HW 6 Regression Fundamentals: BUSN 5000

Let's say you don't omit 𝑥𝑖2xi2, but it is measured with error. Then 𝛽̂2β^2 will be ______ . (unbiased/ biased down/ biased up)

biased down

The coefficient 𝛽1β1 measures the _____ in 𝑦y associated with a unit _____ in 𝑥1x1, holding all of the unobservables constant.

change, change

If we say 𝐸(𝑦|𝑥)=𝛽0+𝛽1𝑥E(y|x)=β0+β1x, where 𝛽0β0 and 𝛽1β1 are population regression _____ and solve the population _____ problem.

coefficients, least-squares

If 𝐸(𝑢𝑖|𝑥𝑖1)=0E(ui|xi1)=0 in (1), the sampling error of 𝛽̂1β^1 converges to 0 and 𝛽̂1β^1 is ______.

consistent

𝑅2R2 measures how much of the variance of the ______ variable is accounted for by the ______ variables.

dependent, independent

If 𝑦𝑖1yi1 is log wage, 𝑥𝑖1xi1 is education and 𝑥𝑖2xi2 is labor market experience, and you omit 𝑥𝑖2xi2 from (2), then 𝛽̂1β^1 will be biased ______ because 𝛽2β2 is _____ and cov(𝑥𝑖1,𝑥𝑖2)cov(xi1,xi2) are _____ correlated.

downward, positive, negatively

The test statistic for whether a explanatory variable has a statistically significant association with the dependent variable is the ratio of the explanatory variable's coefficient ______ to its _____.

estimate, standard error

The modern approach to regression inference allows for the variance of the errors to depend on the ______ variables.

explanatory

The R function lm gives the wrong standard errors, test statistics and confidence intervals because it ignores ______.

heteroscedasticity

The population regression function provides the best ______ to the CEF.

linear approximation

If 𝐸(𝑢𝑖|𝑥𝑖1)=0E(ui|xi1)=0 in (1), 𝑥𝑖1xi1 is _____ of 𝑢𝑖ui and the sampling error of 𝛽̂1β^1 equals ______ on average, which implies that 𝛽̂1β^1 is ______.

mean independent, 0, unbiased

If 𝛽0β0 and 𝛽1β1 solve the population least-squares problem their values ______ the expected value of the _____ difference between the dependent variable and the CEF.

minimize, squared

When the PRF includes more than one 𝑥x, we say that 𝛽1β1 measures the _____ effect of 𝑥1x1 (without necessary giving a causal interpretation).

partial

The result reported in Column [5] is expected because the test score has a ______ effect on wages and is ______ correlated with education.

positive, positively

The modern approach means we should always report _____ standard errors and test statistics.

robust

If there were more than one 𝑥x in (1), then the formula for 𝛽1β1 would be the _____, except 𝑥𝑖1xi1 would be replaced with the _____ from a regression of 𝑥𝑖1xi1 on the other 𝑥xs.

same, residual

The OLS estimator for 𝛽1β1 can be obtained by plugging in the _____ covariance between 𝑥𝑖xi and 𝑦𝑖yi and plugging in the _____ variance for 𝑥𝑖xi.

sample, sample

Basic OLS inference is grounded in the application of the CLT, which says that the ______ distribution of the OLS estimator can be regarded as approximately _____ for large samples.

sampling, normal

The results in Column [2] indicate living in a city is associated with a statistically ______ average wage premium of ______ % (report one decimal place).

significant, 13.6

Larger ______ statistics and smaller ______ values indicate stronger evidence ______ (for/against) the null hypothesis.

t, p, against

If you omit 𝑥𝑖2xi2 from (2), 𝛽̂1β^1 will be biased ______ if 𝛽2β2 and cov(𝑥𝑖1,𝑥𝑖2)cov(xi1,xi2) have the same ______.

upward, sign

The value of 𝛽1β1 that solves the population least-squares problem is:

𝛽1=cov(𝑥𝑖𝑦𝑖)var(𝑥𝑖)

In (2), the test statistic for the null hypothesis that 𝛽2=1β2=1 is ________.

(𝛽̂2−1)/se(𝛽2)

The simple regression of log wages on years of education yields an estimated coefficient of ______ (report 3 decimal places), which suggests an additional year of schooling is associated with a ______ % (report one decimal place) increase in wages. This result ______ (is/ is not) statistically significant at the 5% level.

.052, 5.2, is

Including the IQ test score as an ability proxy decreases the estimated return to schooling by about ______ percentage points (report one decimal place).

0.6

Pro Tips

1. Make sure the first letter in the answer is NOT capitalized. It will say your answer is wrong even if you're right. EXCEPTION: Acronyms are capitalized 2. for questions with multiple answers make sure you put a comma between the answers plus a space after the comma.

On average, men with complete IQ test score data earn approximately $______ more per hour (round to the nearest dime) and have _____ more years of schooling (round to the nearest year).

1.20, 2

Adding the control variables in Column [2] increases the estimated return to schooling by ______ percentage points (report one decimal place).

2.3

On average, men with missing IQ test scores are ______ points _____ (more/less) likely to be from the south and _____ points _____ (more/less) likely to live in a city.

20, more, 9, less

Approximately ______ percent of the sample lives in the South and ______ lives in a city.

40, 71

The average wage of the young men in Card's sample is $ ______ and the standard deviation of wages is $ ______. (Round to the nearest cent.)

5.77, 2.63

The estimated return to schooling obtained from the subsample with complete IQ score data is ______ %, which about _____ percentage points _____ than the return for subsample with missing IQ scores (report one decimal place).

7.6, 1.2, more

Based on the results in Column [2], the estimated return to the first year of experience is about ______ % (report one decimal place).

8.1

The ______ theorem says you can control for other explanatory variables in estimating the effect of an 𝑥x on 𝑦y by either including the other variables directly or regressing 𝑦y on the ______ from a regression of 𝑥x on the other variables.

FWL, residuals

True or false: 𝑅2R2 is centrally important for doing causal inference.

False

PART B: BELOW THIS CARD

PART B: BELOW THIS CARD

As a final exercise, replicate the estimated returns to schooling reported in Column (2) of Card's Table 2 (or your Column [2] above) by first residualizing educ and then regressing lwage on the residualized schooling variable. Report the result "inline" using the summary function

# Estimate regression of educ on the control variables. educ_reg <- lm(educ ~ exper + expersq + black + south + smsa + reg661 + reg662 + reg663 + reg664 + reg665 + reg666 + reg667 + reg668 + smsa66, card) # Construct the residualized educ variable using the `resid` function. (hint: use educ_reg) rhat_educ <- resid(educ_reg) # Estimate the regression of lwage on residualized education # and report the results. summary(lm(lwage ~ rhat_educ, data = card))