Midterm Review Session - Harvard Economics · General Information Review Questions and Answers General Information (2) Suggested Study 1 Review lecture notes and relevant materials

General InformationReview

Questions and Answers

Midterm Review SessionECON 1123 Fall 2008

Dainn [email protected]

Department of EconomicsHarvard University

April 6, 2009

Dainn Wie [email protected] Midterm Review Session



General Information (1)

1 DisclaimerThis is Not an exhaustive list of material-EC1123-Focus on first 8 topics-EC1126-No direction was given

2 You may bring-A calculator-but nothing else is allowed (no cheat sheet!)

3 Test-taking Tips-Time management is important.-Be careful about calculation mistakes.




General Information (2)

Suggested Study1 Review lecture notes and relevant materials in textbook2 Review homeworks and exam questions3 Review previous honor exam questions4 Practice textbook exercise questions

Figure: Study Hard!!




Regression ModelTopics in Regression ModelPanel DataBinary Regression ModelIV Regression

Univariate Regression Model (1)

The regression model

Yi = β0 + β1Xi +ui , i = 1,2, ...,n

1 Why we run regression?⇒ Our purpose is to estimate the causal effect of X (regressor orindependent variable) on Y (dependent variable).

2 What is βo?⇒ β0 is the intercept and the value of Y when X=0.

3 What is β1?⇒ β1 is the slope and is the change in Y associated with a unitchange in X.

4 What is ui?⇒ ui is error term, contains all the other factors besides X thatdetermine Y.





Univariate Regression Model (2)

Example: Do married men make more money than single men?

wagei = β0 + β1marriedi +ui , i = 1,2, ...,n

“wage”=log of monthly wage“married” is a dummy variable.(“married’=1 if married, 0 otherwise.)

1 How would you β0(intercept) and β1(slope)?β0⇒ The average wage of single men.β1⇒ The difference in wage between single and married men.

2 What is ui? In our model which factors are likely to be in theerror term?The error term ui captures all factors other than marital statusthat might affect wage. Education is one potential factor thatwould be included in the error term.Dainn Wie [email protected] Midterm Review Session




Multivariate Regression Model

Let’s consider the following regression model:lwagei = β0 + β1marriedi + β2agei + β3tenurei + β4Southi +ui

Interpretation of β0: Predicted value of Y when all the regressorsare zero. Meaningless in this case.

Interpretation of β1: The average effect of change in maritalstatus on wage, holding other regressors constant.

Interpretation of β2: The average effect of unit change in”age”on wage, holding other regressors constant. (Interpretationof β3 and β4 are similar.

Now we have regression model, but how do we estimate thesecoefficients?⇒ OLS estimation!!





Least Squares Assumptions (1)

What are Least Squares Assumptions?Do the model assumptions hold in our example?

1 (Xi ,Yi ) are iid :random sampling2 ”Large outliers are rare” : If it doesn’t hold, OLS estimator is not

consistent.





Least Squares Assumptions (2)

1 E (ui |Xi ) = 0 : We have endogeneity problem if this assumptionis violated. It means that error term and regressor should not becorrelated. It will be violated in case of OVB, MeasurementError, Simultaneous Causality Bias and Wrong Functional FormSpecification.

2 No Perfect Multicollinearity Condition: The regressors are saidto be perfectly multicollinear if one of the regressors is a perfectlinear function of the other regressors.⇒Dummy Variable Trap: One case of perfect multicollinearity⇒Whenever there is n dummy variables for n category we onlyadd n-1 dummy variables in regression model to avoid dummyvariable trap.





Properties of OLS Estimator (1)

If all four LS assumptions hold, then

β1is an unbiased: On average, it will give us right answer if weevaluated the β1 over repeated trials.⇒E (β1) = β1

Question:Assume that the population is: E (Yi |X ) = 10−7XBut you run following regression to estimate the model.Yi = β0 + β1Xi + β2X

2i + β3X

3i +ui

What can you say about the expected values of β1, β2, β3?⇒





Properties of OLS Estimator (2)

β1 is consistent:The probability that β1 give us right answerbecomes 1 when sample size n is large.

