Econometrics
Test Bank
Course Introduction
Econometrics is a course that introduces students to the application of statistical and mathematical methods to economic data in order to test hypotheses and forecast future trends. Covering both theoretical foundations and practical techniques, the course explores topics such as regression analysis, estimation, hypothesis testing, and model specification. Students learn to interpret and critically evaluate empirical economic research, utilize statistical software, and address common issues like multicollinearity, heteroscedasticity, and autocorrelation in real-world data. Through hands-on assignments and projects, the course prepares students to apply econometric tools for evidence-based analysis in economics and related fields.
Recommended Textbook
Introduction to Econometrics Update 3rd Edition by James H. Stock
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Page 2
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) In a randomized controlled experiment
A)there is a control group and a treatment group.
B)you control for the effect that random numbers are not truly randomly generated C)you control for random answers
D)the control group receives treatment on even days only.
Answer: A
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Chapter 2: Review of Probability
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Sample Questions
Q1) The expected value of a discrete random variable
A)is the outcome that is most likely to occur.
B)can be found by determining the 50% value in the c.d.f.
C)equals the population median.
D)is computed as a weighted average of the possible outcome of that random variable, where the weights are the probabilities of that outcome.
Answer: D
Q2) The correlation between X and Y
A)cannot be negative since variances are always positive.
B)is the covariance squared.
C)can be calculated by dividing the covariance between X and Y by the product of the two standard deviations.
D)is given by corr(X, Y)= \(\frac { \operatorname { cov } ( X , Y ) } { \operatorname { var } ( X ) \operatorname { var } ( Y ) }\)
Answer: C
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Chapter 3: Review of Statistics
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Sample Questions
Q1) The power of the test
A)is the probability that the test actually incorrectly rejects the null hypothesis when the null is true.
B)depends on whether you use \(\bar { Y }\) or \(\bar { Y }\) <sup>2</sup> for the t-statistic.
C)is one minus the size of the test.
D)is the probability that the test correctly rejects the null when the alternative is true.
Answer: D
Q2) Assume that you have 125 observations on the height (H)and weight (W)of your peers in college. Let \({ } ^ { S } H W\) = 68, \({ } ^ { S } H\) = 3.5, \(\text { SW }\) = 29. The sample correlation coefficient is
A)1.22
B)0.50
C)0.67
D)Cannot be computed since males and females have not been separated out.
Answer: C
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Chapter 4: Linear Regression With One Regressor
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Sample Questions
Q1) A necessary and sufficient condition to derive the OLS estimator is that the following two conditions hold: \(\sum _ { i = 1 } ^ { n } \hat { u _ { i } }\) = 0 and \(\sum _ { i = 1 } ^ { n } \hat { u } _ { i } X _ { i }\) = 0. Show that these conditions imply that \(\sum _ { i = 1 } ^ { n } \hat { u _ { i } } Y _ { i }\) = 0.
Q2) The help function for a commonly used spreadsheet program gives the following definition for the regression slope it estimates: \[\frac { n \sum _ { i = 1 } ^ { n
X _
Y _ { i } - \left( \sum _ { i = 1 } ^ { n } X _ { i } \right) \left( \sum _ { i = 1
\right) } { n \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n
} \right) ^ { 2 } }\] Prove that this formula is the same as the one given in the textbook.
Q3) The standard error of the regression (SER)is defined as follows
A) \(\frac { 1 } { n - 2 } \sum _ { i = 1 } ^ { n } \hat { u } _ { l } ^ { 2 }\)
B)SSR
C)1-R<sup>2</sup>
D) \(\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \hat { u } _ { i } ^ { 2 }\)
Q4) Consider the following model:
Y<sub>i</sub> = ?<sub>1</sub>X<sub>i</sub> + u<sub>i</sub>.
Derive the OLS estimator for ?<sub>1</sub>.
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Chapter 5: Regression With a Single Regressor: Hypothesis
Tests and Confidence Intervals
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Sample Questions
Q1) In the presence of heteroskedasticity, and assuming that the usual least squares assumptions hold, the OLS estimator is A)efficient.
B)BLUE.
C)unbiased and consistent.
D)unbiased but not consistent.
Q2) Your textbook discussed the regression model when X is a binary variable Y<sub>i</sub> = ?<sub>0</sub> + ?<sub>1</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 slope coefficient, prove that \(\hat{\beta}_ { 1 }\) is the difference between the average wage for males and the average wage for females.
Q3) Assume that the homoskedastic normal regression assumption hold. Using the Student t-distribution, find the critical value for the following situation:
(a)n = 28, 5% significance level, one-sided test.
(b)n = 40, 1% significance level, two-sided test.
(c)n = 10, 10% significance level, one-sided test.
(d)n = , 5% significance level, two-sided test.
