1. OUTLIER, HETEROSKEDASTICITY,AND NORMALITY
Robust Regression
HAC Estimate of Standard Error
Quantile Regression

8. Robust regression analysis
alternative to a least squares regression　model when fundamental assumptions are unfulfilled by the nature of the data
resistant to the influence of outliers
deal with residual problems
Stata & E-Views

9. Alternatives of OLS A. White’s Standard Errors
OLS with HAC Estimate of Standard Error
B. Weighted Least Squares
Robust Regression
C. Quantile Regression
Median Regression
Bootstrapping

13. If the residual distribution　is normally distributed, the analyst can determine where the level of significance or
rejection regions begin. Even if the sample size is large, the influence of the outlier can
increase the local and possibly even the global error variance. This inflation of error
variance decreases the efficiency of estimation.

15. OLS and Heteroskedasticity
What are the implications of heteroskedasticity for OLS?
Under the Gauss–Markov assumptions (including homoskedasticity), OLS was the Best Linear Unbiased Estimator.
Under heteroskedasticity, is OLS still Unbiased?
Is OLS still Best?

16. A. Heteroskedasticity and Autocorrelation Consistent Variance Estimation
the robust White variance estimator rendered regression resistant to the heteroskedasticity problem.
Harold White in 1980 showed that for asymptotic (large sample) estimation, the sample sum of squared error corrections approximated those of their population parameters under conditions of heteroskedasticity
and yielded a heteroskedastically consistent sample variance estimate of the standard errors

24. Quantile Regression Problem
The distribution of Y, the “dependent” variable, conditional on the covariate X, may have thick tails.
The conditional distribution of Y may be asymmetric.
The conditional distribution of Y may not be unimodal.
Neither regression nor ANOVA will give us robust results. Outliers are problematic, the mean is pulled toward the skewed tail, multiple modes will not be revealed.

25. Reasons to use quantiles rather than means
Analysis of distribution rather than average
Robustness
Skewed data
Interested in representative value
Interested in tails of distribution
Unequal variation of samples
E.g. Income distribution is highly skewed so median relates more to typical person that mean.

26. Quantiles Cumulative Distribution Function Quantile Function
Discrete step function

27. Regression Line

28. The Perspective of Quantile Regression (QR)

31. Optimality Criteria Linear absolute loss Mean optimizes
Quantile τ optimizes
I = 0,1 indicator function

32. Quantile Regression Absolute Loss vs. Quadratic Loss
Quadratic loss penalizes large errors very heavily. When p=.5 our best predictor is the median; it does not give as much weight to outliers. When p=.7 the loss is asymmetric; large positive errors are more heavily penalized then negative errors.

34. Simple Linear Regression
Food Expenditure vs Income
Engel survey of 235 Belgian households
Range of Quantiles
Change of slope at different quantiles?

35. Bootstrapping
When distributional normality and homoskedasticity assumptions are violated,
many researchers resort to nonparametric bootstrapping methods

39. Bootstrap Confidence Limits