1 an Introduction to Panel Data Methods

7/28/2019 1 an Introduction to Panel Data Methods

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[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England

An introduction to Panel datamethods


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Contents

Preliminary background

Static panel data

Dynamic panels

Panel time series


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What is Panel data?Year Firm Investment Value of the firm

1950 1 642.9 3755.6

1951 1 755.9 4833

1952 1 891.2 4924.9

1953 1 1304.4 6241.71954 1 1486.7 5593.6

1950 2 100.66 693.5

1951 2 160.62 809

1952 2 145 727

1953 2 174.93 1001.5

1954 2 172.49 703.2

1950 3 93.5 1610.51951 3 135.2 1819.4

1952 3 157.3 2079.7

1953 3 179.5 2371.6

1954 3 189.6 2759.9

1950 4 32.24 635.2

1951 4 54.38 723.8

1952 4 71.78 864.1

1953 4 90.08 1193.5

1954 4 68.6 1188.9

1950 5 418.8 1677.4

1951 5 588.2 2289.5

1952 5 645.2 2159.4

1953 5 641 2031.31954 5 459.3 2115.5


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What is Panel Data?

Year Firm Investment Value of the firm

1950 1 642.9 3755.6

1951 1 755.9 4833

1952 1 891.2 4924.91953 1 1304.4 6241.7

1954 1 1486.7 5593.6

1950 2 100.66 693.5

1951 2 160.62 809

1952 2 145 727

1953 2 174.93 1001.5

1954 2 172.49 703.2

1950 3 93.5 1610.5

1951 3 135.2 1819.4

1952 3 157.3 2079.7

1953 3 179.5 2371.6

1954 3 189.6 2759.9

1950 4 32.24 635.21951 4 54.38 723.8

1952 4 71.78 864.1

1953 4 90.08 1193.5

1954 4 68.6 1188.9

1950 5 418.8 1677.4

1951 5 588.2 2289.5

1952 5 645.2 2159.41953 5 641 2031.3

1954 5 459.3 2115.5

Time dimension

Cross section dimension


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What is panel data?

Year Firm Investment Value of the firm

1950 1 642.9 3755.6

1951 1 755.9 4833

1952 1 891.2 4924.91953 1 1304.4 6241.7

1954 1 1486.7 5593.6

1950 2 100.66 693.5

1951 2 160.62 809

1952 2 145 727

1953 2 174.93 1001.5

1954 2 172.49 703.2

1950 3 93.5 1610.5

1951 3 135.2 1819.4

1952 3 157.3 2079.7

1953 3 179.5 2371.6

1954 3 189.6 2759.9

1950 4 32.24 635.21951 4 54.38 723.8

1952 4 71.78 864.1

1953 4 90.08 1193.5

1954 4 68.6 1188.9

1950 5 418.8 1677.4

1951 5 588.2 2289.5

1952 5 645.2 2159.41953 5 641 2031.3

1954 5 459.3 2115.5


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What is panel data?

Data for many individuals (cross sections)

across time

Previous slides show the data in stackedformat.

Wh t i th l ti hi b t


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What is the relationship betweeninvestment and value of the

firm? Plot Investment and

value as a scatterplot.

Data is stacked whichimplies that this is therelationship acrosstime and across firms

0

1000

2000

3000

4000

5000

6000

7000

0 200 400 600 8001000 1400

Investment

Valu

e



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firm? The relationship is

positive.

Is this the case for

each firm in ourpanel?

0

1000

2000

3000

4000

5000

6000

7000

8000

0 200 400 600 8001000 1400

Investment

Value



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firm?

