View
118
Download
0
Category
Tags:
Preview:
DESCRIPTION
OUTLIER, HETEROSKEDASTICITY,AND NORMALITY. Robust Regression HAC Estimate of Standard Error Quantile Regression. Robust regression analysis. alternative to a least squares regression model when fundamental assumptions are unfulfilled by the nature of the data - PowerPoint PPT Presentation
Citation preview
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
Review
8
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
Recommended