Testing the New Keynesian Model on U.S. and Euro Area Data

Juselius, Mikael Working Paper Testing the New Keynesian Model on U.S. and Euro Area Data Economics Discussion Papers, No. 2008-23 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Juselius, Mikael (2008) : Testing the New Keynesian Model on U.S. and Euro Area Data, Economics Discussion Papers, No. 2008-23, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/17995 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en D iscussion Papers Discussion Paper 2008-23 May 21, 2008 Testing the New Keynesian Model on U.S. and Euro Area Data Mikael Juselius Swedish School of Economics, and RUESG, University of Helsinki Abstract: I apply the Johansen and Swensen (1999, 2004) method of testing exact rational expectations within the cointegrated VAR (Vector Auto-Regressive) model, to testing the New Keynesian (NK) model. This method permits the testing of rational expectation systems, while allowing for non-stationary data. The NK-model is tested on quarterly U.S. and Euro area time series data. I find that the restrictions implied by the core equations of the NK-model are rejected regardless of sample periods or measures of real marginal costs. I also provide a tentative explanation of the results favored by previous researches. Paper submitted to the special issue “Using Econometrics for Assessing Economic Models” edited by Katarina Juselius. JEL: C32, C52, E31, E52 Keywords: New Keynesian Phillips curve; cointegration; vector autoregressive model Correspondence: Mikael Juselius, University of Helsinki, Department of Economics, P.O. Box 17, FIN – 00014 Helsinki. E-mail: [email protected]. www.economics-ejournal.org/economics/discussionpapers © Author(s) 2008. This work is licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany 1 Introduction The popularity of the New Keynesian model in recent years has led to numerous empirical attempts to evaluate the performance of the model. A recent overview of this literature is provided by Henry and Pagan (2004). The typical study has focused on one of the two structural equations of the NK-model, usually the New Keynesian Phillips (NKP) curve since it captures the relevant features of price stickiness. For instance Gali and Gertler (1999), and Gali et al. (2001) find strong evidence in favor of the Phillips curve, using single equation GMM (General Method of Moments). Sbordone (2002) also reports favorable results by a slightly different approach,1while Fuhrer (1997) obtains less favorable results using ML (Maximum Likelihood). More recent contributions include Matheron and Maury (2004), McAdam and Willman (2004), Roberts (2005), and Nelson and Lee (2007). The second equation, the expectational “IS” curve, has been investigated by Fuhrer (2000) and more recently in Kara and Nelson (2004) and Fuhrer and Rudebusch (2004). Since the early contributions, a number of empirical issues have been raised. The use of single equation estimation procedures has been criticized on the grounds that empirical identification requires a system approach. Furthermore, due to problems such as weak instruments, GMM estimates are likely to be very imprecise. Thorough discussions on these issues can be found in Ma (2002), Mavroeidis (2004), Rudd and Whelan (2005a,b).2These difficulties has led authors such as Linde (2005) and Giordani (2004) to consider a full system approach. Another, largely neglected, issue is the apparent non-stationary behavior of the data. This problem is noted and discussed by Bardsen et al. (2004), among others. If data are non-stationary, we have an additional reason to view the previous results with caution. The aim of this paper is to demonstrate how one can test the validity of the restrictions implied by the NK-model when the key variables are difference stationary. The procedure is illustrated by testing the core equations of the NK-model within a cointegrated VAR (Vector Auto-Regressive) model on 1Sbordone (2002) uses the method of testing present value models, proposed by Campbell and Shiller (1987). She assumes that the data is stationary, although the method also allows for special cases when the data is non-stationary. The Campbell and Shiller (1987) technique with non-stationary data has recently been employed to the NKP-curve by Demery and Duck (2003) and Tillmann (2005). 2However, see also Gali et al. (2005) and Sbordone (2005) for answers to some of this criticism. 