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7/28/2019 1 an Introduction to Panel Data Methods
1/56
The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
An introduction to Panel datamethods
7/28/2019 1 an Introduction to Panel Data Methods
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The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
Contents
Preliminary background
Static panel data
Dynamic panels
Panel time series
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The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
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
7/28/2019 1 an Introduction to Panel Data Methods
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The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
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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The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
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
7/28/2019 1 an Introduction to Panel Data Methods
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The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
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
Wh t i th l ti hi b t
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The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
What is the relationship betweeninvestment and value of the
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
Wh t i th l ti hi b t
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The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
What is the relationship betweeninvestment and value of the
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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[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
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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The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
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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The Bank of England does not accept any liability for misleading or inaccurate information or omissions in the information provided.
[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
Step 1
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
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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[email protected]@bankofengland.co.uk Inflation TargetingInflation Targeting Bank of EnglandBank of England
Step 1
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
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
Heterogeneous panels
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Heterogeneous panels
Fixed effects and GMM biased and
inconsistent as they impose slopehomogeneity
A valid model is the mean group model
Estimate
Heterogeneous panels
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Heterogeneous panels
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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Panel unit root tests
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
Panel unit root tests
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Panel unit root tests
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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Panel unit root tests
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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Re-shaping data into a panel
Re-shaping data into a panel
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Re-shaping data into a panel
Estimation
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Estimation
Once this is done standard equation
commands apply. Goto->quickestimate equation to
estimate model