DISCUSSION PAPER SERIES No. 7537 HOW MUCH NOMINAL RIGIDITY IS THERE IN THE US ECONOMY? TESTING A NEW KEYNESIAN DSGE MODEL USING INDIRECT INFERENCE Vo Phuong Mai Le, Patrick Minford and Michael R. Wickens INTERNATIONAL MACROECONOMICS ABCD www.cepr.org Available online at: www.cepr.org/pubs/dps/DP7537.asp www.ssrn.com/xxx/xxx/xxx ISSN 0265-8003 HOW MUCH NOMINAL RIGIDITY IS THERE IN THE US ECONOMY? TESTING A NEW KEYNESIAN DSGE MODEL USING INDIRECT INFERENCE Vo Phuong Mai Le, University of Cardiff Patrick Minford, University of Cardiff and CEPR Michael R. Wickens, University of Cardiff, University of York and CEPR Discussion Paper No. 7537 November 2009 Centre for Economic Policy Research 53–56 Gt Sutton St, London EC1V 0DG, UK Tel: (44 20) 7183 8801, Fax: (44 20) 7183 8820 Email: cepr@cepr.org, Website: www.cepr.org This Discussion Paper is issued under the auspices of the Centre’s research programme in INTERNATIONAL MACROECONOMICS. Any opinions expressed here are those of the author(s) and not those of the Centre for Economic Policy Research. Research disseminated by CEPR may include views on policy, but the Centre itself takes no institutional policy positions. The Centre for Economic Policy Research was established in 1983 as an educational charity, to promote independent analysis and public discussion of open economies and the relations among them. It is pluralist and nonpartisan, bringing economic research to bear on the analysis of medium- and long-run policy questions. These Discussion Papers often represent preliminary or incomplete work, circulated to encourage discussion and comment. Citation and use of such a paper should take account of its provisional character. Copyright: Vo Phuong Mai Le, Patrick Minford and Michael R. Wickens CEPR Discussion Paper No. 7537 November 2009 ABSTRACT How much nominal rigidity is there in the US Economy? Testing a New Keynesian DSGE model using indirect inference We evaluate the Smets-Wouters model of the US using indirect inference with a VAR representation of the main US data series. We find that the original New Keynesian SW model is on the margin of acceptance when SW's own estimates of the variances and time-series behaviour of the structural errors are used. However when the structural errors implied jointly by the data and the structural model are used the model is rejected. We also construct an alternative (New Classical) version of the model with flexible wages and prices and a one-period information lag. This too is rejected. But when small proportions of both the labour and product markets are assumed to be imperfectly competitive within otherwise flexible markets the resulting ‘weighted' model is accepted. JEL Classification: C12, C32, C52 and E1 Keywords: Bootstrap, DSGE, Grea moderation, indirect inference, New Classical, New Keynesian, regime change, structural break, US model, VAR and Wald statistic Vo Phuong Mai Le Cardiff Business School Cardiff University Aberconway Building Colum Drive CARDIFF CF1 3EU Patrick Minford Cardiff Business School Cardiff University Aberconway Building Colum Drive CARDIFF CF1 3EU Email: LeVP@cardiff.ac.uk Email: minfordp@cf.ac.uk For further Discussion Papers by this author see: For further Discussion Papers by this author see: www.cepr.org/pubs/new-dps/dplist.asp?authorid=164365 www.cepr.org/pubs/new-dps/dplist.asp?authorid=100320 Michael R Wickens Cardiff Business School Cardiff University Aberconway Building Colum Drive CARDIFF CF1 3EU Email: mrw4@york.ac.uk For further Discussion Papers by this author see: www.cepr.org/pubs/new-dps/dplist.asp?authorid=100329 Submitted 02 November 2009 We thank Huw Dixon, Laurence Copeland and David Meenagh for their helpful comments. This work was supported by the UK’s Economic and Social Research Council under grants RES-165-25-0020 and PTA-026-27-1623. A full set of results can be found in the Annexes to this paper at www.cf.ac.uk/carbs/faculty/minfordp/E2008_32Annex.pdf 1 Introduction In this paper we test a dynamic stochastic general equilibrium (DSGE) model of the US economy on the full sample of post-war data using the method of indirect inference. The aim of the tests is to determine the degree of nominal rigidity in the US economy. The extent of nominal rigidity is a major area of disagreement between economists and we believe — and this is con…rmed by our results — that it holds a critical key to the model’s dynamic performance. Our modelling framework is based on that of Smets and Wouters (2007). Smets and Wouters (SW) adopt a New Keynesian (NK) model with sticky prices and wages. We compare this with a New Classical (NC) version of their model which has ‡exible prices and wages and a one-quarter delay for households in receiving macro information. We also consider the possibility that the economy consists of a mixture of the two in which some parts of the economy display nominal rigidities and other parts do not. To anticipate our results, we …nd that for the full sample period a hybrid model in which most of the economy enjoys price and wage ‡exibility but a non-negligible part is subject to nominal contracts, gets closest to matching the data, whereas the NK and NC models are seriously at odds with the data. If, however, we restrict the data to a sub-period from the mid-1980s to the mid-2000s then a model with a high degree of nominal rigidity is then able to match key aspects of the data. Our results suggest that the state-dependency of pricing could dominate its time-dependency for the bulk of the post-war period but that during the later period of the ’great moderation’, when the economy was more stable, time-dependency could have dominated. The SW model contains a full range of structural shocks and nominal and real frictions, and the model is estimated with Bayesian methods. They consider their model satisfactory in a variety of ways. For example, it can compete with standard VAR and BVAR models in forecasting the main US macro variables at business cycle frequencies. In this paper we focus on the model’s dynamic performance within the sample, using a new evaluation procedure based on indirect inference. This exploits the properties of the model’s error processes through bootstrap simulations and analyses whether the simulated data from the structural model can explain the actual data when both are represented by the dynamic behaviour of an auxiliary model. Our test, which is a form of Wald statistic, focuses on the overall capacity of the model to …t the data’s dynamic performance. The idea is to use indirect inference to test the structural model. Indirect inference has been widely used in the estimation of structural models, see Smith (1993), Gregory and Smith (1991,1993), Gourieroux et al. (1993), Gourieroux and Monfort (1995) and Canova (2005). Here we make a di¤erent use of indirect inference as our aim is to evaluate an already estimated or calibrated structural model. The common element is the use of an auxiliary time series model. In estimation the parameters of the structural model are chosen so that when this model is simulated it generates estimates of the auxiliary model similar to those obtained from actual data. The optimal choice of parameters for the structural model are those that minimise the distance between a given function of the two sets of estimated coe¢cients of the auxiliary model. Common choices of this function are the actual coe¢cients, the scores or the impulse response functions. In model evaluation the parameters of the structural model are taken as given. The aim is to compare the performance of the auxiliary model estimated on simulated data derived from the given estimates of a structural model which is taken as a true model of the economy, the null hypothesis - with the performance of the auxiliary model when estimated from actual data. If the structural model is correct then its predictions about the impulse responses, moments and time series properties of the data should statistically match those based on actual data. The comparison is based on the distributions of the two sets of parameter estimates of the auxiliary model, or of functions of these estimates. In