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A general framework for panel data models with an application to Canadian customer-dialed long distance telephone service
Journal of Econometrics 59 (1993) 63-86. North-Holland
A general framework for panel data models with an application to Canadian customer-dialed long distance telephone service * Cheng Hsiao University of California. Irvine, CA 92717, USA University of Southern California. Los Angeles, CA 90089, USA
Trent W. Appelbe and Christopher
R. Dineen
Telecom Canada, Hull, Qu.4.. Canada
We provide a general framework that covers many panel data models as special cases. A Bayes solution to this general framework is derived, and its relation to many widely used estimation and prediction formulae is discussed. Canadian long distance telephone service data are used to illustrate the effects of different ways of controlling unobserved heterogeneity across routes on estimation and prediction.
1. Introduction It is well known that blending characteristics of both cross-sectional and time series data allows us to specify more complicated behavioral hypotheses than could be done by using time series or cross-sectional data alone. Furthermore, panel data offer many more degrees of freedom, and also allow us to control for omitted variable bias and reduce the problem of multicollinearity, hence improving the accuracy of parameter estimates [e.g., Hsiao (1985, 1986, 1991)]. However, an important issue in panel data analysis is how best to control for unobserved individual heterogeneity to avoid biasing the coefficient estimates of
Correspondence to: Cheng Hsiao, Department of Economics, University of Southern California, Los Angeles, CA 90089, USA. * Cheng Hsiao’s work is supported in part by National Science Foundation Grants SES 88-21205 and SES 91-22481 and in part by Irvine Faculty Research Fellowship. We wish to thank Frank Peracchi and a referee for helpful comments and Dennis Bruce and Dave Robinson for empirical work.
0304-4076/93/$06.00 0
1993-Elsevier
Science Publishers B.V. All rights reserved
C. Hsiao et al., General framework for panel data models
64
included explanatory variables and improve the efficiency of parameter estimates. In a linear regression framework, several models have been proposed. These include: (i) a common model for all temporal cross-sectional observations, (ii) different models for different cross-sectional units, (iii) variable intercept models [e.g., Kuh (1963), Nerlove (1965)], (’tv ) error components models [e.g., Balestra and Nerlove (1966), Wallace and Hussain (1969)], (v) random coefficients models [e.g., Hsiao (1974, 1975), Swamy (1970, 1971, 1974)], and (vi) mixed fixed and random coefficients models [e.g., Ghali (1988), Hsiao et al. (1989)], etc. In this paper we provide a basic framework that unifies these models. We also combine route-specific temporal data to estimate the price and income elasticities of demand for Canadian customer-dialed long distance telephone service. In section 2 we set up a unified framework. In section 3 we provide a Bayes solution to the general model which encompasses many of the well-known estimators for each of the special cases. We specify a Canadian customer-dialed long distance telephone service demand model that takes advantage of the information provided by panel data in section 4. Empirical estimates are reported in section 5 and prediction comparisons are provided in section 6. Conclusions are in section 7.
2. A general framework Suppose there are observations of 1 + ki + k2 variables (yit, &, L&) of N cross-sectional units over T time periods, where i = 1, . . . , N and t = 1, . . . , T. Let
z NTx NkZ
=
C. Hsiao et al., General framework for panel data models
65
where
i=l
, . . . . N.
We assume that
(2-l)
y=Xa+Zy+u,
where a and y are Nkl x 1 and Nkz x 1 vectors, respectively,
(2.2) and uf = (Uil, . . . , UiT), We let the NT x NTcovariance matrix C1 be unrestricted to allow for correlations across cross-sectional units and over time. We further assume that the Nkl x 1 vector a satisfies
(2.3)
where ai is a k, x 1 vector, i = 1, . . . , N, A, is an Nkl x m matrix with known elements, 2 is an m x 1 vector of constants, and E -
N(0, C,).
(2.4)
The variance-covariance matrix C2 is assumed to be nonsingular. The Nk2 x 1 vector of y is assumed to satisfy 71 Y=
:
0
YN
=
A,?,
(2.5)
66
C. Hsiao et al.. General framework
for panel data models
where each yi is k2 x 1, A2 is an Nkz x n matrix with known elements, and r is an n x 1 vector of constants. Because A2 is known, (2.1) is formally identical to y=Xa+@+u,
(2.1’)
where 2 = ZAz. However, by postulating (2.5) we allow for various possible fixed parameter configurations. For instance, if we wish to allow for yi to be different across cross-sectional units, we can let A2 = IN @ Ik2, where I, denotes a p x p identity matrix. On the other hand, if we wish to constrain yi = yj, we can let A2 = eN @ Ik2, where eN is an N x 1 vector with each element equal to one. Many of the widely used panel data models that assume that the unobserved heterogeneity is individual-specific but time-invariant, can be treated as special cases of the model (2.1)-(2.5). These include: (i) A common model for all cross-sectional units If there is no interindividual difference in behavioral patterns, we may assume that the regression coefficients are identical for all cross-sectional units at all times. Thus, we have Yit
=
Z1* 7+
i=
Uit,
l,...,
N,
T.
t=l,...,
(2.6)
This model can be obtained from (2.1’) by letting X = 0 and A2 = (Zk,, . . . , Zk2)‘, where Ik, is a k2 row identity matrix. (ii) Diflerent models for diflerent cross-sectional units When each individual unit is viewed as different, we can write the regression model in the form Yit
=
ZftYi
+
i = 1, . . . . N,
Uitr
t = 1, . . . , T,
(2.7)
with yi different for different individuals. Model (2.7) can be obtained from (2.1’) by letting X = 0 and A2 be an Nkz x Nkz identity matrix. (iii) Variable intercept models [e.g., Kuh (1963)] The variable intercept models assume that, conditional on the observed explanatory variables, there are interindividual differences that stay constant through time. These interindividual differences can be captured by allowing the intercept terms of a regression model to vary with i; thus, we have yiz =
yli
+
y2zic2
+
“’
+
Yk,Zirk,
+
uit,
i=
l,...,N.
