Smets and Wouters model estimated with skewed shocks - empirical study of forecasting properties

Transcription

1 COLLEGIUM OF ECONOMIC ANALYSIS WORKING PAPER SERIES Smets and Wouters model estimated with skewed shocks - empirical study of forecasting properties Grzegorz Koloch SGH KAE Working Papers Series Number: 2016/023 December 2016

2 Smets and Wouters model estimated with skewed shocks - empirical study of forecasting properties Grzegorz Koloch December 19, 2016 Abstract In this paper we estimate a Smets and Wouters (2007) model with shocks following a closed skew normal distribution (csn), which was introduced by Gonzalez- Farias et al. (2004) and which nests a normal distribution as a special case. In the paper we present the identication procedure, discuss priors for model parameters, including skewness-related parameters of shocks, i.e. location, scale and skewness parameters. Using data ranging from 1990Q1 to 2012Q2 we estimate the model and recursively verify its out-of sample forecasting properties for time period 2007Q1-2012Q2, therefore including the recent nancial crisis, within a forecasting horizon from 1 up to 8 quarters ahead. Using a RMSE measure we compare the forecasting performance of the model with skewed shocks with a model estimated using normally distributed shocks. We nd that inclusion of skewness can help forecasting some variables (consumption, investment and hours worked), but, on the other hand, results in deterioration in the others (output, ination, wages and the short rate). Keywords: DSGE, Forecasting, Closed Skew-Normal Distribution. JEL: C51, C13, E32 This research was nanced by the National Science Center grant No. 2012/07/E/HS4/ Warsaw School of Economics, Collegium of Economic Analysis, Institute of Econometrics, Decision Analysis and Support Unit, Al. Niepodleglosci 164, Warsaw, Poland. 1

3 1 Introduction In this paper we estimate a Smets and Wouters (2007) model with shocks following a closed skew normal distribution (csn) introduced in Gonzalez-Farias et al. (2004), which nests a normal distribution as a special case. In the paper we discuss the identication procedure, priors for model parameters, including skewness-related parameters of shocks, i.e. location, scale and skewness parameters. Using data ranging from 1990Q1 to 2012Q2 we estimate the model and recursively verify its out-of sample forecasting properties for time period 2007Q1-2012Q2, therefore including the recent nancial crisis, within a forecasting horizon from 1 up to 8 quarters ahead. Using a RMSE measure we compare the forecasting performance of the model with skewed shocks wit a model estimated using normally distributed shocks. We nd that inclusion of skewness can help forecasting some variables (consumption, investment and hours worked), but, on the other hand, results in deterioration in the others (output, ination, wages and the short rate). The paper is organized in the following way. In section two we outline the estimation procedure. Section three presents the Smets and Wouters (2007) model. Section four presents the dataset, a priori assumptions for model parameters and discusses estimation results for the entire data set, i.e. for time span 1990Q1 2012Q2. Section ve discusses forecasting properties of the model with skewed shocks relative to the one estimated using a standard normal distribution based Kalman lter. 2 The skewed Kalman lter In this section we briey describe the CSN distribution, the state-space setting within which the model (see section 3) is formulated and estimated, as well as the ltering distributions and the likelihood function. 2.1 The CSN distribution Let us denote a density function of a p-dimensional normal distribution with mean 1 µ and covariance matrix Σ by φ p (z; µ, Σ). Let us also denote a cumulative distribution 1 All vectors are column vectors in this paper. 2

4 function of a q-dimensional normal distribution with mean µ and covariance matrix Σ by Φ q (z; µ, Σ). We will now dene the closed skew-normal, possibly singular, distribution by means of the moment generating function (mgf) and then, under nonsingularity of the covariance matrix, by means of the probability density function (pdf). For explanation of this distribution we refer to Genton (2004). Denition 2.1. (csn distribution mgf ) Let µ R p and ϑ R q, p, q 1. Let Σ R p p and R q q, Σ 0, > 0, and let D R q p. We say that random variable z has a (p, q)-dimensional closed skew-normal distribution with parameters µ, Σ, D, ϑ and if moment generating function of z, M z (t), is given by: which henceforth will be denoted by: M z (t) = Φ q(dσt; ϑ, + DΣD T ) Φ q (0; ϑ, + DΣD T e tt µ+ 1 2 tt Σt ) z csn p,q (µ, Σ, D, ϑ, ) If Σ > 0, a csn random variable z obtains a probability density function. Denition 2.2. (csn distribution pdf ) If a random variable z follows a (p, q)- dimensional, p, q 1, closed skew-normal distribution with parameters µ, Σ, D, ϑ and, where µ R p, ϑ R q, Σ R p p, Σ > 0, R q q, > 0 and D R q p, than probability density function of z is given by: p(z) = φ p (z; µ, Σ) Φ q(d(z µ); ϑ, ) Φ q (0; ϑ, + DΣD T ) Density function p(z) denes a (p, q)-dimensional nonsingular closed skew-normal distribution in the sense that a random variable has (p, q)-dimensional nonsingular closed skew-normal distribution with parameters µ, Σ, D, ϑ and if and only if its density function for every z R p equals p(z). Parameters µ, Σ and D have interpretation of location, scale and skewness parameters respectively. Parameters ϑ and are articial, but inclusion of them in the denition of p(z) allows for closure of the csn distribution under conditioning and taking marginals respectively. The q-dimension in Φ q is also articial, but it allows for closure of sums of independent random variables and for taking the joint distribution of independent (not necessarily iid) random variables. For D = 0, the csn distribution reduces to a p-dimensional normal one. Dimension q can be interpreted as a skewness related degree of freedom. 3