other words, the testing procedure involves …rst constructing the errors derived from the previously estimated structural model and the actual data. These errors are then bootstrapped and used to generate for each bootstrap new data based on the structural model. An auxiliary time-series model is then …tted to each set of data and the sampling distribution of the coe¢cients of the auxiliary time series model is obtained from these estimates of the auxiliary model. A Wald statistic is computed to determine whether functions of the parameters of the time series model estimated on the actual data lie in some con…dence interval implied by this sampling distribution. This paper builds on Le (2008), who examined the ability of the calibrated model of Canzoneri, Cumby and Diba (2004) to …t the US data. This is a very simple New Keynesian model, designed to investigate 2 certain policy issues by simulation; it contains Calvo contract assumptions in labour and product markets and investment adjustment costs but no other rigidities. It is rejected by the Wald-statistic and other standard ways of measuring the goodness of …t. Furthermore, whatever assumptions were added to the model about indexation and whatever assumptions were made about the degree of nominal rigidity, including moving to a New Classical version, made no impact on the data-acceptability of the model; all versions were rejected at high levels of signi…cance. In particular, the model did not have su¢cient sources of lagged reaction to …t the data well. The SW model for the US economy has many of the features of the Canzoneri et al. model, including Calvo contracts in product and labour markets. But it also embeds backward-looking indexation and real rigidities other than and in addition to investment adjustment costs — both habit formation in consumption and variable capital-utilisation with its own adjustment costs. These are features that New Keynesian economists advocate in order to replicate the persistence of in‡ation and output and supposedly hump shaped responses to monetary shocks — e.g. Christiano et al (2007). We …nd in what follows that neither NK nor NC models can be used to represent the U.S. economy on their own, because both of them fail to satisfy the most basic measures of …tness. However, when a limited degree of nominal rigidity is embedded in the NC model the resulting ‘mixed’ model …ts the data better. This means that, although nominal rigidity is relevant in modelling the economy, its role is not as important as New Keynesian economists propose. Real rigidities on the other hand are highly necessary to replicate the data. We do not directly consider micro data on price-setting such as that examined by Bils and Klenow (2004), but we recognise that work in this …eld is continuing in order to establish how much nominal rigidity is present. We also note that recently Gertler and Leahy (2008) have suggested that state-contingent (Ss) pricing could account both the micro evidence on sticky prices as well as macro price behaviour and that this could look quite like price-‡exibility. The paper is organised as follows. In section 2 we describe the Smets-Wouters model and summarise their main …ndings. We explain the testing procedure in detail in section 3. In section 4 …rst we compare the NK and NC models, and then we compare these with the hybrid model. In section 5 we examine whether changes in monetary regimes are a possible source of misspeci…cation. We summarise our conclusions in section 6. 2 The Smets-Wouters model of the US Smets and Wouters (2007) developed a New Keynesian model and estimated this model on US data covering the period 1966Q1–2004Q4 using Bayesian methods. The model features many nominal and real frictions that create hump-shaped responses of aggregate demand to shocks. The model’s dynamics are driven by seven orthogonal structural shocks: total factor productivity shocks, risk premium shocks, investment-speci…c technology shocks, wage mark-up shocks, price mark-up shocks, exogenous spending shocks and monetary policy shocks. Their model is based on Smets and Wouters (2003) which was estimated on EU data. In the US version there are a few di¤erences. First, the number of structural shocks is reduced from ten to seven. Second, the Dixit-Stiglitz aggregator in the goods and labour markets is replaced by the aggregator developed by Kimball (1995) where the demand elasticity of di¤erentiated goods and labour depends on their relative price. Third, the model features a deterministic growth rate driven by labour-augmenting technological progress which is assumed in order to use the original data without having to detrend them. Their model is estimated by Bayesian methods which combine calibrated parameters with sample information. SW report that the estimated model …ts the US data quite well. This is veri…ed by comparing the marginal likelihood of out-of-sample predictions of the mo del with Bayesian VAR models. Price and wage rigidities are important in explaining the data but the indexation is not. They …nd that demand shocks, such as those to the risk premium and to exogenous spending, and investment speci…c technology shocks explain a signi…cant fraction of the short-run forecast variance in output, but wage mark-up and productivity shocks contribute little to explaining output variation in the medium to long run. They also con…rm that productivity shocks have a signi…cant short-run negative impact on hours worked. In‡ation developments are mostly driven by the price mark-up shocks in the short run and wage mark-up shocks in the long run. 3 The model can capture the cross correlation between output and in‡ation at business cycle frequencies. As an ultimate check of the model’s performance, they estimate the model for two subsamples: the "Great In‡ation" period from 1966Q2 to 1979Q2 and the "Great Moderation" period from 1984Q1 to 2004Q4, and …nd that most of the structural parameters are stable over those periods except for the fall in the standard deviation of the productivity, monetary policy and price mark-up shocks, which re‡ect the decrease in output growth and in‡ation volatility, and for the fall in the monetary policy response to output developments in the second subsample. We note that all of these exercises are carried out in a Bayesian framework and that at no stage is the speci…cation of the model tested. In e¤ect, the Bayesian procedure just improves the …t compared with using only calibrated parameters by employing additional information from the data. Nonetheless, the original calibration dominates the …nal coe¢cient estimates if the priors are too tight, see for example Del Negro and Schorfheide (2007). A full test of the model in our sense does not arise under a Bayesian procedure because the information in the priors is regarded as …xed and known. One reason why we are carrying out such tests is that we do not accept these priors uncritically; in particular, we are concerned about the extent of nominal rigidity assumed in the priors. 3 Model evaluation by indirect inference Our aim is to evaluate an already estimated or calibrated (DSGE) macroeconomic model by indirect inference. By evaluate we mean carry out classical statistical inference on a previously estimated or calibrated model. This is related to, but is di¤erent from, estimating a macroeconomic model by indirect inference. The common feature is the use of an auxiliary model in addition to the structural macroeconomic model. Before considering model evaluation by indirect inference, to set the scene and establish notation, …rst we discuss estimation by indirect inference. 