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BASIC PANEL DATA MODELS
[1] Introduction to panel-data models
(1) Data structure:
Individuals, i = 1, 2, ... , N;
Time, t = 1, 2, ... , T, for each i.
(2) Types of Data:
- large N and small T (most labor data).
- small N and large T (macroeconomic data on G7).
- Both large (farm production).
(3) Balanced v.s. Unbalanced Data:
- Balanced: for any i, there are T observations.
- Unbalanced: T may different over i.
Comment:
- Unbalanced data can be used for regression model, but have some
limitations on analysis of non-linear model such as probit or logit.
- The lecture will focus on balanced data.
(4) Available Panel Data:
- PSID (Panel Study of Income Dynamics)
- Starts in 1968 with 4802 families
- Currently, over than 10,000 families are included.
- Over 5,000 variables.
- Available through the internet.
- NLS (National Longitudinal Surveys of Labor Market Experience)
- Includes five distinct segments of the labor force:
Older men (age between 45 and 49 in 1966)
Young men (between 14 and 24 in 1966)
Mature women (age between 30 and 44 in 1966)
Young women (age between 14 and 21 in 1966)
Youths (age between 14 and 27 in 1979)
- CPS (Current Population Survey)
- Monthly national household survey conducted by Census
Bureau.
- Focuses on unemployment rate and other labor force statistics.
[2] Is Controlling Unobservables Important?
[Example from Stock and Watson, Ch. 10]
- Issue:
- Do alcohol taxes help decrease traffic deaths?
- Data: auto_1.txt
- 48 U.S. states (excluding Alaska and Hawaii): N = 48.
- 1982 -1988: T =7.
Series Descriptions
state State ID (FIPS) Code
year Year
spircons Spirits Consumption
unrate Unemployment Rate
perinc Per Capita Personal Income
emppop Employment/Population Ratio
beertax Tax on Case of Beer
mlda Minimum Legal Drinking Age
vmiles Ave. Mile per Driver
jaild Mandatory Jail Sentence
comserd Mandatory Community Service
allmort # of Vehicle Fatalities (#VF)
mrall Vehicle Fatality Rate (VFR)
(traffic deaths per 10,000 people)
dependent variable: VFR
variable coeff. std. err. t-st beertax 0.1112 0.0624 1. mlda -0.0297 0.0317 -0. jailed 0.1959 0.0723 2. comserd 0.1460 0.0813 1. unrate -0.0227 0.0143 -1. lpinc -1.9018 0.2265 -8. yr83 -0.0900 0.0959 -0. yr84 -0.0648 0.0996 -0. yr85 -0.0783 0.1006 -0. yr86 0.0632 0.1022 0. yr87 0.1032 0.1067 0. yr88 0.1404 0.1107 1. cons 20.7805 2.3157 8.
R-Square = 0.
[Digression]
- Consider a simple multiple regression model:
yi = β 1 + β 2 x2i + β 3 x3i + εi.
- What do β 2 and β 3 measure?
β 2 measures the direct (pure) effect of x2i on yi with x3i held constant.
Similarly, β 3 measures the direct effect of X3i on Yi with X2i held
constant.
- Return to our example:
- Each state would have a different level of preference for alcohol
(say, Pal).
- Pal and Beertax could be positively related (δ 2 > 0).
- Pal would have a positive direct effect on VFR (β 3 > 0).
- The coefficient on Beertax captures the total effect:
β 2 (-) + δ 2 (+)×β 3 (+) = (+).
- How could we control Pal using panel data?
[2] Fixed effects vs. Random effects
(1) Basic Model:
y it = xit ′ (^) β + zi ′ γ + uit = hit ′ δ+ uit ; uit = α i (^) + ε it , (1)
where i = 1, ... , N (cross-section unit), t = 1, ... , T (time),
it it i
x h z
- Assumptions:
- xit: k×1 vector of time-varying regressors.
- zi: g×1 vector of time invariant regressors (overall intercept term
will be included here).
2 ).