5 2.2 State-space setting Let us consider the following state-space model: y t = F x t + Hu t, x t = Ax t 1 + Bξ t, u t N(0, Σ u ), ξ t p ξ (θ) x 0 N(µ x0, Σ x0 ) for t T = {1, 2,..., T }, where x t R p and y t R n denote states and observed variables 2, ξ t R n ξ and u t R nu denote shocks and measurement errors, whereas p ξ (θ) denotes a probability distribution function of martingale dierence shocks ξ t, which depends on a vector of parameters θ. A usual assumption is that p ξ is a multivariate normal distribution, independent across its dimensions. In such a case Kalman lter constitutes an optimal ltration procedure 3, see Simon (2006). If normality assumption is relaxed, Kalman lter remains an optimal linear lter. In this paper, we relax normality assumption and assume that for some values of θ, probability density function p ξ is skewed (asymmetric). In particular, we assume that shocks ξ t follow a closed skew-normal distribution (csn henceforth), which nests the normal distribution as a special case, see Goznalez-Farias et. al. (2006) or Genton (2004). Shocks u t in the measurement equation as well as initial states x 0 are assumed to be normally distributed for simplicity, they could also follow a csn distribution. The csn distribution is chosen because it nests a normal distribution and is analytically trackable. It constitutes a multivariate generalization of a class of one-dimensional skew-normal distributions which, among such generalizations, is closed under most forms of operations imposed on variables in the state-space setting (full rank linear transformations, conditioning, taking joints and marginals of independent variables) 4. It also allows for analytical 2 In general, the term states can refer both to observed and unobserved variables within the statespace setting, but in this paper we make a distinction, so that the term states is reserved to unobserved endogenous variables. Observed variables will also be called observables. 3 In the sense that it minimizes the trace of one-step ahead in-sample forecast errors' covariance matrix. 4 Details are provided in Section 2. 4

6 derivation of ltration and prediction densities as well as of the likelihood function. These features make it technically similar to the normal distribution. To allow for prediction, ltration and estimation, we provide formulae for p(y t Y t 1 ), p(x t Y t 1 ), p(x t Y t ) and for the likelihood function. 2.3 Conditional distributions For t T, let us dene an information set Y t = {y 0, y 1, y 2,..., y t } which consists of observables up to time t. Assume, that the a posteriori distribution of state variables in time t, i.e. distribution of x t 1 conditional on Y t 1, is: x t 1 Y t 1 csn p,qt 1 (µ x,t 1 t 1, Σ x,t 1 t 1, D x,t 1 t 1, ϑ x,t 1 t 1, x,t 1 t 1 ) with parameters µ x,t 1 t 1 R p, R p p Σ x,t 1 t 1 > 0, R q p D x,t 1 t 1 > 0, ϑ x,t 1 t 1 R q 1 and x,t 1 t 1 R q q. If, so, then joint distribution of s t = (x T t 1, ξt t ) T, conditional on Y t 1 is given by: s t Y t 1 csn p+nξ,q t 1 +q ξ (µ s,t t 1, Σ s,t t 1, D s,t t 1, ϑ s,t t 1, s,t t 1 ) where: µ s,t t 1 = (µ T x,t 1 t 1, µt ξ )T, Σ s,t t 1 = Σ x,t 1 t 1 Σ ξ, D s,t t 1 = D x,t 1 t 1 D ξ, ϑ s,t t 1 = (ϑ T x,t 1 t 1, ϑt ξ )T, s,t t 1 = x,t 1 t 1 ξ and, assuming G = [A, B] has a full row rank, the a priori distribution of x t = Ax t 1 + Bξ t = Gs t, conditional on Y t 1 is given by: x t Y t 1 csn p,qt 1 +q ξ (µ x,t t 1, Σ x,t t 1, D x,t t 1, ϑ x,t t 1, x,t t 1 ) where: µ x,t t 1 = Gµ s,t t 1, Σ x,t t 1 = GΣ s,t t 1 G T, D x,t t 1 = D s,t t 1 Σ s,t t 1 G T Σ 1 x,t t 1, ϑ x,t t 1 = ϑ s,t t 1, x,t t 1 = s,t t 1 + D s,t t 1 Σ s,t t 1 D T s,t t 1 D s,t t 1Σ s,t t 1 G T Σ 1 x,t t 1 GΣ s,t t 1D T s,t t 1 Since distribution of measurement errors u t N(0, Σ u ) can be represented as: u t csn nu,n u (0 nu,1, Σ u, 0 nu,n u, 1 nu,1, I nu,n u ) 5