3.1 Estimation Estimation by indirect inference chooses the parameters of the macroeconomic model so that when this model is simulated it generates estimates of the auxiliary model similar to those obtained from the observed data. The optimal choice of parameters for the macroeconomic model are those that minimize the distance between a given function of the two sets of estimated coe¢cients of the auxiliary model. Common choices of this function are (i) the actual coe¢cients, (ii) the scores, and (iii) the impulse response functions. In e¤ect, estimation by indirect inference gives the optimal calibration. Suppose that yt is an m £ 1 vector of observed data, t = 1; :::; T; x t(µ) is an m £ 1 vector of simulated time series generated from the structural macroeconomic model, µ is a k £ 1 vector of the parameters of the macroeconomic model and x t(µ) and yt are assumed to be stationary and ergodic. The auxiliary model is f [yt ; ®]. We assume that there exists a particular value of µ given by µ 0 such that fxt (µ 0 )g Ss=1 and fy tg Tt=1 share the same distribution, i.e. f [xt (µ 0 ); a] = f [yt ; ®] where ® is the vector of parameters of the auxiliary model. The likelihood function for the auxiliary model de…ned for the observed data fy t gTt=1 is L T (y t; ®) = §Tt=1 log f [yt ; ®]: The maximum likelihood estimator of ® is then aT = arg maxLT (y t ; ®): ® The corresponding likelihood function based on the simulated data fxt (µ)gSs= 1 is LS [xt (µ); ®] = §St=1 log f [xt (µ); ®] with aS (µ) = arg maxLS [x t (µ); ®]: a 4 The simulated quasi-maximum likelihood estimator (SQMLE) of µ is µT ; S = arg maxLT [yt ; ®S (µ)]: µ This is the value of µ that produces a value of ® that maximises the likelihood function using the observed data. We suppose that the observed and the simulated data are such that this value of ® satis…es plim aT = plim aS (µ) = ®; hence the assumption that xt (µ) and y t are stationary and ergodic, see Canova (2005). It can then be shown that T 1=2 (aS (µ) ¡ ®) ! N [0; - (µ)] - (µ) = E [¡ 0 @ 2 L[®(µ)] ¡1 @L[®(µ)] @L[®(µ)] @2 L[®(µ)] ¡1 ] E[ ]E [¡ ] : @® 2 @® @® @® 2 The covariance matrix can be obtained either analytically or by bootstrapping the simulations. An alternative to the SQMLE is the extended method of simulated moments estimator (EMSME). This is obtained as follows. Consider the continuous p £ 1 vector of functions g(aT ) and g(® S (µ)) which could, for example, be moments or scores, and let G T (aT ) = T1 §Tt= 1 g(aT ) and G S (®S (µ)) = S1 § Ss=1 g(® S (µ)). We require that aT ! ® S in probability and that GT (aT ) ! G S (® S (µ)) in probability for each µ. The EMSME is µ T ;S = arg min [G T (aT ) ¡ G S (® S (µ))]0 W (µ)[G(aT ) ¡ GS (®S (µ))]: µ 3.2 Model evaluation The parameters of the macroeconomic model and their distributions are taken as given — either estimated or calibrated. The aim is to compare the performance of the auxiliary model based on observed data with its performance based on simulations of the macroeconomic model derived by bootstrapping its structural disturbances. From these simulations we may obtain the joint distribution of the parameters of the auxiliary model and use this to perform a Wald test. This tests whether the estimates of the auxiliary model based on actual data could have come from the particular realisation of the structural model. We choose a VAR as the auxiliary model and base our test on a function of the VAR coe¢cients (augmented by the data variances, as a check for matching variability). We use a VAR(1) on a limited number of key variables. By raising the order of the VAR and increasing the number of variables, the stringency of the overall test of the model is increased. As we …nd that the structural model is already rejected by a VAR(1), we do not proceed to a more stringent test based on a higher order VAR. Non-rejection of the null hypothesis is taken to indicate that dynamic behaviour of the macroeconomic model is not signi…cantly di¤erent from that of the observed data. Rejection is taken to imply that the macroeconomic model is incorrectly speci…ed. Comparison of the impulse response functions of the observed and simulated data should reveal in what respects the macroeconomic model fails to capture the auxiliary model. The Wald test statistic is obtained as follows. We assume that there exists a particular value of µ given by µ0 such that fx t(µ0 )g Ss=1 and fyt gTt= 1 share the same distribution, where S = cT and c ¸ 1. If bµ is the estimated or calibrated value of µ then the null hypothesis can be expressed as H 0 : b µ ! µ0 . Consider again the continuous p £ 1 vector of functions g(aT ); g(® S (µ)); GT (aT ) = T1 §Tt=1 g(aT ) and GS (® S (µ)) = 1 S § g(®S (µ)). The functions g(:) may be impulse response functions. Given an auxiliary model and S s=1 a function of its parameters, our test statistic for evaluating the macroeconomic model is based on the distribution of GT (aT ) ¡ G S (®S (b µ)). The resulting Wald statistic may be written as [G T (aT ) ¡ G S (® S (bµ))]0 W (bµ)[G T (aT ) ¡ GS (® S (bµ))] where the estimate of the optimal weighting matrix is @G(®(b µ)) @G(®(b µ)) 0 ¡1 W (bµ) = f[ ]- (b µ)[ ]g @® @® 5 We obtain the distribution of GT (aT ) ¡ G S (®S (b µ)) and the Wald statistic using the bootstrap. The following steps summarise our implementation of the Wald test by bootstrapping: Step 1: Estimate the errors of the economic model conditional on the observed data and b µ. Estimate the DSGE macroeconomic model’s structural the errors "t given b µ and the observed data. The number of independent structural errors is taken to be less than or equal to the number of endogenous variables. The errors are not assumed to be normally distributed. Step 2: Estimate the empirical distribution of the structural errors On the null hypothesis the f"t g Tt=1 errors are omitted variables. Their empirical distribution is assumed to be given by these structural errors. The simulated disturbances are drawn from these errors. In some DSGE models the structural errors are assumed to be generated by autoregressive processes which under our method we need to estimate. This is the case with the SW model. The model is bootstrapped (drawing these disturbances by time vector to preserve any simultaneity between them) and solved using Dynare (Juillard, 2001). Step 3: Compute the Wald statistic We choose the function of the auxiliary model’s parameters to be the VAR coe¢cients themselves rather than a multi-valued function of them such as the impulse response functions (IRFs). Hence g(aT ) ¡ g(®S (µ)) = aT ¡ ®S (µ) and so GT (aT ) ¡ G S (®S (b µ)) = aT ¡ ® S (bµ) The distribution of aT ¡ ®S (b µ) and its covariance matrix W (bµ) ¡1 are estimated by bootstrapping ® S (bµ). Thus we use the appropriate small-sample distribution rather than the asymptotic distribution that emerges from analytic methods. The bootstrapping proceeds by drawing N bootstrap samples of the structural model, and estimating the auxiliary VAR on each, thus obtaining N aS (bµ): This set of vectors represents the sampling variation implied by the structural model, enabling its mean, covariance matrix and con…dence bounds to be calculated directly. N is generally set to 1000. We can now compute the properties of the model and compare them with those of the data; in particular, we examine the model’s ability to encompass the variances of the data. Assuming the model can do so, we go on to compute the bootstrap Wald statistic [aT ¡ ® S (bµ)]0 W (b µ)[aT ¡ ® S (bµ)]. Figure 1 shows, for just two parameters in the auxiliary equation, the distribution of the statistic and an example of the statistic for two cases — one with a diagonal covariance matrix and one with non-zero covariances. One can think of estimation via indirect inference as changing the parameters of the structural model, thus changing the implied distribution, so as to push the observed data point as far into the centre of the distribution as possible. The test however takes the structural parameters as given and merely notes the position of the observed data point in the distribution. 