Econometrics for Social Sciences
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Course Introduction
Econometrics for Social Sciences introduces students to the statistical methods and tools used to analyze data in economics, political science, sociology, and other social sciences. The course covers core concepts such as regression analysis, hypothesis testing, and causal inference, with a focus on applying econometric techniques to real-world social science questions. Students learn to critically evaluate empirical research, interpret quantitative findings, and use statistical software to manage and analyze datasets. Through hands-on projects and case studies, the course equips students with the skills necessary to conduct rigorous empirical research and informs evidence-based policy discussions in social science 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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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) Analyzing the behavior of unemployment rates across U.S. states in March of 2006 is an example of using A)time series data.
B)panel data.
C)cross-sectional data.
D)experimental data.
Answer: C
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Chapter 2: Review of Probability
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Q1) Let Y be a random variable. Then var(Y)equals
A) \(\sqrt { \left. E \left[ Y - \mu _ { y } \right) ^ { 2 } \right] }\)
B) \(E \left[ \left| \left( Y - \mu _ { y } \right) \right| \right]\)
C) \(E \left[ \left( Y - \mu _ { y } \right) ^ { 2 } \right]\)
D) \(E [ ( Y - \mu ) ]\)
Answer: C
Q2) \(\sum _ { i = 1 } ^ { n } \left( a x _ {i } + b \right)\)
A)n × a × \(\overline { \mathrm { x } }\) <sub> </sub>+ n × b
B)n(a + b)
C) \(\bar { x } + n \times b\)
D) \(n \times a \times \bar { x }\)
Answer: A
Q3) Explain why the two probabilities are identical for the standard normal distribution: Pr(-1.96 X 1.96)and Pr(-1.96 < X < 1.96).
Answer: For a continuous distribution, the probability of a point is zero.
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Chapter 3: Review of Statistics
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Q1) Degrees of freedom
A)in the context of the sample variance formula means that estimating the mean uses up some of the information in the data.
B)is something that certain undergraduate majors at your university/college other than economics seem to have an ? amount of.
C)are (n-2)when replacing the population mean by the sample mean.
D)ensure that \(S _ { y } ^ { 2 } = \sigma _ { y } ^ { 2 }\)
Answer: A
Q2) A scatterplot
A)shows how Y and X are related when their relationship is scattered all over the place.
B)relates the covariance of X and Y to the correlation coefficient.
C)is a plot of n observations on X<sub>i</sub> and Y<sub>i</sub>, where each observation is represented by the point (X<sub>i</sub>, Y<sub>i</sub>).
D)shows n observations of Y over time.
Answer: C
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Chapter 4: Linear Regression With One Regressor
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Q1) In the simple linear regression model, the regression slope
A)indicates by how many percent Y increases, given a one percent increase in X.
B)when multiplied with the explanatory variable will give you the predicted Y.
C)indicates by how many units Y increases, given a one unit increase in X.
D)represents the elasticity of Y on X.
Q2) The OLS slope estimator is not defined if there is no variation in the data for the explanatory variable. You are interested in estimating a regression relating earnings to years of schooling. Imagine that you had collected data on earnings for different individuals, but that all these individuals had completed a college education (16 years of education). Sketch what the data would look like and explain intuitively why the OLS coefficient does not exist in this situation.
Q3) (Requires Appendix)The sample average of the OLS residuals is
A)some positive number since OLS uses squares.
B)zero.
C)unobservable since the population regression function is unknown.
D)dependent on whether the explanatory variable is mostly positive or negative.
Q4) Assume that there is a change in the units of measurement on both Y and X. The new variables are Y<sup>*</sup>= aY and X<sup>*</sup> = bX. What effect will this change have on the regression slope?
