ARMA Estimation Options in Eviews9

[Gelfan]

Hello,

I'm trying to understand the three different ARMA estimation options ML, GLS and CLS listed on the options tab of the Estimate Equation Dialog Box. I have Eviews 9, which I recently upgraded from Eviews 8.

I have the following model:

Y_t = (X_t)' B_t + u_t,
u_t = p*u_{t-1} + e_t
where
e_t ~ n(0,sigma^2)
and
X_t and B_t are n x 1 vectors

i.e. the standard linear regression with first order serial correlation.

To estimate the parameters of this model, I selected LS - Least Squares (NLS and ARMA).

Conditional Least Squares (CLS):

As I understand it, for CLS Eviews does the following:

Write Y_t = (X_t)' B_t + p(Y_{t-1} - (X_{t-1})' B_{t-1} ) + e_t.
Then Eviews finds B* and p* that minimize the sum of squared error e_t based on the nonlinear numerical algorithm the user selected.
Also if my dataset has 100 observations, then differencing leads to 99 observations to be fitted.

This was the default method in Eviews 8. Is this all correct?

Generalized Least Squares (GLS):

Let S be the covariance matrix of the vector u. In matrix form the model is Y = X'B + u where u ~ n(0,S)

In the case of above model which has first order serial correlation (and constant variance), we have:
S = sigma^2 * [1 p p^2 ... p^{n-1}; p 1 p ... p^{n-2}; ... ; p^{n-1} p^{n-2} p^{n-3} ... 1].

Let R = the correlation matrix. Then factor R = A(A') where A is a lower triangular matrix. So in this case
A = [sqrt(1 - p^2) 0 0 ... 0; -p 1 0 ... 0; ... ; 0 0 0 ... -p 1].

Then multiply both sides of the original equation in matrix form to get
AY = AXB + Au

Finally, find B* and p* that minimize (Au')(Au) using the selected numerical algorithm (Gauss-Newton, etc.).

Compared to CLS, GLS treats the first observation Y_1 differently. GLS uses Y_1*sqrt(1 - p^2) where as CLS omits Y_1 if there is not an observation before Y_1 since then the difference can't be calculated.

Is this the correct idea?

Maximum Likelihood (ML):

Take the model in matrix form again: Y = X'B + u where u ~ n(0,S)

Then the log likelihood of Y where Y is an n x 1 matrix is

-(n/2) log(2pi) - (n/2) log(det(S)) - (1/2)(Y-X'B)'(S^-1)(Y-X'B)

Then we find B*, p* and sigma*^2 that minimize the log likelihood function.

Could you please tell me if this is correct and give some more details as to what Eviews does specifically in the ML case? Lastly, regarding MLE with serial correlation is there a reference book that is available?

Sorry for the long post! I appreciate the help.

Thanks,
Gelfan

[EViews Gareth]

Have you read the discussion in the User Guide? It is pretty thorough, and gives further references.

[EViews Glenn]

Just to add to Gareth's comments.

What you have written is correct for the AR(1) case, though the transformations in CLS and GLS are obviously much more complicated for higher order AR with MA terms. As we note in the manual, probably the best way to think of GLS is that it's the same as the ML estimator, ignoring the log(det) term. The manual is a bit more specific about the ML estimator and provides specific references.