6 Correlation=0 2.5 2 1.5 1 1 0.5 0.5 0 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 Correlation=0.9 5 4 3 2 1 1 0.5 1.2 1 0.8 0 0.6 0.4 0.2 0 -0.2 Figure 1: Bivariate Normal Distributions (0:1; 0:9 shaded) with correlation of 0 and 0:9. In addition to our basic Wald statistic we consider a number of related Wald statistics. The basic Wald test is based on the full joint distribution of the VAR coe¢cients as implied by their full covariance matrix, as in the second panel of Figure 1. We refer to this as the Full Wald test; it checks whether the VAR databased coe¢cients lie within the DSGE model’s implied joint distribution and is a test of the DSGE model’s speci…cation in absolute terms. We use the Mahalanobis Distance based on the same joint distribution, normalised as a t-statistic, as an overall measure of closeness between the model and the data- in e¤ect this conveys the same information as in the Wald test but in the form of a t-value. A second Wald test, which we refer to as a ‘Directed Wald statistic’, focuses on speci…c features of the structural model. Here we seek to know how well a particular variable or limited set of variables is modelled and we use the corresponding auxiliary equations for these variables in the VAR as the basis of our test. For example, we may wish to know how well the model can reproduce the behaviour of US and EU output by creating a Wald statistic based on the VAR equation for the two outputs alone. We may also use this Directed Wald test to determine how well the structural model captures the e¤ects of a particular set of shocks. To do this we create the joint distribution of the IRFs for these shocks alone. For example, to determine how well the model deals with supply shocks, we construct the joint distribution of the IRFs for the supply shocks and calculate a Wald statistic for this. Even if a model is mis-speci…ed overall, through these Directed Wald tests we can say whether it is well-speci…ed enough to deal with speci…c 7 aspects of economic behaviour. Finally we should note that we also look at the usual battery of diagnostic statistics for these models, such as ability to match (i.e. embrace within 95% limits) data variances, cross-correlations, and VAR-based IRFs. We attach particular importance to the ability to match data variances, arguing that a failure in this dimension is essentially terminal; for this reason we include the data variances in the full Wald statistic. It should be noted that the cross-correlations (other than contemporaneous) and the IRFs are all derived from the VAR coe¢cients; hence our focus on these rather than the many relationships that can be derived from them. This makes our procedure in many ways quite traditional; our Wald statistics are our main innovation, but these largely summarise the results of these more traditional measures. We are implicitly assuming that the auxiliary model can distinguish between di¤erent structural models. This has been challenged recently by Canova and Sala (2009). They argue that the identi…cation of di¤erent DSGE models is ’weak’ and may give rise to the same VAR. They explain the point analytically by taking a 3-equation (IS-Phillips Curve-Taylor Rule) reduction of a DSGE model, deriving its VAR representation, and pointing out that several parameters cannot be identi…ed. As they note, however, their example is rigged in particular ways. One is that the shocks in the three equations are i.i.d. and there are no lagged endogenous variables (either from adjustment costs or indexation). In DSGE models like the SW, however, shocks are generally autocorrelated and lagged endogenous variables enter widely. As a result, DSGE models like that of SW are substantially over-identi…ed through the rational expectations mechanism and changes in its parameters imply quite di¤erent simulation properties. This is illustrated in our results which attempt to distinguish between DSGE models according to the ’distance’ of their implied VAR from a data-generated VAR - a related approach is due to Del Negro et al (2006). We …nd large changes in the variances implied by the model as its degree of rigidity is changed. These variances can be thought of as providing the elements in the distance function and they show how much it changes as rigidity changes. We also …nd that the Mahalanobis Distance of the model varies with modest changes in model speci…cation. This measure takes account of the joint distribution of all the criterion parameters and so sensitively re‡ects the model’s complete speci…cation. 4 Testing the SW Model using the method of indirect inference We apply the proposed testing procedure to this model for the period of 1947Q1–2004Q4. To do so we need to choose a …ltering method to stationarise the data. We looked at several methods and report basic results on all of them in the Annexes. In all the model used by SW was rejected by the Wald outright when data variances and VAR coe¢cients were included in it. These …lters were: SW’s own …lter (log di¤erencing of all variables except in‡ation, log of hours worked and interest rates which are left in levels- Annex A); di¤erencing all variables (log di¤erencing as SW but di¤erencing the remaining three- Annex B); an HP …lter (Annex C); and linear detrending as followed by SW for their EU work (SW, 2003)- Annex D. In terms of the models’ success in …tting the data in the widest sense there is not much to choose between these …lters, as we have noted. If we consider the model’s ability to replicate the data variances alone, again all are rejected. Filter/Walds Original Di¤erenced HP Linear Detrend Variances 97 100 100 100 Coe¢cients 100 100 100 100 Variances and Coe¢cients 100 100 100 100 Table 1: The Wald statitistics for each …lter on SW’s original model We therefore decided that, in order to carry out our more detailed investigations of the model, we would pick the …lter that extracted the least information from the raw data- linear detrending. It turned out that this …lter was adequate to generate stationarity for all the data. The results reported in the body of the paper that follows all use this …lter. The VAR estimation is performed with …ve main observable variables: output, investment, consumption, the quarterly interest rate and the quarterly in‡ation rate; and we use a VAR of order one. The more 8 variables are added to the VAR and the higher its order, the more detailed properties that the model must match and so the higher is the theshold of the test. As we will see below a VAR(1) on this small list of variables is testing enough. 4.1 Evaluating the SW model using SW’s own assumed error properties First we test the original SW model using their Bayesian estimate means for the error variances and autoregressive coe¢cients. The model is rejected. This is not entirely a surprise since the Bayesian method only updates priors with the data and does not test the model against the dynamics of the data. The model is rejected with the Full Wald test statistic of 100;its normalised Mahalanobis Distance is a massive 98.8, indicating that the data’s dynamic properties are very far indeed from the model’s. This can be explained by a large number (9 out of 25) of the VAR’s parameters that lie outside the 95% model bounds. Based on the their t-stats, some of these coe¢cients lie a long way outside the con…dence intervals; note in particular the partial autocorrelations of consumption and in‡ation where the model’s bounds lie higher than the VAR estimate. One could interpret this as excessive in‡ation and consumption persistence in the model. Furthermore, the IRFs of the VAR (when identi…ed by the model) frequently lie modestly outside the model bounds (from the model bootstrap distribution of VAR coe¢cients). Futhermore the data variances (the bottom 5 entries in Table 2) for nominal variables are too low compared with the data: for interest rates the data variance lies outside the model bounds while for in‡ation it lies just on the top 95% bound. Overall, therefore, even with SW’s own assumed error properties their model is badly out of line with the data. However, as we see in the next section, these properties are by no means the same as those implied by the data under the null that the model holds. 