chapter 8 heteroskedasticity - university of california, davis
TRANSCRIPT
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Chapter 8Heteroskedasticity
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Learning Objectives
• Demonstrate the problem of heteroskedasticity and its implications
• Conduct and interpret tests for heteroscedasticity
• Correct for heteroscedasticity using White’s heteroskedasticity-robust estimator
• Correct for heteroscedasticity by getting the model right
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What is Heteroscedasticity?
• Hetero = different
• Scedastic – from a Greek word meaning dispersion
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CO2 Emissions vs National Income0
2.0e
+06
4.0e
+06
6.0e
+06
8.0e
+06
CO
2(kt
)
0 5.0e+06 1.0e+07 1.5e+07GNI(Millions of US$)
China
USA
JapanGermany
India
Russia
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The Problem• The population model is:
• The errors come from a distribution with constant standard deviation
• What if the standard deviation is not constant? => Heteroskedasticity– Maybe it’s higher for higher levels of X1i,
X2i, or some combination of the two.
0 1 1 2 2β β β ε= + + +i i i iY X X
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The Good News
• As long as you still have a representative sample
• …the error on average is zero and your estimates will still be unbiased
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The Bad News
• …But the errors will not have a constant variance– That could make a big difference when it comes
to testing hypotheses and setting up confidence intervals around your estimates.
• Heteroskedasticity means that some observations give you more information about β than others.
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Example 1: Variance is correlated with RH Variable
Figure 8.1. 2010 CO2 Emissions for 183 Countries. Data Source: World Bank World Development Indicators. http://data.worldbank.org/data-catalog/world-development-indicators
What happens if we drop one of these?
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Using Average GNI per Person
• Still have lots of heteroscedasticity, but the extremes are less extreme than in the first figure
Now what happens if we drop one of these?
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Example 2: Working with Average Data
• The variance is lower for bigger countries (which makes sense, right?)
0 11
1,β β ε ε ε=
= + + = ∑in
i i i i ijj
Y Xn
You estimate (country i):
Model for person j:
2 212
2 2
( )( ) , so ( )
εσ σε σ ε == = = =
∑in
ijj i
ij ii i i
VarnVar Var
n n n
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Example 3: Modeling Probabilities
• Suppose you drop a laptop to see whether it breaks• Bi=1 if laptop breaks,
0 otherwise• p=Prob(breaks)• (1-p) = Prob (doesn’t break)• Variance = p(1-p)• HEIGHT=height you drop
the laptop from
0 1i i iB HEIGHTβ β ε= + +
Low height: never breaks; Large height: always breaks; in between, the variance is greater than zero
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Example 4: Moving Off the Farm (Ch. 7)
• Variance decreases to almost nothing in rich countries (where nobody wants to do hired farm work)
10i i
i
AGLPCYββ ε= + +
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Two Solutions
1. Test and fix after-the-fact (ex-post)
2. Change the model to eliminate the heteroscedasticity
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Testing: Let Every Observation Have Its Own Variance
• Heteroskedasticity means different variances for different observations
• The squared residual is a good proxy for the variance σi
2
– It’s different for each observation– It even looks like a variance!
• So why not regress the squared residuals on all the RH variables and see if there’s a correlation?
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Breusch-Pagan Test
• Estimate the regression:
20 1 1 2 2i i i ie X X uα α α= + + +
0 1 1 2 2i i i iY b b X b X e= + + +
• Save the residuals and square them• Regress the squared residuals on X1 and X2
– For the two RH variable case:
• Under homoscedasticity, NRa2 is distributed as a chi-
squared with df equal to the total number of right-hand variables in the regression (df=2).
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White’s Test (More Flexible than Breusch-Pagan. Why?)
• Estimate the regression:
2 2 20 1 1 2 1 3 2 4 2 5 1 2i i i i i i i ie X X X X X X uα α α α α α= + + + + + +
0 1 1 2 2= + + +i i i iY b b X b X e
• Save the residuals and square them• Regress the squared residuals on X1, X2,
their squares, and their interactions– For the two RH variable case:
• Under homoscedasticity, NRa2 is distributed as a chi-
squared with df equal to the total number of right-hand variables in the White regression, a (here, a=5).