(2.8)
C. Hsiao et al., General framework for panel data models
67
Model (2.8) can be obtained from (2.1)-(2.5) by letting X = 0, the first column of Zi be a T x 1 vector of ones, eT,
P= (Yll ,
. . . . “tlN, 72,
a.3 3 ?k2),
where
I,:_, = k, x (k2 - 1)
(iv) Error components models [e.g., Balestra and Nerlove (1966), Wallace and Hussain (1969)] When the effects of the omitted variables that reflect individual time-invariant differences are treated as random variables (just like the assumption on other components of the random disturbance term of a regression model), we have the error components model Yit = Gt? +
clli
+
(2.9)
42,
with (2.10) if Zit does not include the common intercept term for all cross-sectional units, and (2.11)
EBri = 0 if Zit includes the common intercept term, and
Etali - Eali)talj - Ealj) =
i=j 2’ ifotherwiie
We can obtain (2.9) from (2.1)-(2.4) by letting Xi = eT, a’ = (all, Al
=
(2.12)
. . . , aIN),
eN?
c2 = E&E’= Cr,Z, IN,
(2.13)
68
C. Hsiao et al.. General framework for panel data models
and either rZi = 0 if zit contains the intercept term, or cl1 being an unknown constant if zic does not contain the intercept term. (v) Random coeficients models [Swamy (1970)]
When the heterogeneity among cross-sectional units cannot be captured completely by the time-invariant individual varying constants ali, while the specification of the relationships among variables appears proper given the available data, a natural generalization would be to let the parameters vary across cross-sectional units. In model (2.7) we allow the individual coefficients to be treated as fixed and different. We can also allow the regression coefficients to be treated as random variables with a probability distribution. The random coefficient specification reduces the number of parameters to be estimated substantially, while still allowing the coefficients to differ from unit to unit. Under the assumption that the explanatory variables and coefficients are independent, the Swamy (1970) type random coefficients formulation assumes that yi( =
Xi,(xi +
(2.14)
Uify
(2.15)
Eui = 6,
E(ai- 4(aj - a)’ =
A
if
o
otherwise
i=j,
,
(2.16) (2.17)
Exa(ai - a)’ = 0.
The Swamy type random coefficients model assumes that individual differences satisfy de Finetti’s (1964) exchangeability criterion. That is, the likelihood for the ith individual to have a particular realized value ui is the same as for the jth individual. The individual ai all have the same expected value. The subscript i is purely a labelling device with no substantive content. The observed difference between ai and aj is the work of chance mechanisms only. We can obtain model (2.14)-(2.17) from (2.1’) by letting 2 = 0, Al = eN @ Ik,, a=(a;
,...,
a;V),d=(al-Q
,...,
aN-ti),CZ=ZN@A.
(vi) Mixed jixed and random coejicients models [e.g., Hsiao (1990), Hsiao et al.
(19891 When some of the coefficients are treated as fixed constants and some are treated as random variables, we have the mixed fixed and random coefficients version of model (2.1’). In this case we assume that responses to changes in certain conditions are similar, but responses to changes in other conditions may be individual-specific.
C. Hsiao et al., General framework for panel data models
69
3. A Bayes solution To derive a general solution to the model (2.1)-(2.4), we shall take a Bayesian approach. We take model (2.1’) PS the basic regression model. We assume that the prior distributions of a and 7 are independent. We have no information on 7, hence a diffuse prior is assumed. But assumption (2.3) can be viewed as providing an informative prior on a, namely, a is normally distributed with mean Al 6 and varianceecovariance matrix Cz, although we may not have knowledge on 6. If there is no information on 6, the prior for 6 is assumed to be diffuse and is independent of the prior of 7. To put it formally, we assume: A.l.
The prior distributions of a and 7 are independent, that is, p(a, 7) = p(a) * P(
A.2. A.3.
7).
(3.1)
p(a) - N(A1& G).
(3.2)
There is no information about 5 and 7, that is, p(5) and p( 7) are independent and p(6) cc constant,
(3.3)
p( 7) cc constant.
(3.4)
Theorem 3.1.
Suppose that
y N N(XAlti Under A.I-A.3,
+ .@, Cl + X&X’).
(3.5)
the posterior distribution of ci and ji given y is
(3.6) where
(Cl + X&X’)-‘(XA,,
-I
.?)
1 (C, + X&X’)_ly,
v-1
=
(Cl
+X&X’)-‘(XA,,
(3.7) z”) .