2 quarterly U.S. and Euro area time series data. The sample period is 1960:12005:2 for the U.S. and 1970:1-2003:4 for the Euro area. The restrictions implied by the core equations of the NK-model are tested by the exact linear rational expectations (RE) method proposed by Johansen and Swensen (1999, 2004). The advantage of this method is that it permits formal testing of the restrictions implied by RE systems, at the same time allowing the data to be non-stationary. The tests proposed here are related to the single equation tests of the NKP-curve by Fanelli (2008) and Barkbu and Batini (2005). Fanelli (2008) uses the three-step method of Fanelli (2002), to formally test the NKPcurve within a cointegrated VAR model on Euro area data.3and rejects the NKP-curve specification. Barkbu and Batini (2005) apply the Johansen and Swensen method to the NKP-curve on Euro area data. They obtain favorable results for the NKP-curve using a minimal information set. However, it is doubtful whether the information used by Barkbu and Batini (2005) captures the main features of the inflation process (see discussion by Bardsen et al. (2004)). This paper differs from Fanelli (2008) and Barkbu and Batini (2005) in basically two ways. First, an extended information set is used which combined with the Johansen and Swensen method, allows for testing both core equations of the NK-model instead of only the NKP-curve equation. Second, the model here is also tested on U.S. data. The results suggest that the evidence in favor of the core equations of the NK-model, the IS curve and the new Keynesian Phillips curve, is weak. The restrictions implied by the equations are rejected on both U.S. and Euro area data. Sensitivity analysis of different sample periods and of different measures of marginal costs do not change the results. Furthermore, by only considering the cointegration restrictions implied by the NK-model (cointegration implications, henceforth) a less demanding test of the NK-model is provided . These restrictions form a subset of the complete set of restrictions implied by the NK-model and, hence, constitute a necessary condition for the NK-model. The latter are rejected in most cases. Interestingly, the cointegration implications of the NKP-curve are not rejected on Euro area data when labor’s share is used as a measure of marginal costs. It is precisely for this case that favorable results on the NKP-curve have been reported by, for 3There are some drawbacks to this method. First, it is only possible to test one RE equation and, second, a condition similar to strong exogeneity of the forcing variables is needed in the estimations. 3 instance, Gali and Gertler (1999). But, since the overall restrictions of the equation are rejected, the support for the NKP-curve is nevertheless not overwhelming. Methods that rely on less formal evaluations of the NKP-curve, such as the size and significance of the marginal costs and forward terms, run the risk of claiming success when only the cointegration implications are met. The next section introduces a baseline New Keynesian model. The data and information sets are discussed in section 3, while section 4 introduces the Johansen and Swensen method. The estimation results are presented in section 5 followed by a discussion in section 6. Section 7 concludes. 2 The New Keynesian model This section introduces a standard version of the closed economy New Keynesian model.4The NK-model belongs to a class of “miniature” dynamic stochastic general equilibrium (DSGE) models that are based on optimizing households and firms, rational expectations, and nominal price rigidities. The model consists of three non-linear equations, a forward-looking “IS curve” that relates output to the real rate of interest, a New Keynesian Phillips curve that relates inflation to real marginal costs, and a central bank policy rule for the nominal interest rate. Empirical variants of these equations are obtained by log-linearizing around the steady states of the key variables and adding lags.5The two structural equations, the IS curve and the NKP-curve commonly take the linearized representations ˜yt=ϕ11Et˜yt+1 −ϕ12 (it−Et∆pt+1) + ϕ13 ˜yt−1+υt(1) ∆pt=ϕ21Et∆pt+1 +ϕ22xt+ϕ23∆pt−1(2) 4The standard closed economy model has been extended in several ways, for instance by incorporating labor market imperfections (Erceg et al., 2000) or by accounting for investments in capacity (Razin, 2005). Open economy issues have been investigated by several authors, for example Clarida et al. (2002), Galí and Monacelli (2005), Svensson (2000), Batini et al., 2005, and Matheson (2008). The core equations of the NK-model often have the same form in such extensions of the model. 5The lagged terms can be motivated for example by rule of thumb pricing and habit persistence as in Gali and Gertler (1999) and Fuhrer (2000). Detailed derivations and discussions of the equations are provided by McCallum and Nelson (1999), Clarida et al. (1999), Yun (1996), Walsh (2003), and Woodford (2003), among others. 