An Introduction to Time Series Regression Analysis
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 xt on yt holding exogenous
zt constant in
yt = α0
+ α1xt + α2zt + εt
(1)
where the residual εt is white
noise. Consider (1) a linear first order
approximation of the true economic relationship. Either xt or zt can
represent a vector and there may be multiple equations with a vector of
endogenous variables yt. 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 δyt/δxt, or how the independent
xt affects yt holding zt 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 zt. 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.