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) PremisesOLS Bias QuestionOLS Reliability Question

Scenario 3: Seemingly Unrelated Regressions (SUR)

Standard Ordinary Least Squares (OLS) PremisesError Term Equal Variance Premise: The variance of the error term’s 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 xt’s, are constants; the explanatory variables, the xt’s, are not random variables.Explanatory Variable/Error Term Independence Premise: The explanatory variables, the xt’s, and the error terms, the et’s, are not correlated.

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: Independent Violated: 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 coefficient’s standard error be “trusted?”

Is the OLS estimation procedure for the value of the coefficient BLUE?

Yes

Satisfied

Yes

Yes

Violated

No

No

No

Use a GLS

Approach

Use a IV/RF/TSLS

Approach

PRS 1

PRS 2-3

Panel Data: Time Series and Cross Section Data – Three ScenariosScenario 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.

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 student’s 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 student’s 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 MinutesStudent Week Score SAT Studied Student Week Score SAT Studied Jim 1 18 720 13 Peg 1 31 760 27 Jim 2 20 720 17 Peg 2 32 760 23 Jim 3 24 720 19 Peg 3 28 760 21 Jim 4 16 720 23 Peg 4 22 760 23 Jim 5 8 720 13 Peg 5 22 760 27 Jim 6 18 720 15 Peg 6 31 760 19 Jim 7 27 720 17 Peg 7 26 760 25 Jim 8 15 720 13 Peg 8 24 760 25 Jim 9 14 720 17 Peg 9 25 760 17 Jim 10 11 720 13 Peg 10 24 760 19  Tim 1 15 670 17 Tim 6 12 670 17 Tim 2 5 670 11 Tim 7 12 670 19 Tim 3 14 670 21 Tim 8 17 670 13 Tim 4 13 670 15 Tim 9 11 670 11 Tim 5 14 670 13 Tim 10 10 670 9

The 3 students provide cross section data.The 10 weeks provide time series data.

Quiz Score Model:

Jim’s SAT score equals a constant 720:

Peg’s SAT score equals a constant 760:

Tim’s 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.

Theory

MathMins > 0: Studying more increases a student’s quiz score

Sat > 0: Higher math SAT scores increase a student’s quiz score.

EstMathScore = 73.54 + .118MathSat + .43MathMins

EstMathScoreJim = 73.54 + .118MathSat + .43MathMinsMathSatJim = 720:= 73.54 + .118720 + .43MathMins= 11.42 + .43MathMins

EstMathScoreTim = 73.54 + .118MathSat + .43MathMinsMathSatTim = 670:= 73.54 + .118670 + .43MathMins= 5.52 + .43MathMins

EstMathScorePeg = 73.54 + .118MathSat + .43MathMinsMathSatPeg = 760:= 73.54 + .118760 + .43MathMins= 16.14 + .43MathMins

EViewsLink

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

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

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

MathSati 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 effectsFirst differences

Sat > 0

TypicallyPRS 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

First Differences ApproachFocus 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:

MathSatJim does not varyMathSatPeg does not varyMathSatTim does not vary

Generalizing:

EViewsLink

Dummy Variable/Fixed Effects Approach

Focus on the first student, Jim:

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

Fixed Effects and EViews

Click OK.

Click on MathScore and then while holding the <Ctrl> 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:

MathSatJim does not varyMathSatPeg does not varyMathSatTim does not vary

EViewsLink

12.83 equals the average of Jim’s, Peg’s, and Tim’s intercepts.

Question: How can we obtain the individual intercept estimates themselves?

Click View.Click Fixed/Random EffectsClick Cross-section Effects

Question: Are these the same intercepts? Yes.

Click Open Equation.

Scenario 2: Random Effects Approach

ArtIQi = Mean[ArtIQ] + vi

ArtIQ is an abstract concept and is unobservable. We do know that different students possess different quantities of innate artistic talent.

vi equals the amount by which a student’s innate artistic talent deviates from the meanvi is a random variable

vi

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.

EstArtScore = 40.57 + .40ArtMinsQuestion: 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?

vi up

up

up not effect down

Positive correlation

Negative correlationBiased up Biased down

IndependentUnbiased

Unbiased only if vi, 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 student’s score by 4.0 points.

bArtMins = .40

EViewsLink

ArtIQi upHow are ArtIQi and

related?

ArtIQi = Mean[ArtIQ] + vi

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 1Bob 2

Bob 10

Dan 1Dan 2

Dan 10

Kim 1Kim 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.

Random Effects and EViewsClick on ArtScore and then while holding the <Ctrl> 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 student’s 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.

bArtMins = .81

Click Open Equation.

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 student’s 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

bLabMins = .51

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 week’s 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

Yes

Good news.

Grader unusually generous

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 EViewsClick on LabScore and then while holding the <Ctrl> 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.