SEASONAL DIFFERENCING IN THE BOX JENKINS APPROACH I Using the Autocorrelation Function to determine if there is possible seasonality in your data Case 1 Data are Flat Consider the case where the data are flat as in the influenza data that we previously studied when we were examining the exponential smoothing method of forecasting Look at the SAS program Seas Diff Case 1 sas and run it Here we plot the data and note that it is non trending flat and thus all we need to do is examine the autocorrelation function of the raw data to see if there exist significant autocorrelations at the seasonal lags j 12 24 36 and 48 In fact there appear fairly significant autocorrelations at these lags thus hinting that we need to assume that seasonality is playing a significant role in determining the variation in this data In general when you have flat time series data you can simply plot the sample ACF of the data and see if there are spikes in it at and possibly around the seasonal lags of s 2s 3s 4s etc If there are then more than likely the data has seasonality in it and you should consider some form of seasonal differencing to make your data stationary More will be said below on formal tests for whether you should actually seasonally difference your data or not Case 2 Data has Trend In the case that the time series data at hand has a trend in it we should first difference the data to remove the trend and then consider the autocorrelation function for the differenced data for signs of seasonality at the seasonal lags By first differencing the data we mean forming the series yt yt yt 1 yt and then examining the autocorrelation function of yt For example consider the Plano sales tax revenue data that is contained in the SAS program Seas Diff Case 2 sas Since it has trend in it we should first difference the Plano sales tax revenue data and then inspect the autocorrelation function of the first differenced data for significant spikes at the seasonal frequencies of s 2s 3s and 4s to determine the possible presence or absence of seasonality in the original data In fact there are significant autocorrelations at the 1 seasonal lags 12 24 36 and 48 and thus one should consider the possibility of seasonally differencing the data in order to make it stationary II The general notation for the Multiplicative Seasonal Box Jenkins model 1 1 B s 2 B 2 s L P B Ps 1 1 B 2 B 2 L p B p Ds d1 y t 0 1 1 B s 2 B 2 s L Q B Qs 1 1 B 2 B 2 L q B q at This model is denoted as ARIMA p d q x P D Q s III Using the Hasza and Fuller 1982 and Dickey Hasza Fuller 1984 Seasonal Unit Root tests to determine the appropriate differencing of time series data subject to seasonal variation See the SAS program Plano Unit 2 sas for an example of seasonal unit root testing as applied to the Plano Sales Tax Revenue data References Hasza David P and Fuller Wayne A 1982 Testing for Nonstationary Parameter Specifications in Seasonal Time Series Models Annals of Statistics Vol 10 No 4 Dec 1209 1216 In particular see Table 5 1 p 1214 and the portion of the table associated with the test statistic n3 d 4 Dickey D A Hasza D P and Fuller W A 1984 Test for Unit Roots in Seasonal Time Series Journal of the American Statistical Association Vol 79 No 386 June 355 367 In particular see Table 5 p 361 Invariably in the Box Jenkins approach the two most frequently used transformations for converting time series data that contain seasonality to stationarity are 1 first and seasonal span differencing represented by the differencing operation 1 s yt 1 yt yt s yt yt s yt 1 yt s 1 yt yt 1 yt s yt s 1 s 1 yt or 2 simply seasonal span differencing denoted by s y t yt yt s Here s denotes the frequency of the season s 12 for monthly data s 4 for quarterly data and s 2 for bi annual data Notice in the case of the first and seasonal span differencing the order in which the differencing is performed is of no consequence as the 2 differencing operators 1 and s are commutative If s 12 the first transformation is in words the month to month change in the year over year difference in the data or equivalently the year over year difference in the month to month change in the data while the second transformation is just the year over year difference in the data Of course if one chooses to use the logarithmic transformation of the data where say yt loge zt ln zt is the natural logarithmic transformation of the original data zt then the first transformation is stated as being the month to month change in the yearover year percentage change in the data or equivalently the year over year difference in the month to month percentage change in the data III A Hasza Fuller Test of H 0 1 s yt is appropriate transformation versus H1 1 s yt is not the appropriate transformation The Hasza Fuller test equation in the case of s 12 is yt 1 yt 1 2 yt 1 yt 13 3 yt 12 yt 13 1 12 1 yt 1 L p 12 1 yt p at with a null hypothesis that 12 1 yt is the correct transformation for rendering the seasonal yt to be stationary here tested by the parametric restrictions H 0 1 1 2 0 and 3 1 The alternative hypothesis is that 12 1 yt is not the correct transformation of the data Note the augmenting terms of the test equation are those terms associated with the gamma coefficients The number of augmenting terms p is usually chosen to minimize the AIC or SBC goodness of fit criterion of the test equation III B Dickey Hasza Fuller Test of H 0 s yt is the appropriate transformation versus H 1 s yt is not the appropriate transformation The Dickey Hasza Fuller test equation in the case of s 12 is yt yt 12 1 yt 12 1 12 yt 1 L p 12 yt p at 3 with a null hypothesis that 12 yt is the correct transformation for rendering the seasonal yt to be stationary here tested by the parametric restriction H 0 1 0 The alternative hypothesis is that 12 yt is not the correct transformation of the data Note the augmenting terms of the test equation are those terms associated with the gamma coefficients The number of augmenting terms p is usually chosen to minimize the AIC or SBC goodness of fit criterion of the test equation 4