"Frontmatter". In: Analysis of Financial Time Series

REGRESSION MODELS WITH TIME SERIES ERRORS 71Estimating a regression model with time series errors was not easy before theadvent of modern computers. Special methods such as the Cochrane–Orcutt estimatorhave been proposed to handle the serial dependence in the residuals; see Greene(2000, p. 546). By now, the estimation is as easy as that of other time series models.If the time series model used is stationary and invertible, then one can estimate themodel jointly via the maximum likelihood method. This is the approach we take byusing the SCA package. For the U.S. weekly interest rate data, the fitted version ofmodel (2.42) isc 3t = 0.0002 + 0.7824c 1t + e t , e t = a t + 0.2115a t−1 , ˆσ a = 0.0668, (2.43)with R 2 = 85.4%. The standard errors of the parameters are 0.0018, 0.0077, and0.0221, respectively. The model no longer has a significant lag-1 residual ACF, eventhough some minor residual serial correlations remain at lags 4 and 6. The incrementalimprovement of adding additional MA parameters at lags 4 and 6 to the residualequation is small and the result is not reported here.Comparing the models in Eqs. (2.40), (2.41), and (2.43), we make the followingobservations. First, the high R 2 and coefficient 0.924 of model (2.40) are misleadingbecause the residuals of the model show strong serial correlations. Second, for thechange series, R 2 and the coefficient of c 1t of models (2.41) and (2.43) are close. Inthis particular instance, adding the MA(1) model to the change series only provides amarginal improvement. This is not surprising because the estimated MA coefficientis small numerically, even though it is statistically highly significant. Third, the analysisdemonstrates that it is important to check residual serial dependence in linearregression analysis.Because the constant term of Eq. (2.43) is insignificant, the model shows that thetwo weekly interest rate series are related asr 3t = r 3,t−1 + 0.782(r 1t − r 1,t−1 ) + a t + 0.212a t−1 .The interest rates are concurrently and serially correlated.SummaryWe outline a general procedure for analyzing linear regression models with timeseries errors:1. Fit the linear regression model and check serial correlations of the residuals.2. If the residual series is unit-root nonstationary, take the first difference of boththe dependent and explanatory variables. Go to step 1. If the residual seriesappears to be stationary, identify an ARMA model for the residuals and modifythe linear regression model accordingly.3. Perform a joint estimation via the maximum likelihood method and check thefitted model for further improvement.