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| Title | Estimating robust standard errors in Stata | |
| Author | James Hardin, StataCorp |
The new versions are better (less biased).
In the new implementation of the robust estimate of variance, Stata is now scaling the estimated variance matrix in order to make it less biased.
Estimating robust standard errors in Stata 4.0 resulted in
. hreg price weight displ
Regression with Huber standard errors Number of obs = 74
R-squared = 0.2909
Adj R-squared = 0.2710
Root MSE = 2518.38
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
weight | 1.823366 .7648832 2.384 0.020 .2982323 3.3485
displ | 2.087054 7.284658 0.286 0.775 -12.43814 16.61225
_cons | 247.907 1106.467 0.224 0.823 -1958.326 2454.14
------------------------------------------------------------------------------
and the same model in Stata 5.0 is
. regress price weight displ, robust
Regression with robust standard errors Number of obs = 74
F( 2, 71) = 14.44
Prob > F = 0.0000
R-squared = 0.2909
Root MSE = 2518.4
------------------------------------------------------------------------------
| Robust
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
weight | 1.823366 .7808755 2.335 0.022 .2663446 3.380387
displ | 2.087054 7.436967 0.281 0.780 -12.74184 16.91595
_cons | 247.907 1129.602 0.219 0.827 -2004.454 2500.269
------------------------------------------------------------------------------
Stata 5.0 scales the variance matrix using
n
-----
n - k
for the (unclustered) regression results. To match the previous results, we can undo that scaling
. di .7808755*sqrt(71/74) .76488318 . di 7.436967*sqrt(71/74) 7.284658 . di 1129.602*sqrt(71/74) 1106.4678
Running a robust regression in Stata 4.0 results in
. hreg price weight displ, group(rep78)
Regression with Huber standard errors Number of obs = 69
R-squared = 0.3108
Adj R-squared = 0.2899
Root MSE = 2454.21
Grouping variable: rep78
------------------------------------------------------------------------------
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
weight | 1.039647 .8439705 1.232 0.222 -.6453948 2.724688
displ | 8.887734 7.450619 1.193 0.237 -5.987907 23.76337
_cons | 1234.034 1986.931 0.621 0.537 -2733.002 5201.069
------------------------------------------------------------------------------
The same model run in Stata 5.0 results in
. regress price weight displ, robust cluster(rep78)
Regression with robust standard errors Number of obs = 69
F( 2, 4) = 3.40
Prob > F = 0.1372
R-squared = 0.3108
Number of clusters (rep78) = 5 Root MSE = 2454.2
------------------------------------------------------------------------------
| Robust
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
weight | 1.039647 .9577778 1.085 0.339 -1.619571 3.698864
displ | 8.887734 8.455317 1.051 0.353 -14.58799 32.36346
_cons | 1234.034 2254.864 0.547 0.613 -5026.472 7494.539
------------------------------------------------------------------------------
To match the previous results, the scale factor for clustered data is
n - 1 g
----- x -----
n - k g - 1
so that if we wish to match the previous results we may
. di .9577778*sqrt(4/5)*sqrt(66/68) .84397051 . di 8.455317*sqrt(4/5)*sqrt(66/68) 7.4506198 . di 2254.864*sqrt(4/5)*sqrt(66/68) 1986.9313
Note also that Stata 5.0 includes an F test in the header of the output that is the Wald test based on the robust variance estimate.
There is one final important difference. The hreg command used n-1 as the degrees of freedom for the t tests of the coefficients. This is anticonservative as Stata 5.0 now uses g-1 as the degrees of freedom. The more conservative definition of the degrees of freedom provides much more accurate confidence intervals. So for a dataset with a small number of groups (clusters) and a large number of observations, the difference between regress, robust cluster() and the old hreg will show up in the p-values of the t-statistics as the scale factor will become much less important, but the difference in degrees of freedom will remain important.
