that this result holds

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only when no determinant of p˙ t is excluded from (A.2). The member of this class with the true functional form is given by the coefficients of (A.2) with the correct time profiles because the coefficients of a linear-in-variables representation of a nonlinear equation with fixed parameters will be time-varying, as shown in footnote 8. We simply denote the coefficients of (A.2) with the correct time profiles by αkt∗ , k = 1, . . . , mt , because we do not know the shapes of these time profiles. The NKPC is truly linear if the αkt∗ and p¯˙ t are constant and is truly nonlinear otherwise. The coefficients of (A.2) with the correct time profiles cannot be obtained by increasing the order of the Taylor approximation to Calvo’s optimal pricing equation that is different from (A.2). For the simplicity of exposition, we change the symbols used to denote the explanatory variables of (A.2) other than p¯˙ t and xkt∗ with k ≥ 9 to xkt∗ , k = 1, . . . , 8, respectively. The variables, xkt∗ with k ≥ 4, are called “the excluded variables” because they are excluded from model (2). The steady-state variables in (A.2) are unobservable and their proxies are measured with error. Rather than use some proxies in place of the steady-state variables with the neglect of measurement errors in those proxies, we will do much better to treat those variables as omitted variables. The hybrid NKPC in (2) can be at odds with a rational behavior of households if an incorrect assumption is made about η1t . If we assume that η0t and η1t represent “the” explanatory variables excluded from models (1) and (2), respectively, and assume, in addition, that the explanatory variables included in (1) and (2) are independent of these error terms, respectively, then the assumption will be meaningless. For a proof of this proposition, see Pratt and Schlaifer (1988, p. 34), who showed that the “condition [that the explanatory variables of (1) or (2) be independent of ‘the’ excluded variables themselves] is meaningless unless the definite article is deleted and can then be satisfied only for certain ‘sufficient sets’ of excluded variables . . . ” To find such sufficient sets, we assume that xkt∗ = µ∗0kt + µ∗1kt x1t∗ + µ∗2kt x2t∗ + µ∗3kt x3t∗

(k = 4, . . . , mt ).

(A.3)

Inserting (A.3) into (A.2) gives p˙ t = p¯˙ t + + α3t∗ +

mt k=4 mt

αkt∗ µ∗0kt

+

αkt∗ µ∗3kt x3t∗ .

α1t∗

+

mt k=4

αkt∗ µ∗1kt

x1t∗

+

α2t∗

+

mt

αkt∗ µ∗2kt

x2t∗

k=4

(A.4)

k=4

The expected inflation used in (1) should be consistent with (A.4) because the meaningful assumption in (A.3) leads to (A.4). It is not possible to generate the values of x1t∗ = Et p˙ t+1 from (A.4) because (A.4) has too many unknowns. For this reason, we use models (6)–(9) to generate the proxies for x1t∗ . This procedure and our timing for the variables introduced in the dynamics of the expected inflation in (6)–(9) are valid, provided we take into account the measurement errors contained in these proxies. Let x1t , x2t , x3t denote the available observations on the proxies for x1t∗ , x2t∗ , x3t∗ , respectively. We treat the observed measurements, xkt = xkt∗ + vkt , k = 1, 2, 3, as the sums of (unobserved) “true” values and (unknown) measurement errors. The symbols, x ∗ , with an asterisk denote “true” values and the symbols, x, without an asterisk denote observable variables. The variables, xj t , j = 1, 2, 3, are called “the included explanatory variables” because they are included in

