The Dynamic Stability of AR Models - Tricking EViews

In this post I'm going to focus on understanding the extent to which there's an equivalence between two different ways of estimating an AR(p) model for a time-series, Y_t, using EViews, and to see what information is generated in each case.

In particular, I want to show you how you can "trick" EViews into showing you if your estimated dynamic regression model is "dynamically stable". That is, if the estimated coefficients for the lagged values of Y are such that the model is stationary. If the lag-order is above 2, this isn't something that's always easy to do by just looking at the estimated coefficient values.

By way of example, consider the case of an AR(2) model, with an intercept (or “drift”) term included to allow for a non-zero mean in the series. The results apply equally to the general AR(p) case, as will become apparent in the empirical example that's given below.

AR(2) Model:

 
             Y_t = α + φ₁Y_t-1 + φ₂Y_t-2 + ε_t    ,                (1)

where ε_t is a “white noise” series.

In the case of EViews, we could estimate the model using OLS with the specification:

            Y  C   Y(-1)   Y(-2)

Consider an alternative model for Y:

              Y_t = α + u_t    ;      u_t = φ₁u_t-1 + ε_t .         (2)

That is, we just explain Y in term of a level and a error term that is itself AR(2). From (2), notice that:

             φ₁Y_t-1 = φ₁ α + φ₁u_t-1                           (3)

  

and

                φ₂Y_t-2 = φ₂ α + φ₂u_t-2 .              (4)

So, subtracting (3) and (4) from (2), we have:

              Y_t - φ₁Y_t-1 - φ₂Y_t-2 = α (1- φ₁ - φ₂) + (u_t - φ₁u_t-1 - φ₂u_t-2),   (5)

or, from the definition of in (2):

             Y_t = α (1 - φ₁ - φ₂) + φ₁Y_t-1 + φ₂Y_t-2 + ε_t .             (6)

We see that (1) and (6) (and hence (1) and (2)) are the same, except for the intercept term. To estimate model (2) in EViews we'd use the specification:

                         
           Y    C    AR(1)    AR(2)

We see that if we do this then the estimates of phi1 and phi2 will be same as if used the EViews specification for (1), above. Of course, we could also recover the estimate of α in this second case by dividing the estimated intercept coefficient by the estimate of (1 - φ₁ - φ₂).

What would be the point of using this second specification? We should certainly use it only with care, given that the estimated intercept coefficient has to re-interpreted. However, if we set up this specification with the AR(.) terms in EViews, the package provides us with an extra item in the regression output that we can VIEW – an item that's not available otherwise.

And that's what we're looking for!

Specifically, if there are (any) AR terms in the model specification (even not all successive ones, as in AR(1), AR(4)), then we can VIEW the ARMA STRUCTURE. This then provides information about the dynamic stability of the estimated AR model, in terms of whether or not the inverted roots of the characteristic polynomial lie within the unit circle. In addition, we have the ability to see how the estimated model responds, dynamically, to various types of “shocks” to the data.

The mathematical details relating to the derivation of the stationarity (dynamic stability) conditions for the AR(2) can be found here.

One important thing to notice. We're really interested in a model that is autoregressive in the variable, Y. The second version of the model appears to lack this feature, but have an autoregressive error term. The point is, though, that this second version can be re-written as the first version, except for the intercept - and this intercept doesn't affect the dynamic stability.

An Example:

Consider the following simple autoregressive model for quarterly Japanese consumption data. These data are available in the Data page that goes with this blog; and the EViews file that I have used for the results below is available on the Code page. The model is not intended to be particularly sophisticated - I just want to illustrate simple point here.

The following transformations have been applied to the data:

                 W_t = log(CONS_t)   ;     to linearize the trend

                Y_t = (W_t - W_t-4)  ;     to remove the seasonality.

The second transformation also has the effect of removing most of the trend, as we can see:

Now let’s model the {Y_t} series using an AR(4) model, along the lines of equation (1):

Then, consider the alternative approach, along the lines of equation (2):

Note that, apart from the intercept coefficient, the various estimated coefficients and their standard errors are the same in each output. Also, note the additional information that is provided about the (complex) inverse roots of the characteristic equation at the bottom of this output. Let’s explore this information in more detail. This is what we see if we select VIEW / ARMA Structure:

The estimated AR(4) process is stationary – and that's something that may not be altogether obvious just by looking at the estimated coefficients in the first OLS output.

Returning to the ARMA Structure option, we might also choose the impulse response function. In this case we are going to “shock” the Y_t series by an amount equal to one sample standard deviation, and then see how the predicted series “settle down” over a 24-quarter (6-year) period:

Because the estimated specification is stationary, things settle down to their original state after about 5 years.

So, sometimes with a bit of thought we can "trick" out favourite econometrics package into providing us with information that it might otherwise be reluctant to yield up. This can save us a lot of work!

© 2012, David E. Giles