Lecture 21 Preview: Panel Data

Lecture 21 Preview: Panel Data
Taking Stock of the Ordinary Least Squares (OLS) Estimation Procedure Standard Ordinary Least Squares (OLS) Premises OLS Bias Question OLS Reliability Question Preview: Panel Data Examples First Differences and Cross Section Fixed Effects (Dummy Variables) Period Fixed Effects (Dummy Variables) Random Effects Standard Ordinary Least Squares (OLS) Premises Error Term Equal Variance Premise: The variance of the error term’s probability distribution for each observation , the et, is the same ; all the variances equal Var[e]: Var[e1] = Var[e2] = … = Var[eT] = Var[e] Error Term/Error Term Independence Premise: The error terms , the et’s, are independent. that is, Cov[ei, ej] = 0: Knowing the value of the error term for one observation would not help you predict the error term on any other observation Explanatory Variable/Error Term Independence Premise: The explanatory variables, the xt’s, and the error terms, the et’s, are not correlated. Knowing the value of an observation’s explanatory variable does not help you predict the value of that observation’s error term.

Satisfied: Independent
Taking Stock of the Ordinary Least Squares (OLS) Estimation Procedure OLS Bias Question: Is the explanatory variable/error term independence premise satisfied or violated? Satisfied: Independent Violated: Correlated Is the OLS estimation procedure for the value of the coefficient unbiased? No – Biased Yes – Unbiased OLS Reliability Question: Are the error term equal variance and the error term/error term independence premises satisfied or violated? Satisfied Violated Can the OLS calculation for the coefficient’s standard error be “trusted?” Yes No Is the OLS estimation procedure for the value of the coefficient BLUE? Yes No

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 student’s quiz scores, Math SAT score, and number of minutes each student studied. Quiz Math Minutes Quiz Math Minutes Student Week Score SAT Studied Student Week Score SAT Studied Jim Peg Jim Peg Jim Peg Jim Peg Jim Peg Jim Peg Jim Peg Jim Peg Jim Peg Jim Peg Tim Tim Tim Tim Tim Tim Tim Tim Tim Tim The 10 weeks provide time series (period) data. The 3 students provide cross section data. Scenario 2 - Chemistry Class Panel Data: Two undergraduate students are enrolled in an advanced undergraduate chemistry course. A lab report is due each week. We have weekly data for each student’s lab score and number of minutes each student devoted to the lab. Each week a different graduate student grades the lab reports of the two students. Scenario 3 - 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 student’s project score and number of minutes each student devoted to the project.

Scenario 1 – Math Class Panel Data
Project: Assess the effect of studying on quiz scores. Quiz Score Model: Notation: The subscript denotes the week. The superscript denotes the student. 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. Jim’s SAT score equals a constant 720 in each week: Peg’s SAT score equals a constant 760 in each week : Tim’s SAT score equals a constant 670 in each week : Math SAT scores represent a cross section fixed effect. For each student, MathSat does not vary across time; it varies only by cross section.

Ordinary Least Squares (OLS)
Quiz Score Model: Theory Sat > 0: Higher math SAT scores increase a student’s quiz score. MathMins > 0: Studying more increases a student’s quiz score  EViews Ordinary Least Squares (OLS) Dependent Variable: MathScore Explanatory Variable(s): Estimate SE t-Statistic Prob MathSat 0.0003 MathMins 0.0544 Const  0.0004 Number of Observations 30 Estimated Equation: EstMathScore =  MathSat MathMins bSat = .118 We estimate that a 100 point increase in a student’s math SAT score increases his/her quiz score by an estimated 11.8 points. We estimate that a 10 minute increase in studying increases a student’s quiz score by 4.3 points. bMathMins = .43

EstMathScore = 73.54 + .118MathSat + .43MathMins
MathSatJim = 720: EstMathScoreJim =  MathSat MathMins =   MathMins = MathMins MathSatPeg = 760: EstMathScorePeg =  MathSat MathMins =   MathMins = MathMins MathSatTim = 670: EstMathScoreTim =  MathSat MathMins =   MathMins = MathMins EstMathScore Peg Jim Peg: EstMathScore = MathMins 16.14 Tim Jim: EstMathScore = MathMins 11.42 Slope = .43 Tim: EstMathScore = MathMins 5.52 MathMins

Unobserved Variables: What is privacy concerns did not permit the release of student SAT data? Ordinary Least Squares (OLS) Dependent Variable: MathScore Explanatory Variable(s): Estimate SE t-Statistic Prob MathMins 0.0000 Const 0.8754 Number of Observations 30  EViews Estimated Equation: EstMathScore = MathMins bMathMins = 1.02 We estimate that a 10 minute increase in studying increases a student’s quiz score by points. Question: Might there be a serious econometric problem with using the ordinary least squares (OLS) estimation procedure to estimate this model?

