Statistical Methods for Economics
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Course Introduction
This course provides an introduction to statistical methods and their applications in economics, emphasizing both the theoretical foundations and practical techniques for analyzing economic data. Topics include descriptive statistics, probability distributions, sampling methods, estimation, hypothesis testing, and regression analysis. Students will learn how to interpret and critically evaluate statistical results, as well as apply quantitative tools to real-world economic problems. Throughout the course, emphasis is placed on the use of statistical software for empirical work and on developing the ability to draw meaningful economic inferences from data.
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) Give at least three examples from economics where each of the following type of data can be used: cross-sectional data, time series data, and panel data.
Answer: Answers will vary by student. At this level of economics, students most likely have heard of the following use of cross-sectional data: earnings functions, growth equations, the effect of class size reduction on student performance (in this chapter), demand functions (in this chapter: cigarette consumption); time series: the Phillips curve (in this chapter), consumption functions, Okun's law; panel data: various U.S. state panel studies on road fatalities (in this book), unemployment rate and unemployment benefits variations, growth regressions (across states and countries), and crime and abortion (Freakonomics).
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
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Chapter 2: Review of Probability
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Sample Questions
Q1) Two random variables X and Y are independently distributed if all of the following conditions hold, with the exception of
A)Pr(Y = y \(\mid X\) = x)= Pr(Y = y).
B)knowing the value of one of the variables provides no information about the other.
C)if the conditional distribution of Y given X equals the marginal distribution of Y.
D)E(Y)= E[E(Y \(\mid X\) )].
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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Page 4
Chapter 3: Review of Statistics
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Sample Questions
Q1) When the sample size n is large, the 90% confidence interval for \({ }^{\mu} Y\) is
A) \(\bar { Y }\) ± 1.96SE( \(\bar { Y }\) ).
B) \(\bar { Y }\) ± 1.64SE( \(\bar { Y }\) ).
C) \(\bar { Y }\) ± 1.64 \({ } ^ { \sigma } Y\)
D) \(\bar { Y }\) ± 1.96.
Answer: B
Q2) When you are testing a hypothesis against a two-sided alternative, then the alternative is written as
A)E(Y)> µ<sub>Y,0</sub>.
B)E(Y)= µ<sub>Y,0</sub>.
C) \(\bar { Y }\) ? µ<sub>Y,0</sub>.
D)E(Y)? µ<sub>Y,0</sub>.
Answer: D
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Chapter 4: Linear Regression With One Regressor
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Sample Questions
Q1) 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.
Q2) The following are all least squares assumptions with the exception of:
A)The conditional distribution of u<sub>i</sub> given X<sub>i</sub> has a mean of zero.
B)The explanatory variable in regression model is normally distributed.
C)(X<sub>i</sub>, Y<sub>i</sub>), i = 1,..., n are independently and identically distributed.
D)Large outliers are unlikely.
Q3) Prove that the regression R<sup>2</sup> is identical to the square of the correlation coefficient between two variables Y and X. Regression functions are written in a form that suggests causation running from X to Y. Given your proof, does a high regression R<sup>2</sup> present supportive evidence of a causal relationship? Can you think of some regression examples where the direction of causality is not clear? Is without a doubt?
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Chapter 5: Regression With a Single Regressor: Hypothesis
Tests and Confidence Intervals
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Q1) 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?
Q2) If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal distribution, you can
A)reject the null hypothesis.
B)safely assume that your regression results are significant.
C)reject the assumption that the error terms are homoskedastic.
D)conclude that most of the actual values are very close to the regression line.
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.
Q4) Finding a small value of the p-value (e.g. less than 5%)
A)indicates evidence in favor of the null hypothesis.
B)implies that the t-statistic is less than 1.96.
C)indicates evidence in against the null hypothesis.
D)will only happen roughly one in twenty samples.
Page 7
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Chapter 6: Linear Regression With Multiple Regressors
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Q1) The probability limit of the OLS estimator in the case of omitted variables is given in your text by the following formula: \(\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \hat { \beta } _ { 1 } + \rho _ { \mathrm { X } } \frac { \sigma _ { u } } { \sigma _ { \mathrm { X } } }\) Give an intuitive explanation for two conditions under which the bias will be small.
