"Frontmatter". In: Analysis of Financial Time Series

62 LINEAR TIME SERIES ANALYSIS AND ITS APPLICATIONSparticular, the earning grew exponentially during the sample period and had a strongseasonality. Furthermore, the variability of earning increased over time. The cyclicalpattern repeats itself every year so that the periodicity of the series is 4. If monthlydata are considered (e.g., monthly sales of Wal-Mart Stores), then the periodicity is12. Seasonal time series models are also useful in pricing weather-related derivativesand energy futures.Analysis of seasonal time series has a long history. In some applications, seasonalityis of secondary importance and is removed from the data, resulting in aseasonally adjusted time series that is then used to make inference. The procedureto remove seasonality from a time series is referred to as seasonal adjustment. Mosteconomic data published by the U.S. government are seasonally adjusted (e.g., thegrowth rate of domestic gross product and the unemployment rate). In other applicationssuch as forecasting, seasonality is as important as other characteristics ofthe data and must be handled accordingly. Because forecasting is a major objectiveof financial time series analysis, we focus on the latter approach and discuss someeconometric models that are useful in modeling seasonal time series.2.8.1 Seasonal DifferencingFigure 2.9(b) shows the time plot of log earning per share of Johnson and Johnson.We took the log transformation for two reasons. First, it is used to handle theexponential growth of the series. Indeed, the new plot confirms that the growth islinear in the log scale. Second, the transformation is used to stablize the variabilityof the series. Again, the increasing pattern in variability of Figure 2.9(a) disappearsin the new plot. Log transformation is commonly used in analysis of financial andeconomic time series. In this particular instance, all earnings are positive so that noadjustment is needed before taking the transformation. In some cases, one may needto add a positive constant to every data point before taking the transformation.Denote the log earning by x t . The upper left panel of Figure 2.10 shows the sampleACF of x t , which indicates that the quarterly log earning per share has strong serialcorrelations. A conventional method to handle such strong serial correlations is toconsider the first differenced series of x t [i.e., x t = x t − x t−1 = (1 − B)x t ]. Thelower left plot of Figure 2.10 gives the sample ACF of x t . The ACF is strong whenthe lag is a multiple of periodicity 4. This is a well-documented behavior of sampleACF of a seasonal time series. Following the procedure of Box, Jenkins, and Reinsel(1994, Chapter 9), we take another difference of the data—that is, 4 (x t ) = (1 − B 4 )x t = x t − x t−4 = x t − x t−1 − x t−4 + x t−5 .The operation 4 = (1 − B 4 ) is called a seasonal differencing. In general, for aseasonal time series y t with periodicity s, seasonal differencing means s y t = y t − y t−s = (1 − B s )y t .The conventional difference y t = y t −y t−1 = (1−B)y t is referred to as the regulardifferencing. The lower right plot of Figure 2.10 shows the sample ACF of 4 x t ,