Stochastic Trends in Dynamic Regression Models: An Application to the Employment-Output Equation

Abstract

The question of whether an econometric equation should be formulated in levels or differences is one which has generated a good deal of controversy in recent years; see, for example, Hendry and Mizon (1978). Another important issue concerns the detrending of variables, prior to their inclusion in a regression equation, by regressing each one individually on time. Nelson and Kang (I 984) examine the consequences of such detrending when the appropriate procedure is to estimate the equation in first differences. Since including a time trend in a levels regression is equivalent to prior detrending of the variables, the detrending issue can be regarded as another aspect of the levels versus differences debate. The aim of this paper is to clarify these fundamental issues of dynamic specification. Our approach centres on the introduction of a stochastic trend into a regression model. There are often strong a priori reasons for wanting to include such a term. The application considered here concerns the employment-output relationship, and the role of the stochastic trend in this case is to provide a means of accounting for the underlying productivity trend, part of which derives from technical progress. Technical progress has traditionally been modelled by a deterministic time trend, but as noted by Henry (I979), O'Brien (1983), and Mendis and Muellbauer (1983), equations of this kind failed repeatedly to predict employment satisfactorily after the mid-I970s. One response to these difficulties has been to include ad hoc breaks in the trend and dummy variables; see, for example, HM Treasury (i982). However, we believe that a stochastic trend offers an intuitively more appealing way of modelling variables like productivity and technical progress, and offers a way out of the problems caused by constraining them to be deterministic. From the statistical point of view the key device in modelling a stochastic trend is the state space form. This allows the unknown parameters to be estimated via the prediction error decomposition and for predictiorns to be computed by extending the Kalman filter. In addition, a smoothing algorithm can be used to provide an optimal estimator of the trend at each point throughout the sample. The levels model with a deterministic time trend and the first difference model both emerge as special cases. Section I briefly sets out the economic theory underlying the dynamics of the employment-output equation, while Section II describes the statistical treatment of stochastic trends. The techniques described in Section II are applied to