Quantitative Methods

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Quantitative Methods

• Heteroskedasticity

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Heterskedasticity

• OLS assumes homoskedastic error terms. In OLS,
  the data are homoskedastic if the error term does
  not have constant variance.
• If there is non-constant variance of the error
  terms, the error terms are related to some
  variable (or set of variables), or to case .
  The data is then heteroskedastic.

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Heteroskedasticity

• Example (from wikipedia, I confessit has
  relevant graphs which are easily pasted!)
• Note as X increases, the variance of the error
  term increases (the goodness of fit gets worse)

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Heteroskedasticity

• As you can see from the graph, the b (parameter
  estimate estimated slope or effect of x on y)
  will not necessarily change.
• However, heteroskedasticity changes the standard
  errors of the bsmaking us more or less
  confident in our slope estimates than we would be
  otherwise.

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Heteroskedasticity

• Note that whether one is more confident or less
  confident depends in large part on the
  distribution of the dataif there is relatively
  poor goodness of fit near the mean of X, where
  most of the data points tend to be, then it is
  likely that you will be less confident in your
  slope estimates than you would b otherwise. If
  the data fit the line relatively well near the
  mean of X, then it is likely that you will be
  more confident in your slope estimates than you
  would be otherwise.

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Heteroskedasticity why?

• Learning?either your coders learn (in which case
  you have measurement error), or your cases
  actually learn. For example, if you are
  predicting wages with experience, it is likely
  that variance is reduced among those with more
  experience.

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Heteroskedasticity why?

• Scope of choice some subsets of your data may
  have more discretion. So, if you want to predict
  saving behavior with wealth?wealthier individuals
  might show greater variance in their behavior.

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Heteroskedasticity

• Heteroskedasticity is very common in pooled data,
  which makes sensefor example, some phenomenon
  (i.e., voting) may be more predictable in some
  states than in others.

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Heteroskedasticity

• But note that what looks like heteroskedasticity
  could actually be measurement error (improving or
  deteriorating, thus causing differences in
  goodness of fit), or specification issues (you
  have failed to control for something which might
  account for how predictable your dependent
  variable is across different subsets of data).

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Heteroskedasticity Tests

• The tests for heteroskedasticity tend to
  incorporate the same basic idea of figuring out
  through an auxiliary regression analysis
  whether the independent variables (or case , or
  some combination of independent variables) have a
  significant relationship to the goodness of fit
  of the model.

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Heteroskedasticity Tests

• In other words, all of the tests seek to answer
  the question Does my model fit the data better
  in some places than in others? Is the goodness
  of fit significantly better at low values of some
  independent variable X? Or at high values? Or
  in the mid-range of X? Or in some subsets of
  data?

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Heteroskedasticity Tests

• Also note that no single test is definitivein
  part because, as observed in class, there could
  be problems with the auxiliary regressions
  themselves.
• Well examine just a few tests, to give you the
  basic idea.

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Heteroskedasticity Tests

• The first thing you could do is just examine your
  data in a scatterplot.
• Of course, it is time consuming to examine all
  the possible ways in which your data could be
  heteroskedastic (that is, relative to each X, to
  combinations of X, to case , to other variables
  that arent in the model such as pooling unit,
  etc.)

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Heteroskedasticity Tests

• Another test is the Goldfeld-Quandt. The
  Goldfeld Quandt essentially asks you to compare
  the goodness of fit of two areas of your data.
• Disadvantages?you need to have pre-selected an X
  that you think is correlated with the variance of
  the error term.
• G-Q assumes a monotonic relations between X and
  the variance of the error term.
• That is, is will only work to diagnose
  heteroskedasticity where the goodness of fit at
  the low levels of X is different than the
  goodness of fit of high levels of X (as in the
  graph above). But it wont work to diagnose
  heteroskedasticity where the goodness of fit in
  the mid-range of X is different from the goodness
  of fit at both the low end of X and the high end
  of X.

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Heteroskedasticity Tests

• Goldfeld-Quandt test--steps
• First, order the n cases by the X that you think
  is correlated with ei2.
• Then, drop a section of c cases out of the
  middle(one-fifth is a reasonable number).
• Then, run separate regressions on both upper and
  lower samples. You will then be able to compare
  the goodness of fit between the two subsets of
  your data.