2500

3000

3500

4000

4500

5000

5500

6000

6500

200 400 600 800 1000 1200 1400 1600

Investment

Value

1

400

500

600

700

800

900

1000

1100

20 40 60 80 100 120 140 160 180

Investment

Value

2

1000

1500

2000

2500

3000

0 40 80 120 160 200

Investment

Value

3

0

200

400

600

800

1000

1200

1400

10 20 30 40 50 60 70 80 90 100

Investment

Value

4

1200

1400

1600

1800

2000

2200

2400

2600

2800

200 300 400 500 600 700

Investment

Value

5

Positiveslope

But differentStarting points


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Panel data methods

Most of panel data analysis is concerned

with measuring these two aspects of paneldata relationships:

i,t i Vi,t i,t

Slope

Intercept

i=15 denotes firms

T=1950is time


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Pooled OLS

Model:

To estimate:

Step 1: Stack data

Step 2: Apply OLS formula

i,t Vi,t i,t

Same intercept


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Step1

Year Firm Investment Value of the firm Constant

1950 1 642.9 3755.6 1

1951 1 755.9 4833 1

1952 1 891.2 4924.9 1

1953 1 1304.4 6241.7 1

1954 1 1486.7 5593.6 1


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Step 1


1950 1 642.9 3755.6 1

1951 1 755.9 4833 1

1952 1 891.2 4924.9 1

1953 1 1304.4 6241.7 1

1954 1 1486.7 5593.6 1

1950 2 100.66 693.5 1

1951 2 160.62 809 11952 2 145 727 1

1953 2 174.93 1001.5 1

1954 2 172.49 703.2 1

Data for firms on top of one another


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Step 1


1950 1 642.9 3755.6 1

1951 1 755.9 4833 1

1952 1 891.2 4924.9 1

1953 1 1304.4 6241.7 1

1954 1 1486.7 5593.6 1

1950 2 100.66 693.5 1

1951 2 160.62 809 1

1952 2 145 727 1

1953 2 174.93 1001.5 1

1954 2 172.49 703.2 1

1950 3 93.5 1610.5 11951 3 135.2 1819.4 1

1952 3 157.3 2079.7 1

1953 3 179.5 2371.6 1

1954 3 189.6 2759.9 1

1950 4 32.24 635.2 1

1951 4 54.38 723.8 1

1952 4 71.78 864.1 1

1953 4 90.08 1193.5 11954 4 68.6 1188.9 1

1950 5 418.8 1677.4 1

1951 5 588.2 2289.5 1

1952 5 645.2 2159.4 1

1953 5 641 2031.3 1

1954 5 459.3 2115.5 1

Y variable

Xvariables


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Step 2

Apply OLS formula

, XX1XY

Y Ii,tX Vi,t,constant


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Fixed Effects

Model

To estimate

(a) Stack data(b) Use OLS formula

i,t i Vi,t i,t

Different intercept for each firm


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Step 1: the data

Year Firm Investment Value of the firm Constant for firm 1 Constant for firm2 Constant for firm 3 Constant for firm 4 Constant for firm 5

1950 1 642.9 3755.6 1 0 0 0 0

1951 1 755.9 4833 1 0 0 0 01952 1 891.2 4924.9 1 0 0 0 0

1953 1 1304.4 6241.7 1 0 0 0 0

1954 1 1486.7 5593.6 1 0 0 0 0

1950 2 100.66 693.5 0 1 0 0 0

1951 2 160.62 809 0 1 0 0 0

1952 2 145 727 0 1 0 0 0

1953 2 174.93 1001.5 0 1 0 0 0

1954 2 172.49 703.2 0 1 0 0 0

1950 3 93.5 1610.5 0 0 1 0 01951 3 135.2 1819.4 0 0 1 0 0

1952 3 157.3 2079.7 0 0 1 0 0

1953 3 179.5 2371.6 0 0 1 0 0

1954 3 189.6 2759.9 0 0 1 0 0

1950 4 32.24 635.2 0 0 0 1 0

1951 4 54.38 723.8 0 0 0 1 0

1952 4 71.78 864.1 0 0 0 1 0

1953 4 90.08 1193.5 0 0 0 1 0

1954 4 68.6 1188.9 0 0 0 1 01950 5 418.8 1677.4 0 0 0 0 1

1951 5 588.2 2289.5 0 0 0 0 1

1952 5 645.2 2159.4 0 0 0 0 1

1953 5 641 2031.3 0 0 0 0 1

1954 5 459.3 2115.5 0 0 0 0 1

Y variable

X variables

Dummy for each firm


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Step 2: Use OLS formula

i,t i Vi,t i,t

, 1, 2, 3, 4, 5 X

X1

X

YY Ii,tX Vi,t,5 dummies

We get estimates of the (common) slope

And estimates of the constant for each firm. These areReferred to as fixed effects or individual effects

Thi i i i i l


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This estimation is equivalent

to..(a) Transforming each variable by subtracting the cross-sectional

mean

(b) Use OLS formula on this de-meaned data. This is usedwhen the number of cross sections is very large

(c) Note that this is called the within transformation and

the resulting estimator is called the within estimator(d) Uses only variation through time of each unit, not

between units

Y Ii,tiX Vi,tV i


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Transformation?

