Applied Econometrics
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Course Introduction
Applied Econometrics focuses on the practical application of statistical and econometric methods to economic data in order to test hypotheses and forecast future trends. The course introduces students to the foundational concepts of econometric modeling, estimation, and inference, covering topics such as multiple regression analysis, hypothesis testing, model selection, and interpretation of empirical results. Through hands-on projects and real-world data sets, students develop skills in using econometric software and critically evaluating empirical research. This course equips students with analytical tools essential for data-driven decision making in economics, finance, and related fields.
Recommended Textbook
Introduction to Econometrics Update 3rd Edition by James H. Stock
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Chapter 1: Economic Questions and Data
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Sample Questions
Q1) Ideal randomized controlled experiments in economics are
A)often performed in practice.
B)often used by the Federal Reserve to study the effects of monetary policy.
C)useful because they give a definition of a causal effect.
D)sometimes used by universities to determine who graduates in four years rather than five.
Answer: C
Q2) Econometrics can be defined as follows with the exception of
A)the science of testing economic theory.
B)fitting mathematical economic models to real-world data.
C)a set of tools used for forecasting future values of economic variables.
D)measuring the height of economists.
Answer: D
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Chapter 2: Review of Probability
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Q1) Assume that Y is normally distributed N(?, ?<sup>2</sup>). To find Pr(c<sub>1</sub> ? Y ? c<sub>2</sub>), where c<sub>1</sub> < c<sub>2</sub> and d<sub>i</sub> = \(\frac { c _ { 1 } - \mu } { \sigma }\) , you need to calculate Pr(d<sub>1</sub> ? Z ? d<sub>2</sub>)=
A)?(d<sub>2</sub>)- ?(d<sub>1</sub>)
B)?(1.96)- ?(1.96)
C)?(d<sub>2</sub>)- (1 - ?(d<sub>1</sub>))
D)1 - (?(d<sub>2</sub>)- ?(d<sub>1</sub>))
Answer: A
Q2) var(aX + bY)=
A) \(a ^ { 2 } \sigma \stackrel { 2 } { X } + b ^ { 2 } \sigma \stackrel { 2 } { Y }\)
B) \(a^{2} \sigma_{\mathrm{X}}^{2}+2 a b \sigma_{X Y}+b^{2} \sigma_{Y}^{2} .\)
C) \(\sigma _ { X Y } + \mu _ { X } { } ^ { \mu } Y ^ { * }\)
D) \(a \sigma _ { \mathrm { X } } ^ { 2 } + b \sigma _ { \mathrm { Y } } ^ { 2 }\)
Answer: B
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Chapter 3: Review of Statistics
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Q1) When testing for differences of means, the t-statistic t = \(\frac { \bar { Y } m - \bar { Y } w } { S E ( \bar { Y } m - \overline { \bar { Y } w } ) }\) , where \(S E \left[ \bar { Y } _ { m } - \bar { Y } _ { W } \right) = \sqrt { \frac { s _ { m } ^ { 2 } } { n _ { m } } + \frac { s _ { w } ^ { 2 } } { n _ { w } } }\) has
A)a student t distribution if the population distribution of Y is not normal
B)a student t distribution if the population distribution of Y is normal
C)a normal distribution even in small samples
D)cannot be computed unless \(n _ { w }\) <sub> </sub>= \(n _ { m }\)
Answer: B
Q2) The standard error of \(\bar { Y } , S E ( \bar { Y } ) = \hat { \sigma } \bar { Y }\) is given by the following formula:
A) \(\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 }\)
B) \(\frac { S _ { y } ^ { n } } { n }\)
C) \(S _ { y }\)
D) \(\frac { S _ { y } } { \sqrt { n } }\)
Answer: D
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Chapter 4: Linear Regression With One Regressor
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Sample Questions
Q1) The OLS residuals
A)can be calculated using the errors from the regression function.
B)can be calculated by subtracting the fitted values from the actual values.
