A plot of series g after taking the natural log first

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Chapter 13 / Exercise RQ13-13
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A plot of Series G after taking the natural log, firstdifferencing, and seasonal differencing is shown below.
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Chapter 13 / Exercise RQ13-13
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6.4.4.10. Box-Jenkins Analysis on Seasonal Data[6/27/2012 2:36:41 PM]The number of seasonal terms is rarely more than one. If youknow the shape of your forecast function, or you wish toassign a particular shape to the forecast function, you canselect the appropriate number of terms for seasonal AR orseasonal MA models.The book by Box and Jenkins, Time Series AnalysisForecasting and Control(the later edition is Box, Jenkinsand Reinsel, 1994) has a discussion on these forecastfunctions on pages 326 - 328. Again, if you have only a faintnotion, but you do know that there was a trend upwardsbefore differencing, pick a seasonal MA term and see whatcomes out in the diagnostics.An ACF plot of the seasonal and first differenced natural logof series G is shown below.
6.4.4.10. Box-Jenkins Analysis on Seasonal Data[6/27/2012 2:36:41 PM]The plot has a few spikes, but most autocorrelations are nearzero, indicating that a seasonal MA(1) model is appropriate.Model FittingWe fit an MA(1) model to the data.The model fitting results are shown below.SeasonalEstimate MA(1) MA(1)-------- ------- -------Parameter -0.4018 -0.5569Standard Error 0.0896 0.0731Residual standard deviation = 0.0367 Log likelihood = 244.7AIC = -483.4Test the randomness of the residuals up to 30 lags using theBox-Ljung test. Recall that the degrees of freedom for thecritical region must be adjusted to account for two estimatedparameters.H0: The residuals are random.Ha: The residuals are not random. Test statistic: Q= 29.4935Significance level: α= 0.05Degrees of freedom: h= 30 - 2 = 28Critical value: Χ21-α,h= 41.3371 Critical region: Reject H0if Q> 41.3371Since the null hypothesis of the Box-Ljung test is notrejected we conclude that the fitted model is adequate.ForecastingUsing our seasonal MA(1) model, we forcast values 12
6.4.4.10. Box-Jenkins Analysis on Seasonal Data[6/27/2012 2:36:41 PM]periods into the future and compute 90 % confidence limits.Lower UpperPeriod Limit Forecast Limit------ -------- -------- --------145 424.0234 450.7261 478.4649146 396.7861 426.0042 456.7577147 442.5731 479.3298 518.4399148 451.3902 492.7365 537.1454149 463.3034 509.3982 559.3245150 527.3754 583.7383 645.2544151 601.9371 670.4625 745.7830152 595.7602 667.5274 746.9323153 495.7137 558.5657 628.5389154 439.1900 497.5430 562.8899155 377.7598 430.1618 489.1730156 417.3149 477.5643 545.7760
6.4.5. Multivariate Time Series Models[6/27/2012 2:36:43 PM]6. Process or Product Monitoring and Control6.4. Introduction to Time Series Analysis6.4.5. Multivariate Time Series ModelsIf each timeseriesobservationis a vectorof numbers,you canmodel themusing amultivariateform of theBox-JenkinsmodelThe multivariate form of the Box-Jenkins univariate modelsis sometimes called the ARMAV model, for AutoRegressiveMoving Average Vector or simply vector ARMA process.

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