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 independence premise satisfied or violated? Satisfied: Independent Violated: Correlated Is the OLS estimation procedure for the value of the coefficient unbiased? No – Biased Yes – Unbiased OLS Reliability Question: Are the error term equal variance and the error term/error term independence premises satisfied or violated? Satisfied Violated Can the OLS calculation for the coefficient’s standard error be “trusted?” Yes No Is the OLS estimation procedure for the value of the coefficient BLUE? Yes No

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 independence premise satisfied or violated? Question: What can we do? Satisfied: Independent Violated: Correlated First differences Dummy variable/fixed effects Is the OLS estimation procedure for the value of the coefficient unbiased? Yes – Unbiased No – Biased Question: Do high school students who receive high SAT math scores tend to study more or less than those students who receive low scores? MathSati up positively correlated Sat > 0 Typically More up Question: Would you expect MathSat and MathMins to be correlated? up Yes – Positively correlated Positively correlated Question: Would this cause the ordinary least squares (OLS) estimation procedure for the value of the MathMins coefficient to be biased?  Ordinary least squares (OLS) estimation procedure for the value of the MathMins coefficient is biased upward. Yes – biased upward

First Differences Approach Focus on the first student, Jim: Subtract: Generalizing: Ordinary Least Squares (OLS) Dependent Variable: DifMathScore Explanatory Variable(s): Estimate SE t-Statistic Prob DifMathMins 0.2568 Number of Observations 27 NB: The model includes no constant. bMathMins = .26  EViews Interpretation: We estimate that a 10 minute increase in studying increases quiz score by 2.6 points. MathSatJim does not vary Critical Assumptions: For each student (cross-section) the unobserved (and hence omitted) variable must equal the same value in each week (time period). That is, from week to week: MathSatPeg does not vary MathSatTim does not vary

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

NB: There is no constant. Ordinary Least Squares (OLS) Dependent Variable: MathScore Explanatory Variable(s): Estimate SE t-Statistic Prob MathMins 0.1698 DumJim 0.0058 DumPeg 0.0016 DumTim 0.0492 Number of Observations 30  EViews EstMathScore = 11.86DumJim DumPeg DumTim MathMins bMathMins = .33 EstMathScore Interpretation: We estimate that a 10 minute increase in studying increases quiz score by 3.3 points. Peg Jim Peg: EstMathScore = MathMins 19.10 Tim Jim: EstMathScore = MathMins 11.86 Slope = .33 Tim: EstMathScore = MathMins 7.52 MathMins

 EViews Cross Section Fixed Effects and Statistical Software
Click on MathScore and then while holding the <Ctrl> key down, click on MathMins. Click Open Equation. Click View. Double click the highlighted area. Click Fixed/Random Effects Click the Panel Options tab. Click Cross-section Effects In the Effects specification box, select Fixed from the Cross-section drop down box. Click OK.  EViews Cross ID Fixed Effect 1  2 3  Fixed Effects (FE) Dependent Variable: MathScore Explanatory Variable(s): Estimate SE t-Statistic Prob MathMins 0.1698 Const 0.0050 Number of Observations 30 Cross Sections 3 Periods 10 Question: How does the coefficient estimate compare to the estimate when using dummy variables? Answer: Identical. Question: What does Const, 12.83, represent? Intercept for Jim:  = Answer: The average of three intercepts. Intercept for Peg : = Question: How can we obtain the individual intercept estimates themselves? Intercept for Tim :  = Question: How are these intercepts related to the dummy variable intercepts? Same. MathSatJim does not vary Critical Assumptions: For each student (cross-section) the unobserved (and hence omitted) variable must equal the same value in each week (time period). That is, from week to week: MathSatPeg does not vary MathSatTim does not vary

