OUTLIER, HETEROSKEDASTICITY,AND NORMALITY

Robust Regression HAC Estimate of Standard Error

Quantile Regression

General form of the multiple linear regression model is:

iikiii uxxxfy ),...,,( 21

iikkiii uxxxy ...2211 i = 1,…,n This can be expressed as

i

n

kikki uxy

1

in summation form.

Review

2

Or

uXβy in matrix form, where

nknkn

k

n u

u

xx

xx

y

y

11

1

1111

,,, uβXy

x1 is a column of ones, i.e. TT

nkxx 1111

3

Problems with X

(i) Incorrect model – e.g. exclusion of relevant variables; inclusion of irrelevant variables; incorrect functional form

(ii) There is high linear dependence between two or more explanatory variables

(iii) The explanatory variables and the disturbance term are correlated

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Problems with u

(i) The variance parameters in the covariance-variance matrix are different

(ii) The disturbance terms are correlated (iii) The disturbances are not normally distributed

Problems with

(i) Parameter consistency (ii) Structural change

Review

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

Alternatives of OLS

• A. White’s Standard ErrorsOLS with HAC Estimate of Standard Error

• B. Weighted Least SquaresRobust Regression

• C. Quantile Regression Median RegressionBootstrapping

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?

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

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.

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.

Quantiles• Cumulative Distribution Function

• Quantile Function

• Discrete step function

)Prob()( yYyF

))(:min()( yFyQ

CDF1.0

0.6

0.2

2.01.51.00.50.0

0.4

-0.5-1.0

0.0

0.8

-1.5-2.0

Quantile (n=20)

-1.0

-1.5

1.0

0.0

1.00.8

1.5

0.6

0.5

0.40.2

-0.5

Regression Line

The Perspective of Quantile Regression (QR)

Optimality Criteria• Linear absolute loss

• Mean optimizes

• Quantile τ optimizes

• I = 0,1 indicator function

iymin

ii

ii

ye

eIe )0(min

-1 10

-1 10

1

Quantile RegressionAbsolute Loss vs. Quadratic Loss

0

0.5

1

1.5

2

2.5

3

-2 -1 0 1 2

Quadp=.5p=.7

Simple Linear RegressionFood Expenditure vs Income

Engel 1857 survey of 235 Belgian households

Range of Quantiles

Change of slope at different quantiles?

Bootstrapping

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

Bootstrap Confidence Limits