Question

It's 5 multiple-choice questions of applicated economic, can someone help me? Thank you so much!

Assume that the variable Y is actually determined by the following equation Y; = Bo + B1X1,i+ B2X2,i + Uj additionally assume that corr(X1, X2) = p. The usual assumptions for a linear model hold in this case. You are interested in estimating B1. To accomplish this you collect a sample of the variables Y and X1 and estimate the following model Y; = Yo + 91X1,i+ vi (3) Answer the following questions 6. If p= 0 and B2 > 0. Which of the following options is true? (a) The estimate of 71 in model (3) underestimates B1. (b) The estimate of 71 in model (3) overestimates B1. (c) The estimate of 71 in an unbiased estimator of B1. (d) The estimate of 71 in an unbiased estimator of 71 but not of Bi. 7. If p > 0 and B2 > 0. Which of the following options is true? (a) The estimate of 71 in model (3) underestimates B1. (b) The estimate of 71 in model (3) overestimates B1. (c) The estimate of 71 in an unbiased estimator of B1. (d) The estimate of 71 in an unbiased estimator of 1 but not of B1. 8. If p > 0 and B2 = 0. Which of the following options is true? (a) The estimate of 71 in model (3) underestimates B1. (b) The estimate of 71 in model (3) overestimates Bi. (c) The estimate of 71 in an unbiased estimator of B1. (d) The estimate of 71 in an unbiased estimator of 1 but not of Bi. 9. If p > 0 and 32 <0. Which of the following options is true? (a) The estimate of 71 in model (3) underestimates B1. (b) The estimate of 71 in model (3) overestimates B1. (c) The estimate of 71 in an unbiased estimator of B1. (d) The estimate of 71 in an unbiased estimator of 1 but not of Bi. 10. If p < 0 and B2 <0. Which of the following options is true? (a) The estimate of 71 in model (3) underestimates Bı. (b) The estimate of 71 in model (3) overestimates B1. (c) The estimate of 71 in an unbiased estimator of Bi. (d) The estimate of 71 in an unbiased estimator of 1 but not of B1.

Answer 1

6.

Ans c

As correlation between the two variables is zero, the presence or absence of X2 does not affect the coefficient of X1.

Therefore, $\gamma _{1}$ is an unbiased estimator of $\beta _{1}$

7.

Ans b

As X1 and X2 are positively correlated, a change in X1 also implies a change in the same direction in X2. As a result, $\gamma _{1}$ captures the effect of both, X1 and X2, on the dependent variable. Thus, it will be higher than what it would have been - i.e., it is overestimated.

8.

Ans c

Although X1 and X2 are positively correlated, X2 does not affect the independent variable. Therefore, $\gamma _{1}$ does not capture any effect of X2 on the independent variable. Thus, it is an unbiased estimator.

9.

Ans: a

In an intuition similar to question 7, as there is positive correlation between the two variables and the coefficient of X1 is negative, $\gamma _{1}$ has a negative bias, i.e., it underestimates $\beta _{1}$.

10.

Ans: b

X1 and X2 are negatively correlated, therefore increase in X1 is accompanied by decline in X2. As X2 is omitted, $\gamma _{1}$ captures the effect of X2 as well. Mod $\gamma _{1}$ should have been high but it is being over powered by X2. And given the negative sign, the negative effect must have been larger but it is not shown so.

Therefore, $\gamma _{1}$ overestimates $\beta _{1}$