# Econometric Analysis of Panel Data

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```					Econometric Analysis of Panel Data
• Fixed Effects and Random Effects: Extensions
– Time-invariant Variables
– Two-way Effects
– Nested Random Effects
Time Invariant Variables
• The Model
 β1 
yit  x β  ui  eit  x1
'
it             
'
it   x2     ui  eit
'
 β2
i
 
• Fixed Effects
yit  x1it β1  x2i' β2  ui  eit 
'

  ( yit  yi )  (x1it  x1i' )β1  (eit  ei )
'

yi  x1i β1  x2i β2  ui  ei 
'       '

– b2 can not be identified, thus the individual effects ui can
not be estimated.
Time Invariant Variables
• Fixed Effects: Two-Step Approach
ˆ
(1) yit  x1it β1   i  eit   i  yi  x1i' β1FE
'
ˆ
(2)   x2' β2  w  β2
ˆ                        ˆ
i       i        i        OLS

ˆ
E ( wi )  0, Var ( wi )  Var ( i )   e2 / Ti  x1i' Var (β1) x1i
ˆ
(heteroscedasticity )
ˆ        ˆ
 u  y  x1' β1  x2' β2
ˆ i       i       i   FE    i     OLS
Time Invariant Variables
• Random Effects
yit  xit β  ui  eit , requires Cov(ui , xit )  0
'

– Mundlak’s Approach
If Cov(ui , xit )  0 and yit  x1it β1  x2i' β2  ui  eit ,
'

assume ui  x1i γ1  x2i' γ 2  wi , E ( wi | xit )  0, Var ( wi | xit )   w
2

• Estimate random effects model including group means:
yit  x1it β1  x1i γ1  x2i' δ2  wi  eit ( Note : δ2  β2  γ 2)
'

 ui  x1i γ1  wi
ˆ        ˆ     ˆ
ˆ         ˆ
 y  x1i β1  x2i' δ2
Example: Returns to Schooling
• Cornwell and Rupert Model (1988)
yit  x1it β1  x 2i' β 2  ui  eit
'

• Data (575 individuals over 7 ears)
– Dependent Variable yit:
• LWAGE = log of wage
– Explanatory Variables xit:
• Time-Variant Variables x1it:
– EXP = work experience
WKS = weeks worked
OCC = occupation, 1 if blue collar,
IND = 1 if manufacturing industry
SOUTH = 1 if resides in south
SMSA = 1 if resides in a city (SMSA)
MS = 1 if married
UNION = 1 if wage set by union contract
• Time-Invariant Variables x2i:
– ED = years of education
FEM = 1 if female
BLK = 1 if individual is black
Two Way Effects
• The Model
yit  xit β  ui  vt  eit (t  1, 2,..., Ti )
'

• Assumptions
E (eit | xit )  0
E (ui | xit )  E (vt | xit )  0 (random effects only )
Var (eit | xit )   e2 , Cov(eit , e js | xit , x js )  0
Var (ui | xit )   u2 , Cov(ui , u j | xit , x jt )  Cov(ui , e jt | xit , x jt )  0
Var (vt | xit )   v2 , Cov(vt , vs | xit , xis )  Cov(vt , eis | xit , xis )  0
Cov(ui , vt | xit )  0
Two-Way Effects
• Dummy Variable Representation
yit  xit β  ui  vt  eit (t  1, 2,..., Ti )
'

β 
y i   Xi di     ui iTi  ei  y i  Wi δ  ui iTi  ei
v
(i  1, 2,..., N )
 v1        di1 
v         d 
where v   2  , d   i 2  , d   1 if j  1, 2,..., Ti
  i   ij 0                    otherwise
           
vTi 
           diTi 
 
Two-Way Effects
• Using one-way fixed effects or random effects
model to estimate the dummy variable
representation of two-way effects model.
y i  Wi δ  ui iTi  ei (i  1, 2,..., N )

y  Wδ  u  e
Two-Way Effects
• Two-Way Fixed Effects Model
– Between Estimator
yi  xi' β  ui  v  ei
yt  xt' β  u  vt  et
y  x 'β  u  v  e
– Within Estimator (Group Means Deviations)
yit  xit β  ui  vt  eit (t  1, 2,..., Ti ; i  1, 2,..., N )
'

yit  yi  yt  y  (xit  xi'  xt'  x ' )β  (eit  ei  et  e )
'

 yit  xit β  eit
'
Two-Way Effects
• Two-Way Fixed Effects Model
– OLS
yit  xit β  eit  y i  Xi β  ei  y  Xβ  e
'

