Testing for heteroskedasticity

Saiming Liew

Join Date: Mar 2016

Posts: 10
#1

Testing for heteroskedasticity

16 Apr 2016, 05:14

I am conducting a regression model in stata to determine the impact of paternity leave on several labour market outcomes.

The regression looks like: y = dummy + linear + linear + squared

I would like to test for heteroskedasticity but I am unsure whether a Breusch-Pagan test or a White test would be appropriate in this case. Further, I wish to run regressions without using the squared variable. In that case, would would I switch to the Breusch-Pagan test (if White test is previously appropriate)?
Thank you.
Tags: None
Carlo Lazzaro

Join Date: Apr 2014

Posts: 17372
#2

16 Apr 2016, 06:24

Saiming:
- try -estat hettest- post regression. In general, which one of the two tests you choose doesn't make any substantive difference.(with or without squared terms).
Besides, visual inspection of residual distribution is recommended as well.
As an aside, creating squared terms (as well as categorical variables and interactions is very easy with -fvvarlist-);
- check for omitted variables bias (which is more dangerous than heteroskedasticity) as well, via -estat ovtest-.

Eventually your chance of getting helpful replies is conditional on posting what you typed and what Stata gave you back (as reminded by FAQ #12). Thanks.

Kind regards,
Carlo
(StataNow 18.5)
Comment
Saiming Liew

Join Date: Mar 2016

Posts: 10
#3

17 Apr 2016, 06:58

Thank you for the advice, Carlo.

I ran the regression below:

Code:

regress Genwagegap patleave matparleave c.patleave#c.matparleave c.patleave#c.patleave c.matparleave#c.matparleave

The result was:

Code:

hettest Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance Variables: fitted values of Genwagegap chi2(1) = 5.05 Prob > chi2 = 0.0246

Could you please explain how the Breusch-Pagan test works in this case where there are squared variables? From my understanding, the Breusch-Pagan test works for linear relationships.

Thank you.
Comment

Carlo Lazzaro

Join Date: Apr 2014
Posts: 17372

17 Apr 2016, 08:01

Saiming:
which kind of lnearity do you mean?
Among variables or among regression coefficients?
In OLS the linearity refers to coefficients only.
-estat hettest- checks for violation in the normality of residuals distribution , no matter if you plug in a squared (or higher) term in your equation, as you can see from the following toy-example, where, in both cases, the null is rejected, as evidence of heteroskedasticity:

Code:

. sysuse auto.dta
(1978 Automobile Data)
. reg price mpg
      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(1, 72)        =     20.26
       Model |   139449474         1   139449474   Prob > F        =    0.0000
    Residual |   495615923        72  6883554.48   R-squared       =    0.2196
-------------+----------------------------------   Adj R-squared   =    0.2087
       Total |   635065396        73  8699525.97   Root MSE        =    2623.7
------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -238.8943   53.07669    -4.50   0.000    -344.7008   -133.0879
       _cons |   11253.06   1170.813     9.61   0.000     8919.088    13587.03
------------------------------------------------------------------------------
. estat hettest
Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
         Ho: Constant variance
         Variables: fitted values of price
         chi2(1)      =     7.14
         Prob > chi2  =   0.0075
  
. reg price c.mpg##c.mpg
      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(2, 71)        =     18.28
       Model |   215835615         2   107917807   Prob > F        =    0.0000
    Residual |   419229781        71  5904644.81   R-squared       =    0.3399
-------------+----------------------------------   Adj R-squared   =    0.3213
       Total |   635065396        73  8699525.97   Root MSE        =    2429.9
------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -1265.194   289.5443    -4.37   0.000    -1842.529   -687.8593
             |
 c.mpg#c.mpg |   21.36069   5.938885     3.60   0.001     9.518891    33.20249
             |
       _cons |   22716.48   3366.577     6.75   0.000     16003.71    29429.24
------------------------------------------------------------------------------
. estat hettest
Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
         Ho: Constant variance
         Variables: fitted values of price
         chi2(1)      =     5.04
         Prob > chi2  =   0.0248

Last edited by Carlo Lazzaro; 17 Apr 2016, 08:22.

Kind regards,
Carlo
(StataNow 18.5)

Comment

Saiming Liew

Join Date: Mar 2016

Posts: 10
#5

17 Apr 2016, 08:15

I had interpreted the linearity of Breusch-Pagan tests to be among variables (rather than coefficients) but from what you have said then this is not true. From my understanding, the White test allows for nonlinear relationship between the squared error and the independent variables and so I assumed that this was preferred over the BP test given my model specification.

