"Frontmatter". In: Analysis of Financial Time Series

SIMPLE AUTOREGRESSIVE MODELS 39Model CheckingA fitted model must be examined carefully to check for possible model inadequacy.If the model is adequate, then the residual series should behave as a white noise.The ACF and the Ljung–Box statistics in Eq. (2.3) of the residuals can be used tocheck the closeness of â t to a white noise. For an AR(p) model, the Ljung–Boxstatistic Q(m) follows asymptotically a chi-squared distribution with m − p degreesof freedom. Here the number of degrees of freedom is modified to signify that pAR coefficients are estimated. If a fitted model is found to be inadequate, it must berefined.Consider the residual series of the fitted AR(3) model for the monthly valueweightedsimple returns. We have Q(10) = 15.8 with p value 0.027 based onits asymptotic chi-squared distribution with 7 degrees of freedom. Thus, the nullhypothesis of no residual serial correlation in the first 10 lags is rejected at the 5%level, but not at the 1% level. If the model is refined to an AR(5) model, then we haver t = 0.0092 + 0.107r t−1 − 0.001r t−2 − 0.123r t−3 + 0.028r t−4 + 0.069r t−5 +â t ,with ˆσ a = 0.054. The AR coefficients at lags 1, 3, and 5 are significant at the 5%level. The Ljung–Box statistics give Q(10) = 11.2 with p value 0.048. This modelshows some improvements and appears to be marginally adequate at the 5% significancelevel. The mean of r t based on the refined model is also very close to 0.01,showing that the two models have similar long-term implications.2.4.3 ForecastingForecasting is an important application of time series analysis. For the AR(p) modelin Eq. (2.7), suppose that we are at the time index h and are interested in forecastingr h+l ,wherel ≥ 1. The time index h is called the forecast origin and the positiveinteger l is the forecast horizon.Letˆr h (l) be the forecast of r h+l using the minimumsquared error loss function. In other words, the forecast ˆr k (l) is chosen such thatE[r h+l −ˆr h (l)] 2 ≤ min gE(r h+l − g) 2 ,where g is a function of the information available at time h (inclusive). We referredto ˆr h (l) as the l-step ahead forecast of r t at the forecast origin h.1-Step Ahead ForecastFrom the AR(p) model, we haver h+1 = φ 0 + φ 1 r h +···+φ p r h+1−p + a h+1 .Under the minimum squared error loss function, the point forecast of r h+1 given themodel and observations up to time h is the conditional expectation