Testing a DSGE Model of the EU Using Indirect Inference

Abstract

We use the method of indirect inference, using the bootstrap, to test the Smets and Wouters model of the EU against a VAR auxiliary equation describing their data. We find that their model generates excessive variance compared with the data. But their model fits the dynamic facts quite well if the errors have the properties assumed by SW but scaled down. We compare a New Classical version of the model which also performs reasonably if error properties are chosen using New Classical priors (notably excluding shocks to preferences). Both versions have (different) difficulties fitting the data if the actual error properties are used. A model combining rigid and flexible-wage/price sectors, with a weight of around 5% on the rigid sector, does best in fitting the data.

Appendix: The Smets-Wouters model

1.1 Households

The household utility function is

$$ U\left( c_{it},n_{it},\frac{M_{it}}{P_{t}}\right) =\left[ \frac{\left( c_{it}-hc_{it-1}\right) ^{1-\sigma _{c}}}{1-\sigma _{c}}-\frac{ n_{it}{}^{1+\sigma _{n}}\varepsilon _{t}^{n}}{1+\sigma _{n}}+\frac{\left( \frac{M_{it}}{P_{t}}\right) ^{1-\sigma _{m}}\varepsilon _{t}^{m}}{1-\sigma _{m}}\right] \varepsilon _{t}^{B} $$

where c _it ,n _it , and $\frac{M_{it}}{P_{it}}$ denote the consumption, work and real money balances of the i ^th household, the $\varepsilon _{t}^{i}$, (i = B,n,m) are preference shocks, and P _t is the general price level. The term hc _t − 1 is to capture consumption habits, where c _t is aggregate consumption. The real household budget constraint is

$$ \frac{M_{it}}{P_{t}}+p_{t}^{B}\frac{B_{it}}{P_{t}}=\frac{M_{t-1}}{P_{t-1}}+ \frac{B_{i,t-1}}{P_{t-1}}+y_{it}-c_{it}-i_{it} $$

where bonds B _it are one-period securities with a price of $p_{t}^{B}$. Total household income is

$$ y_{it}=w_{it}n_{it}+a_{it}+r_{t}^{k}z_{it}k_{i,t-1}-\Psi \left( z_{it}\right) k_{i,t-1}+d_{it} $$

where w _it is the real wage rate, k _it is the capital stock, $r_{t}^{k}$ is the rate of return to capital, the term $r_{t}^{k}z_{it}k_{i,t-1}-\Psi (z_{it})k_{i,t-1}$ represents income from capital after depreciation, z _it is capacity utilisation and d _it is dividend income.

The resulting Euler equation is

$$ E_{t}\left[ \beta \frac{\lambda _{t+1}}{\lambda _{t}}\frac{R_{t}P_{t}}{ P_{t+1}}\right] =1 $$

where R _t is the gross nominal rate of return on bonds $\big(R_{t}=\frac{1}{ p_{t}^{B}}\big)$ and λ _t is the marginal utility of consumption:

$$ \lambda _{t}=(c_{t}-hc_{t-1})^{-\sigma _{c}}\varepsilon _{t}^{B} $$

The demand for money is

$$ \left( \frac{M_{t}}{P_{t}}\right) ^{-\sigma _{m}}\varepsilon _{t}^{m}=\left( c_{t}-hc_{t-1}\right) ^{-\sigma _{c}}-\frac{1}{R_{t}} $$

Households are assumed to act as price setters in the labour market. Their nominal wages are given by

$$ W_{it}=\left( \frac{P_{t-1}}{P_{t-2}}\right) ^{\gamma }W_{i,t-1} $$

Households set their nominal wages to maximise their inter-temporal objective function subject to their budget constraint and the demand for labour which is given by

$$ n_{it}=\left( \frac{W_{it}}{W_{t}}\right) ^{-\frac{1+\lambda _{w},t}{\lambda _{w,t}}}n_{t} $$

where n _t the aggregate labour demand and W _t , the aggregate nominal wage, are given by

$$ \begin{array}{rcl} n_{t} &=&\left[ \int\nolimits_{0}^{1}(n_{it})^{\frac{1}{1+\lambda _{w,t}}}di \right] ^{1+\lambda _{w,t}} \\ W_{t} &=&\left[ \int\nolimits_{0}^{1}(W_{it})^{-\frac{1}{\lambda _{w,t}}}di \right] ^{-\lambda _{w,t}} \end{array} $$

and

$$ \lambda _{w,t}=\lambda _{w}+\eta _{t}^{w} $$

and $\eta _{t}^{w}$ is an i.i.d. shock.