4 where constants representing equilibrium values are suppressed, ˜yt=yt−yf t is the flexible price output gap, ytis real output and yf tis the level of output that would prevail under flexible prices, itis the nominal short-term interest rate, ptis the price level, υt=ϕ11Etyf t+1 +ϕ13yf t−1−yf t,xtis real marginal costs, Etis the expectations operator conditional on the agent’s information set at time t, and the coefficients, ϕij ≥0for all iand jare functions of the structural parameters from the underlying theory. The purely forward looking versions of the two structural equations are obtained by setting ϕ11 = 1and ϕ13 =ϕ23 = 0. In addition to equations (1) and (2), a policy rule for the nominal interest rate is usually derived by specifying a policy objective and solving under discretion or commitment. For example, a frequently used specification is the interest rate smoothing Taylor rule it=φ1it−1+ (1 −φ1) (φ2(∆pt−∆p∗) + φ3˜yt+φ4xt) where ∆p∗is a constant inflation target (e.g. Clarida et al., 1999), and φiare parameters. However, there are several problem associated with such policy rules. First, the empirical support for rules with constant targets have been mixed and these rules are generally not robust to small alterations in the specifications (see for example Kozicki, 1999). In contrast, (1) and (2) are more robust to alternative ways of specifying the underlying theoretical structure and have received more support in the literature. Hence, imposing the restrictions from some (empirically incorrect) policy rule may cause rejection of the complete NK-model system even when the structural equations (1) and (2) are data consistent. Second, the optimal inflation target may have varied over time as argued in Cogley and Sargent (2005), Ireland (2007), and Sbordone (2007). Allowing for a time varying policy rule introduces a latent variable into the system, causing a violation of the exactness of the policy equation and thereby rendering the Johansen and Swensen framework inapplicable.6 For these reasons, I do not impose any policy rule restrictions in the empirical analysis. Ignoring these restrictions does not invalidate the tests of (1) and (2), and has the advantage of simplifying the analysis considerably. But, of course, empirical identification of the parameters of the NK-model 6Equations (1) and (2) are still exact even when the optimal inflation target is time varying as noted by Juselius (2008). However, the parameters may not be identifiable in this case. 5 requires that the processes of all forcing variables are specified. Therefore, if the tests of (1) and (2) are not rejected, the next step should be to estimate a reasonable policy rule for the interest rate prior to parameter identification. Exploiting such policy rule restrictions would also increase the degrees of freedom of the test and, hence, its efficiency. The empirical counterpart of the variables in (1) and (2) typically display a high degree of persistence (see for example Bardsen et al. 2004 and Dees et al. 2008) suggesting non-stationarity rather than stationarity. In this case, two important issues that have to be addressed: First, the log-linearization of (1) and (2) is typically achieved by assuming that the key variables are stationary around their (deterministic) steady states. If the variables are non-stationary these derivations are no longer valid.7By linearizing around cointegration relations instead of variables, as in Altug (1989) and Ireland (2004) it is, however, possible to derive (1) and (2) from a DSGE model when the variables are difference stationary.8This possibility is not fully explored here. Instead, I will take the view in Fanelli (2008) and Dees et al. (2008) that (1) and (2) are likely to capture the essential features of inflation and output dynamics regardless of the properties of the individual series. Second, the sources of stochastic trends in (1), (2), and the policy rule need to be specified. The only possible sources of stochastic trends are the exogenous variables (see for instance Framroze Møller, 2008), i.e. xtand yf t. In addition, the nominal interest rate can be an additional source in (1) and (2) if we allow for a time varying inflation target, (∆p)∗ t. Once 7Even if stationarity is assumed, as is common in the literature, the variance of the key variables tend to be very large which makes linearizing around their respective steady states a very poor approximation. Moreover, inference is highly unreliable in small samples with near unit roots as demonstrated by Johansen (2006). 8In fact, this easily achieved for (1). A typical Euler equation for consumption is given by Et(Ct/Ct+1)−λ1=λ2(1 + it)Et(Pt/Pt+1) where Ctis consumption, Ptis the price level, and λiare parameters. If Ct∼I(1), Pt∼I(2), and it∼I(1), and the real interest rate rt= (1 + it)Et(Pt/Pt+1)∼I(0), the Euler equation can be linearized around the log-normally distributed stationary variables Ct/Ct+1 and rt. Using the equilibrium condition Ct=Ytand subtracting yf tfrom both sides of the linearized equation yields the purely forward looking version of (1). Note that subtracting yf tis strictly not needed since we do not linearize around it. A similar derivation for the NKP-curve is much more involved but should in principle yield an equation similar to (2) for some models of price stickiness (e.g. Roberts, 1995). 