Variance of β1becomes smaller⇒when sample size increases⇒when the regressor varies in a wider range⇒when the variance of the error term is smaller

β1 is normally distributed when n is largeBy “Central Limit Theorem”, for large sample size,

β1 ∼ N(β1,σ2β

), and β1−E(β1)√Var(β1)

∼ N(0,1)





Hypothesis Testing

Now we can estimate the model. But how can we conclude thatregressors have causal effect on dependent variable, Y?⇒ Hypothesis TestingH0 : β1 = 0 (Null hypothesis)H1 : β1 6= 0 (Alternative hypothesis)

T-test:t-statistics: β1−0√Var(β1)

∼ N(0,1) Therefore, if |t|>1.96, we

reject null hypothesis at 5% significance level.

P-value: The p-value is the probability of drawing a value of β1

that differs from 0, by at least as much as the value actuallycalculated with your data. If p-value is less than 0.05. then youreject the null hypothesis at the 5% significance level.





Confidence Interval

Confidence Interval?⇒ An interval that contains the true population beta with acertain pre-specified probability. (usually 95%)

Calculation of Confidence Interval[β1−1.96SE (β1)≤ β1 ≤ β1 + 1.96SE (β1)]

Testing H0 : β1 = 0 using confidence interval:If calculated confidence interval contains 0 within it, then wefail to reject null hypothesis. Otherwise, we reject the nullhypothesis.





Single Hypothesis Testing

How would you test whether marital status is a significant factorin determining the log of monthly wage?T-statistics is greater than 1.96 in absolute value and p-value isless than 0.05, the null hypothesis is rejected at 5% (even 1%)significance level.Dainn Wie [email protected] Midterm Review Session




Joint Hypothesis Testing (1)

What if we have more than 2 hypothesis to test?⇒ Use F-test.Distribution of F-test statisticsF − statistic ∼ Fq,n−k−1, whereq=number of hypothesis (restrictions)k=number of regressors under the alternative hypothesis.

As n→ ∞, then the F − statistic ∼ Fn,∞= χ2q/q

We reject null hypothesis if F-stat>critical value found fromF-distribution.





Joint Hypothesis Testing (2)

H0 : βmarried = βage = 0H1 : One or more coefficients is not equal to zero.

1 As sample size is quite big, (935) we can get critical value “3”from F2,∞.⇒ As F-stat 15.64>3, we reject the null hypothesis. At least onecoefficient is significantly different from zero.

2 Or we reject the null hypothesis since reported p-value is lessthan <0.05





Measures of Fit (1)

What is R2? What does it tell us?The R2is the fraction of variation of the dependent variable (lwage)which can be explained by our model. 0.0986 in our example. It meansthat 9.86% of variation in wage is explained by our model.





Measures of Fit (2)

What is Adjusted R2?R2 = 1− n−1

n−k−1(1−R2)

The adjusted R2 increases only if the new added regressorsimproves the model more than would be expected by chance. Inthis case, 0.0947, slightly less than R2.

What is SER? Root Mse?SER≈RMSE when sample size is large.

SER=√

1n−k−1Σu2

i =√

ResidualSSdf =

√149.32930 =0.40and

RMSE=0.4007 in this case.





Heteroskedasticity vs. Homoskedasticity

Homoskedasticity:The error term ui is homoskedastic if thevariance of the conditional distribution of uigiven Xi is constantfor i=1,2,...n. So, it does not depend on Xi .

Heteroskedasticity:Now the variance of the conditionaldistribution of uigiven Xi is different across Xi .

If we have heteroskedasticity...1 Is OLS estimator still unbiased?

Yes. The OLS estimator is still unbiased, consistent, andasymptotically normal.

2 Can we do hypothesis testing as usual?No. Standard error calculated using homoskedasticity-only formulagive us wrong answer. So test-statistic and confidence intervals basedon homoskedasticity-only formula are no longer valid. So, we have touse heteroskedasticity-robust standard errors.





