Contains data from AED Doctor Visits Data.dta
Observations: 5,190 Data for A. Colin Cameron
(2022), ANALYSIS OF ECONOMIC
DATA, Amazon
Variables: 8 22 Jan 2022 15:50
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Variable Storage Display Value
name type format label Variable label
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female float %9.0g Equals 1 if female and 0
otherwise
freepoor float %9.0g Equals 1 if free government
insurance due to low income
freerepa float %9.0g Equals 1 if free government
insurance due to old-age,
disability or veteran
levyplus float %9.0g Equals 1 if private insurance
age float %9.0g Age in years (midpoint of 10 year
age groups)
income float %9.0g Annual income in tens of
thousands of dollars
illness float %9.0g Number of illnesses in past 2
weeks
visits float %9.0g Number of doctor (or specialist)
visits in past 2 weeks
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Sorted by:
Note: Dataset has changed since last saved.ECN 102: Analysis of Economics Data
Final Exam SQ 2025 – Answer Key
This exam consists of 5 short-answer questions (with sub-parts) and 5 multiple choice questions. You will have a maximum of 2 hours 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 scratch page to ensure they are not lost.
This exam is worth 100 points.
For this exam, we will use a dataset on doctor visits that includes age, sex, income, and health statistics for a group of individuals. The variables present in the dataset are listed here.
Question 1: Summary Statistics [16 points]
- How could we classify the data for variable income in this dataset? Explain three ways. [6 points]
Observational (no randomization)
Cross-sectional (no time variation)
[continuous] Numerical (money)
- What is the meaning of an observation with
female==0,freepoor==1in this dataset? [2 points]
A non-female on free low-income government insurance.
- What type of variable is levyplus if we include it in a regression? [2 points]
Dummy variable.
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
income | 5,190 .5831599 .3689067 0 1.5
illness | 5,190 1.431985 1.384152 0 5
visits | 5,190 .3017341 .7981338 0 9- Which variable above has the greatest dispersion? Which has the least? [3 points]
\[CV_{income}=\frac{0.37}{0.58}=0.64\quad CV_{illness}=\frac{1.38}{1.43}=0.965\quad CV_{visits}=\frac{0.80}{0.30}=2.67\]
Visits has the greatest dispersion based on the coefficient of variation (CV); income has the least.
- What is the standard error of illness? (You can leave your answer as a formula) [3 points]
\[s_{illness}=\frac{1.38}{\sqrt{5190}}\]
Question 2: Bivariate Regression [16 points]
Suppose we want to run a regression to predict visits using income with OLS:
- Write down the population regression model we are trying to estimate. [2 points]
\[visits=\beta_1+\beta_2income+u\]
- What will the independence assumption imply for this regression? [2 points]
Error terms are uncorrelated across observations: \(u_i\) is independent from \(u_j\).
- What will the linearity assumption imply for this regression? [2 points]
The true population model is linear: \(visits=\beta_1+\beta_2income+u\).
Source | SS df MS Number of obs = 5,190
-------------+---------------------------------- F(1, 5188) = 81.73
Model | 51.2660961 1 51.2660961 Prob > F = 0.0000
Residual | 3254.2183 5,188 .627258731 R-squared = 0.0155
-------------+---------------------------------- Adj R-squared = 0.0153
Total | 3305.48439 5,189 .637017613 Root MSE = .792
------------------------------------------------------------------------------
visits | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
age | .0048538 .0005369 9.04 0.000 .0038013 .0059063
_cons | .1044828 .0244318 4.28 0.000 .0565861 .1523794
------------------------------------------------------------------------------- Suppose we estimate this regression and find that our 95% confidence interval for \(b_2\) is \((.004,.006)\). What conclusion from the test of association for \(\beta_2\) does this imply? [3 points]
Our coefficient is statistically different from zero at \(\alpha=5\%\) since 0 does not appear in the 95% confidence interval of \(b_2\). We can reject the null hypothesis that \(\beta_2=0\).
