System - non linear ls

[tvonbrasch]

Dear Eviews team

I’m reading the article Co-Integration and Error Correction: Representation, Estimation, and Testing by Engle and Granger.

My problem is how to estimate a system of cointegrating relationships when using the system object in Eviews.

Consider the system of r equations

Beta’*x_t=z_t,

where ‘ is the transpose operator, beta’ is a (r x n) vector (with identifying restrictions imposed) and x_t is a (n x 1) vector of variables and z_t is the (r x 1) vector of cointegrating relationships

According to E&G (p261), the cointegrating relationship can be identified by minimising trace(beta’ M beta), where M is the moment matrix divided by T (= 1/T^2 sum_t x_t x’_t). In other words, by minimising the sum across equations of squared residuals

z^2_1t+z^2_2t+…+z^2_rt.

Given that there are cross equation restrictions on Beta, will the least squares (or non linear least squares) routine do this minimisation. It is not clear to me from the manual what the non-linear least squares routine minimise when applied in a system.

Sinerely
Thomas von Brasch

[tvonbrasch]

Hi again,

My main concern is the minimisation criterion used in the system (non-linear) OLS routine, in particular what is assumed about the variance-covariance matrix of the error terms.

I think my last question was a bit imprecise in this respect. Let me clarify. On p 447 in the user guide: “OLS estimator of the estimated variance matrix of the parameters is valid under the assumption V= sigma kronecker I_T”, i.e., equation (31.12).

I’m wondering if the assumption should be equation (31.10), V=smallsigma*(I_T kronecker I_M), i.e., errors are contemporaneously uncorrelated and homoskedastic both between and within equations?

This is due to how the estimator for beta in (31.14) is the solution to the problem

(1) Min epsilon’ epsilon
= min epsilon_1’ epsilon_1+epsilon_2’epsilon2+…+epsilon_M’epsilonM

where ‘ is the transpose operator, I have used the notation in the manual, i.e., epsilon = (epsilon_1, epsilon_2, …, epsilon_m, …, epsilon_M), and where e.g, the residuals in the m’th equation are defined for T observations as epsilon_m=(epsilon_m0, epsilon_m1, …, epsilon_mT).

In other words, there is no weighting between the squared residuals across equations, i.e., smallsigma is assumed equal in all equations.

To summarise:

a) Is the manual correct, and OLS is valid under (31.12) or is (31.10) the correct assumption?

b) If I have a system with cross equation restrictions, will the least squares (or non-linear least squares) estimator minimise (1) subject to these cross equation restrictions ?

Thomas

[tvonbrasch]

I got this reply from the Eviews team

The documentation is *very* poor for this and I apologize, and I’ll see that it gets fixed. Thanks for pointing things, out, though it is perhaps poor for slightly different reasons than you point out.

The crux of the issue is that saying that you are doing OLS is not a complete description of the data generating process (DGP), so that one cannot compute coefficient covariance matrices without additional assumptions. In the first part of the discussion, we offer a couple of assumptions that one might adopt. In the OLS discussion we make a hash of relating the estimator to the discussion.

There are a few error covariance structures that one might reasonably use for computing the coefficient covariances by default when estimating using system OLS.
Spherical errors
iid, equation heteroskedastic
iid, equation heteroskedastic, contemporaneous correlation

The first corresponds to 31.10, the second to 31.11, and the third to 31.12.

In the OLS discussion we imply that we are using either 1 or 3. As you note, the first sentence of the section implies 3, and the last sentence implies 1. This is almost but not quite the case. What we do is to use 1 in the case where there are cross-equation restrictions, and 2 when there are no restrictions. The latter allows us to match results with equation by equation OLS. Your estimator will therefore minimize the SSR with the cross-equation restrictions imposed, and then will compute the coefficient covariance assuming a single system error variance.

I apologize for the confusion.