Omitted Variable Bias-(1)

Population regression equation (True world):Yi = β1X1i + β2X2i + εi

However assume that we estimate following sample regressionequation by mistake.Yi = β2X2i + εi

Now, OLS estimator is no more unbiased.E (β2) = β2 + δβ1, where δ = ∂X1i

∂X2i





Omitted Variable Bias-(2)

Under what condition, OLS estimator suffers from omittedvarialbe bias?β2 suffers from omitted variable bias because there is omitted variableX1i is both1. A determinant of Y(X1i is part of εi ) and2. Correlated with the regressor X2i (Corr(X1i ,X2i ) 6= 0)

Summary of Bias DirectionCorr(X1iX2i ) > 0 Corr(X1iX2i ) < 0

β1 > 0 positive bias negative biasβ1 < 0 negative bias positive bias





Nonlinear Regression (1)

Polynomials in X

Yi = β0 + β1Xi + β2X2i + · · ·βrX

ri +ui

If r=2, then equation is quadratic regression modelIf r=3, then equation is cubic regression model

1 Linear vs. Qudratic Model? ⇒T-Test on coefficient of X 2

2 Quadratic vs. Cubic Model? ⇒T-test on coefficient of X 3

3 Linear vs. Cubic Model?⇒F-test on coefficients of X 2 and X 3.






Log Transformation

Logarithms and percentages[ln(x +4x)− ln(x)]' 4x

x (when 4xx is small)

Definition of percentage change in X⇒100×4xx

Regression Model Interpretation of β1

Y = β0 + β1ln(X ) +u 1% change in X ⇒0.01β1 change in Y

ln(Y ) = β0 + β1X +u 1 unit change in X ⇒100β1%change in Y

ln(Y ) = β0 + β1ln(X ) +u 1% change in X ⇒ β1% change in Y






Interaction Terms

Assume we have following regression model:Wagei = β0 + β1experi + β2exper2

i +ui

⇒”Wage” is quadratic function of experience.β1 : slope of experience on wageβ2 : change in slope of experience

Question:We want to know whether male and female have differentquadratic function, how do we modify this regression equation?⇒Add interaction terms!






Interaction Terms

Wagei = β0 + β1malei + β2experi + β3exper2i +

+ β4(male× exper)i + β5(male× exper2)i +ui

1 difference in “intercept” across two groups: β1

2 difference in “slope” across two groups: β4

3 difference in “change in slope” across two groups: β5





Regression Assessment

Threats to Internal Validity1 Omitted Variable Bias2 Wrong Functional Form3 Errors-in-Variable Bias4 Sample Selection Bias5 Simultaneous Causality Bias

=⇒In all 5 cases, OLS estimate is Biased!!External Validity Threat: Whether our result could be generalized





Panel Data Analysis (1)

Panel DataNow our data has two dimensions: entity and time.So each observation is observed multiple times.Long format

state year crime rate prisoner

1 90 3 5000

1 91 5 5500

2 90 2 2000

2 91 1 4000

3 90 6 3000

3 91 5 3200






ExampleAssume that we want to know whether number of “prisoner” in eachstate “i” at time”t” has any effect on crime rate of state “i” at year “t”

Crimeit = β0 + β1prisonerit +uit

Since each state is observed multiple times, we can use fixed effectestimation to get rid of omitted variable bias which is invariant withinstate or within year.⇒ Entity Fixed Effect: Cultural and religious characteristics of eachState.⇒ Time Fixed Effect: National level anti-crime policy orexpenditure in each year.






The Fixed Effect Regression Assumptions and Standard Errors1 What is the new LS assumption that we have in Fixed Effect

Regression?⇒ Assumption #5. No serial correlation in error term

2 What happens if the last assumption fails?⇒ Our OLS β1 is stil unbiased, consistent and normallydistributed.⇒ But we have wrong SE!⇒No valid hypothesis testing.

3 What should we do if the last assumption fails?⇒We have to use Heteroskedasticity and AutocorrelationConsistent SE. (HAC)





Binary Regression Model (1)

1 ModelsLPM Pr(Y = 1|X1,...Xk) = β0 + β1X1

Probit Pr(Y = 1|X ) = Φ(β0 + β1X1)

Logit Pr(Y = 1|X ) = F (β0 + β1X1)

2 Interpretation of LPM⇒ One unit increase in X increase probability that Y is equal to1 by β1 % point.

3 Shortcoming of LPM⇒When regressors have extreme value, estimated probabiity canbe less than 0 or greater than 1 which is non-sense.





