ACE 264 Midterm 2

What is the formula for the 95% confidence interval for 𝛽2 from the following regression?hat 𝑌𝑖 = hat 𝛽0 + hat 𝛽1 𝑋1𝑖 + hat 𝛽2 𝑋2𝑖

(hat 𝛽2 - 1.96 * SE(hat 𝛽2), hat 𝛽2 + 1.96 *SE(hat 𝛽2))

Suppose there is a perfectly linear relationship between X and Y in the population. What will bethe R2 value for a correctly specified regression of Y on X?

1

You run an experiment to study the effect of a new type of exercise on cardiovascular health.You divide your group of participants into people who learn the new exercise and those thatdon't. You assign groups randomly, so their use of the exercise (X) is unrelated to all of theirother characteristics. Assigning groups randomly results in which of the following?

1. E(u|X) = 0b. 2. 𝛽 is an unbiased estimator of β 3. This is a causal analysis

Which of the following can be calculated using the sum of the squared residuals (SSR)

1. root mean squared error (RMSE) 2. standard error of the regression (SER) 3. R^2

What percent of the variation in Y is explained by the regression?

69%

What can you conclude from the F-statistic and its p-value

At least one of the estimated β coefficients is jointly different from zero

Say you wanted to create a dummy variable for an overall "old" population, which you define aspopulations with a median age > 50 years. How would you create this variable?

Create a variable called old_pop which takes a value of 1 if the median age is above 50and a value of 0 otherwise.

which statistic has chi-squared distribution

F-statistic

In the regression above, you want to test the null hypothesis that freshmen and sophomoresspend the same amount of time at the library. Which mathematical term represents this nullhypothesis?

H0: β1= β2

For what null hypothesis would you need to use an F-statistic

Ho: B1 = 0 and B2 = 0

How would you interpret the estimates on median age in this regression

Holding all else equal, an increase in median age is correlated with a change in log(GDP) that changes with the value of median age, and is given by: 0.23 - 0.0054*(age_median)

You estimate a model studying the impact of race, gender, and high school education on theprobability of getting an interview for a job posting. The data come from resumes generated byresearchers where all resume characteristics have been randomized.Knowing that the variable high is a dummy variable with a value of 1 if the resume shows at least a highschool education and 0 otherwise, what is the most correct interpretation of 𝛽ℎ𝑖𝑔ℎ ?

Holding all else equal, completing high school causes the probability of a job interview toincrease by about 7% compared to not completing high school, though this is only significantat a 10% confidence level.

Suppose you run a regression between X and Y. Then you divide Y by 10 and run the sameregression. Which will not change?

R^2

Which measures of goodness of fit does not "penalize" you for adding more Xs?

R^2

Say you want to test whether the estimate on female is different from the estimate on black.How would you construct this test? [Note: you should be able to state the null and alternativehypotheses as well.

Run a two-tailed T-test for whether: hat 𝛽𝑓𝑒𝑚𝑎𝑙𝑒 ― hat 𝛽𝑏𝑙𝑎𝑐𝑘 is different from zero

What can you conclude about the SSR/TSS from the R

SSR/TSS = 0.995665

The F-statistic accounts for which of the following in its formula

The B coefficients you want to test (and their t-stats) are correlated with each other

You estimate a regression of the relationship between years in school and time spent at thelibrary (Y). You measure age using dummy variables for first year (𝑦𝑒𝑎𝑟1), second year (𝑦𝑒𝑎𝑟2),third year (𝑦𝑒𝑎𝑟3), and four or more years (𝑦𝑒𝑎𝑟4). The population regression takes the form:𝑌𝑖 = 𝛽0 + 𝛽1𝑦𝑒𝑎𝑟1𝑖 + 𝛽2𝑦𝑒𝑎𝑟2𝑖 + 𝛽3𝑦𝑒𝑎𝑟3𝑖 + 𝑢𝑖. What is the interpretation of 𝛽1?

The average difference in time at the library between first years compared to fourth+ years

10. The assumption E(u|X) = 0 can best be stated in words as:

The average of the population error conditional on X must be zero.

You want to test if the median age in the population has a quadratic relationship with log(GDP).What do you conclude from this regression?

The regression provides evidence that age has a quadratic relationship with log(GDP)

Which of the following best describes the mathematical term hat 𝛽 in words?

The sample estimator of the slope relating two variables, X and Y.

What happens when your regression suffers from imperfect multicollinearity?

The standard errors are large

You want to estimate the impact of democracy on GDP, but you're worried that GDP has a verywide distribution and might have outliers. Which strategy might help handle the issue ofoutliers in this case

Transform GDP using a log transformation - i.e., use log(GDP)

You want to study the effect of salt on the taste of muffins. Muffins taste better with smallamounts of salt, but after this the taste worsens as you add more salt. Which strategy will bestcapture this relationship?

Use a quadratic regression by including salt^2 in your regression

When would be a good cause for using a control variable in a regression

When there is an independent variable that you can't observe directly and that mightbias your estimates if it is omitted.

You want to estimate a regression of how sunlight causes changes in mood. You know sunshineis correlated with ice cream sales. Do ice cream sales cause omitted variable bias and why?

Yes, because ice cream sales are also directly correlated with happiness.

In general, you might want to use a control variable when

You want to advise someone about the effect of changing an independent variable, andyou are worried that 𝐸[𝑢|𝑋] ≠ 0 if the control variable is omitted

Which of the following decreases as the goodness of fit of a regression improves?

the residuals hat (𝑢𝑖)

What is the first step to estimating a non-linear (e.g. polynomial) relationship using regression?

transform the variables (e.g. calculate X^2)

Suppose there is a perfectly linear relationship between X and Y in the population. Which of thefollowing will be exactly equal to zero for all i?

ui

Which of the following mathematical expressions is equivalent to saying that an estimator 𝛽1 isunbiased

𝐸 hat(𝛽1) = 𝛽1

Which of the following describes why you can't include the dummy variable for fourth+ year in the regression?

𝑦𝑒𝑎𝑟1 + 𝑦𝑒𝑎𝑟2 + 𝑦𝑒𝑎𝑟3 + 𝑦𝑒𝑎𝑟4 + would be perfectly multicollinear with the intercept "X0"