Class Outline

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Class Outline

• Generalized Linear Model
• Heteroscedasticity
• OLS under GLS
• Feasible Generalized Least Square Estimator
• Testing for Heteroscedasticity
• Park Test
• Glejser Test
• White Test
• Breusch-Pagan/Godfrey Test
• Goldfeld-Quandt Test
• Estimation and Inference
• Known Variance
• Known Variance Structure
• Proportional to the Square of the explanatory
  variable
• Proportional to the explanatory variable
• Unknown Variance Structure
• Reading Chapter 11

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Heteroscedasticity
In the OLS model we assume that the variance of
the error term is constant (homoscedasticity) B
ut, if we have heteroscedasticity, then
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Heteroscedasticity
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Heteroscedasticity

• Reasons for Heteroscedasticiy
• Error learning models
• Differences across individuals
• Data collecting techniques
• Outliers
• Model misspecification
• Skewness in the distribution of one or more
  regressors
• Incorrect data transformation and incorrect
  functional form

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Heteroscedasticity

• OLS Estimation and Heteroscedasticity
• Assume the following model
• Under the assumption of OLS, the variance for the
  estimator under homoscedasticity is

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Heteroscedasticity

• But, if we have heteroscedasticity, the variance
  is
• In this case, the OLS estimator is still linear
  and unbiased, but it is not the estimator with
  the minimum variance, it is not BLUE.
• The Generalized Least Squares (GLS) model takes
  this into account and produce estimators that are
  BLUE.

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Generalized Linear Model

• Multiple Linear Model
• Assumptions
• E(u)0
• V(u)?2I
• X is a non-stochastic matrix with column rank
  ?(x)K

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Generalized Linear Model

• According to the Gauss-Markov Theorem the least
  squares estimator of ? is the best linear
  unbiased estimator
• Now we will relax the assumption related to the
  variance of u
• We define the variance of u as V(u)?
• We impose two minimum conditions on ?
• ? must be symmetric
• ? must be positive definite

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Generalized Linear Model

• Structure of the Generalized Linear Model

Assumptions E(e)0 V(e) ? X is a non-stochastic
matrix with column rank ?(x)K
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Generalized Linear Model

• Relaxing the assumption of the variance will
  nullify the Gauss-Markov Theorem
• The OLS estimator will no longer be the Best
  Linear Unbiased Estimator
• The optimal strategy is to look for the Best
  Linear Unbiased Estimator for the Generalized
  Linear Model

11
OLS Under the Generalized Linear Model

• If we have heteroscedasticity but we try to
  calculate the Ordinary Least Square (OLS)
  estimator,
• Then,
• ?OLS is still a linear estimator
• ?OLS is still unbiased (we do not need the
  assumption of constant variance to prove
  unbiasedness)
• ?OLS is no longer the BLUE of ?
• Now,
• S2(XX)-1 is no longer an unbiased estimator of
  V(?OLS)

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Feasible Generalized Least Square Estimator

• In most practical cases we do not know ?, so we
  have to use an estimator, which we call
• Then, the estimator that results from replacing ?
  by in the GLS estimator is called the
  feasible generalized least square estimator FGLS

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Feasible Generalized Least Square Estimator

• We are not sure if ?FGLS is a linear estimator
• But, if is a consistent estimator of ?,
  that is, if the sample size is very large, then
  approaches ?, and as a consequence ?FGLS
  approaches ?GLS.
• If is consistent, it can be shown that
  ?FGLS is asymptotically normal, that is, when the
  sample size is very large it has a distribution
  that is very close to the normal distribution.
  Then, inference procedures for the FGLS are valid
  for large samples.

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Testing for Heteroscedasticity

• Park Test
• If the is heteroscedasticity the variance can be
  related to one or more of the independent
  variables
• But we do not know the real value of ?2i. Park
  suggests using as proxies for ui. Then

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Testing for Heteroscedasticity

• Park Test Steps
• Run the original regression
• Obtain the residuals
• Run the regression with the squared residuals
• Test the null hypothesis that ?20. If we found
  this coefficient to be significant then we can
  have heteroscedasticity
• Otherwise, the model is homoscedastic

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Testing for Heteroscedasticity

• Glejser Test
• Similar to the Park test, but considers using the
  absolute value of the error term

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Testing for Heteroscedasticity

• White Test
• Null Hypothesis is that the variance is constant,
  and the Alternative Hypothesis is that there is
  heteroscedasticity
• H0 V(e)?2
• H1 V(e) ?2i i1,2,,n
• Consider the model

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White Test

• Steps for the Whites Test
• 1. Estimate the previous model by OLS and save
  the square of the residuals ( 2)
• 2. Regress 2 on all the variables of the
  previous model, their squares and all possible
  cross-products, that is regress u2 on
• 1 X2 X3 X22 X32 X2X3
• and obtain R2 in this regression
• 3. Under the Null Hypothesis, the statistic nR2
  has the ?2(k-1) distribution

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White Test

• k is the number of variables in this special
  regression
• We reject H0 (homoscedasticity) if nR2 is
  significantly different from zero.
• Problems
• Large Sample
• Does not provide enough information if we do not
  reject the null hypothesis
• If we reject the null, there is no much
  information regarding the causes of it.