C)are unknown since we do not know the population regression function.
D)should not be used in practice since they indicate that your regression does not run through all your observations.
Q2) Imagine that you had discovered a relationship that would generate a scatterplot very similar to the relationship Y<sub>i</sub> = \(x _ { i } ^ { 2 }\) , and that you would try to fit a linear regression through your data points. What do you expect the slope coefficient to be? What do you think the value of your regression R<sup>2</sup> is in this situation? What are the implications from your answers in terms of fitting a linear regression through a non-linear relationship?
Q3) (Requires Appendix material)Which of the following statements is correct?
A)TSS = ESS + SSR
B)ESS = SSR + TSS
C)ESS > TSS
D)R<sup>2</sup> = 1 - (ESS/TSS)
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Chapter 5: Regression With a Single Regressor: Hypothesis
Tests and Confidence Intervals
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Sample Questions
Q1) Under the least squares assumptions (zero conditional mean for the error term, X<sub>i</sub> and Y<sub>i</sub> being i.i.d., and X<sub>i</sub> and u<sub>i</sub> having finite fourth moments), the OLS estimator for the slope and intercept
A)has an exact normal distribution for n > 15.
B)is BLUE.
C)has a normal distribution even in small samples. D)is unbiased.
Q2) The construction of the t-statistic for a one- and a two-sided hypothesis
A)depends on the critical value from the appropriate distribution. B)is the same.
C)is different since the critical value must be 1.645 for the one-sided hypothesis, but 1.96 for the two-sided hypothesis (using a 5% probability for the Type I error).
D)uses ±1.96 for the two-sided test, but only +1.96 for the one-sided test.
Q3) The only difference between a one- and two-sided hypothesis test is
A)the null hypothesis.
B)dependent on the sample size n.
C)the sign of the slope coefficient.
D)how you interpret the t-statistic.
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Chapter 6: Linear Regression With Multiple Regressors
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Sample Questions
Q1) (Requires some Calculus)Consider the sample regression function . \(Y _ { i } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } X _ { 1 i } + \hat { \beta } _ { 2 } X _ { 2 i }\) Take the total derivative. Next show that the partial derivative \(\frac { \Delta Y _ { i } } { \Delta X _ { 1 i } }\) is obtained by holding \(X _ { 2 i }\) constant, or controlling for \(X _ { 2 i }\)
Q2) In a two regressor regression model, if you exclude one of the relevant variables then
A)it is no longer reasonable to assume that the errors are homoskedastic.
B)OLS is no longer unbiased, but still consistent.
C)you are no longer controlling for the influence of the other variable.
D)the OLS estimator no longer exists.
Q3) In a multiple regression framework, the slope coefficient on the regressor X<sub>2</sub><sub>i</sub>
A)takes into account the scale of the error term.
B)is measured in the units of Y<sub>i</sub> divided by units of X<sub>2</sub><sub>i</sub>.
C)is usually positive.
D)is larger than the coefficient on X<sub>1</sub><sub>i</sub>.
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Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple
Regression
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Sample Questions
Q1) Consider the following regression output where the dependent variable is testscores and the two explanatory variables are the student-teacher ratio and the percent of English learners: \(\widehat{\text { TestScore }}\) = 698.9 - 1.10×STR - 0.650×PctEL. You are told that the t-statistic on the student-teacher ratio coefficient is 2.56. The standard error therefore is approximately
A)0.25
B)1.96
C)0.650
D)0.43
Q2) The general answer to the question of choosing the scale of the variables is
A)dependent on you whim.
B)to make the regression results easy to read and to interpret.
C)to ensure that the regression coefficients always lie between -1 and 1.
D)irrelevant because regardless of the scale of the variable, the regression coefficient is unaffected.
