"Frontmatter". In: Analysis of Financial Time Series

42 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSvw-0.2 -0.1 0.0 0.1 0.2••••••••••••o o o o•852 854 856 858 860 862 864monthoo•Figure 2.5. Plot of 1- to 6-step ahead out-of-sample forecasts for the monthly log returns ofthe CRSP value-weighted index. The forecast origin is t = 858. The forecasts are denotedby “o” and the actual observations by a dot. The two dashed lines denote two standard-errorlimits of the forecasts.2.5 SIMPLE MOVING-AVERAGE MODELSWe now turn to another class of simple models that are also useful in modeling returnseries in finance. These models are called moving-average (MA) models. There areseveral ways to introduce MA models. One approach is to treat the model as a simpleextension of white noise series. Another approach is to treat the model as an infiniteorderAR model with some parameter constraints. We adopt the second approach. Asis shown in Chapter 5, the bid-ask bounce in stock trading may introduce an MA(1)structure in a return series.There is no particular reason, but simplicity, to assume a priori that the order ofan AR model is finite. We may entertain, at least in theory, an AR model with infiniteorder asr t = φ 0 + φ 1 r t−1 + φ 2 r t−2 +···+a t .However, such an AR model is not realistic because it has infinite many parameters.One way to make the model practical is to assume that the coefficients φ i s satisfysome constraints so that they are determined by a finite number of parameters. Aspecial case of this idea isr t = φ 0 − θ 1 r t−1 − θ 2 1 r t−2 − θ 3 1 r t−3 −···+a t , (2.14)