Quantitative Methods for Economists
Exam Solutions
Course Introduction
Quantitative Methods for Economists introduces students to the fundamental mathematical and statistical techniques essential for analyzing economic data and solving economic problems. The course covers topics such as linear algebra, calculus, probability theory, and hypothesis testing, all within the context of economic applications. Students learn to formulate and test economic models, interpret empirical results, and utilize quantitative tools to inform decision-making. Emphasis is placed on problem-solving, data analysis, and critical evaluation of quantitative economic research, preparing students for further study and professional work in economics and related fields.
Recommended Textbook
Introduction to Econometrics Update 3rd Edition by James H. Stock
Available Study Resources on Quizplus
18 Chapters
982 Verified Questions
982 Flashcards
Source URL: https://quizplus.com/study-set/3376 Page 2
Chapter 1: Economic Questions and Data
Available Study Resources on Quizplus for this Chatper
17 Verified Questions
17 Flashcards
Source URL: https://quizplus.com/quiz/66988
Sample Questions
Q1) 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
Q2) Most economic data are obtained
A)through randomized controlled experiments.
B)by calibration methods.
C)through textbook examples typically involving ten observation points.
D)by observing real-world behavior.
Answer: D
To view all questions and flashcards with answers, click on the resource link above.
Page 3
Chapter 2: Review of Probability
Available Study Resources on Quizplus for this Chatper
70 Verified Questions
70 Flashcards
Source URL: https://quizplus.com/quiz/66989
Sample Questions
Q1) The systolic blood pressure of females in their 20s is normally distributed with a mean of 120 with a standard deviation of 9. What is the probability of finding a female with a blood pressure of less than 100? More than 135? Between 105 and 123? You visit the women's soccer team on campus, and find that the average blood pressure of the 25 members is 114. Is it likely that this group of women came from the same population?
Answer: Pr(Y<100)= 0.0131; Pr(Y>135)= 0.0478; Pr(105<Y<123)= 0.6784; Pr( \(\bar { Y }\) < 114)= Pr(Z < -3.33)= 0.0004. (The smallest z-value listed in the table in the textbook is -2.99, which generates a probability value of 0.0014.)This unlikely that this group of women came from the same population.
Q2) The probability of an outcome
A)is the number of times that the outcome occurs in the long run.
B)equals M × N, where M is the number of occurrences and N is the population size.
C)is the proportion of times that the outcome occurs in the long run.
D)equals the sample mean divided by the sample standard deviation.
Answer: C
To view all questions and flashcards with answers, click on the resource link above.
Chapter 3: Review of Statistics
Available Study Resources on Quizplus for this Chatper
65 Verified Questions
65 Flashcards
Source URL: https://quizplus.com/quiz/66990
Sample Questions
Q1) A manufacturer claims that a certain brand of VCR player has an average life expectancy of 5 years and 6 months with a standard deviation of 1 year and 6 months. Assume that the life expectancy is normally distributed.
(a)Selecting one VCR player from this brand at random, calculate the probability of its life expectancy exceeding 7 years.
(b)The Critical Consumer magazine decides to test fifty VCRs of this brand. The average life in this sample is 6 years and the sample standard deviation is 2 years. Calculate a 99% confidence interval for the average life.
(c)How many more VCRs would the magazine have to test in order to halve the width of the confidence interval?
Answer: (a)Pr (Y > 7)= Pr(Z > 1)= 0.1587.
(b)6 ± 2.58 × \(\frac { 2 } { \sqrt { 50 } }\) = 6 ± 0.73 = (5.27, 6.73).
(c) \(\frac { 1 } { 2 }\) × (2.58 × \(\frac { 2 } { \sqrt { 50 } }\) )= 2.58 × \(\frac { 1 } { 2 }\) × \(\frac { 2 } { \sqrt { 50 } }\) = 2.58 × \(\frac { 2 } { \sqrt { 4 \times 50 } }\) , or n = 200.