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Heteroskedasticity Tests

• Obtain the residual sum of squares from each
  regression (ESS-1 and ESS-2).
• Then, calculate GQ, which has an F distribution.

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Heteroskedasticity Tests

• The numerator represents the residual mean
  square from the first regressionthat is, ESS-1
  / df. The df (degrees of freedom) are n-k-1.
  n is the number of cases in that first subset
  of data, and k is the of independent variables
  (and then, 1 is for the intercept estimate).

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Heteroskedasticity Tests

• The denominator represents the residual mean
  square from the first regressionthat is, ESS-2
  / df. The df (degrees of freedom) are n-k-1.
  n is the number of cases in that second subset
  of data, and k is the of independent variables
  (and then, 1 is for the intercept estimate).

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Heteroskedasticity Tests

• Note that the F test is useful in comparing the
  goodness of fit of two sets of data.
• How would we know if the goodness of fit was
  significantly different across the two subsets of
  data?
• By comparing them (as in the ratio above), we can
  see if one goodness of fit is significantly
  better than the other (accounting for degrees of
  freedom?sample size, number of variables, etc.)
• In other words, if GQ is significantly greater or
  less than 1, that means that the ESS-1 / df is
  significantly greater or less than the ESS-2 /
  df?in other words, we have evidence of
  heteroskedasticity.

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Heteroskedasticity Tests

• A second test is the Glejser test
• Perform the regression analysis and save the
  residuals.
• Regress the absolute value of the residuals on
  possible sources of heteroskedasticity
• A significant coefficient indicates
  heteroskedasticity

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Heteroskedasticity Tests

• Glejser test
• This makes sense conceptuallyyou are testing to
  see if one of your independent variables is
  significantly related to the variance of your
  residuals.

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Heteroskedasticity Tests

• Whites Test
• Regress the squared residuals (as the dependent
  variables) on...
• All the X variables, all the cross products
  (i.e., possible interactions) of the X variables,
  and all squared values of the X variables.
• Calculate an LM test statistics, which is n
  R2
• The LM test statistic has a chi-squared
  distribution, with the degrees of freedom
  independent variables.

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Heteroskedasticity Tests

• Whites Test
• The advantage of Whites test is that it does not
  assume that there is a monotonic relationship
  between any one X and the variance of the error
  termsthe inclusion of the interactions allows
  some non-linearity in that relationship.
• And, it tests for heteroskedasticity in the
  entire modelyou do not have to choose a
  particular X to examine.
• However, if you have many variables, the number
  of possible interactions plus the squared
  variables plus the original variables can be
  quite high!

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Heteroskedasticity Solutions

• GLS / Weighted Least Squares
• In a perfect world, we would actually know what
  heteroskedasticity we could expectand we would
  then use weighted least squares.
• WLS essentially transforms the entire equation by
  dividing through every part of the equation with
  the square root of whatever it is that one thinks
  the variance is related to.
• In other words, if one thinks ones variance of
  the error terms is related to X1 2, then one
  divides through every element of the equation
  (intercept, each bx, residual) by X1.

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Heteroskedasticity Solutions

• GLS / Weighted Least Squares
• In this way, one creates a transformed equation,
  where the variance of the error term is now
  constant (because youve weighted it
  appropriately).
• Note, however, that since the equation has been
  transformed, the parameter esimates are
  different than in the non-transformed versionin
  the example above, for b2, you have the effect of
  X2/X1 on Y, not the effect of X2 on Y. So, you
  need to think about that when you are
  interpreting your results.

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Heteroskedasticity Solutions

• However...
• We almost never know the precise form that we
  expect heteroskedasticity to take.
• So, in general, we ask the software package to
  give us Whites Heteroskedastic-Constant
  Variances and Standard Errors (Whites robust
  standard errors). (alternatively, less commonly,
  Newey-West is similar.)
• (For those of you who have dealt with
  clusteringthe basic idea here is somewhat
  similar, except that in clustering, you identify
  an X that you believe your data are clustered
  on. When I have repeated states in a
  databasethat is, multiple cases from California,
  etc.I might want to cluster on state (or, if I
  have repeated legislators, I could cluster on
  legislator. Etc.) In general, its a
  recognition that the error terms will be related
  to those repeated observationsthe goodness of
  fit within the observations from California will
  be better than the goodness of fit across the
  observations from all states.)