The result of this maximisation is the following mark-up equation for the re-optimised wage:

$$ \frac{\overset{\symbol{126}}{w}_{t}}{P_{t}}E_{t}\Sigma _{s=0}^{\infty }\beta ^{s}\xi _{w}^{s}\left( \frac{P_{t}/P_{t-1}}{P_{t+s}/P_{t+s-1}}\right) ^{\gamma }\frac{n_{i,t+s}U_{c,t+s}}{1+\lambda _{w,t+s}}=E_{t}\Sigma _{s=0}^{\infty }\beta ^{s}\xi _{w}^{s}n_{i,t+s}U_{n,t+s} $$

where $\overset{\symbol{126}}{w}_{t}$ is the new optimal nominal wage, ξ _w  = 0 if wages are perfectly flexible. The real wage is a mark-up 1 + λ _w,t over the current ratio of the marginal disutility of labour to the marginal utility of an additional unit of consumption. As a result, the aggregate wage satisfies

$$ W_{t}^{-\frac{1}{\lambda _{w}}}=\xi \left[ W_{t-1}\left(\frac{P_{t-1}}{P_{t-2}} \right)^{\gamma }\right] ^{-\frac{1}{\lambda _{w}}}+(1-\xi )\overset{\symbol{126}}{ w}_{t}^{-\frac{1}{\lambda _{w}}} $$

Households, who own firms, choose the capital stock and investment to maximise their inter-temporal subject to their budget constraint and the capital accumulation condition

$$ k_{t}=(1-\delta )k_{t-1}+I\left( \frac{i_{t}\varepsilon _{t}^{i}}{i_{t-1}} \right) i_{t} $$

where $I\big(\frac{i_{t}\varepsilon _{t}^{i}}{i_{t-1}}\big)$ is an adjustment cost function, and $\varepsilon _{t}^{i}$ is an investment shock determined by the autoregression

$$ \varepsilon _{t}^{i}=\rho \varepsilon _{t-1}^{i}+\eta _{t}^{i} $$

The first-order conditions are

$$ \begin{array}{rcl} Q_{t} &=&E_{t}\left[ \beta \frac{\lambda _{t+1}}{\lambda _{t}}\left[ Q_{t+1}(1-\delta )+z_{t+1}r_{t+1}^{k}-\Psi (z_{t+1})\right] \right] \\ 1 &=&Q_{t}I^{\prime }\Bigg( \frac{i_{t}\varepsilon _{t}^{i}}{i_{t-1}}\Bigg) \Bigg( \frac{i_{t}\varepsilon _{t}^{i}}{i_{t-1}}\Bigg) +\beta E_{t}Q_{t+1} \frac{\lambda _{t+1}}{\lambda _{t}}\Bigg( \frac{i_{t+1}\varepsilon _{t+1}^{i} }{i_{t}}\Bigg) \Bigg( \frac{i_{t+1}\varepsilon _{t+1}^{i}}{i_{t}}\Bigg) \Bigg( \frac{i_{t+1}}{i_{t}}\Bigg) \\ r_{t+1}^{k} &=&\Psi ^{\prime }(z_{t}) \end{array} $$

where Q _t is the value of installed capital.

1.2 Firms

It is assumed that there is a single final competitive good and a continuum of monopolistically produced intermediate goods indexed by j, where j is distributed over the unit interval ($j\in \lbrack 0,1]$). The final good is produced by

$$ y_{t}=\left[ \int\nolimits_{0}^{1}y_{t}^{j\frac{1}{1+\lambda _{p,t}}}dj \right] ^{1+\lambda _{p,t}} $$

where $y_{t}^{j}$ is the intermediate good and ν _t is a mark-up generated by

$$ \lambda _{p,t}=\lambda _{p}+\eta _{t}^{p} $$

where $\eta _{t}^{p}$ is an i.i.d. shock. Cost minimisation gives the demand function for intermediate goods as

$$ y_{jt}=\left( \frac{p_{\!jt}}{P_{t}}\right) ^{-\frac{1+\lambda _{p,t}}{\lambda _{p,t}}}y_{t} $$

and the final goods price level which is

$$ P_{t}=\left[ \int\nolimits_{0}^{1}(p_{\!jt})^{-\frac{1}{\lambda _{p,t}}}dj \right] ^{-\lambda _{p,t}} $$

where p _jt are the prices of intermediate goods.

