Alternative specifications of the error process in the stochastic simulation of econometric models

Abstract

SUMMARY This paper analyses the stochastic simulation of econometric models using three different methods for specifying the probability distribution of the structural error terms. The impact of these different assumptions on the simulation bias and model variance is explored empirically. Monte Carlo variance reduction techniques are used to distinguish the effects of the different specifications. A macroeconometric model is a set of behavioural equations which represent endogenous variables as a function of predetermined variables and error terms. The model may be written in the form f(yt, xt) = ut, where yt is a vector of endogenous variables, xt is a vector of predetermined variables, and ut is a vector of stochastic error terms. The parameters of the structural functions f are estimated using econometric methods. When econometric models are used to predict the future, a point prediction is usually obtained with deterministic solution. The deterministic solution is calculated by solving f(yt, xt) = 0 for yt. This is equivalent to assuming that the stochastic terms ut are equal to zero in the future period. It is well known that, if the model is nonlinear, then E(yt I xt) is not equal to the deterministic solution. For example, see, Fisher and Salmon (1986), where large discrepancies between E(yt I xt) arid the deterministic solution are shown in some cases. Alternatively, predictions may be made using stochastic simulation. In stochastic simulation a distribution is assumed for the error vector ut. Then a pseudo-random vector st is drawn from the specified distribution. The model is then solved; the value of Yt which solves f(yt, xt) = st is found. This process is repeated n times and the resulting vectors are averaged. Since this is a Monte Carlo technique, the economist can increase accuracy by performing more replications (increasing n). The results will depend upon the distribution assumed for the error terms ut, and adopted for generating the pseudo-random error terms St. 1 Estimation errors of the structural form parameters could also be considered in simulation studies. However, it was shown in Bianchi and Calzolari (1980) that the impact of errors in estimated coefficients can be treated with analytical or analytical-numerical methods even for nonlinear models, without need of stochastic simulation. Therefore, in this study we shall not deal with errors in coefficients; coefficients will be treated as given.