9 VAR coe¤s AYY AR Y A¼Y AC Y AIY AYR AR R A¼R AC R AIR AY¼ AR ¼ A¼¼ AC ¼ AI¼ AYC AR C A¼C AC C AIC AYI AR I A¼I AC I AII ¾ 2Y ¾ 2R ¾ 2¼ ¾ 2C ¾ 2I Wald stat Actual Estimate 0:99908 0:01503 ¡0:00417 0:10174 0:22591 ¡0:64529 0:85001 0:15154 ¡0:5553 ¡1:7064 0:11612 0:02374 0:59496 ¡0:38833 ¡0:25917 ¡0:08009 ¡0:02553 0:0121 0:78488 ¡0:4296 0:02034 0:01022 0:01159 0:01957 1:02924 18:32858 0:65276 0:44451 10:3888 71:79914 100 Lower Bound Upper Bound T-stats 0:71104 0:96272 2:02349 ¡0:00557 0:04018 0:07322 ¡0:0068 0:0673 ¡1:41540 ¡0:07815 0:02091 5:03459 ¡0:27355 0:18519 2:34051 ¡1:28857 ¡0:40445 0:84625 0:66138 0:86763 1:60632 ¡0:11021 0:18262 1:56321 ¡0:83083 ¡0:2264 ¡0:16999 ¡2:33113 0:39231 ¡1:14775 ¡0:44029 0:32551 0:93503 0:07066 0:26195 ¡2:89958 0:59853 0:85809 ¡2:05169 ¡0:56528 ¡0:04657 ¡0:63310 ¡1:90858 0:40689 0:83059 ¡0:12788 0:08505 ¡0:74255 ¡0:03697 0:00303 ¡0:98493 ¡0:04785 0:01597 1:56170 0:85948 0:95736 ¡5:18686 ¡0:36543 0:08677 ¡2:66676 0:01692 0:08499 ¡1:53642 ¡0:00484 0:00905 2:21138 ¡0:01534 0:00714 2:63929 0:01241 0:04757 ¡1:12305 0:94769 1:08301 0:38801 8:71183 47:42615 ¡0:32176 0:19035 0:56812 3:22089 0:18584 0:46733 1:96505 6:19987 45:31804 ¡0:74834 65:12685 269:8422 ¡1:23001 Mah. Normalised Distance 98:8 Table 2: VAR Parameters, data variances and Model Bootstrap Bounds of the SW Model with SW’s error properties 10 4.2 Evaluating the SW model using actual errors So far we have supplied the SW model with essentially imaginary error properties, chosen by assumption. We now turn to the actual errors derived from estimation on the observed data. We estimate the model’s structural errors, that is, the residual in each structural equation is given by the actual data and the expected variables in it. For this we followed a procedure of robust estimation of the structural residuals along the lines suggested by McCallum (1976) and Wickens (1982) under which the expectations on the right hand side of each equation are generated by an instrumental variable regression that is implied by the model. The instruments chosen are the lagged values of the endogenous variables. Thus, in e¤ect, the generated expectations used in deriving the residuals are the predictions of the data-estimated VAR. Seven behavioural residuals are estimated by this means: consumption, investment, productivity, monetary policy, wage- and price-setting, and one exogenous process, government spending, which enters the goods market clearing condition. These residuals are shown in Figure 2 a. OUTPUT vs. GOVERNMENT SPENDING CONSUMPTION vs. CONSUMPTION EULER RESIDUALS 20 10 data resid 10 0 0 -10 -5 -20 0 50 100 150 200 250 data resid 5 -10 0 INVESTMENT vs. INVESTMENT EULER RESIDUALS 50 100 150 200 250 INTEREST RATE vs. TAYLOR RULE RESIDUALS 40 4 data resid 20 data resid 2 0 0 -20 -40 0 50 100 150 200 250 -2 0 OUTPUT vs. PRODUCTION RESIDUALS 50 100 150 200 250 INFLATION vs. INFLATION RESIDUALS 20 4 data resid 10 data resid 2 0 0 -10 -20 0 50 100 150 200 250 -2 0 50 100 150 200 REAL WAGE vs. REAL WAGE RESIDUALS 10 data resid 0 -10 -20 0 50 100 150 200 250 Figure 2: Single Equation Errors from SWNK model We proceed as though …ve of these residuals follow an AR(1)and the price and wage residuals follow ARM A(1; 1) processes. The standard deviations of the estimated error innovations are in all cases larger 11 250 than those assumed by SW; in the case of investment and the price mark-up they are nearly twice as large (see Table 3). Furthermore, the actual preference, investment and monetary shocks exhibit markedly less persistence than SW assumed. Hence though the properties of the residuals estimated from the data are recognisably similar to those assumed by SW, there are material di¤erences whose e¤ects we go on to investigate in our subsequent bootstrap exercise. We use a vector bootstrap to preserve any dependence between the structural innovations. SW stdev Data stdev SW AR(1) SW M A(1) Estimated AR(1) Estimated M A(1) ¤ Government Spending¤ 0:53 0:673 0:97 Pref Inv Mon Prod 0:23 0:371 0:22 0:45 0:704 0:71 0:24 0:344 0:15 0:45 0:553 0:95 0:944 ¡0:064 0:530 ¡0:062 0:971 Price Mark-up 0:14 0:239 0:89 ¡0:69 0:925 ¡0:709 Wage Mark-up 0:24 0:311 0:96 ¡0:84 0:915 ¡0:848 This includes a response to current productivity Table 3: Standard deviations of innovations and coe¢cients of shocks (actual vs. assumed) 12 The model again fails to capture the scale of the nominal data variances (Table 4, last 5 entries); for the interest rate the data variance is now roughly double the model’s upper bound while for in‡ation it remains around the model upper bound. The results for the VAR coe¢cients are also reported in Table 4. Based on the Full Wald Statistic, for all of the VAR coe¢cients and data variances, the model is strongly rejected on at the 5% level. Seven of the VAR coe¢cients lie outside their 95% bounds, besides the interest data variance. The model’s Mahalanobis Distance is 4.4; notice that this is already substantially better than for the SW model using their assumed error properties, so that it proves helpful in this instance to use the residuals implied by the data.. VAR coe¤s AYY AR Y A¼Y AC Y AIY AYR AR R A¼R AC R AIR AY¼ AR ¼ A¼¼ AC ¼ AI¼ AYC AR C A¼C AC C AIC AYI AR I A¼I AC I AII ¾ 2Y ¾ 2R ¾ 2¼ ¾ 2C ¾ 2I Wald stat Actual Estimate 0:99908 0:01503 ¡0:00417 0:10174 0:22591 ¡0:64529 0:85001 0:15154 ¡0:5553 ¡1:7064 0:11612 0:02374 0:59496 ¡0:38833 ¡0:25917 ¡0:08009 ¡0:02553 0:0121 0:78488 ¡0:4296 0:02034 0:01022 0:01159 0:01957 1:02924 18:32858 0:65276 0:44451 10:3888 71:79914 100 Lower Bound Upper Bound T-stats 0:75267 1:00004 1:52155 0:00444 0:05065 ¡0:85584 ¡0:00459 0:06545 ¡1:69669 ¡0:02606 0:07359 2:97157 ¡0:19814 0:26058 1:59418 ¡1:06427 ¡0:27522 0:12295 0:53595 0:73313 3:90309 ¡0:13061 0:15756 1:94297 ¡0:84942 ¡0:40004 0:60623 ¡1:80976 0:3887 ¡1:85185 ¡0:55029 0:07888 2:23055 0:11413 0:27924 ¡4:05015 0:47347 0:72517 ¡0:16778 ¡0:40915 ¡0:0401 ¡1:78108 ¡2:2204 ¡0:48423 2:53019 ¡0:15347 0:11375 ¡0:78433 ¡0:06976 ¡0:01127 0:87955 ¡0:06965 0:02018 1:70294 0:8106 0:94474 ¡2:99327 ¡0:3507 0:1845 ¡2:51936 ¡0:00146 0:06908 ¡0:48864 ¡0:00373 0:01252 1:44447 ¡0:01151 0:01217 1:76852 0:00317 0:03533 ¡0:02989 0:90238 1:03372 1:86272 9:67374 45:56006 ¡0:41926 0:16837 0:37665 7:52511 0:22269 0:47431 1:84365 4:62427 35:15967 ¡0:46506 63:05612 258:1966 ¡1:24068 Mah. Distance (Normalised) 4:4 Table 4: VAR Parameters, data variances and Model Bootstrap Bounds of the SW Model with Estimated Coe¢cients 4.3 Evaluating the New Classical model using actual errors Next we consider the New Classical version of the SW model proposed above. The results are poor. The main problem is the model’s massive overprediction of in‡ation variance (3rd last entry, Table 5). This occurred regardless of variations in the Taylor Rule; we adopted the NK rule except for setting potential p output, yt ; to a constant. For example a larger in‡ation reaction causes the interest rate variance to blow up but without bringing the in‡ation variance down su¢ciently. Thus the model fails on the basic preliminary test of data variance matching. 13 The model’s Full Wald statistic is again 100. Besides the model’s overprediction of the in‡ation variance, out of 25 VAR coe¢cients, 13 lie outside their 95% bounds. The model wrongly predicts all the partial autocorrelation coe¢cients, except for that of investment. Of the 13 coe¢cients that do not …t, …ve are related to the in‡ation rate. Further, the cross e¤ects from the main macroeconomic variables to the interest, in‡ation rates and consumption are badly predicted. The cross-e¤ect from in‡ation to interest rates in the model is negative; theoretically the interest rate should react to o¤set a rise in the in‡ation rate. The Mahalanobis Distance is 7:1 which is considerably worse than for the New Keynesian version of the SW model. VAR coe¤s AYY AR Y A¼Y AC Y AIY AYR AR R A¼R AC R AIR AY¼ AR ¼ A¼¼ AC ¼ AI¼ AYC AR C A¼C AC C AIC AYI AR I A¼I AC I AII ¾ 2Y ¾ 2R ¾ 2¼ ¾ 2C ¾ 2I Wald stat Actual Estimate 0:99908 0:01503 ¡0:00417 0:10174 0:22591 ¡0:64529 0:85001 0:15154 ¡0:5553 ¡1:7064 0:11612 0:02374 0:59496 ¡0:38833 ¡0:25917 ¡0:08009 ¡0:02553 0:0121 0:78488 ¡0:4296 0:02034 0:01022 0:01159 0:01957 1:02924 18:32858 0:65276 0:44451 10:3888 71:79914 100 Lower Bound Upper Bound T-stats 0:75964 0:98742 1:81743 ¡0:0305 0:06499 ¡0:02425 ¡0:03652 0:25225 ¡1:23410 ¡0:05527 0:07763 2:72888 ¡0:18755 0:24126 1:84048 ¡0:77893 ¡0:00061 ¡1:29619 0:16679 0:55328 4:77475 ¡1:70697 ¡0:51682 4:01721 ¡0:49638 0:04422 ¡2:35582 ¡1:53308 0:39447 ¡2:34629 ¡0:14849 0:09029 2:45278 0:01012 0:12289 ¡1:43555 0:10313 0:43618 3:75306 ¡0:15269 0:00878 ¡7:46875 ¡0:43894 0:13493 ¡0:79497 ¡0:11079 0:13547 ¡1:25499 ¡0:14256 ¡0:02468 1:75061 ¡0:42415 ¡0:06702 2:51545 0:84915 1:00249 ¡3:95797 ¡0:30533 0:16651 ¡2:94383 ¡0:00818 0:06222 ¡0:31562 ¡0:00083 0:03162 ¡0:58207 ¡0:02859 0:07379 ¡0:40644 ¡0:00999 0:03741 0:44958 0:90735 1:03984 1:47843 9:43786 62:14178 ¡0:60189 0:36337 0:76928 1:30337 2:33699 3:60734 ¡7:64773 7:39139 63:64939 ¡1:00088 60:45211 284:8093 ¡1:12994 Mah. Distance (Normalised) 7:1 Table 5: VAR Parameters, data variances and Model Bootstrap Bounds of the NC Model with Estimated Coe¢cients The model’s IRFs also perform poorly (see Annex). The dominant shocks on real variables are productivity and labour supply shocks, and on nominal variables are preference, monetary, productivity and labour supply shocks. The responses of all the variables to these shocks lie outside the model 95% bounds. Furthermore, the model fails to replicate the cross-correlations of many of the main macroeconomic variables; it underpredicts the autocorrelations of interest and in‡ation rates, and their cross-correlations with output; it overpredicts the e¤ect of investment on future output; due to excessive in‡ation variation, it fails to replicate the correlation between in‡ation and output. Overall therefore the New Classical version of the original SW model also fails to match the data in quite serious ways. 14 4.4 Evaluating a hybrid model: a weighted combination of New Keynesian and New Classical models We have analysed two rather di¤erent macroeconomic models with a view to understand the mechanisms behind each of them. The NK model is highly rigid with Calvo price and wage settings, while the NC is a ‡exible wage/price model with only a simple one-period information delay for labour suppliers. In SW’s NK model, because capacity utilisation is fairly ‡exible, output is substantially a¤ected by shocks to demand and this in turn — via the Phillips Curve — moves in‡ation and then — via the Taylor Rule — interest rates. Supply shocks can a¤ect demand directly (e.g. productivity shocks change the return on capital and so a¤ect investment) and also play a role as ‘cost-push’ in‡ation shocks (e.g. price/wage mark-up shocks). Persistent shocks to demand raise ‘Q’ persistently and produce an ‘investment boom’ which, via demand e¤ects, reinforces itself. Thus the model acts as a ‘multiplier/accelerator’ of shocks both on the demand and the supply side. In the NC model an inelastic labour supply causes output variation to be dominated by supply shocks (productivity and labour supply) while investment and consumption to react to output in a standard RBC manner. These reactions, together with demand shocks, create market-clearing movements in real interest rates and — via the Taylor Rule — in in‡ation. Supply shocks are prime movers of all variables in the NC model, while demand shocks add to the variability of nominal variables. In order to mimic real variability and persistence, suitably sized and persistent supply shocks are needed. But to mimic the limited variability in in‡ation and interest rates only a limited variance in demand shocks is required; and to mimic their persistence, the supply shocks must be su¢ciently autocorrelated. We have seen, however, that both the NK and NC versions of the SW model fail to match the data. Essentially, the NK model generates too little nominal variance while the NC model delivers too much. Given that each model fails in an opposite way, we propose a hybrid model that merges the NK and NC models by assuming that wage and price setters …nd themselves supplying labour and intermediate output partly in a competitive market with price/wage ‡exibility, and partly in a market with imperfect competition. We assume that the size of each sector depends on the facts of competition and do not vary in our sample but we allow the degree of imperfect competition to di¤er between labour and product markets.1 We also assume that the monetary authority pursues a Taylor Rule that re‡ects the properties of the hybrid model. In the hybrid model the price and wage setting equations are assumed to be a weighted average of the corresponding NK and NC equations. This weighting process is an informal use of indirect inference, the idea being to …nd the combination of the weights and Taylor coe¢cients that make the combined model perform best when compared with the auxiliary mo del. We …nd that the optimal weights are vw = 0:1 (the NK share for wages) and v p = 0:2 (the NK share for 1 Formally, we model this as follows. We assume that …rms producing intermediate goods have a production function that combines in a …xed proportion labour in imperfect competition (‘unionised’) with labour from competitive markets- thus the labour …rms becomes nt ) = n1t + n2t = (· used by intermediate ¸1+¸w;t h i 1 R1 R1 1+¸w;t di + 0 (n2it)di where n1it is the unionised, n2it the competitive labour provided by the ith 0 (n1it ) household at t; we can think of nt as representing the activities of an intermediary ‘labour bundler’. Note that n1t = vw nt ; n2t = (1 ¡ vw )nt so that Wt = vw W1t +(1 ¡ vw )W2t. Each household’s utility includes the two sorts of labour in the same 1+¾n ² n 1+¾n ² n 1nt 2nt way, that is Uit = ::: ¡ 1it ¡ 2it ::: W1t is now set according to the Calvo wage-setting equation, while W2t is set 1+¾ n 1+¾ n equal to current expected marginal monetary disutility of work; in the latter case a 1-quarter information lag is assumed for current in‡ation but for convenience this is ignored in the usual way as unimportant in the Calvo setting over the whole future horizon. These wages are then passed to the labour bundler who o¤ers a labour unit as above at this weighted average wage. Firms then buy these labour units o¤ the manager for use in the …rm. Similarly, retail output is now made up in a …xed proportion of intermediate goods in an imperfectly competitive market and inte goods sold competitively. 9 Retail output is therefore yt = y1t + y2t = 8rmediate " #1+¸p;t 1 < R = h i R 1 1+¸p;t y dj + 01 yj 2tdj . The intermediary …rm prices y1t according to the Calvo mark-up equation on mar: 0 j1t ; ginal costs, and y2t at marginal costs. Note that y1t = vp yt ; y2t = (1 ¡ vp )yt so that Pt = vpP 1t +(1 ¡ vp )P2t . The retailer combine s these goods as above in a bundle which it sells at this weighted average price. Notice that apart from these equations the …rst-order conditions of households and …rms will be una¤ected by what markets they are operating in. 15 prices). That is, only 10% of labour markets and only 20% of product markets are imperfectly competitive. Therefore, the model requires only a small amount of nominal rigidity in order to match the data. The Taylor rule then becomes: Rt = 0:6Rt¡1 + (1 ¡ 0:6)f2:3¼ t + 0:08yt g + 0:22 (yt ¡ yt¡ 1 ) + "t©: This is a somewhat more ª aggressive response to in‡ation than either the NK (Rt = 0:81Rt¡ 1 + (1 ¡ 0:81) 2:04¼ t + 0:08(y t ¡ ytP + £¡ ¢ ¡ ¢¤ 0:22 y t ¡ ytP ¡ y t¡1 ¡ y P + "rt ) or NC rules (the NC is the same as NK except that it sets ‘potential t¡1 output’ to a constant). Notice that if one substitutes out for the interest rate from a simple money demand function with an exogenous money supply growth process, then one obtains a ‘Taylor Rule’ that has the form ¢Rt = 1¯ f¼ t + °¢yt ) + vt where ¯ is the semi-log interest rate elasticity of money demand, (° is the corresponding income elasticity) and vt is a combination of the money supply growth process and the change in the money demand error. This is fairly close to the rules adopted in these models when the lagged term in interest rates is large and the term in the output gap is small compared with the term in the rate of change of output. The main di¤erence between the hybrid and the NK and NC models is the hybrid model’s ability to reproduce the variances in the data. Using the structural errors from the model and the observed data, we …nd that all the data variances lie within the model’s 95% bounds (Table 6, last 5 entries). Furthermore, only nine of the 25 VAR coe¢cients lie outside their 95% con…dence intervals. While the Full Wald statistic of 100 rejects this model version as it does the others, the