Heteroskedasticity

1
Heteroskedasticity

• Lecture 17

2
Todays plan

• How to test for it graphs, Park and Glejser
  tests
• What we can do if we find heteroskedasticity
• How to estimate in the presence of
  heteroskedasticity

3
Palm Beach County revisited

• How far is Palm Beach an outlier?
• Can the outlier be explained by
  heteroskedasticity?
• If so, what are the consequences?
• Heteroskedasticity will affect the variance of
  the regression line
• It will consequently affect the variance of the
  estimated coefficients and estimated 95 percent
  confidence interval for the prediction (see
  Lecture 10).
• L17.xls provides an example of how to work
  through a problem like this using Excel

4
Palm Beach County revisited (2)

• Palm Beach is a good example to use since there
  are scale effects in the data
• The voting pattern shows that the voting behavior
  and number of registered voters are related to
  the population in each county
• As the county gets larger, voting patterns may
  diverge from what would be assumed given the
  number of registered voters
• Note from the graph as we move away from the
  origin, the difference between registered Reform
  voters and Reform votes cast increases
• Well hypothesize that this will have an affect
  on heteroskedasticity

5
Notation

• Heteroskedasticity is observed as cross-section
  variability in the data
• data across units at point in time
• In our notation, heteroskedasticity is
• E(ei2) ? ?2
• We can also write
• E(ei2) ?i2
• This means that we expect variable variance the
  variance changes with each unit of observation

6
Consequences

• When heteroskedasticity is present
• 1) OLS estimator is still linear
• 2) OLS estimator is still unbiased
• 3) OLS estimator is not efficient - the minimum
  
• variance property no longer holds
• 4) Estimates of the variances are biased
• 5)
• is not an unbiased estimator of sYX2
• 6) We cant trust the confidence intervals or
• hypothesis tests (t-tests F-tests) we
  may draw the
• wrong conclusions

7
Consequences (2)

• When BLUE holds and there is homoskedasticity,
  the first-order condition gives
• With heteroskedasticity, we have
• If we substitute the equation for ci to both
  equations, we find

where
8
Cases

• With homoskedasticity around each point, the
  variance around the regression line is constant
• With heteroskedasticity around each point, the
  variance around the regression line varies with
  each value of the independent variable (with each
  i)

9
Detecting heteroskedasticity

• There are three ways of detecting
  heteroskedastiticy
• 1) Graphically
• 2) Park Test
• 3) Glejser Test

10
Graphical detection

• Graph the errors (or error squared) against the
  independent variable(s). Note you can use either
  e or e2 on the y-axis.
• With homoskedasticity we have E(ei, X) 0
• The errors are independent of the independent
  variables
• With heteroskedasticity we can get a variety of
  patterns
• The errors show a systematic relationship with
  the independent variables

11
Graphical detection (2)

• Using the Palm Beach example (L17.xls), the
  estimated regression equation was

• The errors of this equation, can be graphed
  against the number of registered Reform party
  voters, (the independent variable)
• Graph shows that the errors increasing with the
  number of registered reform voters
• While the graphs may be convincing, we also want
  to use a test to confirm this. We have two

12
Park Test

• Procedure
• 1) Run regression Yi a bXi ei despite the
  heteroskedasticity problem (it can also be
  multivariate)
• 2) Obtain residuals (ei), square them (ei2), and
  take their logs (ln ei2)
• 3) Run a spurious regression
• 4) Do a hypothesis test on with H0 g1 0
• 5) Look at the results of the hypothesis test
• reject the null you have heteroskedasticity
• fail to reject the null homoskedasticity, or
  which is a constant

13
Glejser Test

• When we use the Glejser, were looking for a
  scaling effect
• The procedure
• 1) Run the regression (it can also be
  multivariate)
• 2) Collect ei terms
• 3) Take the absolute value of the errors
• 4) Regress ei against independent variable(s)
• you can run different kinds of regressions

