Bridging DSGE models and the raw data?

4 A method to estimate DSGE models using the raw data is proposed. The approach 5 links the observables to the model counterparts via a ‡exible speci…cation which does 6 not require the model-based component to be solely located at business cycle frequen7 cies, allows the non model-based component to take various time series patterns, and 8 permits model misspeci…cation. Applying standard data transformations induce biases 9 in structural estimates and distortions in the policy conclusions. The proposed ap10 proach recovers important model-based features in selected experimental designs. Two 11 widely discussed issues are used to illustrate its practical use. 12 JEL classi…cation: E3, C3. 13 Keywords: DSGE models, Filters, Structural estimation, Business cycles. 14 I would like to thank Bob King (the editor), two anonymous referees, Frank Schorfheide, Tim Cogley, Giorgio Primiceri, Tao Zha, Chris Sims, Harald Uhlig and the participants of seminars at the Bank of England, Bank of Italy, BIS, UCL, Yale, EUI, University of Modena, University of Sussex, Federal Reserve Bank of New York, University of Zurich, ESSIM, the Fed of Atlanta workshop on Methods of applications for DSGE models; the It workshop in Time Series Econometrics, Zaragoza; and the conference Recent Development of Dynamic Analysis in Economics, Seoul; The Gent workshop in Macroeconomics for comments and suggestions. The …nancial support of the Spanish Ministry of Education, through the grant ECO200908556, and of the Barcelona Graduate School of Economics is gratefully acknowledged. Department of Economics, UPF, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain, [email protected]. 1 INTRODUCTION 1 1 Introduction 15 There have been considerable developments in the speci…cation of DSGE models in the last 16 few years. Steps forward have also been made in the estimation of these models. Despite 17 recent e¤orts, structural estimation of DSGE models is conceptually and practically di¢ 18 cult. For example, classical estimation is asymptotically justi…ed only when the model is the 19 generating process (DGP) of the actual data, up to a set of serially uncorrelated measure20 ment errors, and standard validation exercises are meaningless without such an assumption. 21 Identi…cation problems (see e.g. Canova and Sala, 2009) and numerical di¢ culties are wide22 spread. Finally, while the majority of the models investigators use is intended to explain only 23 the cyclical portion of observable ‡uctuations, both permanent and transitory shocks may 24 produce cyclical ‡uctuations, and macroeconomic data contains many types of ‡uctuations, 25 and some are hardly cyclical. 26 The generic mismatch between what models want to explain and what the data contains 27 creates headaches for applied investigators. Over the last 10 years a number of approaches, 28 re‡ecting di¤erent identi…cation assumptions, have been used: 29 Fit a model driven by transitory shocks to the observables …ltered with an arbitrary 30 statistical device (see Smets and Wouters, 2003, Ireland, 2004a, Rubio and Rabanal, 2005, 31 among others). Such an approach is problematic for at least three reasons. First, since the 32 majority of statistical …lters can be represented as a symmetric, two-sided moving average 33 of the raw data, the timing of the information is altered and dynamic responses hard to 34 interpret. Second, while it is typical to …lter each real variable separately and to demean 35 nominal variables, there are consistency conditions that must hold a resource constraint 36 need not be satis…ed if each variable is separately …ltered and situations when not all 37 nominal ‡uctuations are relevant from the point of view of a model. Thus, speci…cation 38 errors can be important. Finally, contamination errors could be present. For example, a 39 Band Pass (BP) …lter only roughly captures the power of the spectrum at the frequencies 40 1 INTRODUCTION 2 corresponding to cycles with 8-32 quarters average periodicity in small samples and taking 41 growth rates greatly ampli…es the high frequency content of the data. In sum, rather than 42 solving the problem, the approach adds to the di¢ culties applied researchers face. 43 Fit a model driven by transitory shocks to transformations of the observables which, in 44 theory, are likely to be void of non-cyclical ‡uctuations, e.g. consider real ”great ratios”(as 45 suggested in Cogley, 2001, and McGrattan, 2010) or nominal ” great ratios”(as suggested 46 in Whelan, 2005). As Figure 1 shows, such transformations need not resolve the problem 47 because many ratios still display low frequency movements. In addition, since the number 48 and the nature of the shocks driving non-cyclical ‡uctuations needs to be a-priori known, 49 speci…cation errors may be produced. 50 Construct a model driven by transitory and permanent shocks; scale the model by the 51 assumed permanent shocks; …t the transformed model to the observables transformed in the 52 same way (see e.g. Del Negro et al., 2006, Fernandez and Rubio, 2007, Justiniano, et al., 53 2010, among others). Such an approach puts stronger faith in the model than previous ones, 54 explicitly imposes consistency between the theory and the observables, but it is not free of 55 problems. For example, since the choice of which shock is permanent is often driven by 56 computational rather than economic considerations, speci…cation errors could be present. In 57 addition, structural parameter estimates may depend on nuisance features, such as the shock 58 which is assumed to be permanent and its time series characteristics. As Cogley (2001) and 59 Gorodnichenko and Ng (2010), have shown, misspeci…cation of these nuisance features may 60 lead to biased estimates of the structural parameters. 61 Construct a model driven by transitory and permanent shocks; …t the transformed 62 model to the transformed data in the frequency domain (see e.g. Diebold et. al, 1998, Chris63 tiano and Vigfusson, 2003) and select a particular frequency band over which to estimate 64 the structural parameters. This approach is also problematic since it inherits the misspeci65 …cation problems of the previous approach and the …ltering problems of statistically based 66 …ltering approaches. 67 1 INTRODUCTION 3 This paper provides an alternative method to estimate DSGE models. I show …rst that 68 the approach one takes to match the model to the data matters for structural parameter 69 estimation and for economic inference. Unless one has a strong view about what the model 70 is supposed to capture and with what type of shocks, it is di¢ cult to credibly select among 71 various structural estimates (see Canova, 1998). In general, any preliminary data transfor72 mations (should these be statistical or model-based) should be avoided if the observed data 73 is assumed to be generated by rational agents maximizing under constraints in a stochastic 74 environment. Statistical …ltering does not take into account that the data generated by a 75 DSGE model has power at all frequencies and that, if permanent and transitory shocks are 76 present, the permanent and the transitory component of the data will both appear at busi77 ness cycle frequencies. Model based transformations impose tight restrictions on the long 78 run properties of the data. Thus, any deviations from the imposed structure, being these 79 residual low frequency variations, unaccounted or idiosyncratic long run dynamics must be 80 captured by the shocks driving the transformed model. Hence, parameter estimates could 81 be distorted because estimates of income and substitution e¤ects could be biased. 