Sample Question #117 (econometrics – time series – state-space models)

Consider a simple linear Gaussian state-space model where the state variable *μ _{t} *follows a pure random walk. Let

*y*be the observed time series of

_{t}*μ*such that

_{t}*y*, where

_{t}= μ_{t}+ e_{t}*e*is normally distributed with 0 mean and constant variance.

_{t}Show that *y _{t} *is an ARIMA process.

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HINT

The proof is too messy to show on the web. I’ll give you a hint. If you want to know the entire proof, you can find it in chapter 11, section 11.1 of Tsay’s book.

If e(t) has variance=0, it means y(t) = μ(t), which itself is an ARIMA(0,1,0) model. So let’s work with the case where the variance of e(t) > 0. Then since μ(t) is a random walk, we have

μ(t) = (1/1-L)v(t-1), where L is the lag operator and v(t) is the white noise component in μ(t).

Substitute this into y(t) and then multiply both sides by (1-L). Now, can you figure out the rest? The result should show that y(t) is an ARIMA(0,1,1) model.

Comment: in answer a question like this, do as much as you can and try not to worry about getting the right final answer. The interviewer usually just wants to see how much you remember from topics you claim you know well, and how you proceed to solving problems on the spot.