Intermediate goods are produced using the technology

$$ y_{jt}=(z_{t}k_{j,t-1})^{\alpha }N_{\!j,t}^{1-\alpha }\varepsilon _{t}^{a}-\Phi $$

where N _j,t is an index of different types of labour used by firms, Φ is a fixed cost and $\varepsilon _{t}^{a}$ is the productivity shock. Cost minimisation implies that

$$ \frac{W_{t}N_{\!j,t}}{r_{t}^{k}z_{t}k_{j,t-1}}=\frac{1-\alpha }{\alpha } $$

The firm’s marginal cost is

$$ MC_{t}=\frac{1}{\varepsilon _{t}^{a}}W_{t}^{1-\alpha }\left( r_{t}^{k}\right) ^{a}\left[ \alpha ^{-\alpha }(1-\alpha )^{-(1-\alpha )} \right] $$

which is independent of the intermediate good produced. Nominal firm profits are

$$ \pi _{j,t}=(p_{\!jt}-MC_{t})\left( \frac{p_{\!jt}}{P_{t}}\right) ^{-\frac{ 1+\lambda _{p,t}}{\lambda _{p,t}}}y_{t}-MC_{t}\Phi $$

Firms are assumed to be able to re-optimise their price randomly with probability 1 − ξ _p as in the Calvo model. The optimal price $\overset{ \symbol{126}}{p_{t}}$ is obtained from the first-order condition

$$ E_{t}\Sigma _{s=0}^{\infty }\beta ^{s}\xi _{p}^{s}\lambda _{t+s}y_{j,t+s} \left[ \frac{\overset{\symbol{126}}{p_{t}}}{P_{t}}\left( \frac{ P_{t+s-1}/P_{t-1}}{P_{t+s}/P_{t}}\right) ^{\gamma }-(1+\lambda _{p,t+s}) \frac{MC_{t+s}}{P_{t+s}}\right] =0 $$

which shows that the optimal price is a function of future marginal costs, and is a mark-up over them unless λ _p  = 0. The general price index therefore satisfies

$$ P_{t}^{-\frac{1}{\lambda _{p,t}}}=\xi _{p}\left( P_{t-1}\left(\frac{P_{t-1}}{ P_{t-2}}\right)^{\lambda _{p}}\right) ^{-\frac{1}{\lambda _{p,t}}}+(1-\xi _{p}) \overset{\symbol{126}}{p_{t}}^{-\frac{1}{\lambda _{p,t}}} $$

1.3 Market equilibrium

Final goods market equilibrium satisfies the national income constraint

$$ y_{t}=c_{t}+i_{t}+g_{t}+\Psi (z_{t})k_{t-1} $$

1.4 Solution

We solve the model in its log-linearised form with DYNARE. The errors named in the text are given as follows

$$ \label{TableKey copy(4)} \begin{array}{rclrcl} \epsilon _{t}^{B}&=& \mbox{Preference Shock}\qquad\quad{\kern17pt} \eta _{t}^{n}&=& \mbox{Labour Preference Shock} \\ \epsilon _{t}^{i}&=& \mbox{Investment Shock}\qquad\quad{\kern14pt} \eta _{t}^{w}&=& \mbox{Wage Mark-up Shock} \\ \eta _{t}^{Q}& =& \mbox{Equity Shock}\qquad\quad{\kern36.2pt} \epsilon _{t}^{g}&=& \mbox{Government Spending Shock} \\ \epsilon _{t}^{a}&=& \mbox{Productivity Shock}\quad\qquad{\kern12.5pt} \bar{\pi}_{t}&=& \mbox{Inflation Objective Shock} \\ \eta _{t}^{p}&=& \mbox{Price Mark-up Shock}\qquad\quad \eta _{t}^{R}&=& \mbox{Monetary Shock} \end{array} $$

1.5 Log-linearised model

For the empirical analysis by SW the model is log-linearised around its non-stochastic steady-state. Denoting log-deviations about equilibrium by a caret $\symbol{94}$, and noting that variables dated t + 1 are rational expectations, the log-linearised model is