Scenario 2 – Period Fixed Effects (Dummy Variables)
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 student’s lab score and number of minutes each student devoted to the lab. Each week a different graduate student grades the lab reports of the two students. In the first week, both Ted’s and Sue’s lab reports are graded by one graduate student. In the second week, both lab reports are graded by a different graduate student. Project: Assess the effect of time devoted on project scores. Model: Applying the model to each student, Ted and Sue: Since Ted’s report and Sue’s report are both graded by the same graduate student in each week, we can drop the student superscript in GraderGenerosity:

Model: Applying this to each week: Unobserved Variable: The explanatory variable GraderGenerosity is unobservable. It must be omitted from the regression.

Ordinary Least Squares (OLS) Pooled Regression Model: Ordinary Least Squares (OLS) Dependent Variable: LabScore Explanatory Variable(s): Estimate SE t-Statistic Prob LabMins 0.0086 Const 0.0006 Number of Observations 20  EViews Estimated Equation: EstLabScore = LabMins bLabMins: We estimate that an additional 10 minutes of time devoted to the lab increases the lab score by 5.1 points Question: But might there be a serious econometric problem with using the ordinary least squares (OLS) estimation procedure to estimate this model?

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 independence premise satisfied or violated? Satisfied: Independent Violated: Correlated Is the OLS estimation procedure for the value of the coefficient unbiased? No – Biased Yes – Unbiased OLS Reliability Question: Are the error term equal variance and the error term/error term independence premises satisfied or violated? Satisfied Violated Can the OLS calculation for the coefficient’s standard error be “trusted?” Yes No Is the OLS estimation procedure for the value of the coefficient BLUE? Yes No

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 independence premise satisfied or violated? Satisfied: Independent Violated: Correlated Is the OLS estimation procedure for the value of the coefficient unbiased? No – Biased Yes – Unbiased Grader unusually generous unaffected  OLS estimation procedure for the coefficient value is unbiased

Question: Might there be a serious econometric problem with using the ordinary least squares (OLS) estimation procedure to estimate this model? Question: Since each week’s lab report is graded by a different graduate student would you expect some graduate students to be more demanding than others? OLS Bias Question: Is the explanatory variable/error term independence premise satisfied or violated? Satisfied: Independent Is the OLS estimation procedure for the value of the coefficient unbiased? Yes – Unbiased Yes Question: In each week, would you expect error terms for the two students to be correlated? OLS Reliability Question: Are the error term equal variance and the error term/error term independence premises satisfied or violated? Yes We can think of the residuals as the estimated errors. Satisfied Violated Can the OLS calculation for the coefficient’s standard error be “trusted?” Yes No Is the OLS estimation procedure for the value of the coefficient BLUE? Yes No

Period Fixed Effects (Dummy Variables)
Focus on Week 1: Fold the constant and GraderGenerosity terms into a new constant. NB: The new constant term, 1, for both Ted and Sue is identical because both their lab reports are graded by the same graduate student. For each week, we have folded the generosity of the grader into the constant. In each week the constant is identical for both students because the same graduate student grades both lab reports. We now have ten new constants for each week, one constant for each of the ten weeks. Period fixed effects estimates the values of parameters.

Period Fixed Effects (FE)
Period Fixed Effects and Statistical Software  EViews Period Fixed Effects (FE) Dependent Variable: LabScore Explanatory Variable(s): Estimate SE t-Statistic Prob LabMins 0.0121 Const 0.0000 Number of Observations 20 Cross Sections 2 Periods 10 Question: What does the estimate of the constant, 63.27, represent? Answer: The average of the weekly time constants. Estimated Equation: EstLabScore = LabMins bLabMins = .37: We estimate that an additional 10 minutes of time devoted to the lab increases the lab score by 3.7 points Period ID Fixed Effect 1 2 3  4 5  6  7  8 9  10 The period fixed effects suggest that the graduate student who graded for week 8 was the most generous. the graduate student who graded the lab reports for week 9 was the toughest. Period Dummy Variable/Fixed Effects Critical Assumption: For each week (time period) the omitted variable must equal the same value for each student (cross section).