To view all questions and flashcards with answers, click on the resource link above. Page 5
Chapter 4: Linear Regression With One Regressor
Available Study Resources on Quizplus for this Chatper
65 Verified Questions
65 Flashcards
Source URL: https://quizplus.com/quiz/66991
Sample Questions
Q1) You have obtained a sub-sample of 1744 individuals from the Current Population Survey (CPS)and are interested in the relationship between weekly earnings and age. The regression, using heteroskedasticity-robust standard errors, yielded the following result: \(\widehat { E a m }\) = 239.16 + 5.20 × Age, R<sup>2</sup> = 0.05, SER = 287.21., where Earn and Age are measured in dollars and years respectively.
(a)Interpret the results.
(b)Is the effect of age on earnings large?
(c)Why should age matter in the determination of earnings? Do the results suggest that there is a guarantee for earnings to rise for everyone as they become older? Do you think that the relationship between age and earnings is linear?
(d)The average age in this sample is 37.5 years. What is annual income in the sample?
(e)Interpret the measures of fit.
Q2) The regression R<sup>2</sup> is a measure of
A)whether or not X causes Y.
B)the goodness of fit of your regression line.
C)whether or not ESS > TSS.
D)the square of the determinant of R.
To view all questions and flashcards with answers, click on the resource link above. Page 6
Chapter 5: Regression With a Single Regressor: Hypothesis
Tests and Confidence Intervals
Available Study Resources on Quizplus for this Chatper
59 Verified Questions
59 Flashcards
Source URL: https://quizplus.com/quiz/66992
Sample Questions
Q1) When estimating a demand function for a good where quantity demanded is a linear function of the price, you should A)not include an intercept because the price of the good is never zero.
B)use a one-sided alternative hypothesis to check the influence of price on quantity. C)use a two-sided alternative hypothesis to check the influence of price on quantity.
D)reject the idea that price determines demand unless the coefficient is at least 1.96.
Q2) Your textbook discussed the regression model when X is a binary variable Y<sub>i</sub> = ?<sub>0</sub> + ?<sub>i</sub>D<sub>i</sub> + u<sub>i</sub>, i = 1,..., n Let Y represent wages, and let D be one for females, and 0 for males. Using the OLS formula for the intercept coefficient, prove that \(\widehat { \beta 0 }\) is the average wage for males.
Q3) Carefully discuss the advantages of using heteroskedasticity-robust standard errors over standard errors calculated under the assumption of homoskedasticity. Give at least five examples where it is very plausible to assume that the errors display heteroskedasticity.
To view all questions and flashcards with answers, click on the resource link above. Page 7
Chapter 6: Linear Regression With Multiple Regressors
Available Study Resources on Quizplus for this Chatper
65 Verified Questions
65 Flashcards
Source URL: https://quizplus.com/quiz/66993
Sample Questions
Q1) Under the least squares assumptions for the multiple regression problem (zero conditional mean for the error term, all X<sub>i</sub> and Y<sub>i</sub> being i.i.d., all X<sub>i</sub> and u<sub>i</sub> having finite fourth moments, no perfect multicollinearity), the OLS estimators for the slopes and intercept
A)have an exact normal distribution for n > 25.
B)are BLUE.
C)have a normal distribution in small samples as long as the errors are homoskedastic. D)are unbiased and consistent.
Q2) Consider the multiple regression model with two regressors X<sub>1</sub> and X<sub>2</sub>, where both variables are determinants of the dependent variable. When omitting X<sub>2</sub> from the regression, then there will be omitted variable bias for \(\widehat { \beta 1 }\)
A)if X<sub>1</sub> and X<sub>2</sub> are correlated
B)always
C)if X<sub>2</sub> is measured in percentages
D)if X<sub>2</sub> is a dummy variable
Q3) In the multiple regression with two explanatory variables, show that the TSS can still be decomposed into the ESS and the RSS.
To view all questions and flashcards with answers, click on the resource link above.
