Introductory Econometrics
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Course Introduction
Introductory Econometrics provides students with a foundational understanding of the methods and tools used to analyze economic data. The course covers the theory and application of regression analysis, illustrating how econometric techniques can be used to test economic theories and evaluate policy impacts. Core topics include simple and multiple linear regression, hypothesis testing, model specification, and issues such as multicollinearity, heteroskedasticity, and autocorrelation. Students gain practical experience through hands-on exercises and real-world data analysis using statistical software, empowering them to interpret results critically and make informed decisions based on quantitative evidence.
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) 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
Q2) Analyzing the effect of minimum wage changes on teenage employment across the 48 contiguous U.S. states from 1980 to 2004 is an example of using A)time series data.
B)panel data.
C)having a treatment group vs. a control group, since only teenagers receive minimum wages.
D)cross-sectional data.
Answer: B
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Chapter 2: Review of Probability
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Q1) When there are ? degrees of freedom, the t<sub>?</sub> distribution
A)can no longer be calculated.
B)equals the standard normal distribution.
C)has a bell shape similar to that of the normal distribution, but with "fatter" tails.
D)equals the \(X _ { \infty } ^ { 2 }\) distribution.
Answer: B
Q2) Two variables are uncorrelated in all of the cases below, with the exception of A)being independent.
B)having a zero covariance.
C) \(|{}^{\sigma } X Y | \leq \sqrt { \sigma _ { x } ^ { 2 } \sigma _ { y } ^ { 2 } }\)
D)E(Y \(\mid X\) )= 0.
Answer: C
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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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.
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Chapter 4: Linear Regression With One Regressor
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Sample Questions
Q1) You have analyzed the relationship between the weight and height of individuals. Although you are quite confident about the accuracy of your measurements, you feel that some of the observations are extreme, say, two standard deviations above and below the mean. Your therefore decide to disregard these individuals. What consequence will this have on the standard deviation of the OLS estimator of the slope?
Q2) (Requires Appendix)The sample regression line estimated by OLS
A)will always have a slope smaller than the intercept.
B)is exactly the same as the population regression line.
C)cannot have a slope of zero.
D)will always run through the point ( \(\bar { X }\) , \(\bar { Y }\) ).
Q3) E(u<sub>i</sub> <sub> </sub> | X<sub>i</sub>)= 0 says that
A)dividing the error by the explanatory variable results in a zero (on average).
B)the sample regression function residuals are unrelated to the explanatory variable.
C)the sample mean of the Xs is much larger than the sample mean of the errors.
D)the conditional distribution of the error given the explanatory variable has a zero mean.
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Chapter 5: Regression With a Single Regressor: Hypothesis
Tests and Confidence Intervals
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Q1) You extract approximately 5,000 observations from the Current Population Survey (CPS)and estimate the following regression function: \(\widehat { \text { ahe } }\) = 3.32 - 0.45 \(\times\) Age, R<sup>2</sup>= 0.02, SER = 8.66 (1.00)(0.04)
Where ahe is average hourly earnings, and Age is the individual's age. Given the specification, your 95% confidence interval for the effect of changing age by 5 years is approximately
A)[$1.96, $2.54]
B)[$2.32, $4.32]
C)[$1.35, $5.30]
D)cannot be determined given the information provided
Q2) In order to formulate whether or not the alternative hypothesis is one-sided or two-sided, you need some guidance from economic theory. Choose at least three examples from economics or other fields where you have a clear idea what the null hypothesis and the alternative hypothesis for the slope coefficient should be. Write a brief justification for your answer.
Q3) In many of the cases discussed in your textbook, you test for the significance of the slope at the 5% level. What is the size of the test? What is the power of the test? Why is the probability of committing a Type II error so large here?
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Chapter 6: Linear Regression With Multiple Regressors
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Sample Questions
Q1) The intercept in the multiple regression model
A)should be excluded if one explanatory variable has negative values.
B)determines the height of the regression line.
C)should be excluded because the population regression function does not go through the origin.
D)is statistically significant if it is larger than 1.96.
Q2) Imperfect multicollinearity
A)is not relevant to the field of economics and business administration B)only occurs in the study of finance
C)means that the least squares estimator of the slope is biased D)means that two or more of the regressors are highly correlated
Q3) Give at least three examples from macroeconomics and three from microeconomics that involve specified equations in a multiple regression analysis framework. Indicate in each case what the expected signs of the coefficients would be and if theory gives you an indication about the likely size of the coefficients.
