Quiz 9: The Box-Jenkins Arima Methodology

Business

a. Compute the 95% confidence interval for the autocorrelation coefficient at any lag. Consider the confidence level is 95%, and img . The formula for confidence interval is given below: img Here, the confidence level is 0.95. For two tailed test, img From Table B-2, the required img value for 95% confidence level is 1.96. Thus, img . Substitute 1.96 for img , 100 for n in the above confidence interval formula. img Thus, the 95% confidence interval for the autocorrelation coefficient at any lag is img . b. Conclusion: Here, the series is random, because there is no particular pattern occurs in the plot and also all the autocorrelation coefficients lie within 95% confidence intervals. c. Conclusion: The series may possibly stationary autoregressive process or non-stationary autoregressive process, because the inference depends on how rapid the autocorrelations decline to zero for the given time period. d. Conclusion: Here, the series is seasonal with the period is 4, because the autocorrelations img , img , and img are significantly different from zero and also the process is quarterly.

Compute the forecasts for periods 5, 6, 7. From the given information, the first four observations are img , img , img and img . Assume that img and img . The model is given below: img The forecast value for periods 1, 2, 3, and 4 are calculated as follows: img The forecast value for period 5 is, img Thus, the forecast value for period 5 is img . The forecast value for period 6 is, img Thus, the forecast value for period 6 is img . The forecast value for period 7 is, img Thus, the forecast value for period 7 is img .

a. Compute forecasts for periods 61, 62, and 63 from origin 60. Assume that img and img . The time series model is given below: img The forecast value for period 61 is, img Thus, the forecast value for period 61 is img . The forecast value for period 62 is, img Thus, the forecast value for period 62 is img . The forecast value for period 63 is, img Thus, the forecast value for period 63 is img . b. Compute forecasts for periods 62 and 63. Assume that img and img . The time series model is given below: img The forecast value for period 62 is, img Thus, the forecast value for period 62 is img . The forecast value for period 63 is, img Thus, the forecast value for period 63 is img . c. Compute the 95% prediction interval about the forecast for period 61. From part a., img . The formula for prediction interval is given below: img Substitute 75.65 for img and 3.2 for img in the above prediction interval formula. img Thus, the 95% prediction interval about the forecast for period 61 is img .