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Chapter 5: Regression With a Single Regressor: Hypothesis
Tests and Confidence Intervals
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Q1) Consider the following two models involving binary variables as explanatory variables: \(\widehat { \text { Wage } }\) = \(\widehat { \beta 0 }\) + \(\widehat { \beta 1 }\) DFemme and \(\widehat { \text { Wage } }\) = \(\widehat { \phi _ { 1 } }\) <sub> </sub>DFemme + \(\widehat { \phi _ { 2 } }\) <sub> </sub>Male where Wage is the hourly wage rate, DFemme is a binary variable that is equal to 1 if the person is a female, and 0 if the person is a male. Male = 1 - DFemme. Even though you have not learned about regression functions with two explanatory variables (or regressions without an intercept), assume that you had estimated both models, i.e., you obtained the estimates for the regression coefficients. What is the predicted wage for a male in the two models? What is the predicted wage for a female in the two models? What is the relationship between the ? s and the ?s? Why would you prefer one model over the other?
Q2) Your textbook states that under certain restrictive conditions, the t- statistic has a Student t-distribution with n-2 degrees of freedom. The loss of two degrees of freedom is the result of OLS forcing two restrictions onto the data. What are these two conditions, and when did you impose them onto the data set in your derivation of the OLS estimator?
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Chapter 6: Linear Regression With Multiple Regressors
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Q1) Under imperfect multicollinearity
A)the OLS estimator cannot be computed.
B)two or more of the regressors are highly correlated.
C)the OLS estimator is biased even in samples of n > 100.
D)the error terms are highly, but not perfectly, correlated.
Q2) One of the least squares assumptions in the multiple regression model is that you have random variables which are "i.i.d." This stands for
A)initially indeterminate differences.
B)irregularly integrated dichotomies.
C)identically initiated deltas (as in changes).
D)independently and identically distributed.
Q3) 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).
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Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple
Regression
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Q1) Consider a regression with two variables, in which X<sub>1i</sub> is the variable of interest and X<sub>2i</sub> is the control variable. Conditional mean independence requires
A)E(u<sub>i</sub>|X<sub>1i</sub>, X<sub>2i</sub>)= E(u<sub>i</sub>|X<sub>2i</sub>)
B)E(u<sub>i</sub>|X<sub>1i</sub>, X<sub>2i</sub>)= E(u<sub>i</sub>|X<sub>1i</sub>)
C)E(u<sub>i</sub>|X<sub>1i</sub>)= E(u<sub>i</sub>|X<sub>2i</sub>)
D)E(u<sub>i</sub>)= E(u<sub>i</sub>|X<sub>2i</sub>)
Q2) At a mathematical level, if the two conditions for omitted variable bias are satisfied, then
A)E(u<sub>i</sub> | X<sub>1</sub><sub>i</sub>, X<sub>2</sub><sub>i</sub>,..., X<sub>ki</sub>)? 0.
B)there is perfect multicollinearity.
C)large outliers are likely: X<sub>1</sub><sub>i</sub>, X<sub>2</sub><sub>i</sub>,..., X<sub>ki</sub> and Y<sub>i</sub> and have infinite fourth moments.
D)(X<sub>1</sub><sub>i</sub>, X<sub>2</sub><sub>i</sub>,..., X<sub>ki</sub>,Y<sub>i</sub>), i = 1,..., n are not i.i.d. draws from their joint distribution.
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Chapter 8: Nonlinear Regression Functions
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Sample Questions
Q1) Assume that you had estimated the following quadratic regression model \(\widehat{\text { TestScore }}\) = 607.3 + 3.85 Income - 0.0423 Income<sup>2</sup>.
If income increased from 10 to 11 ($10,000 to $11,000), then the predicted effect on testscores would be
A)3.85
B)3.85-0.0423
C)Cannot be calculated because the function is non-linear
D)2.96
Q2) Give at least three examples from economics where you expect some nonlinearity in the relationship between variables. Interpret the slope in each case.
Q3) A nonlinear function
A)makes little sense, because variables in the real world are related linearly.
B)can be adequately described by a straight line between the dependent variable and one of the explanatory variables.
C)is a concept that only applies to the case of a single or two explanatory variables since you cannot draw a line in four dimensions.
D)is a function with a slope that is not constant.
