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Lecture 4.7 Preview: Panel Data Taking Stock of the Ordinary Least Squares (OLS) Estimation Procedure Panel Data: Three Scenarios Scenario 1: First Differences and Dummy Variable/Fixed Effects (FE) Scenario 2: Random Effects (RE) Standard Ordinary Least Squares (OLS) Premises OLS Bias Question OLS Reliability Question Scenario 3: Seemingly Unrelated Regressions (SUR) Standard Ordinary Least Squares (OLS) Premises Error Term Equal Variance Premise: The variance of the error terms probability distribution for each observation is the same. Error Term/Error Term Independence Premise: The error terms are independent. Explanatory Variable Constant Premise: The explanatory variables, the x t s, are constants; the explanatory variables, the x t s, are not random variables. Explanatory Variable/Error Term Independence Premise: The explanatory variables, the x t s, and the error terms, the e t s, are not correlated.

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Taking Stock of the Ordinary Least Squares (OLS) Estimation Procedure OLS Bias Question: Is the explanatory variable/error term independence premise satisfied or violated? Is the OLS estimation procedure for the value of the coefficient unbiased? Satisfied: IndependentViolated: Correlated OLS Reliability Question: Are the error term equal variance and the error term/error term independence premises satisfied or violated? Can the OLS calculation for the coefficients standard error be trusted? Is the OLS estimation procedure for the value of the coefficient BLUE? Yes Satisfied Yes Violated No Use a GLS Approach Use a IV/RF/TSLS Approach PRS 1 PRS 2-3

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Panel Data: Time Series and Cross Section Data – Three Scenarios Scenario 1 - Math Class Panel Data: Three college students enrolled in a math class: Jim, Peg, and Tim. A quiz is given in each week. We have weekly data for each students quiz scores, Math SAT score, and number of minutes each student studied. Scenario 2 - Studio Art Class Panel Data: Three college students are randomly selected from a heavily enrolled art class. An art project is assigned each week. We have weekly data for each students project score and number of minutes each student devoted to the project. Scenario 3 - Chemistry Class Panel Data: Two students are enrolled in an advanced undergraduate chemistry course. A lab report is due each week. We have weekly data for each students lab score and number of minutes each student devoted to the lab. Each week the two lab reports are graded by a different graduate student. Quiz Math Minutes Quiz Math Minutes Student Week Score SAT Studied Jim 11872013 Peg 1 3176027 Jim 22072017 Peg 2 3276023 Jim 32472019 Peg 3 2876021 Jim 41672023 Peg 4 2276023 Jim 5 872013 Peg 5 2276027 Jim 61872015 Peg 6 3176019 Jim 72772017 Peg 7 2676025 Jim 81572013 Peg 8 2476025 Jim 91472017 Peg 9 2576017 Jim 101172013 Peg 10 2476019 Tim 11567017 Tim 6 1267017 Tim 2 567011 Tim 7 1267019 Tim 31467021 Tim 8 1767013 Tim 41367015 Tim 9 1167011 Tim 51467013 Tim 10 10670 9 The 3 students provide cross section data.The 10 weeks provide time series data.

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Quiz Score Model: Jims SAT score equals a constant 720: Pegs SAT score equals a constant 760: Tims SAT score equals a constant 670: Since the MathSat variable only depends on the student and does not depend on the week, we can drop the time subscript t for the MathSat variable, but of course we must retain the individual student superscript i: Scenario 1 – Math Class Panel Data Project: Assess the effect of studying on quiz scores. Math SAT scores are a cross section fixed effect. For each student, MathSat does not vary across time.

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Theory MathMins > 0: Studying more increases a students quiz score Sat > 0: Higher math SAT scores increase a students quiz score. EstMathScore = 73.54 +.118MathSat +.43MathMins EstMathScore Jim = 73.54 +.118MathSat +.43MathMins MathSat Jim = 720: = 73.54 +.118 720 +.43MathMins = 11.42 +.43MathMins EstMathScore Tim = 73.54 +.118MathSat +.43MathMins MathSat Tim = 670: = 73.54 +.118 670 +.43MathMins = 5.52 +.43MathMins EstMathScore Peg = 73.54 +.118MathSat +.43MathMins MathSat Peg = 760: = 73.54 +.118 760 +.43MathMins = 16.14 +.43MathMins EViewsLink

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Jim: EstMathScore = 11.42 +.43MathMins Peg: EstMathScore = 16.14 +.43MathMins Tim: EstMathScore = 5.52 +.43MathMins EstMathScore MathMins 16.14 11.42 5.52 Peg Jim Tim Slope =.43

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Unobserved Variables: What is privacy concerns did not permit the release of student SAT data? Interpretation: EstMathScore =.59 + 1.02MathMins Question: Might there be a serious econometric problem with using the ordinary least squares (OLS) estimation procedure to estimate this model? EViewsLink

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Question: Do high school students who receive high SAT math scores tend to study more or less than those students who receive low scores? Question: Would you expect MathSat and MathMins to be correlated? up positively correlated MathSat i up Positively correlated up Ordinary least squares (OLS) estimation procedure for the MathMins coefficient value is biased upward. OLS Bias Question: Is the explanatory variable/error term independence premise satisfied or violated? More Yes – Positively correlated Question: What can we do? Dummy variable/fixed effects First differences Sat > 0 Typically PRS 4-5 Question: Might there be a serious econometric problem with using the ordinary least squares (OLS) estimation procedure to estimate this model? PRS 6 Question: Would this cause the ordinary least squares (OLS) estimation procedure for the MathMins coefficient to be biased? Yes – biased upward