In that case, can I just check that Breusch-Pagan test (rather than the White test) is suitable for the specification I have chosen?

Thank you.
Comment

Carlo Lazzaro

Join Date: Apr 2014
Posts: 17372

17 Apr 2016, 08:36

Saiming:
- you may want to compare the results of both -estat hettest- and -estat imtest, White- on your data; otherwise, you can impose -vce(robust)- standard errors.
However, I would also consider different test apart those focused on heteroskedasticity, like in the elaboration of the following toy-example:

Code:

. sysuse auto.dta
(1978 Automobile Data)

. reg price mpg

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(1, 72)        =     20.26
       Model |   139449474         1   139449474   Prob > F        =    0.0000
    Residual |   495615923        72  6883554.48   R-squared       =    0.2196
-------------+----------------------------------   Adj R-squared   =    0.2087
       Total |   635065396        73  8699525.97   Root MSE        =    2623.7

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -238.8943   53.07669    -4.50   0.000    -344.7008   -133.0879
       _cons |   11253.06   1170.813     9.61   0.000     8919.088    13587.03
------------------------------------------------------------------------------

. estat hettest

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

         chi2(1)      =     7.14
         Prob > chi2  =   0.0075

. estat imtest, white

White's test for Ho: homoskedasticity
         against Ha: unrestricted heteroskedasticity

         chi2(2)      =      7.31
         Prob > chi2  =    0.0259

Cameron & Trivedi's decomposition of IM-test

---------------------------------------------------
              Source |       chi2     df      p
---------------------+-----------------------------
  Heteroskedasticity |       7.31      2    0.0259
            Skewness |       6.51      1    0.0107
            Kurtosis |       1.10      1    0.2935
---------------------+-----------------------------
               Total |      14.92      4    0.0049
---------------------------------------------------

. reg price c.mpg##c.mpg

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(2, 71)        =     18.28
       Model |   215835615         2   107917807   Prob > F        =    0.0000
    Residual |   419229781        71  5904644.81   R-squared       =    0.3399
-------------+----------------------------------   Adj R-squared   =    0.3213
       Total |   635065396        73  8699525.97   Root MSE        =    2429.9

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -1265.194   289.5443    -4.37   0.000    -1842.529   -687.8593
             |
 c.mpg#c.mpg |   21.36069   5.938885     3.60   0.001     9.518891    33.20249
             |
       _cons |   22716.48   3366.577     6.75   0.000     16003.71    29429.24
------------------------------------------------------------------------------

. estat hettest

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

         chi2(1)      =     5.04
         Prob > chi2  =   0.0248

. estat imtest, white

White's test for Ho: homoskedasticity
         against Ha: unrestricted heteroskedasticity

         chi2(4)      =      2.73
         Prob > chi2  =    0.6038

Cameron & Trivedi's decomposition of IM-test

---------------------------------------------------
              Source |       chi2     df      p
---------------------+-----------------------------
  Heteroskedasticity |       2.73      4    0.6038
            Skewness |       2.93      2    0.2316
            Kurtosis |       1.05      1    0.3053
---------------------+-----------------------------
               Total |       6.71      7    0.4599
---------------------------------------------------

. estat vif

    Variable |       VIF       1/VIF 
-------------+----------------------
         mpg |     34.69    0.028824
 c.mpg#c.mpg |     34.69    0.028824
-------------+----------------------
    Mean VIF |     34.69

As you can see, when a squared term was added, heteroskedasticity is still highlighted by -estat hettest-, whereas the null is not rejected by -estat imtest,white-.
However, the regression outcome is not reliable, due to a quite impressive mean VIF, that makes me think that the model is basically misspecified (as it is), also due to omitted variable bias:

Code:

 Ramsey RESET test using powers of the fitted values of price
       Ho:  model has no omitted variables
                  F(3, 68) =      3.98
                  Prob > F =      0.0113

Kind regards,
Carlo
(StataNow 18.5)

Comment

Annur Wijayakusuma

Join Date: Mar 2020

Posts: 31
#7

02 Apr 2020, 17:01

Hi Carlo,

Thank you for your great explanation. I have been looking for this explanation for so long and you nail it.

Appreciate it

Regards,

Annur
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Uyi Erhabor

Join Date: Dec 2020

Posts: 5
#8

11 Jan 2021, 05:26

Hi everyone hope you are all well,

xtreg DiffMeanHourlyPercent Year2019 Year2020, fe - I am trying to test the heteroskedasticity assumption before running this fixed regression model, but I am not sure which test I should use as my independent variables are year dummies.

Kind regards,

Uyi Erhabor
(Stata 16.1 SE)
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