Contains data from /Applications/Stata/ado/base/a/auto.dta
Observations: 74 1978 automobile data
Variables: 12 13 Apr 2022 17:45
(_dta has notes)
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Variable Storage Display Value
name type format label Variable label
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make str18 %-18s Make and model
price int %8.0gc Price ($)
mpg int %8.0g Mileage (mpg)
rep78 int %8.0g Repair record 1978
headroom float %6.1f Headroom (in.)
trunk int %8.0g Trunk space (cu. ft.)
weight int %8.0gc Weight (lbs.)
length int %8.0g Length (in.)
turn int %8.0g Turn circle (ft.)
displacement int %8.0g Displacement (cu. in.)
gear_ratio float %6.2f Gear ratio
foreign byte %8.0g origin Car origin
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Sorted by: foreign
Note: Dataset has changed since last saved.ECN 102: Analysis of Economics Data
Final Exam SS1 2025
This exam consists of 5 short-answer questions (with sub-parts) and 5 multiple choice questions. You will have a maximum of 100 minutes to complete this exam. No additional time may be taken without prior accommodation. Show all work to receive full credit. You may use only the calculators provided by the instructor. For questions requiring computation, it suffices to express your final answer with 2 decimal places. There is a formula sheet and scratch paper provided at the end of this exam. You may remove these pages and discard them after the exam. If you plan to use any of these extra pages for your final answers, please write your name and student ID at the top of each to ensure they are not lost.
This exam is worth 100 points.
Name:
Student ID:
Question 1: Ordinary Least Squares (OLS) Estimator [15 points]
Suppose we run a multivariate regression of an outcome \(y\) on \(k-1>1\) regressors and a constant for a sample of size \(n\). We will refer to our general OLS estimator as \(b_j\), with \(j=2,...,k\) (we do not care about the constant term but still need to estimate it). Please answer the following:
- What does \(b_j\) represent in words? Please be as specific as possible. [2 points]
- What is the difference between \(x_j\) and \(\tilde{x}_j\)? [2 points]
- What does it mean for our estimator \(b_j\) to be unbiased? Explain in words. [3 points]
- What does it mean for our estimator \(b_j\) to be consistent? Explain in words. [3 points]
- What does it mean for our estimator \(b_j\) to be the Best Linear Unbiased Estimator (BLUE)? Explain in words. [3 points]
- If we know our regressor \(x_j\) is not normally distributed, should we still expect \(\beta_j\) to be normally distributed? Why or why not? [2 points]
Question 2: OLS Regression Output [20 points]
We will work with the 1978 automotive dataset. Variables in the dataset and their labels are given below:
Suppose we now estimate a regression model and obtain the following results:
Source | SS df MS Number of obs = 74
-------------+---------------------------------- F(3, 70) = 11.09
Model | 204556469 3 68185489.6 Prob > F = 0.0000
Residual | 430508927 70 6150127.53 R-squared = 0.3221
-------------+---------------------------------- Adj R-squared = 0.2931
Total | 635065396 73 8699525.97 Root MSE = 2479.9
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price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
mpg | -56.19416 85.07654 -0.66 0.511 -225.874 113.4856
weight | 2.061945 .6586383 3.13 0.003 .748332 3.375557
headroom | -675.5962 392.3504 -1.72 0.090 -1458.115 106.922
_cons | 3158.306 3617.449 0.87 0.386 -4056.468 10373.08
------------------------------------------------------------------------------- Write down the population regression model we are estimating using variable names. [2 points]
- Write down the sample regression model we have estimated above using variable names. [2 points]
- Interpret the coefficient on mpg in words. [3 points]
- We see the 95% confidence interval for headroom includes zero and that
P>|t|for headroom \(=0.090\). Do these results surprise you? Why or why not? [3 points]
- Interpret the p-value for weight. What null and alternate hypotheses does this p-value correspond to? What is the name of this test? State your full conclusion in words for this test. [4 points]
- Interpret the \(R^2\) of our regression in words. [2 points]
- Interpret
Prob>Ffrom the table. What null and alternate hypotheses does this p-value correspond to? What is the name of this test? State your full conclusion in words for this test. [4 points]
Question 3: Intuition for Inference [16 points]
Here we will ask questions about regression inference (bivariate or multivariate).