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) Using the 420 observations of the California School data set from your textbook, you estimate the following relationship: \(\widehat{\text { TestScore }}\) = 681.440.61LchPct
n=420, R<sup>2</sup>=0.75, SER=9.45 where TestScore is the test score and LchPct is the percent of students eligible for subsidized lunch (average = 44.7, max = 100, min = 0).
a. Interpret the regression result.
b. In your interpretation of the slope coefficient in (a)above, does it matter if you start your explanation with "for every x percent increase" rather than "for every x percentage point increase"?
c. The "overall" regression F-statistic is 1149.57. What are the degrees of freedom for this statistic?
d. Find the critical value of the F-statistic at the 1% significance level. Test the null hypothesis that the regression R<sup>2</sup>= 0.
e. The above equation was estimated using heteroskedasticity robust standard errors. What is the standard error for the slope coefficient?
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Chapter 8: Nonlinear Regression Functions
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Sample Questions
Q1) You have been told that the money demand function in the United States has been unstable since the late 1970. To investigate this problem, you collect data on the real money supply (m=M/P; where M is M<sub>1</sub> and P is the GDP deflator), (real)gross domestic product (GDP)and the nominal interest rate (R). Next you consider estimating the demand for money using the following alternative functional forms: (i)m = <sub>0</sub> + <sub>1 </sub>× GDP + <sub>2 </sub>x R+ u (ii)m = <sub>0</sub> × \(G D P ^ { \beta 1 }\) x \( { R } ^ { \beta 2 }\) × e<sup>u</sup> (iii)m = <sub>0</sub> × \(G D P ^ { \beta 1 }\) x \(1 + R ^ { \beta _ { 2 } }\) × e<sup>u</sup>
Give an interpretation for <sub>1</sub> and <sub>2</sub> in each case. How would you calculate the income elasticity in case (i)?
Q2) The following are properties of the logarithm function with the exception of
A)ln(1/ x)= -ln(x).
B)ln(a + x)= ln(a)+ ln(x).
C)ln(ax)= ln(a)+ ln(x).
D)ln(x<sup>a</sup>)a ln(x).
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Chapter 9: Assessing Studies Based on Multiple Regression
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Q1) Errors-in-variables bias
A)is present when the probability limit of the OLS estimator is given by \(\hat { \beta } _ { 1 }\stackrel { p } { \longrightarrow } \beta _ { 1 } + \frac { \sigma _ { x } ^ { 2 } } { \sigma _ { x } ^ { 2 } + \sigma _ { w } ^ { 2 } }\)
B)arises when an independent variable is measured imprecisely.
C)arises when the dependent variable is measured imprecisely.
D)always occurs in economics since economic data is never precisely measured.
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) Panel data estimation can sometimes be used
A)to avoid the problems associated with misspecified functional forms.
B)in case the sum of residuals is not zero.
C)in the case of omitted variable bias when data on the omitted variable is not available.
D)to counter sample selection bias.
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Chapter 10: Regression With Panel Data
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Q1) Give at least three examples from macroeconomics and five from microeconomics that involve specified equations in a panel data analysis framework. Indicate in each case what the role of the entity and time fixed effects in terms of omitted variables might be.
Q2) Your textbook modifies the four assumptions for the multiple regression model by adding a new assumption. This represents an extension of the cross-sectional data case, where errors are uncorrelated across entities. The new assumption requires the errors to be uncorrelated across time, conditional on the regressors as well (cov(u<sub>it</sub>, u<sub>is</sub> | X<sub>it</sub>, X<sub>is</sub>)= 0 for t s.).
(a)Discuss why there might be correlation over time in the errors when you use U.S. state panel data. Does this mean that you should not use OLS as an estimator?
(b)Now consider pairs of adjacent states such as Indiana and Michigan, Texas and Arkansas, New York and Connecticut, etc. Is it likely that the fifth assumption will hold here, even though the "contemporaneous" errors are correlated? If not, can you still use OLS for estimation?
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Chapter 11: Regression With a Binary Dependent Variable
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Q1) (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)= \(\frac { 1 } { 1 + e ^ { - \left( \beta _ { 0 } + \beta _ { 1 } X \right) } }\)
D)Y = AK<sup>? L?u</sup>.
Q2) (Requires Appendix material and Calculus)The logarithm of the likelihood function (L)for estimating the population mean and variance for an i.i.d. normal sample 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 <sup>2</sup>)- \(\frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \mu _ { Y } \right) ^ { 2 }\) Derive the maximum likelihood estimator for the mean and the variance. How do they differ, if at all, from the OLS estimator? Given that the OLS estimators are unbiased, what can you say about the maximum likelihood estimators here? Is the estimator for the variance consistent?
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Chapter 12: Instrumental Variables Regression
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Sample Questions
Q1) Write a short essay about the Overidentifying Restrictions Test. What is meant exactly by "overidentification?" State the null hypothesis. Describe how to calculate the J-statistic and what its distribution is. Use an example of two instruments and one endogenous variable to explain under what situation the test will be likely to reject the null hypothesis. What does this example tell you about the exactly identified case? If your variables pass the test, is this sufficient for these variables to be good instruments?
Q2) 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?
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.