- Suppose our R-squared for this regression is 0.0155. What does this mean in words? [2 points]
\(0.0155\times100=1.55\)% of the variation in visits can be explained by variation in income.
Suppose instead we estimated a regression of the log of visits on the log of income:
- What would we call this type of regression? [2 points]
Log-log regression.
- How would we interpret \(b_2\) from this regression in words? [3 points]
A 1% change in income is associated with a \(b_2\)% change in visits.
Question 3: Multivariate Regression [23 points]
We now estimate \(visits=\beta_1+\beta_2income+\alpha_1freepoor+\alpha_2(income\times freepoor)+u\) with OLS:
Source | SS df MS Number of obs = 5,190
-------------+---------------------------------- F(3, 5186) = 15.25
Model | 28.9007111 3 9.63357037 Prob > F = 0.0000
Residual | 3276.58368 5,186 .631813282 R-squared = 0.0087
-------------+---------------------------------- Adj R-squared = 0.0082
Total | 3305.48439 5,189 .637017613 Root MSE = .79487
------------------------------------------------------------------------------
visits | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
income | -.1804711 .0305348 -5.91 0.000 -.2403322 -.1206101
freepoor | -.1235766 .0947101 -1.30 0.192 -.3092484 .0620951
incomeXfre~r | -.2656318 .2516047 -1.06 0.291 -.7588831 .2276195
_cons | .4156881 .0214031 19.42 0.000 .373729 .4576472
------------------------------------------------------------------------------- What type of variable is freepoor in this regression? [2 points]
Dummy variable.
- Which population parameter suggests that those on low-income government insurance have 0.12 fewer doctor visits on average compared to those not on low-income government insurance? [2 points]
\[\alpha_1\]
- Which population parameter suggests that those on low-income government insurance have 0.265 fewer doctor visits per unit of income compared to those not on low-income government insurance? [2 points]
\[\alpha_2\]
- Which population parameter suggests that those not on low-income government insurance have 0.18 fewer doctor visits per unit of income? [2 points]
\[\beta_2\]
- What conclusion should we draw from the reported F-statistic and its associated p-value? [3 points]
Since Prob > F \(\approx0.000<\alpha=0.05\), we can reject the null of our test of overall significance and say that at least one of our coefficients is non-zero for predicting our outcome variable. Our regressors jointly predict variation in our outcome variable.
- Which of our coefficients are statistically different from zero at \(\alpha=5\)%? Which are not? [3 points]
The coefficient on income is statistically different from zero at 5%; those on freepoor and the interaction term incomeXfreepoor are not different from zero.
- If our four population assumptions hold, how would we expect \(b_2\) to be distributed? Your answer should take the form \(b_2\sim\_\_(\_\_,\_\_)\). Use Greek letters when appropriate. [3 points]
\[b_2\sim N(\beta_2,\sigma^2_{b_2});\left[\sigma^2_{b_2}=\frac{\sigma^2_u}{\sum_{i=1}^n\tilde{x}_{2i}^2}\right]\]
- Find an expression for the marginal effect of income in this regression. [3 points]
\[\frac{\partial\widehat{visits}}{\partial income}=b_2+a_2freepoor\]
- If we want to compare model fit of this regression to that of our previous bivariate model, should we use R-squared or Adj R-squared? Why? [3 points]
We should use adjusted R-squared, since our models contain different numbers of regressors. Adjusted R-squared penalizes the addition of new regressors while R-squared does not.
Question 4: Multivariate Regression [18 points]
For a multivariate OLS regression with sample size \(n\), \(k-1\) regressors \(x_2,...,x_k\), \(k\) estimated coefficients \(b_1,...,b_k\), and \(j=1,...,k\) as our index of coefficients:
- What is the difference between \(x_j\) and \(\tilde{x}_j\)? [3 points]
\(x_j\) is all the variation of our \(j^{th}\) regressor while \(\tilde{x}_j\) is the unique variation in \(x_j\) not correlated with/independent of all other regressors in our model. We use \(\tilde{x}_j\) instead of \(x_j\) in a multivariate regression.