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Chapter 9: Assessing Studies Based on Multiple Regression
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Q1) A definition of internal validity is
A)the estimator of the causal effect being unbiased and consistent
B)the estimator of the causal effect being efficient
C)inferences and conclusions being generalized from the population to other populations
D)OLS estimation being available in your statistical package
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) 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.
Q4) The question of reliability/unreliability of a multiple regression depends on
A)internal but not external validity
B)the quality of your statistical software package
C)internal and external validity
D)external but not internal validity
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Chapter 10: Regression With Panel Data
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Q1) If X<sub>it</sub> is correlated with X<sub>is</sub> for different values of s and t, then
A)X<sub>it</sub> is said to be autocorrelated
B)the OLS estimator cannot be computed
C)statistical inference cannot proceed in a standard way even if clustered standard errors are used
D)this is not of practical importance since these correlations are typically weak in applications
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 entity fixed effects, you should calculate the F-statistic and compare it to the critical value from your F<sub>q</sub><sub>,</sub><sub> </sub> distribution, where q equals A)48.
B)54.
C)7.
D)47.
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Chapter 11: Regression With a Binary Dependent Variable
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Q1) When estimating probit and logit models,
A)the t-statistic should still be used for testing a single restriction.
B)you cannot have binary variables as explanatory variables as well.
C)F-statistics should not be used, since the models are nonlinear.
D)it is no longer true that the \(\bar { R }\) <sup>2</sup> < R<sup>2</sup>.
Q2) The logit model derives its name from
A)the logarithmic model.
B)the probit model.
C)the logistic function.
D)the tobit model.
Q3) 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.
Q4) Probit coefficients are typically estimated using A)the OLS method
B)the method of maximum likelihood
C)non-linear least squares (NLLS)
D)by transforming the estimates from the linear probability model
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Chapter 12: Instrumental Variables Regression
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Q1) The IV estimator can be used to potentially eliminate bias resulting from A)multicollinearity.
B)serial correlation.
C)errors in variables.
D)heteroskedasticity.
Q2) 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.
Q3) 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.
Q4) 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?
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Chapter 13: Experiments and Quasi-Experiments
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Q1) Present alternative estimators for causal effects using experimental data when data is available for a single period or for two periods. Discuss their advantages and disadvantages.
Q2) Failure to follow the treatment protocol means that
A)the OLS estimator cannot be computed.
B)instrumental variables estimation of the treatment effect should be used where the initial random assignment is the instrument for the treatment actually received.
C)you should use the TSLS estimator and regress the outcome variable Y on the initial random assignment in the first stage to get predicted values of the outcome variable.
D)the Hawthorne effect plays a crucial role.
Q3) The following does not represent a threat to internal validity of randomized controlled experiments:
A)attrition.
B)failure to follow the treatment protocol.
C)experimental effects.
D)a large sample size.
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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Q1) You have collected data for real GDP (Y)and have estimated the following function: ln \(\hat { Y }\) <sub>t</sub> = 7.866 + 0.00679×Zeit (0.007)(0.00008)
t = 1961:I - 2007:IV, R<sup>2</sup> = 0.98, SER = 0.036 where Zeit is a deterministic time trend, which takes on the value of 1 during the first quarter of 1961, and is increased by one for each following quarter.
a. Interpret the slope coefficient. Does it make sense?
b. Interpret the regression R<sup>2</sup>. Are you impressed by its value?
c. Do you think that given the regression R<sup>2</sup>, you should use the equation to forecast real GDP beyond the sample period?
Q2) The time interval between observations can be all of the following with the exception of data collected
A)daily.
B)by decade.
C)bi-weekly.
D)across firms.