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Chapter 13: Experiments and Quasi-Experiments
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Q1) Your textbooks gives several examples of quasi experiments that were conducted. The following is not an example of a quasi experiment:
A)labor market effects of immigration.
B)effects on civilian earnings of military service.
C)the effect of cardiac catheterization.
D)the effect of unemployment on the inflation rate.
Q2) Threats to internal validity of quasi-experiments include
A)failure of randomization.
B)failure to follow the treatment protocol.
C)attrition.
D)all of the above with some modifications from true randomized controlled experiments.
Q3) 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.
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) The forecast is
A)made for some date beyond the data set used to estimate the regression.
B)another word for the OLS predicted value.
C)equal to the residual plus the OLS predicted value.
D)close to 1.96 times the standard deviation of Y during the sample.
Q2) The formulae for the AIC and the BIC are different. The A)AIC is preferred because it is easier to calculate
B)BIC is preferred because it is a consistent estimator of the lag length
C)difference is irrelevant in practice since both information criteria lead to the same conclusion
D)AIC will typically underestimate p with non-zero probability
Q3) Stationarity means that the A)error terms are not correlated.
B)probability distribution of the time series variable does not change over time.
C)time series has a unit root.
D)forecasts remain within 1.96 standard deviation outside the sample period.
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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Q1) Money supply is linked to the monetary base by the money multiplier. Macroeconomic textbooks tell you that the central bank cannot control the money supply, but it can control the monetary base. As a result, you decide to specify a distributed lag equation of the growth in the money supply on the growth in the monetary base. One of your peers tells you that this is not a good idea for modeling the relationship between the two variables. What does she mean?
Q2) Given the relationship between the two variables, the following is most likely to be exogenous:
A)the inflation rate and the short term interest rate: short-term interest rate is exogenous
B)U.S. rate of inflation and increases in oil prices: oil prices are exgoneous
C)Australian exports and U.S. aggregate income: U.S. aggregate income is exogenous
D)change in inflation, lagged changes of inflation, and lags of unemployment: lags of unemployment are exogenous
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Chapter 16: Additional Topics in Time Series Regression
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Q1) Volatility clustering
A)is evident in most cross-sections.
B)implies that a series is serially correlated.
C)can mostly be found in studies of the labor market.
D)is evident in many financial time series.
Q2) Consider the GARCH(1,1)model \(\sigma _ { t } ^ { 2 }\) = ?<sub>0</sub> + ?<sub>1</sub>
<sub> </sub> \(u _ { t - 1 } ^ { 2 }\) + ?<sub>1</sub>
<sub> </sub> \(\sigma _ { t - 1 } ^ { 2 }\) Show that this model can be rewritten as \(\sigma _ { t } ^ { 2 }\) = \(\frac { \alpha _ { 0 } } { 1 - \phi _ { 1 } }\) + ?<sub>1</sub>( \(u _ { t - 1 } ^ { 2 }\) + ?<sub>1</sub>
<sub> </sub> \(u _ { t - 2 } ^ { 2 }\) + \(\phi _ { 1 } ^ { 2 }\) \(u _ { t - 3 } ^ { 2 }\) + \(\phi _ { 1 } ^ { 3 }\) \(u _ { t - 4 } ^ { 2 }\) + ...). (Hint: use the GARCH(1,1)model but specify it for \(\phi _ { t - 1 } ^ { 2 }\) ; substitute this expression into the original specification, and so on.)Explain intuitively the meaning of the resulting formulation.
Q3) "Heteroskedasticity typically occurs in cross-sections, while serial correlation is typically observed in time-series data." Discuss and critically evaluate this statement.
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Chapter 17: The Theory of Linear Regression With One Regressor
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Q1) "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.
Q2) 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.
Q3) The following is not part of the extended least squares assumptions for regression with a single regressor:
A)var(u<sub>i</sub> | X<sub>i</sub>)= \(\sigma _ { u } ^ { 2 }\)
B)E(u<sub>i</sub> | X<sub>i</sub>)= 0.
C)the conditional distribution of u<sub>i</sub> given X<sub>i</sub> is normal.
D)var(u<sub>i</sub> | X<sub>i</sub>)= \(\sigma _ { u , i } ^ { 2 }\)
Q4) "One should never bother with WLS. Using OLS with robust standard errors gives correct inference, at least asymptotically." True, false, or a bit of both? Explain carefully what the quote means and evaluate it critically.
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Chapter 18: The Theory of Multiple Regression
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Q1) 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.
Q2) 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
Q3) The presence of correlated error terms creates problems for inference based on OLS. These can be overcome by
A)using HAC standard errors.
B)using heteroskedasticity-robust standard errors.
C)reordering the observations until the correlation disappears.
D)using homoskedasticity-only standard errors.
Q4) Write an essay on the difference between the OLS estimator and the GLS estimator.
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