An Introduction to Time Series Regression Analysis

Henry Thompson

Auburn University

This webpage contains the introduction, section headings, and conclusion of a tutorial on time series regression analysis. Email thomph1@auburn.edu for information on the full text.

This introduction to applied time series econometrics focuses on time series regression to estimate a model with OLS. The stochastic linear model focuses on the effect of x_t on y_t holding exogenous z_t constant in

y_t = α₀ + α₁x_t + α₂z_t + ε_t (1)

where the residual ε_t is white noise. Consider (1) a linear first order approximation of the true economic relationship. Either x_t or z_t can represent a vector and there may be multiple equations with a vector of endogenous variables y_t. Rely on theory to suggest which variables are endogenous and exogenous.

The goal is to interpret theory in terms of the estimated coefficient α₁, the partial derivative δy_t/δx_t, or how the independent x_t affects y_t holding z_t constant. Begin with a theoretical model in general functional form, translate into linear functions, and derive the estimating equation (1) as reduced form equation. Parameters of the theoretical model can then be derived from the estimated parameters in (1). Theory provides suggestions about which variables to use for the control variable z_t. The estimated parameter α₁ is.

The ultimate form of the regression is typically not so simple as (1) since OLS is based on normally distributed variables and time series variables may have trends or structural breaks, or be random walks.

Stationarity is a critical concept in applied time series. The assumption is that the observed process has a long history converging to its steady state. Variables in an OLS regression are assumed to be stochastic, and while stationarity is a weaker condition it has intuitive appeal and can lead to reliable OLS estimation.

SECTIONS

Endogenous and exogenous variables

Predictions and economic models

Model transformation

White noise

Stationarity

Stationarity with a structural break

Difference stationarity

Unit root with a structural break

Error correction model ECM

The lagged transformation model

Detrending

Structural Breaks

Other Models: 2SLS, VAR, Causality, Conditional Mean and Variance

CONCLUSION

Stationary series can enter OLS regressions in applied time series analysis. The data may dictate a regression with variables in different forms including levels, differences, or white noise residuals from previous regressions. If there is no transformation of a variable to make it stationary, estimate a model with the best possible specification and variables as close to stationarity as possible noting the lack of confidence. Report the best possible model with the lowest autocorrelation.

Use the estimated parameters to interpret economic theory working through the algebra of differences, residuals, or lags. Relate empirical results directly to economic theory suggesting ways to improve theory based on empirical results. Economic theory evolves as evidence accumulates.

Advanced techniques deal with optimal lag structures across variables, endogenous influences across processes, simultaneous equations, simultaneous estimation of time varying variance, unit roots in the presence of endogenous structural breaks, and unit roots with lagged instrumental variables. Successful applied time series analysis requires good intuition about economic theory as well as reliable time series analysis.