1
(34
C. Hsiao et al., General framework for panel data models
70
The proof of this theorem follows straightforwardly of a linear regression model of
from the Bayes solution
y = XAla + z”y + u*,
(3.9)
with diffuse prior for a and 7, where (3.10)
u* = u + XE [e.g., Zellner (1971)]. Theorem 3.2.
Suppose that given a and f,
y - N(Xa + gf, Cl),
(3.11)
and given 6, a w N(A, a, C,). Then, under A.l-A.3, where
(3.12)
(i) the posterior distribution of a given 6 and y is N(a*, DI),
a* = {X’[C;’
- C~12(zlC~‘Z)-‘z”fC~‘]X
x (X’[C;’
- C;l~(z,C;lz”)-‘.?C;‘]y
Dl = {X’[C;’
- C;‘~(z”rC;‘z”)-‘z,C;l]X
+ CT;‘}-’ + C;‘AIE},
+ Cl’}+;
(3.13) (3.14)
(ii) the (unconditional) posterior distribution of a given y is N(a**, D:), where a** = DF*{X’[C;’
- c;1z”(zlc;‘Z)-‘~c;‘]y},
D; = {X’[C;’
- C;1z(z”‘C;‘2)-‘Z’C;‘]X
+ C;’
- C;‘Al(A;C;‘Al)-‘A;C;‘}-‘;
(3.15)
(3.16)
(iii) the posterior distribution of 7 given y and a is N(E DJ), where f=
(.?C;‘z)-‘Z’C;‘(y
6, = (pC;‘z)-1;
- Xa),
(3.17) (3.18)
C. Hsiao et al., General framework for panel data models
(iv) the posterior
distribution
f* = D:z”f([C;l
of 7 given y and ti is N( f*, OS), where
- C;‘X(X’C;‘X
- c;‘x(x’c;‘x 0;
= {z’[C;’
+ CT’)-‘X’C;‘]y
+ C;‘)-‘C;‘Ala),
- C;‘X(X’C;‘X
(v) the (unconditional)
posterior
-** = D;* S?{c;’ Y
(3.19)
+ C;‘)-‘X’C;‘]z)-‘;
distribution
off
- c;‘x[x’c;‘x
(3.20)
given y is N( 1)**, Dz*), where + c;’ (3.21)
- C;‘A1(A;C;‘Al)A;C;‘]-‘X’C;‘)-‘y, D:* = [_@C;’
11
- C;‘X[X’C;‘X
+ C;’ (3.22)
- C;‘Al(A;C;‘A~)-‘A;C;‘I-‘X’C;‘)ZI-’.
The derivation of these results are in the appendix. The Corollaq 3.1. **, Dz), where N(a Ii** =
(unconditional)
posterior
distribution
B2{A;C;1(X’[C;1 - c;‘z”(z”lc;‘2)-‘z’c;1]x - c;~z”(zlc;‘~)-‘z”,c;‘lu),
x x’[c;’ D2 = {A; cc;’ x
z’c,‘x
= {A;(& x
- c;‘(x’c;‘x
of ii given y is
+ CT’)_’ (3.23)
- xC;‘.z(~c;‘z”)-’
+ C;‘)-‘C;‘]Al}-l
+ [x’(c;’
-
2c;1)x]-‘)-‘Al}-‘.
c;1.2(.Fc;‘2)-’ (3.24)
The derivation of (3.23) and (3.24) follows straightforwardly from the joint distribution of 6 and jJ [(3.6)] is equal to the product of the conditional distribution of 7 given ti C(3.19) and (3.20)] and the marginal distribution of 6. Remark
1. The Bayes estimators of a, a, and 7 are derived conditional on Cl and C2. Of course, unconditional Bayes estimators can be derived by introducing the prior distributions of Cl and Cz. However, with unrestricted Cl and Cz, the computation can be burdensome with little gain in insight.
72
C. Hsiao et al., General framework
for panel data models
Therefore, in the event that Ci and Cz can be consistently estimated, we shall confine our analysis with the assumption that Ci and C2 are known. Remark 2.
In a classical framework, it makes no sense to predict the independently drawn random variable ai. However, in panel data, we actually operate with two dimensions - a cross-sectional dimension and a time series dimension. Even though ai is an independently distributed random variable across i, once a particular ai is drawn, it stays constant over time. Therefore, it makes sense to predict ai. The classical predictor of ai is the generalized least squares estimator of the model (2.1’). The Bayes predictor (3.13) or (3.15) is the weighted average between the generalized least squares estimator of a for the model (2.1’) and the overall mean A 1ti if 6 is known [ (3.13)] or A 1ti ** if ti is unknown, ,**
= D1(x’[c;’
-
c;‘z(z”lc;l~)-lz”~c~l]x~
+ C;‘A,a**}, (3.25)
where B = {x’[c;’ x {x’[c;’
- c;‘z(z”,c;‘2)-‘z”lc;‘]x}~ -
c;‘z”(z”‘c;‘z)-‘z”lc;‘ly).
