Bias

Question 1

What are three things that bias standard errors?

Answer 1

• Heteroskedasticity (error not constant) – standard errors too small
• Multicollinearity – inflates standard errors
• Inclusion of covariates not correlated with outcome

Question 2

If errors are iid, but very large variance, are coefficients biased?

Answer 2

No

Question 3

If errors are iid, but very large variance, are standard errors biased?

Answer 3

No

Question 4

If errors are iid, but very large variance, are standard errors precise?

Answer 4

No

Question 5

If Covariance between X1 and U is positive, is coefficent biased?

Answer 5

Yes, because unbiasedness of Beta1 depends on E(U | X1, X2) = 0. If Cov(X1, U) != 0, it implies E(U | X1, X2) != 0 and therefore Beta1 is biased.

Question 6

If you Omit explanatory variable uncorrelated with other independent variables (but correlated with Y), is coefficient biased?

Answer 6

no

Question 7

If you Omit explanatory variable uncorrelated with other independent variables (but correlated with Y), are standard errors biased?

Answer 7

no

Question 8

If you omit explanatory variable uncorrelated with other independent variables (but correlated with Y), wha happens to precision of standard errors?

Answer 8

less precise

Question 9

If X1 is correlated with error term and Y, is coefficient biased?

Answer 9

Yes, because independent variable is endogenous.

Question 10

Error terms not normally distributed, are coefficients biased?

Answer 10

No

Question 11

Error terms not normally distributed, are coefficients precise?

Answer 11

No

Question 12

Error terms not normally distributed, can you conduct significant tests?

Answer 12

Significance tests incorrect, but as sample gets bigger this issue will become less severe (by CLT)

Question 13

If Var(u | X1, X2) depends on X1 or X2, are standard errors precise?

Answer 13

SE will be too small; cannot resolve by increasing sample size; need Weighted Least Squares or robust standard errors

Question 14

If Covariance (X1, X2) is positive, are coefficients biased?

Answer 14

No, as long as X1 and X2 are not perfectly collinear

Question 15

If Y is continuous but not normally distributed, are coefficients biased?

Answer 15

no

Question 16

If Y is continuous but not normally distributed, are standard errors biased?

Answer 16

no, with large enough sample size (based on CLT)

Question 17

If Y is continuous but not normally distributed, are standard errors precise?

Answer 17

no

Question 18

Does Y needed to be normally distributed for OLS to be BLUE?

Answer 18

No–but distribution of errors is relevant for statistical inference (for t-statistics to be coming from t-distribution, etc.). Even if we don’t think the errors in the population are normally distributed, we can assume asymptotic normality of OLS—if the sample size is large enough we can invoke CLT and assume normality, as long as other G-M assumptions are satisfied.

Question 19

If X1 and X2 are highly correlated, will coefficients be baised?

Answer 19

No–This is multicollinearity. Estimates will be highly unstable, but not biased per se.

Question 20

If X1 and X2 are highly correlated, will standard errors be precise?

Answer 20

No, they will be highly inflated

Question 21

If X1 and X2 correlated and X2 is omitted from the model. Assume both related to Y. Will coefficients be biased?

Answer 21

Yes (OVB)

Question 22

If X1 and X2 correlated. X2 has no effect on Y. (X2 is included in model but is extraneous.), will coefficients be biased?

Answer 22

No

Question 23

X1 and X2 correlated. X2 has no effect on Y. (X2 is included in model but is extraneous.), what happens to standard errors?

Answer 23

Less precise

Question 24

X1 is not correlated with X2 but is correlated with Y and is included-what happens to precision of standard errors?

Answer 24

Improves precision

Question 25

If X1 is correlated with X2 but not correlated with Y, and X2 is covariate of interest, what happens to precision of estimates?

Answer 25

Would reduce variation in the covariate of interest, which would make estimates less precise

Question 26

If X1 is correlated with X2 and Y, and X2 is the covariate of interest, and X1 is endogenous, what happens to coefficient?

Answer 26

If X1 is correlated with error term even after including all other controls, it will be biased and transmit bias to correlated variables.

Question 27

If X1 is correlated with X2 and Y, and X2 is the covariate of interest, and X1 is exogenous, what happens to coefficient?

Answer 27

If X1 is exogenous, though, won’t bias coefficient estimate and will partial out influence of this variable from the relationship between the key covariate and the dependent variable.

Question 28

If X1 is correlated with X2 and Y, and X2 is the covariate of interest, what happens to precision of standard errors?

Answer 28

Due to multicollinearity (if VIF > 4), standard errors will be inflated

Question 29

What happens if Covariate X1 is measured with error

Answer 29

If this is classical (random) measurement error, there is a possibility of attenuation bias (i.e., bias towards 0). This is because variance in X1 increases (denominator in beta1 estimate) without a corresponding increase in the covariance of x and y (numerator for beta1 estimate). The bias in X1 will also be transmitted to the coefficients of other covariates that are correlated with X1.

Question 30

Failure to include X2 that affects Y and is correlated with X1, will coefficients be biased?

Answer 30

Yes. Violates zero conditional mean. Omitted variable bias.

Question 31

If you Include irrelevant X2 that’s not correlated with anything, will coefficients be biased?

Answer 31

no--This is overspecification. Not biased, but fewer degrees of freedom, so significance tests will be affected

Question 32

If you Include irrelevant X2 that’s not correlated with anything, will standard errors be biased?

Answer 32

no--This is overspecification. Not biased, but fewer degrees of freedom, so significance tests will be affected

Question 33

If X2 is correlated with error term, will coefficients be biased?

Answer 33

Yes, coefficient on X2 will be biased because X2 no longer exogenous (violates zero conditional mean assumption). Coefficient on X1 might also be biased if X1 and X2 are correlated.

Question 34

If X2 is correlated with X1 and X2 is correlated with Y and X2 has a direct impact on Y and is exogenous, what happens to coefficients for X1 if you drop X2

Answer 34

If X2 has a direct impact on Y and is exogenous, dropping X2 will result in bias on X1 coefficient (OVB).

Question 35

If X2 is correlated with X1 and X2 is correlated with Y and is endogenous and correlated with Y only through a third variable that impacts both Y and X2, what happens to coefficients for X1 if you drop X2

Answer 35

If X2 endogenous and correlated with Y only through a third variable that impacts both Y and X2, then no bias on X1. Actually, including X2 would have been a way for bias from third omitted variable to be transferred to X1.

Question 36

If X2 is correlated with X1 and X2 is correlated with Y and X2 has both direct impact on Y and correlated with Y through some omitted third variable, what happens to coefficients for X1 if you drop X2

Answer 36

If X2 has both direct impact on Y and correlated with Y through some omitted third variable, no way to avoid bias unless you can add in the third variable.