8
Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple
Regression
Available Study Resources on Quizplus for this Chatper
64 Verified Questions
64 Flashcards
Source URL: https://quizplus.com/quiz/66994
Sample Questions
Q1) The formula for the standard error of the regression coefficient, when moving from one explanatory variable to two explanatory variables, A)stays the same.
B)changes, unless the second explanatory variable is a binary variable. C)changes.
D)changes, unless you test for a null hypothesis that the addition regression coefficient is zero.
Q2) The F-statistic with q = 2 restrictions when testing for the restrictions ?<sub>1</sub> = 0 and ?<sub>2</sub> = 0 is given by the following formula: \[F = \frac { 1 } { 2 } \left( \frac { t _ { 1 } ^ { 2 } + t _ { 2 } ^ { 2 } - 2 \hat { \rho } _ {
\hat { \rho } _ { t _ { 1 } , t _ { 2 } } ^ { 2 } } \right)\] Discuss how this formula can be understood intuitively.
Q3) Trying to remember the formula for the homoskedasticity-only F-statistic, you forgot whether you subtract the restricted SSR from the unrestricted SSR or the other way around. Your professor has provided you with a table containing critical values for the F distribution. How can this be of help?
To view all questions and flashcards with answers, click on the resource link above.
9
Chapter 8: Nonlinear Regression Functions
Available Study Resources on Quizplus for this Chatper
63 Verified Questions
63 Flashcards
Source URL: https://quizplus.com/quiz/66995
Sample Questions
Q1) Give at least three examples from economics where you expect some nonlinearity in the relationship between variables. Interpret the slope in each case.
Q2) The best way to interpret polynomial regressions is to
A)take a derivative of Y with respect to the relevant X.
B)plot the estimated regression function and to calculate the estimated effect on Y associated with a change in X for one or more values of X.
C)look at the t-statistics for the relevant coefficients. D)analyze the standard error of estimated effect.
Q3) In the regression model Y<sub>i</sub> = <sub>0</sub> + <sub>1</sub>X<sub>i </sub>+ <sub>2</sub>D<sub>i</sub> + <sub>3</sub>(X<sub>i</sub> × D<sub>i</sub>)+ u<sub>i</sub>, where X is a continuous variable and D is a binary variable, to test that the two regressions are identical, you must use the
A)t-statistic separately for <sub>2</sub> = 0, <sub>2 </sub>= 0.
B)F-statistic for the joint hypothesis that <sub>0 </sub>= 0, <sub>1 </sub>= 0.
C)t-statistic separately for <sub>3 </sub>= 0.
D)F-statistic for the joint hypothesis that <sub>2 </sub>= 0, <sub>3</sub>= 0.
To view all questions and flashcards with answers, click on the resource link above.
Page 10
Chapter 9: Assessing Studies Based on Multiple Regression
Available Study Resources on Quizplus for this Chatper
65 Verified Questions
65 Flashcards
Source URL: https://quizplus.com/quiz/66996
Sample Questions
Q1) Correlation of the regression error across observations
A)results in incorrect OLS standard errors.
B)makes the OLS estimator inconsistent, but not unbiased.
C)results in correct OLS standard errors if heteroskedasticity-robust standard errors are used.
D)is not a problem in cross-sections since the data can always be "reshuffled."
Q2) Misspecification of functional form of the regression function
A)is overcome by adding the squares of all explanatory variables.
B)is more serious in the case of homoskedasticity-only standard error.
C)results in a type of omitted variable bias.
D)requires alternative estimation methods such as maximum likelihood.
Q3) Sample selection bias occurs when
A)the choice between two samples is made by the researcher.
B)data are collected from a population by simple random sampling.
C)samples are chosen to be small rather than large.
D)the availability of the data is influenced by a selection process that is related to the value of the dependent variable.
Q4) Think of three different economic examples where cross-sectional data could be collected. Indicate in each of these cases how you would check if the analysis is externally valid.
