Stochastic and deterministic trends in state economies.

Introduction

Identifying the fundamental nature of growth paths in regional economies is essential for effective analyses of past performance, future prospects, and policy effects. Seldom seen among traditional economic activity measures are time paths that meander gently around a constant level. Rather, time paths tend to be directional and punctuated by episodes of departure from what may appear to be an orderly drift, or trend, in the economic variable of interest. When a variable progresses through time in this manner, and many do, it is said to be nonstationary. Two very different types of trend growth, deterministic and stochastic, serve as sources of nonstationary behavior in economic variables. The question of which type of trend growth is most common in state economies is taken up in this paper.

Contemporary directions in macroeconomic analysis have been strongly influenced by concerns over nonstationary behavior in macroeconomic variables. Stochastic trends (unit roots), deterministic trends, cointegration, and error-correction models for economic variables have been firmly established in the vocabulary of macroeconomics. Granger (1986), Hendry (1986), Stock and Watson (1988), and Chinn (1991) provide useful reviews of the above issues and methods. Recognition and appropriate treatment of nonstationary behavior by variables in regional economies has been slower to emerge than in macroeconomics. Yet, the same issues surrounding nonstationary time series that have been dealt with at the national and international levels must be addressed at the regional level.(1)

This paper examines the nonstationary behavior of several variables considered to be fundamental indicators of economic activity at the state level and then assesses linkages to national counterparts. First, the trend characteristics (stochastic vs. deterministic) of the broadest of state activity indicators, gross state product, are examined using the "one-digit" industrial level of detail.(2) Two tests for unit roots (stochastic trends) using opposite prior beliefs about the presence of unit roots will be used with time series data from the 50 states.(3) The null hypotheses of the tests are the presence of a unit root and the presence of a deterministic trend (no unit root), respectively. After establishing the trend characteristics of state level activity, consideration is given to cointegrating relationships (stable long-term linkages) between state industries and corresponding U.S. industries possessing stochastic trends. The results show a general lack of cointegrating relationships between state and U.S. industries. Finally, total gross product and personal income for the states and nation are analyzed. Cointegrating relationships at the economy-wide level between states and the nation are also rare.

Patterns and Detection of Nonstationary Behavior

Establishing the trend characteristics of key economic variables in regions is an important task. At stake is the proper characterization of how the variables grow or decline over time. Stochastic and deterministic trends are the two major, and very different, descriptions of nonstationary behavior. A variable with a deterministic trend can be represented as

[Y.sub.t] = [[Delta].sub.0] + [[Delta].sub.1]t + [e.sub.t]

where the [e.sub.t] represent deviations from trend and are assumed to be a stationary stochastic process with zero mean. Shocks to the variable transmitted through the [e.sub.t] provide transitory impacts upon [Y.sub.t] but they soon dissipate and [Y.sub.t] tends to return to the trend growth (or decline) path. Accordingly, variables having deterministic trends can show strong cyclical patterns over time as the movement around trend proceeds.

In contrast, a variable with a stochastic trend (unit root) behaves according to

[Y.sub.t] - [Y.sub.t-1] = [Alpha] + [e.sub.t],

and each current and past error term is permanently incorporated into the current level of [Y.sub.t]. The growth pattern embodied in a stochastic trend clearly differs from that of a deterministic trend. Shocks to the variable through the [e.sub.t] do not dissipate over time and there is no tendency for the variable to return toward a deterministic trend. Therein lies one of the major growth implications of stochastic trend behavior, the permanency of shocks to the variable. To illustrate the importance of distinguishing between these behaviors, suppose an economic development package was being aimed at a particular industry. If the gross product of the industry had a stochastic trend, then the desired positive shock from the package would be permanently impounded in the industry product and not ultimately diluted by an eventual return to long-term trend. Industries where output exhibits a stochastic, rather than deterministic, trend may be preferred targets to receive promotional shocks. However, variables with stochastic trends can be very erratic and negative shocks provided by uncontrollable forces are permanently incorporated on a continual basis as well.(4)

Identification of stochastic versus deterministic trend behavior is a key element in the specification of econometric models involving regional economic variables. Chinn (1991) has summarized the actions that are to be taken when specifying models involving nonstationary variables. If the variables in a model possess deterministic trends, a time trend should be included as an additional independent variable. If the variables in a model have stochastic trends, one must test for a cointegrating relationship. (see "Cointegration Tests for State and U.S. Industries" below. If such a relationship is present, the model should be specified in an error-correction format. If a cointegrating relationship is not found, the model should be specified in period-to-period changes (first differences) of the variables so that the stochastic trends have been removed.

Regional models that do not follow the above guidelines are likely to be misspecified and offer descriptions of relationships that in reality may only be mirages. Estimation of models that relate variables containing stochastic trends can easily produce the illusion of significant relationships when in fact the variables bear no relationship to one another. The prospect of these spurious relationships was examined by Granger and Newbold (1974). A tour through some of the classic regional models listed and summarized in Bolton (1985) reveals many equations that are specified in levels or log levels of variables. If the variables in level or log level form have stochastic trends, equations containing them are incorrect representations of regional economic relationships.

Two tests are used to discern whether a stochastic trend or deterministic trend characterizes a variable. The Augmented …