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− ∞(38) ˆ = ˆ 1 hC − − ∑ ˆt hCt1 ( Rt+i −πˆt+1+i)σ c i=0Our estimates ofσ c and h thus imply that an expected one percent increase in the short-terminterest rate for four quarters has an impact on consumption of about 0.30.The estimate of the adjustment cost parameter is very similar to the one estimated in CEE (2001). 30It implies that investment increases by about 0.2 percent following a one percent increase in thecurrent price of installed capital. Also the estimates for the fixed cost parameter and the elasticity ofthe cost of adjusting capacity utilisation are in line with the results in CEE (2001). The estimate ofσ l is around 2.5, implying an intermediate estimate of the elasticity of labour supply. However,this estimate did not prove to be very robust across specifications.Finally, our estimation delivers plausible parameters for the long and short-run reaction function ofthe monetary authorities, broadly in line with those proposed by Taylor (1993). Obviously, as therewas no single monetary policy in the euro area over most of the estimation period, these resultsneed to be taken with a grain of salt. The estimates imply that in the long run the response ofinterest rates to inflation was greater than one, thereby satisfying the so-called Taylor principle.Also the response to output is similar to the one suggested by Taylor (1993). In addition, we alsofind a significant positive short-term reaction to the current change in inflation and the output gap.Finally, in agreement with the large literature on estimated interest rate rules, we also find evidenceof a substantial degree of interest rate smoothing.3.3 Assessing the empirical performance of the estimated DSGE model3.3.1 Comparing the estimated DSGE model with VARsThe discussion in the previous section shows that the model is able to deliver reasonable andsignificant estimates of the model parameters. In this section, we analyse how well our estimatedmodel does compared to a-theoretical VAR models estimated on the same data set. As discussed inGeweke (1999), the Bayesian approach used in this paper provides a framework for comparing andchoosing between fundamentally misspecified models on the basis of the marginal likelihood of themodel. 31The marginal likelihood of a model A is defined as:(39) M = ∫ p(θθ A )p(Y T θ , A ) dθ3031Table 1 reports 1 ϕ = S".See also Landon-Lane (1998) and Schorfheide (2002).20 NBB WORKING PAPER No.35 - October 2002
where p(θ A ) is the prior density for model A and p( Y Tθ , A)is the probability density functionor the likelihood function of the observable data series, YT, conditional on model A and parametervector θ . By integrating out the parameters of the model, the marginal likelihood of a model givesan indication of the overall likelihood of the model given the data.The Bayes factor between two models i and j is then defined as(40)Bij =MMijMoreover, prior information can be introduced in the comparison by calculating the posterior odds:p(41) = iMiPO i∑ j p jMjwhere p iis the prior probability that is assigned to model i. If one is agnostic about which of thevarious models is more likely, the prior should weight all models equally.The marginal likelihood of a model (or the Bayes factor) is directly related to the predictive densityor likelihood function of a model, given by:T+mT + m+,θt=T + 1(42) pˆT 1= ∫ p( θ YT, A ) ∏ p( ytYT,ϑ,A ) d θasTpˆ0 = M T .Therefore, the marginal likelihood of a model also reflects its prediction performance. Similarly, theBayes factor compares the models’ ability to predict out of sample.Geweke (1998) discusses various ways to calculate the marginal likelihood of a model. 32 Table 2presents the results of applying some of these methods to the DSGE model and various VARs. Theupper part of the Table compares the DSGE model with three standard VAR models of lag orderone to three, estimated using the same seven observable data series. The lower part of Table 2compares the DSGE model with Bayesian VARs estimated using the well-known Minnesota prior. 33In both cases, the results show that the marginal likelihood of the estimated DSGE model is very3233If, as in our case, an analytical calculation of the posterior distribution is not possible, one has to be able to makedrawings from the posterior distribution of the model. If the distribution is known and easily drawn from, independentdraws can be used. If that is not possible, various MCMC methods are available. Geweke (1998) presents differentposterior simulation methods (acceptance and importance sampling, Gibbs sampler and the Metropolis-Hastingsalgorithm used in this paper). Given these samples of the posterior distribution, Geweke (1998) also proposes differentmethods to calculate the marginal likelihood necessary for model comparison (a method for importance sampling andfor MH algorithm, a method for the Gibbs sampler, and the modified harmonic mean that works for all samplingmethods). Schorfheide (1999) also uses a Laplace approximation to calculate the marginal likelihood. This methodapplies a standard correction to the posterior evaluation at the posterior mode to approximate the marginal likelihood.So it does not use any sampling method but starts from the evaluation at the mode of the posterior. Furthermore, in thecase of VAR-models the exact form of the distribution functions for the coefficients and the covariance matrix isknown, and exact (and Monte Carlo integration) recursive calculation of the posterior probability distribution and themarginal likelihood using the prediction error decomposition is possible.See Doan, Litterman and Sims (1984).NBB WORKING PAPER No. 35 - October 2002 21
Page 3 and 4: AbstractThis paper develops and est
Page 5 and 6: TABLE OF CONTENTS:1. INTRODUCTION..
Page 7 and 8: 1. IntroductionIn this paper we pre
Page 9 and 10: shock leads to a gradual increase i
Page 12 and 13: 2.1.2 Labour supply decisions and t
Page 14 and 15: The first-order conditions result i
Page 16 and 17: jt⎛⎜ p− MCtt )⎜⎝Ptp,t(24)
Page 18 and 19: where β = 1/(1−τ+ r k ) . The c
Page 20: price and wages in the absence of t
Page 23 and 24: whereÊ t denotes the number of peo
Page 25: curve for a normal distribution wit
Page 29 and 30: 3.3.2 Comparison of empirical and m
Page 31 and 32: the effects on marginal cost and in
Page 33 and 34: 4.3 Historical decompositionGraphs
Page 35 and 36: very much in response to the variou
Page 37 and 38: estimate of potential output should
Page 39 and 40: Table 1: Parameter estimatesPrior d
Page 41 and 42: Table 3: Variance decompositionC I
Page 43 and 44: 300020001000persistence productivit
Page 46 and 47: Graph 3: Productivity shock (1)0.50
Page 48 and 49: Graph 5: Wage mark-up shock0.050yc0
Page 50 and 51: Graph 7: Preference shock1.210.8yc0
Page 52 and 53: Graph 9: Equity premium shock0.30.2
Page 54 and 55: Graph 11: Monetary policy shock0-0.
Page 56 and 57: Graph 13: Inflation decomposition -
Page 58 and 59: Graph 16: Labour supply shock: flex
Page 60 and 61: Graph 18: Investment shock: flexibl
Page 62 and 63: Graph 20: Natural Output Gap105Natu
Page 64 and 65: ReferencesAbel A.D. (1990), “Asse
Page 66 and 67: Geweke J. (1999), “Computational
Page 68 and 69: NATIONAL BANK OF BELGIUM - WORKING
Page 70 and 71: 24. "The impact of uncertainty on i

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