8
Jun 17

Autoregressive processes

Autoregressive processes: going from the particular to the general is the safest option. Simple observations are the foundation of any theory.

Intuition

Electricity demand in France and Britain

Figure 1. Electricity load in France and Great Britain for 2001 to 2006

If you have only one variable, what can you regress it on? Only on its own past values (future values are not available at any given moment). Figure 1 on electricity demand from a paper by J.W. Taylor illustrates this. A low value of electricity demand, say, in summer last year, will drive down its value in summer this year. Overall, we would expect the electricity demand now to depend on its values in the past 12 months. Another important observation from this example is that probably this time series is stationary.

AR(p) model

We want a definition of a class of stationary models. From this example we see that excluding the time trend increases chances of obtaining a stationary process. The idea to regress the process on its own past values is realized in

(1) y_t=\mu+\beta_1y_{t-1}+...+\beta_py_{t-p}+u_t.

Here p is some positive integer. However, both this example and the one about random walk show that some condition on the coefficients \mu,\beta_1,...,\beta_p will be required for (1) to be stationary. (1) is called an autoregressive process of order p and denoted AR(p).

Exercise 1. Repeat calculations on AR(1) process to see that in case p=1 for (1) the stability condition |\beta_1|<1 is sufficient for stationarity (that is, the coefficient \mu has no impact on stationarity).

Question. How does this stability condition generalize to AR(p)?

Characteristic polynomial

Denote L the lag operator defined by Ly_t=y_{t-1}. More generally, its powers are defined by L^ky_t=y_{t-k}. Then (1) can be rewritten as

y_t=\mu+\beta_1Ly_t+...+\beta_pL^py_t+u_t.

Whoever first did this wanted to solve the equation for y_t. Sending all terms containing y_t to the left we have

y_t-(\beta_1Ly_t+...+\beta_pL^py_t)=\mu+u_t.

The identity operator is defined by Iy_t=y_t, so y_t=Iy_t. Factoring out y_t we get

(2) (I-\beta_1L-...-\beta_pL^p)y_t=\mu+u_t.

Finally, formally solving for y_t we have

(3) y_t=(I-\beta_1L-...-\beta_pL^p)^{-1}(\mu+u_t).

Definition 1.  In I-\beta_1L-...-\beta_pL^p replace the identity by 1 and powers of the lag operator by powers of a real number x to obtain the definition of the characteristic polynomial:

(3) p(x)=1-\beta_1x-...-\beta_px^p.

p(x) is a polynomial of degree p and by the fundamental theorem of algebra has p roots.

Definition 2. We say that model (1) is stable if its characteristic polynomial (3) has roots outside the unit circle, that is, the roots are larger than 1 in absolute value.

Under this stability condition the passage from (2) to (3) can be justified. For AR(1) process this actually has been done.

Example 1. In case of a first-order process, p(x)=1-\beta_1x has one root x=1/\beta_1 which lies outside the unit circle exactly when |\beta_1|<1.

Example 2. In case of a second-order process, p(x) has two roots. If both of them are larger than 1 in absolute value, then the process is stable. The formula for the roots of a quadratic equation is well-known but stating it here wouldn't add much to what we know. Most statistical packages, including Stata, have procedures for checking stability.

Remark. Hamilton uses a different definition of the characteristic polynomial (linked to vector autoregressions), that's why in his definition the roots of the characteristic equation should lie inside the unit circle.

26
May 17

Stationary processes 1

Along with examples of nonstationary processes, it is necessary to know a couple of examples of stationary processes.

Example 1. In the model with a time trend, suppose that there is no time trend, that is, b=0. The result is white noise shifted by a constant a, and it is seen to be stationary.

Example 2. Let us change the random walk slightly, by introducing a coefficient \beta for the first lag:

(1) y_t=\beta y_{t-1}+u_t

where u_t is, as before, white noise:

(2) Eu_t=0Eu_t^2=\sigma^2 for all t and Eu_tu_s=0 for all t\ne s.

This is an autoregressive process of order 1, denoted AR(1).

Stability condition|\beta|<1.

By now you should be familiar with recurrent substitution. (1) for the previous period looks like this:

(3) y_{t-1}=\beta y_{t-2}+u_{t-1}.

Plugging (3) in (1) we get y_t=\beta^2y_{t-2}+\beta u_{t-1}+u_t. After doing this k times we obtain

(4) y_t=\beta^ky_{t-k}+\beta^{k-1}u_{t-k+1}+...+\beta u_{t-1}+u_t.

To avoid errors in calculations like this, note that in the product \beta^{k-1}u_{t-k+1} the sum of the power of \beta and the subscript of u is always t.

Here the range of time moments didn't matter because the model wasn't dynamic. In the other example we had to assume that in (1) t takes all positive integer values. In the current situation we have to assume that t takes all integer values, or, put it differently, the process y_t extends infinitely to plus and minus infinity. Then we can take advantage of the stability condition. Letting k\rightarrow\infty (and therefore t-k\rightarrow-\infty) we see that the first term on the right-hand side of (4) tends to zero and the sum becomes infinite:

(5) y_t=...+\beta^{k-1}u_{t-k+1}+...+\beta u_{t-1}+u_t=\sum_{j=0}^\infty\beta^ju_{t-j}.

We have shown that this representation follows from (1). Conversely, one can show that (5) implies (1). (5) is an infinite moving average, denoted MA(\infty).

It can be used to check that (1) is stationary. Obviously, the first condition of a stationary process is satisfied: Ey_t=0. For the second one we have (use (2)):

(6) Var(y_t)=Ey_t^2=E(...+\beta^{k-1}u_{t-k+1}+...+\beta u_{t-1}+u_t)(...+\beta^{k-1}u_{t-k+1}+...+\beta u_{t-1}+u_t)

=...+\beta^{2k-2}Eu^2_{t-k+1}+...+\beta^2Eu^2_{t-1}+Eu^2_t=(...+\beta^{2k-2}+...+\beta^2+1)\sigma^2=\frac{\sigma^2}{1-\beta^2},

which doesn't depend on t.

Exercise. To make sure that you understand (6), similarly prove that

(7) Cov(y_t,y_s)=\beta^{|t-s|}\frac{\sigma^2}{1-\beta^2}.

Without loss of generality, you can assume that t>s. (7) is a function of the distance in time between t,s, as required.