1. A special medical test always detects the presence of a disease if a person has it; however, 5% of perfectly healthy people will test positive as well (there is a 5% "false positive" rate). Suppose that .1% of people actually have the disease, and that members of the population are tested at random.

True, False, and Explain: The approximate conditional probability of having the disease given the fact that you test positive is 95%.

FALSE. The probability of testing positive given you have the disease is 1; the probability of having the disease is .001; the probability of testing positive given not having the disease is .05; the probability of not having the disease is .999. Applying the conditional probability formula:

2. You have the following information for a regression of Y on a constant and an independent variable, X:

There are 27 data points. Designate your estimate of error variance s².

True, False, and Explain: A 95% confidence interval for the true value of the coefficient on X is .5± 2.06*s/10

TRUE. b=50/100=.5. The critical value is 2.06. Your standard error is .

3. Tenure is a dummy variable that =1 if a person gets tenure, and 0 if they don't. Female is a dummy variable that =1 if a person is female, and 0 if they are male. Your data set contains equal numbers of men and women. You estimate that:

Tenure=.4-.1*Female

True, False, and Explain:

In your data set, 4 men get tenure for every 3 women.

The regression results imply that the conditional probability of getting tenure if you are male =.4, and =.3 if you are female. Since the problem specifies that you have equal number of observations for men and women, their relative proportions will be 4:3.

i indexes the person, t the year; N=200:

4. True, False, and Explain: If the R² for (3) has been correctly calculated, then some part of (2) must be wrong.

FALSE. (2) has a higher R² than (3), but it also has a different variable, first difference of education. Also, note that including the first difference of education is equivalent to having both education and lagged education as explanatory variables, with a restriction imposed that their coefficients be equal and opposite in sign. Therefore, the R² for (2) could be lower OR higher; it has one more variable, but adds a restriction as well.

5. True, False, and Explain: Assuming (1) and (3) have been correctly calculated, you can reject the null that the true coefficient on lagged education equals 0 at the 1% level.

TRUE. Just compute the test statistic: 200-3/1*(.33-.3)/(1-.33)=197*.03/.67=8.82. The critical value is 6.76; 8.82>6.76, so you can indeed reject the null.

6. You estimate the following production function:

You want to test the hypothesis that the capital and labor elasticities of output are equal. You have thousands of observations.

True, False, and Explain: You can always accept the hypothesis if the covariance between the coefficients on ln L and ln K is positive.

FALSE. You are testing the hypothesis b_L=b_K; in other words, H₀: b_L-b_K=0. Just set up the appropriate t-test:

Note that as the covariance goes to .025, the denominator goes to infinity, allowing you to reject the hypothesis for some positive values of the covariance.

The first person says: "We should just regress the Pass dummy (=1 if you passed) on a constant and the Midterm dummy (=1 if A or B), then just look at the coefficient on Midterm."

The second person says: "There is no need to do a regression. Just look at the pass rate for people who got an A or B."

True, False, and Explain: Both people are wrong.

FALSE. The first person correctly explains the best way to calculate the conditional probabilities you are interested in; the second person computes the average pass rate for the A&B group, without determining the marginal impact. (The second person also throws out data that could be used to get the standard errors down).

TRUE. The first person ignores several possible omitted variable problems (excluding hw, attendance, etc.); the second person computes the average pass rate for the A&B group, without determining the marginal impact. (The second person also throws out data that could be used to get the standard errors down).

8. An economist studying gender discrimination estimates (where Male =1 if a person is male):

Income=$30,000+$2000*Experience+$5000*Male

True, False, and Explain: It will be impossible to estimate the equation Income=b1+b2*Experience+b3*Experience²+b4*Male+b5*Male²

TRUE. Male²=Male, since 1²=1 and 0²=0, so you would violate the full rank condition.

9. An economist investigates the impact of the minimum wage on employment, but fails to ask about the level of worker safety in each store.

True, False, and Explain: It is reasonable to think that omitting a measure of worker safety (where hours of employment is the dependent variable) will not bias the coefficient on the variable that measures the impact of the minimum wage.

FALSE. As Card and Krueger discuss (along with many other people), one way to get around the minimum wage is to reduce non-wage benefits - such as safety. Excluding a safety measure thus tends to make your estimate of the impact of the minimum wage on employment less negative (or more positive).

1. Look at the two regressions on the following page. Briefly explain TWO ways to test the hypothesis that the true coefficient on LAGUN equals zero. Do not actually calculate the test statistics.

Method number one: Just do a t-test on the coefficient on LAGUN in equation #2.

Method number two: Plug the R²'s for the restricted regression (#1) and the unrestricted regression (#2) into the change-in-R² formula, and apply an F-test.

2. Given the identical Y=C+S, and the regression estimate (in deviation form) c=.9y (R²=.9), estimate the regression (in deviation form) y=b_ss.

(Hint: First estimate the regression y=b_cc).

Applying the "reverse-regression" formula b_xyb_yx=R², you can immediately infer that .9b_c=.9, so b_c=1. (If you got that answer or close, you got 10 points).

Once you get there, this is just like problem ?? on the homework. You want to find out:

. Since S=Y-C,

You know that:

and

Plugging in:

.

3. Card and Krueger's method of collecting data (telephone surveys) has been criticized. What forms of measurement error would NOT tend to change their final results? What forms of measurement error would? How would you characterize the problems they are in fact likely to have encountered, and why?

Errors that don't matter: (a) Measurement error that merely adds or subtracts a CONSTANT from each observation would not change the estimated coefficients. So if the same error happened again and again, having the same effect each time, it would make no difference. (This just comes from the rules for linear transformations).

(b) In contrast, random errors (not constants) lead to attenuation bias, biasing your coefficients towards zero.

(c) Errors that tend increase some values (e.g. measured employment in NJ) without increasing other values (e.g. measured employment in Pennsylvania) clearly tend to bias the coefficients, possibly changing the sign.

It is hard to see what errors would tend to add a constant to any variable. What process would e.g. make each respondent add two to his estimated number of workers? Random errors leading to attenuation bias definitely happen, as C&K's own results report. But this just means that if everything else was done properly, the effect of the minimum wage is more positive than they find! The errors that tend to actually reverse the results would have to be of type (c). Many of these were discussed in class - if for example employers had already adjusted their workforces before the law kicked in, then the pre-minimum-wage-hike level of employment in NJ was understated, possibly masking the effect of the increase.