To view all questions and flashcards with answers, click on the resource link above. Page 11
Chapter 10: Regression With Panel Data
Available Study Resources on Quizplus for this Chatper
50 Verified Questions
50 Flashcards
Source URL: https://quizplus.com/quiz/66997
Sample Questions
Q1) Consider estimating the effect of the beer tax on the fatality rate, using time and state fixed effect for the Northeast Region of the United States (Maine, Vermont, New Hampshire, Massachusetts, Connecticut and Rhode Island)for the period 1991-2001. If Beer Tax was the only explanatory variable, how many coefficients would you need to estimate, excluding the constant?
A)18
B)17
C)7
D)11
Q2) It is advisable to use clustered standard errors in panel regressions because A)without clustered standard errors, the OLS estimator is biased
B)hypothesis testing can proceed in a standard way even if there are few entities (n is small)
C)they are easier to calculate than homoskedasticity-only standard errors
D)the fixed effects estimator is asymptotically normally distributed when n is large
To view all questions and flashcards with answers, click on the resource link above.
12
Chapter 11: Regression With a Binary Dependent Variable
Available Study Resources on Quizplus for this Chatper
50 Verified Questions
50 Flashcards
Source URL: https://quizplus.com/quiz/66998
Sample Questions
Q1) (Requires Appendix material and Calculus)The log of the likelihood function (L)for the simple regression model with i.i.d. normal errors is as follows (note that taking the logarithm of the likelihood function simplifies maximization. It is a monotonic transformation of the likelihood function, meaning that this transformation does not affect the choice of maximum):
L = - \(\frac { n } { 2 }\) log(2 )- \(\frac { n } { 2 }\) log <sup>2</sup> - \(\frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i } \right) ^ { 2 }\) Derive the maximum likelihood estimator for the slope and intercept. What general properties do these estimators have? Explain intuitively why the OLS estimator is identical to the maximum likelihood estimator here.
Q2) Nonlinear least squares
A)solves the minimization of the sum of squared predictive mistakes through sophisticated mathematical routines, essentially by trial and error methods.
B)should always be used when you have nonlinear equations.
C)gives you the same results as maximum likelihood estimation.
D)is another name for sophisticated least squares.
To view all questions and flashcards with answers, click on the resource link above.
13
Chapter 12: Instrumental Variables Regression
Available Study Resources on Quizplus for this Chatper
50 Verified Questions
50 Flashcards
Source URL: https://quizplus.com/quiz/66999
Sample Questions
Q1) Studies of the effect of minimum wages on teenage employment typically regress the teenage employment to population ratio on the real minimum wage or the minimum wage relative to average hourly earnings using OLS. Assume that you have a cross section of United States for two years. Do you think that there are problems with simultaneous equation bias?
Q2) To calculate the J-statistic you regress the A)squared values of the TSLS residuals on all exogenous variables and the instruments. The statistic is then the number of observations times the regression R<sup>2</sup>.
B)TSLS residuals on all exogenous variables and the instruments. You then multiply the homoskedasticity-only F-statistic from that regression by the number of instruments. C)OLS residuals from the reduced form on the instruments. The F-statistic from this regression is the J-statistic.
D)TSLS residuals on all exogenous variables and the instruments. You then multiply the heteroskedasticity-robust F-statistic from that regression by the number of instruments.
To view all questions and flashcards with answers, click on the resource link above.
Page 14
Chapter 13: Experiments and Quasi-Experiments
Available Study Resources on Quizplus for this Chatper
50 Verified Questions
50 Flashcards
Source URL: https://quizplus.com/quiz/67000
Sample Questions
Q1) Specify the multiple regression model that contains the difference-in-difference estimator (with additional regressors). Explain the circumstances under which this model is preferable to the simple difference-in-difference estimator. Explain how the W's can be used to test for randomization. How does the interpretation of the W variables change compared to the differences estimator with additional regressors?