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Page 7
Chapter 6: Linear Regression With Multiple Regressors
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Sample Questions
Q1) The OLS residuals in the multiple regression model
A)cannot be calculated because there is more than one explanatory variable. B)can be calculated by subtracting the fitted values from the actual values.
C)are zero because the predicted values are another name for forecasted values. D)are typically the same as the population regression function errors.
Q2) Your textbook extends the simple regression analysis of Chapters 4 and 5 by adding an additional explanatory variable, the percent of English learners in school districts (PctEl). The results are as follows: \(\widehat{\text { TestScore }}\) = 698.9 - 2.28 × STR and \(\widehat{\text { TestScore} }\) = 698.0 - 1.10 × STR - 0.65 × PctEL
Explain why you think the coefficient on the student-teacher ratio has changed so dramatically (been more than halved).
Q3) (Requires Calculus)For the case of the multiple regression problem with two explanatory variables, show that minimizing the sum of squared residuals results in three conditions: \[\sum _ { i = 1 } ^ { n } { \hat u_ { i } } = 0 ; \sum _ { i = 1 } ^ { n } \hat { u _ { i } } X _ { 1 i } = 0 ; \sum _ { i = 1 } ^ { n } \hat { u _ { i } } X _ { 2 i } = 0\]
Q4) In the multiple regression with two explanatory variables, show that the TSS can still be decomposed into the ESS and the RSS.
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Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple
Regression
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Sample Questions
Q1) Consider the following multiple regression model Y<sub>i</sub> = ?<sub>0</sub> + ?<sub>1</sub>X<sub>1</sub><sub>i</sub> + ?<sub>2</sub>X<sub>2</sub><sub>i</sub> + ?<sub>3</sub>X<sub>3</sub><sub>i</sub> + u<sub>i</sub>
You want to consider certain hypotheses involving more than one parameter, and you know that the regression error is homoskedastic. You decide to test the joint hypotheses using the homoskedasticity-only F-statistics. For each of the cases below specify a restricted model and indicate how you would compute the F-statistic to test for the validity of the restrictions. (a)?<sub>1</sub> = -?<sub>2</sub>; ?<sub>3 </sub>= 0 (b)?<sub>1</sub> + ?<sub>2 </sub>+ ?<sub>3 </sub>= 1 (c)?<sub>1</sub> = ?<sub>2</sub>; ?<sub>3 </sub>= 0
Q2) Explain carefully why testing joint hypotheses simultaneously, using the F-statistic, does not necessarily yield the same conclusion as testing them sequentially ("one at a time" method), using a series of t-statistics.
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) (Requires Calculus)In the equation \(\widehat{\text { TestScore }}\) = 607.3 + 3.85 Income - 0.0423Income<sup>2</sup>, the following income level results in the maximum test score
A)607.3.
B)91.02.
C)45.50.
D)cannot be determined without a plot of the data.
Q2) In the model Y<sub>i</sub> = ?<sub>0</sub> + ?<sub>1</sub>X<sub>1</sub><sub> </sub>+ ?<sub>2</sub>X<sub>2</sub> + ?<sub>3</sub>(X<sub>1</sub> × X<sub>2</sub>)+ u<sub>i</sub>, the expected effect \(\frac { \Delta Y } { \Delta X _ { 1 } }\) is
A)?<sub>1</sub><sub> </sub>+ ?<sub>3</sub>X<sub>2</sub>.
B)?<sub>1</sub>.
C)?<sub>1</sub><sub> </sub>+ ?<sub>3</sub>.
D)?<sub>1</sub><sub> </sub>+ ?<sub>3</sub>X<sub>1</sub>.
Q3) In the log-log model, the slope coefficient indicates A)the effect that a unit change in X has on Y. B)the elasticity of Y with respect to X.
C)?Y / ?X.
D) \(\frac { \Delta Y } { \Delta X }\) × \(\frac { Y } { X }\)
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Chapter 9: Assessing Studies Based on Multiple Regression
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Sample Questions
Q1) A statistical analysis is internally valid if
A)all t-statistics are greater than |1.96|
B)the regression R<sup>2</sup> > 0.05
C)the population is small, say less than 2,000, and can be observed
D)the statistical inferences about causal effects are valid for the population studied
Q2) In the case of errors-in-variables bias, the precise size and direction of the bias depend on
A)the sample size in general.
B)the correlation between the measured variable and the measurement error.
C)the size of the regression R<sup>2</sup>.
D)whether the good in question is price elastic.
Q3) Simultaneous causality bias
A)is also called sample selection bias.
B)happens in complicated systems of equations called block recursive systems.
C)results in biased estimators if there is heteroskedasticity in the error term.
D)arises in a regression of Y on X when, in addition to the causal link of interest from X to Y, there is a causal link from Y to X.
