Week 2

Question 1

Unbiasedness

Answer 1

This concept says that we will estimate the true population coefficient on average, since our distribution will be normally distributed and centered around the true value, even though we sometimes make over/underestimations.
E(B-hatt)=B.
We say that the estimator is an unbiased.
(Holds under OLS1-4)

Question 2

Consistency

Answer 2

As the sample size increases, we observe decreased variability of the estimates. Becomes more concentrated around the true value –> so for very large samples we can say that B-hatt=B. This is called consistency. B-hat is consistent for B.
(Holds under OLS1-4)

Question 3

The power of the test

Answer 3

It is the ability to detect false hypotheses by triggering rejections. Detect deviations from the null hypothesis. If the difference between B-hat and B (which is the numerator in the T-statistic) is positive this means a shift of the N(0,1) to the right - IF this movement increases the probability (the gained critical region in the positive side>the loss in the neg region) it means that the overall prob of rejection increases. –> Assuring that more rejections are done if H0 is wrong –> POWER.

Question 4

How to get around the linearity assumption when using non-linear functions?

Answer 4

We create a new variable for the squared x1, so that x2=x1^2. This will let our regression model to look linear and we can therefore estimate our betas. But we must be carfeul when interpreting the betas, because we cannot use the ceteris paribus - cannot hold x2 still while changing x1 because x2 contains x1 as well.

Question 5

Interpret:

• level-level
• level-log
• log-level
• log-log

Answer 5

• Level-level: a 1 unit change in x leads to B change in Y.
• Level-log: a 1% change in x leads to B/100 change in Y.
• Log-level: a 1 unit change in x leads to B*100 change in Y.
• Log-log: a 1% change in x leads to B% change in Y (elasticity!!)

Question 6

Dummy variable

Answer 6

Random variable that takes the value 1 for a subpopulation satisfying a certain property and the value 0 for everyone else. Used to model group membership. The zero group is called the base or benchmark group.

Also common to use with interaction term when the marginal effects are different in the different groups.

Question 7

Interpret the coefficient for the dummy variable?

Answer 7

It is the difference between the intercepts corresponding to the two groups. The curve will shift with this parameter.

Question 8

Interpret the coefficient for the dummy variable when it is multiplied by another regressor?

Answer 8

Then we have a dummy variable that also changes the effect/slope of that regressor. When taking the partial derivative w/r to that regressor, say x1, we will see that the the parameter will give us the difference in the marginal effect of x1.

Question 9

Modeling complementarities between regressors:

Answer 9

It is the same as for dummy variables, we multiply two regressors. The difference here is that both variables are measured on a real scale (dummy only 0 or 1). This will give us a coefficient measuring how they work together: one direct effect of x1 and one indirect effect of x1 that goes through this third combined regressor.

Question 10

Asymptotic approximation

Answer 10

The statements are true only in very large samples.

Question 11

3 things that will reduce the var(B-hat1):

Answer 11

• decreasing the variance of U
• increasing the variance of x1
• decreasing the correlation between x1 and other regressors.

All of these will help making better predictions of U-hat and therefore reducing the variance in our estimates.

Question 12

Decision rule in hypothesis testing with a test statistic:

Answer 12

If the test statistic in absolute value is larger than the critical value calculated from the level of significance (within the critical region), we reject the null hypothesis and stop believing it’s true. Instead we make a statement about the population that is in line with the alternative hypothesis. This rule is equivalent to finding out that the p-value is smaller than alpha.

Question 13

Important difference between F-test and two-sided t-test:

Answer 13

That the t-test is two sided so you have to take into account that the alpha (probability of type-1 error) must be divided on the two sides, while in the F-test we only consider the right hand side as it is skewed to the right.

Question 14

Confidence interval for a coefficient:

Answer 14

A random set of possible values for B such that the true value is contained in this set with a pre-specified probability. That probability is called the “confidence level”.

Question 15

What is the numerator/denominator degrees of freedom?

Answer 15

The numerator is q=the number of tested coefficients. The denominator is n-(k+1).

Question 16

Problem with p-values:

Answer 16

That it is so easy to use that people without statistical knowledge think that they can use them, but they can’t. They are prone to manipulate the test by changing the level of significance afterwards just to reach a statement…But the sample is only random before drawing it, after it is realized and then we can no longer make probabilistic statements. Cannot set up alpha=0.05 but then find p-value =0.09 and say that the test rejects at 10%…

Question 17

Short vs long regression:

Answer 17

The long regression regress Y on X1 and X2 while the short regression is one that recover the estimates B for a regression that only regesses on X1. The coefficient in the short regression will combine the direct effect of X1 (as measured by B1 in the long one) plus the indirect effect (as measured by B2 in the long) which will be the scaled ability of X1 to capture var in X2. If no co-movements, no contribution.

Question 18

Specification

Answer 18

The variables that we include in the regression, also called the dgp.

Question 19

Why do we transform regressors?

Answer 19

Because we think that the true effect of the variable is different than linear, so we can for ex transform into a log or a quadratic term.

Question 20

Interaction term

Answer 20

A regressor that is defined as the product of two variables.

Question 21

Does our estimator B-hat following a normal dist?

Answer 21

Yes it does, if we transform it like this:
(B-hat - B)/sd(B-hat).
This gives us N(0,1). Mean 0, sd=1.

Question 22

Standard error

Answer 22

SE is the estimator of the standard deviation of an OLS estimator.

Question 23

Robust standard error

Answer 23

A standard error that will “work” if the unobserved component is heteroscedastic. It includes outliers at a larger extent and therefore the SE is expected to increase in this model.

Question 24

Null hypothesis

Answer 24

A claim about our model. We are trying to disprove this claim.

Question 25

Type-I error:

Answer 25

Rejecting the null hypothesis even though it is true. The probability of doing this error is equal to a. Ex 5%.

Question 26

Type-II error:

Answer 26

Not rejecting the null hypothesis even though it is false.

Question 27

What is the prob of doing type-II error?

Answer 27

It depends on a lot of things and there is no clear answer, depends on the way in which the H0 is violated.