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Chapter 15: Estimation of Dynamic Causal Effects
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Q1) In your intermediate macroeconomics course, government expenditures and the money supply were treated as exogenous, in the sense that the variables could be changed to conduct economic policy to influence target variables, but that these variables would not react to changes in the economy as a result of some fixed rule. The St. Louis Model, proposed by two researchers at the Federal Reserve in St. Louis, used this idea to test whether monetary policy or fiscal policy was more effective in influencing output behavior. Although there were various versions of this model, the basic specification was of the following type: ln(Y<sub>t</sub>)= <sub>0</sub> + <sub>1</sub> ln m<sub>t</sub> + ... + <sub>p</sub> ln m<sub>t</sub><sub>-</sub><sub>p</sub><sub>-</sub><sub>1</sub> + <sub>p</sub><sub>+</sub><sub>1</sub> ln G<sub>t</sub> + ... + <sub>p</sub><sub>+</sub><sub>q</sub> ln G<sub>t</sub><sub>-</sub><sub>q</sub><sub>-</sub><sub>1</sub> + u<sub>t</sub>
Assuming that money supply and government expenditures are exogenous, how would you estimate dynamic causal effects? Why do you think this type of model is no longer used by most to calculate fiscal and monetary multipliers?
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Chapter 16: Additional Topics in Time Series Regression
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Q1) You have collected time series for various macroeconomic variables to test if there is a single cointegrating relationship among multiple variables. Formulate the null hypothesis and compare the EG-ADF statistic to its critical value.
(a)Canadian unemployment rate, Canadian Inflation Rate, United States unemployment rate, United States inflation rate; t = (-3.374).
(b)Approval of United States presidents (Gallup poll), cyclical unemployment rate, inflation rate, Michigan Index of Consumer Sentiment; t = (-3.837).
(c)The log of real GDP, log of real government expenditures, log of real money supply (M2); t = (-2.23).
(d)Briefly explain how you could potentially improve on VAR(p)forecasts by using a cointegrating vector.
Q2) The EG-ADF test
A)is the similar to the DF-GLS test
B)is a test for cointegration
C)has as a limitation that it can only test if two variables, but not more than two, are cointegrated
D)uses the ADF in the second step of its procedure
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Chapter 17: The Theory of Linear Regression With One Regressor
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Q1) One of the earlier textbooks in econometrics, first published in 1971, compared "estimation of a parameter to shooting at a target with a rifle. The bull's-eye can be taken to represent the true value of the parameter, the rifle the estimator, and each shot a particular estimate." Use this analogy to discuss small and large sample properties of estimators. How do you think the author approached the n condition? (Dependent on your view of the world, feel free to substitute guns with bow and arrow, or missile.)
Q2) Consider the model Y<sub>i</sub> = <sub>1</sub>X<sub>i</sub> + u<sub>i</sub>, where u<sub>i</sub> = c \(x _ { i } ^ { 2 }\) e<sub>i</sub> and all of the X's and e's are i.i.d. and distributed N(0,1).
(a)Which of the Extended Least Squares Assumptions are satisfied here? Prove your assertions.
(b)Would an OLS estimator of <sub>1</sub> be efficient here?
(c)How would you estimate <sub>1</sub> by WLS?
Q3) (Requires Appendix material)State and prove the Cauchy-Schwarz Inequality.
Q4) "I am an applied econometrician and therefore should not have to deal with econometric theory. There will be others who I leave that to. I am more interested in interpreting the estimation results." Evaluate.
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Chapter 18: The Theory of Multiple Regression
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Q1) Write the following three linear equations in matrix format Ax = b, where x is a 3×1 vector containing q, p, and y, A is a 3×3 matrix of coefficients, and b is a 3×1 vector of constants.
q = 5 +3 p - 2 y
q = 10 - p + 10 y p = 6 y
Q2) Let there be q joint hypothesis to be tested. Then the dimension of r in the expression R = r is
A)q × 1.
B)q × (k+1).
C)(k+1)× 1.
D)q.
Q3) Your textbook derives the OLS estimator as \(\hat { \beta }\) = \(\left( X ^ { \prime } \right.\) X)<sup>-</sup><sup>1</sup> <sup> </sup> \(X ^ { \prime }\) <sup>Y</sup>.
Show that the estimator does not exist if there are fewer observations than the number of explanatory variables, including the constant. What is the rank of \(X ^ { \prime }\) X in this case?
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