Stochastic Dynamic Properties of Linear Econometric Models

Previous work on this topic goes back to Wicksell (1907)¹⁾ who takes into consideration uncorrelated random numbers in order to explain business cycle motions. Later, Yule (1927) and Slutzky (1937) apply an autoregressive and a moving average scheme, respectively. They show that these series have many of the apparent cyclic properties which characterize economic time series.

The covariance matrices (1.18) contain all information about the time paths and the lead-lag relationships of the endogenous variables. But, as is well known (see e.g. König and Wolters (1972a, p.56)), these figures are, in general, difficult to interpret. Moreover, the variances and covariances do not give direct measures of the intensity and of the connection of different cycles respectively.

Until now, in analyzing dynamic properties of econometric models the usual procedure was to treat the estimated regression coefficients as fixed parameters according to the classical theoretical specification of these models. In the sequel an approach which explicitly considers the stochastic nature of the estimated regression coefficients in order to test the Frisch hypothesis is applied. Hence, besides the uncertainty originated by the specification of equations which is considered by regarding the influence of the residuals, the risk of the estimation procedure is now taken into account. Studying the stochastic nature of the estimated regression coefficients gives information about the sensitivity and the stability of cycles with regard to variations in the structural parameters as, in the following, all estimated second-order moments will be taken as fixed. According to this approach, the spectral matrices in (2.8) or (2.14) contain random variables. Hence, one has to derive from these relationships the distributions or first- and second-order moments of the spectra given the distributions or first- and second-order moments of the structural parameters.

Adequate modelling of the exogenous variables is a very important task for the evaluation of econometric models with respect to both the forecasting properties (Menges (1977)) and the judgement of policy measures. For instance, Goldfeld and Blinder (1972) discussed whether policy variables, generally treated as exogenous, should be treated as endogenous assuming the policymaker pursues an active counter-cyclical policy during the period of investigation. The analysis of the operation of historical stabilization policies requires the estimation and inclusion of systematic reaction patterns in the econometric model. Otherwise, results with regard to the quantitative importance of policy measures may be seriously biased. As Goldfeld and Blinder (1972) stress, the proper inclusion of policy reaction functions can be considered to be more important for judging the relative effectiveness of policies than any structural estimation problem.

Frisch states that business cycles can be generated in stable linear dynamic economic models only through the influence of stochastic shocks on the model solution. This hypothesis serves as a basis for analyzing econometric models in this study. The influences of stochastic elements on the dynamic properties of econometric models are treated in a systematic manner — in contrast to many other applications of econometric models for policy questions. The original hypothesis is enlarged in several directions: The residuals follow linear stationary stochastic processes. It is no longer assumed that exogenous variables are fixed, but their deviations from trend are modeled by linear stationary stochastic processes. Approaches are discussed which abandon the stability assumption and allow stochastic variations in the estimated regression coefficients of econometric models.