Seemingly Unrelated Regressions (SUR) with equivalent of the White or Newey-West covariance matrix?

From http://www.mathworks.com/examples/econometrics/mw/econ-ex83533901-implement-seemingly-unrelated-regression-analyses: "SUR accommodates intra-period innovation heteroscedasticity and correlation, but inter-period innovation independence and homoscedasticity".

I would like to estimate an SUR system which is also robust to heteroscedasticity and serial correlation within each panel, i.e. inter-period. How can I incorporate the White or better yet Newey-West covariance matrix for SUR in Matlab?

So far I have followed http://www.mathworks.com/examples/econometrics/mw/econ-ex83533901-implement-seemingly-unrelated-regression-analyses#9 and have a working multivariate time series model where all predictors are used for each response series.

Setting:

%n is the Number of response series

%k is the Number of exogenous series

%T = time series dimension

%Y is a Txn matrix

%X is a Txk matrix

n=3

T=100

k=4

X = randn(100,k)

Y = randn(100,n)

%sample code for SUR in MATLAB based on the link above:

ExpandX = kron(X,eye(n));

nk = size(ExpandX,2); % Number of regression variables

CellX = mat2cell(ExpandX,n*ones(T,1));

Mdl1 = vgxset('n',n,'nX',nk,'Constant',true);

[Mdl1_Est,Mdl1_SE,~,Mdl1_W] = vgxvarx(Mdl1,Y,CellX);

vgxdisp(Mdl1_Est,Mdl1_SE)

Thanks in advance for any help with getting White or HAC-style standard errors in SUR.

Answer 1

[Hang Qian]

Hi Ilona,

Suppose that we have a SUR with n equations and T periods.

First, estimate a SUR system using the function VGXVARX(...), which returns both estimated coefficients and the n-by-n intra-period covariance matrix S. It should be a consistent estimator even if the inter-period correlations are not addressed in this step.

Second, transform the data: multiply both y and X by chol(inv(S)). The transformed data do not have intra-period correlations any more.

Third, pool the transformed y and X as if they were stacked as a single-equation with n*T observations, and call the function hac(...), which returns White and Newey-West standard errors. It helps to fight inter-period heteroscedasticity and serial correlation.

Regards,

Hang Qian

[Ilona Mostipan]

Thank you Hang Qian.

As you suggested, in the first step, I used the n by n innovations covariance matrix as the S.

After the second step, I obtained transformed data: y as T by n matrix, X as a n*T by n*K matrix, where K is the number of exogenous explanatory variables in my SUR. I notice that the transformed X has a lower triangular sub structure, whereas the original X was diagonal in sub structure.

This is where I get stuck implementing the third step of pooling the transformed data to call hac(...). How can I stack the data as a single-equation given my dimensions for y and X?