Random Walk

Study guides: Exam SRM, Exam MAS-II

A random walk is the non-stationary series whose change at each step is white noise: $Y_t = Y_{t-1} + \varepsilon_t$ . It is ARIMA $(0,1,0)$ , the boundary case of an AR(1) at $\phi = 1$ , and the null hypothesis a unit-root test is testing.

$Y_t = Y_{t-1} + \varepsilon_t = Y_0 + \sum_{i=1}^{t}\varepsilon_i$

$\mathrm{Var}(Y_t) = t\sigma^{2}, \qquad \mathrm{Corr}(Y_t, Y_{t+k}) = \sqrt{\frac{t}{t+k}}$

The variance grows without bound in $t$ , so the series is not stationary and has no mean to revert to — it wanders
Its ACF decays extremely slowly and near-linearly, the classic visual signature of a unit root
First differencing makes it white noise: $\nabla Y_t = \varepsilon_t$ . That is the whole content of the "I" in ARIMA
The $h$ -step forecast is the last observation, $\hat Y_{t+h} = Y_t$ , for every $h$ — and the forecast interval widens with $\sqrt{h}$
A random walk with drift, $Y_t = \delta + Y_{t-1} + \varepsilon_t$ , adds a linear trend of slope $\delta$ — a stochastic trend, not a deterministic one

Forecasting a Random Walk with Drift A quarterly loss index follows $Y_t = 1.4 + Y_{t-1} + \varepsilon_t$ with $\sigma = 3$ . The latest value is $Y_{40} = 218$ . Forecast three quarters ahead and give a 95% interval.

Answer $\hat Y_{43} = Y_{40} + 3(1.4) = 218 + 4.2 = 222.2$ The forecast error is $\varepsilon_{41}+\varepsilon_{42}+\varepsilon_{43}$ , with variance $3\sigma^2 = 27$ and SD $\sqrt{27} = 5.20$ : $222.2 \pm 1.96(5.20) = (212.0,\ 232.4)$ The interval widens with $\sqrt{h}$ , not $h$ — six quarters out the SD would be $3\sqrt{6} = 7.35$ , not double.