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)
-------------------------------------------------------------------------------
Variable Storage Display Value
name type format label Variable label
-------------------------------------------------------------------------------
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
-------------------------------------------------------------------------------
Sorted by: foreign
Note: Dataset has changed since last saved.ECN 102: Analysis of Economics Data
Final Exam SS1 2026
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 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 final answers 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 these extra pages for your final answers, please write your name and student ID at the top to ensure they are not lost.
This exam is worth 100 points.
Name:
Student ID:
For this exam, we will use the 1978 automotive dataset from Stata. The variables present in the dataset are listed here.
Question 1: Summary Statistics [16 points]
- Is this sample of data likely to be observational or experimental? Explain. [2 points]
- Is this sample of data likely to be cross-section, time series, panel, or repeated cross-section? Explain. [2 points]
- Suppose we run
summarize headroom, detailand find \(kurtosis=2.2\). Interpret this value in words. [2 points]
- What type of variable is foreign if we include it in a regression? Why? [2 points]
- Which of the variables below has the greatest dispersion? The least? [3 points]
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
price | 74 6165.257 2949.496 3291 15906
weight | 74 3019.459 777.1936 1760 4840
mpg | 74 21.2973 5.785503 12 41- What is the standard error of price? What is this an estimator for? [3 points]
- What is the meaning of an observation with
foreign==0in this dataset? [2 points]
Question 2: Standard Errors [15 points]
Suppose we are considering different choices of standard errors for OLS regression.
- Which OLS assumption(s) are we concerned may be violated if we are considering using robust standard errors? State the assumption(s) precisely in math and words. [2 points]
- If we do switch from default standard errors to robust standard errors, how would our regression coefficients change? Explain. [2 points]
- If we do switch from default to robust standard errors, would our confidence intervals generally get wider, narrower, or stay the same? Explain. [3 points]
- In what setting should we use clustered standard errors? Give a specific economic example and explain which assumption(s) are being addressed. [4 points]
- If we knew that errors were correlated across clusters but not within them, would clustered standard errors be appropriate? Explain. [2 points]
- Under what circumstance(s) should we use bootstrapped standard errors? [2 points]
Question 3: Multivariate Regression [23 points]
We obtain the following results after running gen mpgXforeign = mpg * foreign:
Source | SS df MS Number of obs = 74
-------------+---------------------------------- F(3, 70) = 9.48
Model | 183435281 3 61145093.6 Prob > F = 0.0000
Residual | 451630115 70 6451858.79 R-squared = 0.2888
-------------+---------------------------------- Adj R-squared = 0.2584
Total | 635065396 73 8699525.97 Root MSE = 2540.1
------------------------------------------------------------------------------
price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
mpg | -329.2551 74.9854 -4.39 0.000 -478.8088 -179.7013
foreign | -13.5874 2634.6636 -0.01 0.996 -5.27e+03 5241.0835
mpgXforeign | 78.8883 112.4812 0.70 0.485 -145.4485 303.2250
_cons | 1.26e+04 1527.8882 8.25 0.000 9553.2609 1.56e+04
------------------------------------------------------------------------------(Hint: review variable descriptions on page 2)
- Write down the population model with proper variable names. [2 points]
- Interpret the coefficient \(b_2\) on mpg in words. (Hint: think about what
foreign==0means.) [2 points]
- Interpret the coefficient \(a_1\) on foreign in words. [2 points]
- Interpret the coefficient \(a_2\) on mpgXforeign in words. [2 points]
- What conclusion can we draw from the reported F-statistic and
Prob>Ffor the test of overall significance? What does this tell us about \(F^*_{3,70,0.05}\)? [3 points]
- Which of our regression coefficients are statistically different from zero at \(\alpha=5\%\)? Which are not? What is the name for these tests? [3 points]
- Write down the fitted regression equation for domestic cars only. Write down the fitted regression equation for foreign cars only. [3 points]
- Find an expression for the marginal effect of mpg in this regression. [3 points]
- Compute \(ResSS\) for a regression of price on just a constant term. [3 points]
