Contains data from SalaryCameron.dta
Observations: 776 Data for A. Colin Cameron
(2022), ANALYSIS OF ECONOMIC
DATA, Amazon
Variables: 12 22 Jan 2022 15:50
-------------------------------------------------------------------------------
Variable Storage Display Value
name type format label Variable label
-------------------------------------------------------------------------------
salary float %9.0g Annual Salary in dollars
lnsalary float %9.0g Natural Logarithm of Annual
Salary
satverb int %32.0g SAT test verbal score (highest
self-reported score as of 2007)
satmath int %32.0g SAT test mathematics score
(highest self-reported score as
of 2007)
highgrade byte %8.0g Highest grade ever completed
age float %9.0g Age in years
sex byte %14.0g 1 = female 0 = male
minority float %9.0g 1 = minority 0 = not minority
height float %9.0g Height in inches
weight int %8.0g Weight in pounds
genhealth byte %9.0g Health status: 1=Excellent
2=VeryGood 3=Good 4=Fair 5=Poor
actscore float %32.0g ACT test score (highest
self-reported score as of 2007)
-------------------------------------------------------------------------------
Sorted by:
Note: Dataset has changed since last saved.ECN 102: Analysis of Economics Data
Final Exam FQ 2024 – 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. Make sure to read every question thoroughly and carefully. Make sure to answer all parts of all questions completely and fully explain when asked. Show all work to receive full credit.
You may use only the calculators provided by the instructor. Graphing calculators, phone calculators, and other non-standard calculators may not be used. For questions requiring computation, it suffices to express your final answer with 2 significant figures.
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, you must write your name and student ID at the top of each scratch page to ensure it is not lost.
This exam is worth 100 points.
For this exam, we will use a dataset that includes annual salary, demographic characteristics, and other metrics for a sample of individuals. The variables present in the dataset are listed here, along with their format and labels.
Question 1: Summary Statistics [16 points]
- Is this sample of data likely to be observational or experimental? Explain. [2 points]
Observational: no randomization is taking place, we are merely observing a sample of individuals.
- Is this sample of data likely to be cross-section, time series, panel, or repeated cross-section? Explain. [2 points]
Cross-section: we observe different units of observation (people) at a single point in time; we have no year, month, day variable.
- How else could we classify data for our variable salary? Explain. [2 points]
Numerical or continuous numerical.
- What type of variable is minority if we include it in a regression? Explain. [2 points]
This is a dummy variable because it takes only values 1 or 0 if someone is or is not a minority.
- Are any other variables the same type as our variable minority when used in regression? [2 points]
Sex is also a dummy variable that takes value 1 for female and 0 for male.
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
height | 776 67.68814 4.198084 50 83
weight | 776 171.3015 43.5782 92 475
salary | 776 36184.23 24426.81 300 130254- Which variable above has the greatest dispersion? Which has the least? [3 points]
\[CV_{height}=\frac{4.20}{67.69}=0.06\]
\[CV_{weight}=\frac{43.58}{171.30}=0.25\]
\[CV_{salary}=\frac{24426.81}{36184.23}=0.68\]
Salary has the highest dispersion; height has the least.
- What would we expect the standard error of height to be? What is this an estimator for? [3 points]
\[se_{height}=\frac{4.20}{\sqrt{776}}=0.15\]
This is an estimator for the population standard deviation of the sample mean of height.
Question 2: Bivariate Regression [16 points]
Suppose we want to run a regression to predict salary using age with OLS.
- Write down the population regression model we are trying to estimate. [2 points]
\[\widehat{salary}=\beta_1+\beta_2age\text{ or}\] \[salary=\beta_1+\beta_2age+u\]
- What will the homoskedasticity assumption imply for this regression? [2 points]
\(Var(u)=\sigma^2_u\) or our errors have a constant variance across values of age.
- What will the linearity assumption imply for this regression? [2 points]
Our true population model is \(salary=\beta_1+\beta_2age+u\) (we do not have any omitted covariates or nonlinear relationship).