(3.26)
[For details, see Hsiao (1991).] Remark 3. Many of the widely used panel data model estimators are special cases of the Bayes estimators. For instance, the generalized least squares estimator for the error components model is the Bayes estimator (3.21) after substituting Xi = eT, Ai = eN and Cl = crzINT, Cz = ozlZN into (3.21). The Swamy (1971) random coefficients model estimator or Lindley-Smith (1972) linear hierarchy model estimator can be obtained from (3.23) by letting Zi = 0, A1=eN@Ik,,a=(a; ,..., &),&‘=(ai-ti )..., aN-6),Cz=IN@d. Remark 4. Given that most panel data involves a large number of individuals, the Bayes estimators for the completely stacked model (3.9) and (2.1’) may involve the inversion of matrices of very high order. Computation can be simplified by operating with lower-dimension matrices. Eqs. (3.13) to (3.22) and (3.23) and (3.24) provide formulae to convert operations involving matrices of high dimensions to operations with matrices of much smaller dimensions. Remark 5. There appears no straightforward generalization of the above results for the single equation models to the simultaneous equations models. However, there are many ways the above results can still be applied to simultaneous
73
C. Hsiao et al., General framework for panel data models
equations models. For instance, consider a simultaneous equations model of the form KBi = XiRi + Zi& + Ui,
i=
l,...,N,
(3.27)
where x is the T x G matrix of endogenous variables, Xi and Zi are T x kl and T x k2 matrices of the exogenous variables of the ith cross-sectional unit, respectively; Bi, Ri, and fi are G x G, kl x G, and k2 x G matrices of coefficients; and Ui is a T x G matrix of errors. We assume that Bi and 4 are matrices of fixed constants, and Ri is random with Ri =
R+
(3.28)
Cip
where J? and &iare kl x G matrices of fixed constants and random variables, respectively, with E(ei) = 0, and the variancecovariance matrix of Eidefined by COV(&i)= E [(vet ei)(vec ei)‘] = CZ,
vi.
(3.29)
The reduced form of (3.27) is of the form Yj = XiIZil + Zis?7i, + 6,
(3.30)
where
iii,
= ly;‘.
(3.31)
Then, nia is fixed and nil is random with mean I?ii and variance-covariance matrix of the form COV($)
=
E[vec($)vec($)‘]
= Ci*,.
(3.32)
Ignoring the possible restrictions on pi,, ITizp and Cz, (3.30) is in the form of a multivariate regression model analogous to (2.1’). Considering one equation of (3.30) at a time, the results of this section can be applied straightforwardly. Of course, one can also derive the analogous results for the multivariate regression model (3.30) directly by following the procedures outlined here. If one wishes to take into account the prior restrictions on r’iil, Z7iz, and Cg explicitly, restricted reduced form estimators will have to be derived. The computation, though, can be very laborious.
74
C. Hsiao et al., General framework for panel data models
If the interest is in the structural form parameters, a computationally manageable approach appears to take the limited information method conditioning on the estimated reduced form parameters. More specifically, consider the first equation in the model (3.27) Yil
=
FlBil
+
Xilail
+
Zil
yil
+
Uil)
i=
l,...,N,
(3.33)
where yil and x denote the TX 1 and T x Cl matrices of the included endogenous variables with the coefficient Of yii set equal to one; Xii and Zil are T x kr and T x kt included exogenous variables, respectively; Uil denotes the TX 1 vector of disturbance term; Bil, ail, and yil are G, x 1, k: x 1, and k: x 1 vectors of coefficients with pii and yii fixed and ail random. Let the reduced form of x1 be expressed as
Substituting (3.34) into (3.33), we have Yil =
Xilail
+
zilYil
+
El/&l
+
Vi13
(3.35)
where vii = uil + Kl Bii. Conditional on the estimated fli: and I?i*,we can treat ZI, and 6, as the set of exogenous variables with fixed coefficients and Xii as the set of exogenous variables with random coefficients, and utilize the results of this section. Of course, one should keep in mind that the resulting Bayes estimator is a conditional posterior mean given the estimated fiz and ni*,.
4. A point-to-point
telephone service demand model
When aggregate data are used to estimate the demand for long distance telephone service, a common specification is to regress quantity on price, income, and other socio-demographic variables [e.g., Appelbe et al. (1991), de Fontenay et al. (1990), Taylor (1980)]. When panel data are used, because they contain information on specific routes, we can model a unique feature of telephone service. That is, a long distance call requires the interaction of two economic agents, both of whom derive utility from it, but only one, the originator, bears the cost. However, an implicit repayment agreement may exist in the form of an obligation on the part of the receiver of the call to make a return call at some future time.’ The joint consumption nature of calling is further illustrated by the observation that it is not primarily from the call itself that agents derive utility, but ‘See Larson, Lehman, and Weisman (1990) for details on the theoretical underpinnings of this type of model.