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Chapter 10: Regression With Panel Data
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Sample Questions
Q1) Consider a panel regression of unemployment rates for the G7 countries (United States, Canada, France, Germany, Italy, United Kingdom, Japan)on a set of explanatory variables for the time period 1980-2000 (annual data). If you included entity and time fixed effects, you would need to specify the following number of binary variables: A)21.
B)6.
C)28.
D)26.
Q2) In the panel regression analysis of beer taxes on traffic deaths, the estimation period is 1982-1988 for the 48 contiguous U.S. states. To test for the significance of time fixed effects, you should calculate the F-statistic and compare it to the critical value from your F<sub>q,</sub><sub> </sub> distribution, where q equals A)6.
B)7.
C)48.
D)53.
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12
Chapter 11: Regression With a Binary Dependent Variable
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Sample Questions
Q1) To measure the fit of the probit model, you should:
A)use the regression R<sup>2</sup>.
B)plot the predicted values and see how closely they match the actuals.
C)use the log of the likelihood function and compare it to the value of the likelihood function.
D)use the fraction correctly predicted or the pseudo R<sup>2</sup>.
Q2) The binary dependent variable model is an example of a
A)regression model, which has as a regressor, among others, a binary variable.
B)model that cannot be estimated by OLS.
C)limited dependent variable model.
D)model where the left-hand variable is measured in base 2.
Q3) (Requires Appendix material)Briefly describe the difference between the following models: censored and truncated regression model, count data, ordered responses, and discrete choice data. Try to be specific in terms of describing the data involved.
Q4) The logit model derives its name from
A)the logarithmic model.
B)the probit model.
C)the logistic function.
D)the tobit model.
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Chapter 12: Instrumental Variables Regression
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Sample Questions
Q1) 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.
Q2) The two conditions for instrument validity are corr(Z<sub>i</sub>, X<sub>i</sub>) 0 and corr(Z<sub>i</sub>, u<sub>i</sub>)= 0. The reason for the inconsistency of OLS is that corr(X<sub>i</sub>, u<sub>i</sub>) 0. But if X and Z are correlated, and X and u are also correlated, then how can Z and u not be correlated? Explain.
Q3) Endogenous variables
A)are correlated with the error term.
B)always appear on the LHS of regression functions.
C)cannot be regressors.
D)are uncorrelated with the error term.
Q4) Write an essay about where valid instruments come from. Part of your explorations must deal with checking the validity of instruments and what the consequences of weak instruments are.
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Chapter 13: Experiments and Quasi-Experiments
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Sample Questions
Q1) If the causal effect is different for different people, then the population regression equation for a binary treatment variable X<sub>i</sub>, can be written as A)Y<sub>i</sub> = <sub>0</sub> + <sub>1</sub>X<sub>i</sub> + u<sub>i</sub>.
B)Y<sub>i</sub> = <sub>0</sub> + <sub>1</sub><sub>i</sub>X<sub>i</sub> + u<sub>i</sub>.
C)Y<sub>i</sub> = <sub>0</sub><sub>i</sub> <sub>1</sub><sub>i</sub>X<sub>i</sub> + u<sub>i</sub>.
D)Y<sub>i</sub> = <sub>0</sub> + <sub>1</sub>G<sub>i</sub> + <sub>2</sub>D<sub>t</sub> + u<sub>i</sub>.
Q2) In a quasi-experiment
A)quasi differences are used, i.e., instead of ?Y you need to use ( \(\bar { Y } \text { after }\) - ? × \(\bar { Y } \text { before }\) ), where 0 < ? < 1.
B)randomness is introduced by variations in individual circumstances that make it appear as if the treatment is randomly assigned.
C)the causal effect has to be estimated through quasi maximum likelihood estimation.
D)the t-statistic is no longer normally distributed in large samples.
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Chapter 14: Introduction to Time Series Regression and Forecasting
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Sample Questions
Q1) You want to determine whether or not the unemployment rate for the United States has a stochastic trend using the Augmented Dickey Fuller Test (ADF). The BIC suggests using 3 lags, while the AIC suggests 4 lags.
(a)Which of the two will you use for your choice of the optimal lag length?
(b)After estimating the appropriate equation, the t-statistic on the lag level unemployment rate is (-2.186)(using a constant, but not a trend). What is your decision regarding the stochastic trend of the unemployment rate series in the United States?
(c)Having worked in the previous exercise with the unemployment rate level, you repeat the exercise using the difference in United States unemployment rates. Write down the appropriate equation to conduct the Augmented Dickey-Fuller test here. The t-statistic on relevant coefficient turns out to be (-4.791). What is your conclusion now?
Q2) One reason for computing the logarithms (ln), or changes in logarithms, of economic time series is that
A)numbers often get very large.