Q3) Adding the Percent of English Speakers (PctEL)to the Student Teacher Ratio (STR)in your textbook reduced the coefficient for STR from 2.28 to 1.10 with a standard error of 0.43. Construct a 90% and 99% confidence interval to test the hypothesis that the coefficient of STR is 2.28.
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Chapter 8: Nonlinear Regression Functions
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Sample Questions
Q1) You have estimated an earnings function, where you regressed the log of earnings on a set of continuous explanatory variables (in levels)and two binary variables, one for gender and the other for marital status. One of the explanatory variables is education.
(a)Interpret the education coefficient.
(b)Next, specify the binary variables and an equation, where the default is a single male, without allowing for interaction between marital status and gender. Indicate the coefficients that measure the effect of a single male, single female, married male, and married female.
(c)Finally allow for an interaction between the gender and marital status binary variables. Repeat the exercise of writing down the various effects based on the female/male and single/married status. Why is the latter approach more general than the former?
Q2) In the model ln(Y<sub>i</sub>)= ?<sub>0</sub> + ?<sub>1</sub>X<sub>i</sub> + u<sub>i</sub>, the elasticity of E(Y|X)with respect to X is A)?<sub>1</sub>X
B)?<sub>1</sub>
C) \(\frac { \beta _ { 1 } X } { \beta _ { 0 } + \beta _ { 1 } X }\)
D)Cannot be calculated because the function is non-linear
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Chapter 9: Assessing Studies Based on Multiple Regression
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Sample Questions
Q1) Your textbook compares the results of a regression of test scores on the student-teacher ratio using a sample of school districts from California and from Massachusetts. Before standardizing the test scores for California, you get the following regression result: \(\widehat{\text { TestScr }}\) <sup>=</sup><sup> 698.9 </sup><sup>-</sup><sup> 2.28</sup><sup>×</sup><sup>STR</sup>
n = 420, R2 = 0.051, SER = 18.6 <sub>In addition, you are given the following information: the sample mean of the student</sub><sub>-</sub><sub>teacher ratio is 19.64 with a standard deviation of 1.89, and the standard deviation of the test scores is 19.05.</sub> <sub>a.</sub><sub> </sub><sub>After standardizing the test scores variable and running the regression again, what is the value of the slope? What is the meaning of this new slope here (interpret the result)?</sub> <sub>b.</sub><sub> </sub><sub>What</sub><sub> will be the new intercept? Now that test scores have been standardiz</sub><sub>ed, should you interpret the intercept?</sub>
c. Does the regression R2 change between the two regressions? What about the t-statistic for the slope estimator?
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Chapter 10: Regression With Panel Data
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Sample Questions
Q1) A pattern in the coefficients of the time fixed effects binary variables may reveal the following in a study of the determinants of state unemployment rates using panel data:
A)macroeconomic effects, which affect all states equally in a given year.
B)attitude differences towards unemployment between states.
C)there is no economic information that can be retrieved from these coefficients.
D)regional effects, which affect all states equally, as long as they are a member of that region.
Q2) Consider the special panel case where T = 2. If some of the omitted variables, which you hope to capture in the changes analysis, in fact change over time, then the estimator on the included change regressor
A)will be unbiased only when allowing for heteroskedastic-robust standard errors.
B)may still be unbiased.
C)will only be unbiased in large samples.
D)will always be unbiased.
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Chapter 11: Regression With a Binary Dependent Variable
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Sample Questions
Q1) (Requires Appendix material)The following are examples of limited dependent variables, with the exception of
A)binary dependent variable.
B)log-log specification.
C)truncated regression model.
D)discrete choice model.
Q2) (Requires Advanced material)Only one of the following models can be estimated by OLS:
A)Y = AK<sup>?</sup>L<sup>?</sup> + u.
B)Pr(Y = 1 | X)= ?(?<sub>0</sub> + ?<sub>1</sub>X)
C)Pr(Y = 1 | X)= F(?<sub>0</sub> + ?<sub>1</sub>X)= \(\frac
_ { 0 } + \beta _ { 1 } X \right) } }\)
D)Y = AK<sup>? L?u</sup>.
