Method of moments for an AR(1) persistence

Question

A mean-zero series follows a first-order autoregression, $X_t = \phi X_{t-1} + \varepsilon_t$ , with independent shocks $\varepsilon_t$ and unknown persistence $\phi$ . From your data the sample variance is $1.0$ and the sample lag-one autocovariance (the average of $X_t X_{t-1}$ ) is $0.6$ .

Use the method of moments to estimate $\phi$ .

Accepted Answer

Which moment identifies \phi. The natural moment here is not the mean (which is zero) but a lagged one: how much today co-moves with yesterday. Multiply the model equation by X{t-1} and take expectations. Because the shock \varepsilont is independent of the past, \mathbb{E}[\varepsilont X{t-1}] = 0, leaving \mathbb{E}[Xt X{t-1}] = \phi\,\mathbb{E}[X{t-1}^2]. In words, the lag-one autocovariance equals \phi times the variance. So the persistence is the lag-one autocorrelation: \phi = \frac{\mathbb{E}[Xt X{t-1}]}{\mathbb{E}[X{t-1}^2]}. Match sample to theory. Replace the two theoretical moments…

Method of moments for an AR(1) persistence

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