- αi varies over i but constant over time (individual effects).
- Matrix Notation:
- Define:
1 1 1 : ; : ; : ; :
i i i
i i i i
iT (^) iT iT iT
y x u
y X u
y (^) x u
ε i 1
⎛ ⎞ ⎛^ ′⎞ ⎛ ⎞ ⎛
⎜ ⎟ ⎜^ ⎟ ⎜ ⎟ ⎜
QT = IT - PT =
T
T T T
T
T T T
T
T T T
⎜ −^ − ⎟
⎜ −^ − ⎟
(T×T deviation-from-mean operator);
PTQT = 0T×T; PTeT = eT ; QTeT = 0T× 1.
Example:
Let yi =. Then, P
1
2 :
i
i
iT
y
y
y
Tyi =^ :
i
i
i
y
y
y
; QTyi =
1
2 :
i i
i i
iT i
y y
y y
y y
PT(eTzi) = eTzi ; QT(eTzi) = 0T×g , where
y i (^) ty T
= Σ (^) it.
(2) Fixed Effects Model:
- Assumptions:
a) The αi are treated as parameters (1980, JEC, Kiefer)
(i.e., different intercepts for individuals)
b) The αi are random variables which are correlated with all the
regressors.
- Mundlak (1978, ECON): a) and b) are equivalent.
- Within Estimation (Least Square Dummy Variables (LSDV))
yi = X i β+ ( e zT i ′ )^ γ + eT α i + ε i = X i β + eT ( zi ′ γ + α i )+ ε i
i
Observe:
Q yT i = Q XT i β+ Q eT T ( zi ′ γ + α i )+ QT ε i = Q XT i β + QT ε. (4)
ˆ β W = OLS on (4) = (^) ( )
1 i X Q Xi T i i X Q yi T
− Σ ′^ Σ ′ i
= OLS on (3) with dummy variables for individuals.
⎟ ⎟ →
11 12
1
1 2
:
... : ...
:
T
N N
NT
y y y y y y y ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ = ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
11 1 12 2
1 1 1 2
1 2
:
... : : ...
:
T T T V
T N N N N N
NT N
y y y y
y y Q y Q y Q y
Q y y y y y
y y
⎛ − ⎞ ⎜ (^) − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (^) − ⎟ ⎛ ⎞ ⎜^ ⎟ ⎜ ⎟ ⎜^ ⎟ ⎜ ⎟ ⎜^ ⎟ = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜^ ⎟ ⎝ ⎠ ⎜^ ⎟ ⎜ − ⎟ ⎜ (^) − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (^) − ⎟ ⎝ ⎠
:
... : : ...
:
T T V
T N N N
N
y y
y P y P y P y
P y y y
y
⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎜^ ⎟ ⎜ ⎟ ⎜^ ⎟ ⎜ ⎟ ⎜^ ⎟ = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜^ ⎟ ⎝ ⎠ ⎜^ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
• PVQV = 0TN×TN; QVV = 0NT×NT.
End of Digression
ˆ
β W = {OLS on QVy = QVXβ + QVε} = (X′QVX)
-1X′Q
Vy.
→ OLS on yit − yi = ( xit − xi ) ′ β + (ε it − ε i ).
- Properties of the within estimator:
- unbiased.
- consistent as either T → ∞ or N → ∞.
( ) (^ )
2 1 1 1 (^1 ) 2 2 1
( ˆ ) N^ T ( )( )
W i t it i it i
N i i T i V
Cov s x x x x
s X Q X s X Q X
− = = − (^) − =
= Σ ′^ = ′
where s^2 = SSE from within estimation /{N(T-1)-k}.
→ s
2 is a consistent estimator of σε
2 .
- Why is s^2 consistent?
