TS Module 16: ARIMA Forecasting HW

(The attached PDF file has better formatting.)

Homework assignment: ARIMA(0,1,1) process

The estimated (forecast) and actual values for Periods 48, 49, and 50 of an ARIMA(0,1,1) process are shown below.

Forecasts are one period ahead forecasts:  ŷ₄₈(1) for Period 49 and ŷ₄₉(1) for Period 50.

           The estimated and actual values are for the ARIMA(0,1,1) time series.

           The values of ì and è₁ are for the ARMA model of first differences. (Cryer and Chan use è for an ARMA process, not è₁.)

To solve the homework assignment, use the following steps:

           Determine the residual for each period for the ARIMA(0,1,1) model.

           These are also the residuals for the ARMA process of first differences.

           Determine the forecasts and actual values for the MA(1) process of first differences for the last two periods.

           Write equations for these forecasts in terms of the mean, the residual in the previous period, and è₁. Remember that è₁ is the negative of the moving average parameter.

           You have a pair of linear equations with two unknowns, ì and è₁.

           Solve for  ì and è₁ and verify that these values give the forecasts in the table.

           Use the residual for Period 50 and the values of ì and è₁ to forecast Period 51.

           Derive the forecast for the original ARIMA time series for Period 51.