# 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 . The formula for confidence interval is given below: Here, the confidence level is 0.95. For two tailed test, From Table B-2, the required value for 95% confidence level is 1.96. Thus, . Substitute 1.96 for , 100 for n in the above confidence interval formula. Thus, the 95% confidence interval for the autocorrelation coefficient at any lag is . 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 , , and 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 , , and . Assume that and . The model is given below: The forecast value for periods 1, 2, 3, and 4 are calculated as follows: The forecast value for period 5 is, Thus, the forecast value for period 5 is . The forecast value for period 6 is, Thus, the forecast value for period 6 is . The forecast value for period 7 is, Thus, the forecast value for period 7 is .

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