In time series analysis, an AR$(1)$ model takes the form:

$$x_t = \beta_0 + \beta_1 \cdot x_{t-1} + w_t,$$

where $w_t$ is the white noise term.

In order for the model to be stationary and to converge to the mean, $\beta_1$ must be less than one. However, I don't get why such a model converges to B0 as opposed to $\beta_0 + \beta_1 \cdot \beta_0$. Surely, $\beta_1 \cdot \beta_0$ is large enough to matter even if $\beta_1 < 1$. I can see how the rest of the terms eventually approach zero as you recursively keep on subbing in $x_t$ into $x_{t-1}$ in order to calculate $x_{t+1}$, but even if you keep on doing this to calculate $x_{t+\infty}$ there is always a $\beta_0 \cdot \beta_1$ term.

Is my math wrong?

  • 2
    $\begingroup$ it converges to B0/(1-B1). Which is basically B0 + B0*B1 + B0*B1^2 + B0*B1^3 + .... $\endgroup$ – Julius Apr 8 '18 at 21:54

Taking expectations of both sides and letting $\mu \equiv \mathbb{E}(X_t)$ be the stationary mean, you have:

$$\mu = \beta_0 + \beta_1 \mu.$$

Rearranging then gives:

$$\mu = \frac{\beta_0}{1-\beta_1}.$$


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