"Frontmatter". In: Analysis of Financial Time Series

66 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSpoint forecasts are shown by dots, and the dashed lines show 95% interval forecasts.The forecasts show a strong seasonal pattern and are close to the observed data.When the seasonal pattern of a time series is stable over time (e.g., close to adeterministic function), dummy variables may be used to handle the seasonality. Thisapproach is taken by some analysts. However, deterministic seasonality is a specialcase of the multiplicative seasonal model discussed before. Specifically, if = 1,then model (2.37) contains a deterministic seasonal component. Consequently, thesame forecasts are obtained by using either dummy variables or a multiplicative seasonalmodel when the seasonal pattern is deterministic. Yet use of dummy variablescan lead to inferior forecasts if the seasonal pattern is not deterministic. In practice,we recommend that the exact likelihood method should be used to estimate a multiplicativeseasonal model, especially when the sample size is small or when there isthe possibility of having a deterministic seasonal component.2.9 REGRESSION MODELS WITH TIME SERIES ERRORSIn many applications, the relationship between two time series is of major interest.The Market Model in finance is an example that relates the return of an individualstock to the return of a market index. The term structure of interest rates is anotherexample in which the time evolution of the relationship between interest rates withdifferent maturities is investigated. These examples lead to the consideration of alinear regression in the formr 1t = α + βr 2t + e t , (2.39)where r 1t and r 2t are two time series and e t denotes the error term. The least squares(LS) method is often used to estimate model (2.39). If {e t } is a white noise series,then the LS method produces consistent estimates. In practice, however, it is commonto see that the error term e t is serially correlated. In this case, we have a regressionmodel with time series errors, and the LS estimates of α and β may not be consistent.Regression model with time series errors is widely applicable in economics andfinance, but it is one of the most commonly misused econometric models becausethe serial dependence in e t is often overlooked. It pays to study the model carefully.We introduce the model by considering the relationship between two U.S. weeklyinterest rate series:1. r 1t : The 1-year Treasury constant maturity rate.2. r 3t : The 3-year Treasury constant maturity rate.Both series have 1967 observations from January 5, 1962 to September 10, 1999 andare measured in percentages. The series are obtained from the Federal Reserve Bankof St Louis. Figure 2.12 shows the time plots of the two interest rates with solidline denoting the 1-year rate and dashed line the 3-year rate. Figure 2.13(a) plots r 1tversus r 3t , indicating that, as expected, the two interest rates are highly correlated.