"Frontmatter". In: Analysis of Financial Time Series

WHITE NOISE AND LINEAR TIME SERIES 27(a) Simple returnsacf-0.2 -0.1 0.0 0.1 0.20 20 40 60 80 100lag(b) Log returnsacf-0.2 -0.1 0.0 0.1 0.20 20 40 60 80 100lagFigure 2.2. Sample autocorrelation functions of monthly simple and log returns of the valueweightedindex of U.S. Markets from January 1926 to December 1997. In each plot, the twohorizontal lines denote two standard-error limits of the sample ACF.if r t is normally distributed with mean zero and variance σ 2 , the series is called aGaussian white noise. For a white noise series, all the ACFs are zero. In practice,if all sample ACFs are close to zero, then the series is a white noise series. Basedon Figures 2.1 and 2.2, the monthly returns of IBM stock are close to white noise,whereas those of the value-weighted index are not.The behavior of sample autocorrelations of the value-weighted index returns indicatesthat for some asset returns it is necessary to model the serial dependence beforefurther analysis can be made. In what follows, we discuss some simple time seriesmodels that are useful in modeling the dynamic structure of a time series. The conceptspresented are also useful later in modeling volatility of asset returns.Linear Time SeriesA time series r t is said to be linear if it can be written asr t = µ +∞∑ψ i a t−i , (2.4)i=0