Time Series errors correlated across equations

[tahir]

Dear All

I am estimating 3 equations simultaneously through GMM. Time series data, of course, for a single country. Don’t confuse it with panel or cross section. Estimation for a single country time series.
My question is

(1) do I need to check that error term of the first equation is correlated with the error terms of other equations or not. That is, errors are correlated across equations. If yes what is the test. IS it Hauseman?????

Or the possibility of cross sectional dependence in errors is only in cross- sectional or panel data and it is not the issue in Time series.

Thanks in advance

[TomDoan]

One thing to note is that the difference between "panel data" and "a set of time series" is often just one of the techniques applied.

If you're doing GMM, then you have a set of moment conditions. If the moment conditions hold, you can get consistent estimates of the parameters even if the moment conditions are serially correlated. There are three ways to handle serial correlation:

1. You can adjust the weight matrix on the conditions to take the serial correlation into account.
2. You can use the standard cross sectional weights on the conditions and adjust the covariance matrix to take the serial correlation into account.
3. You can redo the moment conditions by quasi-differencing the model to eliminate the serial correlation.

(3) is rarely done. It can be quite complicated in a multivariate setting, since it requires eliminating joint serial correlation, and in most cases, the new moment conditions (with economic or financial data) aren't likely to hold anyway. (This is the reason you rarely see AR1 estimation in regression models nowadays).

The choice between (1) and (2) can be a tricky one in a multivariate setting because the number of orthogonality conditions can be fairly large. The problem in (1) is that long-run covariance matrix has to get inverted. If the number of system-wide orthogonality conditions is a high percentage of the number of observations, the accuracy of the inverted long-run covariance matrix can be suspect.

This is an example of (2) --- the combination of ROBUST, LAGS and LWINDOW=NEWEY estimates using the inverse of sigma (the covariance matrix of the residuals) to weight the conditions (it would ordinarily by inv(sigma (x) Z'Z) but here Z is just constant), then corrects the covariance matrix for up to four lags of (joint) serial correlation.

Code: Select all

nonlin(parmset=meanparms) m1 m2 m3 m4
frml f1 = s{0}-m1
frml f2 = (s{0}-m1)^2-m2
frml f3 = (s{0}-m1)^3-m3
frml f4 = (s{0}-m1)^4-m4
*
instruments constant
nlsystem(robust,lags=4,lwindow=newey,parmset=meanparms,inst) 2 * f1 f2 f3 f4

If you want to test for joint serial correlation (the above allow for this whether it's there or not), you can use @MVQSTAT.

[tahir]

Thank you Sir