Q3) The following problems could be analyzed using probit and logit estimation with the exception of whether or not
A)a college student decides to study abroad for one semester.
B)being a female has an effect on earnings.
C)a college student will attend a certain college after being accepted.
D)applicants will default on a loan.
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Chapter 12: Instrumental Variables Regression
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Sample Questions
Q1) You have been hired as a consultant to estimate the demand for various brands of coffee in the market. You are provided with annual price data for two years by U.S. state and the quantities sold. You want to estimate a demand function for coffee using this data. What problems do you think you will encounter if you estimated the demand equation by OLS?
Q2) In the case of the simple regression model Y<sub>i</sub> = <sub>0</sub> + <sub>1</sub>X<sub>i</sub> + u<sub>i</sub>, i = 1, , n, when X and u are correlated, then
A)the OLS estimator is biased in small samples only.
B)OLS and TSLS produce the same estimate.
C)X is exogenous.
D)the OLS estimator is inconsistent.
Q3) In the case of exact identification
A)you can use the J-statistic in a test of overidentifying restrictions.
B)you cannot use TSLS for estimation purposes.
C)you must rely on your personal knowledge of the empirical problem at hand to assess whether the instruments are exogenous.
D)OLS and TSLS yield the same estimate.
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Chapter 13: Experiments and Quasi-Experiments
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Sample Questions
Q1) With panel data, the causal effect
A)cannot be estimated since correlation does not imply causation.
B)is typically estimated using the probit regression model.
C)can be estimated using the "differences-in-differences" estimator.
D)can be estimated by looking at the difference between the treatment and the control group after the treatment has taken place.
Q2) Causal effects that depend on the value of an observable variable, say W<sub>i</sub>,
A)cannot be estimated.
B)can be estimate by interacting the treatment variable with W<sub>i</sub>.
C)result in the OLS estimator being inefficient.
D)requires use of homoskedasticity-only standard errors.
Q3) A potential outcome
A)is the outcome for an individual under a potential treatment.
B)cannot be observed because most individuals do not achieve their potential.
C)is the same as a causal effect.
D)is none of the above.
Q4) Describe the major differences between a randomized controlled experiment and a quasi-experiment.
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Chapter 14: Introduction to Time Series Regression and Forecasting
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Sample Questions
Q1) The random walk model is an example of a
A)deterministic trend model.
B)binomial model.
C)stochastic trend model.
D)stationary model.
Q2) The Times Series Regression with Multiple Predictors
A)is the same as the ADL(p,q)with additional predictors and their lags present.
B)gives you more than one prediction.
C)cannot be estimated by OLS due to the presence of multiple lags.
D)requires that the k regressors and the dependent variable have nonzero, finite eighth moments.
Q3) You should use the QLR test for breaks in the regression coefficients, when
A)the Chow F-test has a p value of between 0.05 and 0.10.
B)the suspected break data is not known.
C)there are breaks in only some, but not all, of the regression coefficients.
D)the suspected break data is known.
Q4) (Requires Appendix material): Show that the AR(1)process Y<sub>t</sub> = a<sub>1</sub>Y<sub>t</sub><sub>-</sub><sub>1</sub> + e<sub>t</sub>; \(\left| a _ { 1 } \right|\) < 1, can be converted to a MA( )process.
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Chapter 15: Estimation of Dynamic Causal Effects
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Sample Questions
Q1) Autocorrelation of the error terms
A)makes it impossible to calculate homoskedasticity only standard errors.
B)causes OLS to be no longer consistent.
C)causes the usual OLS standard errors to be inconsistent.
D)results in OLS being biased.
Q2) GLS is consistent and BLUE if
A)X is predetermined.
B)the error process is AR(1).
C)X is strictly exogenous.
D)all the roots are inside the unit circle.
