ROBUST STANDARD ERRORS
Hainan Sheng
Artem Meshcheryakov
Nga Trinh
OLS MODEL ASSUMPTIONS
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IF THERE EXISTS HETEROSCEDASTICITY,
• Estimators of parameters for OLS are still unbiased
and consistent, but the standard errors are not
efficient
• GLS model is BLUE, but we have to assume its
covariance matrix
• If the standard errors are not adjusted for
heteroscedasticity, we cannot use the usual t
statistics or F statistics for testing our hypothesis
ROBUST STANDARD ERRORS
•
ROBUST STANDARD ERRORS
•
WIDELY USED ROBUST STANDARD
ERRORS
Articles between 2009 and 2012.9 that used some
types of regression analysis and reported robust
standard errors : (King and Roberts, 2012)
•
•
•
•
International Organization: 73%
American Political Science Review: 66%
American Journal of Political Science: 45%
Across all academic fields: Google Scholar finds
55,800 articles using “ robust standard errors”
ROBUST S.E. IN SAS
• What data can exhibit heteroscedasticity?
• Expenses & Income:
• R&D expenses & Net Income
• Species characteristics & their areal:
• Frog size in North America, vs Russia, vs Asia and vs Australia
• Stocks Returns & Trading Volume
• What data can exhibit autocorrelation?
• Time series data
TESTS FOR HETEROSCEDASTICITY
TESTS FOR HETEROSCEDASTICITY
• Breusch-Pagan test
• Includes regressors
• H0: Homoscedasticity, H1: Heteroscedasticity
• White test allows for non-linearities
• Includes regressors, crosss-products, squares of regressors
• H0: Homoscedasticity, H1: Heteroscedasticity
TESTS FOR AUTOCORRELATION
• Durbin-Watson statistic
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0 < H-stat < 4
H-stat < 1 – serious problems with autocorrelation
H-stat closer to 2 and greater than 2 – no autocorrelation
Can detect only AR(1) process
• Breusch-Godfrey test
• More general than Durbin-Watson statistic
• Statistically more powerful
• H0: No autocorrelation H1: Autocorrelation
HETEROSCEDASTICITY &
AUTOCORRELATION CORRECTION
• Heteroscedasticity-consistent standard errors –
White’s standard errors
• Newey-West standard errors - correction of
standard errors for heteroscedasticity and
autocorrelation.
CONCERNS
• When classical and robust standard errors differ, the
model might be misspecified
• Even though robust standard errors of MLE are
efficient, model misspecification might lead to
biased estimators
• The reason is the difference between the model
fitted to the data and the process that generates
data
CONCERNS
• The specification error in the model leads to errors in the
likelihood function and we try to fit the incorrect
likelihood function to the data
• Estimators obtained from misspecified model might be
invalid and the inference from these estimates is
misleading
• So even though the variance is fixed, the bias due to
specification error might be large, and bias might be of
greater interest than variance
CONCERNS
• In summary, why do we care about the variance of
the estimators when they are incorrect and biased
due to specification error in the model?
• If the model we use to fit the data looks like the
process that generates data, we do not have
specification error and hence we do not need to
use robust standard errors (Freedman, 2006)
RECOMMENDATIONS
• If researchers believe that the model is misspecified,
they should try to improve the model rather than
just using robust standard errors (Leamer, 2010)
• Comparing classical and robust standard errors
might help detect misspecification
• However, no one type of robust standard errors is
consistent under all types of misspecification
RECOMMENDATIONS
• Non-significant difference between classical and robust
standard errors does not guarantee that the model is
correctly specified
• Researchers should use different tests and diagnostic
procedures to ensure that model’s assumptions are
consistent with the data
• White (1980) states that robust standard error “does not
relieve the investigator of the burden of carefully
specifying his models”
REFERENCES
• Freedman, David A. (2006). “On the so-called “Huber Sandwich
Estimator” and Robust Standard Errors.”The American Statistician
60 (4), 299-302
• King, Gary and Margaret Roberts (2012). “How robust standard
errors expose methodological problems they do not fix?”Working
paper
• Leamer, Edward E. (2010). “Tantalus on the road to
asymptopia.”The Journal of Economic Perspectives 24 (2, Spring),
31-46
• White, Halbert. (1980). “A heteroskedasticity-consistent
covariance matrix estimaror and a direct test for
heteroskedasticity.”Econometrica 48 (4, May), 817-838