Applied Time Series Analysis for the Social Sciences

2Foundations

Chapter 1 hinted that the interpretation of the estimated coefficients in a time series model with a lagged dependent variable is not straightforward. A prerequisite to understanding time series analysis is recognizing the mathematical equivalence of time series models that seem to suggest very different causal mechanisms. This chapter lays the groundwork.

• The first section focuses on the intuition using simple algebra – the method of successive substitutions.
• In the next set of sections, lag operator algebra is introduced as a tool for making these equivalences more apparent and for simplifying proofs. It is no exaggeration to say that lag operator algebra is as useful for time series analysts as matrix algebra is to those studying systems of linear equations.
• The following section provides an example. The derivation presented in the beginning of the chapter using the method of successive substitutions is derived using lag operator algebra.
• The final section provides a brief aside on difference equations and their solutions. Solving difference equations is a special case of identifying mathematically equivalent equations which appear to suggest very different causal processes.

2.1 Multiple Interpretations: Some Intuition

Consider a simple model of approval of the US President:

(2.1)

where is the Gallup poll approval rating of the president, CDI is the average ...