- To make sense of this, let qi = Q uT i = QT ε i. Then we can
observe:
2 1 2 2
T t it i i i T i T i i
T
E q E q q E Q E tr Q
ε tr Q^ T ε
Σ = = ′^ = ′^ = ε′
Then, by the central limit theorem,
1
2
lim ( ) ( 1)
N i i i
p i i i
q q q q N T N T
E q q N T
=
[VFR example from Stock and Watson, Ch. 10]
- Data: auto_1.txt
- 48 U.S. states (excluding Alaska and Hawaii): N = 48.
- 1982 -1988: T =7.
- We can think of Pal as αi. Pal would be time-invariant.
- Within estimation results:
dependent variable: VFR
variable coeff. std. err. t-st beertax -0.4768 0.1657 -2. mlda -0.0019 0.0178 -0. jailed 0.0147 0.1201 0. comserd 0.0345 0.1377 0. unrate -0.0629 0.0111 -5. lpinc 1.7964 0.3625 4. yr83 -0.0972 0.0322 -3. yr84 -0.2812 0.0371 -7. yr85 -0.3745 0.0389 -9. yr86 -0.3376 0.0422 -8. yr87 -0.4347 0.0481 -9. yr88 -0.5213 0.0537 -9.
R-Square = 0.
- Now, the estimated coeff. on Beertax has the expected sign and is
significant!
- Other estimators:
- OLS of yit on xit and zi, that is, OLS on y = Xβ + VZγ + u,
→ biased and inconsistent.
- "Between" estimator of δ (β and γ) = OLS of yi on xi and zi
= OLS on PTyi = PTXiβ + eTzi′γ + PTui = PTHiδ + PTu
= OLS on PVy = PVXβ + VZγ + PVu = PVHδ + PVu
= (H′PVH)-1H′PVy.
→ Biased and inconsistent.
- Assume ε and α are uncorrelated:
Cov(ui) = Cov(eTαi+εi) = Cov(eTαi) + Cov(εi)
= eTv(αi)eT′ + Cov(εi) = σα^2 eTeT′ + σε^2 IT
= σα
2 eTeT′ + σε
2 IT = Tσα
2 eT(eT′eT)
2 IT
= Tσα
2 PT + σε
2 IT = σε
2 [(Tσα
2 /σε
2 )PT+IT]
= σε^2 [{(Tσα^2 +σε^2 )/σε^2 }PT + QT]
≡ σε
(^2) (θ-2P T + QT)^ ≡^ σε
Cov(u) = σε
2 Ω, Ω = θ
- PV + QV.
- Σ ≠ IT unless θ = 1 (that is, σα
2 = 0).
2 PT + QT → Σ
-1/ = θPT + QT.
Ω-1^ = θ^2 PV + QV → Ω-1/2^ = θPV + QV.
- Whitening the error in the model:
Σ-1/2yi = Σ-1/2Hiδ + Σ-1/2ui (5)
→ Cov(Σ
-1/ ui) = σε
2 IT.
Ω
-1/ y = Ω
-1/ Hδ + Ω
-1/ u
→ Cov(Ω-1/2u) = σε^2 INT.
ˆ
δ GLS = OLS on (5) = (ΣiHi′Σ
-1H
i)
iHi′Σ
-1y i
= (H′Ω
where Ω = IN ⊗ Σ.
- A practical way to obtain GLS
- (^) (1 ) ; * (^) (1 ) ; * (1 )
yit = yit − − θ yi xit = xit − − θ xi zi = zi − − θ zi = θ zi.
(quasi-differenced data)
yi^ = Σ-1/2yi = yi-(1-θ)PTyi ; Xi^ = Σ-1/2Xi = Xi-(1-θ)PTXi;
eTzi*′^ = Σ-1/2eTzi′ = θeTzi′.
y
= Ω
-1/ y = y-(1-θ)PVy; X
= Ω
-1/ X = X-(1-θ)PVX;
Ω
-1/ VZ = θVZ.
- GLS estimator of δ = (β′,γ′)′
= OLS on yit^ = xit′β + zi*′γ + error
= OLS on yi^ = Xiβ + eTzi*′γ + error
= OLS on Ω
-1/ y = Ω
-1/ Hδ + error.