Source | SS df MS Number of obs = 776
-------------+---------------------------------- F(1, 774) = 5.78
Model | 3.4281e+09 1 3.4281e+09 Prob > F = 0.0164
Residual | 4.5899e+11 774 593010915 R-squared = 0.0074
-------------+---------------------------------- Adj R-squared = 0.0061
Total | 4.6242e+11 775 596669065 Root MSE = 24352
------------------------------------------------------------------------------
salary | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
age | 2605.833 1083.809 2.40 0.016 478.28 4733.387
_cons | -34210.21 29291.25 -1.17 0.243 -91709.92 23289.5
------------------------------------------------------------------------------- Suppose we estimate this regression and find that our 95% confidence interval for \(b_2\) is \((478.28,4733.39)\). What conclusion about our population coefficient \(\beta_2\) on age does this imply? [3 points]
Since 0 is not included in our 95% CI, we can reject the null for the test of association and conclude that \(\beta_2\neq0\) or that age is associated with salary.
- Suppose our R-squared for this regression is 0.0074. What does this mean in words? [2 points]
\(0.0074\times100=0.74\%\) of the variation in salary is explained by age.
Suppose instead we estimated a regression of salary on the log of age.
- What would we call this type of regression? [2 points]
Linear-log regression
- How would we interpret \(b_2\) for this regression in words? [3 points]
A 1% change in age is associated with a \(b_2/100\)-unit (dollar) change in salary.
Question 3: Multivariate Regression [23 points]
We now estimate \(\widehat{salary}=\beta_1+\beta_2age+\alpha_1sex+\alpha_2(age\times sex)\) with OLS and obtain the following:
Source | SS df MS Number of obs = 776
-------------+---------------------------------- F(3, 772) = 4.32
Model | 7.6433e+09 3 2.5478e+09 Prob > F = 0.0049
Residual | 4.5478e+11 772 589087072 R-squared = 0.0165
-------------+---------------------------------- Adj R-squared = 0.0127
Total | 4.6242e+11 775 596669065 Root MSE = 24271
------------------------------------------------------------------------------
salary | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
age | 1706.862 1579.951 1.08 0.280 -1394.648 4808.373
sex | -53271.99 58521.46 -0.91 0.363 -168152.1 61608.08
ageXsex | 1807.953 2165.534 0.83 0.404 -2443.081 6058.986
_cons | -7612.026 42668.87 -0.18 0.858 -91372.78 76148.73
------------------------------------------------------------------------------(Hint: review variable descriptions on page 2)
- What type of variable is sex in this regression? [2 points]
Dummy variable
- Interpret the value of \(b_2\) in words. [2 points]
Being one year older on average is associated with a $1706.86 higher salary for men (\(sex=0\)).
- Interpret the value of \(a_1\) in words. [2 points]
Women make, on average, $53,271.99 less than men (or the difference in intercept between men and women).
- Interpret the value of \(a_2\) in words. [2 points]
Women make $1807.95 more for each additional year older they are compared to men. Being one year older is associated with a \(1706.86+1807.95=\$3514.81\) higher salary for women (\(sex=1\)) (or difference in slopes between men and women).
- What conclusion should we draw from the reported F-statistic and its associated p-value? [3 points]
\(\Pr(F > 4.32) = 0.0049 < \alpha = 0.05\): reject the null hypothesis of the default test of overall significance. At least one of our regressors/our model explains variation in our outcome variable statistically significantly.
- Which of our coefficients are statistically different from zero at \(\alpha=5\)%? Which are not? [3 points]
All regressors have a p-value above \(\alpha=0.05\): none are statistically significantly 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]
\[b_2\sim N\left(\beta_2,\frac{\sigma^2_u}{\sum_{i=1}^n\tilde{x}_{2i}^2}\right)\text{ or }b_2\sim N(\beta_2,\sigma^2_{b_2})\]
- Find an expression for the marginal effect of age in this regression. [3 points]
\[\frac{\partial\widehat{salary}}{\partial age}=b_2+a_2sex\]
- Why are our values for R-squared and Adj R-squared different in this regression? Why might we prefer one instead of the other for multivariate regressions? [3 points]
Adjusted R-squared imposes a penalty for adding additional regressors, here \(k=4\), while regular R-squared does not. We may want a measure of model fit that penalizes endlessly adding regressors.
