TS Module 18: Forecast updates and weights HW

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

An ARIMA(0,1,1) model for a time series of 100 observations, y_t, t = 1, 2, …, 100, has θ₁ = 0.4.

           The forecast of the next observation, y₁₀₁, is 25.

           The actual value of y₁₀₁ is 26.

           The forecast of the next observation, y₁₀₂, is 26.

           The actual value of y₁₀₂ is 26.

We continue to use the same ARIMA model. That is, we don’t re-estimate the parameters with the additional data. We forecast y₁₀₃, the ARIMA value in the next period.

A.     From the actual and forecasted values of y₁₀₁, derive the residual for the ARMA model of the first differences.

B.     From the actual value of y₁₀₁ and the forecasted value of y₁₀₂, derive the forecasted value for Period 102 for the ARMA model of the first differences.

C.    This forecasted value for Period 102 is a function of ì, è₁, and the residual for Period 101. Derive the ì (mean) of the ARMA model of first differences.

D.    From the actual and forecasted values of y₁₀₂, derive the residual for the ARMA model of the first differences for Period 102.

E.     Using this residual, determine the forecasted first difference for the next period. 

F.     From the forecasted first difference, derive the forecasted value of the original time series.