Q2) You have collected data on individuals and their attributes. Consequently you have generated several binary variables, which take on a value of "1" if the individual has that characteristic and are "0" otherwise. One example is the binary variable DMarr which is "1" for married individuals and "0" for non-married variables. If you run the following regression: ahe<sub>i</sub>= <sub>0</sub> + <sub>1</sub>×educ<sub>i</sub> + <sub>2</sub>×DMarr<sub>i </sub>+ u<sub>i</sub>
a. What is the interpretation for <sub>2</sub>?
b. You are interested in directly observing the effect that being non-married ("single")has on earnings, controlling for years of education. Instead of recording all observations such that they are "1" for a not married individual and "0" for a married person, how can you generate such a variable (DSingle)through a simple command in your regression program?
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Page 8
Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple
Regression
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Sample Questions
Q1) If you wanted to test, using a 5% significance level, whether or not a specific slope coefficient is equal to one, then you should
A)subtract 1 from the estimated coefficient, divide the difference by the standard error, and check if the resulting ratio is larger than 1.96.
B)add and subtract 1.96 from the slope and check if that interval includes 1.
C)see if the slope coefficient is between 0.95 and 1.05.
D)check if the adjusted R<sup>2</sup> is close to 1.
Q2) In the multiple regression model, the t-statistic for testing that the slope is significantly different from zero is calculated
A)by dividing the estimate by its standard error.
B)from the square root of the F-statistic.
C)by multiplying the p-value by 1.96.
D)using the adjusted R<sup>2</sup> and the confidence interval.
Q3) For a single restriction (q = 1), the F-statistic
A)is the square root of the t-statistic.
B)has a critical value of 1.96.
C)will be negative.
D)is the square of the t-statistic.
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Chapter 8: Nonlinear Regression Functions
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Sample Questions
Q1) Choose at least three different nonlinear functional forms of a single independent variable and sketch the relationship between the dependent and independent variable.
Q2) Your task is to estimate the ice cream sales for a certain chain in New England. The company makes available to you quarterly ice cream sales (Y)and informs you that the price per gallon has approximately remained constant over the sample period. You gather information on average daily temperatures (X)during these quarters and regress Y on X, adding seasonal binary variables for spring, summer, and fall. These variables are constructed as follows: DSpring takes on a value of 1 during the spring and is zero otherwise, DSummer takes on a value of 1 during the summer, etc. Specify three regression functions where the following conditions hold: the relationship between Y and X is (i)forced to be the same for each quarter; (ii)allowed to have different intercepts each season; (iii)allowed to have varying slopes and intercepts each season. Sketch the difference between (i)and (ii). How would you test which model fits the data the best?
Q3) Sketch for the log-log model what the relationship between Y and X looks like for various parameter values of the slope, i.e., <sub>1</sub> > 1; 0 < <sub>1</sub> < 1; <sub>1 </sub>= (-1).
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Chapter 9: Assessing Studies Based on Multiple Regression
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Sample Questions
Q1) In the simple, one-explanatory variable, errors-in-variables model, the OLS estimator for the slope is inconsistent. The textbook derived the following result \(\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }\) Show that the OLS estimator for the intercept behaves as follows in large samples: \(\hat { \beta } _ { 1 } \stackrel { p } { \rightarrow } \beta _ { 0 } + \mu \widetilde { \mathrm { X } } \frac { \sigma _ { w } ^ { 2 } } { \sigma _ { \mathrm { X } } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }\) where \(\bar { X }\) \(\stackrel { p } { \rightarrow }\) \({ } ^ { { } ^ { \mu } } \widetilde { X }\)
Q2) To analyze the situation of simultaneous causality bias, consider the following system of equations:
Y<sub>i</sub> = <sub>0</sub> + <sub>1</sub>X<sub>i</sub> + u<sub>i</sub> X<sub>i</sub> = \(\gamma _ { 0 }\) + \(\gamma _ { 1 }\) Y<sub>i</sub> + v<sub>i</sub>
Demonstrate the negative correlation between X<sub>i</sub> and \(\gamma _ { 1 }\) for \(\gamma _ { 1 }\) < 0 , either through mathematics or by presenting an argument which starts as follows: "Imagine that u<sub>i</sub> is negative."
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Chapter 10: Regression With Panel Data
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Sample Questions
Q1) (Requires Appendix material)When the fifth assumption in the Fixed Effects regression (cov (u<sub>it</sub>, u<sub>is</sub> | X<sub>it</sub>, X<sub>is</sub>)= 0 for t ? s)is violated, then
A)using heteroskedastic-robust standard errors is not sufficient for correct statistical inference when using OLS.
B)the OLS estimator does not exist.
C)you can use the simple homoskedasticity-only standard errors calculated in your regression package.
D)you cannot use fixed time effects in your estimation.
Q2) Consider the regression example from your textbook, which estimates the effect of beer taxes on fatality rates across the 48 contiguous U.S. states. If beer taxes were set nationally by the federal government rather than by the states, then
A)it would not make sense to use state fixed effect.