7 and observation equation of the state-space setting, i.e. a likelihood equation, reads y t = F x t + u t, then a posteriori distribution of x t, i.e. distribution of x t conditional on Y t 1 and y t = α, is as follows: x t Y t = (x t y t = α) Y t 1 csn p,qt 1 +q ξ (µ x,t t, Σ x,t t, D x,t t, ϑ x,t t, x,t t ) where: µ x,t t = µ x,t t 1 + K(α F µ x,t t 1 ) Σ x,t t = Σ x,t t 1 KF Σ x,t t 1 D x,t t = D x,t t 1Σ x,t t 1 D x,t t 1Σ x,t t 1 F T (F Σ x,t t 1 F T + Σ T u ) 1 F Σ x,t t 1 Σ 1 x,t t 0 0 ϑ x,t t = ϑ x,t t 1 0 DΣ x,t t 1F T 0 x,t t = x,t t 1 + D x,t t 1 Σ x,t t 1 Dx,t t 1 T 0 (F Σ x,t t 1 F T + Σ u ) 1 F Σ x,t t 1 (α F µ x,t t 1 ) + D x,t t 1Σ x,t t 1 F T 0 (F Σ x,t t 1 F T +Σ u ) 1 0 I nu,n u D x,t t 1Σ x,t t 1 F T 0 T D x,t t Σ x,t t Dx,t t T for K = Σ x,t t 1 F T (F Σ x,t t 1 F T + Σ u ) 1. Notice, that in the formula for D x,t t one requires to perform inversion of Σ x,t t and in the formula for D x,t t 1 one requires to perform inversion of Σ x,t t 1, therefore, for the csn distribution to propagate through the state-space setting, matrices Σ x,t t and Σ x,t t 1 must be of full rank. However, Σ x,t t is updated in every iteration of the ltering routine according to Σ x,t t = Σ x,t t 1 KF Σ x,t t 1, so that it can become singular for some t > 1. Since update of Σ x,t t in the csn lter is, technically, the same as in case of the Kalman lter, this can also be the case under assumption of normality. The dierence, however, is, that in the standard Kalman lter inversion not of Σ x,t t is involved, but of (F Σ x,t t 1 F T + Σ u ) 1, which is kept positive denite throughout the lter's iterations. Possible singularity of Σ x,t t precludes, in general case, propagation of the csn distribution within the state-space. Moreover, since Σ x,t t 1 = GΣ s,t t 1 G T = G(Σ x,t 1 t 1 Σ ξ )G T, singularity of Σ x,t t, even if G is full row rank, can translate into singularity of Σ x,t+1 t. 6

8 Expression D x,t t 1 = D s,t t 1 Σ s,t t 1 G T Σ 1 x,t t 1 results as a solution, with respect to D x,t t 1, of a matrix equation D x,t t 1 Σ x,t t 1 = D s,t t 1 Σ s,t t 1 G T. Therefore, in cases when Σ x,t t 1 is not positive denite, it can be subject to Tikhonov regularization, analogically to the way in which regularization is applied to the system of normal equations in the least squares method (ridge regression) or in which Levenberg-Marquardt correction is applied within a Rapshon-Newton algorithm, hence D x,t t 1 can be replaced by an optimal solution of the regularized system in the sense of Tikhonov. In particular, we use a L 2 regularization to matrix Σ x,t t 1 when calculating D x,t t 1 and to matrix Σ x,t t when calculating D x,t t. Notice, that the regularization does not aect (directly) location and scale parameters of the distribution. Note, that when we add two csn variables we have to add their q-dimensions, so that the q-dimension of x t is the sum of q-dimensions of x t 1 and ξ t, therefore contribution of ξ t to the q-dimension of x t in every period is equal to the size of ξ t (i.e. n ξ ), hence the q-dimension of x t quickly increases with t and so does the q-dimension of y t. As a result, dimension of normal integrals involved within cumulative probability distribution functions in the formula of the likelihood function quickly becomes intracktable. To our best knowledge there is no ecient way of calculating such integrals with an arbitrary correlation structure, see (Genz and Bretz (2009)). Therefore, to make the estimation operational, we work with the following approximation: Φ q (z, µ, Σ) q j=1 Φ 1(z j, µ j, Σ jj ), which eliminates the curse of dimensionality. Accuracy of this approximation depends on the correlation structure implied by the covariance matrix Σ. In particular, if Σ is diagonal, the result is exact. 2.4 Likelihood function To get the likelihood function we need to pin down distribution of y t conditional on Y t 1. Since x t Y t 1 csn p,qt 1 +q ξ (µ x,t t 1, Σ x,t t 1, D x,t t 1, ϑ x,t t 1, s,t t 1 ) 7

