Panel Random-Coefficient
Model (xtrc)
경제학과 박사과정
이민준
Contents
1.
2.
3.
4.
Model & Methodology
Literature Using This Model
STATA Manual
Estimation Example and Simulation
1. Model & Methodology
• What is a random-coefficient model?
• Fixed(constant)-coeff. Model
: yi    xi  ui
• Here,
(i) The marginal effect of x on y is assumed to be
constant over all i.
(ii) The effect of unobservable factors is captured by the
error-term.
1. Model & Methodology
• We, instead, may want to assume that the parameters
of the equation is not constant.
• Random Coeff. Model
: yi   i  i xi  ui
• Some structures should be imposed on the parameters,
since otherwise this is not estimable (and is also
meaningless).
• Based on the assumptions on the structure, this model
can be further classified.
1. Model & Methodology
• The Usual assumption is that,
i   i , i    i ,i ~ (0,2 ), i ~ (0,2 )
, where  i , i are independent of x.
2
2
ˆ
ˆ

,

,

,

• Then the main goal is to estimate

 and then
to predict the values ˆ i , ˆi for each i using this.
• cf. Under the above assumption, estimating ˆ , ˆ
is not different from estimating a fixed-coeff. model
with hetero-skedasticity.
1. Model & Methodology
• Random Coefficient models with panel data
• Why we use R.C. model of panel data?
 “(Conventional models) does not allow the interaction of
the individual specific and/or time varying differences with
the included explanatory variables.”
(Hsiao and Pesaran (2004))
• The general form of the model is,
: yit  xit it  uit
• Again, we need to assume some structures of it
1. Model & Methodology
• Hsiao(1974) Model
: it    it    i  t
where,
'
E
(

)

E
(

)