ˆ                        ˆ ˆ
βOLS  ( X' X) 1 X' y, Var (βOLS )   e2 ( X' X) 1
ˆ
ˆˆ
 2  e'e / ( NT  N  T  1  K )
ˆe

ˆ        ˆ
e  y  Xβ
– Estimated Individual and Time Effects
ˆ
ui  ( yi  y )  ( xi'  x ' )β  u
ˆ
ˆ
v  ( y  y )  ( x '  x ' )β  v
t      t            t
Two-Way Effects
• Two-Way Random Effects Model
– Partial Group Means Deviations
yit  xit β  ui  vt  eit (t  1, 2,..., Ti ; i  1, 2,..., N )
'

yit  i yi  t yt  it y
 (xit  i xi'  t xt'  it x ' )β  (eit  i ei  t et  it e )
'

 e2                          e2
where i  1                             , t  1                    ,
  Ti
2
e
2
u                  N t
2
e
2
v

 e2
it  i  t  1 
 e2  Ti u2  Nt v2
Two-Way Effects
• Two-Way Random Effects Model
– Consistent estimates of s are derived from:
• e2 asym. var. of two-way fixed effects model
• u2 asym. var. of between (individual) effects model or
one-way fixed (individual) effects model
• v2 asym. var. of between (time) effects model or one-
way fixed (time) effects model
– For improved efficiency, iterate the consistent
estimation until convergence.
Nested Random Effects
• Three-Level Model
yijt  xijt β   ijt  xijt β  (ui  wij  eijt )
'                '

(i  1, 2,..., M ; j  1, 2,..., N i ; t  1, 2,..., Tij )
• Assumptions
– Each successive component of error term is imbedded or
nested within the preceding component
E (eijt | X)  E ( wij | X)  E (ui | X)  0
Var (eijt | X)   e2 ,Var ( wij | X)   w , Var (ui | X)   u
2                    2

Cov(eijt , wij | X)  Cov(eijt , ui | X)  Cov( wij , ui | X)  0

• Model Estimation
– GLS, ML, etc.
Example: U. S. Productivity
• The Model (Munnell [1988])
– Two-level model
ln( gsp jt )  b0  b1 ln(cap jt )  b 2 ln(hwy jt )  b3 ln( waterjt )
 b 4 ln(util jt )  b5 ln(emp jt )  b6 ln(unemp jt )  u j  e jt

– Three-level model
ln( gspijt )  b0  b1 ln(capijt )  b 2 ln(hwyijt )  b3 ln( waterijt )
 b 4 ln(utilijt )  b5 ln(empijt )  b6 ln(unempijt )  ui  wij  eijt
– See, B.H. Baltagi, S.H. Song, and B.C. Jung, The Unbalanced Nested
Error Component Regression Model, Journal of Econometrics, 101,
2001, 357-381.
Example: U. S. Productivity
• Description
– i=1,…,9 regions; j=Ni states
• 6. Gulf: AL, FL, LA, MO
• Mid West: IL, IN, KY, MI, MN, OH, WI
• Mid Atlantic: DE, MD, NJ, NY, PA, VA
• 8. Mountain: CO, ID, MT, ND, SD, WY
• 1. New England: CD, ME, MA, NH, RI, VT
• South: GA, NC, SC, TN, WV
• 7. Southwest: AZ, NV, NM, TX, UT
• Tornado Alley: AK, IA, KS, MS, NE, OK
• 9. West: CA, OR, WA
– t=1970-1986 (17 years)
Example: U. S. Productivity
• Productivity Data
– 48 Continental U.S. States, 17 Years:1970-1986
•   STATE = State name,
•   ST_ABB = State abbreviation (Region = 1, . . . , 9),
•   YR = Year (1970, . . . ,1986),
•   PCAP = Public capital,
•   HWY = Highway capital,
•   WATER = Water utility capital,
•   UTIL = Utility capital,
•   PC = Private capital,
•   GSP = Gross state product,
•   EMP = Employment,
•   UNEMP = Unemployment

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