Multicollinearity

Question 1

Perfect Multicollienearity

Answer 1

R^2 = 1 perfect linear relationship between the explanatory variables.

Question 2

What happens to the estimators when multicollinearity exists?

Answer 2

Cannot identify unique estimates for parameters and therefore cannot draw any statistical inferences (ie hypothesis testing) about a sample.

Question 3

Near, Imperfect Multicollinearity

Answer 3

Two or more explanatory variables aren’t exactly linearly related.

Question 4

Inferior Good

Answer 4

Income increases but demand for the good declines

Question 5

r

Answer 5

Coefficient of Correlation; used to determine the strength of or degree of collinearity. May not be adequate if multiple variables are involved.

Question 6

Ordinary Least Squares (OLS)

Answer 6

OLS produces estimators with the smallest variances; they are BLUE: Best, linear, unbiased, estimate. OLS remain BLUE even when one of the partial regression coefficients are statistically insignificant.

Question 7

Unbiasedness

Answer 7

the estimator provides you with the correct parameter coefficient; converges, because it is a repeated sampling property, with the true population value of the estimates

Question 8

Consequences of Multicollinearity

Answer 8

1. OLS doesn’t destroy minimum variance property; however the numerical value of the variance does not have to be small.
2. Large variances and standard error, wider confidence intervals
3. Small t-value; insignificant t-ratios
4. Failure to reject the null, resulting in a type 2 error
5. Cannot estimate Xs influence on Y
6. High R^2 but few statistically significant t ratios
7. OLS estimators and their standard errors become more sensitive to small changes in the data.
8. Wrong signs for regression coefficients
9. difficulty in assessing the individual contributions of explanatory variables to the explained sum of squares or R^2. –> because the variables are so collinear when one moves the other does also, making it impossible to separate them.

Question 9

Why kind of problem is Multicollinearity?

Answer 9

It is a sampling (regression) phenomenon; some samples can have Xs that are so collinear, it messes up the regression analysis. Xs might not be linear in the population. This problem occurs because data is typically nonexperimental; observed as they occur.

Question 10

t critical vs t value

Answer 10

t critical are the values that determine the critical region under the normal distribution, outside of which we would reject the null hypothesis at a selected level of confidence. The t value is the value we calculate using the standard error.

Question 11

Sample Specific

Answer 11

1. matter of degree, and not of presence or absence of multicollinearity/
2. condition of the X variables that are non-stochastic; feature of the sample and not the population

Question 12

Indicators of Multicollinearity

Answer 12

1. High R^2 but few significant t ratios
2. High pairwise correlation among explanatory variables. >.8, there is possibility of multicollinearity
3. Examination of partial correlations
4. Subsidiary or auxiliary regressions
5. Variance Inflation Factor

Question 13

Partial correlation coefficient

Answer 13

correlation btw two variables, holding the influence of the other x variables constant.

Question 14

Auxiliary Regression

Answer 14

regressing each x variable on the remaining X’s to compute the corresponding R^2; these regressions are considered “subsidiary or auxiliary to the main regression. want to find coefficient of determination; then determine if it the R^2 for each is statistically significant using F test. – high R^2 can be a surface indicator

Question 15

Variance Inflation Factor

Answer 15

VIF = 1/ (1- R^2); as R^2 increases the variance and standard error increase or inflated. Undefined if perfect collinearity (R^2 =1) and 1 perfect collinearity.

Question 16

Additional takeaways

Answer 16

R^2 can be counterbalanced by a low variance and sum of differences X, doesn’t have to.
Multicollinearity by itself need not cause high standard errors.

Question 17

When is Multicollinearity bad/not so bad?

Answer 17

1. when using as as predictor, and the same relationship is expected to continue into the future (big if)
2. If objective is to predict & reliable estimation of the individual parameters of the chosen model, then MC is bad news.

Question 18

Correlation Diagnostics

Answer 18

1. Correlation Matrix

2. Auxiliary Regression

Question 19

Correlation Matrix

Answer 19

Getting the pairwise correlation for all of the various explanatory variables.

Question 20

Issues with Solutions

Answer 20

no sure fire way to cure MC; plus the problem remains with the sample, and may not reflect the population, OLS estimators retain their BLUE property,

Question 21

Solutions to Multicollinearity

Answer 21

1. Dropping the Variable from the model
2. New data or new sample
3. Rethinking the model
4. Prior Information about Some Parameters
5. Transformation of Variables

Question 22

Dropping a variable from the model

Answer 22

Might want to remove, but always return to theory first. because economically the regression might be appropriate.–can lead to model specification error which will result in biased ; don’t drop a variable from an economically viable model just because of collinearity.

Question 23

Acquiring Additional Data or a New Sample

Answer 23

the new sample might not display as much MC as your original dataset. A sample with a larger n may reduce some of the MC.

Question 24

Rethinking the model

Answer 24

1. Did you omit any variables?
2. Is it the correct functional form?
3. Does it align with theory?

Question 25

Prior Information about some Parameters

Answer 25

use coefficient obtain in another study that you believe still holds (tall order) and apply it to the regression; subtract the coefficient *Variable from the y and then just regress for the remaining variables. This method is also difficult because you must rely on extraneous prior info.

Question 26

Transformation of Variables

Answer 26

ex. aggregate values are transformed to per capita; this reduced some of the multicollinearity.