Year Firm Investment Cross Section Mean of Investme

1950 1 642.9 1016.22

1951 1 755.9 1016.22

1952 1 891.2 1016.22

1953 1 1304.4 1016.22

1954 1 1486.7 1016.22

1950 2 100.66 150.74

1951 2 160.62 150.74

1952 2 145 150.74

1953 2 174.93 150.74

1954 2 172.49 150.74

1950 3 93.5 151.02

1951 3 135.2 151.021952 3 157.3 151.02

1953 3 179.5 151.02

1954 3 189.6 151.02

1950 4 32.24 63.416

1951 4 54.38 63.416

1952 4 71.78 63.4161953 4 90.08 63.416

1954 4 68.6 63.416

1950 5 418.8 550.5

1951 5 588.2 550.5

1952 5 645.2 550.5

1953 5 641 550.51954 5 459.3 550.5

MeanFor firm i acrosstime


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Before transformation

-3 -2 -1 0 1 2 3

0

20

40

60

80

100

Y

X

Firm 1

-1 0 1 2 3 4 5

0

20

40

60

80

100

Y

X

Firm 2

-1 0 1 2 3 4

0

20

40

60

80

100

Y

X

Firm 3

-4 -3 -2 -1 0 1 2 3

0

20

40

60

80

100

Y

X

Firm 4

Sameslope

DifferentIntercepts!


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Between estimator

The fixed effects estimator uses

information on variation through time The between estimator uses only

information across cross-sections OLS regression on averages i.e.

Y iX V i


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Generalised Least Squares

Note that for any heteroskedastic model:

Estimation by GLS exploits the fact that:

VEXy

2][

=

+=

VVV

VVEE

Xy

VXVyV

22/12/12

2/12/1**

***

2/12/12/1

][]'[

==

=

+=

+=

Square

rootofcovariancematrix


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Random Effects

Model:

Variation across cross sections is random

i,t Vi,t i,t i

Unit specific shock

Ei,t 0,Ei 0

Ei,t2 2,Ei2 2

Ei,t,j 0 i,j 0


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Random Effects

Lets consider the ith cross-section

The covariance matrix of w is

It Vt t i

Wt t i

EWt2

2

2

EWtWs 2 , t s


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Random Effects

introduces a particular (heteroscedastic)

structure

+

++

=

2222

2222

2222

uuu

uuu

uuu

L

MOMM

L

L

VARWt


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Random Effects

Covariance matrix block diagonal

=

= I

L

MOMML

L

00

00

00


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Estimation

As with FE, estimation is carried out by

(a) transforming the variables to remove thecovariance structure.

(b) Using OLS on the transformed variables

This is an example of generalised leastsquares


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Estimation

We need

This is a function of

GLS is equivalent to the following

transformation of Y and X

2/12/1 ][ = I

2,

2

1/2Yit 1

Yit Yi

1

T u2

2

If this is equal to zeroFE and RE the same


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Estimation

We need estimates of

Consider

And

Deviations removes heterogeneity

,

it Vit it i

i Vi i i

Iit i Vit Vi it i


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Estimation

So FE can be used to estimate

Consider the between regression

e i i Vi i i

vare i

2

T 2

2 varei

2

T


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Fixed or Random

Type of dataset

Hausman test. Tests the assumptions

needed for random effects to be consistent

If null hypothesis is rejected use fixedeffects

i,t,j 0


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Hausman test

Call the FE estimator with estimatedcovariance matrix ; and the RE estimatorwith

If the RE model is correct, is consistent and

efficient, and so > If the RE model is wrong, then the RE estimates

are inconsistent, but the FE estimates are still

consistent

The Hausman test uses the test statistic:

FR)(

FV

)( RV

)(~)

()]

()

([)

(

21

kVV

RFRFRF

R

)(F

V )(R

V


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Pooled OLS?