0
,
E
(


(i) i
t
i t)  0
(ii) E(i x'it )  0, E(t x'it )  0
(iii)
and
1. Model & Methodology
• Swamy(1970) Model
: it    i
where,
(i) E ( i )  0
(ii) E ( i x'it )  0
(iii)
• This is what STATA runs with ‘xtrc’ command.
1. Model & Methodology
• But in this case, why do not we estimate separate
equations for each i ?
• According to Hsiao et al. (1989),
(i) By reducing the # of parameters to be estimated,
this improves efficiency.
(ii) It reduces the multicollinearity problem due to the
co-movement over time among the explanatory
variables, by appealing to the panel-specific
differences.
1. Model & Methodology
• In addition, when the assumption that the individual
differences( i ) are randomly distributed is true, there
is no aggregation bias.
1. Model & Methodology
• Estimation method of the Swamy(1970) model
: yi  xi i   i  xi (  i )   i  xi   ( xi vi   i )  xi   wi
Then,
E (wi )  xi E (vi )  E ( i )  0
E ( wi w'i )  E ( i 'i )  xi E (vi v'i ) x'i   i2 I  xi x'i   i
Stacking the equations, we have y  x  w
where   E (ww' ) is a block diagonal matrix with
 i along the main diagonal.
1. Model & Methodology
• GLS estimator of ˆ
m
1
1
1
: ̂  (i x'i  i xi ) i x'i  i yi  Wibi
i 1
where m
Wi  { (  Vi ) 1}1 (  Vi ) 1 , bi  ( x'i xi ) 1 x'i yi , Vi   i2 ( x'i xi ) 1
i 1
m
and Var (ˆ )   (  Vi ) 1
i 1
• Here we can see that ˆ is a matrix-weighted average
of the panel-specific OLS estimator( bi ). (The weight
is inversely proportional to their covariant matrix.)
1. Model & Methodology
• To estimate the above, we have to know ,Vi . For this,
we use two-step approach, following Swamy(1970).
• Here,
bi  OLS Panel specific estimator
ˆi ' ˆi , ˆ
2
ˆ i 
Vi  ˆ i2 ( xi ' xi )1
ni  k
m
m
1
1
( bi bi 'mb b ' )   Vˆi
Finally, ˆ 
m  1 i 1
m i 1
1 m
where b   bi
m i 1
1. Model & Methodology
• Prediction of ̂i (Swamy and Mehta(1975))
: ˆi  ˆ  ˆ xi ' ( xi ˆ xi 'ˆ i2 I )1 ( yi  xi ˆ )
1
1
 (ˆ 1  Vˆi ) 1 (ˆ 1ˆ  Vˆi bi )
: This is reported by the option, betas
• Test of parameter constancy (Swamy(1970))
 H0:  i   , i  1,..., m
m
2
(bi   *)'Vˆi 1 (bi   *)
 Test statistic:  k ( m1)  
i 1
m
m
where  *  (Vˆi 1 ) 1  Vˆi 1bi
i 1
i 1
2. Literature Using This Model
• Swamy(1970)
• Revisits Grunfeld(1958)’s firm investment function
yit   0i  1i x1i ,t 1   2i x2i ,t 1  uit
where y is gross invest., x1 is value of share and x2 is
capital stock.
• Test of parameter constancy: rejected.
• Estimation result: 1  0.0843(0.0104),  2  0.1961(0.0412)
• We will return to this example in section 4.
2. Literature Using This Model
• Hendricks et al. (1979)
• Goal: to estimate the level and shape of the electricity
demand function.
• Data: control/test individuals during Connecticut
Peak Load pricing experiment. (for 3 months)
• Individual demand function (daily/periodical)
: q jt  X jt  j   jt
where X includes time-variable (knot variable) and
dummy for the experiment group.
2. Literature Using This Model
• As a result, we get an estimate of each consumer’s
demand function
2. Literature Using This Model
• But what determines  j ?
• This paper assumes that  j is a function of exogenous
variables related to each j.
8
      ki Z kji   ij
i
j
i
k 1
where Z includes ownership of some appliances, # of
people, climate factors, etc.
• In general, Z explains  very well!
• With this estimation result, we can reconstruct the
demand function for each profile-group.
2. Literature Using This Model
2. Literature Using This Model
• Hsiao et al. (1989)
• Goal: to estimate the regional electricity demand in
the state of Ontario
• Model: log-adjustment structure
ln yit   i ln yi ,t 1   i ' ln xit   i ' ln wit   i ' ln zi  uit
where x is the set of economic factors, w of climate
factors and z of regional & seasonal specific factors.
(the unit is municipality)
2. Literature Using This Model
• The parameter constancy test rejects H0.
• Using pure random coeff. model: cannot represent the
difference in parameters which results from the
region-specific factors.
• Hence this paper uses a mixed fixed-random coeff.
model (assuming that the coeff. of z are fixed and
others are randomly distributed).
• Using the root mean square error of the predicted
values, we can show that this model performs well.
2. Literature Using This Model
2. Literature Using This Model
• It is interesting to see that the mixed model
outperforms the region-specific estimations in
predicting regional demand behaviors.
