Quantitative Methods

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Quantitative Methods
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Multicollinearity

• What is multicollinearity?
• Multicollinearity is shared variance across
  independent variables.
• An examplepredicting an individuals view on
  tax cuts with both years of education and annual
  income.

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Multicollinearity

• What is the consequence of multicollinearity?
• Multicollinearity makes it more difficult to
  come up with robust (or efficient, or stable)
  parameter estimates. In other words, its more
  difficult to determine what the effect is of
  income (on attitudes toward tax cuts) separate
  from the effect of education (on attitudes toward
  tax cuts).
• This makes senseimagine if you go out into the
  quad, and just ask undergraduate students their
  views on Governor Jindal. Will you be able to
  easily separate out the effect of age from the
  effect of year at LSU? There will be some
  variancebecause there are older students, and
  because only some students graduate in 4 years.
  The two variables are not perfectly collinear.
  But it will be difficult to figure out whether
  age has an effect, independent of year at LSU.

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Multicollinearity

• Multicollinearity across variables means that you
  are less sure of your estimates of the effects
  (of age and year in school for instance)and so
  you expect those estimates could potentially be
  much different from sample to sample...
• And so the standard errors are potentially larger
  than they would be otherwise.

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Multicollinearity

• What does that mean in terms of OLS?
• Recall that to calculate out estimates in
  multivariate OLS, you are using the information
  left over in Y after you have predicted Y with
  all the other independent variables as well as
  the information left over in that particular
  after you have predicted that particular X with
  all the other independent variables.

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Multicollinearity

• And, recall that the standard error of b taps
  into the stability, robustness, efficiency of the
  parameter estimate b. In other words, the
  standard error represents the degree to which you
  expect that the estimate remains stable over
  repeated sampling.

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Multicollinearity

• What do large standard errors mean?
• Smaller t scores (because the t is calculated by
  dividing the parameter estimate by the standard
  error)
• Larger confidence intervals (recall that if you
  took an infinite number of samples, and
  calculated out a 95 confidence intervals for
  each one, you would expect the true slope to
  fall in the 95 of those confidence intervals).
• Less statistically significant results
• But note that the bs the actual parameter
  estimatesare not biased. In other words, over
  repeated sampling, youd expect that your
  parameter estimates would average out to the
  true slope.

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Multicollinearity

• How do we test for multicollinearity?
• Calculate out auxiliary R-squareds. First, take
  each independent variable, and regress it on all
  the other independent variablesand save the
  R-squareds. These are the auxiliary
  R-squaredswhich represent the variance in each
  independent variable that is explained (or used
  up) by all the others.

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Multicollinearity

• If the auxiliary R-squareds are high, that means
  that for those variables, there isnt much
  information (variance) left over to estimate the
  effect of that variable.
• So, for example, back to the opinion of Governor
  Jindal. Imagine your independent variables were
  gender, race, political party, ideology, age,
  science major dummy variable, humanities major
  dummy variable, social science major dummy
  variable, business major dummy variable, and year
  in school.

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Multicollinearity

• Which subsets of the following variables would
  seem to covary together?
• gender, race, political party, ideology, age,
  science major dummy variable, humanities major
  dummy variable, social science major dummy
  variable, business major dummy variable, and year
  in school
• If you regress year in school (as the dependent
  variable) on all the other independent variables,
  would you expect a high or low auxiliary
  R-squared?

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Multicollinearity

• What SAS, Stata, or SPSS will give you are
  actually VIF scores. VIF scores are calculated
  by calculating a tolerance figure by
  subtracting the auxiliary R-squared from 1, then
  dividing 1 by that tolerance figure.
• Anything above a .75 auxiliary R-squared is
  relatively highso anything below a .25 tolerance
  is considered a sign of potentially high
  multicollinearity. And therefore, a VIF score
  greater than 4 is considered a sign of high
  multicollinearity.

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Multicollinearity

• What can one do about multicollinearity?
• First, if you have significant effects,
  multicollinearity isnt a problem (as a practical
  matter)once you have concluded that you can
  reject the null hypothesis of no effect it
  might be nice to be even more confident in that
  conclusionbut rejection is rejection, and
  asterisks are asterisks.