Can test to see if fixed effects are

significant and should be included.

Pooled estimator not appropriate if fixedeffects important


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Dynamic Panel data

Model

Lets estimate using FE

This is because the average contains

i,t i Vi,t i,t1 i,t

i,t i Vi,t V i i,t1 t1,i i,t i

Correlated !Implies inconsistency

it it1


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Anderson Hsiao IV estimator

Step 1: Take first differences

Step 2: use instruments for the RHSvariables

Valid instruments include

Level instruments generally preferred

i,t i,t1 Vi,t Vi,t1 i,t1 Ii,t2 i,t i,t1

Zit Ii,t2 , Ii,t2

Zit Zit1Zit Iit

d i


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Anderson Hsiao

Consistent estimator, but does not exploit allmoment conditions

In period 3 our equation is:

Instruments

In period 4

Instruments and so on

i,3 i,2 Vi,3 Vi,2 i,2 i,1 i,3 i,2Vi,1, i,1

i,4 i,3 Vi,4 Vi,3 i,3 i,2 i,4 i,3

Vi,2, i,2,Vi,1, i,1

A l d B d E i


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Arrelano and Bond Estimator

AB propose using these extra moment

conditions

W

Zi1Zi1 ,Zi2

.

.

Zi1 ,Zi2 . . .Zit2

W

it i 0

A l d B d E ti t


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Arrelano and Bond Estimator

One and two step estimator

Second step standard errors biaseddownwards so inference based on firststep

1step XitWV01WXit1XitWV0

1WIit

Preliminary estimate of weighting matrix

2step Xit

WV11

WXit1

Xit

WV11

WIit

Based on residuals from the first step

V lid I t t


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Valid Instruments

Test for second order serial correlation for

validity of GMM Sargan test

itit2 0

21

1

~)())((' Kp

asy

i

N

iiii WWWs

=

Second step residuals Number of columns of W

P l Ti S i


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Panel Time Series

Panels with long T readily available.

Raises some special issues for panel data

analysis.

Heterogeneity

H t l


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Heterogeneous panels

it i Bix i,t iyit1 v t

Change across

Cross-sections

Bi B i

i i

y it i Bx it yit1 e te t ix it iy it1 v it

If homogeneity assumed Serially correlated error term



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Fixed effects and GMM biased and

inconsistent as they impose slopehomogeneity

A valid model is the mean group model

Estimate



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Estimate

y1t 1 B1x1,t 1y1t1 v1t

y2t 2 B2x2,t 2y2t1 v2t

.

.

yNt N BNxN,t NyNt1 vNt

1Ni

B 1NBi

1

N

i

Means

Panel unit root tests


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Levin and Lin (1992) and Levin Lin and Chu(2002). Model is:

Test

The equation can be augmented to accountfor serial correlation

Assume that all series have the sameunderthe alternative hypothesis

y it i t y it1 c it v it

H0 : 0 vsH0 : 0



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Instead, Im, Pesaran and Shin (1992-

2003) Model is:

Testbased on the averageT-ratio across i

Maddala and Wu 1999 suggest the teststatistic

y it i t iy it1 c it v it

0 : i 0 vs H0 : i 0

2i1N lnp i~22N

P-value for ith ADF regression

Panel Unit Root tests


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Panel Unit Root tests

These unit root tests assume cross

sections are uncorrelated. Pesaran (2007) proposes the following

modification to the IPS test

yit i t y it1 y t1 c it v it

Average across cross sectionsProxy for common component



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Interpretation can be difficult

If the null hypothesis is rejected it does notmean that all series are stationary

It may simply mean some of the series inthe panel are stationary

Panel data in Eviews


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Panel data in Eviews

Organising data

Date and

id


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Panel data in Eviews: reshaping


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data

Re-shaping data into a panel


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Estimation


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Estimation

Once this is done standard equation

commands apply. Goto->quickestimate equation to

estimate model

Documents

1 an Introduction to Panel Data Methods