9 and u t N(0, Σ u ), assuming that F is full row rank, distribution of F x t conditional on Y t 1 is: F x t Y t 1 csn n,qt 1 +q ξ (µ F x,t t 1, Σ F x,t t 1, D F x,t t 1, ϑ F x,t t 1, F x,t t 1 ) where: µ F x,t t 1 = F µ x,t t 1, Σ F x,t t 1 = F Σ x,t t 1 F T, D F x,t t 1 = D x,t t 1 Σ x,t t 1 F T Σ 1 F x,t t 1, ϑ F x,t t 1 = ϑ x,t t 1, F x,t t 1 = x,t t 1 + D x,t t 1 Σ x,t t 1 D T x,t t 1 D x,t t 1Σ x,t t 1 F T Σ 1 F x,t t 1 F Σ x,t t 1D T x,t t 1 and distribution of y t Y t 1 = F x t + u t Y t 1 is: y t Y t 1 csn n,qt 1 +q ξ (µ yt Y t 1, Σ yt Y t 1, D yt Y t 1, ϑ yt Y t 1, yt Y t 1 ) where: µ yt Y t 1 = µ F x,t t 1 Σ yt Y t 1 = Σ F x,t t 1 + Σ u D yt Y t 1 = D F x,t t 1 Σ F x,t t 1 (Σ F x,t t 1 + Σ u ) 1 ϑ yt Y t 1 = ϑ F x,t t 1 yt Yt 1 = F x,t t 1 + (D F x,t t 1 (I Σ F x,t t 1 (Σ F x,t t 1 + Σ u ) 1 ))Σ F x,t t 1 DF T x,t t 1 therefore: Φ qt 1 +p p(y t Y t 1 ) = φ n (y t ; µ yt Yt 1, Σ yt Yt 1 ) ξ (D yt Yt 1 (y t µ yt Yt 1 ); ϑ yt Yt 1, yt Yt 1 ) Φ qt 1 +p ξ (0; ϑ yt Yt 1, yt Yt 1 + D yt Yt 1 Σ yt Yt 1 Dy T t Y t 1 ) 3 The model In this section we present equations of the model, which is then used for forecasting experiment. Outlined model structure is in line with Smets and Wouters (2007). It presents a medium-scale DSGE economy with price and wage rigidities, partial indexation of not re-ooptimized prices and wages to lagged ination, and with stochastic price and wage mark-ups. The model also involves habit formation in consumption, a distinction between installed capital and capital services, stochastic capital utilization as well as exogenous government spending. Monetary policy is assumed to be governed by a Taylor type of policy rule, in which apart from ination and output 8

10 gap - i.e. a dierence between output and its potential level, also a rst dierence of output gap is involved. Potential output is dened as an hypothetical output level obtained without nominal rigidities and without mark-up wedges in prices and wages. The model involves 7 shocks, which are: a TFP shock (η a ), a risk premium (η b ) an exogenous spending shock (η g ), an investment-specic technology shock (η I ), a monetary policy shock (η r ) and two mark-up shocks - a shock in a price mark-up (η p ) and a shock in a wage mark-up (η w ). The model is estimated using 7 observed variables, which are consumption, investment, output, hours worked, ination rate, real wages and short-term nominal interest rate. 3.1 Aggregate resource constraint The aggregate resource constraint is give by: y t = c y c t + i y i t + z y z t + ɛ g t where y t denotes output, c t consumption, i t - investment, z t level of capital utilization, and ɛ g t exogenous spending, e.g. government spending, for: c y = 1 g y i y denoting a steady-state share of consumption in output, where g y and i y are steadystate ratios of exogenous spending to output and of investment to output, wherein: i y = (γ 1 + δ)k y where γ is a steady-state growth rate, δ denotes depreciation rate, and k y stands for a steady-state ratio of capital to output. Finally, z y = R k k y where R k represents a steady-state value of capital rental rate. Exogenous spending is assumed to follow: ɛ g t = ρ gɛ g t 1 + ηg t + ρ gaηt a 3.2 Dynamics of consumption Consumption dynamics is governed by: c t = c 1 c t 1 + (1 c 1 )E t (c t+1 ) + c 2 (l t E t (l t+1 )) c 3 (r t E t (π t+1 ) + ɛ b t) 9

11 where: c 1 = λ γ 1 + λ γ c 2 = (σ c 1) W H L C σ c (1 + λ γ ) c 3 = 1 λ γ (1 + λ γ )σ c where λ = 0 implies no external habit formation and σ c = 1 implies logarithmic utility. The shock to required return on assets follows: ɛ b t = ρ b ɛ g t 1 + ηb t 3.3 Dynamics of investment Investment dynamics is governed by: i t = i 1 i t + (1 i 1 )E t (i t+1 ) + i 2 q t + ɛ i t where: i 2 = 1 i 1 = 1 + βγ 1 σc 1 (1 + βγ 1 σc )γ 2 φ for φ denoting a steady-state elasticity of capital adjustment cost function and β is household's discount factor and q t denoting a real value of capital stock. The shock to the investment specic technology is assumed to follow: ɛ i t = ρ i ɛ i t 1 + η i t Arbitrage equation for the real value of capital stock q t is give by: q t = q 1 E t (q t+1 ) + (1 q 1 )E t (r k t+1) (r t E t (π t+1 ) + ɛ b t) where: for r k t q 1 = βγ σc (1 δ) = being a rental rate on capital. 1 δ R k + 1 δ 10

