# Question

Many auto regressions are mean-reverting. Mean-reverting means that the forecasts eventually tend back to (revert to) the mean of the time series. For example, a manager uses an AR(1) model to predict sales next week using sales in recent weeks. Sales typically run about $250,000 per week. The estimating equation (with sales in thousands of dollars) is

ŷt = 50 + 0.8 y t–1

(a) If sales this week are $250,000 (the mean level), what does the equation forecast for next week?

(b) If sales this week are $300,000 ($50,000 above the mean), does the equation forecast sales to increase farther above the mean or to return toward the mean?

(c) In general, when does a first-order auto regression with positive slope (0 < b1 < 1) predict an increase in the time series? A decrease?

ŷt = 50 + 0.8 y t–1

(a) If sales this week are $250,000 (the mean level), what does the equation forecast for next week?

(b) If sales this week are $300,000 ($50,000 above the mean), does the equation forecast sales to increase farther above the mean or to return toward the mean?

(c) In general, when does a first-order auto regression with positive slope (0 < b1 < 1) predict an increase in the time series? A decrease?

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