Econometrics 2 Combine cross sections obtained at different points - - PowerPoint PPT Presentation

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Pooled Cross Sections and Panel Data 1

Econometrics 2

Pooled Cross Sections and Panel Data II

Pooled Cross Sections and Panel Data 2

Pooled Cross Sections and Panel Data

• Last time: Pooling independent cross sections across time (13.1-2).

Combine cross sections obtained at different points in time. ”Partial” pooling: Allow the coefficients of some variables to

change between time periods.

Include time dummies and interaction effects. Wage equation example (data in CPS78_85, see homepage):

• Significant change in the ”return to education” from 1978 to

1985.

• No significant change in the ”gender gap” between 1978 and

1985.

Policy analysis: Locating a garbage incinerator:

• Significantly negative causal effect on the prices of nearby houses.
• Diff-in-diff approach: Differences in space of differences over time

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Pooled Cross Sections and Panel Data 3

Pooled Cross Sections and Panel Data

Today: Two-period panel data: Follow the same individuals

• ver two periods (13.3-4)

Unobserved effects model: Time-invariant and

idiosyncratic effects

Omitted variables bias (heterogeneity bias) First-difference estimation Policy analysis with two-period panel data

Pooled Cross Sections and Panel Data 4

Data structure

Panel data: Same n individuals in period 1 and period 2.

Period 1: Period 2: Total of 2n observations on n individuals Period 2 could be some years (months, weeks, …) after period 1

Also called longitudinal data. Simple case: One regressor. Simply want to estimate the

effect of x on y.

1 11 12 1

( , , ,..., ), 1,2,...,

i i i i k

y x x x i n =

2 21 22 2

( , , ,..., ), 1,2,...,

i i i i k

y x x x i n =

3

Pooled Cross Sections and Panel Data 5

Unobserved effects model

Model: Time dummy: Same values for all individuals Composite error term: Unobserved fixed effect (unobserved heterogeneity):

Time-invariant Specific to each individual

Idiosyncratic error:

Varies over individuals and time: ”Regular” error term

1

2

i t it i it

y d x a u β δ β = + + + +

2t

d

it i it

v a u = +

i

a

it

u

Pooled Cross Sections and Panel Data 6

Assumptions on the composite error term

Composite error term: Assume that (conditional on the regressors): Note: We will make no assumption on

(for now).

it i it

v a u = +

( ) 0, 1,2,..., , 1,2 ( ) 0, for all , 1,2 ( ) 0, 1,2,..., , 1,2 ( ) 0, for all , and s, 1,2

it it jt it i it jt

E u i n t E u u i j t E u a i n t E u x i j t = = = = ≠ = = = = = =

( )

i it

E a x

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Pooled Cross Sections and Panel Data 7

Correlated unobserved heterogeneity

Unobserved time-invariant effect

could well be correlated with the observed variable:

Pooling the observations and estimating the model by OLS:

Will result in inconsistent estimates.

Problem cannot be solved if the available data is just a single

cross section of information on and

Fixed effect panel data solution: Estimate a model in which: The parameter of interest, , is identified The fixed effects, , does not appear.

One such method is first-differencing.

it

x

i

a

it

y

1

β

i

a ( )

i it

E a x ≠

Pooled Cross Sections and Panel Data 8

First-difference estimation

• Model:
• The unobserved fixed effect

is ”differenced” away.

• We have a cross section of first differences that allows us to

estimate consistently (given the assumptions on ).