12 3.4 Production technology Production technology is give by: y t = φ p (αk s t + (1 α)l t + ɛ a t ) where y t denotes output, which is produced using capital services k s t and labour supply (i.e. hours worked) l t. Parameter α denotes a share of capital in production and parameter φ p denotes one plus the share of xed costs in production. Total factor productivity shock is assumed to follow: ɛ a t = ρ a ɛ a t 1 + η a t It is assumed that capital installed becomes eective with a lag of one quarter, so that capital services used in production equal capital installed in the previous period, denoted by k t 1, plus the rate of capital utilization, denoted by z t : k s t = k t 1 + z t where: with: z t = z 1 r k t z 1 = 1 Ψ Ψ for Ψ [0, 1] denoting a positive function of the elasticity of capital utilization adjustment cost. The law of motion of capital stock is given by: k t = k 1 k t 1 + (1 k t )i t + k 2 ɛ i t where: k 2 = (1 k 1 = 1 δ γ (1 δ) )(1 + βγ 1 σc )γ 2 φ γ 3.5 Prices A stochastic price mark-up (marginal product of labour less real wage) is given by: µ p t = α(ks t l t ) + ɛ a t w t 11

13 where w t denotes a real wage. Prices are assumed to be sticky and partial indexation of not re-optimized prices to the level of lagged ination is assumed. Ination is governed by a New-Keynesian Philips curve of the form: π t = π 1 π t 1 + π 2 E t (π k t+1) π 3 µ p t + ɛp t where: π 3 = π 1 = π 2 = ι p 1 + βγ 1 σc ι p βγ 1 σc 1 + βγ 1 σc ι p 1 (1 βγ 1 σc ξ p )(1 ξ p ) 1 + βγ 1 σc ι p ξ p ((φ p 1)ɛ p + 1) for ι p denoting a degree of indexation of not re-optimized prices to past ination (ι p = 0 denotes no indexation), ξ p denotes a degree of price stickiness (ξ p = 0 denotes no stickiness), ɛ p denotes a parameter of curvature of the Kimball goods market aggregator. Price mark-up shock is assumed to follow: ɛ p t = ρ pɛ p t 1 + ηp t µ pη p t 1 Rental rate of capital is governed by: r k t = (k t l t ) + w t A stochastic wage mark-up (real wage less marginal rate of substitution between working and consuming) is given by: µ w t = w t (σ l l t λ (c t λc t 1 )) where σ L denotes an elasticity of labour supply with respect to the real wage and λ denotes a habit formation parameter for consumption. Nominal wages, like prices, are also assumed to be sticky and not re-optimized wages are indexed to past ination. Real wages are governed by: w t = w 1 w t 1 + (1 w 1 )(E t (w t+1 + E t (π t+1 ) w 2 π t + w 3 π t 1 w 4 µ w t + ɛ w t where: w 1 = βγ 1 σc w 2 = 1 + βγ1 σc ι w 1 + βγ 1 σc 12

14 w 4 = w 3 = ι w 1 + βγ 1 σc 1 (1 βγ 1 σc ξ w )(1 ξ w ) 1 + βγ 1 σc ξ w ((φ w 1)ɛ w + 1) for ι w denoting a degree of indexation of not re-optimized wages to past ination (ι w = 0 denotes no indexation), ξ w denotes a degree of price stickiness (ξ w = 0 denotes no stickiness), ɛ w denotes a parameter of curvature of the Kimball labour market aggregator. Price mark-up shock is assumed to follow: ɛ w t = ρ w ɛ w t 1 + η w t µ w η w t 1 Finally, monetary policy equation is give by: r t = ρr t 1 (r π π t + r y (y t y p t )) + r y((y t y p t ) (y t 1 y p t 1 )) + ɛr t where y p t denotes a potential product, which is dened as a product that would prevail under no price and wage rigidity and without mark-up shocks (to prices and wages). Monetary policy shock is assumed to follow: ɛ r t = ρ r ɛ r t 1 + η r t 4 Estimation In this section we present estimates of the Smets and Wouters (2007) model, rst under assumption of normally distributed shocks and then under assumption that shocks follow a skewed distribution. We start by presenting how priors for the estimation were assumed. As far as structural parameters and shocks' persistence parameters are concerned, in both cases prior assumptions are the same and follow Smets and Wouters (2007). As for shocks scale parameters, i.e. the σ's, in the rst case these are shocks' standard deviations whereas in the second case these are shocks scale parameters, which, along with their skewness parameters imply values of shocks' standard deviations. Therefore, prior assumptions regarding σ's are dierent in the two discussed cases. Moreover, in the second case shocks' skewness parameters emerge, i.e. the d's, which were not present in the rst case. On the other hand, the rst case can be thought of as a special case of the second one, just with shocks' skewness parameters set at zero. In such a situation shocks' scale parameters become their standard deviations. 13

