Interview Question: Time Series Equation

Sample Question #108 (econometrics – time series)

Consider the process

yt = εt + βεt-1εt-2,    t = 1, …, T

where εt is a sequence of i.i.d. random variables with mean zero and constant variance, and ε0 = ε-1 = 0.

Show that yt is white noise. What is the MMSE (minimum mean squared-error estimator) of yT+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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One Response to Interview Question: Time Series Equation

  1. Brett says:

    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. 

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