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First Differences Approach Focus on the first student, Jim: Subtract: Interpretation: DifEstMathScore =.26DifMathMins Critical Assumptions: For each student (cross-section) the omitted variable must equal the same value in each week (time period). That is, from week to week: MathSat Jim does not vary MathSat Peg does not vary MathSat Tim does not vary Generalizing: EViewsLink

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Dummy Variable/Fixed Effects Approach Focus on the first student, Jim:

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Jim: EstMathScore = 11.86 +.33MathMins Peg: EstMathScore = 19.10 +.33MathMins Tim: EstMathScore = 7.52 +.33MathMins EstMathScore MathMins 19.10 11.86 7.52 Peg Jim Tim EstMathScore = 11.86DumJim1 + 19.10DumPeg1 + 7.52DumTim1 +.33MathMins EViewsLink Slope =.33

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Fixed Effects and EViews Click OK. Click on MathScore and then while holding the key down, click on MathMins. Click the Panel Options tab. Double click the highlighted area. In the Effects specification box, select Fixed from the Cross-section drop down box. Intercept for Jim: 12.83.97 = 11.86 Intercept for Peg : 12.83 + 6.27 = 19.10 Intercept for Tim : 12.83 5.31 = 7.52 Critical Assumptions: For each student (cross-section) the omitted variable must equal the same value in each week (time period). That is, from week to week: MathSat Jim does not vary MathSat Peg does not vary MathSat Tim does not vary EViewsLink 12.83 equals the average of Jims, Pegs, and Tims intercepts. Question: How can we obtain the individual intercept estimates themselves? Click View. Click Fixed/Random Effects Click Cross-section Effects Question: Are these the same intercepts?Yes. Click Open Equation.

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Scenario 2: Random Effects Approach ArtIQ i = Mean[ArtIQ] + v i ArtIQ is an abstract concept and is unobservable. We do know that different students possess different quantities of innate artistic talent. v i equals the amount by which a students innate artistic talent deviates from the mean v i is a random variable vivi Randomly select three students, Bob, Dan, and Kim, from a large studio art class. Model: Project: Assess the effect of time devoted on project scores.

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EstArtScore = 40.57 +.40ArtMins Question: Might there be a serious econometric problem with using the ordinary least squares (OLS) estimation procedure to estimate this model? In the context of this model, are the explanatory variable, and the error term,, correlated? v i up up not effect down Positive correlation Negative correlation Biased upBiased down Independent Unbiased Unbiased only if v i, the random variable reflecting artistic talent, is not correlated with the number of minutes a student spends on the project. OLS Bias Question: Is the explanatory variable/error term premise satisfied for violated? AIQ > 0 We estimate that a 10 minute increase devoted to an art project increases a students score by 4.0 points. b ArtMins =.40 EViewsLink ArtIQ i up How are ArtIQ i and related? ArtIQ i = Mean[ArtIQ] + v i

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For purposes of illustration, assume that the explanatory variable and error term are uncorrelated. Satisfied: All is well. Violated: Cannot trust the standard errors and OLS is not BLUE. Individual Week Bob 1 Bob 2 Bob 10 Dan 1 Dan 2 Dan 10 Kim 1 Kim 2 Kim 10 The random effects estimation procedure exploits this error term pattern to calculate better estimates. OLS Reliability Question: Are the error term equal variance and the error term/error term independence premises satisfied or violated? NB: This is critical because if it were not true, the OLS estimation procedure for the coefficient value would be biased.

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Random Effects and EViews Click on ArtScore and then while holding the key down, click on ArtMins. Click the Panel Options tab. In the Effects specification box, select Random from the Cross-section drop down box. We estimate that a 10 minute increase devoted to an art project increases a students score by 8.1 points. Intuition: We can exploit the additional information about the error terms to improve the estimation procedure. Click OK. Using more information is a good thing. Random Effects Critical Assumption: The omitted variable and the included variable are independent. Double click the highlighted area. b ArtMins =.81 Click Open Equation.

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Scenario 3: Seemingly Unrelated Regressions Two students, Ted and Sue, are enrolled in an advanced undergraduate chemistry course. A lab report is due each week. We have weekly data for each students lab score and number of minutes each student devoted to the lab. Each week the two lab reports are graded by a different graduate student. Project: Assess the effect of time devoted on project scores. Model: EstLabScore = 52.7 +.51LabMins We estimate that an additional 10 minutes of time devoted to the lab increases the lab score by 5.1 points b LabMins =.51

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Question: Might there be a serious econometric problem with using the ordinary least squares (OLS) estimation procedure to estimate this model? OLS Bias Question: Is the explanatory variable/error term premise satisfied for violated? Model: unaffected OLS estimation procedure for the coefficient value is unbiased Satisfied: All is well. Violated: Cannot trust the standard errors and OLS is not BLUE. OLS Reliability Question: Are the error term equal variance and the error term/error term independence premises satisfied or violated? Question: Since each weeks lab report is graded by a different graduate student would you expect some graduate students to be more demanding than others? Question: In each week, would you expect error terms for the two students to be correlated? Yes Good news. Grader unusually generous

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General Question: How are the generalized least squares estimation procedure for heteroskedasticity, generalized least squares estimation procedure for autocorrelation, random effects, and seemingly unrelated regressions similar? Generalized least squares, random effects, and seemingly unrelated regressions are apply information we have about the error terms to improve the estimation procedure. Intuition: The application of additional information improves the results. Seemingly Unrelated Regression and EViews Click on LabScore and then while holding the key down, click on LabMins. Click the Panel Options tab. In the Weights specification box, select Cross-section SUR in the GLS Weightsdrop down box. Click OK. Double click the highlighted area. Click Open Equation.

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