HETEROSCEDASTICITY

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HETEROSCEDASTICITY

• The assumption of equal variance Var(ui) s2,
  for all i, is called homoscedasticity, which
  means equal scatter (of the error terms ui
  around their mean 0)

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• Equivalently, this means that the dispersion of
  the observed values of Y around the regression
  line is the same across all observations
• If the above assumption of homoscedasticity does
  not hold, we have heteroscedasticity (unequal
  scatter)

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• Consequences of ignoring
• heteroscedasticity during the OLS procedure
• The estimates and forecasts based on them will
  still be unbiased and consistent
• However, the OLS estimates are no longer the best
  (B in BLUE) and thus will be inefficient.
  Forecasts will also be inefficient

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• The estimated variances and covariances of the
  regression coefficients will be biased and
  inconsistent, and hence the t- and F-tests will
  be invalid

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• Testing for heteroscedasticity
• 1. Before any formal tests, visually examine the
  models residuals ûi
• Graph the ûi or ûi2 separately against each
  explanatory variable Xj, or against Yi, the
  fitted values of the dependent variable

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• 2. The Goldfeld-Quandt test
• Step 1. Arrange the data from small
• to large values of the indp variable Xj
• Step 2. Run two separate regressions,
• one for small values of Xj and one for
• large values of Xj, omitting d middle
• observations (app. 20), and record
• the residual sum of squares RSS for
• each regression RSS1 for small
• values of Xj and RSS2 for large Xjs.

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• Step 3. Calculate the ratio
• F RSS2/RSS1, which has an F distribution with
• d.f. n d 2(k1)/2 both in the numerator
  and the denominator, where n is the total of
  observations, d is the of omitted observations,
  and k is the of explanatory variables.

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• Step4. Reject H0 All the variances si2 are equal
  (i.e., homoscedastic) if F gt Fcr, where Fcr is
  found in the table of the F distribution for
• n-d-2(k1)/2 d.f. and for a predetermined
  level of significance a, typically 5.

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• Drawbacks of the the Goldfeld-Quandt test
• It cannot accommodate situations where several
  variables jointly cause heteroscedasticity
• The middle d observations are lost

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• 3. Lagrange Multiplier (LM) tests (for large
  ngt30)
• The Breusch-Pagan test
• Step 1. Run the regression of ûi2
• on all the explanatory variables. In
• our example (CN p. 37), there is
• only one explanatory variable, X1,
• therefore the model for the OLS
• estimation has the form
• ûi2 a0 a1X1i vi

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• Step 2. Keep the R2 from this regression. Lets
  call it Rû22 Calculate either
• (a) F (Rû22/k)/(1-Rû22)/(n-(k1), where k is
  the of explanatory variables the F statistic
  has an F distribution with d.f. k, n-(k1)
• Reject H0 All the variances si2 are equal
  (i.e., homoscedastic) if F gtFcr

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• or
• (b) LM n Rû22, where LM is called the
  Lagrangian Multiplier (LM) statistic and has an
  asymptotic chi-square (?2) distribution with d.f.
  k
• Reject H0 All the variances si2 are equal
  (i.e., homoscedastic)
• if LMgt ?cr2

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• Drawbacks of the Breusch-Pagan test
• It has been shown to be sensitive to any
  violation of the normality assumption
• Three other popular LM tests the Glejser test
  the Harvey-Godfrey test, and the Park test, are
  also sensitive to such violations (wont be
  covered in this course)

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• One LM test, the White test, does not depend on
  the normality assumption therefore it is
  recommended over all the other tests

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• The White test
• Step 1.The test is based on the regr.
• of û2 on all the explanatory varia-
• bles (Xj), their squares (Xj2), and
• all their cross products. E.g., when
• the model contains k 2 explanat.
• variables, the test is based on an estim. of the
  model û2 ß0 ß1X1 ß2X2ß3X12ß4X22 ß5X1X2
  v

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• Step 2. Compute the statistic
• ?2 nRû22, where n is the sample size and Rû22
  is the unadjusted R-squared from the OLS
  regression in Step 1. The statistic ?2 nRû22,
  has an asymptotic chi-square (?2) distrib. with
  d.f. k, where k is the of ALL explanatory
  variables in the AUXILIARY model.
• Reject H0 All the variances si2 are equal
  (i.e., homoscedastic) if ?2 gt ?cr2

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• Estimation Procedures when H0 is rejected
• 1. Heteroscedasticity with a known proportional
  factor
• If it can be assumed that the error variance is
  proportional to the square of the indep. variable
  Xj2, we can correct for heteroscedasticity by
  dividing every term of the regression by X1i and
  then reestimating the model using the transformed
  variables. In the two-variable case, we will have
  to reestimate the following model (CN, p. 39)
• Yi/X1i ß0/X1i ß1 ui/X1i

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• 2. Heteroscedasticity consistent covariance
  matrix (HCCM)
• As we know, the usual OLS inference is faulty in
  the presence of hetero-scedasticity because in
  this case the estimators of variances Var(bj) are
  biased. Therefore, new ways have been developed
  for estimation of hetero-scedasticity-robust
  variances.
• The most popular is the HCCM procedure proposed
  by White.

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• The heteroscedasticity consistent covariance
  matrix (HCCM) procedure.
• Lets consider the model Yi ß0 ß1X1i ß2X2i
  ... ßkXki ui
• Step 1. Estimate the initial model by the OLS
  method. Let ûi denote the OLS residuals from the
  initial regression of Y on X1, X2, .., Xk

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• Step 2. Run the OLS regression of Xj (each time
  for a different j) on all other independent
  variables. Let wij denotes the ith residual from
  regressing Xj on all other independent variables.

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• Step 3. Let RSSj be the residual sum of squares
  from this regression RSSj SXjXj(1-R2).
• RSSj can also be calculated as
• RSSj n-(k1)SER2, where SER is the standard
  error of regression and can easily be found in
  the Excels OLS solution.

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• Step 4. The heteroscedasticity-robust variance
  Var(bj) can be calculated as follows
• Var(bj) Swij2ûi2/RSSj2.
• The square root of Var(bj) is called the
  heteroscedasticity-robust standard error for bj.
• Example CN, p. 44.

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• 3. Feasible Generalized Least Squares (FGLS)
  method
• Step 1. Compute the residuals ûi from the OLS of
  the initial regression model

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• Step 2. Regress ûi2 against a constant term and
  all the explanatory variables from either the
  Breusch-Pagan test for heteroscedasticity (e.g.,
  when k 2
• ûi2 a0 a1X1i a2X2i vi )
• or the White test for heteroscedasticity
• ûi2 a0 a1X1i a2X2i a3X1i2 a4X2i2
  a5X1i X2i vi

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• Step 3. Estimate the original model by OLS using
  the weights zi 1/si, where si2 are the
  predicted values of the dependent variable (the
  ûi2) in the Breusch-Pagan (or White) model. Note
  the model must be estimated without a constant
  term.
• Such OLS procedure is called
• WLS (weighted least squares).

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• It may happen that the predicted values si2 of
  the dependent variable may not be positive, so we
  cannot calculate the corresponding weights zi
  1/si. If this situation arises for some
  observations, then we can use the original ûi2
  and take their positive square roots.