15 4.1 Estimation with normally distributed shocks A priori distributions of structural parameters were assumed as shown in Table (1). A priori distributions of disturbances' persistence parameters are provided in Table (2). A priori distributions of shocks' volatilities (standard deviations) are provided in Table (3). 4.2 Estimation with skewed shocks In what follows, we assume that shocks η a, η b, η g, η I, η r, η p and η w follow a closed skew-normal distribution: (η A,t, η b,t, η g,t, η I,t, η r,t, η p,t, η w,t ) csn 7,7 (µ η, Σ η, D η, ϑ η, η ) where: µ η = (µ a, µ b, µ g, µ I, µ r, µ p, µ w ) Σ η = diag (σ a, σ b, σ g, σ I, σ r, σ p, σ w ) D η = diag (d a, d b, d g, d I, d r, d p, d w ) ϑ η = (0, 0, 0, 0, 0, 0, 0) η = diag (1, 1, 1, 1, 1, 1, 1) where µ a, µ b, µ g, µ I, µ r, µ p, µ w denote location parameters of shocks, σ a, σ b, σ g, σ I, σ r, σ p, σ w denote scale parameters of shocks (which are not their variances), and nally d a, d b, d g, d I, d r, d p, d w denote skewness parameters of shocks. Since we want shocks to have a mean value equal to zero, value of µ η = (µ a, µ b, µ g, µ I, µ r, µ p, µ w ) T is set at: where: 2 Σd µ η = π 1 + d T Σd Σ = diag(σ 2 a, σ 2 b, σ2 g, σ 2 I, σ 2 r, σ 2 p, σ 2 w) and: d = (d a, d b, d g, d I, d r, d p, d w ) T Note that for d a = d b = d g = d I = d r = d p = d w = 0 the normal distribution case is also contained in the above specication. 14

16 parameter distribution mean std. dev. ψ Normal σ c Normal h Beta ξ w Beta σ L Normal ξ p Beta ι w Beta ι p Beta Ψ Beta Φ Normal r π Normal ρ Beta r y Normal r y Normal π Gamma (β 1 1) Gamma L Normal γ Normal α Normal Table 1: Priors for estimation of structural parameters. 15

17 parameter distribution mean std. dev. ρ a Beta ρ b Beta ρ g Beta ρ I Beta ρ r Beta ρ p Beta ρ w Beta µ p Beta µ w Beta Table 2: Priors for estimation of disturbances' persistence parameters. parameter distribution mean std. dev. σ a Inv. Gamma σ b Inv. Gamma σ g Inv. Gamma σ I Inv. Gamma σ r Inv. Gamma σ p Inv. Gamma σ w Inv. Gamma Table 3: Priors for estimation of shocks' volatilities. 16

18 parameter distribution mean std. dev. σa Inv. Gamma σb Inv. Gamma σg Inv. Gamma σi Inv. Gamma σr Inv. Gamma σp Inv. Gamma σw Inv. Gamma Table 4: Priors for estimation of shocks' volatilities. To set priors for scale and skewness parameters (location parameters is implied 2 Σd via a relation µ η = π ), we rst run estimation with normal shocks 1+d T Σd using a standard Kalman lter based maximum likelihood estimation procedure and then t ltered (identied) shocks with univariate closed skew-normal distributions. Estimates of scale and skewness parameters of these univariate distributions provide modes for a priori distributions for skew-normal lter based estimation. A priori distributions of shocks' scale and skewness parameters are provided in Table (4) and Table (5) respectively. In Table (4) square roots of scale parameters were presented to make them at the level comparable with levels of standard deviations. For skewness parameters relatively at priors were assumed. This was the case, because we wanted the lter to identify skewness parameters without restrictions imposed on them a priori. Posterior modes for structural parameters, shocks' persistence parameters, shocks' standard deviations (normal distribution case) and scale parameters (skewed case), and of shocks' skewness parameters are presented in Tables (6) - (9) respectively. Letter 'N' denotes, that results refer to the normal distribution case, whereas letter 'S' refers to the skewed case. In case of skewness parameters (Table (9)) no abbrevations were used. 17

19 parameter mode-n std-n mode-s std-s ψ σ c h ξ w σ L ξ p ι w ι p Ψ Φ r π ρ r y r y π (β 1 1) L γ α Table 5: Modes of estimation of structural parameters (N-normal shocks) and scale parameters (S-skewed shocks). 18