82 The paper proposes to estimate structural parameters by creating a ‡exible link between 83 the DSGE model and the raw data that allows model based and non-model based compo84 nents to have power at all frequencies. The methodology can be applied to models featuring 85 transitory or transitory and permanent shocks and only requires that interesting features of 86 the data are left out from the model these could be low frequency movements of individual 87 series, di¤erent long run dynamics of groups of series, etc.. Since the non-model based com88 ponent can endogenously capture aspects of the data the model is not designed to explain, 89 researchers need not to take a stand on what is left out from the model, or on its time series 90 representation, and therefore shields the analysis from important speci…cation errors. More91 over, because the information present at all frequencies is used in the estimation, …ltering 92 distortions are eliminated and ine¢ ciencies minimized. The setup has two other advantages 93 over competitors: structural estimates re‡ect the uncertainty present in the speci…cation 94 1 INTRODUCTION 4 of non-model based features; what the model leaves out at interesting frequencies is easily 95 quanti…able. Thus, R-squared type measures can be built to "test" the structure and to 96 evaluate the explanatory power of additional shocks. 97 The approach is related to work by Del Negro et al. (2006), in that certain cross equation 98 restrictions that the DGP may impose on the data are not used in estimation, and to the 99 work of Ireland (2004b), in that a non-structural part is added to a structural model prior to 100 estimation and, crucially, it does not substitute for theoretical e¤orts designed to strengthen 101 the ability of DSGE models to account for all observable ‡uctuations. But it can …ll the gap 102 between what is nowadays available and such a worthy long run aspiration, giving researchers 103 a rigorous tool to address policy questions. 104 Using a simple experimental design and two practically relevant cases, the paper doc105 uments the biases that standard transformations produce, interprets them using the tools 106 developed in Hansen and Sargent (1993), and shows that crucial parameters are better esti107 mated with the proposed procedure. To highlight how the approach can be used in practice, 108 the paper …nally examines two questions greatly discussed in macroeconomics: the time vari109 ations in the policy activism parameter and the sources of output and in‡ation ‡uctuations. 110 To focus attention on the issues of interest, two simplifying assumptions are made: (i) the 111 estimated DSGE model features no missing variables or omitted shocks and (ii) the number 112 of structural shocks equals the number of endogenous variables. While omitted variables 113 and singularity issues are important in practice, and the semi-structural methods suggested 114 in Canova and Paustian (2011) produce more robust inference when they are present, it 115 is useful to sidestep them because the problems discussed here occur regardless of whether 116 (i)-(ii) are present or not 1. 117 The rest of the paper is organized as follows. The next section presents estimates of 118 the structural parameters of a simple model when number of statistical and model based 119 1As a referee has pointed out the approach can be used to estimate singular structural models as long as the non-model based component has the same rank as the dimension of the observable variables. Such an extension is not pursued here. 2 ESTIMATION WITH TRANSFORMED DATA 5 transformations are employed. Section 3 discusses the alternative methodology. Section 4 120 compares approaches using a simple experimental design. Section 5 examines two economic 121 questions. Section 6 concludes. 122 2 Estimation with transformed data 123 To show how estimates of the structural parameters of a DSGE model depend on the prelim124 inary transformation employed, this section considers a textbook small scale New-Keynesian 125 model, where agents face a labor-leisure choice, production is stochastic and requires labour, 126 there is external habit in consumption, an exogenous probability of price adjustments, and 127 monetary policy is conducted with a conventional Taylor rule. Details on the structure are 128 in the on-line appendix. 129 The model features a technology disturbance zt, a preference disturbance t, a monetary 130 policy disturbances t; and a markup disturbance t. The latter two shocks are assumed to 131 be iid. Depending on the speci…cation zt; are either both transitory, with persistence z 132 and respectively, or one of them is permanent. The structural parameters to be estimated 133 are: c, the risk aversion coe¢ cient, n, the inverse of the Frisch elasticity, h the coe¢ cient 134 of consumption habit, 1 , the share of labor in production, r, the degree of interest rate 135 smoothing, and y, the parameters of the monetary policy rule, 1p, the probability of 136 changing prices. The auxiliary parameters to be estimated are: ; z, the autoregressive 137 parameters of transitory preference and technology shocks, and z; ; r; the standard 138 deviations of the four structural shocks. The discount factor and the elasticity among 139 varieties are not estimated since they are very weakly identi…ed from the data. 140 Depending on the properties of the technology and of the preference shocks, the optimality 141 conditions will have a log-linear representation around the steady state or a growth path, 142 driven either by the technology or by the preference shock, see table 1. Four observable 143 variables are used in the estimation. When the model features transitory shocks, parameter 144 estimates are obtained applying four statistical …lters (linear detrending (LT), Hodrick and 145 2 ESTIMATION WITH TRANSFORMED DATA 6 Prescott …ltering (HP), growth rate …ltering (FOD) and band pass …ltering (BP)) to output, 146 the real wage, the nominal interest rate and in‡ation. Moreover, three data transformations 147 are employed. In the …rst, the log of labour productivity, the log of real wages, the nominal 148 rate and the in‡ation rate, all demeaned, are used as observables (Ratio 1). In the second 149 the log ratio of output to the real wage, the log of hours worked, the nominal rate and the 150 in‡ation rate, all demeaned, are used as observables (Ratio 2). In the third, the log of the 151 labor share, the log ratio of real wages to output, the nominal interest rate and the in‡ation 152 rate all demeaned, are used as observables (Ratio 3). When the model features a trending 153 TFP (TFP trend), the linear stochastic speci…cation zt = bt+ t ; is used and the observables 154 for the transformed model are linearly detrended output, linearly detrended wages, demeaned 155 in‡ation and demeaned interest rates. When the model features trending preferences shocks 156 (Preference trend), the unit root speci…cation, t = t 1+ t is employed and the observables 157 for the transformed model are the demeaned growth rate of output, demeaned log of real 158 wages, demeaned in‡ation and demeaned interest rates. Finally, when the model feature 159 a trending TFP, the likelihood function of the transformed model is approximated as in 160 Hansen and Sargent (1993) and only the information at business cycle frequencies ( 32 ; 8 ) is 161 used in the estimation (TFP trend, frequency domain). 