6 stochastic trends in the exogenous variables are permitted, restrictions on the parameters of (1) and (2) must be considered on order to ensure consistency.9 Consider first the restriction ϕ21 +ϕ23 = 1, which is frequently imposed on the NKP-curve. When xtcontains a stochastic trend, this restriction implies the implausible result that ∆pt∼I(2) unless ϕ22 = 0. Thus, to avoid this problem, ϕ21 +ϕ23 <1must be imposed instead. Similarly, the restriction ϕ11 +ϕ13 = 1 is often imposed in the literature. Under this restriction, (1) can expressed in terms of ∆˜ytand υtis stationary. In this case, the real interest rate, rt=it−Et∆pt+1, must be stationary if the NK-model is true, implying that both itand ∆ptmust share the stochastic trend in xt. On the other hand, the case ϕ11 +ϕ13 <1is not possible unless ytis stationary, since otherwise the non-stationary latent variable υtdoes not cancel in (1). Hence, the restriction ϕ11 +ϕ13 = 1 must be applied when (1) is linearized around a non-stationary flexible price level of output. An alternative version of (1), in terms of ytrather than ˜yt, can be obtained by linearizing around the difference of ytor even around a stationary nonlinear combination of ytand the real interest rate. In this case ϕ11 +ϕ13 <1 can be permitted allowing for more interesting dynamics. For example, if the inflation target is time varying, the real interest rate can be non-stationary and still be consistent with the NK-model. Another advantage is that no measure of yf tis needed in the analysis. For these reasons, the specifications considered in the empirical analysis are yt=Etyt+1 −ϕ32 (it−Et∆pt+1)(3) ∆pt=ϕ41Et∆pt+1 +ϕ42xt.(4) corresponding to the purely theoretical NK-model, and yt=ϕ51Etyt+1 −ϕ52 (it−Et∆pt+1) + ϕ53yt−1(5) ∆pt=ϕ61Et∆pt+1 +ϕ62xt+ϕ63∆pt−1(6) corresponding to the hybrid version of the model. Two popular measures of xthave been used in empirical work; labor’s share of income, i.e. xt=wtnt/ytpt, where wtis wages and ntis the number of employed, and the output gap, i.e. xt=yt−yn t, where yn tis some measure 9Here, the discussion is restricted to such restrictions that have direct implications for the stochastic trends. For a more general discussion of restrictions on the parameters of the NK-model, see for instance Bardsen et al. (2004). 7 of potential output.10 The output gap should, in principle, be stationary by construction. But this implies that both inflation and nominal interest rates should be stationary as well, making it hard to reconcile (3) and (4) with the high degree of persistence typically found in empirical studies of these variables. This may also help to explain why ϕ42 or ϕ62 have been found to be insignificant in previous empirical applications with the output gap as a measure of marginal costs (see for example Gali and Gertler, 1999). However, the empirical results of section 5 show that unit-roots cannot be rejected in output gap measures for samples of up to 30 years of quarterly data. Thus, (3) and (4) may still be reasonable descriptions of short-run to medium-run price and output dynamics, even when output gaps are used as real marginal costs. Finally, it should be noted that money is not absent from the New Keynesian model. But as long as money stock is completely determined by the relationship mt−pt=ϕ51yt−ϕ52it and the central bank is targeting the interest rate, money has no interesting role to play in the model. This ’unimportance of money’ assumption will also be tested empirically. 3 Data and information This section introduces the data and discusses potential information sets that can be used to evaluate the NK-model. The data consists of quarterly U.S. and Euro area time series on the following variables (in logs): a price index, pt, a nominal short-run interest rate, it, a real money aggregate, mt, real output, yt, potential output, yn t, and real aggregate wages, wt. The Euro area data spans the years 1970:01-2003:04 and the U.S. data 1960:01-2005:02 (apart from a production function based measure of potential output which spans 1973:02-2003:04 and 1964:2-2005:02 respectively). Figure 1 plots Euro area and U.S. inflation rates. Detailed descriptions of the data are provided in appendix A. 10The implicit assumption here is that the flexible price output gap is approximately equal to some measure of the gap between output and its potential yn t. This may of course be incorrect. 8 IEU 11 , 1973:3-2003:4 IEU 12 , 1970:1-2003:4 r λitrace trace95 p-value λitrace trace95 p-value 0 0.41 158.42** 88.55 0.00 0.34 152.30** 88.55 0.00 1 0.27 95.83** 63.66 0.00 0.32 96.15** 63.66 0.00 2 0.25 57.76** 42.77 0.00 0.14 45.34* 42.77 0.03 3 0.14 23.43 25.73 0.10 0.12 25.50 25.73 0.05 4 0.04 5.18 12.48 0.58 0.07 9.08 12.48 0.18 Table 1: The rank test statistic (trace test) for the full sample Euro area data. In the table, λiare the eigenvalues from the reduced rank regression (see Johansen, 1995). Trace95 are the 95%-quantiles of the trace distribution and (**) denotes rejection at the 1% significance level and (*) denotes rejection at the 5% significance level. Standard misspecification tests indicated some deviations from normality in both models,15 as well as minor problems with autocorrelation and ARCH in the share model. None of the variables were