Heteroskedasticity

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Heteroskedasticity
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What is Heteroskedasticity?

• One of the assumptions of the CLR model
  (assumption V) is that the error term in the
  linear model is drawn from a distribution with a
  constant variance
• When this is the case, we say that the errors are
  homoskedastic
• If this assumption does not hold, we run into the
  problem of heteroskedasticity
• Heteroskedasticity, as a violation of the
  assumptions of the CLR model, causes the OLS
  estimates to lose some of their nice properties

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What is Heteroskedasticity?

• Example Suppose we take a cross-section sample
  of firms from a certain industry and we want to
  estimate a model with sales as the dependent
  variable
• We may suspect that sales of larger firms will
  vary more than those of smaller firms, implying
  that there will be heteroskedasticity
• Heteroskedasticity is common in cross-section
  data and needs to be identified and corrected
  (firms, banks, insurance companies, mutual funds,
  real estate, etc.)

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The Heteroskedasticity Problem

• Heteroskedasticity can be distinguished between
  two versions
• Pure Heteroskedasticity
• Impure Heteroskedasticity
• Pure heteroskedasticity refers to
  heteroskedasticity that occurs in a correctly
  specified model
• This term can be used to distinguish it from the
  case of impure heteroskedasticity caused by a
  specification error, such as an omitted variable
  bias
• Typically, when we refer to heteroskedasticity,
  we imply the pure version

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The Heteroskedasticity Problem

• If heteroskedasticity is present in a correctly
  specified model, then the variance of the error
  term is not constant
• for i 1, 2, , n
• Implication Rather than being constant across
  observations, the variance of the error term
  changes depending on the observation

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The Heteroskedasticity Problem

• Heteroskedasticity is common in cross-section
  data where there is a wide disparity between
  large and small observed values of the variables
• The larger is this disparity, the higher is the
  chance that the error term will not have a
  constant variance
• In the most frequent model of heteroskedasticity,
  the variance of the error term (?i) depends on an
  exogenous variable (Zi)

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The Heteroskedasticity Problem

• We write
• The variable Z, called the proportionality
  factor, may or may not be one of the explanatory
  variables in the regression model
• The higher is the value of Z, the higher is the
  variance of the error term for observation i

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The Heteroskedasticity Problem

• Example Trying to explain firm sales with a
  cross-section sample of firms, the variable Zi
  may be the asset size of firm i
• This would imply that the larger the asset size
  of firm i the more volatile will be that firms
  sales
• This example shows that heteroskedasticity is
  likely to occur in a cross-section sample because
  of the larger variability in the observations of
  the dependent variable
• Heteroskedasticity may also occur in time series
  data (more rarely) or when the quality of data
  changes in the sample

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The Heteroskedasticity Problem

• If there is specification error in our model,
  such as omitted variable bias, the errors may
  exhibit impure heteroskedasticity
• Example In the two-factor model of stock returns
  estimated without the variable (INF), the errors
  may exhibit heteroskedasticity
• The error term now includes the variable (INF)
  and the error term may be larger for larger
  values of the variable (INF)

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The Impact of Heteroskedasticity on the OLS
Estimates

• Heteroskedasticity has the following implications
  for the OLS estimates
• OLS estimates are still unbiased
• OLS estimates do not have the minimum variance
  anymore (not efficient)
• Heteroskedasticity causes OLS to underestimate
  the variances and standard errors of the
  estimated coefficients
• This implies that the t-test and F-test are not
  reliable
• The t-statistics tend to be higher leading us to
  reject a null hypothesis that should not be
  rejected

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Testing for the Presence of Heteroskedasticity

• There are several tests that can be used to
  detect the presence of heteroskedasticity in the
  data
• We will cover two such tests
• The Breusch-Pagan Test
• The White Test
• These test the following null hypothesis against
  the alternative that it is not true

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Testing for the Presence of Heteroskedasticity

• The steps of the Breusch-Pagan test are as
  follows
• Estimate the original regression model by OLS and
  obtain the squared OLS residuals (one for each
  observation)
• Run a new linear regression of the squared OLS
  residuals on all the explanatory variables
• Obtain the R-sq of this regression
• Calculate the following F-statistic using the
  above R-sq

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Testing for the Presence of Heteroskedasticity

• The above F-statistic follows an F distribution
  with k degrees of freedom in the numerator and (n
  k 1) degrees of freedom in the denominator
• Reject the null hypothesis that there exists no
  heteroskedasticity if the F-statistic is greater
  than the critical F-value at the selected level
  of significance
• If the null cannot be rejected, then there exists
  heteroskedasticity in the data and an alternative
  estimation method to OLS must be followed

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Testing for the Presence of Heteroskedasticity

• The steps of the White test are as follows
• Estimate the original regression model by OLS and
  obtain the squared OLS residuals (one for each
  observation)
• Run a new regression of the squared OLS residuals
  on each explanatory variable X, the square of
  each explanatory variable X, and the product of
  each variable X times every other variable X
• Example If our original model has two
  explanatory variables, then we would run the
  following regression at the second stage

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Testing for the Presence of Heteroskedasticity

• Obtain the R-sq of this regression
• Calculate the test statistic nR2 where n is the
  sample size and R-sq is that obtained in the
  previous step
• This statistic follows a chi-square distribution
  with K degrees of freedom (K number of
  variables included in the second stage
  regression)
• In our example above, there are five explanatory
  variables, so the test statistic nR2 will follow
  a chi-square distribution with five degrees of
  freedom
• Reject the null hypothesis that there exists no
  heteroskedasticity if the test-statistic is
  greater than the critical ?2-value at the
  selected level of significance

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Correcting the Problem of Heteroskedasticity

• If heteroskedasticity is detected, an alternative
  estimation approach to OLS must be used that
  corrects this problem
• Two commonly-used approaches are
• The method of Weighted Least Squares (WLS)
• Obtaining heteroskedasticity-corrected standard
  errors
• The WLS method transforms the original regression
  model in order to make the errors have the same
  variance
• After this transformation takes place, the model
  can be estimated by OLS since the
  heteroskedasticity problem has been corrected

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Correcting the Problem of Heteroskedasticity

• Example Suppose we want to estimate the
  following model
• and that the variance of the error term takes
  the following form
• where ?2 is the assumed constant variance and Zi
  is the proportionality factor

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Correcting the Problem of Heteroskedasticity

• If we divide the model by the proportionality
  factor Zi we obtain the following model
• The error term of the transformed model (ui) has
  now a constant variance and, thus, the model can
  be estimated by OLS
• Note If the proportionality factor (or weight
  variable) in the above regression is NOT any of
  the explanatory variables, then we must include a
  constant in the above model, otherwise a constant
  is already included

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Correcting the Problem of Heteroskedasticity

• The heteroskedasticity-corrected standard errors
  is the most popular method to correct for
  heteroskedasticity
• This approach improves the estimation of the
  models standard errors (SE) without having to
  transform the estimated model
• Given that these SE are more accurate, they can
  be used for t-tests and other hypotheses tests
• Typically, the corrected SE will be larger
  leading to lower t-statistics
• This approach works better in large samples and
  some software packages do include it (such as
  SPSS ?)