162 The data comes from the FRED database at the Federal Reserve Bank of St. Louis 163 and Bayesian estimation is employed. Since some of the statistical …lters are two-sided, a 164 recursive LT …lter and a one-sided version of the HP …lter have also been considered. The 165 qualitative features of the results are unchanged by this re…nement. 166 Table 2 shows that the posterior distribution of several parameters depend on the prelim167 inary transformation used (see e.g. the risk aversion coe¢ cient c, the Frisch elasticity 1 n , 168 the interest smoothing coe¢ cient r; and persistence and the volatility of the shocks). Since 169 posterior standard deviations are tight, except when estimation is conducted in frequency 170 domain, di¤erences across columns are a-posteriori signi…cant. Posterior di¤erences are also 171 economically relevant. For example, the volatility of markup shocks in the LT, the Ratio 172 2 ESTIMATION WITH TRANSFORMED DATA 7 1 and the Preference trend economies is considerably larger and, perhaps unsurprisingly, 173 risk aversion stronger. In addition, when a frequency domain approach is used, the Frisch 174 elasticity is estimated to be very small. 175 Di¤erences in the location of the posterior of the parameters translate into important 176 di¤erences in the transmission of shocks. As shown in Figure 2, the magnitude of the impact 177 coe¢ cient and of the persistence of the responses vary with the preliminary transformation 178 employed and, for the …rst few horizons, di¤erences are statistically signi…cant. Furthermore, 179 in the case of technology shocks, the sign of some of the responses is a¤ected. 180 Why are parameter estimates so di¤erent? The …rst four transformations only approx181 imately isolate business cycle frequencies, leaving measurement errors in the transformed 182 data. In addition, di¤erent approaches spread the measurement error across di¤erent fre183 quencies: the LT transformation leaves both long and short cycles in the …ltered data; the 184 HP transformation leaves high frequencies variability unchanged; the FOD transformation 185 emphasizes high frequency ‡uctuations and reduces the importance of cycles with business 186 cycle periodicity; and even a BP transformation induces signi…cant small sample approxima187 tion errors (see e.g. Canova, 2007). Since the magnitude of the measurement error and its 188 frequency location is transformation dependent, di¤erences in parameter estimates are likely 189 to be important. An approach which can reduce the problematic part of the measurement 190 error is in Canova and Ferroni (2011). More importantly, …ltering approaches neglect the 191 fact that the spectral properties of a DSGE model are di¤erent from the output of a sta192 tistical …lter. Data generated by a DSGE model driven by transitory shocks has power at 193 all frequencies of the spectrum and if shocks are persistent most of the power will be in the 194 low frequencies. Thus, concentrating on business cycles frequencies may lead to ine¢ cien195 cies. Furthermore, when transitory and permanent shocks are present, the transitory and 196 the permanent components of the model will jointly appear in any frequency band and it 197 is not di¢ cult to build examples where, e.g. permanent shocks dominate the variability at 198 business cycle frequencies (see Aguiar and Gopinath, 2007). Hence, the association between 199 3 THE ALTERNATIVE METHODOLOGY 8 the solution of the model and the …ltered observables is generally incorrect and biases likely 200 to be generalized. 201 Implicit or explicit model-based transformations avoid these problems by specifying a 202 permanent and a transitory component of the data with power at all frequencies of the spec203 trum. However, since speci…cation problems are present (should we use a unit root process 204 or a trend stationary process? Should we allow trending preferences or trending technol205 ogy?), particular choices lead to nuisance parameters problems (the model estimated with a 206 trending TFP has MA components which do not appear when the preferences are trending, 207 see table 1), and to particular cointegration relationships in the observables, inference de208 pends on the assumptions made and any deviation of the observed data from the assumed 209 structure leads to biases. Finally, frequency domain estimation is ine¢ cient, since most of 210 the variability the model produces is in the low frequencies. Furthermore, while frequency 211 estimation can help to tone down the importance of aspects of the model researchers do not 212 trust, as suggested in Hansen and Sargent (1993), it can not de-emphasize the importance 213 of what the model leaves out at the frequencies of interest. 214 3 The alternative methodology 215 Start from the assumption that the observable data has been generated by rational expec216 tation agents, optimizing their objective functions under constraints in a stochastic environ217 ment. Assume that the econometrician knows the data generating process for a portion of 218 the data but she is unsure about the transmission produced by certain shocks (e.g. those in219 ducing permanent e¤ects) or how to capture aspects of the data (e.g. those with medium-long 220 period of oscillation). Thus, she is aware that the model used for inference is misspeci…ed. 221 Rather than trying to …lter out from the data what the model is unsuited to explain or add 222 ad-hoc features to the model to reduce the misspeci…cation, I will assume that the investi223 gator takes the misspeci…ed structure as given, because it is unclear how to model all the 224 ‡uctuations present in the data or because the available short cuts are unlikely to satisfac225 3 THE ALTERNATIVE METHODOLOGY 9 torily account for its complexity. To estimate the parameters of the model she uses the raw 226 data and disregards certain cross equations restrictions present in the DGP but builds a 227 link between the misspeci…ed structural model and the raw data which is su¢ ciently ‡exible 228 to capture what the model is unsuited to explain and allows model and non-model based 229 components to jointly appear at all frequencies of the spectrum. 