Lecture 19 Box-Jenkins Seasonal Models

Transcription

1 Lecture 19 Box-Jenkins Seasonal Models If the time series is nonstationary with respect to its variance, then we can stabilize the variance of the time series by using a pre-differencing transformation. This can be illustrated by following example: Example: Monthly sales for a souvenir shop on the wharf at a beach resort town in Queensland, Australia. Jan 197-Dec Source: Makridakis, Wheelwright and Hyndman (199), Exercise 5., p.3. Table 19.1 Monthly sales Monthly sales Monthly sales Monthly sales Monthly sales Time Series Plot of Monthly sales Quarterly sales Quarters Figure 19.1 Instructor: Ms. Azmat Nafees Time Series Analysis -Nov- 1

2 Figure 19.1 shows that the monthly sales follow a strong trend, and that they have a seasonal pattern with one major peak and several minor peaks during the year. It also appears that the amount of seasonal variation is also increasing with the level of the time series. This suggests that a predifferencing transformation should be used in order to obtain a transformed series that displays constant seasonal variation. Figure 19. gives a plot of the square roots of the monthly sales. This plot suggests that the square root transformation is not strong enough to equalize the seasonal variation. 350 Time Series Plot of Square root of sales 300 Square root of sales Index Figure 19.3 gives a plot of the natural logarithms of the monthly sales. It might be concluded that this transformation has also equalized the seasonal variation. 1 Time Series Plot of natural log of sales 11 natural log of sales Index Instructor: Ms. Azmat Nafees Time Series Analysis -Nov-

3 Each figure shows a nonstationary situation of time series. We now see how to use differencing to transform seasonal nonstationary time series values into stationary time series values. Box-Jenkins modeling indicates that if we are analyzing a seasonal time series, then one of the four differencing transformations illustrated in table 19. will usually produce stationary time series values. Original predifferenced values First regular differenced = First seasonal differenced = L = for quarterly & L = 1 for monthly First regular differenced and first seasonal differenced =( ) ( ).. = =. = = =. = =( ) ( ) =( ) ( ).. =( ) ( ) To determine whether the particular transformation is appropriate, we examine the behavior of the Autocorrelation of the values produced by the transformation at the nonseasonal level and at the seasonal level. The behaviors displayed by the ACF and PACF at the seasonal level are same as defined for nonseasonal time series, that is, a spike exists at a nonseasonal lag k (1,, 3,., L 3) in the ACF if > and in the PACF if >. Similarly, for seasonal time series, we say that a spike exists at a seasonal lag k in the ACF if > and in the PACF if >. We first define the classification of seasonal lags in ACF and PACF. They are: 1) Exact seasonal lags: These lags are defined as L, L, 3L, and L. For instance, if considering monthly time series (L = 1) then exact seasonal lags will be 1,, 3, and. ) Near seasonal lags: These lags are defined as (L ), (L 1), (L + 1), (L + ), (L ), (L 1), (L + 1), (L + ), (3L ), (3L 1), (3L + 1), (3L + ), (L ), (L 1), (L + 1), and (L + ). We say that ACF (or PACF) cuts off after lag k at the seasonal level if there are no spikes at exact seasonal lags or near seasonal lags greater than lag k in this function. In general, it can be shown that if the ACF of the transformed time series values does both of the following: 1) Cuts off fairly quickly or dies down fairly quickly at the nonseasonal level. ) Cuts off fairly quickly or dies down fairly quickly at the seasonal level. Then these transformed values should be considered stationary. Otherwise, these values should be considered nonstationary. If the SAC for seasonal differences indicate that the seasonal differences are stationary, then there is no need to do a regular difference on the stationary seasonal differences. In the above example, we will first follow the transformation rules, i.e. i. z =y, ii. z =y y, iii. z =y y, e.g. z =y y, z =y y, and so on (for L = 1). iv. z =(y y ) (y y ), e.g. z =(y y ) (y y ), z =(y ) (y y ), and so on (for L = 1). y Instructor: Ms. Azmat Nafees Time Series Analysis -Nov- 3

4 Then we plot ACF for each rule to determine stationarity. Figures 19. shows dies down extremely both at nonseasonal and seasonal level (L and L), then the values should be considered nonstationary. In figure 19.5, the values obtained by using transformation z =y y should be considered nonstationary as the ACF shows dies down slowly at nonseasonal and extremely slowly at seasonal level (L and L). Figure 19. ACF of z =y Autocorrelation Function for Monthly sales (with 5% significance limits for the autocorrelations) Autocorrelation Figure 19.5 ACF of z =y y Autocorrelation Function for Zt (with 5% significance limits for the autocorrelations) Autocorrelation Instructor: Ms. Azmat Nafees Time Series Analysis -Nov-

5 Next we consider the ACF and PACF of z =y y in figure 19.. Figure 19.(a) ACF of seasonal differences Autocorrelation Function for Zt-L (with 5% significance limits for the autocorrelations) Autocorrelation Figure 19.(b) PACF of seasonal differences Partial Autocorrelation Function for Zt-L (with 5% significance limits for the partial autocorrelations) Partial Autocorrelation Both of the figures show at the nonseasonal level, the PACF has spikes at lags 1 and 3, and cuts off after lag 3 and the ACF dies down fairly quickly. Next, at the seasonal level figure 19.(a) shows that the ACF has spike at lag equal to or nearly equal to 1 and cuts off after lag 1 (since no spike at lag equal or nearly equal to ). The PACF at seasonal level dies down fairly quickly since the spikes at lags 1 and are of decreasing size. Therefore, we say that the values obtained by using the transformation z =y y should be considered stationary. We further do not need to use regular differencing for transformation to reach stationarity. Instructor: Ms. Azmat Nafees Time Series Analysis -Nov- 5

6 Box-Jenkins Seasonal model 1) The seasonal Moving Average model of order q can be expressed as: = +,,, For example, if the ACF has a spike at lag L and cuts off after lag L and the PACF dies down at seasonal level, then we might tentatively conclude that the time series values are described by the seasonal moving average model of order 1. It can be written as = +, ) The seasonal AutoRegressive model of order p can be expressed as: = + +, +, + +, For example, if the PACF has a spike at lag L and cuts off after lag L and the ACF dies down at seasonal level, then we might tentatively conclude that the time series values are described by the seasonal autoregressive model of order 1. It can be written as = + +, Since the ACF and PACF of stationary seasonal time series values often exhibit behavior at both seasonal and nonseasonal level, we must then follow three-step procedure for tentatively identifying a model describing these values: Step 1: Use the behavior of the ACF and PACF at the nonseasonal level to tentatively identify a nonseasonal model describing the time series values. For example, at the nonseasonal level the PACF has spikes at lags 1, 3, and 5 and cuts off after lag 5 and ACF dies down. The tentatively identified nonseasonal model will be AR(5) given as: = Step : Use the behavior of the ACF and PACF at the seasonal level to tentatively identify a seasonal model describing the time series values. For example, at the seasonal level the ACF has a spike at lag 1 and cuts off after lag 1 and the PACF dies down. The tentatively identified seasonal model will be MA(1) given as: = +, Step 3: Combine the models obtained in step 1 and to arrive at an overall tentatively identified model. For example, combining the tentative models defined in step 1 and, we get the overall tentatively identified model as: = , Instructor: Ms. Azmat Nafees Time Series Analysis -Nov-