• Also note that the pure random-coeff. model is the
worst in prediction.
• Conclusion: “Neither should we pool the data w/o
taking account of heterogeneity across units, nor
should we simply treat all regional heterogeneity as
random draws from a common population.”
3. STATA Manual
4. Estimation Example and
Simulation
• Estimation example: revisit Grunfeld(1958)
• Data: invest2 data (invest. data for 5 firms)
. use "D:\work\¹ÎÁØ\¼ö¾÷\¼ö¾÷-2010³â 1Çбâ\°è·®°æÁ¦ÇÐ ¼¼¹Ì³ª\panel random coeff\invest2.dta", clear
. xtset company time
panel variable:
time variable:
delta:
company (strongly balanced)
time, 1 to 20
1 unit
. xtrc invest market stock, betas
Random-coefficients regression
Group variable: company
Number of obs
Number of groups
=
=
100
5
Obs per group: min =
avg =
max =
20
20.0
20
Wald chi2(2)
Prob > chi2
invest
Coef.
market
stock
_cons
.0807646
.2839885
-23.58361
Test of parameter constancy:
Std. Err.
.0250829
.0677899
34.55547
z
3.22
4.19
-0.68
chi2(12) =
P>|z|
0.001
0.000
0.495
603.99
=
=
17.55
0.0002
[95% Conf. Interval]
.0316031
.1511229
-91.31108
.1299261
.4168542
44.14386
Prob > chi2 = 0.0000
4. Estimation Example and
Simulation
j
Group-specific coefficients
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
Group 1
market
stock
_cons
.1027848
.3678493
-71.62927
.0108566
.0331352
37.46663
9.47
11.10
-1.91
0.000
0.000
0.056
.0815062
.3029055
-145.0625
.1240634
.4327931
1.803978
.084236
.3092167
-9.819343
.0155761
.0301806
14.07496
5.41
10.25
-0.70
0.000
0.000
0.485
.0537074
.2500638
-37.40575
.1147647
.3683695
17.76707
.0279384
.1508282
-12.03268
.013477
.0286904
29.58083
2.07
5.26
-0.41
0.038
0.000
0.684
.0015241
.0945961
-70.01004
.0543528
.2070603
45.94467
.0411089
.1407172
3.269523
.0118179
.0340279
9.510794
3.48
4.14
0.34
0.001
0.000
0.731
.0179461
.0740237
-15.37129
.0642717
.2074108
21.91034
.147755
.4513312
-27.70628
.0181902
.0569299
42.12524
8.12
7.93
-0.66
0.000
0.000
0.511
.1121028
.3397506
-110.2702
.1834072
.5629118
54.85766
Group 2
market
stock
_cons
Group 3
market
stock
_cons
Group 4
market
stock
_cons
Group 5
market
stock
_cons
4. Estimation Example and
Simulation
• Seemingly Unrelated Regression (for comparison)
. reshape wide invest market stock, i(time) j(company)
(note: j = 1 2 3 4 5)
Data
Number of obs.
Number of variables
j variable (5 values)
xij variables:
long
->
wide
100
5
company
->
->
->
20
16
(dropped)
invest
market
stock
->
->
->
invest1 invest2 ... invest5
market1 market2 ... market5
stock1 stock2 ... stock5
. sureg (invest1 market1 stock1) (invest2 market2 stock2)
> 4) (invest5 market5 stock5)
(invest3 market3 stock3)
Seemingly unrelated regression
Equation
Obs
Parms
RMSE
"R-sq"
chi2
P
invest1
invest2
invest3
invest4
invest5
20
20
20
20
20
2
2
2
2
2
84.94729
12.36322
26.46612
9.742303
95.85484
0.9207
0.9119
0.6876
0.7264
0.4220
261.32
207.21
46.88
59.15
14.97
0.0000
0.0000
0.0000
0.0000
0.0006
(invest4 market4 stock
4. Estimation Example and
Simulation
• d
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
invest1
market1
stock1
_cons
.120493
.3827462
-162.3641
.0216291
.032768
89.45922
5.57
11.68
-1.81
0.000
0.000
0.070
.0781007
.318522
-337.7009
.1628853
.4469703
12.97279
invest2
market2
stock2
_cons
.0695456
.3085445
.5043112
.0168975
.0258635
11.51283
4.12
11.93
0.04
0.000
0.000
0.965
.0364271
.2578529
-22.06042
.1026641
.3592362
23.06904
invest3
market3
stock3
_cons
.0372914
.130783
-22.43892
.0122631
.0220497
25.51859
3.04
5.93
-0.88
0.002
0.000
0.379
.0132561
.0875663
-72.45443
.0613268
.1739997
27.57659
invest4
market4
stock4
_cons
.0570091
.0415065
1.088878
.0113623
.0412016
6.258805
5.02
1.01
0.17
0.000
0.314
0.862
.0347395
-.0392472
-11.17815
.0792788
.1222602
13.35591
invest5
market5
stock5
_cons
.1014782
.3999914
85.42324
.0547837
.1277946
111.8774
1.85
3.13
0.76
0.064
0.002
0.445
-.0058958
.1495186
-133.8525
.2088523
.6504642
304.6989
4. Estimation Example and
Simulation
• Simulation
• N=5, T=20, Iteration=1000
• Generation: yit  i xit   it
where i ~ N (3,1) and xit ,  it ~ N (0,1)
• We will do:
(i) xtrc regress (ii) prediction for each ̂i
(iii) Pooled OLS (iv) parameter constancy test (reject %)
(v) calculate ˆi  i (vi) SUR for comparison
4. Estimation Example and
Simulation
• Result
. su
Variable
Obs
Mean
b_rc
b_po
test
b_pre1
b_pre2
1000
1000
1000
1000
1000
3.017712
3.015449
.967
3.002296
3.006243
b_pre3
b_pre4
b_pre5
err_pre1
err_pre2
1000
1000
1000
1000
1000
err_pre3