- Suppose we add a new regressor \(x_{k+1}\) to our model. What do we expect to happen to the R-squared of our regression? [3 points]
We expect R-squared to weakly increase (get larger or stay the same) since adding regressors to our model can never decrease our model fit.
- Suppose we add a new regressor \(x_{k+1}\) to our model and the standard error of \(b_j\) decreases. What might this suggest about our model? [3 points]
This suggests that \(x_{k+1}\) improves our model fit and decreases RMSE in the numerator of \(s_{b_j}\).
- Suppose instead we add a new regressor \(x_{k+1}\) to our model and the standard error of \(b_j\) increases. What might this suggest about our model? [3 points]
This suggests that \(x_{k+1}\) is correlated with \(x_j\), since this would decrease \(\tilde{x}_j\) in the denominator of \(s_{b_j}\).
- Name the two models that are compared when computing the statistic for the default test of overall significance. Write down the population model for both. [6 points]
\[\text{Unrestricted: } y=\beta_1+\beta_2x_2+...+\beta_kx_k+u\]
\[\text{Restricted: }y=\beta_1+\nu\]
Question 5: Miscellaneous [17 points]
- Compute the following sum [3 points]: \[6+\sum_{i=3}^{5}\left(\frac{60}{i}\right)(i-2)\]
\[\begin{aligned} &= 6 + \left[\frac{60}{3}(3-2) + \frac{60}{4}(4-2) + \frac{60}{5}(5-2)\right] \\ &= 6 + 20(1) + 15(2) + 12(3) \\ &= 6 + 86 \\ &= 92 \end{aligned}\]
- If the default p-values reported for all coefficients from a multivariate regression of \(y\) on \(x_2,...,x_k\) are above 0.05 but we observe
Prob > F = 0.035, what can we conclude about our regressors? [4 points]
All of our regressors are not statistically significantly different from zero individually but do jointly explain variation in our outcome variable.
- If we choose to use robust standard errors in our model and then find out the homoskedasticity assumption is not violated, should we be concerned? Why or why not? [3 points]
No. Robust standard errors are correct for both homoskedastic and heteroskedastic errors.
- If we are told to use clustered standard errors in a regression model but know that population assumptions 1, 2, and 4 hold for our data, should we cluster or not? Why? [3 points]
No. We know by assumption 4 that our error terms are independent, meaning there is no need to cluster.
- If to determine the impact a certain variable (not regressor) has on our multivariate regression model we must use an F test/cannot use a t-test, what does this tell us about this variable? [4 points]
This means our variable must appear in multiple regressors rather than just one (e.g. income above).
Multiple Choice [2 points each]
Only one answer is correct for each question. Read carefully and choose the best possible answer.
MC 1 & 2
Suppose the probability of rejecting \(H_0\) is 0.05 when \(\beta_j=0\) and 0.95 when \(\beta_j\neq0\) for a test on OLS \(b_j\).
The size of this test will be equal to:
a) 5%
95%
100%
0%
The power of this test will be equal to:
- 5%
b) 95%
100%
0%
MC 3
A trial judge says he believes he made a Type I error when sentencing a defendant. The precedent in court is that individuals are innocent until proven guilty. If the judge is correct, which of the following took place?
The defendant was truly guilty and was found guilty
The defendant was truly guilty but was found innocent
The defendant was truly innocent and was found innocent
d) The defendant was truly innocent but was found guilty
MC 4
If \(\frac{Q-4}{4}\sim N(0,1)\), it must be true that:
\(E[Q] = 0\)
\(V[Q] = 4\)
c) \(Q\sim N\)
\(\sigma_Q=16\)
None of the above
MC 5
In which of the following scenarios did we make a mistake?
- We claimed that we could not compare the R-squared of a regression of y on x and a regression of ln(y) on x.
b) We claimed that we could not compare the R-squared of a regression of y on x and a regression of y on ln(x).
We claimed that the critical value approach and p-value approach for a given test would always give us the same answer of whether to reject the null hypothesis or not.
We claimed that the standard error of X will always be smaller than the standard deviation of X for \(n>1\).
None of the above