A stationary series follows X_t = −0.6 X_(t−1) + ε_t, where ε_t is white noise with variance 32. What is Var(X_t)? Give a number.

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Variance with a negative coefficient

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A stationary series follows $X_{t} = - 0.6 X_{t - 1} + ϵ_{t}$ , where $ϵ_{t}$ is white noise with variance $32$ . What is $Var (X_{t})$ ? Give a number.

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Hint

Take variances on both sides. The new shock is independent of the previous value.

Solution

Taking variances gives $V = 0.36 V + 32$ , because $- 0.6 \times - 0.6 = 0.36$ : the sign of the coefficient does not matter.
So $1 - 0.36 = 0.64$ and $V = \frac{32}{0.64} = 50$ .

Answer: $50$

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