230 Let the (log)-linearized stationary solution of a DSGE model be of the form: 231 x2t = A( )x1t 1 +B( ) t (1) x1t = C( )x1t 1 +D( ) t (2) where A( ); B( ); C( ); D( ) depend on the structural parameters , x1t (log ~ x1t log x1t) 232 includes exogenous and endogenous states, x2t = (log ~ x2t log x2t) all other endogenous 233 variables, t the shocks and x2t; x1t are the long run paths of ~ x2t and ~ x1t. 234 Let xt ( ) = R[x1t; x2t] 0 be an N 1 vector, where R is a selection matrix picking out 235 of x1t and x2t variables which are observable and/or interesting from the point of view of 236 the researcher and let xt ( ) = R[ x1t; x2t] 0. Let xt = log ~ x d t E(log ~ xt ) be the log demeaned 237 N 1 vector of observable data. The speci…cation for the raw data is then: 238 xt = ct( ) + x nm t + x m t ( ) + ut (3) where ct( ) = log xt ( ) E(log ~ xt ), ut is a iid (0; u) (proxy) noise, x t ; xt and ut are 239 mutually orthogonal and x t is given by: 240 x t = 1x nm t 1 + wt 1 + et et iid (0; e) wt = 2wt 1 + vt vt iid (0; v) (4) where 1 = diag( 11; ::: 1N); 2 = diag( 21; ::: 2N); 0 < 1i; 2i 1; i = 1; :::N . To under241 stand what the speci…cation for x t implies, notice that when 1 = 2 = I, and et; vt are 242 uncorrelated (4) is the locally linear trend speci…cation used in state space models, see e.g. 243 Gomez (1999). In addition, if 1 = 2 = I; e and v are diagonal, vi = 0, and ei > 0; 8i, 244 3 THE ALTERNATIVE METHODOLOGY 10 x t is a vector of I(1) processes while if vi = ei = 0; 8i, x t is deterministic. When 245 instead 1 = 2 = I, and vi and ei are functions of , (4) approximates the double ex246 ponential smoothing setup used in discounted least square estimation of state space models, 247 see e.g. Delle Monache and Harvey (2010). Thus, if xt ( ) = x ( );8t, the observable xt can 248 display any of the typical structures that motivates the use of the statistical …lters. Further249 more, as Delle Monache and Harvey (2010) have emphasized, (4) is robust against several 250 types of misspeci…cation of the time series properties of what the model does not explain. 251 Note also, whenever v is not constrained to be zero, the growth rates of the endogenous 252 variables may display persistent deviations from their mean, a feature that characterizes 253 many real macroeconomic variables, see e.g. Ireland (2010). Finally, when xt ( ) is not 254 constant, and 1i and 2i are complex conjugates for some i, the speci…cation can capture 255 residual low frequency variations with power at frequency !. To see this notice that when 256 N=1, (4) implies that (1 2L)(1 1L)x t = (1 2L)et+ vt 1 (1 L) t. If the roots 257 1 1 ; 1 2 of the polynomial 1 ( 1 + 2)z + 1 2z = 0 are complex, they can be written as 258 1 1 = r(cos!+ i sin!); 1 2 = r(cos! i sin!), where r = p 1 2 and ! = cos [ 1+ 2 2 p 1 2 ] and 259 (4) is x t = P j r sin!(j+1) sin! (1 L) t, whose period of oscillation is p = 2 ! = 2 cos 1[ 1+ 2 2 p 1 2 ] . 260 Thus, given r and p, there exists 1; 2 that produce x nm t with the required properties. 261 Given (1)-(4), the data will endogenously select the speci…cation for the non-model based 262 component which is more appropriate for each series and this will be done jointly with the 263 estimation of the structural parameters . Identi…cation of the structural parameters is 264 achieved via the cross equation restrictions that the model imposes on the data. Estimates 265 of the non-structural parameters are implicitly obtained from the portion of the data the 266 model can not explain. 267 The speci…cation has a number of advantages over existing approaches. One does not 268 need to take a stand on the time series properties of the non-model based component and on 269 the choice of …lter to tone down its importance and this shields researchers from important 270 speci…cation and …ltering errors. As shown in Ferroni (2011), the setup can be used to …nd 271 3 THE ALTERNATIVE METHODOLOGY 11 the most appropriate speci…cation of the non-model based component and, if a researcher 272 is interested in doing so, to perform Bayesian averaging over di¤erent types of non-model 273 based speci…cations, which is not possible in standard setups. Furthermore, as shown below, 274 all components in (3) may have power at all frequency. Finally, since joint estimation is 275 performed, structural parameter estimates re‡ect the uncertainty present in the speci…cation 276 of the non-model based component. 277 3.1 Two special cases 278 It is useful to consider two special cases of the setup to give a sense of what the approach 279 does. Suppose …rst that the model features only transitory shocks while the data may 280 display common or idiosyncratic long run drifts, low frequency movements and business 281 cycle ‡uctuations. Here xt ( ) = x ( );8t, are the steady states of the model and, if the 282 model is correctly speci…ed on average, ct( ) = 0. Assume that no proxy errors are present. 283 Then (3) is 284 xt = x nm t + x m t ( ) (5) and x t captures the features of x d t that the stationary model does not explain. Depending 285 on the speci…cation of 1 and 2, these include long run drifts, both of common and idio286 syncratic types, and those idiosyncratic low and business cycle movements the model leaves 287 unexplained. In this setup, x t has two interpretations: As in Altug (1989), McGrattan 288 (1994) and Ireland (2004b), it can be thought of as a measurement error added to the struc289 tural model. However, rather than being iid or AR(1), it has the richer representation (4). 290 Alternatively, it can be thought as a reduced form representation for the components of the 291 data the investigator is unsure how to model. Thus, as in Del Negro et al. (2006), x t 292 relaxes the cross equations restrictions that the DGP implies and captures what the model 293 can not explain via the ‡exible parameterization (4). 294 Suppose, alternatively, that the model features transitory shocks and one or more per295 manent shocks. In this case xt ( ) represents the (stationary) solution in deviation from 296 3 THE ALTERNATIVE METHODOLOGY 12 the permanent shocks and xt ( ) the model based component generated by the permanent 297 shocks. Suppose again that there are no proxy errors. In that case (3) reduces to 298 xt = ct( ) + x ;nm t + x m t ( ) (6) where x ;nm t captures the features of x d t which neither the transitory portion x m t ( ) nor the 299 permanent portion ct( ) of the model explains. These may include, idiosyncratic long run 300 patterns (such as diverging trends), idiosyncratic low frequency movements, or unaccounted 301 cyclical ‡uctuations. Comparing (5) and (6), one can see that x t = ct( ) + x ;nm t . Thus, 302 the setup can be used to measure how much of the data the model leaves unexplained and 303 to evaluate whether certain shocks may reduce the discrepancy. For example, one could 304 start from a model featuring a few transitory shocks and measure the relative importance 305 of x t at a particular set of frequencies. If it is large, one could add additional transitory 306 shocks and see how much the relative importance of x t has fallen at those frequencies. 307 Alternatively, one could add a permanent shock and compare the magnitude of x ;nm t and 308 x t at a particular set of frequencies. By comparing the outcomes of the two exercises, one 309 can also assess whether the addition of a permanent or a transitory shock is more bene…cial. 310 The same logic can be used to evaluate the model when, e.g. the permanent shock takes 311 the form of a stochastic deterministic trend (as in the case of labor augmenting technological 312 progress), when it is represented with a unit root, or when all long run paths are left unmod313 elled. Hence, the approach naturally provides a setup to judge the goodness of …t of a model 314 and to evaluate the contribution of certain features to the understanding of economic phe315 nomena. It does so by giving researchers a constructive criteria to increase the complexity 316 of models; and an integrated framework to examine the sensitivity of the estimation results 317 to the speci…cation of nuisance features, both of which are absent from existing methods. 