Hello again everybody,

We’re getting towards the final straight line before the exam, and I will post here the content of all the little flash cards that I created for myself.

Starting back where I left, in the Quantitative Methods this post will be about Autoregressive Processes sometime denoted AR.

These processes are of the following form:

$$x_{t+1} = b_0 + b_1 x_t$$

Where $x_t$ is the value of process $X$ at time $t$, and $b_0$ and $b_1$ are the parameters we are trying to estimate.

To estimate the parameters $b_0$ and $b_1$, we proceed as follows:

- Estimate the parameters using linear regression
- Calculate the auto-correlations of the residuals
- Test whether these auto-correlations are significant:

Note that we cannot use the Durbin-Watson test we used previously in this section of the CFA curriculum; we will be using a t-test that works this way:

$$t = \frac{\rho_{\epsilon_t, \epsilon_{t+k}}}{\frac{1}{\sqrt{T}}}=\rho_{\epsilon_t, \epsilon_{t+k}} \sqrt{T}$$

Where $\epsilon_t$ is the residual term of the regression at time $t$, and $T$ is the number of observation. The t statistic has $T-2$ degrees of freedom. If they are statistically significant, then we cannot continue our analysis because of reasons I’ll explain a bit later in the post.

With AR processes, you are trying to actually predict the next values of a given process using a linear relationship between successive values an by applying simple linear regression. The thing is, if you want to be able to trust your estimated $b_0$ and $b_1$ parameters, you need the process to be **covariance-stationary**.

Now, a bit of math. *If a process has a finite mean-reverting level, then it is covariance-stationary*. What is the mean-reverting level? Well it simple the value $x_t$ at which $x_{t+1}=x_t$. So, let’s write this in an equation:

$$x_{t+1} = x_t = b_0 + b_1 x_t$$

$$ (1-b_1) x_t = b_0 \iff x_t=\frac{b_0}{1-b_1}$$

So, $X$ is covariance stationary if $b_1 \neq 1$.

The test for auto-correlations we did in the point 3) guarantees that the process is covariance-stationary if the auto-correlations are not statistically significant.

What if the process X is not covariance-stationary? Well you create a new process $Y$ where:

$$y_t = x_t – x_{t-1}$$

So, you have a new model

$$y_t = b_0 + b_1 y_{t-1} + \epsilon_t$$

which models the next change in the process X which is then covariance stationary. You can use that for the analysis.

This little “trick” is called *first differencing*.

That’s it, stay tuned for more soon!