20 parameter distribution mean std. dev. d a Normal d b Normal d g Normal d I Normal d r Normal d p Normal d w Normal Table 6: Priors for estimation of shocks' skewness parameters. parameter mode-n std-n mode-s std-s ρ a ρ b ρ g ρ I ρ r ρ p ρ w µ p µ w Table 7: Modes of estimation of disturbances' persistence parameters (N-normal shocks) and scale parameters (S-skewed shocks). 19

21 parameter mode-n std-n mode-s std-s σ a σ b σ g σ I σ r σ p σ w Table 8: Modes of estimation of shocks' volatilities (N-normal shocks) and scale parameters (S-skewed shocks). parameter mode std d a d b d g d I d r d p d w Table 9: Modes of estimation of shocks' skewness parameters 20

22 4.3 Forecasting experiment The forecasting experiment was conducted for the time period starting in 2007Q1 and ending in 2012Q2, so that the recent nancial crisis is involved in the sample. Forecasting properties of the Smets and Wouters (2007) model are well known, but in the original paper the experiment was conducted up till 2004Q4, so that our results extend the sample by almost 8 years. Importantly, the experiment in the original paper does not involve the nancial crisis. We start by presenting graphs of out-of-sample forecasts obtained using a model with normally distributed shocks, which are presented on Figures (1) - (7), and of forecasts obtained using a model with skewed shocks, which are presented on Figures (8) - (14). Figure 1: Recursive forecasts of output - model with normal shocks. Table (10) presents Root Mean Square Errors (RMSE) for observed variables of forecasts obtained from the model with normally distributed shocks. The following Table (11) presents analogical results of forecasts obtained from the model with skewed shocks. To make the comparison easier two following tabes, i.e. Table (12) and Table (13), present dierence in respective RMSE, i.e. RMSE of forecasts with normal shocks less RMSE of forecasts with of skewed shocks, and percentage gain of 21

23 Figure 2: Recursive forecasts of consumption - model with normal shocks. Figure 3: Recursive forecasts of investment - model with normal shocks. 22

24 Figure 4: Recursive forecasts of hours worked - model with normal shocks. Figure 5: Recursive forecasts of ination - model with normal shocks. 23

25 Figure 6: Recursive forecasts of real wages - model with normal shocks. Figure 7: Recursive forecasts of short rate - model with normal shocks. 24

26 Figure 8: Recursive forecasts of output - model with skewed shocks. Figure 9: Recursive forecasts of consumption - model with skewed shocks. 25

27 Figure 10: Recursive forecasts of investment - model with skewed shocks. Figure 11: Recursive forecasts of hours worked - model with skewed shocks. 26

28 Figure 12: Recursive forecasts of ination - model with skewed shocks. Figure 13: Recursive forecasts of real wages - model with skewed shocks. 27

29 Figure 14: Recursive forecasts of short rate - model with skewed shocks. the model with skewed shocks relative to the one with normally distributed shocks. From the tables it can be seen, that inclusion of skewness resulted in improvement of the forecasting performance of some variables - of consumption, investment and hours worked, but, on the other hand, reduced accuracy in case of the others - output, ination, wages and the short rate. In other words, we do not nd a uniform improvement over all the variables, but we nd and improvement over some of them. Largest improvement is observed in the series of hours worked and it concerns, quite uniformly, all the forecasting horizons (increase in accuracy of 15% 19%). In case of consumption, improvement is more pronounced in the rst three quarters (14% 15%) and then falls to the level of 9% 10%. As far as investment is concerned, the improvement takes a U-shaped form, since in the rst quarter it reaches 5% and then, over quarters from Q2 to Q5 falls to 1% 3% and nally reaches higher levels again (5% 8%). Deterioration of the forecasting performance is most pronounced in case of the short rate ( 25% - 14%) and, apart from the rst quarter, is quite uniform at the level of 25% - 22%. For wages and ination, the level of deterioration takes an inverted U-shape (apart from an outlier in Q3 for wages), reaching a through at the level of 20% in Q4 in case of ination and the 28

30 Forecast horizon 1Q 2Q 3Q 4Q 5Q 6Q 7Q 8Q Output Consumption Investment Hours worked Ination Wages Short rate Table 10: RMSE for forecasts of observed variables for normally distributed shocks. Forecast horizon 1Q 2Q 3Q 4Q 5Q 6Q 7Q 8Q Output Consumption Investment Hours worked Ination Wages Short rate Table 11: RMSE for forecasts of observed variables for skewed shocks. level of 7% in Q2 in case of wages. For wages in Q3 and in Q7 - Q8 we observe an improvement in fact. For output, level of forecasting performance deterioration increases with the forecasting horizon, starting from 4% in the rst quarter and reaching 25% in the last one. 29