Mahalanobis Distance of 3:1 implies that the hybrid model is substantially closer to the data. VAR coe¤s AYY AR Y A¼Y AC Y AIY AYR AR R A¼R AC R AIR AY¼ AR ¼ A¼¼ AC ¼ AI¼ AYC AR C A¼C AC C AIC AYI AR I A¼I AC I AII ¾ 2Y ¾ 2R ¾ 2¼ ¾ 2C ¾ 2I Wald stat Actual Estimate 0:99908 0:01503 ¡0:00417 0:10174 0:22591 ¡0:64529 0:85001 0:15154 ¡0:5553 ¡1:7064 0:11612 0:02374 0:59496 ¡0:38833 ¡0:25917 ¡0:08009 ¡0:02553 0:0121 0:78488 ¡0:4296 0:02034 0:01022 0:01159 0:01957 1:02924 18:32858 0:65276 0:44451 10:3888 71:79914 100 Lower Bound Upper Bound T-stats 0:76761 0:99753 1:58317 ¡0:04394 0:01945 1:55148 ¡0:02898 0:06556 ¡0:76799 ¡0:03905 0:10284 2:05348 ¡0:2093 0:28019 1:59406 ¡0:97523 ¡0:14419 ¡0:40543 0:49302 0:75838 3:23533 ¡0:27545 0:1136 2:20900 ¡0:69948 ¡0:10079 ¡1:14167 ¡1:69603 0:44879 ¡1:94286 ¡0:29996 0:30247 0:72094 0:07486 0:27241 ¡2:85460 0:51488 0:78029 ¡0:76721 ¡0:27412 0:18435 ¡2:98726 ¡1:31646 0:30961 0:54641 ¡0:142 0:09477 ¡0:82072 ¡0:06465 0:00228 0:19880 ¡0:09455 0:00559 2:01856 0:81382 0:96991 ¡2:92486 ¡0:36452 0:1334 ¡2:63914 ¡0:00329 0:07082 ¡0:56989 0:00293 0:02417 ¡0:45314 ¡0:01199 0:02095 0:83386 ¡0:00749 0:03827 0:27887 0:89898 1:04153 1:49582 9:69749 61:85333 ¡0:61346 0:29191 0:76451 1:58414 0:43895 0:89102 ¡1:65685 7:30487 72:01693 ¡0:99793 61:41478 301:772 ¡1:17817 Mah. Distance (Normalised) 3:1 Table 6: VAR Parameters, data variances and Model Bootstrap Bounds of the Weighted Model with Estimated Coe¢cients 16 Shocks Y R ¼ C I Govt. Spending 2:6796 11:8312 2:0282 5:0587 11:298 Pref Inv Mon Prod 0:9823 16:2245 7:0541 1:0009 0:0508 1:9547 17:4343 3:7657 1:7749 28:0270 0:6995 2:2156 33:3303 0:6429 0:1305 48:2598 15:3872 17:6394 34:3637 32:6701 Price mark-up 0:5086 3:5352 4:9596 0:34915 0:2853 Wage mark-up 0:00003 0:000695 0:000769 0:00004 0:00001 Labour supply 44:9154 33:3713 31:2218 56:8097 27:5383 Total 100 100 100 100 100 Table 7: Variance Decompositions of the weighted Model with estimated rhos Since the optimal combination indicates that the ma jority of the market participants behave in a competitive manner, it is not a surprise that the variance decomposition (Table 7) shows that the supply shocks - (productivity and labour supply shocks - explain most of the movements of the real variables. They also explain a large part of the nominal variables. While the demand shocks also contribute quite a lot to movements in the interest rate, they do so less for movements in in‡ation. So why are these results di¤erent from those of the NK and NC models? The hybrid model mostly acts like the NC model, where the supply shocks explain most of the variation and demand shocks play a small in part in the variability of real variables due to one period information lag and they add to the variability of nominal variables. Since, however, some economic agents behave in the New Keynesian manner, aggregate supply and labour supply are more elastic, demand shocks have a greater impact on real variables. Most importantly, in‡ation variability is dampened down to encompass actual data variability. It is remarkable how large the reduction in the lower bound is by the introduction of only small Calvo shares (10% in wages, 20% in prices — or 30% rigidity overall); the lower bound of in‡ation’s standard deviation falls no less than 57%. The reason appears to be that the variability of in‡ation also reacts to the variability of expected in‡ation. Thus, as the Calvo element rises, expected in‡ation varies less which, in turn, reduces the variability of actual in‡ation and, again in turn, reduces the variability of expected in‡ation, and so on in a sort of ‘multiplier’ process. This is an e¤ect anticipated by Dixon (1992,1994). Now we investigate the VAR impulse response functions to three main shocks: investment, labour supply, and productivity shocks. The main di¤erences from the data are in the long-run responses of interest and in‡ation rates to the shocks; also the response of consumption is much more aggressive in the data than in the model. Nonetheless, these responses lie only just outside the 95% bounds. We can therefore say that the performance of the hybrid model, based on the IRFs, is relatively good compared to the NK and NC models. 17 Output Interest Rate 1 0.2 0 0 -1 0 5 10 Inflation 15 20 -0.2 0 5 -1 0 5 0.1 1 0 0 -0.1 0 5 0 5 10 Investment 15 20 15 20 10 15 Consumption 10 20 15 20 15 20 15 20 15 20 15 20 2 0 -2 10 Figure 3: Investment Shock Output Interest Rate 1 0.2 0.5 0 0 0 5 10 15 20 -0.2 0 5 Inflation 0.5 1 0 0.5 -0.5 0 5 10 10 Consumption 15 20 15 20 0 0 5 10 Investment 2 0 -2 0 5 10 Figure 4: Productivity Shock Output Interest Rate 0 0.5 -0.5 0 -1 0 5 10 15 -0.5 0 20 5 Inflation 0.5 0 0 -0.5 -0.5 0 5 10 Investment 10 Consumption 15 20 15 20 -1 0 5 10 5 0 -5 0 5 10 Figure 5: Labour Supply Shock 18 The cross-correlations are accepted in a number of cases. The actual autocorrelations and cross-correlations of the variables lie within the model’s bounds, though the correlation of investment with future output lies outside the bound. The performance of the cross-correlations among the nominal variables is, however, poor. The autocorrelations of interest and in‡ation rates are underpredicted by the model, even though the di¤erences are much smaller than those for the NK and NC models. These failures are consistent with the overall rejection of the hybrid model. Y v. Y(-i) R v. R(-i) 0.9 P v. P(-i) 0.8 C v. C(-i) 0.9 0.6 0.8 0.4 0.6 2 4 6 8 10 0.4 0.5 0 0 0.3 0.6 0.2 0.2 0.4 0.6 0.7 0.4 0.5 0.8 0.8 0.6 0.7 I v. I(-i) 2 4 6 8 10 0.2 0.4 2 R v. Y(-i+1) 4 6 8 10 2 P v. Y(-i+1) 4 6 8 10 2 C v. Y(-i+1) 4 6 8 10 I v. Y(-i+1) 0.2 0.9 0.2 0.8 0.8 0 0 0.6 0.7 -0.2 0.6 -0.2 0.4 0.5 -0.4 0.2 0.4 -0.4 0.3 -0.6 2 4 6 8 10 2 R v. Y(+i-1) 4 6 8 10 0 2 P v. Y(+i-1) 4 6 8 10 C v. Y(+i-1) 0.1 -0.3 0.75 -0.4 0.7 -0.5 2 4 6 8 10 10 8 10 0.8 0.8 -0.2 -0.6 8 0.9 0.85 -0.1 -0.4 6 0.9 0 -0.2 4 I v. Y(+i-1) 0.95 0 2 0.7 0.6 0.65 2 4 6 8 10 0.5 2 4 6 8 10 2 4 6 Figure 6: Cross-Correlations for Weighted Model (with estimated coe¢cients) We now consider the model’s performance for particular aspects of the data, using the Directed Wald test. Our method is to focus …rst on individual variables and then in groups by estimating the best ARMA(i,j) in the case of a single variable and a VAR(1) for a group of variables. We then apply the Directed Wald test. To assess the individual shocks we take the IRFs (we use the IRF average) of the shock for the variables where they have a major impact and generate the model-implied joint distribution of these IRFs, computing the Wald statistic for the joint values in the data. We also look at the joint distribution of the variances to con…rm our earlier judgement from the individual variances. Table 8 below reports these Wald statistics. First, the model does …t the data variances jointly but only at the 99% level. Second, the real variables …t the data taken as a group, though again only at the 99% level, as do nominal variables taken as a group. When, however, nominal and real variables are combined the dynamic …t deteriorates sharply and the model is rejected at the 99% level; only if we restrict ourselves to output and in‡ation does the model pass this Wald test at the 99% level. This is mirrored in the individual shocks; the responses to both productivity and labour supply, the two key shocks in this model, are borderline 19 rejected at 99%. For individual variables, the responses of all are accepted at the 99% level; in‡ation is accepted at the 95% level. As observed earlier, many of the VAR coe¢cients involving interest rates are rejected individually. It therefore seems clear that this is the area to look for better speci…cation of the model. Variable combinations Y; C; IN V Y; C; IN V; R Y; C; IN V; ¼ Y ; R; ¼ Y; ¼ R; ¼ Y (AR (3)) R (ARM A (1; 1)) ¼ (AR (3)) C (AR (3)) IN V (AR (2)) Direct Wald 98:3 99:0 100 99:4 97:6 96:2 96:2 98:4 90:3 98:8 95:2 Table 8: Directed Wald statistics BY VARIABLE COMBINATIONS Shocks P rod LabSup ¾ 2Y Variables Y; R; ¼; C; IN V Y; R; ¼; C; IN V Variances ; ¾ 2R ; ¾ 2¼ ; ¾ 2C ; ¾ 2I Directed Wald 98:2 99:1 Directed Wald 97 Table 9: Directed Wald statistics 5 Regime change as a possible source