Old Stuff New Stuff on Exam 3 Econometrics Course Review Jeff Borowitz Georgia State University Jeff Borowitz Econometrics Review 1 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing Logistics We will have no final exam in this class - just a third exam on Thursday, December 5 The third exam will not be cumulative Material since (and including) dummy variables There will be no 7th Problem Set Jeff Borowitz Econometrics Review 2 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing Plan for Day Review of Course Topics Go over answers exam 2 Next class, we will do an applied problem like for the other exams Jeff Borowitz Econometrics Review 3 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing Ordinary Least Squares (Single/Multi-Variate) Gauss Markov Assumptions 1 2 3 4 5 6 Linear in parameters Data is a random sample No multicollinearity Zero conditional mean of variance Homoskedasticity Normality Formulas to calculate βˆ in univariate case Ceteris paribus coefficient interpretation Jeff Borowitz Econometrics Review 4 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing Analysis of Variance SSE , SST , SSR R2 Fraction of explained variation ¯2 Adjusted R 2 : R Larger SST means parameters are estimated with less noise Jeff Borowitz Econometrics Review 5 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing Understanding Bias/Variance Tradeoff Where does OLS coefficient variance come from? Increase in error variance Decrease in sample size (SST ) Amount of independent variation (1-Rj2 ) Omitted variable bias What effects on parameters of interest? Bias/variance tradeoff Jeff Borowitz Econometrics Review 6 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing Hypothesis Testing: Steps Test whether βˆ = 3 against the alternative that βˆ = 3 at the two-sided α = .05 confidence level 1 State null/alternative hypothesis: H0 :βˆ = 3 H1 :βˆ = 3 2 Form t-statistic: t= 3 4 βˆ − 3 ˆ se(β) Look up tcrit from a table based on degrees of freedom, α, and number of sides Reject if magnitude of t is bigger than tcrit (if two-sided) Jeff Borowitz Econometrics Review 7 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing Confidence Intervals Since βˆ is unbiased, the true β is normally distributed around βˆ To calculate the α confidence interval 1 2 3 Find c as the critical t value in a two sided test at the α level ˆ Upper bound is βˆ + cse(β) ˆ ˆ Lower bound is β − cse(β) There is a 1 − α chance that the true β is in this range If a value a lies in the confidence interval, you would not reject the test of βˆ = a Jeff Borowitz Econometrics Review 8 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing Units Analyzed how our estimation depends on units Scaling RHS variable by α: Scales β, se(β) by 1/α t, p, R 2 stay the same Other coefficients also unchanged Scaling LHS variable by α Scales all β’s, se(β)’s by α t, p, R 2 stay the same Jeff Borowitz Econometrics Review 9 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing F-Testing Compare nested models y =β0 + β1 x1 + β2 x2 + u vs. y =β0 + β1 x1 + β2 x2 + β3 x3 + β4 x4 + u Test hypotheses like: β3 = β4 = 0 Calculate the special F statistic for the test of all non-intercept coefficients in the model Jeff Borowitz Econometrics Review 10 / 22 Old Stuff New Stuff on Exam 3 Ordinary Least Squares Hypothesis Testing Interpreting Models with Different Functional Forms Logged left and right hand side variables, and their interpretations Including quadratics Compute marginal effects Determine and interpret turn around point Interaction terms Compute Marginal Effects Interpret coefficients Jeff Borowitz Econometrics Review 11 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments New Stuff The stuff from this point on could be on the test Jeff Borowitz Econometrics Review 12 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments Dummy Variables This will include some slightly more complicated dummy variable stuff Dummy/continuous interactions Where is there equivalence between the groups? Dummy-dummy interactions: e.g. the interpretation of gender/race and interacted coefficients Be able to determine the difference in outcomes between black males and white females, or any other combination: y = β0 + δ0 female + δ1 black + δ2 female ∗ black+ u Jeff Borowitz Econometrics Review 13 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments Interaction Terms We didn’t really interpret interaction terms very heavily on the last test Returns to education vary with experience Or returns to experience vary with education. . . log(wage) =β0 + β1 educ + β2 exper + β3 educ · exper + u Returns to education vary with a dummy variable: log(wage) =β0 + β1 educ + β2 female + β3 educ · female + u Jeff Borowitz Econometrics Review 14 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments Linear Probability Models Just a regression where LHS variable is binary Would be a big problem if you have lots of units predicted to have probabilities not from 0 to 1 yˆ has the interpretation as a probability Changes in x have the interpretation of increasing the probability that y = 1. Jeff Borowitz Econometrics Review 15 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments Heteroskedasticity What is it? Different points have different error variances How to tell if it is a problem in your data? Breusch-Pagan test: regress uˆ2 on X ’s White test: Regress uˆ2 on yˆ and higher powers What are the problems? Standard errors are invalid Use White standard errors instead Can improve on power using WLS/GLS/FGLS How could I ask about it? Walk through applying the test or FGLS Find the modified WLS model as in problem set 5 Show some R results where models necessary to calculate BP/White tests are shown, and you decide whether to reject and why. Jeff Borowitz Econometrics Review 16 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments Weighted Least Squares Weighted Least Squares We can transform a heteroskedastic model so that it is homoskedastic Like on the last problem set. . . Feasible Generalized Least Squares If you don’t know the form of heteroskedasticity, you estimate it uˆ2 = α0 + α1 x1 + . . . + ε ˆ2 Then divide each observation by: uˆ You should be able to do a WLS transformation, and describe how you would do FGLS specifically Jeff Borowitz Econometrics Review 17 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments Miscellaneous Stuff Proxy variables What makes a good proxy (correlated with important part of unobservable)? Ramsey RESET test Use an F-test for whether yˆ and higher powers help explain y in initial regression Measurement error Measurement error on LHS is no big deal, just increases error variance Measurement error on RHS biases coefficients towards 0 Missing data If data is missing at random, or in a way that just depends on x’s that’s OK If data is missing in a way that depends on y , that’s bad Jeff Borowitz Econometrics Review 18 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments Time Series What are the new Gauss-Markov assumptions Strict exogeneity: xt is uncorrelated with future and past ut This rules out e.g. police levels responding to murder rates No serial correlation: if murder is high for some reason today, that can’t make it more likely to be high tomorrow too Finite distributed lag models What does the lag mean? Impact propensity: contemporaneous coefficient Long run propensity: sum of coefficients Jeff Borowitz Econometrics Review 19 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments Time Series: Seasonality and Trends If variables are trending together this will bias coefficients in a time series regression. You can detrend a variable by regressing it on t and taking residuals You can get the same regression results by first detrending the variables or by controlling for t in a regression You can seasonally adjust a variable by regressing it on a set of dummy variables for each season and taking residuals. Jeff Borowitz Econometrics Review 20 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments More Advanced Methods Difference in differences Explain how to use treatment group and post event dummies to get difference in differences estimate y =β0 + β1 post + β2 T + β3 post · T + u Interpret the interaction coefficient as the effect of treatment Fixed effects Use variation only within an individual: how does joining a union affect wages compared to an individual’s history Instrumental Variables A good instrument affects x but not y (proximity to college affects college attendance but not wages) Example question: would IQ make a good instrument for education? Why or why not? Jeff Borowitz Econometrics Review 21 / 22 Old Stuff New Stuff on Exam 3 Heteroskedasticity Miscellaneous Issues Time Series/Panel/Instruments Perspective Econometrics are only going to get more important in economics and in the world, as more stuff get measured OLS is a useful framework to understand a range of econometric issues Bias/variance tradeoff Hypothesis testing Heteroskedasticity Time series Jeff Borowitz Econometrics Review 22 / 22
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