318 3 THE ALTERNATIVE METHODOLOGY 13 3.2 Estimation 319 Estimation of the structural parameters can be carried out with both classical and Bayesian 320 methods. (1)-(4) can be cast into the linear state space system: 321 st+1 = Fst +G!t+1 !t (0; !) (7) xt = ct( ) +Hst (8) where st = x t wt x m t ( ) ut 0 ; !t+1 = (et+1; vt+1; ut+1; t+1) 0, H = I 0 I I ; 322 F = BB@ 1 I 0 0 0 2 0 0 0 0 R[A C]0 0 0 0 0 0 CCA ; G = BB@ I 0 0 0 0 I 0 0 0 0 0 R[B D]0 0 0 I 0 CCA : Hence, the likelihood can 323 be computed with a modi…ed Kalman …lter (accounting for the possibility of di¤use initial 324 observations) for a given # = ( ; 1; 2; e; v; u) and maximized using standard tools. 325 When a Bayesian approach is preferred, one can obtain the non-normalized posterior of 326 #, using standard MCMC tools. For example, the estimates presented in this paper are 327 obtained with a Metropolis algorithm where, given initial # 1 and a prior g(#), candidate 328 draws are obtained from # = # 1 + ; where is distributed t(0; ; 5) and is a tuning 329 parameter, and the draw accepted if the ratio g(# jy) g(# 1jy)exceeds a uniform random variable, 330 where g(#ijy) = g(#i)L(yj#i), i = ; 1, and L(yj#i) is the likelihood of #i, . Iterated a 331 large number of times, for appropriately chosen, the algorithm ensures that the limiting 332 distribution of # is the target distribution (see e.g. Canova, 2007). 333 3.3 The relationship with the existing literature 334 Apart from the work of Del Negro et al. (2006) and of Altug (1989), McGrattan (1994) and 335 Ireland (2004b) already mentioned, the procedure is related to a number of existing works. 336 First, the state space setup (7)-(8) is similar in spirit to the one suggested by Harvey 337 and Jeager (1993), even thought these authors consider only univariate processes and do not 338 use a structural model to explain the observables. It also shares important similarities with 339 the one employed by Cayen et al. (2009), who are interested in forecasting trends. Two 340 3 THE ALTERNATIVE METHODOLOGY 14 are the most noticeable di¤erences. First, these authors use a two-step estimation approach, 341 conditioning on …ltered estimates of the parameters of the DSGE model, while here a one 342 step approach is employed. Second, all the deviations from the model are bundled up in the 343 non-model speci…cation while here it is possible to split them into model interpretable and 344 model non-interpretable parts. 345 The contribution of the paper is also related to two distinct branches of the macroeco346 nomic and macroeconometric literature. The …rst attempts to robustify inference when the 347 trend properties of data are misspeci…ed (see Cogley, 2001, and Gorodnichenko and Ng, 348 2010). I share with the …rst author the idea that economic theory may not have much to say 349 about certain types of ‡uctuations but rather than distinguishing between trend stationary 350 and di¤erence stationary cycles, the paper wants to design an estimation procedure which 351 deals with the mismatch between theoretical and empirical concepts of ‡uctuations without 352 taking a stand on the time series properties of what the model leaves unexplained. The idea 353 of jointly estimating structural and auxiliary parameters without fully specifying the DGP 354 is also present in Gorodnichenko and Ng. However, a likelihood based estimator, as opposed 355 to a minimum distance estimator, is used here because it works regardless of the time series 356 properties of the raw data. In addition, rather than assuming that the model is the DGP, the 357 procedure assumes that the DSGE model is misspeci…ed a much more useful assumption 358 in practice. 359 The second branch points out that variations in trend growth are as important as cyclical 360 ‡uctuations in explaining the dynamics of macroeconomic variables in emerging markets 361 (see e.g. Aguiar and Gopinath, 2007, and Andrle, 2008). While the …rst paper characterizes 362 di¤erences between emerging and developing economies, the latter is concerned with the 363 misuse of models driven by transitory shocks in policy analyses for developing countries. 364 This paper shows that the problems they highlight are generic and that policy analyses with 365 misspeci…ed models are possible without imposing controversial assumptions about what the 366 model is not designed to explain. 367 3 THE ALTERNATIVE METHODOLOGY 15 3.4 Setting the priors for e and v 368 If the number of observable variables is small and the number of data points large, one can 369 easily obtain estimates of from(7)-(8). If the number of observables is large or the sample 370 size limited, weak identi…cation problems and small sample biases may become relevant. 371 Note, in fact, that in (4) there are 2N +2N non-structural parameters to be estimated and 372 that it may be di¢ cult to distinguish variations in the level from variations in the growth 373 rates of the variables. Thus, it may be worth to impose some structure on v and e; 374 if information about what the model leaves out is available, and shrewdly cut down on the 375 dimensionality of the non-structural parameter space. For example, one may want to assume 376 that v and e are diagonal (so that the non-model based component is series speci…c), and 377 of reduced rank (the non-model based component is common across (groups of) series); that 378 they have only sparse non-zero elements on the diagonal (the non-model based component 379 exists only in a number of observables) or that they are proportional to each other (shocks to 380 the level and the growth rate are related). Some a-priori restrictions appear to be necessary 381 also because given a DSGE structure, the decomposition of the data in model based and non382 model based components depends on the strength of the shock signals. Thus, the procedure 383 de…nes a family of decompositions, indexed by the relative intensity of the shocks driving the 384 model and the non-model based components. Given that it is typically di¢ cult to estimate 385 this intensity parameter unrestrictedly in small samples, and that unrestricted estimates may 386 imply non-model based components with undesirable high frequency variability, a sensible 387 smoothness prior for e and v is needed. 388 The restrictions which we recommend to be used, and are employed in the two appli389 cations described below, involves making e and v diagonal, of reduced rank, sparse, and 390 function of the structural shocks. As mentioned, it is possible to approximate the double 391 exponential smoothing restrictions used in discounted least square estimation of state space 392 models by selecting e.g. ei = q 2 and vi = q 2 (4 )2 ; where i indicates the non-zero ele393 ments of the matrices, t is one of structural shocks and a smoothing parameter. Thus, 394 4 THE PROCEDURE IN A CONTROLLED EXPERIMENT 16 given a prior for t and , a prior for all non-zero elements of e and v is automatically gen395 erated. The speci…cation is attractive because it is parsimonious and considerably reduces 396 the number of non-structural parameters to be estimated. Since has the same interpreta397 tion as in the HP …lter, an agnostic prior for could be centered at 400 with uniform range 398 over [4,6400], which allows for very smooth as well as relatively jagged non-model based com399 ponents 2. When the likelihood for this parameter is ‡at, one could alternatively calibrate 400 to di¤erent values and, in models driven by transitory shocks, eliminate candidates produc401 ing non-model based estimates which are not su¢ ciently smooth. Since one of the structural 402 shocks needs be selected to form the prior for e and v, one could also experiment choosing 403 the disturbance with, potentially, the largest or the smallest variance to calibrate the prior. 