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Breusch-Pagan/Godfrey Test

• This test could be more informative about the
  causes of the heteroscedasticity
• We will check if certain variables are the causes
  of heteroscedasticity
• Consider the general model
• where the uis are normally distributed with
  E(u)0 and

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Breusch-Pagan/Godfrey Test

• In this equation, note that when
• which is a constant. Then, the test is

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Breusch-Pagan/Godfrey Test

• Steps to calculate this test
• Estimate the original model and calculate the
  squared residuals ui2.
• Obtain an estimator for the variance
• 3. Construct the variable pi, defined as

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Breusch-Pagan/Godfrey Test

• 4. Regress pi on the Zik variables k2,,g and
  obtain the Explained Sum of Squares (SSR). The
  test statistic is
• the test statistic has an asymptotic ?2
  distribution with p-1 degrees of freedom

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Goldfeld-Quandt Test

• This test is useful if we believe that
  heteroscedasticity can be attributed to a
  specific variable
• Consider the simple linear model
• where it is thought that ?2iV(ui) is related to
  the variable xi.

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Goldfeld-Quandt Test

• Steps for the test
• 1. Sort the observations according to the value
  of xi
• 2. Eliminate C central observations
• 3. Run two separate regressions using the first
  and the last (n-C)/2 observations and compute the
  sum of square residuals for each of these
  regressions. Call SSE1 to the first and SSE2 to
  the second
• 4. Divide by n-k these two numbers give unbiased
  estimates of the variances of the error terms of
  each regression

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Goldfeld-Quandt Test

• Then under the null hypothesis of
  homoscedasticity they should be equal, or
  alternatively SSE1/SSE2 should be very close to
  1. Then, a simple test F of variance equality can
  be used,

Note Econometricians do not agree on the value
of C. Goldfeld and Quandt suggest that C8 if
n30 and c16 if n60. But, Judge et. al. note
that C4 if n30 and c10 if n60.
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Estimation and Inference Under Heteroscedasticity

• Remember that under heteroscedasticity, the
  variance of u is,
• The best linear unbiased estimator under
  heteroscedasticity is the Generalized Least
  Square Estimator (GLS)
• With XPX, YPY and uPu, and PP?-1

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Estimation and Inference Under Heteroscedasticity

• In this particular case, when we know the
  variance, P is equal to,
• Then, multiplying by P has the effect of dividing
  each observation by its standard deviation.
  Hence, GLS estimation proceeds by performing OLS
  on these transformed variables best linear
  unbiased estimator under heteroscedasticity is
  the Generalized Least Square Estimator (GLS)

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Estimation and Inference Under Heteroscedasticity

• The problem is that in practice we rarely know
  the ?is.
• Any attempt to estimate them requires to estimate
  Kn estimators (the coefficients of the
  regression plus the n unknown variances). This is
  impossible to do with just n observations.
• Options
• If we have some information about the variances,
  we could look for a Generalized Least Square
  Estimator
• Given that the least square estimator is unbiased
  though no efficient, we could try to look for a
  valid estimator for the variance matrix of the
  estimators

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Known Variance Structure

• Consider the simple two-variable case
• If we do not know the variances of the uis we
  can assume some particular form for the
  heteroscedasticity

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Known Variance Structure

• Variance proportional to the square of the
  explanatory variable
• To obtain a GLS estimator for the model, divide
  all observations by xi,

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Known Variance Structure

• Note that
• In this case we do not need to know the variance
  of the error term
• This strategy provides a GLS instead of a FGLS
  (this is consequence of the assumption about the
  variance)
• Note that now the intercept of the transformed
  model is the slope of the original model and the
  slope of the transformed model is the intercept
  in the original model

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Known Variance Structure

• Variance proportional to the explanatory variable
• Then, the transformed model is,

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Unknown Variance Structure

• In many cases we do not know the variance
  structure. This alternative strategy consists in
  retaining the OLS estimator (which is unbiased)
  and look for a valid estimator for its variance.
• Remember, the variance of the OLS estimator under
  heteroscedasticity is,
• Where ? is the variance matrix. In order to
  estimate this variance we need to estimate n
  parameters.
• White showed that a consistent estimator for X
  ?X is given by XDX, where D is a nxn diagonal
  matrix with the typical element equal to the
  square of the residuals of the OLS regression for
  the original model

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Unknown Variance Structure
Note that D is not a consistent estimator of ?,
but XDX is a consistent estimator of X ?X.
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Unknown Variance Structure
Then, a heteroscedasticity consistent estimator
of the variance matrix is given by The
strategy consists in using OLS to estimate B, but
replace the variance of OLS by Whites consistent
estimator for the variance. This provides an
unbiased estimator of ?, consistent estimation of
the V(BOLS) and a valid inference framework for
large samples