14
Glejser Test (2)

• 4) continued
• If heteroskedasticity takes one of these forms,
  this will suggest an appropriate transformation
  of the model
• The null hypothesis is still H0 g1 0 since
  were testing for a relationship between the
  errors and the independent variables
• We reach the same conclusions as in the Park Test

15
A cautionary note

• The errors in the Park Test (vi) and the Glejser
  Test (ui) might also be heteroskedastic.
• If this is the case, we cannot trust the
  hypothesis test H0 g1 0 or the t-test
• If we find heteroskedastic disturbances in the
  data, what can we do?
• Estimate the model Yi a bXi ei using
  weighted least squares
• Well look at two examples of weighted least
  squares one where we know the true variance, and
  one where we dont

16
Correction with known ?i2

• Given that the true variance is known and our
  model is
• Yi a bXi ei
• Consider the following transformation of the
  model

• In the transformed model, let
• So the expected value of the error squared is

17
Correction with known ?i2 (2)

• Given that there is heteroskedasticity, E(ei2)
  ?i2
• thus
• In this simplistic example, we re-weighted model
  by the constant ?i
• What this example shows when the variance is
  known, we must transform our model to obtain a
  homoskedastic error term.

18
Correction with unknown ?i2

• Given an unknown variance, we need to state the
  ad-hoc but plausible assumptions with our
  variance ?i2 (how the errors vary with the
  independent variable)
• For example we can assert that E(ei2) ?2Xi
• Remember Glejser Test allows us to choose a
  relationship between the errors and the
  independent variable

19
Correction with unknown ?i2 (2)

• In this example you would transform the
  estimating equation by dividing through by
  to get

• Letting
• The expected value of this error squared is

20
Correction with unknown ?i2 (3)

• Recalling an earlier assumption, we find

• When we dont know the true variance we re-scale
  the estimating equation by the independent
  variable

21
Returning to Palm Beach

• On L17.xls we have presidential election data by
  county in Florida
• To get a correct estimating equation, we can run
  a regression without Palm Beach if we think its
  an outlier.
• Then we can see if we can obtain a prediction for
  the number of reform votes cast in Palm Beach
• We can perform a Glejser Test for the regression
  excluding Palm Beach
• We run a regression of the absolute value of the
  errors (ei)against registered Reform voters (Xi)

22
Returning to Palm Beach (2)

• The t-test rejects the null
• this indicates the presence of heteroskedasticity
• We can re-scale the model in different ways or
  introduce a new independent variable (such as the
  total number of registered voters by county)
• Keep transforming the model and running the
  Glejser Test
• When we fail to reject the null there is no
  longer heteroskedasticity in the model

23
Robust estimation

• Heteroskedastic tests not used any more. Most
  software reports robust standard errors. Note
  that this is also the approach of the text book.
• Have looked at tests for heteroskedasticity to
  get you used to weighted least squares.
  Important for the topics to come.
• Robust standard errors report approximations to
  the estimation of the variance for the
  coefficient when there is a non-constant
  variance. It only holds for large samples.
• Know that for a homoskedastic error term
  Var(uiXi) s2
• Var(b) s2/Sxi2

24
Robust estimation (2)

• Using analogous arguments, we can state that for
  the heteroskedastic case Var(uiXi) si2
• Var(b) si2 Sxi2 /(Sxi2)2
• This can be approximated (in the bi-variate model
  case) by
• Var(b) Sxi2ui2 /(Sxi2)2
• See L17_robust.xls and hetero.pdf to compare the
  results from calculating the robust standard
  error on the spreadsheet using EXCEL and the
  results from STATA for robust estimation.

25
Summary

• Even with re-weighted equations, we might still
  have heteroskedastic errors
• so we have to rerun the Glejser Test until we
  cannot reject the null
• If we cannot reject the null, we may have to
  rethink our model transformation
• if we suspect a scale effect, we may want to
  introduce new scaling variables
• Variables from the re-scaled equation are
  comparable with the coefficients from the
  original model