404 For the applications in section 5, which structural disturbance is employed to calibrate the 405 prior is irrelevant. 406 In sum, the approach is easy to implement it requires only a few additional lines in 407 an existing computer code, requires some ingenuity to decrease the dimensionality of the 408 parameter space when the sample is small, but it is otherwise fully operational in practice 409 and, as shown below, it has nice properties in a simple experimental design. 410 4 The procedure in a controlled experiment 411 To examine the properties of the procedure and to compare them to those of standard 412 transformations, I use the same setup employed in section 2 and simulate 150 data points 413 assuming that the preference shock has a transitory and a permanent component. Thus, 414 t = 1t + 2t; 1t = 1t 1 + T t and 2t = 2t 1 + P t . This speci…cation is chosen since 415 Chang et al. (2007) have indicated that a model with permanent preference shocks can 416 capture well low frequency variations in hours worked. In this setup, the data will display 417 stationary ‡uctuations driven by four transitory shocks (which we correctly capture with 418 2It is worth noting that selecting the signal to noise ratio is much less demanding than assuming a particular format for the drifts the data displays or selecting a shock which drives them. 4 THE PROCEDURE IN A CONTROLLED EXPERIMENT 17 a model) and important non-stationary ‡uctuations driven the permanent preference shock 419 (which we will either try to …lter out, eliminate with certain data transformations, or account 420 with a non-model based component) making the design relevant for practical purposes. The 421 estimated model is misspeci…ed relative to the DGP in that the permanent component due 422 to the preference shock is left out, but all the other features are correctly represented. 423 Furthermore, since the permanent component of the preference shock is orthogonal to all 424 transitory shocks, the design …ts the setup of section 3. 425 The structural parameters will be estimated using the proposed approach and the same 426 transformations used of section 2 in the most ideal situations one could consider these 427 include priors centered at the true parameter vector and initial conditions equal to the true 428 parameter vector. When the approach of section 3 is used, the non-model based component 429 is restricted to have a double exponential smoothing format and, consistently with the DGP 430 (see appendix) is allowed to enter only in output and the real wage. The true values of 431 the structural parameters are in table 3. In the estimation the same prior distributions 432 for the structural parameters displayed in table 2 are used. Two cases are examined: one 433 where the permanent disturbance has relative high variability p = 1:50 and one where it 434 has relative low variability p = 0:15. In the …rst case, the contribution of the permanent 435 component to the spectrum of the series is of the same order of magnitude as the contribution 436 of the transitory component at almost all frequencies. Thus, both …ltering and speci…cation 437 errors are present with standard transformations. In the second case, the contribution of 438 the permanent component to the spectrum of the series is everywhere small. Here, standard 439 transformations will only produce …ltering errors and, in a large sample, the BP …lter provides 440 a consistent although ine¢ cient estimator of model based ‡uctuations. 441 As table 3 shows, the distortions produced by standard approaches are important. Apart 442 from producing estimates of utility and technology parameters which are biased and very 443 much …lter dependent, the persistence of the preference and of the technology shocks ; z 444 and the standard deviations of the preference and the markup shocks and are gen445 4 THE PROCEDURE IN A CONTROLLED EXPERIMENT 18 erally distorted. In comparison, estimates of utility and technology parameters reported in 446 the column labelled ”Flexible”are closer (in a MSE sense) to the true values and both the 447 persistence and the standard deviations of the shocks are better captured. Matching the 448 persistence and the volatility of the shocks is important since conditional and unconditional 449 moments crucially depend on these parameters. Note also that while with standard transfor450 mations, estimates depend on the relative intensity of the permanent and transitory signals, 451 this is much less the case for the procedure this paper suggests. 452 To understand the nature of the distortions produced by standard transformations, 453 note that the log-likelihood of the data can be represented as L( jyt) = [A1( ) + A2( ) + 454 A3( )jy], see Hansen and Sargent (1993), where A1( ) = 1 P !j log detG (!j), A2( ) = 455 1 P !j trace [G (!j) F (!j)], A3( ) = (E(y) ( ))G (!0) (E(y) ( )), !j = j T ; j = 456 0; 1; : : : ; T 1. G (!j) is the model based spectral density matrix of yt, ( ) the model based 457 mean of yt, F (!j) is the data based spectral density and E(y) the unconditional mean of yt. 458 A2( ) and A3( ) are penalty functions: A2( ) sums deviations of the model-based from the 459 data-based spectral density over frequencies; A3( ) weights deviations of model-based from 460 data-based means with the spectral density matrix of the model at frequency zero. 461 Suppose the data is transformed so that the zero frequency is eliminated and the low 462 frequencies de-emphasized. Then, the log-likelihood consists of A1( ) and of A2( ) = 463 1 P !j trace [G (!j)] F (!j) , where F (!j) = F (!j)I!j and I!j is a function describing 464 the e¤ect of the …lter at frequency !j. Suppose that I! = I[!1;!2], i.e. an indicator function 465 for the business cycle frequencies, as in an ideal BP …lter. Then A2( ) matters only at 466 business cycle frequencies. Since at these frequencies [G (!j)] < F (!j) , A2( ) and A1( ) 467 enter additively L( jyt), two types of biases will be present. Since estimates F̂ (!j) only 468 approximately capture the features of F (!j) , Â2( ) has smaller values at business cycle 469 frequencies and a nonzero value at non-business cycle ones. Moreover, in order to reduce the 470 contribution of the penalty function to the log-likelihood, parameters are adjusted so that 471 [G (!j)] is close to F̂ (!j) at those frequencies where F̂ (!j) is not zero. This is done by 472 4 THE PROCEDURE IN A CONTROLLED EXPERIMENT 19 allowing …tting errors, (a larger A1( )), at frequencies where F̂ (!j) is zero in particular, 473 the low frequencies. Hence, the volatility of the structural shocks will be overestimated (this 474 makes G (!j) close to F̂ (!j) at the relevant frequencies), in exchange for misspecifying 475 their persistence. These distortions a¤ect agents’decision rules. Higher perceived volatility, 476 for example, implies distortions in the risk aversion coe¢ cient. Inappropriate persistence 477 estimates, on the other hand, imply that perceived substitution and income e¤ects are dis478 torted with the latter typically underestimated. When I! is not the indicator function, the 479 derivation of the size and the direction of the distortions is more complicated but the same 480 logic applies. Clearly, di¤erent I! produce di¤erent F̂ (!j) and thus di¤erent distortions. 