1

β

1

2

i t it i it

y d x a u β δ β = + + + +

2 1 2 2 1 1 1 1 2 1 1 2 1 2 1 2 1 2 2

Period 2: ( ) Period 1: First-differencing: ( )

i i i i i i i i i i i i i i i i i

y x a u y x a u y y x x u u y x u β δ β β β δ β δ β = + + + + = + + + − = + − + − ⇔ ∆ = + ∆ + ∆

it

u

i

a

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Pooled Cross Sections and Panel Data 9

First-difference estimation

More general case: Several observed regressors, some

may be time-invariant

1 2

Example: Wage equation for prime-age male workers the log of wage for worker in period local unemployment rate for worker in period experience (months working) for worker in peri

it it it

y i t x i t x i

3 2 1 12 2 22 2

• d

number of years of education for worker (time-invariant) "ability" for worker (time-invariant) First-differenced model:

i i i i i i

t x i a i y x x u δ β β ∆ = + ∆ + ∆ + ∆

Pooled Cross Sections and Panel Data 10

First-difference estimation

2 1 12 2 22 2

Example: Wage equation for prime-age male workers First-differenced model: Note: * The time-invariant variable, years of education, has been differenced out along

i i i i

y x x u δ β β ∆ = + ∆ + ∆ + ∆

3 22 2

with the fixed effect. Cannot estimate . * The variable will be equal to 12 for most workers, less than 12 if individual has been unemployed. If little variation over workers then

i

x β β ∆ will be imprecisely estimated (large standard errors). * First-differenced estimates could be very different from pooled cross-sectional estimates: Indicates important heterogeneity bias.

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Pooled Cross Sections and Panel Data 11

Policy analysis with panel data (treatment effects)

Panel data even more useful for policy analysis than a

time series of cross sections.

Program evaluation:

Want to measure the causal effect of an individual participating in

some programme

• ”Active labour market policy” programme
• Subsidies to firms to make them innovate, become more productive,

export, ….

Potential problem:

• Individuals select into the program
• Or they are assigned to the program

based on individual characteristics that are related to the outcome variable.

Outcome measures: Post-programme wage, R&D expenses,

productivity, export intensity, …

Pooled Cross Sections and Panel Data 12

Policy analysis with panel data

Model: Note:

Similar to model used for independent cross sections Panel data allows error component structure: Control for time-invariant characteristics of

• participants ( ) and
• non-participants ( )

including variables that are likely to affect the participation decision.

it i it

v a u = + 1

it

prog =

it

prog =

1

2

i t it it

y d prog v β δ β = + + +

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Pooled Cross Sections and Panel Data 13

Policy analysis with panel data

• First-differenced model:
• If participation only in period 2 (”before-after”) the OLS estimate

becomes simply

• Diff-in-diff estimate.
• Panel structure: No assumption needed on
• Still need to assume that

and are uncorrelated for consistency.

• Review the incinerator example.

2 1 2 2

i

i i

y prog u δ β ∆ = + + ∆

1

ˆ

part non part

y y β

−

= ∆ − ∆

i

a

it

u ∆

it

prog

Pooled Cross Sections and Panel Data 14

Policy analysis with panel data: Example

Example: The effect of a grant to firms for job training. Aim of program: Enhance the productivity of workers in

the firm.

Effect measure: ”Scrap rate” (proportion of produced

items that have defects):

Many defects = low average level of productivity in the firm Few defects = high productivity.

Model: How can we obtain a consistent estimate of any causal

effect, ?

1

88

it t it i it

scrap d grant a u β δ β = + + + +

1

β

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Pooled Cross Sections and Panel Data 15

Policy analysis with panel data: Example

Problem:

Participation may be related to unobserved firm effects (worker

and manager ability, the amount of capital available,…).

Unobserved effects likely to be directly related to productivity.

OLS on pooled set of observations: Diff-in-diff approach:

• 2

log( ) 0.597 0.189 88 0.057 108, 0.0034 (0.205) (0.328) (0.431)

it t it

scrap d grant n R = − + = =

• 2

log( ) 0.057 0.317 54, 0.067 (0.097) (0.164)

it it

scrap grant n R ∆ = − − ∆ = =

Pooled Cross Sections and Panel Data 16

Policy analysis with panel data: Example

Questions: Are there indications of heterogeneity bias

here?

What is the likely direction of any bias? How do firms select into the job training

program?

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Pooled Cross Sections and Panel Data 17

Next time

Thursday this week! Panel data with several observations over time for the

same individual units.

W sec. 13.5, 14.1. Exercises start this week!