OLS with robust standard errors

[Mary Anne]

Hello,

I want to perform a multiple linear regression in my model, OLS with robust standard errors. Due to this last part, I am a little confused with the statistical tests that I can do, since some of the most known ones are not possible to perform, such as the hettest.

Thank you for your time

[Maxence Morlet]

Please could describe your data further? Are they cross-sectional, longitudinal, time series?

Intuitively, if you're using OLS, you will want to test all Gauss-Markov assumptions. There are tests specific to panel data and tests specific to time-series data, which is why I ask about the nature of your data.

[Carlo Lazzaro]

Mary Anne:
as an aside to Maxence's helpful advice, please note that repeating -estat hettest- after invoking -robust- to take heteroskedasticity into account, is pointless, as outcome is expected to report (again) the evidence of heteroskedasticity (that's why Stata throws an error message in that instance).
This happens because -estat hettest- focuses on residual distribution (that remains unchanged after invoking the -robust- option, which, in turn, affects standard errors, with no bearing on residual distribution).
Conversely, you can run:
- estat ovtest- (with the usual caution in interpreting its output, as it is not that white/black magic able to spot what predictors are missing in the right-hand side of your regression equation, as it is often supposed/hoped to be);
-linktest- (that I prefer to -estat ovtest-);
-estat imtest-.

What above is (hopefully) made clearer in the following toy-example:

. use "C:\Program Files\Stata16\ado\base\a\auto.dta"
(1978 Automobile Data)

. regress price mpg trunk i.foreign

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(3, 70)        =      9.68
       Model |   186257112         3  62085704.1   Prob > F        =    0.0000
    Residual |   448808284        70  6411546.91   R-squared       =    0.2933
-------------+----------------------------------   Adj R-squared   =    0.2630
       Total |   635065396        73  8699525.97   Root MSE        =    2532.1

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -261.9892   64.91261    -4.04   0.000    -391.4533   -132.5251
       trunk |   83.64631   86.50052     0.97   0.337    -88.87354    256.1662
             |
     foreign |
    Foreign  |   1887.461   711.4158     2.65   0.010     468.5866    3306.335
       _cons |   10033.08   2256.685     4.45   0.000     5532.264     14533.9
------------------------------------------------------------------------------

. estat hettest

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
         Ho: Constant variance
         Variables: fitted values of price

         chi2(1)      =     4.73
         Prob > chi2  =   0.0297

. regress price mpg trunk i.foreign, vce(robust)

Linear regression                               Number of obs     =         74
                                                F(3, 70)          =      10.77
                                                Prob > F          =     0.0000
                                                R-squared         =     0.2933
                                                Root MSE          =     2532.1

------------------------------------------------------------------------------
             |               Robust
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -261.9892   67.65337    -3.87   0.000    -396.9196   -127.0588
       trunk |   83.64631   65.27092     1.28   0.204    -46.53244    213.8251
             |
     foreign |
    Foreign  |   1887.461    629.807     3.00   0.004     631.3504    3143.571
       _cons |   10033.08   2168.228     4.63   0.000     5708.686    14357.47
------------------------------------------------------------------------------

. estat hettest
hettest not appropriate after robust cluster()
r(498);

. linktest

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(2, 71)        =     23.41
       Model |   252361179         2   126180590   Prob > F        =    0.0000
    Residual |   382704217        71  5390200.24   R-squared       =    0.3974
-------------+----------------------------------   Adj R-squared   =    0.3804
       Total |   635065396        73  8699525.97   Root MSE        =    2321.7

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        _hat |  -2.423435   .9922671    -2.44   0.017     -4.40196   -.4449106
      _hatsq |   .0002969   .0000848     3.50   0.001     .0001278    .0004659
       _cons |   9075.642   2808.766     3.23   0.002     3475.122    14676.16
------------------------------------------------------------------------------

. estat ovtest

Ramsey RESET test using powers of the fitted values of price
       Ho:  model has no omitted variables
                  F(3, 67) =      4.29
                  Prob > F =      0.0079

. estat imtest

Cameron & Trivedi's decomposition of IM-test

---------------------------------------------------
              Source |       chi2     df      p
---------------------+-----------------------------
  Heteroskedasticity |       8.64      8    0.3733
            Skewness |       5.44      3    0.1426
            Kurtosis |       1.34      1    0.2472
---------------------+-----------------------------
               Total |      15.42     12    0.2194
---------------------------------------------------

.

Eventually, please note that -linktest- can also be run manually with very similar results (in this toy-example, OLS - or, more technically speaking, the functional form of the regressand- is misspecified, as both -_hatsq- and -sq_fitted- have explanatory power):

. use "C:\Program Files\Stata16\ado\base\a\auto.dta"
(1978 Automobile Data)

. regress price mpg trunk i.foreign, vce(robust)

Linear regression                               Number of obs     =         74
                                                F(3, 70)          =      10.77
                                                Prob > F          =     0.0000
                                                R-squared         =     0.2933
                                                Root MSE          =     2532.1

------------------------------------------------------------------------------
             |               Robust
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -261.9892   67.65337    -3.87   0.000    -396.9196   -127.0588
       trunk |   83.64631   65.27092     1.28   0.204    -46.53244    213.8251
             |
     foreign |
    Foreign  |   1887.461    629.807     3.00   0.004     631.3504    3143.571
       _cons |   10033.08   2168.228     4.63   0.000     5708.686    14357.47
------------------------------------------------------------------------------

. predict fitted, xb

. g sq_fitted=fitted^2

. regress price fitted sq_fitted, robust

Linear regression                               Number of obs     =         74
                                                F(2, 71)          =      26.28
                                                Prob > F          =     0.0000
                                                R-squared         =     0.3974
                                                Root MSE          =     2321.7

------------------------------------------------------------------------------
             |               Robust
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      fitted |  -2.423435   .6867943    -3.53   0.001    -3.792864   -1.054006
   sq_fitted |   .0002969   .0000654     4.54   0.000     .0001664    .0004273
       _cons |   9075.642   1586.624     5.72   0.000     5912.003    12239.28
------------------------------------------------------------------------------

[Mary Anne]

Hello,

Maxence, I forgot to mention that my data is in cross sectional data. I am sorry

Carlo, thank you so much for your help!