481 Since estimates of F (!j) are imprecise, even for large T , there are only two situations 482 when estimation biases are small. First, the permanent component has low power at business 483 cycle frequencies in this case, the distortions induced by the penalty function are limited. 484 This occurs when transitory volatility dominates (as in the second panel of table 3). Second, 485 when Bayesian estimation is performed, the prior is selected to limit the distortions induced 486 by the penalty function. This is very unlikely, however, since priors are not elicited with 487 such a scope in mind. 488 If instead one …ts a transformed version of the model to transformed data, as it is done 489 in model based approaches, the log-likelihood is composed of A1( ) = 1 P !j log jG (!j)I!j j 490 and A2( ) since the actual and model data are …ltered in the same way, the …lter does not 491 a¤ect the penalty function. Suppose that I! = I[!1;!2]. Then A1( ) matters only at business 492 cycle frequencies while the penalty function is present at all frequencies. Therefore, parame493 ter estimates are adjusted so as to reduce the misspeci…cation at all frequencies. Since the 494 penalty function is generally more important at the low frequencies, parameters are selected 495 to make [G (!j)] close to F̂ (!j) at those frequencies and large …tting errors are permitted 496 at medium and high frequencies. Consequently, the volatility of the shocks will be generally 497 underestimated in exchange for overestimating their persistence somewhat paradoxically, 498 this procedure implies that the low frequency components of the data are those that matter 499 5 TWO APPLICATIONS 20 most for estimation. Cross frequency distortions imply that the econometrician recovers 500 an economy which di¤ers substantially from the true one. For example, since less noise is 501 perceived, agents decision rules imply a higher degree of data predictability, and higher per502 ceived persistence implies that perceived substitution and income e¤ects are distorted with 503 the latter overestimated. 504 To further highlight the properties of the proposed approach, the top row of …gure 3 505 reports estimates of the permanent and transitory components of output obtained with the 506 Kalman …lter and either the true parameters or the median estimates presented in the top 507 panel of table 3. The bottom two rows of …gure 3 compare the autocorrelation function and 508 the spectral density of the true and estimated components of output. 509 The true and the estimated components of output display similar volatility properties. 510 In addition, the rate of decay of the autocorrelation functions of the true and the estimated 511 components is practically identical. Finally, as anticipated, the two estimated components 512 have power at all frequencies of the spectrum, and at business cycle frequencies (indicated 513 by the vertical bars in the last row of graphs) the permanent component is more important 514 than the transitory component. 515 The conditional dynamics in response to transitory shocks are also well captured. Figure 516 4, which presents impulse responses obtained with true and estimated parameters, indicates 517 that the sign and the persistence of the responses are well matched. Magnitudes are occa518 sionally imprecisely estimated this problem would remain even if we double the sample size 519 but overall, the approach does a good job in reproducing the main qualitative features of 520 the DGP. Thus, economic inference is less prone to ”mismatch”distortions. 521 5 Two applications 522 This section shows how the proposed approach can be used to inform researchers about two 523 questions which have received a lot of attention in the literature: the time variations in the 524 policy activism parameter and the sources of output and in‡ation ‡uctuations. The …rst 525 5 TWO APPLICATIONS 21 question is analyzed with the model presented in section 2. The second, with a medium 526 scale model widely used in academic and policy circles. 527 5.1 The policy activism parameter 528 What are the features of the monetary policy rule in place during the ”Great In‡ation”of 529 the 1970s and the return to norm of the 1980s and 1990s? This question has been extensively 530 studied in the literature following Clarida et al. (2000). One synthetic way to summarize 531 the information contained in the data is to compute the policy activism parameter y 1 , 532 which gives a sense of the relative importance of the output and the in‡ation stabilization 533 objectives of the Central Bank. The conventional wisdom suggests that the absolute value of 534 this parameter has declined over time, re‡ecting changes in the preferences of the monetary 535 authorities, but most of the available evidence is obtained either with reduced form methods 536 or, when structural method are used, with …ltered data. Are the results to be trusted? Is the 537 characterization o¤ered by the approach of this paper di¤erent? Figure 5 plots the posterior 538 density of the policy activism parameter obtained when the data is linearly detrended (top 539 left box) or HP …ltered (top right box) before estimation and when the approach of this 540 paper is employed (lower left box) for the samples 1964:1-1979:4 and 1984:1-2007:4. The 541 prior for the structural and auxiliary parameters is the same as in table 1. In the ‡exible 542 approach, and given the short subsamples, e and v are assumed to be diagonal, a common 543 non-model based component is assumed for all the variables, the signal-to-noise ratio in the 544 four series is captured by a single parameter , a-priori uniformly distributed over [100, 545 6400], 1= 2 = I and the proxy error is set to zero. 546 The posterior density of the policy activism parameter shifts to the left in the second 547 sample when HP …ltered data is used and, for example, the posterior median moves from 548 -0.23 in the …rst sample to -0.33 in the second. This left shift of the posterior density is 549 absent when LT data is used and the median of the posterior in the second sample moves 550 closer to zero (from -0.38 to 0.12) care should be exercised here since the median is not a 551 5 TWO APPLICATIONS 22 good estimator of the central tendency of the posterior for the 1984-2007 sample. In both 552 cases, the Kolmogorov-Smirnov statistic rejects the null that the posterior distributions are 553 the same in the two samples. Thus, standard approaches con…rm the existence of a break 554 in the conduct of monetary policy, although it is not clear in which direction the movement 555 is: with HP …ltered data, output gap considerations have become relatively more important; 556 with LT …ltered data, the opposite appears to be true. 557 When the approach of section 3 is used, the posterior density of y 1 in the two samples 558 overlaps considerably. Interestingly, both the location and the shape of the density in the 559 two samples are very similar and the Kolmogorov-Smirnov statistic does not reject the null 560 that the posterior distributions in the two samples are the same. Thus, evidence in favor of 561 a structural break in the conduct of monetary policy is much weaker in this case. 562 Why are the results di¤erent? As mentioned, the non-model based component soaks 563 up all the features that the model is not designed to explain. Thus, in principle, it could 564 absorb the changes present in the endogenous variables. This, however, does not seem to 565 be the case: the median estimate of is around 3200 in both samples, making the non566 model based component quite smooth relative to the model based component (see on-line 567 appendix for plots of the two components of the four variables) and essentially time invariant. 