Sample Question #108 (econometrics – time series)

Consider the process

y1, …, T_{t}= ε_{t}+ βε_{t-1}ε_{t-2}, t =

where *ε _{t}* is a sequence of i.i.d. random variables with mean zero and constant variance, and

*ε*=

_{0}*ε*

_{-1}= 0.

Show that *y _{t}* is white noise. What is the MMSE (minimum mean squared-error estimator) of

*y*

_{T+1}? What about the MMSLE (minimum mean squared-error linear estimator)?

[Taken from exercise 6.10 of Harvey, *Forecasting, Structural Time Series Models and the Kalman Filter*]

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ANSWER

E(y_t) = 0, Var(y_t) = Var(ε_t)+β^2 * Var(ε_t)^2. It’s easy to see that {y_t} is serially independent across time. So y_t is white noise since it’s i.i.d. with finite and constant mean and variance.

The MMSE of y_(T+1) is β * ε_T * ε_(T+1), while the MMSLE is 0.