Q2) Experimental effects, such as the Hawthorne effect, A)generally are not germane in quasi-experiments. B)typically require instrumental variable estimation in quasi-experiments. C)can be dealt with using binary variables in quasi-experiments. D)are the most important threat to internal validity in quasi-experiments.
Q3) You want to study whether or not the use of computers in the classroom for elementary students has an effect on performance. Explain in some detail how you would ideally set up such an experiment and what threats to internal and external validity there might be.
Q4) (Requires Appendix material)Discuss how the differences-in-differences estimator can be extended to multiple time periods. In particular, assume that there are n individuals and T time periods. What do the individual and time effects control for?
To view all questions and flashcards with answers, click on the resource link above. Page 15
Chapter 14: Introduction to Time Series Regression and Forecasting
Available Study Resources on Quizplus for this Chatper
50 Verified Questions
50 Flashcards
Source URL: https://quizplus.com/quiz/67001
Sample Questions
Q1) (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.
Q2) The Bayes-Schwarz Information Criterion (BIC)is given by the following formula
A)BIC(p)= ln [ \(\frac { \operatorname { SSR } ( p ) } { \mathrm { T } }\) ] + (p+1)
\(\frac { \ln ( T ) } { T }\)
B)BIC(p)= ln [ \(\frac { \operatorname { SSR } ( p ) } { \mathrm { T } }\) ] + (p+1)
\(\frac { 2 } { T }\)
C)BIC(p)= ln [ \(\frac { \operatorname { SSR } ( p ) } { \mathrm { T } }\) ] - (p+1)<sub> </sub> <sub> </sub> \(\frac { \ln ( T ) } { T }\)
D)BIC(p)= ln [ \(\frac { \operatorname { SSR } ( p ) } { \mathrm { T } }\) ] × (p+1)<sub> </sub> <sub> </sub> \(\frac { \ln ( T ) } { T }\)
Q3) The BIC is a statistic
A)commonly used to test for serial correlation
B)only used in cross-sectional analysis
C)developed by the Bank of England in its river of blood analysis
D)used to help the researcher choose the number of lags in an autoregression
To view all questions and flashcards with answers, click on the resource link above.
Page 16
Chapter 15: Estimation of Dynamic Causal Effects
Available Study Resources on Quizplus for this Chatper
50 Verified Questions
50 Flashcards
Source URL: https://quizplus.com/quiz/67002
Sample Questions
Q1) Your textbook used a distributed lag model with only current and past values of X<sub>t</sub><sub>-1</sub> coupled with an AR(1)error model to derive a quasi-difference model, where the error term was uncorrelated. (a)Instead use a static model Y<sub>t</sub> = <sub>0</sub> + <sub>1</sub>X<sub>t</sub> + u<sub>t</sub> here, where the error term follows an AR(1). Derive the quasi difference form. Explain why in the case of the infeasible GLS estimators you could easily estimate the s by OLS.
(b)Since <sub>1</sub> (the autocorrelation parameter for u<sub>t</sub>)is unknown, describe the Cochrane-Orcutt estimation procedure.
(c)Explain how the iterated Cochrane-Orcutt estimator works in this situation. Iterations stop when there is "convergence" in the estimates. What do you think is meant by that?
(d)Your textbook has pointed out that the iterated Cochrane-Orcutt GLS estimator is in fact the nonlinear least squares estimator of the model. Given that -1 < <sub>1</sub> < 1, suggest a "grid search" or some strategy to "nail down" the value of \(\hat { \phi }\) <sub>1</sub> which minimizes the sum of squared residuals. This is the so-called Hildreth-Lu method.
To view all questions and flashcards with answers, click on the resource link above. Page 17
Chapter 16: Additional Topics in Time Series Regression
Available Study Resources on Quizplus for this Chatper
50 Verified Questions
50 Flashcards
Source URL: https://quizplus.com/quiz/67003
Sample Questions
Q1) Think of at least five examples from economics where theory suggests that the variables involved are cointegrated. For one of these cases, explain how you would test for cointegration between the variables involved and how you could use this information to improve forecasting.
