cluster-robust standard errors vs. robust standard errors in a cross-sectional setting


I have an cross-sectional individual-level data which could be structured as a panel data, but for now let's ignore this potential panel data structure . Based on this I want to allow for correlations among individuals at the state (U.S. states) level (the most aggregate level) so that I am wondering whether you could please illustrate how to compute the one-way cluster-robust covariance matrix (clustering by state) for a linear model in the cross-sectional context.


2 Answers


The one-way cluster robust standard errors can be computed using the "sandwich" estimator method for covariance:

VCE( β ) = (X'X)-1Ω(X'X)-1

In the case of panel series where we have N groups and T time periods per a group
NT*Ω is found by summing i from 1 to N

NT*Ωi = Xi'uiui'Xi


ui ≡ (ui1 . . . uiT)
Xi ≡ (xi1 . . . xiT).

Thinking in a pooled ols framework, ui is a T x 1 vector of pooled ols residuals for group i.

In GAUSS, this can be achieved using the olsmt procedure and the results stored in the olsmtOut structure member. Within the olsmtOut structure named oOut the member oOut.resid houses estimate residuals.

A complete tutorial on using both structures and the olsmt procedure can be found on the Aptech tutorial page click here.

This guidance in finding the cluster-robust covariance matrix in GAUSS. However, I will update this post with a complete example of code that can be used to compute the one-way cluster-robust covariance matrix for panel data within the next 24 hours.


Data Generation

This example demonstrates the use of the "sandwich" covariance method to compute a one-way cluster robust convariance matrix in GAUSS. The randomly generated data used in this example can be replicated using the rndseed and the rndn procedure. To generate a panel of 6 independent regressors across 30 groups each having 20 observations each:

rndseed 1046823;
n_groups = 30;
n_obs_each = 20;
n_variables = 6;

//Generate pooled Y data
y = rndn(n_obs_each*n_groups,1);

//Generate pooled X data
x = rndn(n_obs_each*n_groups,6);

Model Estimation

Once the data is generated the pooled ols model is estimated using olsmt:

//Set up olsmt
struct olsmtControl oc0;
oc0 = olsmtControlCreate;
//Compute residuals

//Print output

//Turn constant off

//Output structure
struct olsmtOut oOut;

oOut = olsmt(oc0 , 0 , y , x);

One-way Cluster Robust Covariance

For clarity, this example uses a GAUSS for loop to calculate the the one-way cluster robust covariance matrix. This is done by first creating a vector of group indicators. Since the generated data has n_groups, first construct a sequential vector ranging from one to n_groups using seqa:

//Generate sequential vector
group = seqa(1 , 1 , n_groups);

Next, reshape group such that there are n_obs_each in each group:

//Reshape group vector
group = vec(reshape(group , n_obs_each , n_groups));

Next, using a for loop, the within group covariance matrix is calculated from the residuals from the olsmtOut structure, oOut.resid. These covariances are stored in a three dimensional GAUSS array, Vcx_i:

//Initialize the variance-covariance matrix
Vcx_i = arrayinit(n_groups|n_vars|n_vars , 0);

//Loop through groups
for i(1 , n_groups , 1);
     x_i = selif(x , group .== i);
     u_i = selif(oOut.resid, group .== i);
     Vcx_i[i,.,.] = x_i'*(u_i*u_i')*x_i;

Sum across the groups:

//Sum across the groups (note groups are held in the 3 dimension)
Vcx_tot = arraytomat(asum(Vcx_i , 3));

Finally, using the "sandwich" method find panel covariance matrix and scale:

//Find avar_beta
avar_beta2 = inv(X'X)*Vcx_tot*inv(X'X);

//Small sample correction
c = (n_groups/(n_groups-1))*((t-1)/(t-n_vars));
avar_oneway = c*avar_beta2;

Eliminating the for Loop

Though the for loop is intuitive, replacing the loop with matrix operations can be much more computationally efficient. The first step is to create an indicator matrix to separate group :

//Create giant block X matrix
Eg = ones(t,t);
Sg = eye(n_groups).*.Eg;

The indicator matrix can then be used to compute a "giant" matrix of group-specific covariances:

//Giant matrix of covariances
sig = (oOut.resid*oOut.resid').*Sg;

Finally, the "sandwich" equation and the small sample correction constant are used to calculate the one-way cluster robust covariance matrix:

//Find covariance matrix
//Sandwich "bread"
inv_xx = (inv(X'X));

//Sandwich "meat"
x_sig_x = (X'*sig*X);

//Small sample correction
c = (n_groups/(n_groups-1))*((t-1)/(t-n_vars));

//Cluster robust standard error
avar_beta = c*((inv_xx*x_sig_x)*inv_xx);

//Print solution
print "One-way cluster robust covariance matrix with small sample correction:";
print avar_oneway;
print ;

print "One-way cluster robust se:";
print sqrt(diag(avar_oneway));
print ;

print "Non-adjusted se:";
print sqrt(diag(;

Creating a GAUSS Procedure

One final step that can be taken is to create GAUSS procedure for finding the one-way cluster robust covariance matrix. The code below creates and demonstrates the use of the clusterCov procedure which can be called either from the command line or from within a program file. This procedure is to be used post-estimation and requires four inputs: residuals (pooled_res), a "giant" stacked matrix of independent regressors (bigX), the number of groups (n_groups), and the number of observations per a group (t):

//Setup example
//Use ols residuals from past example
pooled_res = oOut.resid;

//Call function
avar_beta3 = covCluster(pooled_res , x , n_groups , t);

//Declare procedure
proc(1) = covCluster(pooled_res , big_x , n_groups , t);
local Eg ,Sg , sig , xxi , x_sig_x , avar_beta , avar_oneway , c , n_vars;

  n_vars = cols(big_x);

  //Create giant block X matrix
  Eg = ones(t , t);
  Sg = eye(n_groups) .*. Eg;

  //Group specific covariance matrix
  sig = (pooled_res*pooled_res') .* Sg;
  xxi = (inv(big_x'big_x));
  x_sig_x = (big_x'*sig*big_x);
  avar_beta = (inv_xx*x_sig_x)*inv_xx;

  //Small sample correction
  c = (n_groups/(n_groups-1))*((t-1)/(t-n_vars));
  avar_oneway = c*avar_beta;

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