Heteroskedasticity

Heteroskedasticity

1. Consequences of heteroskedasticity for OLS

   • Unaffected
     • Biasedness of β
     • Consistency of β
     • R^2
       • Both variances in the population R2 are unconditional variances
   • Affects
     • Estimators of variances, Var(βhat), biased
       • Standard errors no longer valid for t statistics and intervals
         • OLS t statistics no longer have a t distribution if there is heteroskedasticity.  Not solved by a larger sample
     • Estimators no longer BLUE

2. Heteroskedasticity-robust inference

   • We want to adjust the standard errors and t, F and LM statistics so that they are valid in the presence of heteroskedasticity of unknown form. This allows us to still use OLS.
   • Heteroskedasticity-robust procedures are valid (in large samples) whether or not the errors have constant variance. We do not need to know which is the case. 
   • Robust Variance
     • Robust standard errors are often found to be larger than the usual standard errors
     • Formula p.245
   • Robust t-statistic
     • t = (estimate - hypothesised value) / standard error
   • Why use OLS standard errors at all?
     • Robust se valid whether homoskedasticity or not
       • However, they require large sample sizes
         • Small sample sizes could lead to robust t-statistics that have distributions not very close to the t distribution. This would throw off inference
         • Fine to report only robust se in large samples
     • If homoskedasticity holds and errors are normally distributed, usual t-statistics have exact t-distribution, regardless of sample size
   • Chow test
     • The usual chow test using F statistics is not valid if there is heteroskedasticity. 
     • Include dummy variable distinguishing the two groups along with interactions between that dummy variable and all other explanatory variables
       • Test whether there is no difference in the two regression functions
         • Test that the coefficients on the dummy variable and all interactions are zero
         • Test whether the slopes are all the same

3. Testing for heteroskedasticity

   • Breusch-Pagan
     • Obtain residuals from original estimation, then regress the squared residuals on the explanatory variables
     • Then test H_0:δ_1=δ_2=…=δ_k=0 using an F test     
   • White
     • regresses u ̂^2 on all explanatory variables, squared variables and interactions     If K=2:  u ̂^2=δ_0+δ_1 x_1+δ_2 x_2+δ_3 x_1^2+δ_4 x_2^2+δ_5 x_1 x_2+u.       Then test if coefficients are equal

4. Weighted Least Squares

   • If there is heteroskedasticity and we can specify the form of the variance, WLS is more efficient than OLS
   • Variance is σ2h(i)
     • h(i) is some functional form that determines heteroskedasticity. This changes for across observations
     • If we know h, WLS is BLUE
   • Minimse the weighted sum of squared residuals
     • Less weight given to observations with higher error variance
     • Each squared residuals weighted by 1/(h(i))
       • Specify weights proportional to the inverse of the variance
   • Feasible GLS estimator
     • estimates h (functional form of variance for observations) 
     • 1. Run the regression of y on xi and obtain residuals
       • 2. Create log(u(hat)2) by first squaring the OLS residuals and then taking the natural log
         • 3. Regress log(u(hat)2) on xi and obtain fitted values g(hat)
           • 4. Estimate the normal MLR by WLS using weights (1/h(hat))
     • estimates no longer BLUE if we don't know h
       • FGLS estimates are biased
         • Asymptotically unbiased in a large sample = consistent
       • As N goes to infinity, variance of FGLS is less than that of OLS
   • WLS vs OLS
     • If they give statistically significant estimates of different signs
       • Zero conditional mean assumption misspecified
       • Misspecified variance function
         • WLS standard errors and statistics no longer valid
           • Fix with robust standard errors
         • Better to use the wrong form of heteroskedasticity in WLS than to ignore it in OLS

5. Linear probability model

   • The binary dependent variable automatically leads to heteroskedasticity unless all slope parameters are zero. 
   • 1. Estimate the model by OLS and obtain the fitted values
     • 2. Determine whether all the fitted values are inside the unit interval. If so, proceed to step 3
       • if not, some adjustment is needed to bring all fitted values into unit interval
       • 3. Construct the estimated variances: h(hat) = y(hat)(1-yhat)
         • 4. Estimate the MLR by WLS, using weights (1/h hat)