Q2) A VAR with five variables, 4 lags and constant terms for each equation will have a total of
A)21 coefficients.
B)100 coefficients.
C)105 coefficients.
D)84 coefficients.
Q3) A VAR with k time series variables consists of
A)k equations, one for each of the variables, where the regressors in all equations are lagged values of all the variables
B)a single equation, where the regressors are lagged values of all the variables
C)k equations, one for each of the variables, where the regressors in all equations are never more than one lag of all the variables
D)k equations, one for each of the variables, where the regressors in all equations are current values of all the variables
To view all questions and flashcards with answers, click on the resource link above.
18
Chapter 17: The Theory of Linear Regression With One Regressor
Available Study Resources on Quizplus for this Chatper
49 Verified Questions
49 Flashcards
Source URL: https://quizplus.com/quiz/67004
Sample Questions
Q1) Consider the model Y<sub>i</sub> - <sub>1</sub>X<sub>i</sub> + u<sub>i</sub>, where the X<sub>i</sub> and u<sub>i</sub> the are mutually independent i.i.d. random variables with finite fourth moment and E(u<sub>i</sub>)= 0.
(a)Let \(\hat { \beta }\) <sub>1</sub> denote the OLS estimator of <sub>1</sub>. Show that \(\sqrt { n }\) ( \(\hat { \beta }\) <sub>1</sub>- <sub>1</sub>)= \(\frac { \frac { \sum _ { i = 1 } ^ { n } X _ { i } t _ { i } } { \sqrt { n } } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } }\) (b)What is the mean and the variance of \(\frac { \sum _ { i = 1 } ^ { n } X _ { i } u _ { i } } { \sqrt { n } }\) ? Assuming that the Central Limit Theorem holds, what is its limiting distribution?
(c)Deduce the limiting distribution of \(\sqrt { n }\) ( \(\hat { \beta }\) <sub>1</sub><sub>1</sub>)? State what theorems are necessary for your deduction.
Q2) (Requires Appendix material)If X and Y are jointly normally distributed and are uncorrelated,
A)then their product is chi-square distributed with n-2 degrees of freedom
B)then they are independently distributed
C)then their ratio is t-distributed
D)none of the above is true
To view all questions and flashcards with answers, click on the resource link above. Page 19
Chapter 18: The Theory of Multiple Regression
Available Study Resources on Quizplus for this Chatper
50 Verified Questions
50 Flashcards
Source URL: https://quizplus.com/quiz/67005
Sample Questions
Q1) Define the GLS estimator and discuss its properties when is known. Why is this estimator sometimes called infeasible GLS? What happens when is unknown? What would the matrix look like for the case of independent sampling with heteroskedastic errors, where var(u<sub>i</sub> | X<sub>i</sub>)= ch(X<sub>i</sub>)= <sup>2</sup> <sup> </sup> \(x _ { 1 i } ^ { 2 }\) ? Since the inverse of the error variance-covariance matrix is needed to compute the GLS estimator, find <sup>-</sup><sup>1</sup>. The textbook shows that the original model Y = X + U will be transformed into \(\widetilde { Y } = \widetilde { X } \beta + \widetilde { U } , \text { where } \widetilde { Y } = F Y , \widetilde { X } = F X \text {, and } \widetilde { U }\) = FU, and \(F\) F = <sup>-</sup><sup>1</sup>. Find F in the above case, and describe what effect the transformation has on the original data.
Q2) The formulation R = r to test a hypotheses
A)allows for restrictions involving both multiple regression coefficients and single regression coefficients.
B)is F-distributed in large samples.
C)allows only for restrictions involving multiple regression coefficients.
D)allows for testing linear as well as nonlinear hypotheses.
To view all questions and flashcards with answers, click on the resource link above.
Page 20