Time series Mod 17 moving average forecasts
(The attached PDF file has better formatting.)
** Exercise 17.1: MA(1) Forecast
A time series of 60 interest rate observations yt, t = 1, 2, …, 60, is represented by an MA(1) model, with
ì = 8.55%, è = –0.50, and ó2å = 0.04.
What is the variance of the one period ahead forecast?
What is the variance of the two periods ahead forecast?
What is the variance of the three periods ahead forecast?
Part A:
The only random variable in the one period ahead forecast is the standard error, so the variance is
ó2å = 0.04.
Part B:
The two periods ahead forecast is
ì + åt+2 + 50% × åt+1.
The residuals are independent, so the variance of a sum of residuals is the sum of their variances.
If the variance of random variable Y is
ó2, the variance of á × Y is á2 × ó2.
The variances of both residuals are
ó2, so the combined variance is ó2 × (1 + 0.52) = 0.04 × (1.25) = 0.05.
Part C:
The three periods ahead forecast is
ì + åt+3 + 50% × åt+2.
For more than one period ahead, the variance of the forecast error is
(1 +
è2) × ó2å = (1 + 0.25) × 0.04 = 0.050.