B)you can test state fixed effects using homoskedastic-only standard errors.
C)the OLS estimator will be biased.
D)you should not use time fixed effects since beer taxes are the same at a point in time across states.
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Chapter 11: Regression With a Binary Dependent Variable
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Sample Questions
Q1) Your task is to model students' choice for taking an additional economics course after the first principles course. Describe how to formulate a model based on data for a large sample of students. Outline several estimation methods and their relative advantage over other methods in tackling this problem. How would you go about interpreting the resulting output? What summary statistics should be included?
Q2) 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>.
Q3) In the linear probability model, the interpretation of the slope coefficient is
A)the change in odds associated with a unit change in X, holding other regressors constant.
B)not all that meaningful since the dependent variable is either 0 or 1.
C)the change in probability that Y=1 associated with a unit change in X, holding others regressors constant.
D)the response in the dependent variable to a percentage change in the regressor.
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Chapter 12: Instrumental Variables Regression
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Q1) Two Stage Least Squares is calculated as follows; in the first stage:
A)Y is regressed on the exogenous variables only. The predicted value of Y is then regressed on the instrumental variables.
B)the unknown coefficients in the reduced form equation are estimated by OLS, and the predicted values are calculated. In the second stage, Y is regressed on these predicted values and the other exogenous variables.
C)the exogenous variables are regressed on the instruments. The predicted value of the exogenous variables is then used in the second stage, together with the instruments, to predict the dependent variable.
D)the unknown coefficients in the reduced form equation are estimated by weighted least squares, and the predicted values are calculated. In the second stage, Y is regressed on these predicted values and the other exogenous variables.
Q2) 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.
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Chapter 13: Experiments and Quasi-Experiments
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Q1) A causal effect for a single individual
A)can be deduced from the average treatment effect.
B)cannot be measured.
C)depends on observable variables only.
D)is observable since it is used as part of calculating the mean of individual causal effects.
Q2) In the context of a controlled experiment, consider the simple linear regression formulation Y<sub>i</sub> = ?<sub>0</sub> + ?<sub>1</sub>X<sub>i</sub> + u<sub>i</sub>. Let the Y<sub>i</sub> be the outcome, X<sub>i</sub> the treatment level when the treatment is binary, and u<sub>i</sub> contain all the additional determinants of the outcome. Then calling \(\hat \beta _ { 1 }\) a differences estimator
A)makes sense since it is the difference between the sample average outcome of the treatment group and the sample average outcome of the control group.
B)and \(\hat \beta _ { 0 }\) the level estimator is standard terminology in randomized controlled experiments.
C)does not make sense, since neither Y nor X are in differences.
D)is not quite accurate since it is actually the derivative of Y on X.
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Chapter 14: Introduction to Time Series Regression and Forecasting
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Q1) The ADL(p,q)model is represented by the following equation A)Y<sub>t</sub> = <sub>0</sub> + <sub>p</sub>Y<sub>t</sub><sub>-</sub><sub>p</sub> + <sub>q</sub>X<sub>t</sub><sub>-</sub><sub>q</sub> + u<sub>t</sub>.
B)Y<sub>t</sub> = <sub>0</sub> + <sub>1</sub>Y<sub>t</sub><sub>-</sub><sub>1</sub> + <sub>2</sub>Y<sub>t</sub><sub>-</sub><sub>2</sub> + ... + <sub>p</sub>Y<sub>t</sub><sub>-</sub><sub>p</sub> + <sub>q</sub>u<sub>t</sub><sub>-</sub><sub>q</sub>. C)Y<sub>t</sub> = <sub>0</sub> + <sub>1</sub>Y<sub>t</sub><sub>-</sub><sub>1</sub> + <sub>2</sub>Y<sub>t</sub><sub>-</sub><sub>2</sub> + ... + <sub>p</sub>Y<sub>t</sub><sub>-</sub><sub>p</sub> + <sub>0</sub> + <sub>1</sub>X<sub>t</sub><sub>-</sub><sub>1</sub> + u<sub>t</sub><sub>-</sub><sub>q</sub>.
D)Y<sub>t</sub> = <sub>0</sub> + <sub>1</sub>Y<sub>t</sub><sub>-</sub><sub>1</sub> + <sub>2</sub>Y<sub>t</sub><sub>-</sub><sub>2</sub> + ... + <sub>p</sub>Y<sub>t</sub><sub>-</sub><sub>p</sub> + <sub>1</sub>X<sub>t</sub><sub>-</sub><sub>1</sub> + <sub>2</sub>X<sub>t</sub><sub>-</sub><sub>2</sub> + ... + <sub>q</sub>X<sub>t</sub><sub>-</sub><sub>q</sub> + u<sub>t</sub>.
