Multicollinearity & Micronumerosity

Transcription

1 Multicollinearity & Micronumerosity Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10

2 Objectives By the end of this meeting, participants should be able to: Explain why OLS cannot be estimated in the case of perfect multicollinearity or exact micronumerosity. Diagnose the presence of (imperfect) multicollinearity. Respond thoughtfully to the presence of multicollinearity or micronumerosity. Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10

3 What Kind of Problem Is Posed by Multicollinearity and Micronumerosity? We cannot estimate a regression model with OLS in a case of either: Perfect multicollinearity, or one predictor being a perfect function of one or more other predictors. Exact micronumerosity, or having fewer observations than parameters to be estimated. We have relatively large standard errors with either: Near multicollinearity, which is high but imperfect. In other words, one covariate is predicted very well by the others, but not perfectly. Near micronumerosity, which means the number of observations barely exceeds the number of parameters to be estimated. OLS is still BLUE under multicollinearity or micronumerosity, though. These issues have nothing to do with statistical assumptions, but rather are data features. (See Achen quote on p. 326.) Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10

4 State Policy as a Function of Opinion, Conditioned by Initiatives An Example of Assessing Multicollinearity Input Variable Estimate Std. Err. VIF Proportion economic Public ideology Initiative usage Proportion economic init. usage Public ideology init. usage Intercept N = 48, R 2 = , F 5,42 = 15.6 (p <.0001). Citation: Abbreviated example from: Monogan, James E., III, Virginia Gray, & David Lowery Public Opinion, Organized Interests, & Policy Congruence in Initiative & Noninitiative U.S. States. State Politics & Policy Quarterly 9(3): Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10

5 U.S. Preferences on Defense Spending, (Differenced Net Support for More Spending) Data Analysis with Micronumerosity Citation: Input Variable Estimate Std. Err. Intercept Net dislike of the Soviet Union t Net dislike of the Soviet Union t Defense Appropriations t N = 15, R 2 = 0.84, ˆσû = 6.84, Durbin-Watson d = Wlezien, Christopher The Public as Thermostat: Dynamics of Preferences for Spending. American Journal of Political Science 39(4): Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10

6 Diagnosing Multicollinearity 1 High R 2 but few significant t ratios. 2 High pair-wise correlations among regressors. 3 Examination of partial correlations. 4 Auxiliary regressions. 5 Eigenvalues and condition index. 6 Tolerance and variance inflation factor. For variable X j, we have: TOL j = 1 VIF j = (1 R 2 j ). 7 Scatterplot. Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10

7 Responding to Multicollinearity Some of these are better ideas than the others... 0 Do nothing. 1 A priori information. (You can test the restrictions.) 2 Combining cross-sectional and time series data. 3 Dropping variables and specification bias. 4 Transformation of variables. (E.g., first differences.) 5 Additional or new data. 6 Reducing collinearity in polynomial regressions. 7 Multivariate measurement: factor analysis, principal components, NOMINATE, etc. 8 Alternative estimation: ridge regression (not a great idea). Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10

8 Responding to Micronumerosity 0 Do nothing. 1 A priori information. (You can test the restrictions.) 2 Dropping variables and specification bias. (Parsimony rules the roost here.) 3 Additional or new data. 4 Multivariate measurement: factor analysis, principal components, NOMINATE, etc. 5 Alternative analyses. Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10

9 Diagnosing Multicollinearity Problems with Software R Code library(car) data(ericksen) summary(ericksen) mod.census<-lm(undercount~minority+crime+poverty+ language+highschool+housing+as.factor(city)+conventional, data=ericksen) summary(mod.census) vif(mod.census) #requires car Stata Code use summarize regress crime pctmetro poverty single vif Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10

10 Final Schedule Dec. 4 Reading day; optional review session 4:00-6:00pm, Baldwin 301 Dec. 10 Final examination (cumulative), 7:00-10:00pm, Baldwin 301 Review the course goals and weekly objectives. You may bring one page of notes, front and back. (No magnifying devices.) Bring a calculator. Bring loose leaf paper (ruled or unruled) and writing instruments. Coffee, Red Bull, 5 Hour Energy are permitted. (Though I will check the containers for more notes.) Jamie Monogan (UGA) Multicollinearity & Micronumerosity POLS / 10