Economics 105: Statistics

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Economics 105 Statistics

• Any questions?

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Multicollinearity

• Multicollinearity typically refers to severe,
  but imperfect multicollinearity
• Matter of degree, not existence
• Consequences
• Estimates of the coefficients are still unbiased
• Std errors of these estimates are increased
• t-statistics are smaller
• Estimates are sensitive to changes in
  specification (i.e., which variables are included
  in the model)
• R2 largely unaffected

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Multicollinearity

• Detection
• calculate all the pairwise correlation
  coefficients
• .7 or .8 is some cause for concern
• VIFs can also be calculated
• Hallmark is high R2 but insignificant
  t-statistics
• Remedy
• Do nothing
• Drop a variable
• Transform multicollinear variables
• need to have same sign and magnitudes
• Get more data (i.e., increase the sample size)

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Causes of Heteroskedasticity

• Violation of (3)
• Causes of heteroskedasticity
• Learning
• Discretion
• Outliers
• Model misspecification
• should include quadratic
• incorrect functional form (ln-ln, ln-linear)
• Skewed explanatory variables
• income, wage, education

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Causes of Heteroskedasticity

• Violation of (3)
• Causes of heteroskedasticity
• Learning

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Causes of Heteroskedasticity

• Violation of (3)
• Causes of heteroskedasticity
• Discretion

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Causes of Heteroskedasticity

• Violation of (3)
• Causes of heteroskedasticity
• Learning
• Discretion
• Outliers
• Model misspecification
• should include quadratic
• incorrect functional form (ln-ln, ln-linear)
• Skewed explanatory variables
• income, wage, education

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Consequences of Heteroskedasticity

• Unbiased estimators are produced when OLS is used
  when the errors are heteroskedastic
• However, the standard errors are incorrect and
  OLS is no longer BLUE
• Any hypothesis tests conducted could yield
  erroneous results

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Weighted Least Squares

• If OLS estimators are not BLUE in the presence of
  heteroskedasticity, what are the best estimators?
  
• Can weight the observations so that more weight
  is put on observations associated with levels of
  X having a smaller error variance
• Transform the model so that the errors no longer
  exhibit heteroskedasticity
• The basic model with heteroskedasticity is
• Yi ?0 ?1Xi ?i
• Var(?i) ?i2

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Weighted Least Squares

• Dividing each observation by the associated
  standard deviation of the error transforms the
  model

• OLS estimators for this model are BLUE in the
  presence of heteroskedasticity

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Weighted Least Squares

• The error term in the transformed model is no
  longer heteroskedastic

• The transformed model is called a weighted least
  squares
• Each observation is now weighted by the inverse
  of the standard deviation of the error
• Major difficulty in estimating weighted least
  squares
• Dont observe
• So we assume and transform
  model

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Detection of Heteroskedasticity

• Most tests to detect heteroskedasticity share a
  similar approach
• Look for some association between some function
  of the errors and some function of the
  explanatory variable(s)
• OLS estimates are constructed such that there
  will be no correlation between the errors and the
  explanatory variablessuggesting we work with
  some function of the errors
• One approach is to regress ei2 on the
  explanatory variables and all combinations of the
  explanatory variables
• Looking to see if larger values of the error are
  associated with some function of the explanatory
  variables

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Detection of Heteroskedasticity

• Informal test Graph the residuals
• ei versus predicted values of Y (Y-hat)
• ei versus each of the Xs (one at a time)
• look for patterns
• Formal tests
• White
• Breusch-Pagan
• Goldfeld-Quandt
• Park

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Breusch-Pagan test

• Estimate the model by OLS
• Obtain the squared residuals,
• Run
• keeping the R2 from this regression
• 4. Do the whole model F-test, rejection indicates
  heteroskedasticity. Assumes
• Breusch, T.S. and A.R. Pagan (1979), A Simple
  Test for Heteroskedasticity and Random
  Coefficient Variation, Econometrica 50, pp. 987
  - 1000.

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Breusch-Pagan test (not needing )

• Estimate the model by OLS
• Obtain the squared residuals,
• Run
• keeping the R2 from this regression, call it
• Test statistic
• Rejection indicates heteroskedasticity. .

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White test

• Estimate the model by OLS
• Obtain the squared residuals,
• Run
• keeping the R2 from this regression
• 4. Do the whole model F-test, rejection indicates
  heteroskedasticity
• White, H. (1980), A Heteroskedasticity-Consisten
  t Covariance Matrix Estimator and a Direct Test
  for Heteroskedasticity, Econometrica 48, pp. 817
  - 838.

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White test

• Adds squares cross products of all Xs
• Advantages
• no assumptions about the nature of the het.
• Disadvantages
• Rejection (a statistically significant White
  test statistic) may be caused by het or it may be
  due to specification error its a nonconclusive
  test
• Number of covariates rises quickly
• so could also run
• since the predicted values are functions of
  the Xs (and the estimated parameters) and do
  F-test