568 Thus, variations in the time series properties of the endogenous variables are not captured 569 by the non-model based component. What instead happens, is that structural non-policy 570 parameters change to accommodate for the changes in the time series properties of in‡ation 571 and interest rate over time. Interestingly, the explanatory power of the model increases in 572 the second sub-sample: on average, at business cycle frequencies, the model explains 40 per 573 cent of output variations in the …rst sample and 55 per cent in the second sample. For 574 in‡ation and interest rates, the increase is smaller (from 40 to 50 percent). 575 Since about 50 percent of the variability observables at business cycle frequencies is not 576 captured by the model in both samples, it is worth investigating how the …t can be improved 577 by altering its structure, keeping the number of observables and the estimation approach 578 5 TWO APPLICATIONS 23 unchanged. One device that the literature has employed to improve the …t of this kind of 579 models is to allow for a time varying in‡ation target in the policy rule, see e.g. Ireland 580 (2007). The target is assumed to be driven by a permanent shock and enters only in the 581 interest rate equation. Thus, the estimated speci…cation moves from (5) to (6), where now 582 ct( ) appears only in the interest rate equation. What would this modi…cation do to the 583 posterior distribution of the policy activism parameter? 584 The last box of …gure 5 indicates that adding a time varying in‡ation target reduces the 585 spread of the posterior distributions. Hence, the shift to the right in the posterior in the 586 second sub-sample becomes statistically signi…cant even though ,e.g., the median value of 587 the two distributions is close in absolute value. Adding an in‡ation target improves the …t 588 for the interest rate at business cycle frequencies (the proportion of the variance explained 589 increase to 57 percent in the …rst sample and to 68 percent in the second); for in‡ation, 590 instead, the explanatory power of the model is unchanged in the …rst sub-sample and worsen 591 considerably in the second (the variance share explained at business cycle frequencies is now 592 only 28 percent). Hence, adding a time varying in‡ation target does not seem to be a very 593 promising way to improve our understanding of how in‡ation ‡uctuations are generated. 594 5.2 Sources of output and in‡ation ‡uctuations 595 The question of what drives output and in‡ation ‡uctuations has a long history in macro596 economics. In standard medium scale DSGE models, like the one employed by Smets and 597 Wouters (2003) and (2007), output and in‡ation ‡uctuations tend to be primarily explained 598 by markup shocks. Since these shocks are an unlikely source of cyclical ‡uctuations, Chari at 599 al (2009) have argued that misspeci…cation is likely to be present (see Justiniano et al., 2010, 600 for an alternative interpretation). Researchers working in the area use …ltering devices to …t 601 the model to the data (as in Smets and Wouters (2003)), arbitrarily data transformations (as 602 in Smets and Wouters, 2007) or build a permanent component in the model (as in Justiniano 603 et al., 2010) and use model-consistent data transformations to estimate the structural para604 5 TWO APPLICATIONS 24 meters. What would the approach of this paper tell us about sources of cyclical ‡uctuations 605 in output and in‡ation? How much of the variability of the observables at business cycle 606 frequencies is explained by the model? To answer this question, the same model and the 607 same data set used in Smets and Wouters (2007) are employed but a more standard setup 608 is employed. In particular, no MA terms for the price and wage markup disturbances are 609 assumed all shocks have a standard AR(1) structure; the model is solved in deviations from 610 the steady state, rather than in deviation from the ‡exible price equilibrium; and the policy 611 rule does not include a term concerning output growth. 612 Table 4 reports results obtained eliminating a linear trend from the variables; taking 613 growth rates of the real variables and demeaning nominal ones; and using the approach 614 suggested in this paper. When a linear trend is removed, the forecast error variance decom615 position of output at the …ve years horizon is indeed primarily driven by price markup shocks, 616 with a considerably smaller contribution of investment speci…c and preference shocks. For 617 in‡ation, price markup shocks account for almost 90 percent of the forecast error variability 618 at the …ve years horizon. When the model is instead …tted to growth rates, price markup 619 shocks account for over 90 percent of the variability of both output and in‡ation at the …ve 620 years horizon. Thus, even without some of the standard bells and whistles, the conclusion 621 that markup shocks dominate remains. Why are price markup shocks important? Since, 622 compared to other shocks, they are relatively unrestricted in the model, they tend to absorb 623 any misspeci…cation the model has and any measurement error that the …lters leave in the 624 transformed data. Furthermore, since the combined speci…cation and measurement errors 625 are unlikely to be iid, the role of markup shocks is overestimated. When the bridge suggested 626 in this paper is used, the non-model based component of real variables is restricted to have a 627 common structure (there are only two parameters simultaneously controlling the non-model 628 based component of output, consumption, investment), 1= 2 = I , and a proxy error is 629 allowed in each equation, the picture is quite di¤erent. Output ‡uctuations at the …ve year 630 horizon are driven almost entirely by preference disturbances, while in‡ation ‡uctuations are 631 6 CONCLUSIONS 25 jointly accounted for by wage markup, TFP and price markup disturbances. More interest632 ingly, the model explains only 20 percent of the output and in‡ation ‡uctuations at business 633 cycle frequencies. Thus, it is seems premature to use it to evaluate policy alternatives. 634 It is useful to characterize the properties of the non-model based component to evaluate 635 the theoretical modi…cations that are needed to capture what the current model leaves out. 636 The non-model component is well represented by the speci…cation employed and restrictions 637 on the representation used assuming, for example, no or only one unit root are all rejected 638 in formal testing (log Bayes factor exceeding 10 in both cases). Thus, if shocks are to be 639 added to the model, it is important that they have permanent features and display persistent 640 deviations from a balanced growth path. Ireland (2010) has suggested one such speci…cation. 641 Others, which allow both TFP and investment shocks to have these features, are also possible. 642