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Module 17 forecast confidence intervals practice problems (The attached PDF file has better formatting.) [Know the relations for confidence intervals of autoregressive and moving average processes. Final exam problems are multiple choice.] ** Exercise 17.1: Width of Confidence Interval for AR(1) process We use an AR(1) process to model a time series of N observations, yt, t = 1, 2, …, N, for which we estimate ì, ö, and ó2å. W = the width of the P% confidence interval for the k periods ahead forecast, k 2. Note: For a 95% confidence interval (a significance level of 5%), P = 95%, not 5%. What is the effect on W as The estimate of ì increases? The estimate of ö increases? The estimate of ó2å increases? k increases? P increases? N increases? Part A: The mean ì is a location parameter. It does not affect the shape of the time series or the width of the confidence interval for forecasts. Part B: W depends on ö2, so it depends on the absolute value of ö. The variance of the forecast is ó2å × (1 – ö2k) / (1 – ö2) = ó2å × (1 + ö2 + ö4 + … + ö2k-2). As |ö| increases, the variance increases, so the standard deviation increases and the width of the confidence interval increases. Part C: The variance of the forecast is proportional to ó2å, so as ó2å increases, the variance of the forecast increases, the standard deviation of the forecast increases, and the confidence interval becomes wider. Part D: The variance of the forecast increases with k, reaching a limit of ó2å / (1 – ö2) as k . Part E: The z-value (or t-value if the parameters are estimated) increases with P. The confidence interval is the mean of the forecast ± (t-value) × the standard deviation of the forecast, so as the t-value increases, the confidence interval becomes wider. Part F: As N increases, the t-value has more degrees of freedom, so it is smaller, and the confidence interval becomes narrower. Jacob: The textbook does not mention the relation of the t-value to N. Rachel: This topic is covered in the regression analysis course; it will not be tested on the time series final exam. The time series textbook does not want to get into discussions of the degrees of freedom. Take heed: For a moving average process, the width of the confidence interval reaches a maximum for lags > q (the moving average order). If the ARIMA process has an autoregressive part, the width of the confidence interval widens as k increases. The increase is asymptotic to ã0. ** Exercise 17.2: Width of Confidence Interval for MA(1) process We use an MA(1) process to model a time series of N observations, yt, t = 1, 2, …, N, for which we estimate ì, è, and ó2å. W = the width of the P% confidence interval for the k periods ahead forecast, k 2. Note: For a 95% confidence interval (a significance level of 5%), W = 95%, not 5%. What is the effect on W as The estimate of ì increases? The estimate of ö increases? The estimate of ó2å increases? k increases? P increases? N increases? Part A: The mean ì is a location parameter. It does not affect the shape of the time series or the width of the confidence interval for forecasts. Part B: W depends on è2, so it depends on the absolute value of è. The variance of the forecast is ó2å × (1 + è2). As |ö| increases, the variance increases, so the standard deviation increases and the width of the confidence interval increases. Part C: The variance of the forecast is proportional to ó2å, so as ó2å increases, the variance of the forecast increases, the standard deviation of the forecast increases, and the confidence interval becomes wider. Part D: The variance of the forecast does not change with k, as long as k 2. Part E: The z-value (or t-value if the parameters are estimated) increases with P. The confidence interval is the mean of the forecast ± (t-value) × the standard deviation of the forecast, so as the t-value increases, the confidence interval becomes wider. Part F: As N increases, the t-value has more degrees of freedom, so it is smaller, and the confidence interval becomes narrower. Take heed: For a moving average process, the width of the confidence interval reaches a maximum for lags > q (the moving average order). If the ARIMA process has an autoregressive part, the width of the confidence interval widens as k increases. The increase is asymptotic to ã0. ** Exercise 17.3: Width of Confidence interval A time series of 250 observations is modeled by an AR(1) process with ì = 0. If the parameters ö and ó2å are known (not estimated from the data), what is the 95% confidence interval? What is the width of this confidence interval? If the parameters ö and ó2å are estimated from the data, is the 95% confidence interval wider or narrower? Part A: The forecast is yT × ö. The variance of this forecast is ó2å × (1 + ö2). The standard deviation of this forecast is óå × (1 + ö2)0.5. The z-value for a two-sided 95% confidence interval is 1.960. The 95% confidence interval is yT × ö ± 1.960 × óå × (1 + ö2)0.5. [Note: Final exam problems give the z-values for significance levels.] Part B: The width of the confidence interval is 2 × 1.960 × óå × (1 + ö2)0.5. The width depends on the standard deviation of the forecast, not on the expected value. Part C: If ö is estimated from the data, the forecast is uncertain. It may be higher or lower, so the confidence interval is wider. If ó2å is estimated from the data, the variance of the forecast is uncertain. It may be higher or lower, so the confidence interval is wider.
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17.3 Problem does not explicitly state that we are looking at the variance of a k>=2 period ahead forecast which would change the variance/SD of the confidence interval. Are we to assume that this is just a continuation?
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