- If we write down \(\alpha=0.05\) for a test of inference, what could this quantity represent? [2 points]
(Hint: You should have two answers)
- Given our answer above, why would we not want to set \(\alpha=0\)? Why would we not want to set \(\alpha=1\)? Please give a short explanation for each. [4 points]
- Given our standard choice of \(\alpha=0.05\), what are three ways we can maximize the power of our inference? Please give a short explanation for each. [6 points]
Suppose we have a test of association for our regression coefficient \(\beta_j\) with standard null and alternate hypotheses. For this test:
- What would it mean to commit a Type I error? [2 points]
- What would it mean to commit a Type II error? [2 points]
Question 4: Errors and Prediction [14 points]
Suppose we are worried our data violates homoskedasticity and so choose to use robust standard errors:
- Will this cause our standard errors to grow, shrink, or stay the same size (compared to using default standard errors)? You do not need to explain. [2 points]
- Will this cause our t-statistics to grow, shrink, or stay the same size (compared to using default standard errors)? Please explain. [3 points]
- Will this cause our regression coefficients to grow, shrink, or stay the same size (compared to using default standard errors)? Please explain. [3 points]
- Will this cause our confidence intervals to get wider, narrower, or stay the same size (compared to using default standard errors)? Please explain. [3 points]
- Suppose we run
scatter y xand confirm our data indeed looks heteroskedastic. Please draw an example of what this scatterplot might look like below. [3 points]
Question 5: OLS and Logs [25 points]
Suppose we run the following regression using the dataset described above:
Source | SS df MS Number of obs = 74
-------------+---------------------------------- F(1, 72) = 27.92
Model | 177453129 1 177453129 Prob > F = 0.0000
Residual | 457612267 72 6355725.93 R-squared = 0.2794
-------------+---------------------------------- Adj R-squared = 0.2694
Total | 635065396 73 8699525.97 Root MSE = 2521.1
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price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
ln_mpg | -5992.728 1134.137 -5.28 0.000 -8253.588 -3731.868
_cons | 24290.48 3442.733 7.06 0.000 17427.51 31153.44
------------------------------------------------------------------------------- What name could we give this type of regression? [2 points]
- How would we interpret the \(b_2\) coefficient we estimate with this regression? [3 points]
- Give an expression for the marginal effect of mpg in this regression. [3 points]
- Find the MEM of mpg if \(\overline{mpg}=21.30\), \(\overline{price}=6165.3\). [3 points]
- Find the MER of mpg in this regression for \(mpg^*=25\), \(price^*=6000\). [3 points]
Suppose we now estimate the following model:
Source | SS df MS Number of obs = 74
-------------+---------------------------------- F(1, 72) = 31.00
Model | 3.37819527 1 3.37819527 Prob > F = 0.0000
Residual | 7.84533782 72 .108963025 R-squared = 0.3010
-------------+---------------------------------- Adj R-squared = 0.2913
Total | 11.2235331 73 .153747029 Root MSE = .3301
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ln_price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
ln_mpg | -.826847 .1484986 -5.57 0.000 -1.122873 -.5308204
_cons | 11.14146 .4507755 24.72 0.000 10.24286 12.04007
------------------------------------------------------------------------------- What name could we give this new type of regression? [2 points]
- How would we interpret the \(b_2\) coefficient we estimate with this regression? [3 points]
- Give an expression for the marginal effect of mpg in this regression. [3 points]
- Give an expression for the MEM of mpg in this regression if \(\overline{mpg}=21.30\), \(\overline{price}=6165.3\). [3 points]
Multiple Choice [2 points each]
Here we will think about different types of data. Only one answer is correct. Choose the best answer.
MC 1
If we collect experimental data on a new campus initiative from UC Davis students, which of the following must be true?
We randomized the days of the week we interviewed students about the initiative
We randomized the parts of campus where we interviewed students about the initiative
We randomized the time of day we interviewed students about the initiative
We did not know whether students had taken part in the initiative or not
None of the above
MC 2
If we collect ordered, categorical data on student class year, which of the following is true?
We can include the variable for class year directly in a regression
We must assign each unique value of class year a number before we can include this in a regression
We must create a different dummy variable for each unique value of class year before we can include these in a regression
We cannot include class year in a regression
None of the above
MC 3
If we collect unordered, categorical data on student major, which of the following is true?
We can include the variable for major directly in a regression
We must assign each unique value of major a number before we can include this in a regression
We must create a different dummy variable for each unique value of major before we can include these in a regression
We cannot include major in a regression
None of the above
MC 4
If we survey students passing by the MU on their GPA (once per student) on the first day of fall quarter, which of the following is true?
Our data will be experimental, numerical, time-series
Our data will be experimental, numerical, cross-sectional
Our data will be observational, categorical, panel
Our data will be observational, numerical, cross-sectional
None of the above
MC 5
When considering a sample from a population, which of the following is true?
A representative sample will lead to proper inference while a non-representative sample may lead to biased inference
It is often unfeasible or impossible to survey an entire population, so we choose a sample size \(n<N\) to be well-powered for our inference
We often consider hypothetical or future subjects as members of our population, such as future students or those who almost took a class but dropped last minute
While we estimate results from our sample, our ultimate goal is to generalize these results back to the population
All of the above