C. Hsiao et al., General framework
for panel data models
15
rather from the exchange of information facilitated by the call. Thus, for a route with endpoint areas A and B, an agent in A derives utility from calls originating in both A and B, and similarly an agent in B derives utility from calls originating in both areas. As a result of these demand characteristics, it is assumed that calling in one direction affects return calling through a reciprocal calling effect. Hence, the demand function of an individual in A for calls from A to B includes calls from B to A as an argument along with the usual determinants of demand such as price and income. Analogously, the agent in B has calls from A to B in his demand function for calls from B to A. Therefore, a pair of demand equations from area A to B and area B to A are postulated: 1%
QA
=
uOA
+
log
QB
=
BOB
+
+
&Al%
pA
a,A1og~A
+
JIB
+
log
+
a2A
USADA
PB
+
b2s
1%
+
log
XA
+
a3A
1%
Qr, (4.1)
#A,
XB
hi 1%ZB + ~SB DB + %,,
+
b3B
log
QA (4.2)
where QA denotes the demand from A to B, PA denotes real price from A to B, XA denotes real income in A, ZA denotes other socio-economic factors such as market size, postal strikes, etc. in A, QB denotes the demand from B to A, PB denotes the real price from B to A, Xa denotes real income in B, Za denotes other factors in B (including market size, postal strikes, etc.), DA and DB denote the seasonal dummies, and UAand aa are the unobservable error terms in eqs. (4.1) and (4.2), respectively. The point-to-point panel data model (4.1) and (4.2) allows us to derive much more information than aggregate models and open up new possibilities for the evaluation of the impacts of price changes on demand and revenues. For example, it provides unique information on uni-directional price elasticities (the elasticity of demand for company A with respect to a price change for traffic originating in company A), cross-price elasticities between companies (the elasticity of demand for company A with respect to a price change for traffic originating in company B), bi-directional price elasticities (the elasticity of demand for company A with respect to simultaneous equal percentage price changes for traffic originating in both company A and company B), and the reciprocal calling coefficient (the responsiveness of calling from company A to company B due to a change in calling from company B to company A).2 Elasticities may be similarly defined with respect to other determinants of demand such as income and market size. With this additional information, the ‘For
detailed
derivation,
see Appelbe
et al. (1992)
76
C. Hsiao et al., General framework for panel dara models
separate effects of a price change for traffic originating in company A on traffic from A to B and from B to A may be calculated using the appropriate uni-directional and cross-company elasticities. Or, in the case of an across the board price change such as the recent Canadian federal goods and services tax (GST), the bi-directional elasticity may be employed. To fit this basic model to panel data, we consider several different ways of modeling unobserved heterogeneity. These include: Model 1.
Fully Constrained Model.
responses
across
%A
a4A
=
bm
Model 2.
=
We assume there are no differences in Thus, we have aOA= bOB, aIA = bIs, aZA = bzB, and asA = bsB, VA, B.
rOUteS. b4B,
Fully Unconstrained Model.
We allow the response coefficients to
be different across routes. Model 3. Generalized Variable Intercept Model (or Partially Constrained Model). We assume there are route-specific effects that do not vary over time.
In other words, we let the coefficients of the intercept and seasonal dummies vary across rOUteS, but we COILStrain alA = blB, aZA = bzB, asA = baB, VA, B. In some cases, we let the coefficients of some of the variables (e.g., market size) differ across cross-sections to take into account regional effects such as different company sizes, geographic characteristics, and socio-economic factors. Model 4. Mixed Fixed and Random CoefJicients Model. We invoke the representative consumer argument in assuming that consumers respond in more or less the same way to price and income changes. The coefficients of these variables across routes are considered random draws from a common population with constant mean and variance-covariance matrix. However, we also assume that there are route-specific effects, and these effects are more appropriately captured by fixed and different coefficients for route-specific seasonal dummy variables and, occasionally, the market size variable.
5. Estimates of the demand for Canadian long distance service We use data on the demand for customer-dialed long distance service (DDD) between the nine major telephone companies in Canada. These are British Columbia Telephone Company (BCT), Alberta Government Telephones (AGT), Saskatchewan Telecommunications (SASK), Manitoba Telephone System (MTS), Bell Canada (BELL), New Brunswick Telephone Company (NBT), Maritime Telegraph and Telephone Company (MTT), The Island Telephone Company (ITC), and Newfoundland Telephone Company (NTC). Bell is split into Ontario region and Quebec region.
C. Hsiao et al., General framework
77
for panel data mo&ls