found to be stationary, nor long-run excludable in the two models. The results of these tests are reported in appendix C. Finally, recursive tests for constant parameters were also performed on both models.16 The results from these tests indicated two possible structural breaks, one in the early 1980’s and one at around the middle of 1993. For this reason, separate analyzes for the full sample and for the subsample 1982:1-2003:4 were conducted.17 Similar breaks have been found, for example, by Batini (2006) and Barkbu and Batini (2005). Table 2 reports the rank test statistic for the subsample 1982:1-2003:4. Again, the appropriate choice of rank seems to be three in the gap model, though r= 2 was almost accepted. For the share model the rank test suggests r= 1. Based on additional information in the model we find that r= 2 is the appropriate choice.18 Sensitivity analysis was conducted with respect 15This is mainly due to some large outliers in the turbulent seventies. These outliers can be accounted for by dummy variables, but doing so does not change the results. 16The full description of these recursive test can be found in Hansen and Johansen (1999) and include two tests for the constancy of the β-vectors, a test for the constancy of the log-likelihood, a fluctuation test of the eigenvalues, among others. These results are available upon request. 17The subsamples 1970:1-1981:4 and 1993:2-2003:4 are too small for reliable estimation and hence not considered in the main text. However, the latter subsample is discussed in appendix C. 18The trace test point unambiguously toward r= 2 if wtis excluded from the model. Since additional information should not in principle reduce the CI rank, this provides an 15 IEU 11 , 1982:1-2003:4 IEU 12 , 1982:1-2003:4 rλitrace trace95 p-value λitrace trace95 p-value 0 0.41 133.54** 88.55 0.00 0.45 112.46** 88.55 0.00 1 0.27 88.67** 63.66 0.00 0.24 60.50 63.66 0.09 2 0.25 46.32* 42.77 0.02 0.19 37.04 42.77 0.17 3 0.14 21.03 25.73 0.18 0.11 18.58 25.73 0.31 4 0.04 7.17 12.48 0.34 0.09 8.40 12.48 0.23 Table 2: The rank test statistic (trace test) for the 1982:1-2003:2 Euro area data. In the table, λiare the eigenvalues from the reduced rank regression (see Johansen, 1995). Trace95 is the 95%-quantiles of the trace distribution and (**) denotes rejection at the 1% significance level and (*) denotes rejection at the 5% significance level. to these choices but it did not change any of the results significantly. Note that the finding of r= 2 is the share model implies three common stochastic trends, which is inconsistent with the NK-model in section 2. No serious misspecification was detected in either model, except for some small deviations from normality. As before, stationarity was rejected for all variables in both models, but now long-run exclusion of wtcould not be rejected with a p-value of 0.51. Finally, recursive tests of parameter stability were performed and did not signal parameter instability over the period. An interesting additional result is that the money stock is needed in the information sets. Long-run exclusion was rejected for this variable and removing it from the information sets considerably worsened the fit of each model. Moreover, the hypothesis that money has a unit vector in αwas rejected (formally tested in appendix C), implying that money has some explanatory power over the other variables in the system. Thus, it appears that money is important, at least when M3 is used as the money stock measure. The results of testing the restrictions implied by the core equations of the NK-model are reported in table 3. The details of the estimations are provided in appendix B. The single equation restrictions are first considered separately. As can be seen from table 3, almost all restrictions are strongly rejected. Furthermore, the coefficient estimates are clearly not sensible within the NK-model. For instance, in the cases of the NKP-curve, equations (4) and additional reason for maintaining r= 2 despite the evidence for r= 1 in table 2. It seems that the inclusion of wtin the information set “muddles the water”. 16 ITEqu iϕi1ϕi2ϕi3−2lnQ df p-value 73:33 – -0.18 – 49.68 13 0.00 “ 4 1.11 -0.06 – 48.98 12 0.00 “ 5 0.79 -0.09 0.21 41.10 12 0.00 IEU 11 “ 6 1.79 -0.11 -0.66 30.44 11 0.01 82:13 – -0.18 – 58.45 13 0.00 “ 4 1.16 -0.05 – 41.27 12 0.00 “ 5 0.83 -0.14 0.17 54.52 12 0.00 “ 6 1.96 -0.10 -0.87 20.86 11 0.04 70:13 – -0.33 – 55.83 13 0.00 “ 4 1.49 -0.05 – 55.72 12 0.00 “ 5 0.81 -0.20 0.19 48.73 12 0.00 IEU 12 “ 6 2.12 -0.04 -0.72 40.69 11 0.00 82:13 – -0.22 – 44.10 13 0.00 “ 4 2.13 -0.10 – 14.28 12 0.28 “ 5 0.83 -0.17 0.17 40.24 12 0.00 “ 6 2.98 -0.11 -0.72 7.22 11 0.78 Table 3: Tests of the restrictions implied by the core equations of the NK-model (3)-(6) on Euro area data. The column “Equ i” indicates that the restrictions implied by equation (i) is being tested and ϕij are the corresponding estimates. In equation (5) we have the additional restriction ϕ51 +ϕ53 = 1 (hence, we have 12 degrees of