Question 4: OLS and Logs [18 points]
Suppose we obtain the following results after running gen ln_price = ln(price):
Source | SS df MS Number of obs = 74
-------------+---------------------------------- F(1, 72) = 22.87
Model | 2.70578153 1 2.70578153 Prob > F = 0.0000
Residual | 8.51775155 72 .118302105 R-squared = 0.2411
-------------+---------------------------------- Adj R-squared = 0.2305
Total | 11.2235331 73 .153747029 Root MSE = .34395
------------------------------------------------------------------------------
ln_price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
mpg | -0.0333 0.0070 -4.78 0.000 -0.0471 -0.0194
_cons | 9.3493 0.1535 60.91 0.000 9.0434 9.6553
------------------------------------------------------------------------------- What name do we give this type of regression? [2 points]
- How should we interpret the \(b_2\) coefficient from this regression in words? [3 points]
- Write an expression for the marginal effect of mpg on price in this regression. [3 points]
- Find the MEM of mpg if \(\overline{price}=6165.26\). [3 points]
Suppose we obtain the following results after also running gen ln_mpg = ln(mpg):
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
------------------------------------------------------------------------------
ln_price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
ln_mpg | -0.8268 0.1485 -5.57 0.000 -1.1229 -0.5308
_cons | 11.1415 0.4508 24.72 0.000 10.2429 12.0401
------------------------------------------------------------------------------- What name do we give this new type of regression? [2 points]
- How should we interpret the \(b_2\) coefficient from this regression in words? [3 points]
- Find the MER of mpg at \(mpg^*=25\) and \(price^*=\$4{,}500\). [2 points]
Question 5: F-Testing [18 points]
Refer to the regression from Question 3. Suppose we want to determine whether the variable country of origin has any effect on a car’s price.
- Write down the null and alternate hypotheses for this test. How many linear restrictions (\(q\)) would such an F-test impose? [3 points]
- Write down the unrestricted model and restricted model for this test. [4 points]
- Suppose we conduct the test above in Stata and find
Prob > F = 0.0387. What should we conclude about country of origin from this result? [2 points]
- How is this test different from the default t-test of regressor foreign? [3 points]
- Now consider a new, hypothetical regression with \(n=74\). A joint F-test on this regression yields \(ResSS_r=480{,}000{,}000\) and \(ResSS_u=420{,}000{,}000\), where \(k=4\) and \(q=2\). Compute the F-statistic. (Hint: you may cancel out the same number of trailing zeros from both \(ResSS\) values before computing; this does not affect the result.) [4 points]
- Using your F-statistic from part (e), identify the correct critical value at \(\alpha=5\%\) from the Stata output below, and state whether you reject or fail to reject \(H_0\). [2 points]
invFtail(1,70,0.05) = 3.9778
invFtail(2,70,0.05) = 3.1277
invFtail(2,70,0.025) = 3.8903
invFtail(3,70,0.05) = 2.7355Multiple Choice [2 points each]
Only one answer is correct for each question. Choose the best possible answer.
MC 1
What is true about the chance of a Type I error when conducting multiple hypothesis tests?
The total chance of a Type I error decreases with the number of tests
The total chance of a Type I error increases with the number of tests
The total chance of a Type I error is always equal to \(\alpha\)
None of the above
MC 2
What is true about the power of a test (\(1-P[\text{Type II}]\)) when conducting inference?
We should only run tests with a power of \(1\) to avoid any chance of a Type II error
Power will decrease as we increase the sample size \(n\) while keeping all regressors fixed
Power will increase when the true population parameter is closer to the null value
The OLS estimator already maximizes power among all linear unbiased estimators
MC 3
Suppose we have data on vehicle body type which takes exactly three mutually exclusive and exhaustive values: sedan, hatchback, and wagon. We want to include body type in a regression that also has a constant term so create a dummy variable for each type: \(d_{sed}\), \(d_{hatch}\), and \(d_{wag}\). Which of the following is true?
We can safely include all three dummy variables along with the constant
We can include at most two of the three dummy variables along with the constant
We must include all three dummy variables and the constant to obtain unbiased estimates
We must encode the three body types (1, 2, 3) and include them as a single variable
None of the above
MC 4
What would be a valid way to increase the precision of our OLS estimator \(b_j\) on a regressor \(x_j\) in a multivariate regression with \(j=2,\ldots,k\)?
