MultiCollinearity

1. MultiCollinearity

2. The Nature of the Problem OLS requires that the explanatory variables are independent of error term • But they may not always be independent of each other. • Multicollinearity: data on explanatory variables for sample are perfectly or highly correlated • Perfect colinearity is when tow of the X variables are the same • STATA automatically controls for this by dropping one of them • Perfect Collinearity: Coefficients are indeterminate; SE are infinite • High degree of collinearity: large SE, imprecise coefficients, large interval estimation • Multiple Regression analysis: Cannot isolate independent effects i.e. hold one variable constant while changing the other. • OLS estimators still BLUE • Implications: • Large Variances and Covariances of estimators: • Large confidence intervals • Insignificant t-statistics • Non-rejection of zero-coefficient hypothesis • P(Type II error) large • F-Tests fail to reject joint insignificance • R2 high • Estimators and SE sensitive to few obs/data points • Detecting Multicollinearity • High R2/insignificant t-tests/significant F-Tests • High correlation coefficients between sample data on variables • Solve Multicollinearity: • Impose Economic Restrictions e.g CRS in CD production function • Improve Sample data • Drop variable (risk of mis-specifying model and having omitted variable bias) • Use rates of change of variables (impact on error term)

3. The Consequences • This is not a violation of the GM theorem • OLS is still BLUE and consistent • The standard errors and hypothesis tests are all still valid • So what is the problem? • Imprecision

4. Imprecision • Because the variables are correlated the move together • Therefore OLS cannot determine the partial effect with much reliability • Difficult to isolated specific effect of one variable when the tend to move together • This manifests itself as high standard errors • Equivalently the confidence intervals are very wide • Recall: CI=b+/-t*se

5. Perfect MC • Extreme example • Attempt a regression where one variable is perfectly correlated with another • Standard errors would be infinite because it would be impossible to separate the independent effects of the two variables which move exactly together • Stata will spot perfect multicolineararity and drop one variable • Can also happen if one variable is linear combination of others

6. Perfect Multicolinearity gen x2=inc_pc (1 missing value generated) . regress price inc_pchstock_pc x2 Source | SS df MS Number of obs = 41 -------------+------------------------------ F( 2, 38) = 210.52 Model | 6.7142e+11 2 3.3571e+11 Prob > F = 0.0000 Residual | 6.0598e+10 38 1.5947e+09 R-squared = 0.9172 -------------+------------------------------ Adj R-squared = 0.9129 Total | 7.3202e+11 40 1.8301e+10 Root MSE = 39934 ------------------------------------------------------------------------------ price | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- inc_pc | 16.15503 2.713043 5.95 0.000 10.66276 21.6473 hstock_pc | 124653.9 389266.5 0.32 0.751 -663374.9 912682.6 x2 | (dropped) _cons | -190506.4 80474.78 -2.37 0.023 -353419.1 -27593.71 ------------------------------------------------------------------------------

7. Imperfect Multicolinearity • If two (or more) x variables are highly correlated – but not perfectly correlated, stata wont drop them but the standard errors will be high • The implications • CI wider • more likely to not reject null hypothesis • Variables will appear individually statistically insignificant • But they will be jointly significant (F-test)

8. Detecting MC • Low t-statistics for individual tests of significance • High F-statistic for test of joint significance • High R2 • All these signs suggest that the variables matter collectively but it is difficult to distinguish their individual effects.

9. An Example regress lnQlnKlnL Source | SS df MS Number of obs = 33 -------------+------------------------------ F( 2, 30) = 33.12 Model | 3.11227468 2 1.55613734 Prob > F = 0.0000 Residual | 1.4093888 30 .046979627 R-squared = 0.6883 -------------+------------------------------ Adj R-squared = 0.6675 Total | 4.52166348 32 .141301984 Root MSE = .21675 ------------------------------------------------------------------------------ lnQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lnK | .4877311 .7038727 0.69 0.494 -.9497687 1.925231 lnL | .5589916 .8164384 0.68 0.499 -1.108398 2.226381 _cons | -.1286729 .5461324 -0.24 0.815 -1.244024 .9866783 ------------------------------------------------------------------------------

10. The Example • Production function example • K and L tend to increase over time together • Economically they have independent effects • But we cannot estimate their separate effects reliably with this data • individually insignificant • Nevertheless, K and L matter jointly for output • High R2 • High F statistic: can reject the null of joint insignificance

11. What to Do about it? • Maybe nothing • OLS is still BLUE • Individual estimates imprecise but model could still good at prediction • Add more data in the hope of getting more precise estimates • Making use of consistency • The distribution gets narrower as sample size rises • Drop variable • Risk omitted variable bias