Q3) Estimation of dynamic multipliers under strict exogeneity should be done by
A)instrumental variable methods.
B)OLS.
C)feasible GLS.
D)analyzing the stationarity of the multipliers.
Q4) The Cochrane-Orcutt iterative method is
A)a special case of GLS estimation.
B)a method to compute HAC standard errors.
C)a special case of maximum likelihood estimation.
D)a grid search for the autoregressive parameters on the error process.
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Chapter 16: Additional Topics in Time Series Regression
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Sample Questions
Q1) The error term in a multiperiod regression
A)is serially correlated.
B)causes OLS to be inconsistent.
C)is serially correlated, but less so the longer the forecast horizon.
D)is serially uncorrelated.
Q2) The coefficients of the VAR are estimated by
A)using a simultaneous estimation method such as TSLS.
B)maximum likelihood.
C)panel methods.
D)estimating each of the equations by OLS.
Q3) The following is not a consequence of X<sub>t</sub> and Y<sub>t</sub> being cointegrated:
A)if X<sub>t</sub> and Y<sub>t</sub> are both I(1), then for some , Y<sub>t</sub>X<sub>t</sub> is I(0).
B)X<sub>t</sub> and Y<sub>t</sub> have the same stochastic trend.
C)in the expression Y<sub>t</sub> - X<sub>t</sub> , is called the cointegrating coefficient.
D)if X<sub>t</sub> and Y<sub>t</sub> are cointegrated then integrating one of the variables gives you the same result as integrating the other.
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Chapter 17: The Theory of Linear Regression With One Regressor
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Q1) The WLS estimator is called infeasible WLS estimator when
A)the memory required to compute it on your PC is insufficient.
B)the conditional variance function is not known.
C)the numbers used to compute the estimator get too large.
D)calculating the weights requires you to take a square root of a negative number.
Q2) All of the following are good reasons for an applied econometrician to learn some econometric theory, with the exception of
A)turning your statistical software from a "black box" into a flexible toolkit from which you are able to select the right tool for a given job.
B)understanding econometric theory lets you appreciate why these tools work and what assumptions are required for each tool to work properly.
C)learning how to invert a 4×4 matrix by hand.
D)helping you recognize when a tool will not work well in an application and when it is time for you to look for a different econometric approach.
Q3) What does the Gauss-Markov theorem prove? Without giving mathematical details, explain how the proof proceeds. What is its importance?
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Chapter 18: The Theory of Multiple Regression
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Sample Questions
Q1) Give several economic examples of how to test various joint linear hypotheses using matrix notation. Include specifications of R = r where you test for (i)all coefficients other than the constant being zero, (ii)a subset of coefficients being zero, and (iii)equality of coefficients. Talk about the possible distributions involved in finding critical values for your hypotheses.
Q2) One of the properties of the OLS estimator is
A)X \(\hat \beta\) = 0<sub>k</sub><sub>+</sub><sub>1</sub>.
B)that the coefficient vector \(\hat \beta\) has full rank.
C) \(X ^ { \prime }\) (Y - X \(\hat \beta\) )= 0<sub>k</sub><sub>+</sub><sub>1</sub>.
D)( \(X ^ { \prime }\) X)<sup>-</sup><sup>1</sup>= \(X ^ { \prime }\) Y
Q3) The TSLS estimator is
A)(X'X)<sup>-</sup><sup>1</sup> X'Y
B)(X'Z(Z'Z)<sup>-</sup><sup>1 </sup>Z'X)<sup>-</sup><sup>1</sup> X'Z(Z'Z)<sup>-</sup><sup>1 </sup>Z' Y
C)(X <sup>-</sup><sup>1</sup><sup>X</sup>)<sup>-</sup><sup>1</sup>(X <sup>-</sup ><sup>1</sup><sup>Y</sup>)
D)(X'P<sub>z</sub>)<sup>-</sup><sup>1</sup><sup>P</sup><sub>z</sub>Y
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