Binary Regression Model (2)

Interpretation of Probit and LogitNow, one unit increase in X increases insider of Φ or F function by β1

So when X increases from a to b⇒ Probit Model:Pr(Y |X = b)−Pr(Y |X = a) = Φ(β0 + β1b)−Φ(β0 + β1a)⇒ Logit Model:Pr(Y |X = b)−Pr(Y |X = a) = F (β0 + β1b)−F (β0 + β1a)





IV Regression Model (1)

IV Regression ModelYi = β0 + β1Xi + β2Wi +ui

X : endogenous variable

W: exogenous variable (or control variable)

Assume there is variable Z which can be excluded from themodel, but correlated with X.

=⇒IV Regression breaks X into a part that is correlated with errorterm ui and a part that is not, using instrument variable Z. And we canestimated unbiased β1 by only using part of X that is not correlatedwith error term.





IV Regression (2)

The concept of IV using simple diagram





IV Regression (3)

Two conditions of IV1 Relevance Condition: Corr(X ,Z ) 6= 0

2 Exogeneity Condition: Corr(Z ,u) = 0

Estimation of IV Regression (2SLS)IV regression model: Yi = β0 + β1Xi + β2Wi +ui

1 Regress X on Z to isolate exogenous part of XXi = π0 + π1Zi + π2Wi + vi

And get predicted value Xi = π0 + π1Zi + π2Wi , i=1,...,n2 Replace Xiby Xi in the regression of interest:

Yi = β0 + β1Xi +Wi +ui

Now β1can be estimated by OLS and estimated βTSLS isconsistent.





IV Regression (4)

Checking Relevance Condition: Corr(X ,Z ) 6= 0⇒First Stage F-statistics

1 Run the first stage regressionXi = π0 + π1Zi + π2Wi + vi

2 Get F-statistics testing the following hypothesis.H0 : π1 = 0

3 Use Rule of ThumbIf the first stage F-statistic is greater than 10, then the set ofour instruments is OK.





IV Regression (5)

Checking Exogeneity Condition: Corr(Z ,u) = 0⇒Overidentification Test

When number of endogenous variable is kand number of instrument is m,We say IV estimator (β1) is

Exactlly Identified if m=k

Overidentified if m>k

Underidentified if m<k

⇒We can estimate IV estimator β2SLS only when model is exactlyidentified or overidentified!⇒But we can check exogeneity condition using J-test only whenmodel is overidentified!





IV Regression (6)

Overidentification Test: J-test⇒Testing Exogeneity

1 Estimate the IV regression and get residuals ui

2 Run following regression:ui = δ0 + δ1Zi + δ2Wi + νi

3 Compute the F-statistic testing the hypothesis that the coefficients onAll instruments are zero. (δ1 in this case)

4 The J-statistic is J=mF.5 J-statistic has a chi-squared distribution with m-k degree of freedom.

J ∼ χ2m−k under the null hypothesis that all the instruments are

exogenous.6 We reject the null hypothesis that IVs are exogenous if J-stat is

significantly large.





Program Evaluation (1)

Ideal experiment

An ideal experiment randomly assigns subjects to treatment level X.⇒ Eliminate “sample selection bias” by randomization.⇒ Empirical Strategy: Yi = β0 + β1Xi +ui

β1 is the “difference estimator” and is effect of treatment

However, ideal experiment is not feasible in most caseswe⇒We use “quasi-Experiment” or “natural experiment” for research.

In a Quasi-experiment also called a natural experiment,randomness is introduced by variations in individualcircumstances that make it appear as if the treatment is randomlyassigned.






Program Evaluation and Diff-in-Diff






Before Program After Program After-Before

Treatment α α + β + γ β + γ

Control 0 0 + β β

Treatment-Control α α + γ γ

Regression ModelYit = α(Treatment)it + β (After)it + γ(Treatmentit ∗Afterit) +ui

Treatmentit = 1 for treatment group, 0 for control groupAfterit = 1 if after experiment, 0 if before experiment⇒ γ give us estimation of DD estimator




Q & A

Good luck!


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Midterm Review Session - Harvard Economics · General Information Review Questions and Answers General Information (2) Suggested Study 1 Review lecture notes and relevant materials