freedom). (6), the coefficients on the forward terms, ϕ41 and ϕ61 are above one and the coefficient on the forcing variable is small and negative regardless of the measure used for marginal costs. Also, for the IS curve, the coefficient on the real interest rate has the wrong sign. Only the coefficients of the forward and backward terms can be considered sensible. In the few cases where the restrictions are not rejected, the coefficients are not plausible. If the coefficients are restricted to the unit interval in these cases, the restrictions are strongly rejected. Finally, the restrictions from both (3)-(4) and (5)-(6) where tested simultaneously on all periods and all information sets. These restrictions were strongly rejected in all cases, as should be expected, given the rejection of the single equation restrictions above. These results imply that the evidence for the IS curve and the New Keynesian Phillips curve on Euro area data must be considered weak. The results 17 IUS 11 , 1964:2-2005:2 IUS 12 , 1960:1-2005:2 rλitrace trace95 p-value λitrace trace95 p-value 0 0.24 114.22** 88.55 0.00 0.22 108.36** 88.55 0.000 1 0.19 70.06* 63.66 0.01 0.11 64.04* 63.66 0.046 2 0.13 36.86 42.77 0.18 0.09 42.66 42.77 0.051 3 0.06 14.55 25.73 0.62 0.07 24.40 25.73 0.074 4 0.03 5.15 12.48 0.58 0.06 10.18 12.48 0.121 Table 4: The rank test statistic (trace test) for the full sample U.S. data. In the table, λiare the eigenvalues from the reduced rank regression (see Johansen, 1995). Trace95 is the 95%-quantiles of the trace distribution and (**) denotes rejection at the 1% significance level and (*) denotes rejection at the 5% significance level. of testing the NKP-curve are similar to those of Fanelli (2008) in this respect. In section 6 we discuss some reasons for this failure of the model. 5.2 U.S. data Initial modeling of the two information sets for U.S. data suggested k= 3 that a linear trend should be restricted to the cointegration space in both models. The rank test statistic reported in Table 4 suggest that the rank is two in the first model. However, r= 1 is close to acceptance in the share model and there is also some uncertainty between the choice of r= 2 and r= 3. It appears that the inclusion of wtin the information set again “muddles the water”. The choice r= 3 can be disregarded if one takes into account other information in the model as done before. Thus, only the results for r= 2 are reported below (see footnote 14). Again, this result implies three common stochastic trends which is inconsistent with the NK-model. Standard misspecification tests indicated deviations from normality, due to some very large outliers, and problems with ARCH, stemming from the short-term interest rate series, in both models. None of the variables were found to be stationary nor long-run excludable in the two models. Finally, recursive tests for constant parameters were also performed. Both models showed evidence of a structural break at around 1979, marking the beginning of the Volcker-Greenspan era. Similar structural breaks have previously been found in the empirical literature, for example in Roberts (2005) and Romer and Romer (2004). Because the ARCH problems in the short-run interest rate disappeared after 1982, the sample was split at that point. Roberts 18 IUS 11 , 1982:1-2005:2 IUS 12 , 1982:1-2005:2 rλitrace trace95 p-value λitrace trace95 p-value 0 0.53 147.41** 88.55 0.00 0.51 137.34** 88.55 0.00 1 0.35 78.80** 63.66 0.00 0.36 73.83** 63.66 0.00 2 0.21 39.61 42.77 0.10 0.17 34.01 42.77 0.29 3 0.10 18.71 25.73 0.31 0.15 17.44 25.73 0.39 4 0.09 8.92 12.48 0.19 0.03 2.80 12.48 0.88 Table 5: The rank test statistic (trace test) for the subsample, 1982:1-2005:2, U.S. data. In the table, λiare the eigenvalues from the reduced rank regression (see Johansen, 1995). Trace95 is the 95%-quantiles of the trace distribution and (**) denotes rejection at the 1% significance level and (*) denotes rejection at the 5% significance level. (2005) considers a similar split, but leaves out the years 79-83 corresponding to the Volcker disinflation era. A sensitivity analysis of this choice showed that the main results did not change. Thus, separate analyzes of the full sample and of the subsample 1982:1-2005:2 were conducted. In the share model, there was some evidence of a structural break around 1993. Table 5 reports the rank test statistic for the sample 1982:1-2005:2. The appropriate choice of rank is two in both models. There was no serious misspecification in the gap model, apart from some small deviations from normality, while there were evidence of small problems with ARCH, autocorrelations, and deviations from normality in the second model. Stationarity was rejected for all variables in both models and again the long-run exclusion of wtcould not be rejected (p-value 0.40) in the second model. Finally, recursive tests for parameter stability were re-performed for the models. The tests did not show any serious parameter instability over the period in the gap model, while there were