Question 4: Hypothesis Testing [18 points]
For a general multivariate OLS regression with sample size \(n\), \(k-1\) regressors (\(x_2,...,x_k\)) and a constant, and \(j=1,...,k\) as our index of coefficients:
- What are the null and alternate hypotheses for the default test of association for a single \(b_j\)? [2 points]
\[H_0:\beta_j=0\] \[H_A:\beta_j\neq0\]
- Suppose we reject this null hypothesis for the default test of association for \(b_j\) above. How would we interpret this conclusion in words? [4 points]
There is a statistically significant association between the unique variation in our regressor \(x_j\) and our outcome variable \(y\).
- 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]
\[\text{Unrestricted: } \hat{y}=\beta_1+\beta_2x_x+...+\beta_kx_k\text{ ; Restricted:}\hat{y}=\beta_1+0+...0\]
- If the p-value for the default test of association for \(b_j\) is 0.035, what would we expect the p-value to be for an F-test of the single regressor \(b_j\)? [3 points]
\(p=0.035\) (the same as for the test of association)
- If the t-statistic for the default test of association for \(b_j\) is 2.12, what would we expect the F-statistic to be for an F-test of the single regressor \(b_j\)? [3 points]
\(F=(t)^2=2.12^2=4.49\) (the square of the t-statistic for the test of association)
Question 5: Miscellaneous [17 points]
- 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.
- What assumption or assumptions are we concerned may be violated if we choose to use robust standard errors in our regression? [3 points]
Homoskedasticity assumption (no explanation needed)
- What assumption or assumptions are we concerned may be violated if we choose to use clustered standard errors in our regression? [3 points]
Independence and homoskedasticity assumptions (no explanation needed)
- If we suspect but are not sure that our data is homoskedastic, what type of standard errors should we use? Are we concerned that doing so may lead to problems? Why or why not? [4 points]
We should use robust standard errors. We are not concerned; using robust standard errors would not be wrong even if our data is homoskedastic; NOT using robust standard errors would be wrong if our data is NOT homoskedastic.
- Compute the following sum [3 points]: \[\sum_{i=2}^3(3i)^{i-1}-i\]
\[\sum_{i=2}^3(3i)^{i-1}-i=6^{2-1}-2+9^{3-1}-3=6+81-5=82\]
Multiple Choice [2 points each]
Only one answer is correct for each question. Read carefully and choose the best possible answer.
MC 1 and 2
Suppose the probability of not rejecting \(H_0\) is 0.95 when \(\beta_j=0\) and 0.3 when \(\beta_j\neq0\) for a test on OLS \(b_j\).
The power of this test will be equal to:
0.05
0.95
c) 0.7
- 0.3
The size of this test will be equal to:
a) 0.05
0.95
0.7
0.3
MC 3
A trial judge says he believes he made a Type II 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
b) The defendant was truly guilty but was found innocent
The defendant was truly innocent and was found innocent
The defendant was truly innocent but was found guilty
MC 4
If \(\frac{Q-7}{3}\sim (0,1)\), it must be true that:
\(E[Q] = 0\)
\(V[Q] = 3\)
\(Q\sim N\)
\(\sigma_Q=9\)
e) None of the above
MC 5
In which of the following scenarios did we make a mistake?
a) We claimed that \(R^2\) and \(\bar{R}^2\) should always be the same for a bivariate regression.
We claimed that the null hypothesis for a computed statistic should be rejected at a significance level of \(\alpha\) if the \(100(1-\alpha)\)% confidence interval for that statistic does not include the null value.
We claimed that the sample standard deviation of X will always be weakly greater (\(\geq\)) than the standard error of \(\bar{X}\) for any sample size.
We claimed that the total effect of \(b_j\) from multivariate OLS of y on \(x_2,...,x_k\) will always be equal to the partial effect if \(x_j\) is independent from all other regressors.
None of the above