Thirty-two of the possible point-to-point intercompany routes are modeled. The various modeled routes were divided into three groupings. These pointto-point groupings were formed by considering the average mileage of the calls made over a specific route. Group 1 (short haul) has the majority of its calls in the O-680 miles range and has an average haul length of approximately 340 miles. Group 2 (medium haul) has the majority of its calls in the 291-1200 miles range and has an average haul length of approximately 775 miles. Group 3 (long haul) has the majority of its calls over 920 miles and has an average haul length of approximately 1990 miles. The routes forming the three groups are: Group 1: BCT-AGT, AGT-BCT, AGT-SASK, SASK-AGT, SASK-MTS, MTS-SASK, QUE-NBT, NBTQUE, NBT-MTT, MTT-NBT, MTT-ITC, ITC-MTT; Group 2: BCT-SASK, SASK-BCT, AGT-MTS, MTS-AGT, MTS-ONT, ONT-MTS, ONT-NBT, NBT-ONT, ONT-MTT, MTT-ONT; Group 3: BCT-QUE, QUE-BCT, BCT-MTS, MTS-BCT, BCT-ONT, ONT-BCT, SASK-ONT, ONT-SASK, ONT-NTC, NTC-ONT. The traffic is further divided according to full-rate (peak) and discount-rate (off-peak). The full-rate category consists of calls placed between 8 a.m. and 6 p.m. Monday to Saturday. The discount-rate category consists of all other calls. The quantity demanded is represented by price deflated revenues (PDR). The explanatory variables include own price, the level of income of economic activity, the size of the market, seasonality, special or atypical events (i.e., postal strikes and data problems), and traffic in the reverse direction for each route. The price variable is a chained Laspeyres index deflated by the relevant Consumer Price Index (CPI). Income is represented by Total Wages and Salaries or Retail Sales deflated by CPI or Employment. Market size is represented by the number of telephone lines in service (NAS) in the originating region only. The reciprocal calling variable is represented by the price deflated revenues for the traffic in the reverse direction of a specific route. We use quarterly data from 1980.1 to 1989.IV to estimate the various versions of the basic panel data model C(4.1)and (4.2)]. However, because (4.1) and (4.2) is a pair of simultaneous equations, the full information approach of deriving the route-specific coefficients estimates for the mixed fixed and random coefficients model (model 4) can be computationally unwieldy. Therefore, we take a limited information approach as outlined in remark 5 of section 3. That is, we substitute the reduced form estimates of log QB and log QA into the righthand side of (4.1) and (4.2), respectively, and treat the resulting system as a seemingly unrelated regression model. While this procedure may not be necessary for other models, it is also followed to facilitate comparison. The error terms UAitand aair are assumed to follow the pattern
UAit
=
PAUAi,t-
1 +
&Air,
UBir
=
PBUBi,t-1
+
&Bit,
(5.1)
78
C. Hsiao et al., General framework for panel data models
where &Aitand &Bitare assumed to be independently distributed over time, but are COntemporaneOuSly correlated, that is, E(&Air&Bjt) = Iterations between the regression coefficients and error term parameters are applied until the estimates converge. Results for the price and income coefficients of the off-peak models are reported in tables l-3. For confidentiality reasons, the routes reported in these tables are not in the same order as listed in paragraph 3 of this section. We note that the estimates of price and income coefficients are indeed sensitive to the way the unobserved heterogeneity is being modelled. For instance, the point estimates of the short-, medium-, and long-haul off-peak price coefficients are - 0.01, - 1.1, - 0.38, respectively, for fully constrained DABije
Table 1 Short-haul regression coefficients.” Fully constrained
Partially constrained
Unconstrained
Mixed coefficients
Price 1 2 3 4 5 6 7 8 9 10 11 12
-
0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102
Average
- 0.0102
(- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28)
-
0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38)
-
0.1203 ( 0.34) 0.2915 ( 0.98) 0.4910 ( - 3.87) 0.0069 ( - 0.03) 0.0634 ( - 0.43) 1.1448 ( - 4.72) 0.3697 ( - 3.48) 0.8745 ( - 4.40) 0.2372 ( - 2.14) 0.1728 ( - 2.05) 0.3670 ( - 6.33) 0.6379 ( - 4.69) N/A
- 0.3091
- 0.0385 (N/A) 0.0659 (N/A) - 0.4684 (N/A) - 0.2452 (N/A) - 0.0993 (N/A) - 0.4479 (N/A) - 0.2926 (N/A) - 0.2618 (N/A) - 0.2421 (N/A) - 0.2447 (N/A) - 0.4294 (N/A) - 0.4308 (N/A) - 0.2612
Income 1 2 3 4 5 6 7 8 9 10 11 12
0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906
Average
0.0906
( ( ( ( ( ( ( ( ( (
( (
3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40)
at-statistics are in parentheses.
0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028
24.81) 24.81) 24.8 1) 24.8 1) 24.81) 24.81) 24.81) 24.8 1) 24.81) 24.81) 24.81) 24.81)
0.65) 0.2168 ( 0.38) 0.0693 ( 5.18) 0.6144 ( 0.16) 0.0313 ( 2.52) 0.3188 ( 4.59) 1.3680 ( 2.52) 0.4308 ( - 0.1451 ( - 1.08) 0.4383 ( 6.63) 2.25) 0.2413 ( 0.3691 ( 12.13) 0.7948 ( 13.30)
WA
0.2895 0.1457 0.5406 0.2482 0.3390 0.4156 0.3064 0.2709 0.3947 0.3492 0.4105 0.3858 0.3413
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
C. Hsiao et al.. General framework for panel dara models
79
Table 2 Medium-haul regression coefficients. Fully constrained
Partially constrained
Mixed coefficients
Unconstrained
Price 1 2 3 4 5 6 7 8 9 10
-
l.lOOO(l.lOOO(l.lOOO(l.lOOO(1.1000 (1.1000 (l.lOOO(l.lOOO(l.lOOO(l.lOOO(-
Average
- 1.1000
32.17) 32.17) 32.17) 32.17) 32.17) 32.17) 32.17) 32.17) 32.17) 32.17)
- 0.2147 - 0.2147 - 0.2147 - 0.2147 - 0.2147 - 0.2147 -L 0.2147 - 0.2147 - 0.2147 - 0.2147
( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30)
- 0.2147
0.0534 - 0.0758 0.0376 - 0.9261 - 0.3328 0.1713 - 0.9495 - 0.8326 - 0.1955 - 0.4106
( ( ( ( ( ( ( (
0.35) - 0.34) 0.13) - 3.21) - 3.23) 1.02) - 4.47) - 3.71) ( - 2.23) ( - 3.70)
-
0.0113 0.1827 0.0391 0.4698 0.1888 0.0498 0.7456 0.6994 0.2014 0.4345