B)economic variables are hardly ever negative.
C)they often exhibit growth that is approximately exponential.
D)natural logarithms are easier to work with than base 10 logarithms.
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Page 16
Chapter 15: Estimation of Dynamic Causal Effects
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Sample Questions
Q1) GLS involves
A)writing the model in differences and estimating it by OLS, using HAC standard errors.
B)truncating the sample at both ends of the period, then estimating by OLS using HAC standard errors.
C)checking the AIC rather than the BIC in choosing the maximum lag-length of the regressors.
D)transforming the regression model so that the errors are homoskedastic and serially uncorrelated, and then estimating the transformed regression model by OLS.
Q2) Your textbook presents as an example of a distributed lag regression the effect of the weather on the price of orange juice. The authors mention U.S. income and Australian exports, oil prices and inflation, monetary policy and inflation, and the Phillips curve as other potential candidates for distributed lag regression. You are considering estimating the effect of minimum wages on teenage employment (employment population ratio)using a time series of U.S. data. Write a short essay on whether a distributed lag model would be a suitable tool to figure out dynamic causal effects in this case.
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Chapter 16: Additional Topics in Time Series Regression
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Sample Questions
Q1) The BIC for the VAR is
A)BIC(p)= ln[det ( \(\hat \Sigma\) <sub>u</sub>)] + k(kp+1) \(\frac { 2 } { T }\)
B)BIC(p)= ln[det ( \(\hat \Sigma\) <sub>u</sub>)] + k(p+1) \(\frac { \ln ( T ) } { T
C)BIC(p)= ln[det ( \(\hat \Sigma\) <sub>u</sub>)] + k(kp+1) \(\frac { \ln ( T ) } { T
D)BIC(p)= ln[SSR(p)] + k(p+1) \(\frac { \ln ( T ) } { T }\)
Q2) You can determine the lag lengths in a VAR
A)by using confidence intervals.
B)by using critical values from the standard normal table.
C)by using either F-tests or information criteria.
D)with the help from economic theory and institutional knowledge.
Q3) A VAR allows you to test joint hypothesis that involve restrictions across multiple equations by
A)computing a z-statistic.
B)computing the BIC but not the AIC.
C)using a stability test.
D)computing an F-statistic.
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Chapter 17: The Theory of Linear Regression With One Regressor
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Q1) You need to adjust \(S _ {\hat{ u }} ^ { 2 }\) by the degrees of freedom to ensure that \(S _ {\hat{ u }} ^ { 2 }\) is
A)an unbiased estimator of \(\sigma _ { u } ^ { 2 }\)
B)a consistent estimator of \(\sigma _ { u } ^ { 2 }\)
C)efficient in small samples.
D)F-distributed.
Q2) Slutsky's theorem combines the Law of Large Numbers
A)with continuous functions.
B)and the normal distribution.
C)and the Central Limit Theorem.
D)with conditions for the unbiasedness of an estimator.
Q3) If the variance of u is quadratic in X, then it can be expressed as A)var(u<sub>i</sub>|X<sub>i</sub>)= \(\theta _ { 0 } ^ { 2 }\)
B)var(u<sub>i</sub>|X<sub>i</sub>)= ?<sub>0</sub><sub> </sub>+ ?<sub>1</sub> <sub> </sub> \(x _ { i } ^ { 1 / 2 }\)
C)var(u<sub>i</sub>|X<sub>i</sub>)= ?<sub>0</sub><sub> </sub>+ ?<sub>1</sub> <sub> </sub> \(x _ { i } ^ { 2 }\)
D)var(u<sub>i</sub>|X<sub>i</sub>)= \(\sigma _ { \mathrm { u } } ^ { 2 }\)
Q4) (Requires Appendix material)State and prove the Cauchy-Schwarz Inequality.
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Chapter 18: The Theory of Multiple Regression
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Q1) The OLS estimator
A)has the multivariate normal asymptotic distribution in large samples.
B)is t-distributed.
C)has the multivariate normal distribution regardless of the sample size.
D)is F-distributed.
Q2) The assumption that X has full column rank implies that A)the number of observations equals the number of regressors.
B)binary variables are absent from the list of regressors.
C)there is no perfect multicollinearity.
D)none of the regressors appear in natural logarithm form.
Q3) One implication of the extended least squares assumptions in the multiple regression model is that
A)feasible GLS should be used for estimation.
B)E(U|X)= I<sub>n</sub>.
C) \(X ^ { \prime }\) X is singular.
D)the conditional distribution of U given X is N(0<sub>n</sub>, I<sub>n</sub>).
Q4) Using the model Y = X + U, and the extended least squares assumptions, derive the OLS estimator \(\hat {\beta}\) Discuss the conditions under which \(X ^ { \prime }\) X is invertible.
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