Methods in Applied Econometrics

REGRESSION ANALYSIS

• Estimate the population regression function (PRF) on the basis of the sample regression function (SRF) as accurately as possible

• The two-variable PRF:

Yi = β1 + β2Xi + ui

• The PRF is not directly observable. We estimate it from the SRF:

Yi = 𝛽1 + 𝛽2Xi + ui = 𝑌i + ui

• Here 𝑌i is the estimated (conditional mean) value of Yi

ORDINARY LEAST SQUARES (OLS)• Choose the SRF in such a way that the sum of squares of the residuals, ∑ 𝑢𝑖

2 = ∑(Yi − 𝑌i)2

is as small as possible.

• Differentiating ∑ 𝑢𝑖2partially with respect to 𝛽1and 𝛽2, we obtain

• Setting these equations to zero, after algebraic simplification and manipulation,

• Solving these two equations

BINARY RESPONSE REGRESSION MODELS

• There are three approaches to developing a probability model for a binary response variable:

1. The linear probability model (LPM)

2. The logit model

3. The probit model

• All the conclusions will be valid if the sample is reasonably large.

• Therefore, in small samples, the estimated results should be interpreted carefully

LPM

• The linear probability model (LPM) parameters are estimated based on OLS method

• Run the OLS regression and obtain 𝑌i= estimate of the true E(Yi | Xi).

• Then obtain 𝑤i= 𝑌i (1 − 𝑌i), the estimate of wi.

• Use the estimated wi to transform the data as shown in equation below and estimate the transformed equation by OLS (i.e., weighted least squares).

LOGIT

• For each sample X, compute the probability of occurance as 𝑃i=ni/Niwhere Ni is the sample space and ni is the number of occurrences.

• For each Xi , obtain the logit as 𝐿i= 𝑃i /(1- 𝑃i)

• Then obtain 𝑤i= Ni 𝑃i /(1 − 𝑃i), the estimate of wi

• Use the estimated wi to transform the data as shown in equation below and estimate the transformed equation by OLS (i.e., weighted least squares).

PROBIT

• The estimating model that emerges from the normal CDF (cumulative distribution function) is popularly known as the probit model, although sometimes it is also known as the normit model.

• We express the utility index Ii as a function of explanatory variables Xias Ii = β1 + β2Xi and the CDF for Ii is

• Now the data regression is done as

REGRESSION DISCONTINUITY DESIGN• The Parametric/Global Strategy

Y i =α +β0Ti + f (ri ) + εi

where

• α = the average value of the outcome for those in the treatment group after controlling for the rating variable;

• Y i = the outcome measure for observation i;

• Ti = 1 if observation i is assigned to the treatment group and 0 otherwise;

• ri = the rating variable for observation i, centered at the cut-point

• εi = a random error term for observation i, which is assumed to be independently and identically distributed.

• The coefficient, β0 for treatment assignment represents the marginal impact of the program at the cut-point

REGRESSION DISCONTINUITY DESIGN

• The function f (ri ) represents the relationship between the rating variable and the outcome.

• A variety of functional forms can be tested to determine which fits the data best, so that bias will be minimized.

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DIFFERENCE IN DIFFERENCE MODEL

• Basic two-way fixed effects model• Cross section and time fixed effects

• Use time series of untreated group to establish what would have occurred in the absence of the intervention

• Key concept: can control for the fact that the intervention is more likely in some types of states

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DIFFERENCE IN DIFFERENCE MODEL

• Three different representations• Tabular

• Graphical

• Regression equation

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DIFFERENCE IN DIFFERENCE MODEL

Before

Change

After

Change Difference

Group 1

(Treat)

Yt1 Yt2 ΔYt

= Yt2-Yt1

Group 2

(Control)

Yc1 Yc2 ΔYc

=Yc2-Yc1

Difference ΔΔY

ΔYt – ΔYc

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time

Y

t1 t2

Yt1

Yt2

treatment

control

Yc1

Yc2

Treatment effect=(Yt2-Yt1) – (Yc2-Yc1)

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DIFFERENCE IN DIFFERENCE MODEL

• Three key variables• Tit =1 if obs i belongs in the state that will eventually be treated

• Ait =1 in the periods when treatment occurs

• TitAit -- interaction term, treatment states after the intervention

• Yit = β0 + β1Tit + β2Ait + β3TitAit + εit

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Yit = β0 + β1Tit + β2Ait + β3TitAit + εit

Before

Change

After

Change Difference

Group 1

(Treat)

β0+ β1 β0+ β1+ β2+ β3 ΔYt

= β2+ β3

Group 2

(Control)

β0 β0+ β2 ΔYc

= β2

Difference ΔΔY = β3

INSTRUMENTAL VARIABLE APPROACH

Consider treatment assignment (dummy variable) X and outcome Y

Regress Y on X

Yi = β0 + β1Xi + εi

The estimate of β1 is just the difference between the mean Y for X = 1 (the treatment group) and the mean Y for X = 0 (the control group)

Thus the OLS estimate is

= β1 +

Y β β ε

Y β ε

1 0 1 1

0 0 0

1 0Y Y 1 0

INSTRUMENTAL VARIABLE APPROACH

If the treatment is randomly assigned, then X is uncorrelated with ε (X is exogenous)

If X is uncorrelated with ε if and only if

But if , then the mean difference is

= β1 + = β1

This implies that standard methods (OLS) give an unbiased estimate of β1, which is the average treatment effect

That is, the treatment-control mean difference is an unbiased estimate of β1,

1 0

1 0

1 0Y Y 1 0

GENERALISED METHOD OF MOMENTS

• Why use GMM?• Nonlinear estimation

• Structural estimation

• ‘Robust’ estimation

• Models estimated using GMM• Many….

• Rational expectations models • Euler Equations

• Non-Gaussian distributed models

GENERALISED METHOD OF MOMENTS

• Simple moment conditions

Population Sample

0ˆ0],cov[

0ˆ0][

'1

1

ttT

tT

XX

E

GENERALISED METHOD OF MOMENTS

• OLS as a MM estimator

• Moment conditions:

• MM estimator

ˆˆ that so , XyXy

0)ˆ(0][1

1

T

t

ttTXyE

0)ˆ('0]'[ XyXXE

yXXXXXyX ''ˆ0ˆ''1