err_pre4
err_pre5
err_sur1
err_sur2
err_sur3
err_sur4
err_sur5
Std. Dev.
Min
Max
.4522505
.4609577
.1787259
.9737398
.964548
1.390671
1.292617
0
-.3319063
-.0110268
4.638842
4.890881
1
6.029431
6.223836
3.025869
2.992999
3.061155
.0072867
.0112834
1.01018
.9496223
1.008083
.2316447
.2398169
-.5807662
.2628025
-.1997216
-.716162
-.8152806
5.976103
6.106585
6.570777
.7287912
1.272547
1000
1000
1000
1000
1000
-.0049967
.0067088
.0050713
.0046177
.0145143
.2396529
.2329368
.2404257
.2556837
.2653535
-.823391
-.8310407
-1.037565
-.9312465
-.8621137
.946359
.7822555
.9830741
.7962257
1.377899
1000
1000
1000
-.0003351
.0072393
.0064375
.2705998
.2564982
.2650744
-.8326595
-.8609526
-.9207019
1.070353
.8850923
.9107749
4. Estimation Example and
Simulation
• Result
. su
Variable
Obs
Mean
b_rc
b_po
test
b_pre1
b_pre2
1000
1000
1000
1000
1000
3.017712
3.015449
.967
3.002296
3.006243
b_pre3
b_pre4
b_pre5
err_pre1
err_pre2
1000
1000
1000
1000
1000
err_pre3
err_pre4
err_pre5
err_sur1
err_sur2
err_sur3
err_sur4
err_sur5
Std. Dev.
Min
Max
.4522505
.4609577
.1787259
.9737398
.964548
1.390671
1.292617
0
-.3319063
-.0110268
4.638842
4.890881
1
6.029431
6.223836
3.025869
2.992999
3.061155
.0072867
.0112834
1.01018
.9496223
1.008083
.2316447
.2398169
-.5807662
.2628025
-.1997216
-.716162
-.8152806
5.976103
6.106585
6.570777
.7287912
1.272547
1000
1000
1000
1000
1000
-.0049967
.0067088
.0050713
.0046177
.0145143
.2396529
.2329368
.2404257
.2556837
.2653535
-.823391
-.8310407
-1.037565
-.9312465
-.8621137
.946359
.7822555
.9830741
.7962257
1.377899
1000
1000
1000
-.0003351
.0072393
.0064375
.2705998
.2564982
.2650744
-.8326595
-.8609526
-.9207019
1.070353
.8850923
.9107749
4. Estimation Example and
Simulation
• Result
. su
Variable
Obs
Mean
b_rc
b_po
test
b_pre1
b_pre2
1000
1000
1000
1000
1000
3.017712
3.015449
.967
3.002296
3.006243
b_pre3
b_pre4
b_pre5
err_pre1
err_pre2
1000
1000
1000
1000
1000
err_pre3
err_pre4
err_pre5
err_sur1
err_sur2
err_sur3
err_sur4
err_sur5
Std. Dev.
Min
Max
.4522505
.4609577
.1787259
.9737398
.964548
1.390671
1.292617
0
-.3319063
-.0110268
4.638842
4.890881
1
6.029431
6.223836
3.025869
2.992999
3.061155
.0072867
.0112834
1.01018
.9496223
1.008083
.2316447
.2398169
-.5807662
.2628025
-.1997216
-.716162
-.8152806
5.976103
6.106585
6.570777
.7287912
1.272547
1000
1000
1000
1000
1000
-.0049967
.0067088
.0050713
.0046177
.0145143
.2396529
.2329368
.2404257
.2556837
.2653535
-.823391
-.8310407
-1.037565
-.9312465
-.8621137
.946359
.7822555
.9830741
.7962257
1.377899
1000
1000
1000
-.0003351
.0072393
.0064375
.2705998
.2564982
.2650744
-.8326595
-.8609526
-.9207019
1.070353
.8850923
.9107749
4. Estimation Example and
Simulation
• Result
. su
Variable
Obs
Mean
b_rc
b_po
test
b_pre1
b_pre2
1000
1000
1000
1000
1000
3.017712
3.015449
.967
3.002296
3.006243
b_pre3
b_pre4
b_pre5
err_pre1
err_pre2
1000
1000
1000
1000
1000
err_pre3
err_pre4
err_pre5
err_sur1
err_sur2
err_sur3
err_sur4
err_sur5
Std. Dev.
Min
Max
.4522505
.4609577
.1787259
.9737398
.964548
1.390671
1.292617
0
-.3319063
-.0110268
4.638842
4.890881
1
6.029431
6.223836
3.025869
2.992999
3.061155
.0072867
.0112834
1.01018
.9496223
1.008083
.2316447
.2398169
-.5807662
.2628025
-.1997216
-.716162
-.8152806
5.976103
6.106585
6.570777
.7287912
1.272547
1000
1000
1000
1000
1000
-.0049967
.0067088
.0050713
.0046177
.0145143
.2396529
.2329368
.2404257
.2556837
.2653535
-.823391
-.8310407
-1.037565
-.9312465
-.8621137
.946359
.7822555
.9830741
.7962257
1.377899
1000
1000
1000
-.0003351
.0072393
.0064375
.2705998
.2564982
.2650744
-.8326595
-.8609526
-.9207019
1.070353
.8850923
.9107749
5. References
•
•
•
•
•
•
•
•
Grunfeld, Y. (1958), “The Determinants of Corporate Investment” Unpublished
Ph.D. Thesis, University of Chicago.
Grunfeld, Y. and Z. Griliches (1960), “Is Aggregation Necessarily Bad?” The
Review of Economics and Statistics Vol. 42 No.1
Hendricks, W., R. Koenker and D. Poirier (1979), “Residentia Demand for
Electricity” Journal of Econometrics (9)
Hsiao, C., D. Mountain, M. Chan and K. Tsui (1989), “Modeling Ontario
Regional Electricity System Demand Using a Mixed Fixed and Random
Coefficients Approach” Regional Science and Urban Economics (19)
Hsiao, C. and M, Pesaran (2004), “Random Coefficient Panel Data Models”
CESifo Working Paper No. 1233
Swamy, P. (1970), “Efficient Inference in a Random Coefficient Regression
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