Chapter 8: Heteroskedasticity

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Chapter 8 Heteroskedasticity
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• 1. Introduction

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Introduction

• A CLRM assumption that the disturbances ui are
  homoscedastic
• They all have the same variance.
• So var(ui) ?2 a constant
• Suppose the variance varies across observations
• Then we have heteroskedasticity.
• So var(ui) (?i)2 which varies with each
  observation

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Example

• Average wages rise with the size of firm. Suppose
  wages look like this

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Example

• Can we expect the variance of wages to be
  constant?
• The variance increases as firm size increases.
• So larger firms pay more on average, but there is
  more variability in their wages.

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Savings Example

• Savings increases with income, so does the
  variability of savings or spending
• As incomes grow, people have more discretionary
  income, so more scope for choice about how to
  dispose of it.

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Overall

• Heteroskedasticity is more likely in cross
  sectional than time-series data.

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• 2. Consequences of
• Heteroskedasticity

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Consequences

• If we have heteroskedasticity , what happens to
  our estimator?
• Still linear
• Still unbiased
• Not the most efficient - it does not have minimum
  variance.
• So it is not BLUE.

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Consequences

• If we use usual variance formulas, they will be
  biased
• This is because the estimator is not an unbiased
  estimator of ?2
• So F tests and t tests are unreliable

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• 3. Detecting Heteroskedasticity

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Detecting Heteroskedasticity

• Past research indicates it
• Know that scale effects will exist
• Ex spending patterns in relation to income
• Firm profitability or investment spending in
  relation to the size of the firm.

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Detecting Heteroskedasticity

• Examine residuals
• Assume no heteroskedasticity and run OLS and then
  look at estimated residuals
• In 2-variable model
• Plot squared residuals against the independent
  variable.
• Plain residual has no correlation with X or Y

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Detecting Heteroskedasticity

• In multivariate model, do against different Xs,
  or against the predicted value of Y.
• Predicted Y is a linear combination of the Xs.
• Graph could show linear or quadratic relationship
• plots provide clues as to the nature of the
  heteroskedasticity and how we might transform
  variables.

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• 4. Park Test

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Park Test

• If we find some evidence of heteroskedasticity by
  looking at the residuals we can do an explicit
  test.
• Regress the variance on the X variables.
• Ln(?i)2b1 b2lnXi vi

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Park Test

• Dont know the variance
• Use squared residuals as a proxy
• Run ln(ei)2b1 b2lnXi vi
• For a multivariate model, run the squared
  residuals against each X variable, or against the
  predicted Y.
• If b2 is significantly different from 0, then we
  have heteroskedasticity.

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• 4. Glejser Test

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Glejser test

• Similar test to the Park test.
• Regress the absolute values of ei on X.
• The form of the regression may vary
• Can run on square root of X or 1/X etc.
• If significant ts, then heteroskedasticity

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• 5. Goldfeld-Quandt Test

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Goldfeld-Quandt test

• Order the observations according to the magnitude
  of the X thought to be related.
• Divide observations into two groups, one with low
  values of X and one with high, omitting some
  central observations.

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Goldfeld-Quandt test

• Run two separate regressions
• Calculate F test
• FESSlarge X/df/ESSsmall X/df
• Should be unity for homoskedasticity.

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• 6. Remedial Measures When ? is Known

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Remedies

• OLS estimators are not efficient under
  heteroskedasticity, though they are unbiased and
  consistent.
• We can transform the model to get rid of the
  heteroskedasticity.
• If we know ?2 then use the method of weighted
  least squares.

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Weighted Least Squares

• Suppose have heteroskedasticity as in the firm
  data example
• Wages increase with size of firm, but also the
  variance increases.
• To correct for heteroskedasticity Give less
  weight to data points from populations with
  greater variability and more weight to those that
  have smaller variability.

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Weighted Least Squares

• OLS gives equal weight to all observations.
• Weighting observations is called weighted least
  squares
• A subset of generalized least squares
• Using this method leads to BLUE estimators.

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Weighted Least Squares

• Start with basic model
• Yi b1 b2Xi ui
• Y wages and X firm size
• Assume the true error variance (?i)2 is known for
  each observation.
• Divide through by the standard deviation
• Y/ ?i b1(1/ ?i) b2Xi/ ?i ui/?i

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Weighted Least Squares

• Look at the error term.
• Let vi ui/?i
• If is vi homoskedastic then OLS on this model
  will give us BLUE estimators
• Square vi
• (vi)2 (ui)2/(?i)2
• E(vi)2 E(ui)2/(?i)2

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Weighted Least Squares

• Since (?i)2 is known, E(vi)2 becomes 1/(?i)2
  E(ui)2
• But E(ui)2 (?i)2
• So this becomes 1/(?i)2 (?i)2 1
• So the transformed error terms is heteroskedastic
• Estimate this model by OLS to get BLUE
  estimators.
• So WLS is OLS on the transformed variables.

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Weighted Least Squares

• In OLS, minimize
• (ei)2 (Y- b1 - b2X2)2
• In GLS, minimize
• w(Y- b1 - b2X2)2
• where w 1/?i

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Weighted Least Squares

• In GLS we minimize a weighted sum of squares
• The weights we are using are inversely
  proportional to ?i
• observations from a population with large
  variance will get smaller weights and vice versa.

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Weighted Least Squares

• OLS minimizes the sum of squared residuals
• But these residuals are very large for the
  observations with a large variance, so it is
  giving these more weight.
• GLS corrects for this

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• 7. Remedial Measures When ? is not Known

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Remedies

• We have to make assumptions about (?i)2 if we
  dont know it.
• Plot residuals against X and find a cone shape
• This indicates that the error variance is
  linearly related to X.
• Now transform the model by dividing Y, the
  intercept, X and the error term by square root of
  X.

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Remedies
It can be proved that the error variance in this
model is homoskedastic and we can estimate by OLS
(actually its a form of GLS).
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Remedies

• Plot residuals against X and find a trumpet shape
• This indicates that the error variance increases
  proportional to the square of X.
• Now transform the model by dividing Y, the
  intercept, X and the error term by X.

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Remedies
Slope has become intercept and intercept becomes
slope. But this changes back when we multiply
out by X
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Remedies

• Log transformation of both sides
• Log transformation compresses the scales in which
  the variables are measured
• This reduces the differences.
• Cannot do this if some Y and X values are zero or
  negative.