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Chapter 15: Estimation of Dynamic Causal Effects
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Q1) Ascertaining whether or not a regressor is strictly exogenous or exogenous ultimately requires all of the following with the exception of A)economic theory.
B)institutional knowledge.
C)expert judgment.
D)use of HAC standard errors.
Q2) Consider the distributed lag model Y<sub>t</sub> = <sub>0</sub> + <sub>1</sub>X<sub>t</sub> + <sub>2</sub>X<sub>t</sub><sub>-</sub><sub>1</sub> + <sub>3</sub>X<sub>t</sub><sub>-</sub><sub>2</sub> + + <sub>r</sub><sub>+</sub><sub>1</sub>X<sub>t</sub><sub>-</sub><sub>r</sub> + u<sub>t. </sub>The dynamic causal effect is
A) <sub>0</sub> + <sub>1</sub>
B) <sub>1</sub> + <sub>2</sub>+ + <sub>r</sub><sub>+</sub><sub>1</sub>
C) <sub>0</sub> + <sub>1</sub>+ + <sub>r</sub><sub>+</sub><sub>1</sub>
D) <sub>1</sub>
Q3) Infeasible GLS
A)requires too much memory even for today's PCs.
B)uses complicated interative techniques.
C)cannot be calculated since it also uses quasi differences for X<sub>t</sub>.
D)assumes the parameters of the error autocorrelation process to be known.
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Chapter 16: Additional Topics in Time Series Regression
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Q1) The dynamic OLS (DOLS)estimator of the cointegrating coefficient, if Y<sub>t</sub> and X<sub>t</sub> are cointegrated,
A)is efficient in large samples
B)statistical inference about the cointegrating coefficient is valid
C)the t-statistic constructed using the DOLS estimator with HAC standard errors has a standard normal distribution in large samples
D)all of the above
Q2) The biggest conceptual difference between using VARs for forecasting and using them for structural modeling is that
A)you need to use the Granger causality test for structural modeling.
B)structural modeling requires very specific assumptions derived from economic theory and institutional knowledge of what is exogenous and what is not.
C)you can no longer use the information criteria to decide on the lag length.
D)structural modeling only allows a maximum of three equations in the VAR.
Q3) Some macroeconomic theories suggest that there is a short-run relationship between the inflation rate and the unemployment rate. How would you go about forecasting these two variables? Suggest various alternatives and discuss their advantages and disadvantages.
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Chapter 17: The Theory of Linear Regression With One Regressor
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Q1) Consider estimating a consumption function from a large cross-section sample of households. Assume that households at lower income levels do not have as much discretion for consumption variation as households with high income levels. After all, if you live below the poverty line, then almost all of your income is spent on necessities, and there is little room to save. On the other hand, if your annual income was $1 million, you could save quite a bit if you were a frugal person, or spend it all, if you prefer. Sketch what the scatterplot between consumption and income would look like in such a situation. What functional form do you think could approximate the conditional variance var(u<sub>i</sub> | Inome)?
Q2) Asymptotic distribution theory is
A)not practically relevant, because we never have an infinite number of observations. B)only of theoretical interest.
C)of interest because it tells you what the distribution approximately looks like in small samples.
D)the distribution of statistics when the sample size is very large.
Q3) "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) The difference between the central limit theorems for a scalar and vector-valued random variables is
A)that n approaches infinity in the central limit theorem for scalars only.
B)the conditions on the variances.
C)that single random variables can have an expected value but vectors cannot.
D)the homoskedasticity assumption in the former but not the latter.
Q2) The GLS assumptions include all of the following, with the exception of A)the X<sub>i</sub> are fixed in repeated samples.
B)X<sub>i</sub> and u<sub>i</sub> have nonzero finite fourth moments.
C)E(U \(U'\) \(|X\) )= ?(X), where ?(X)is n × n matrix-valued that can depend on X.
D)E(U \(|X\) )= 0<sub>n</sub>.
Q3) The heteroskedasticity-robust estimator of \(\sum \sqrt { n ( \hat { \beta } - \beta ) }\) is obtained
A)from ( \(X ^ { \prime }\) X)<sup>-</sup><sup>1</sup> <sup> </sup> \(X ^ { \prime }\) <sup>U</sup>.
B)by replacing the population moments in its definition by the identity matrix.
C)from feasible GLS estimation.
D)by replacing the population moments in its definition by sample moments.
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