Deck 14: Advanced Ols

Question 1

We derive the B1hat equation by setting the sum of squared residuals equation to 0 and solving for B1hat.

Answer 1

False

Question 2

If the errors are not homoscedastic and uncorrelated with each other, then OLS estimates are biased and but the easy-to-use standard OLS equation for the variance of B1hat is still appropriate.

Answer 2

False

Question 3

The conditions for omitted variable bias can be derived by substituting the true value of Y into the B1hat equation for the model where X2 is omitted.

Answer 3

True

Question 4

Even if we don't observe a variable, we can make informed speculations about the effect of omitting the variable on coefficients on variables we do observe.

Question 5

A single poorly measured independent variable will not cause other coefficients to be biased.

Question 6

Which assumption is necessary for the expected value of b1hat to equal b1?

A) Errors are homoscedastic.
B) Errors are not correlated with each other.
C) The error is uncorrelated with the independent variable.
D) The error is uncorrelated with the dependent variable.

Question 7

Which assumption is not necessary for the following equation to correctly characterize the standard error of $\beta$ 1hat?

A) Errors are homoscedastic.
B) Errors are not correlated with each other.
C) Errors are heteroscedastic.
D) The independent variable varies.

Question 8

In which of the following will omitted variable bias be the most severe? Treat X2 as the omitted variable in a bivariate model of
Y_i = $\beta$ ₀ + $\beta$ ₁X_1i + $\varepsilon$ _i

A) X2 is weakly related to X1, but strongly related to Y.
B) X2 is unrelated to X1, but strongly related to Y.
C) X2 is strongly related to X1, but not related to Y.
D) X2 is strongly related to X1 and strongly related to Y.

Question 9

Suppose we omit X₂ from the following in a bivariate model
Y_i = $\beta$ ₀ + $\beta$ ₁X_1i + $\varepsilon$ _i
And suppose X₂ is positively related to X₁ and $\beta$ ₂ is positive, while $\beta$ ₁ is negative. What is the likely effect of omitting X₂ on our estimate of $\beta$ ₁?

A) No effect; $\beta$ ₁ hat will be unbiased.
B) $\beta$ ₁ hat will be more negative than it should be.
C) $\beta$ ₁ hat will be less negative than it should be.
D) $\beta$ ₁ hat will be biased toward zero.

Question 10

Suppose we omit X₂ from the following in a bivariate model
Y_i = $\beta$ ₀ + $\beta$ ₁X_1i + $\varepsilon$ _i
And suppose X₂ is negatively related to X₁ and $\beta$ ₂ is positive, while $\beta$ ₁ is positive. What is the likely effect of omitting X₂ on our estimate of $\beta$ ₁?

A) No effect; $\beta$ ₁hat will be unbiased.
B) $\beta$ ₁hat will be more positive than it should be.
C) $\beta$ ₁hat will be less positive than it should be.
D) $\beta$ ₁hat will equal zero.

Question 11

Suppose we omit X₃ from the following in a bivariate model
Y_i = $\beta$ ₀ + $\beta$ ₁X_1i + $\beta$ ₂X_2i + $\varepsilon$ _i
And suppose each of the independent variables is completely uncorrelated with the other independent variables (i.e., they have been randomly chosen in a randomized experiment). What is the consequence of omitting X₃ on $\beta$ ₁hat?

A) No effect; $\beta$ ₁hat will be unbiased.
B) $\beta$ ₁hat will be biased toward zero.
C) $\beta$ ₁hat will be more positive than it should be.
D) $\beta$ ₁hat will be less positive than it should be.

Question 12

Suppose we omit X₃ (a variable that actually affects Y) from the following in a bivariate model
Y_i = $\beta$ ₀ + $\beta$ ₁X_1i + $\beta$ ₂X_2i + $\varepsilon$ _i
And suppose each of the independent variables are correlated with the other independent variables. What is the consequence of omitting X₃ on $\beta$ ₁hat?

A) No effect; $\beta$ ₁hat will be unbiased.
B) The bias will depend only on the correlation of X₁ and X_3.
C) The bias will depend on the correlations of all the independent variables_.
D) The bias will depend only on the correlation of X₁ and X_2.

Question 13

Suppose we omit X₃ (a variable that does not affect Y) from the following in a bivariate model
Y_i = $\beta$ ₀ + $\beta$ ₁X_1i + $\beta$ ₂X_2i + $\varepsilon$ _i
And suppose each of the independent variables are correlated with the other independent variables. What is the consequence of omitting X₃ on $\beta$ ₁hat?

A) No effect; $\beta$ ₁hat will be unbiased.
B) The bias will depend only on the correlation of X₁ and X_3.
C) The bias will depend on the correlations of all the independent variables_.
D) The bias will depend only on the correlation of X₁ and X_2.

Question 14

Suppose that X₁ is measured with error.
Y_i = $\beta$ ₀ + $\beta$ ₁X_1i + $\varepsilon$ _i
X_i = X*_1i + $\nu$ _i
In which of the following cases will the bias be most severe?

A) The variance of the independent variable ( $\sigma$ _x*) is much smaller than the variance of the error measurement error ( $\sigma$ _$\nu$ ).
B) The variance of the independent variable ( $\sigma$ _x*) is much larger than the variance of the error measurement error ( $\sigma$ _$\nu$ ).
C) The variance of the error measurement error ( $\sigma$ _$\nu$ ) is zero.
D) There will be no bias.

Question 15

Suppose that X₁ is measured with error.
Y_i = $\beta$ ₀ + $\beta$ ₁X_1i + $\varepsilon$ _i
X_i = X*_1i + $\nu$ _i
Which of the following is most accurate?

A) The estimate of $\beta$ ₁hat will be larger than it should be.
B) The estimate of $\beta$ ₁hat will be smaller than it should be.
C) $\beta$ ₁hat will be biased toward zero.
D) We know $\beta$ ₁hat will be biased, but do not know which direction.

Question 16

Show the steps in involved in deriving the OLS estimates for B1hat, and describe the assumption that is necessary for B1hat to be an unbiased estimator of B1.

Question 17

Please explain, in words and using equations, why B1hat is a random variable.

Question 18

Derive the equation for the omitted variable bias condition.

Question 19

Since a lot of times we do not have the values for X2, and X2 could be correlated both with X1 and Y, we will have to anticipate the sign of the bias, but we won't necessarily know the magnitude of the bias. Provide a list/table that gives the anticipated sign of the bias based on the relationship between X2 and X1 as well as the relationship between X2 and Y.

Question 20

Derive the equation for attenuation bias due to a measurement error in the independent variable in a case where there is only one independent variable.