still some evidence of a break around 1993 in the share model. The results of testing the restrictions implied by the core equations of the NK-model on the U.S. data are reported in table 6. Almost all restrictions are rejected, as can be seen from table 6. Furthermore, the coefficient estimates are very similar to those of the Euro area data. Finally, the restrictions from both (3)-(4) and (5)-(6) were tested simultaneously on both periods and both models. These restrictions were strongly rejected in all cases. Hence, the evidence in favor of the NK-model on U.S data must also be considered weak. 19 ITEqu iϕi1ϕi2ϕi3−2lnQ df p-value 64:23 – -0.40 – 86.41 18 0.00 “ 4 1.13 -0.03 – 49.13 17 0.00 “ 5 0.80 -0.28 0.20 75.01 17 0.00 IUS 11 “ 6 1.52 -0.04 -0.54 24.43 16 0.08 82:13 – 0.06 – 72.63 18 0.00 “ 4 1.60 -0.02 – 50.71 17 0.00 “ 5 0.80 -0.27 0.20 75.53 17 0.00 “ 6 1.98 -0.07 -0.75 39.02 16 0.00 60:13 – -0.43 – 73.71 18 0.00 “ 4 1.24 -0.05 – 45.02 17 0.00 “ 5 0.80 -0.29 0.20 61.34 17 0.00 IUS 12 “ 6 1.66 -0.04 -0.42 32.94 16 0.01 82:13 – 0.02 – 57.65 18 0.00 “ 4 1.66 -0.02 – 44.17 17 0.00 “ 5 0.70 0.00 0.30 38.54 17 0.00 “ 6 2.27 -0.01 -0.55 38.04 16 0.00 Table 6: Tests of the restrictions implied by the core equations of the NK-model (3)-(6) on the U.S. data. The column “Equ i” indicates that the restrictions implied by equation (i) are being tested and ϕij are the corresponding estimates. In equation (5) we have the additional restriction ϕ52 +ϕ53 = 1 (hence, we have 17 degrees of freedom). 6 Explaining the results The reasons for the empirical failure of the NK-model are investigated in this section. We begin by discussing a particular condition on cointegration for the NK-model. This condition is then tested and interpreted in light of previous findings in the literature. The estimated coefficients in tables 3 and 6 are also given an interpretation. 6.1 Cointegration implications of the NK-model under I(1) data It is easy to provide a necessary condition on cointegration implied by the NK-model, provided that the data is non-stationary and well described by model (8). In this case inflation must be cointegrated with the measure 20 of marginal costs and output must be cointegrated with the real rate of interest.19 This can be seen directly, by observing that if the first restriction β∗α0c1=d∗ 1(13) in (11) holds, then d∗ 1∈sp(β∗). That d∗ 1∈sp(β∗)is only a necessary condition is clear, since (11) includes several other restrictions. The advantage of the condition d∗ 1∈sp(β∗)is that it is very easy to verify on data. For each of equations (3)-(6), d∗ 1will take an explicit form (the d∗ 1corresponding to equation iis denoted by d∗ i1). Thus, if xt=yt−yn twe get d∗ 31 = (−ϕ32, ϕ32,0,0,0,0)0 d∗ 41 = (1 −ϕ41,0,0,−ϕ42, ϕ42,0)0 d∗ 51 = (−ϕ52, ϕ52,0,1−ϕ51 −ϕ53,0,0)0 d∗ 61 = (1 −ϕ61 −ϕ63,0,0,−ϕ62, ϕ62,0)0 and if xt=wt−yt, the signs on the coefficients ϕ42 and ϕ62 are changed. Assuming ϕij >0for all iand j, it can be seen that d∗ 31 is nested in d∗ 51 by the restriction ϕ51 +ϕ53 = 1, and that d∗ 41 and d∗ 61 are similar. Note also the theoretically interesting cases ϕ41 6= 1 and ϕ61 +ϕ63 = 1. Table 7 reports the results of testing whether these relations are in the estimated cointegration space. Attention here is restricted to the subsample starting in 1982:1. It can be seen from the table, that the cointegration implications of the “IS” curve, the rows of d∗ 31 and d∗ 51, are rejected in all cases. Hence, it is not surprising that the corresponding hypotheses in tables 3 and 6 were rejected. Table 7 also reveals some interesting facts about the NKP-curve (d∗ 41 and d∗ 61). The cointegration implications of the NKP-curve on Euro area data are rejected if we use the output gap as a measure of marginal cost. Furthermore, it can easily be seen that the signs of the estimates are wrong in this case. However, if labor’s share is used as a measure instead, the condition holds and the signs on the coefficients are correct and of reasonable magnitude. This, then, provides a possible explanation for the success of the labor’s share measure in the previous literature. In essence, since the data has been assumed to be stationary, what has been estimated is the necessary condition on cointegration implied by the NKP-curve. However, this finding is not sufficient to conclude that the NKP-curve is a good description of 19A similar necessary condition on cointegration for present value models is discussed by Campbell and Shiller (1987). 