Time Trends

BIBLIOGRAPHY

A time trend, or time index, is the ordered set of natural numbers, for example, t = (1, 2, 3, 4 …), that measures the time span between observations. The slope of a time-trend line represents the growth of a variable. For example, a time-trend line may be used to illustrate growth in production or industry earnings. To predict or explain economic variables, regression equations often use time trends. There are three main reasons for introducing time trends in regression equations that use time-series data. One reason is that a time trend captures the trajectory of the variable over time, providing forecasts of an economic variable. A second reason is that a time trend captures the effect of relevant variables in the regression equation that change over time and are not directly measurable. For example, in the estimation of production functions, Thomas Cooley and Edward Prescott (1973) use a trend variable as a proxy for technological change. A third advantage is that the time trend may capture specification error in regression equations that stem from functional form choice or variable aggregation.

Both linear and nonlinear time trends may be used in regression equations. The assumption in the linear-trend model is that changes will continue into the future at the same or similar rate. This assumption is particularly restrictive when only more recent observations contribute to explaining the future. A more flexible form is a linear trend under a spline-functional form (nonlinear time trend). The spline function jointly determines the trajectory and the memory of the series by allowing the slope of the time trend to vary across time. For example, the rate of change in the price of gasoline may vary across time. In this case, a time trend in a spline function allows the forecasting model to switch the slope parameters with the current economic regime. Both linear and nonlinear time-trend functions may be used in forecasting economic series such as prices, inventories, productivity, and consumption.

In addition to its uses in forecasting, a time trend serves as a proxy for nonmeasurable variables when explaining economic relationships, and it is commonly used in consumption models as well as in models that explain production, employment, and other factors of production. For example, food consumption is often specified as a function of personal disposable income, the price of food, the price of other goods, and a time trend. The time trend captures changes in consumer preferences.

A time trend also captures omitted information from specification error that stems from functional form choice or variable aggregation in the regression equation. The effectiveness of a time trend as a proxy for the omitted information from specification error depends on its correlation to included and excluded information in the regression equation, as well as the choice of functional form for the time trend in the equation. In a simulation study, Camilo Sarmiento and Richard Just (2005) provide evidence that a time trend is able to capture variation of the aggregation error (a special form of specification error) in aggregate consumption functions more effectively than methods based on conceptually accurate, time-specific approximations.

Clive Granger (2001) indicates the potential and unexplored uses of varying coefficients to approximate functional form in applications that use time series data. A time trend as an interaction variable in the model can be used to introduce time-varying coefficients and, thus, approximate unobserved functional structure in economic models, while reducing dimensionality issues in the specification of the functional form from the m -dimensional space (number of explanatory variables in the model) to the simpler one-dimensional space. The functional form for the time trend as an interactive variable in the regression equation could be linear or nonlinear. Significant empirical work and simulation analysis is needed in this area.

The popularity of a time trend in many economic models stems from its simplicity and intuitive interpretation. Statistical tests may be used to evaluate the effectiveness of time trends in forecasting and regression equations. Effectiveness depends largely on the application. The choice of time trend as a tool in model building involves not only whether to include a time trend and its functional form (spline function), but also whether to include it as an interactive variable.

BIBLIOGRAPHY

Cooley, Thomas F., and Edward C. Prescott. 1973. Systematic (Non-random) Varying Parameter Regression: A Theory and Some Applications. Annals of Economic and Social Measurement 16: 463–474.

Granger, Clive W. J. 2001. Macroeconomics: Past and Future. Journal of Econometrics 100: 17–19.

Sarmiento, Camilo, and Richard E. Just. 2005. Empirical Modelling of the Aggregation Error in the Representative Consumer Model. Applied Economics 37: 1163–1175.

Camilo Sarmiento