Analysis of Financial Time Series, Third Edition

2.9 Regression Models with Time Series Errors

In many applications, the relationship between two time series is of major interest. An obvious example is the market model in finance that relates the excess return of an individual stock to that of a market index. The term structure of interest rates is another example in which the time evolution of the relationship between interest rates with different maturities is investigated. These examples lead naturally to the consideration of a linear regression in the form

2.43

where yt and xt are two time series and et denotes the error term. The least-squares (LS) method is often used to estimate model (2.43). If {et} is a white noise series, then the LS method produces consistent estimates. In practice, however, it is common to see that the error term et is serially correlated. In this case, we have a regression model with time series errors, and the LS estimates of α and β may not be consistent.

A regression model with time series errors is widely applicable in economics and finance, but it is one of the most commonly misused econometric models because the serial dependence in et is often overlooked. It pays to study the model carefully.

We introduce the model by considering the relationship between two U.S. weekly interest rate series:

1. r1t: the 1-year Treasury constant maturity rate

2. r3t: the 3-year Treasury constant maturity rate ...