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Multicollinearity

• Second, if you dont have significant effects,
  unfortunately, there is no way to know for sure
  whether it is because of multicollinearityit
  could just be that even if you had much more
  information to work with, you would still have
  insignificant effects.
• However, its reasonable to note to your reader
  that collinearity does existand its reasonable
  to point out the magnitude of your parameter
  estimates (perhaps by presenting predicted
  outcomes). We talked about the importance of
  predicted outcomes when we discussed
  non-linearity relationships between x and y, but
  in general, talking about the actual predictions
  based on hypothetical cases is a pretty useful
  way to show your reader the substantive magnitude
  of the effects.

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Multicollinearity

• There arent that many solutions to
  multicollinearity.
• Sometimes, its appropriate to combine
  independent variables into an index (i.e.,
  socioeconomic status).
• And theres the option of gathering more
  datamulticollinearity is essentially an
  information problem, and so more information is
  always useful. However, that obviously isnt
  always feasible.

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Dummy Variables

• Why would we use dummy variables?
• Examplereligionor major--why cant we use an
  interval level variable?
• Rule 1 with dummy variableswe need to leave a
  comparison category out. (Why? Why cant we
  include male and female in the same
  equation?)

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Dummy Variables

• Rule 2 with dummy variablesall the parameter
  estimates are interpreted relative to that
  omitted comparison category.
• Rule 3 with dummy variablesthe parameter
  estimates essentially work to change the
  intercept.

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Interactions

• Interactions are useful when you believe that the
  effect of one variable depends on the value of
  another.
• So, for example, if you believe that gender
  influences the types of bills that legislators
  introducebut that the effect of gender becomes
  less pronounced as the number of women in the
  legislature increases.

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Interactions

• In that case, you would include in a model gender
  (probably coded as 1female), the (or number)
  of women in the legislature, and an interaction
  that is calculated by
• women in legislature multiplied by gender

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Interactions

• Lets think about this in terms of the actual
  equation.
• X1 gender
• X2 women in legislature
• X3 interaction (gender women in
  legislature)
• Y a b1 x1 b2 X2 b3 x3 e
• Predicted Y a b1 x1 b2 x2 b3 x3

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Interactions

• Y a b1 x1 b2 X2 b3 x3 e
• Predicted Y a b1 x1 b2 x2 b3 x3
• What is predicted Y if the legislator is male?
  (What is the only variable that really factors
  into the prediction, for male legislators?)
• What is predicted Y if the legislator is female?
• What is the slope for male legislators? What is
  the slope for female legislators?
• What is the intercept for male legislators? What
  is the intercept for female legislators?

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Interactions

• So, the b3 the estimated effect for the
  interaction variables represents the additional
  effect of percentage women in the legislature
  for women.
• And the b1 the estimated effect for the
  dummy variable for female represents the change
  in intercept for women (compared to men)

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Interactions

• A couple of practical points, and then an
  example....
• First, there are no legislatures with 0 women.
  Therefore, the dummy variable for women can never
  be interpreted in isolation of the interaction.
  And, the interaction cant really be interpreted
  in isolation of the dummy variable.
• Given that, there are joint tests of
  significancethat well talk about next week,
  based on our example.

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Interactions

• Second, could you just have a model with only the
  interaction term, or perhaps only the interaction
  term plus one of the comonent variables, if your
  theory indicated that the other variable(s) X1
  and / or X2 were not relevant?
• I would not advise it. Its possible that X1 and
  X2 are just randomly significantin which case
  the interaction will look significant, but it
  might not be if you include X1 in the model.
• You really dont know if the interaction is
  significant unless you control for X1 and X2.

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Interactions

• Third, you do have a lot of multicollinearity
  with interactions. So, if you suspect that
  multicollinearity is a problem, its sometimes
  useful to tell your reader (in a footnote, or in
  additional columns in a table) what the results
  looked like if you dropped one or both of the
  variables, or dropped the interaction.

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Interactions

• And now the example.
• Theres been a lot of debate over the selection
  system for judgespartisan election versus
  non-partisan election versus appointment.

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Interactions

• And in the paper Im handing out, the authors
  examine whether individuals in a state with a
  partisan election system are less confident in
  courtsand whether the effect of the election
  system depends on the persons education level.
  (Essentially arguing that the partisan election
  system only has an effect among those who are
  well educatedand actually know what selection
  system is used0.

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Interactions

• Additional work has examined whether the effect
  of partisan election system on confidence in
  the courts depends on the degree to which the
  selection system produces a court that is
  unbalanced (skewed Democratic or Republican) or
  not representative of the states partisan
  breakdown.
• So lets look at the tables.