31 Forecast horizon 1Q 2Q 3Q 4Q 5Q 6Q 7Q 8Q Output Consumption Investment Hours worked Ination Wages Short rate Table 12: Dierence in RMSE for forecasts of observed variables obtained using a model with normally distributed shocks and using a model with skewed shocks. Forecast horizon 1Q 2Q 3Q 4Q 5Q 6Q 7Q 8Q Output Consumption Investment Hours worked Ination Wages Short rate Table 13: Relative improvement of RMSE of forecasts of observed variables obtained when using a model with skewed shocks (positive value represents an improvement in relative terms). 30

32 After having investigated the RMSE statistics we move on to log predictive scores of the forecasts. The state-space form of the reduced, estimated Smets and Wouters (2007) model, takes the following form: y t = F x t x t = Ax t 1 + Bξ t for t T = {1, 2,..., T }, where x t R p and y t R n denote states and observed variables respectively and ξ t follows a 7-dimensional either normal or csn distribution. Let p N (ξ t Y t 1 ) denote a normal distribution of ξ t with parameters estimated using observed variables up to period t 1 and, analogically, let p S (ξ t Y t 1 ) denote a csn distribution of ξ t with parameters estimated up to period t 1. Let also A t 1 and B t 1 denote model matrices with parameters estimated using observed variables up to period t 1 and let p x (x t Y t 1 ) and p y (x t Y t 1 ) denote distributions of states and observables, as estimated using data up to period t 1. In order not to abuse notation in what follows, let empirical observations be denoted in this section by v t, t T. Let τ 1 denote a rst period of the rst forecasting horizon, i.e. 2007Q1 and let τ 2 denote a rst period of the last forecasting horizon, i.e. 2003Q1. Using a state-space model form, we evaluate p x (x Y t 1 ), x = x t, x t+1,..., x t+h 1, and p y (y t Y t 1 ), y = y t, y t+1,..., y t+h 1, for t = τ 1, τ 1 + 1,..., τ 2, where h = 8 denotes the forecasting horizon, in two cases - when ξ t follows either distribution p N or distribution p S. This is achieved by means of a Monte Carlo simulation of draws from distribution p S (ξ t Y t 1 ) and from distribution p N (ξ t Y t 1 ), respectively, for t = τ 1, τ 1 + 1,..., τ 2, τ 2 + 1,..., τ 2 + h 1, with 10 4 draws for each t. Result of a Monte Carlo draw of each ξ t is then propagated according to x t = A t x t 1 + B t ξ t, for t, t + 1,..., t + h 1, which in turn is plugged into y t = F x t, for t, t + 1,..., t + h 1. This results in Monte Carlo estimation of distributions p x (x t Y t 1 ) and p y (y t Y t 1 ) for t = τ 1, τ 1 + 1,..., τ 2, τ 2 + 1,..., τ 2 + h 1. Using simulated distribution p y (y t Y t 1 ) we evaluate a version of a log-predictive score statistic of forecasts using data v t for t = τ 1, τ 1 + 1,..., τ 2, τ 2 + 1,..., τ 2 + h 1, for each variable i = 1, 2,..., 7, i.e. for each dimension of y t, and for each forecast 31

33 horizon j = 1, 2,..., 8, according to: lps i,j = τ 2 t=τ 1 ln( (v t+j 1, i) p y (v t+j 1 Y t 1 )d(v t+j 1, i)) where (v t, i) denotes observed variable v t without the i-th dimension. Table (14) and Table (15) presents the results. What can be observed is, that in terms of the score, normal distribution consequently tends to dominate the csn, even for variables and horizons, for which csn consequently produces better RMSE statistics. The latter one achieves better score only for terminal horizons in case of output, consumption, investment and hours worked. 5 Concluding remarks In this paper we conducted a forecasting experiment which consists in estimating a medium scale DSGE economy, allowing the structural shocks to follow a skewed distribution. In particular, since we work within a state-space setting (a rst order approximation around a steady-state), we used a closed skew-normal distribution, which is closed under most state-space setting transformations. Using data ranging from 1990Q1 to 2012Q2 we estimate the model and recursively verify its out-of sample forecasting properties for time period 2007Q1-2012Q2. Results are mixed in the sense, that inclusion of skewness via a CSN distribution can help forecasting some variables (consumption, investment and hours worked), but, on the other hand, results in deterioration in the others (output, ination, wages and the short rate). Moreover, the log predictive score of a model with normally distributed shocks tends to uniformly dominate the one with skewed shocks. References [Genz and Bretz (2009)] Genz, A., Bretz, F. (2009): Computation of Multivariate Normal and t Probabilities., Springer. [Genton (2004)] Genton, M. G. (ed.): Skew-elliptical distributions and their applications. A journel beyond normality., A CRC Press Company (2004). 32

34 [González-Farías et al. (2004)] González-Farías, G., Domínguez-Molina, J., Gupta, A. (2004): Additive properties of skew normal random vectors., Journal of Statistical Planning and Inference, Vol. 126, pp [Smets and Wouters (2007)] Smets, F., Wouters, R.: Shocks and frictions in US business cycles. A Bayesian approach., ECB Working Paper Series No 722 (2007). 33