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I am working through slides hosted at Basic Time Series Models, and am not sure how to mathematically derive a closed-form expression of the autocovariance of the "random walk with drift" model.

The model specification is $x_t = \delta + x_{t-1} + w_t$ with $E[w_t] = 0$, $var(w_t) = \sigma^2 < \infty$, and $x_0 = 0$.

Autocovariance is given as $\gamma(t, t + h) = E[(x_t - \mu_t)(x_{t+h} - \mu_{t+h})]$.

After algebraic manipulation, I am able to represent autocovariance as $E[(x_tx_{t+h} - x_t(t+h)\delta - x_{t+h}t\delta + (t^2+th)\delta]$, but I don't see how to reach the suggested closed-form expression $\gamma(t, t + h) = (t+1)\sigma^2$.

NOTE: This is not homework. I am attempting to learn time series modeling and forecasting on my own time.

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  • $\begingroup$ In the title you have closed form expression for autocovariance but in the body closed-form expression of the "random walk with drift" model. What are you really after? $\endgroup$ – Richard Hardy Mar 8 '15 at 8:02
  • $\begingroup$ Apologies. I am after a closed form expression for autocovariance and have updated the body accordingly. $\endgroup$ – Jubbles Mar 8 '15 at 8:04
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Try using the linearity of expectation to turn the autocovariance from one term into four terms (actually fewer if you combine some of them) and then noting that you can rewrite $x_{t+h}$ as

$\delta + x_{t+h-1} + w_{t+h} = \delta + \left(\delta + x_{t+h-2} + w_{t+h-1}\right) + w_{t+h} = \dots = h\delta + x_t + \sum_{i = t+1}^{t+h}w_i$.

This will allow you to write

$\gamma(t,t+h) = E[x_tx_{t+h}] - E[x_t](t+h)\delta - E[x_{t+h}]t\delta+(t^2+th)\delta^2$

which can then be written as

$\gamma(t,t+h) = E[x_t(h\delta + x_t + \sum_{i = t+1}^{t+h}w_i)] - t\delta\cdot(t+h)\delta - (t+h)\delta \cdot t\delta + (t^2+th)\delta^2 $.

You can continue along this path to get the desired result.

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