Application procedure

The steps undertaken to perform an indirect inference estimation using a di- rected Wald is as follows:

A global optimisation simulated annealing algorithm is used to search for parameter values within a prede…ned set of upper and lower bounds. The bounds are set using economic data and intuition to prevent the algorithm from searching in areas which would either not …nd an optimal set or produce illogical results.

For every given set of coe¢ cients, the residuals from the model’s equations are calculated.

These residuals are then …tted to equations that most likely represents their time series properties. In the case of stationary data a simple AR(1) is used. In the case of nonstationary data, the residuals are assumed to be either

trend stationary with a drift or are characterised by a unit root.

Using the residuals, the innovation series are obtained, the i.i.d shocks. These series represent the structural shocks of the model.

The innovations are then used to calculate the bootstrap simulations, under the null hypothesis that the model is correct; in this and in the subsequent chapter the number of bootstraps is set to 1000, i.e. this step creates 1000 ’arti…cial’data sets.

The simulated series are then …tted to an auxiliary model. In the case of stationary data, a vector autoregressive (VAR) model is used and in the case of nonstationary data a vector error correction (VEC) model is used. These models are widely used in the testing and estimation of DSGE models since the reduced form of a DSGE model can be represented as a VAR

Collect the coe¢ cients from each simulation (the constant term is not in- cluded) to construct a distribution containing the sampling properties of the coe¢ cients of the auxiliary model

Collect the variances as well. The inclusion of variances in the calculation of the Wald increases the probability of rejection since more criteria is imposed on the model.

sent a function of the parameter estimates (coe¢ cients and variances) from the auxiliary model used on simulated data.

Calculate the Wald statistic using the same auxiliary model and actual data

Compare the Wald from actual data to the distribution of the Wald using simulated data

Check if the Wald in the actual data lies within the 95% con…dence interval.

To achieve that, construct the Transformed Mahalanobis Distance (TMD). This takes the Wald, which is a 2 _{chi-squared statistic, and converts it to a} normal distribution which would allow an easier interpretation of the results, then calculate the t-statistic. The 95th percentile is 1.645

If the value of TMD is less than 1.645 - the test does not reject the null hypothesis, i.e. that the model is a close approximation to the true data generating process. If the answer is no, the model is rejected.

Collect the TMD value and the parameter set that generated the simulated data.

Choose another set of parameter values and repeat these steps from the beginning

The intuition behind this method is as follows. A theoretical model is used to generate time series data, called ‘arti…cial’data. This data is uniquely de…ned by the equations of the model and the chosen parameter set. Actual time series data is also collected. The procedure checks how close the two data sets are. The best estimates of the parameter values are those that create arti…cial data which most resembles the actual data.

This procedure can be used for both purposes - to test and to estimate pa- rameters of the DSGE model. If the procedure is used for testing purposes the steps are performed only once on a given set of parameter values. If it is used to obtain parameter estimates that enable the model to …t the data, the procedure is repeated as many times as the economist decides using a simulated annealing algo- rithm. In the following estimation estimations, the global optimisation algorithm repeats the test up to 100 times. The best estimates of the model parameters are those that generated the lowest TMD. There are several bene…ts from this proce- dure. Using the innovations calculated from the model residuals does not require knowledge of the actual distribution of structural shocks, Minford (2015). Le et al. (2015b) evaluated the small sample properties using Monte Carlo analysis and concluded that indirect inference testing is more powerful than an LR test in small samples. Last but not least, indirect inference uses the same speci…cation of the auxiliary model when applied to both simulated and actual data. Therefore, even

if the auxiliary model is not the best representation of the time series, e.g. VAR(1) is used when VARMA(1,1) is most suitable, it would still generate robust results since it is applied uniformly on the two types of data.

It is important to point out that the approach here is de…ned as directed Wald as opposed to full Wald. In the full Wald the auxiliary model includes all endogenous variables from the DSGE model. This would most certainly lead to a rejection of the model since it is a mere simpli…ed approximation of reality. As a rule of thumb, the more variables and the more lags are included the higher the chance that the model will be rejected. That is why the directed Wald is used. In this case the testing/estimation procedure is focused on a few variables at any given time that are of interest to the researcher. The next section presents the results from testing the model using initial parameter values and stationary data.