ArtIQi = Mean[ArtIQ] + vi
Scenario 3: Random Effects Randomly select three students, Bob, Dan, and Kim, from a large studio art class. Project: Assess the effect of time devoted on project scores. Model: ArtIQ is an abstract concept and is unobservable. In general, what does ArtIQ represent? We do know that different students possess different quantities of innate artistic talent. vi An unobserved (and hence omitted) variable. ArtIQi = Mean[ArtIQ] + vi vi equals the amount by which a student’s innate artistic talent deviates from the mean vi is a random variable

Model: Ordinary Least Squares (OLS) Dependent Variable: ArtScore Explanatory Variable(s): Estimate SE t-Statistic Prob ArtMins 0.2522 Const 0.0000 Number of Observations 30  EViews Estimated Equation: EstArtScore = ArtMins bArtMins = .40: We estimate that a 10 minute increase devoted to an art project increases a student’s score by 4.0 points. Question: Might there be a serious econometric problem with using the ordinary least squares (OLS) estimation procedure to estimate this model?

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 independence premise satisfied or violated? Satisfied: Independent Violated: Correlated Is the OLS estimation procedure for the value of the coefficient unbiased? Yes – Unbiased No – Biased OLS Reliability Question: Are the error term equal variance and the error term/error term independence premises satisfied or violated? Satisfied Violated Can the OLS calculation for the coefficient’s standard error be “trusted?” Yes No Is the OLS estimation procedure for the value of the coefficient BLUE? Yes No

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 independence premise satisfied or violated? Satisfied: Independent Violated: Correlated Is the OLS estimation procedure for the value of the coefficient unbiased? Yes -Unbiased No – Biased ArtIQi = Mean[ArtIQ] + vi ArtIQi up vi up In the context of this model, are the explanatory variable, and the error term, , correlated? How are ArtIQi and related? AIQ > 0 up NB: Unbiased only when the unobserved (and hence omitted) variable, ArtIQ, and the included explanatory variable, ArtMins, are not correlated. Positively correlated Independent Negatively correlated up not affected down  Positive correlation  Independent  Negative correlation Biased up Unbiased Biased down

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 independence premise satisfied or violated? Satisfied: Independent Violated: Correlated Is the OLS estimation procedure for the value of the coefficient unbiased? Yes – Unbiased No – Biased NB: For purposes of illustration, assume that the unobserved (and hence omitted) variable, ArtIQ, and the included explanatory variable, ArtMins, are not correlated: OLS Reliability Question: Are the error term equal variance and the error term/error term independence premises satisfied or violated? Satisfied Violated Can the OLS calculation for the coefficient’s standard error be “trusted?” Yes No That is, we will assume that ArtIQi and not correlated Is the OLS estimation procedure for the value of the coefficient BLUE? Yes No  Independent

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? Violated: Cannot “trust” the standard errors and OLS is not BLUE. Satisfied: All is well. Individual Week Residual Bob Bob Bob Dan Dan Dan Kim Question: Does it appear that the ordinary least squares (OLS) calculation for the coefficient’s standard error can be “trusted” and that the estimation procedure for the coefficient value the best linear unbiased estimation procedure (BLUE)? Kim Kim No. The random effects estimation procedure exploits this error term pattern to calculate “better” estimates.

Period Random Effects (RE)
Random Effects and Statistical Software Click on ArtScore and then while holding the <Ctrl> key down, click on ArtMins. Double click the highlighted area. Click Open Equation. Click the Panel Options tab. In the Effects specification box, select Random from the Cross-section drop down box. Click OK.  EViews Period Random Effects (RE) Dependent Variable: ArtScore Explanatory Variable(s): Estimate SE t-Statistic Prob ArtMins 0.0000 Const 0.0007 Number of Observations 30 Cross Sections 3 Periods 10 bArtMins = .81 We estimate that a 10 minute increase devoted to an art project increases a student’s score by 8.1 points. Intuition: We can exploit the additional information about the error terms to improve the estimation procedure. Using more information is a “good” thing. Random Effects Critical Assumption: The unobserved (and hence omitted) variable, ArtIQ, and the included explanatory variable, ArtMins, are independent.

Lecture 21 Preview: Panel Data

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