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t ∗ ∗ − vkt for xkt∗ in (A.4) gives equation (4) with γ0t = (p¯˙ t + m (2). Substituting xkt k=4 αkt µ0kt ) mt ∗ ∗ ∗ and γj t = (αj t + k=4 αkt µj kt )(1 − (vj t /xj t )), j = 1, 2, 3. The intercept, γ0t , contains an additional term if the available data on p˙ t contain measurement errors. The following interpretations of the terms of γ s may be given: t ∗ ∗ ˙ t of the portions of excluded 1. The term, m k=4 αkt µ0kt , of γ0t is the joint effect on p variables remaining after the effects of the “true” values of the included explanatory variables have been removed. This interpretation shows that in (4), γ0t = 0 even when p¯˙ t = 0. This disproves the claim that having a nonzero γ0t in (4) and a nonzero NKPC. The hybrid NKPC π00 in (5) amounts to having in (4) something that is not a t ∗ ∗ in (2) is misspecified because η1t is not equal to (p¯˙ t + m k=4 αkt µ0kt ) and ωf , λ2 , ωb are not equal to γ1t , γ2t , γ3t , respectively. The coefficient, ωf , does not become a function of the measurement error, p˙ t+1 − Et p˙ t+1 , when Et p˙ t+1 is replaced by p˙ t+1 , as under GMM estimation of (2). The correct interpretation of γ0t suggested by (A.4) results in the inclusion of an intercept in (4), (5), and Tables 2 and 3, and it is the misinterpretation of η0t and η1t that led to the elimination of a constant term from (1), (2), andTable 1. t ∗ ∗ 2. The term, m k=4 αkt µj kt , of γj t with j > 0 measures a omitted-variables bias as a result of excluded variables. This term is nonzero when the j th included variable acts partly as a stand-in variable for some of the excluded variables. It can be interpreted as the “indirect” effect because xj t affects some of omitted variables, which affect p˙ t . The omitted-variables bias can be time-varying even when the NKPC is truly linear ∗ because some of the µt j kt ∗can∗be time-varying. 3. The term, −(αj∗t + m k=4 αkt µj kt )(vj t /xj t ), of γj t with j > 0 measures a measurementerror bias as a result of mismeasuring xj t . It can be time-varying even when the NKPC is truly linear because the (vj t /xj t ) are time-varying and some of the µ∗j kt and mt can be time-varying. Typically, the output gap used in (7)–(9) is known to be measured with large error, and probably also the wage inflation used in (6)–(9). Consequently, the measurement error, v1t , in our proxy, x1t , for expected inflation can be large in magnitude. This does not imply a large absolute value t for ∗the∗ measurement-error bias component of the coefficient γ1t of x1t if |α1t∗ + m k=4 αkt µ1kt | is small and |x1t | is large relative to |v1t |. Similarly, the conditions under which the magnitude of the measurement-error bias component of the coefficient, γ2t , of rulct is small can be derived. 4. The term, αj∗t , is the direct effect of xj t on p˙ t and it is devoid of omitted-variable and measurement-error biases. This direct effect is constant if the NKPC is truly linear and is time-varying otherwise. If it is zero for all t, then the nonzero correlation between p˙ t and xj t is spurious [Swamy, Tavlas, and Mehta (2007)]. This definition is different from that of Granger and Newbold (1974) and applies to both linear and nonlinear models, whereas Granger and Newbold’s definition applies only to linear models. [Swamy and Tavlas (2007, pp. 299 and 301) proved that the nonzero values of α3t∗ are spurious.] The estimates in Table 2 support this analytical result. It should be noted that the direct effect, α2t∗ , on p˙ t of the (log of) average real marginal cost, st , (the driving process) does not become zero when γ0t is not set equal to zero. It may appear to be zero if it is confused with γ2t . 5. When the interpretations of γ s given here are adopted, model (4) coincides with a NKPC based on an open-economy specification without the assumption of a zerosteady-state inflation rate. Therefore, it is incorrect to say that in (4), we chose a

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NKPC based on a closed-economy specification with a zero-steady-state inflation rate. 6. The coefficients of model (4) may not follow random walks because they are the sums of three terms, each of which may follow a different stochastic process. Without realizing this, if we assume incorrectly that the coefficients of (4) follow random walks, then the components of γ s are not identifiable and the following comment by Pratt and Schlaifer (1988) applies: “If a statistician with a Bayesian computer program treats the likelihood function of c as if it were the likelihood function of b, as he must if he supplies no proper prior distribution of b given c, then what his printout will contain will be neither beans nor corn but succotash” (p. 49). They further point out that rather than assess a judgmental distribution of “proxy effects” for excluded variables based on mere guesses about their signs and magnitudes, a Bayesian will do much better to search like a non-Bayesian for concomitants that absorb them. Using equation (5), we are able to identify the components of γ s and do exactly as Pratt and Schlaifer suggest. The coefficient drivers included in (5) are our concomitants. They are used to decompose the coefficients of (4) into their respective components. There is no such thing as measurement errors in coefficient drivers. The only property of coefficient drivers that is important here is the closeness of the correspondence of their time variations to time variations in the components of γ s. We judge that time variations in the coefficient drivers included in (5) correspond closely to time variations in the components of the coefficients of (4).