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Set up your Breusch-Pagan or White test
2 20 1 0: for all vs : not σ σ=iH i H H
Test statistic:
Reject the null hypothesis if: 22aaNR χ>
2aNR
Rejecting the null implies that have heteroskedasticity
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White Test of Moving Off the Farm
• White’s test shows a quadratic relationship between PCY and the variance
• exceeds the critical Chi-Square value, so we reject the null of homoskedasticity
2 20 1 2i i i ie PCY PCY uα α α= + + +
Variables in White's Auxiliary
Regression
Estimated Coefficient
Standard Error
PCY -38.27 12.41 PCY-squared 0.91 0.37
Constant 418.11 61.47 R-squared 0.14
N 95.00 N*R-squared 13.09 Critical Chi-
Square at α=.05 5.99
2aNR
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Fixing the Problem at Its Source
• If your data are averages:
0 1i i iY Xβ β ε= + +2
( )ii
Varnσε =
• We know the cause of the heteroskedasticity, so transform the regression equation by multiplying through by in
0 1i i i i i i in Y n n X nβ β ε= + +
2 22
2( ) ii i
i
nVar nnσε σ= =
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Fix It By Getting the Model Right
• Fit a line instead of a cubic and it LOOKS like the variance is related to Q (heteroskedasticity)
• Fit a cubic and you should get rid of the problem
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What If the Model’s Right and There’s Still Heteroskedasticity?
• White’s solution: Replace s2 withwhen calculating the variance
Before:
White’s Method(For simple regression;for multiple, use v method):
2ˆie
1
2 22
2 12
22
11
N
ii
b NN
iiii
x sss
xx
=
==
= =
∑
∑∑
2 2
11 2
2
1
N
i ii
N
ii
x eVb
x
=
=
=
∑
∑
Heteroskedasticity-Consistent Estimator, HCEStata: reg y x1 x2, robust
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Moving Off the Farm With and Without White’s Correction
• Here, the heteroskedasticity is not enough to change hypothesis test results, but it has a bigeffect on confidence intervals– The corrected standard error is almost 50% larger!
• In other cases, test results can change, too
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White-corrected Confidence Intervals
01.871.6598.1*04.471.65 ±=±Uncorrected Regression:
Robust Regression:
61.1171.6598.1*86.571.65 ±=±
In this example, we get a confidence interval that is quite a bit smaller than it should be when we ignore heteroskedasticity
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Log Regression Gets Rid of Heteroskedasticity in CO2 Study
Taking the log compresses the huge differences in income evident in figure 8.2. Country GNIs range from $0.19 billion to $15,170 billion. The log ranges from –1.66 to 9.63.
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Heteroskedasticity-Robust Standard ErrorsUncorrected Robust
2 12282.71 0.44(36713.99) (0.03)
i i iCO GNI e= + +
Sample Size = 182
R-squared = 0.61
White test: NR2 = 50.84
2 12282.71 0.44(16752.29) (0.12)
i i iCO GNI e= + +
Sample Size = 182
R-squared = 0.61
Uncorrected Robust
ln( 2 ) 1.09 1.01ln( )(0.27) (0.026)
i i iCO GNI e= − + +
Sample Size = 182
R-squared = 0.90
White test: NR2 = 0.73
ln( 2 ) 1.09 1.01ln( )(0.25) (0.023)
i i iCO GNI e= − + +
Sample Size = 182
R-squared = 0.90
5% critical value for χ2
(2) = 5.99
Success!
Uncorrected Robust
2 2155.05 0.20
(458.28) (0.02)
i ii
i i
CO GNI ePOP POP
= + +
Sample Size = 182
R-squared = 0.33
White test: NR2 = 16.52
2 2155.05 0.20
(407.25) (0.04)
i ii
i i
CO GNI ePOP POP
= + +
Sample Size = 182
R-squared = 0.33
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What We Learned• Heteroskedasticity means that the error variance is
different for some values of X than for others; it can indicate that the model is misspecified.
• Heteroskedasticity causes OLS to lose its “best” property and it causes the standard error formula to be wrong (i.e., estimated standard errors are biased).
• The standard errors can be corrected with White’s heteroskedasticity-robust estimator.
• Getting the model right by, for example, taking logs can sometimes eliminate the heteroskedasticity problem.