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
- 0.3000
N/A Income
1 2 3
Average
0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244
( ( ( (
( ( ( ( (
(
6.71) 6.71) 6.71) 6.71) 6.71) 6.71) 6.71) 6.71) 6.71) 6.71)
0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779
( ( ( ( (
( ( ( ( (
14.47) 14.47) 14.47) 14.47) 14.47) 14.47) 14.47) 14.47) 14.47) 14.47)
1.1559 ( 0.2382 ( 0.7090 ( 1.1264 ( 0.3084 ( 0.2810 ( 0.7673 ( 0.1274 ( 0.3566 ( 0.1475 ( N/A
5.47) 1.43) 2.84) 3.46) 3.49) 1.98) 4.98) 1.10) 10.51) 0.95)
0.4956 0.4081 0.6557 0.6244 0.3722 0.4346 0.5803 0.1118 0.3620 0.2560
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
0.4301
model (model 1) and - 0.309, - 0.215, - 0.235, respectively, for partially constrained model (model 3). This suggests that perhaps there are route-specific effects which should not be ignored. In fact, tests for the homogeneity of the seasonal dummy coefficients across the routes are rejected by the conventional F-statistics. A further test of the homogeneity of the price and income coefficients across routes was also rejected by the data. Therefore, a fully unconstrained model (model 2) is estimated. Unfortunately, because of the multicollinearity among the variables and the shortage of degrees of freedom, the individual route estimates are quite diverse. Many of them are either not statistically significant and/or have the wrong signs. Column 4 of tables 1-3 reports the alternative estimates when the price and income coefficients are assumed to be randomly distributed with common mean and constant varianceecovariance matrix (model 4). It is apparent from the comparison of the estimates between models 2 and 4 that the estimates of model 4 are much more concentrated around the
80
C. Hsiao et al., General framework for panel data models
Table 3 Long-haul Fully constrained
regression
coefficients.
Partially constrained
Mixed coefficients
Unconstrained
Price 1 2 3 4 5 6 7 8 9 10
-
Average
- 0.3848
0.3848 0.3848 0.3848 0.3848 0.3848 0.3848 0.3848 0.3848 0.3848 0.3848
1 2 3 4 5 6 7 8 9 10
0.5433 0.5433 0.5433 0.5433 0.5433 0.5433 0.5433 0.5433 0.5433 0.5433
Average
0.5433
( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78)
-
0.2360 0.2360 0.2360 0.2360 0.2360 0.2360 0.2360 0.2360 0.2360 0.2360
( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84)
- 0.2360
( ( ( ( ( ( ( ( ( (
22.59) 22.59) 22.59) 22.59) 22.59) 22.59) 22.59) 22.59) 22.59) 22.59)
0.4076 0.4076 0.4076 0.4076 0.4076 0.4076 0.4076 0.4076 0.4076 0.4076
- 0.0712(0.1694 ( - 1.0142 (- 0.4874 (- 0.3190 (0.0365 ( - 0.3996 (- 0.1033 (- 0.3965 (- 0.6187(-
0.15) 0.44) 5.22) 2.29) 2.71) 0.20) 3.92) 0.95) 4.22) 4.82)
11.31) 11.31) 11.31) 11.31) 11.31) 11.31) 11.31) 11.31) 11.31) 11.31)
0.4076
1.4301 - 0.0348 0.3698 0.2497 0.5556 0.1119( 0.9197 0.3886 0.6688 0.1928
N/A
0.2875 0.0220 0.7743 0.1686 0.2925 0.0568 0.3881 0.2504 0.2821 0.5934
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
- 0.3116
N/A
( ( ( ( ( ( ( ( ( (
-
( (( ( ( ( ( ( (
3.07) 0.09) 1.95) 0.70) 2.71) 0.95) 8.10) 3.88) 6.16) 2.39)
0.4740 0.2679 0.3394 0.3145 0.3501 0.1344 0.5342 0.5255 0.5648 0.2574
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
0.3762
mean, and many of the routes that have unreasonable unconstrained estimates now have the correct sign.3 6. A prediction comparison We have seen in the last section that the parameter estimates are very sensitive to the way the unobserved heterogeneity is modelled. To select a representative formulation for the demand for Canadian long distance telephone service we note that ‘a severe test for an economic theory, the only test and the ultimate test is its ability to predict’ [Klein (1988, p. 21)] and ‘the prediction principle that predictive performance is central in evaluating hypothesis is of key importance
3This is because, as pointed model is the weighted average
out in remark 2 of section between the unconstrained
3, the individual estimates of the mixed estimates and the overall mean.
81
C. Hsiao et al., General framework for panel data models
in econometrics’ [Zellner (1988, p. 31)]. We therefore provide a predictive performance comparison of the four models in this section. We use the data from 1988.1 to 1990.111 to evaluate the root mean square percentage prediction error,
of the four models for each route. The predicted value is generated from the reduced form of (4.1) and (4.2) by treating the estimated structural form parameters as if they were true and explicitly taking account of the first-order autoregressive structure of the error terms. The results are reported in tables 4-6. In general the fully constrained model performs very badly. That is, the homogeneity across routes assumption is not supported by the data. As for the prediction performance of the other three models, the results vary. However, it does appear that on average the mixed model (model 4) has a slight edge. In addition, we note: (i) The unconstrained coefficients for seasonal dummies are much more spread out than the price and income coefficient. Models 2, 3, and 4 all allow the coefficients of seasonal dummies to vary across routes. (ii) Model 4 estimates of price and income coefficients for each route appear to be more reasonable on a priori grounds than fully unconstrained estimates (model 2). (iii) Similarity of the constrained price and income coefficient estimates of model 3 and their respective mean estimates from model 4. (iv) Under a mixed fixed and random coefficients formulation (model 4), in principle, we
Table 4 Short-haul
root mean square
percentage
prediction
error.