21 IEU ˆ β∆pˆ βiˆ βmˆ βyˆ βyn/w ˆ βtp-value d∗ 31 -1 1 0 0 0 0 0.00 IEU 11 d∗ 41 1 0 0 7.78 -7.78 0 0.01 d∗ 51 -5.17 5.17 0 1 0 0 0.00 d∗ 61 0 0 0 1 -1 0 0.02 d∗ 31 -1 1 0 0 0 0 0.00 IEU 12 d∗ 41 1 0 0 0.08 -0.08 0 0.37 d∗ 51 -26.42 26.42 0 1 0 0 0.03 d∗ 61 0 0 0 -1 1 0 0.00 d∗ 31 -1 1 0 0 0 0 0.00 IUS 11 d∗ 41 1 0 0 -0.18 0.18 0 0.33 d∗ 51 -41.10 41.10 0 1 0 0 0.00 d∗ 61 0 0 0 1 -1 0 0.10 d∗ 31 -1 1 0 0 0 0 0.00 IUS 12 d∗ 41 1 0 0 0.09 -0.09 0 0.00 d∗ 51 -43.54 43.54 0 1 0 0 0.02 d∗ 61 0 0 0 -1 1 0 0.00 Table 7: Tests of the cointegration implications of the NK-model. Estimated βcoefficients are denoted by ˆ βx, where xindicates the variable. The hypotheses in d∗ 61 are derived under the additional restriction ϕ61 +ϕ63 = 1. inflation. Methods that rely on less formal tests of the NKP-curve, and more generally the NK-model, run the risk of claiming success when only the cointegration implications are met. Generally, cointegration between the key variables of any expectational equation of the form (9), is a necessary condition when the data is I(1). Hence, investigating cointegration between the variables in the system provides valuable information for future theoretical developments. This, potentially interesting avenue, is not explored further in the present paper. The opposite finding holds on U.S. data. The cointegration implications are not rejected when the output gap measure is used, but rejected when the labor’s share measure is used. This result may account for the poor performance of the labor’s share based NKP-curve on U.S. data that has been previously observed. Finally, note that d∗ 61 in table 7 essentially tests if the output gap or labor’s share is stationary. The stationarity of the measures is rejected in 22 most cases, with the exception of the U.S. output gap. However, output gap measures should be structurally stationary by construction and it is puzzling that stationarity is rejected for the European output gap. It appears that the persistence of the European business cycle creates a near unit-root problem in the output gap series.20 6.2 Coefficient estimates and solution of the RE system The fact that neither ∆ptand xtnor ytand rt=it−∆ptare cointegrated in most cases, may account for the implausible and strange coefficients in tables 3 and 6. For instance, note that the coefficients on xtin the NKP-curve are consistently small compared to the coefficients on ∆pt. If the coefficients on xtare not statistically different from zero, it seems reasonable that the coefficients capture the unit root behavior of inflation, rather than being meaningful in terms of the NKP-curve. Such results were found by Bardsen et al. (2004) on Euro area data. The stability of the RE system can be investigated by the method proposed by Blanchard and Kahn (1980). To this end, (3)-(4) and (5)-(6), with xt=yt−yn t, are written in the form (Xt+1 EtPt+1 )=A(Xt Pt)+γZt(14) where γZtcollects the exogenous variables yn tand it.21 In terms of (14), (3) and (4) are represented by Xt=∅,Pt= (yt,∆pt)0, and A=(1−ϕ32ϕ42 ϕ41 −ϕ32 ϕ41 −ϕ42 ϕ41 1 ϕ41 ).(15) Likewise, for equations (5) and (6) we have Xt= (yt−1,∆pt−1)0,Pt= (yt,∆pt)0, and A=      0 0 1 0 0 0 0 1 −ϕ53 ϕ51 ϕ52ϕ63 ϕ51ϕ61 1 ϕ51 +ϕ52ϕ62 ϕ51ϕ61 −ϕ52 ϕ51ϕ61 0−ϕ63 ϕ61 −ϕ62 ϕ61 1 ϕ61       .(16) 20The unit-root in the output gap variable is rejected if the full sample, 1973:2-2003:4, is investigated. 21Alternatively, both yn tand itcould be treated as predetermined with roots less or equal to one in absolute value. Doing so does not add anything to the analysis. 23 The corresponding Amatrices when xt=wt−ytare similar apart from some changes in the signs. Using the values from tables 3 and 6, the roots of (15) and (16) can be calculated. For all cases in the tables, one root is very close to unity, while the remaining roots are within the unit circle. In the different gap models the roots cluster around 1.02 and for the share models around 0.99. This suggest that there is no unique stable forward solution to the system and, since it is unlikely that we could reject the unit root statistically, that the solution is non-stationary. Hence, the proper interpretation of the estimates in tables 3 and 6 is that they confirm that the data is non-stationary and that the cointegration implications do not hold. 7 Conclusions This paper applies the Johansen and Swensen (1999, 2004) method of testing linear rational expectations models, to testing the New Keynesian Model on U.S. and Euro area data. The tests were conducted on both the individual equations separately and on system as a whole. The NK-model was rejected on both U.S. and Euro area data. Several sensitivity analyzes with respect to the choice of measures, sample periods, etc. were also preformed but they did not change the results. Hence, the evidence for the NK-model must be considered weak. Some potential reasons for the empirical shortcomings of the model were also discussed. Much of the previous literature has assumed stationarity on behalf of the key variables in the NK-model. However, this is empirically implausible, as shown in this paper among others. When non-stationarity is allowed, the equations of the NK-model do not satisfy, in most cases, a particular necessary condition, namely that the key variables must be cointegrated. Interestingly, the cointegration implications are satisfied when labor’s share is used as a measure of marginal costs on Euro area data. This might explain the success of the NKP-curve previously reported for this measure and data. 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