"Frontmatter". In: Analysis of Financial Time Series

38 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSto suggest that one approach outperforms the other in a real application. Substantiveinformation of the problem under study and simplicity are two factors that also playan important role in choosing an AR model for a given time series.Parameter EstimationFor a specified AR(p) model in Eq. (2.7), the conditional least squares method,which starts with the (p + 1)th observation, is often used to estimate the parameters.Specifically, conditioning on the first p observations, we haver t = φ 0 + φ 1 r t−1 +···+φ p r t−p + a t ,t = p + 1,...,T,which can be estimated by the least squares method. Denote the estimate of φ i by ˆφ i .The fitted model isˆr t = ˆφ 0 + ˆφ 1 r t−1 +···+ ˆφ p r t−pand the associated residual isâ t = r t −ˆr t .The series {â t } is called the residual series, from which we obtainˆσ 2 a = ∑ Tt=p+1 â 2 tT − 2p − 1 .For illustration, consider an AR(3) model for the monthly simple returns of the valueweightedindex in Table 2.1. The fitted model isr t = 0.0103 + 0.104r t−1 − 0.010r t−2 − 0.120r t−3 +â t , ˆσ a = 0.054.The standard errors of the coefficients are 0.002, 0.034, 0.034, and 0.034, respectively.Except for the lag-2 coefficient, all parameters are statistically significant atthe 1% level.For this example, the AR coefficients of the fitted model are small, indicating thatthe serial dependence of the series is weak, even though it is statistically significantat the 1% level. The significance of ˆφ 0 of the entertained model implies that theexpected mean return of the series is positive. In fact, ˆµ = 0.0103/(1 − 0.104 +0.010 + 0.120) = 0.01, which is small, but has an important long-term implication.It implies that the long-term return of the index can be substantial. Using the multiperiodsimple return defined in Chapter 1, the average annual simple gross return is[ ∏ 864t=1 (1 + R t)] 12/864 − 1 ≈ 0.1053. In other words, the monthly simple returns ofthe CRSP value-weighted index grew about 10.53% per annum from 1926 to 1997,supporting the common belief that equity market performs well in the long term. Aone-dollar investment at the beginning of 1926 would be worth about $1350 at theend of 1997.