$$ \begin{array}{rcl} {\hat{c}}_{t} &=&\frac{h}{1+h}{\hat{c}}_{t-1}+\frac{1}{1+h}{\hat{c}}_{t+1}- \frac{1-h}{(1+h)\sigma _{c}}\big[ \big( {\hat{R}}_{t}-{\hat{\pi}} _{t+1}\big) +\left( {\hat{\varepsilon}}_{t}^{b}-{\hat{\varepsilon}} _{t+1}^{b}\right) \big] \\ {\hat{\imath}}_{t} &=&\frac{1}{1+\beta }{\hat{\imath}}_{t-1}+\frac{\beta }{ 1+\beta }{\hat{\imath}}_{t+1}+\frac{\varphi }{1+\beta }{\hat{Q}}_{t}+\beta { \hat{\varepsilon}}_{t}^{i}-{\hat{\varepsilon}}_{t}^{i} \\ {\hat{Q}}_{t} &=&-\big( {\hat{R}}_{t}-{\hat{\pi}}_{t+1}\big) +\frac{ 1-\delta }{1-\delta +\overset{\_}{r}^{k}}{\hat{Q}}_{t+1}+\frac{\overset{\_}{r }^{k}}{1-\delta +\overset{\_}{r}^{k}}{\hat{r}}_{t}^{k}+\hat{\eta}_{t}^{Q} \\ {\hat{\pi}}_{t} &=&\frac{\nu }{1+\beta \gamma _{p}}{\hat{\pi}}_{t-1}+\frac{ \beta }{1+\beta \gamma _{p}}{\hat{\pi}}_{t+1}\\ &&+\,\frac{(1-\beta \xi _{p})(1-\xi _{p})}{(1+\beta \gamma _{p})\xi _{p}}\left[ \alpha {\hat{r}} _{t}^{k}+(1-\alpha ){\hat{w}}_{t}-{\hat{\varepsilon}}_{t}^{a}+\eta _{t}^{p} \right] \\ {\hat{w}}_{t} &=&\frac{1}{1+\beta }{\hat{w}}_{t-1}+\frac{\beta }{1+\beta }{ \hat{w}}_{t+1}+\frac{\gamma _{w}}{1+\beta }{\hat{\pi}}_{t-1}-\frac{1+\beta \gamma _{w}}{1+\beta }{\hat{\pi}}_{t}+\frac{\beta }{1+\beta }{\hat{\pi}} _{t+1} \\ &&-\,\frac{(1-\beta \xi _{w})(1-\xi _{w})}{(1+\beta )\big[ 1+\frac{(1+\lambda _{w})\sigma _{n}}{\lambda _{w}}\big] \xi _{w}}\left[ {\hat{w}}_{t}-\sigma _{n}{\hat{N}}_{t}-\frac{\sigma _{c}}{1-h}({\hat{c}}_{t}-h{\hat{c}}_{t-1})- \hat{\varepsilon}_{t}^{n}-\eta _{t}^{w}\right] \\ {\hat{N}}_{t} &=&-{\hat{w}}_{t}+(1+\Psi ){\hat{r}}_{t}^{k}+{\hat{k}}_{t-1} \\ {\hat{y}}_{t} &=&(1-\delta k_{y}-g_{y}){\hat{c}}_{t}+\delta k_{y}{\hat{\imath }}_{t}+g_{y}\hat{\varepsilon}_{t}^{g}=\phi \lbrack {\hat{\varepsilon}} _{t}^{g}+\alpha {\hat{k}}_{t-1}+\alpha \psi {\hat{r}}_{t}^{k}+(1-\alpha ){ \hat{N}}_{t} \\ {\hat{R}}_{t} &=&\rho {\hat{R}}_{t-1}+(1-\rho )\left[ \overset{\_}{\pi } _{t}+r_{\pi }\left( {\hat{\pi}}_{t-1}-\overset{\_}{\pi }_{t}\right) +r_{y}{ \hat{y}}_{t}\right] +r_{\Delta \pi }\left( {\hat{\pi}}_{t}-{\hat{\pi}} _{t-1}\right) \\&&+\,r_{\Delta y}\left( {\hat{y}}_{t}-{\hat{y}}_{t-1}\right) -r_{a}\eta _{t}^{a}-r_{n}\eta _{t}^{n}+\eta _{t}^{R} \end{array} $$

where φ = I ^{″ − 1}, $\beta =\big(1-\delta +\overset{\_}{r} ^{k}\big)^{-1}$, $\psi =\frac{\Psi ^{\prime }(1)}{\Psi ^{\prime \prime }(1)}$, $\overset{\_}{\pi }_{t}$ is the inflation target and the equations include various parameters which are long-run average values. Thus, there are nine endogenous variables and ten independent shocks. Five of the shocks arise from technology and preferences $\big(\varepsilon _{t}^{a},\varepsilon _{t}^{i},\varepsilon _{t}^{b},\varepsilon _{t}^{n},\varepsilon _{t}^{g}\big)$ which are generated by first-order autogressive processes, three are cost-push shocks $\big(\eta _{t}^{w},\eta _{t}^{p},\eta _{t}^{Q}\big)$ which are i.i.d. shocks and two are monetary shocks $\big(\overset{\_}{\pi }_{t}, \eta _{t}^{R}\big)$.