Introduce a new regressor \(x_{k+1}\) that is correlated with \(y\) but uncorrelated with \(x_j\) and all other regressors (keeping sample size fixed)
Introduce a new regressor \(x_{k+1}\) that is correlated with \(x_j\) but uncorrelated with \(y\) and all other regressors (keeping sample size fixed)
Remove regressors from the model that are correlated with \(y\) but uncorrelated with \(x_j\) (keeping sample size fixed)
Decrease the sample size \(n\) while keeping all regressors fixed
None of the above
MC 5
In which of the following scenarios did we make a mistake?
We claimed that adding a new regressor to a model can never decrease \(R^2\) but can decrease \(\bar{R}^2\)
We claimed that \(b_j\) from a multivariate regression of \(y\) on \(x_2,\ldots,x_k\) will be equal to \(b_2\) from a bivariate regression of \(y\) on \(x_j\) if \(x_j\) is uncorrelated with all other regressors
We claimed that heteroskedasticity in our data is problematic because it causes our OLS coefficient estimates to be biased
We claimed that for a single-restriction F-test of \(\beta_j=0\), the F-statistic would equal the square of the t-statistic from the test of association for \(x_j\) on \(y\)
None of the above
(Do not miss the BONUS on the next page!)
BONUS [up to 10 extra points]
In order to prove that \(\bar{X}\sim (\mu,\sigma^2/n)\) we needed to make three assumptions about our random variable \(X\). Please name each of these three assumptions and describe in detail what they mean for \(X\) (e.g. what it would mean for them to be true versus not true). You may find it helpful to use a specific example of a r.v. \(X\) in your answer, such as a coin toss or die roll.
Note: you do not need to replicate the proofs for \(\mu_{\bar{X}},\sigma^2_{\bar{X}}\)
Scratch paper
Formula sheet
Univariate Data
\[\bar{x}=\frac1n\sum_{i=1}^nx_i\quad s_x^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2\quad se_x=\frac{s_x}{\sqrt{n}}\]
\[\bar{x}\pm t^*_{\alpha/2;n-1}\times(s_x/\sqrt{n})\quad t=\frac{\bar{x}-\mu_0}{s_x/\sqrt{n}}\]
\[ttail(df,t)=P[T>t]\quad invttail(df,p)\rightarrow t^*:P[T>t^*]=p\]
Bivariate Data
\[r_{xy}=\frac{s_{xy}}{s_x s_y}\quad\hat{y}=b_1+b_2x\quad b_1=\bar{y}-b_2\bar{x}\quad b_2=\frac{\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n(x_i-\bar{x})^2}=r_{xy}\frac{s_y}{s_x}\]
\[TSS=\sum(y_i-\bar{y})^2\quad ResSS=\sum(y_i-\hat{y}_i)^2\quad R^2=1-\frac{ResSS}{TSS}\]
\[b_2\pm t^*_{\alpha/2;n-2}\times s_{b_2}\quad t=\frac{b_2-\beta_2}{s_{b_2}}\quad s_{b_2}=\frac{s_e}{\sqrt{\sum(x_i-\bar{x})^2}}\quad s_e=\sqrt{\frac{ResSS}{n-2}}\]
Multivariate Data
\[\hat{y}=b_1+b_2x_2+\ldots+b_kx_k\quad b_j=\frac{\sum_{i=1}^n\tilde{x}_{ji}(y_i-\bar{y})}{\sum_{i=1}^n\tilde{x}_{ji}^2}\]
\[b_j\pm t^*_{\alpha/2,n-k}\times s_{b_j}\quad t=\frac{b_j-\beta_j}{s_{b_j}}\quad s_{b_j}=\frac{s_e}{\sqrt{\sum_{i=1}^n\tilde{x}_{ji}^2}}\quad s_e=\sqrt{\frac{ResSS}{n-k}}\]
\[\bar{R}^2=1-\frac{ResSS/(n-k)}{TSS/(n-1)}\quad F_{q,n-k}=\frac{(ResSS_r-ResSS_u)/q}{ResSS_u/(n-k)}\]
\[invFtail(q,n-k,p)\rightarrow F^*:P[F>F^*]=p\]
Scratch paper