of mis-speci…cation In view of the apparently crucial role of interest rates in the hybrid model, the implication is that the problem could lie in the speci…cation of monetary policy, and in particular the use of one monetary regime for the whole sample from 1950s to the 2000s. We therefore tested for structural change during this period following the procedure of Perron and Wu (2007) designed to test for multiple breaks in VAR parameters; we found evidence of parameter breaks in two places: 1965 and 1984. The estimated breaks are: The 95% C.I. for the 1st break is The 95% C.I. for the 2nd break is 1965.02 1984.02 (1964.04;1965.04) (1983.02-1985.02) Table 10: Perron-Wu Multivariate Structural Break Test These are natural places to …nd such breaks due to changes that occurred in the monetary regime. The earler break is associated with the emergence of serious in‡ation for the …rst time; the later break is associated with the shift towards interest rate setting that followed from the adoption of (implicit) in‡ation targeting. In moving to three sub-periods we tripled the size of our testing problem. Furthermore linear detrending no longer proved su¢cient to make the data stationary; we therefore used a Hodrick-Prescott …lter. So far we have been unable to locate acceptable versions of the model for the …rst two sub-periods. However for the third and latest sub-period (1984.03-2004.02), we found good results when we shifted the weights in the hybrid model greatly towards the New Keynesian end of the spectrum (0:8; 0:8). It may well be that in the 20 ’great moderation’ price-setting was far less disturbed by shocks to the state and was dominated instead by time dependence. The model is still rejected on the full VAR with a Wald of 100 but its Mahalanobis Distance is the lowest to this point, at 2:9:While it is also rejected for interest rates alone for its best Ar(2) representation, for the combined variable set of output, in‡ation and interest rates it is now jointly accepted at the 99% level and nearly accepted at the 95% level. Signi…cantly, this is the …rst time that any model we have examined over the full data period has passed the test embracing real GDP and both nominal variables. Actual AYY 0:92801 AR 0:06743 Y A¼Y 0:05812 AYR ¡0:47031 AR 0:75002 R A¼R 0:09010 AY¼ 0:13438 AR 0:12020 ¼ A¼¼ 0:00798 V ar(Y ) 0:91360 V ar(R) 0:06939 V ar(¼) 0:03058 Wald 96:8 Lower Upper State 0:61496 0:94141 T RU E ¡0:04377 0:04308 F ALSE ¡0:03036 0:06507 T RU E ¡0:91141 0:53397 T RU E 0:51648 0:87073 T RU E ¡0:06550 0:38117 T RU E ¡1:06434 0:62982 T RU E ¡0:17840 0:29051 T RU E ¡0:12482 0:35663 T RU E 0:54450 2:08374 T RU E 0:03334 0:11078 T RU E 0:02884 0:05445 T RU E M-distance (Normalised) 1:8 Table 11: VAR coe¢cients and variances Y (AR (3)) R(AR(1)) R (AR (2)) ¼ (AR (1)) Y; R; ¼ var(Y ); var(R); var(¼) Direct Walds 54:7 97:2 100 69:6 96:6 60:5 Table 12: Direct Walds for di¤erent combinations of output, in‡ation and interest rate 6 Conclusion We have used the method of indirect inference to test a well-known DSGE model of the New Keynesian type on its dynamic performance for US post-war data. We compared this model with a ‡exible wage/price version with a short information lag (New Classical) and found that if we use the structural errors jointly implied by each model and the data, then neither model can …t the data variances. The NK produces too little variation in interest rates and the NC model generates an excessive variation in in‡ation rates. But when the two models are combined in a weighted combination to give a hybrid model which is a mixture of imperfectly competitive and ‡exible-price markets, then with 90% ‡exibility in the product market and 80% in the labour market the hybrid model comes much closer to matching the data, even though it too is rejected as mis-speci…ed especially in respect of interest rate behaviour. One possible reason is monetary regime change, for which there is evidence in the mid-1960s and the mid-1980s. When we examined the period since 1984 we found that in respect of output and nominal variables the data did not reject a model with quite a high degree of nominal rigidity (around 90% in both goods and labour markets). This suggests that the situation with regard to price or wage rigidity in the US economy may have changed over time; during most of the period state dependence in pricing could have dominated time dependence due to the economy’s large ‡uctuations, but, during the later period, the ’great moderation’, time dependence seems to have dominated. 21 References [1] Bils, M. and P. J. Klenow, 2004, "Some Evidence on the Importance of Sticky Prices," Journal of Political Economy, 112 (5), 947–985. [2] Canova, F., 2005. Methods for Applied Macroeconomic Research, Princeton University Press, Princeton. [3] Canova, F. and L. Sala (2009) ’Back to square one: Identi…cation issues in DSGE models’, of Monetary Economics, 56, 431–449. Journal [4] Canzoneri, M. , R. Cumby, and B. 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(1982) "The e¢cient estimation of econometric models with rational expectations", Review of Economic Studies, 49, 55-67. 23 Appendix A Listing of models — SWNK and SWNC SWNK MODEL Consumption Euler equation à ! 0: 71 1 ¡ 1:0043 (1:3952 ¡ 1) 0:83192 ¡ ¢ ¢ ct = (lt ¡ E t lt+1 )¡ ¡ (rt ¡ Et p t+1 )+ebt 0:71 E t ct +1 + 0:71 + 1:0043 1 + 1:0:71 1:38 1 + 1:0043 1:38 0043 (1) Investment Euler equation 0: 71 1:0043 ct¡1 + 1 + 1:0:71 1 0043 1 1 0:99(1:0043) 1 innt¡ 1 + Et innt+1 + qqt + einnt 1 + 0:99(1:0043) 1 + 0:99(1:0043) (1 + 0:99(1:0043)) (1:0043 2 ) 5:74 (2) Tobin Q equation innt = qqt = 1 ¡ 0:025 0:032649 Et qqt+ 1 + E t rkt+1 ¡ (rt ¡ Et p t+1 ) + 1 ¡ 0:025 + 0:032649 1 ¡ 0:025 + 0:032649 1 0:71 1¡ 1:0043 (1+ 0:71 1:0043 ebt (3) )1: 38 Capital Accumulation equation kt = µ 0:025 1¡ 1:0043 ¶ µ ¶ ¡ ¢ 0:025 0:025 kt¡1 + innt + 1 ¡ (1 + 1:0043 (0:99)) 1:00432 (5:74) (ennt ) 1:0043 1:0043 Price Setting equation 2 pt = 4 ³ 0:99x1:0043 0:24 1 E tp t+1 + 1+ 0:99£1:0043x0:24 p t¡1 ¡ 1+0:99x1:0043£0:24 1+0:99x1:0043£0:24 ³ ´ (1¡0:99(1: 0043)(0: 66))( 1¡0: 66) (0:19rkt + (1 ¡ 0:19) w t ¡ eat) 0: 66((1:6¡ 1)(10)+1) ´ 3 Wage Setting equation 2 6 wt = 6 4 5 + ep t 0:99x1:0043 1 0:99x1:0043 99x1: 0043£0: 58 Et p t+1 ¡ 1+0: pt 1+0:99x1:0043 Et w t+1 + 1+0:99x1:0043 w t¡1 + 1+0:99x1:0043 1+0:99x1:0043 ³ ´ ( 1¡0: 99x1: 0043x0:7)(1¡0:7) 0:58 1 + 1+0:99x1:0043 p t¡1 ¡ 1+0:99x1:0043 (1+(10)(1:5¡1)) 0:7 Labour demand ³ w t ¡ 1:83lt ¡ ³ 1 0:71 1¡ 1:0043 ´¡ ct ¡ 0:71 c 1:0043 t¡1 ¢´ 3 7 7 + ew t 5 µ ¶ 1 ¡ 0:54 lt = ¡w t + 1 + rkt + kt¡1 0:54 (4) (5) (6) (7) Market Clearing condition in goods market yt = 0:64ct + 0:17innt + 0:19 1 ¡ 0:54 rkt + eg t 0:54 (8) Aggregate Production equation rkt = 1 1:6 (0:19) 1¡0:54 0:54 (y t ¡ 1:6 (0:19) kt¡1 ¡ 1:6 (1 ¡ 0:19) lt ¡ 1:6ea t) (9) Taylor Rule ³ ³ ´´ ³³ ´ ³ ´´ f f f rt = 0:81rt¡ 1 + (1 ¡ 0:81) 2:03p t + 0:08 yt ¡ y t + 0:22 yt ¡ y t ¡ y t¡1 ¡ y t¡1 + ert 24 (10) SWNC MODEL Consumption Euler equation ct = 0: 71 1:0043 ct¡1 + 1 + 1:0:71 0043 1 (1:3952 ¡ 1) 0:83192 E tct+1 + ¡ ¢ (lt ¡ E t lt+1 )¡ 0:71 0:71 1 + 1:0043 1 + 1:0043 1:3952 à Investment Euler equation 0:71 1 ¡ 1:0043 ¡ ¢ 0: 71 1 + 1:0043 1:3952 ! (rt) +ebt (11) 1 0:99(1:0043) 1 innt¡ 1 + Et innt+1 + qqt + einnt 1 + 0:99(1:0043) 1 + 0:99(1:0043) (1 + 0:99(1:0043)) (1:0043 2 ) 5:74 (12) Tobin Q equation innt = qqt = 1 ¡ 0:025 0:032649 E tqqt+1 + Et rkt+ 1 ¡ (rt ) + 1 ¡ 0:025 + 0:032649 1 ¡ 0:025 + 0:032649 1 0:71 1¡ 1:0043 0:71 (1+ 1:0043 )1:3952 ebt Capital accumulation equation µ ¶ µ ¶ ¡ ¢ 0:025 0:025 0:025 kt = 1 ¡ kt¡1 + innt + 1 ¡ (1 + 1:0043 (0:99)) 1:0043 2 (5:74) (ennt) 1:0043 1:0043 1:0043 (13) (14) Marginal Product of Labour 0:19rkt + (1 ¡ 0:19) wt = eat Labour supply w t = 1:83lt + Labour Demand ³ 1 0:71 1¡ 1:0043 ´¡ ¢ 0:71 ct ¡ 1:0043 ct¡ 1 ¡ (¼ t ¡ E t¡1 ¼ t ) µ ¶ 1 ¡ 0:54 lt = ¡w t + 1 + rkt + kt¡1 0:54 (15) (16) (17) Market clearing condition yt = 0:64ct + 0:17innt + 0:19 1 ¡ 0:54 rkt + eg t 0:54 (18) Production function rkt = 1 1:6 (0:19) 1¡0:54 0:54 (y t ¡ 1:6 (0:19) kt¡1 ¡ 1:6 (1 ¡ 0:19) lt ¡ 1:6ea t) (19) Taylor Rule rt = 0:81rt¡ 1 + (1 ¡ 0:81) (2:03pt + 0:08yt ) + 0:22 (y t ¡ yt¡ 1 ) + ert 25 (20)