Fully constrained
Partially constrained
1 2 3 4 5 6 7 8 9 10 11 12
0.10826 0.07539 0.09788 0.07222 0.10707 0.10357 0.07358 0.08434 0.06095 0.09667 0.11890 0.07325
0.10280 0.03288 0.03565 0.07217 0.06121 0.04126 0.05825 0.05572 0.10561 0.10209 0.03322 0.06256
0.04371 0.04085 0.02508 0.02772 0.05982 0.04862 0.04576 0.03640 0.08456 0.08598 0.04524 0.10759
0.06808 0.03282 0.02749 0.05285 0.09564 0.05355 0.05766 0.05118 0.10481 0.10321 0.06457 0.03958
Average
0.08934
0.06362
0.05428
0.06262
0.08748
0.06590
0.07272
0.06540
Revenue-weighted
average
Unconstrained
Mixed coefficients
82
C. Hsiao et al., General framework for panel data models
can generate predictions using the overall mean 6 rather than using individual ai. A priori, one would expect that these predictions on average should perform reasonably well. In this sense, we may view the prediction generated by model 3 as using ti instead of ai, except that under the mixed coefficients framework the h is an inefficient estimate. Tables 4-6 show that the difference in predictive performance between models 3 and 4 is on average not that large, except for the long-haul off-peak model. In view of these observations, we would be in favor of selecting a mixed fixed and random coefficients formulation in which the coefficients of price and income variables are assumed to be random with
Table 5 Medium-haul
root mean square
percentage
prediction
error.
Fully constrained
Partially constrained
1 2 3 4 5 6 I 8 9 10
0.05244 0.10361 0.08650 0.04167 0.09534 0.07910 0.11093 0.08284 0.09650 0.15154
0.07276 0.05566 0.03421 0.02623 0.04215 0.05730 0.10649 0.06750 0.12336 0.13097
0.04858 0.02937 0.04173 0.11346 0.04814 0.10088 0.09737 0.05463 0.11992 0.10765
0.09223 0.0603 1 0.04570 0.04755 0.04369 0.06124 0.10073 0.05348 0.12347 0.11940
Average
0.09005
0.07166
0.07617
0.07478
0.08723
0.06573
0.07562
0.07039
Revenue-weighted
average
Unconstrained
Mixed coefficients
Table 6 Long-haul
root mean square
percentage
prediction
error. Mixed coefficients
Fully constrained
Partially constrained
1 2 3 4 5 6 7 8 9 10
0.11236 0.07635 0.15800 0.06146 0.05981 0.09863 0.10154 0.13085 0.05136 0.12980
0.05764 0.00966 0.09175 0.07964 0.06273 0.10256 0.07203 0.08542 0.12341 0.09124
0.15916 0.07221 0.07625 0.03852 0.05282 0.11077 0.04639 0.02724 0.03882 0.04634
0.01744 0.08550 0.08579 0.06567 0.08538 0.13918 0.02608 0.03406 0.07537 0.07347
Average
0.09802
0.07761
0.06685
0.06879
0.10612
0.07834
0.05443
0.05220
Revenue-weighted
average
Unconstrained
C. Hsiao el al., General framework for panel data models
83
a common mean to invoke a representative consumer argument and the coefficients of seasonal dummies to be fixed and different across routes to capture the fixed route-specific effects.
7. Conclusions In this paper we provide a unified framework to link various panel data models. A Bayes solution is derived which can be specialized to many specific estimators discussed in the literature. We illustrate the application of this framework by estimating a point-to-point Canadian long distance telephone service demand model. It does appear that the approach offers wide possibilities for controlling unobserved heterogeneity. However, many issues remain unresolved. In particular, the choice between random versus fixed parameters formulations and a more manageable solution for simultaneous equations models.
Appendix
In this appendix we sketch the derivation of Theorem 3.2. Let y* = c;‘Py,
x* = c;‘Px,
z”* = c-r/zz, 1
** = c;‘/Zu,
(A-1)
and f* = [I _ ~*(~*~~*)-‘z*~]x*.
(A.2)
We can rewrite (2.1’) as y* =
_f*a+ z”*7 + u*,
(A.3)
where z”*1d* = 0,
(A.4)
jj = ji + (Z*‘Z”*)-‘Z”*‘X*a,
(A.5)
and Eu* = 0,
Eu*u*’ = 1.
(A@
By Bayes theorem,
(A-7)
84
C. Hsiao et al., General framework for panel data models
where, under A.2-A.3, P(a) cc P(a 1G)P(ii).
(A.8)
The product on the right-hand where Q is given by
side of (A.7) is proportional
to exp{ - f Q>,
Q = QI + Q2 + QS + Q4,
(A.9)
with Q1 =
{a- (z*‘x”* + C;')-'(r?*'y* x
+
{a- (r?*Tz* + c;y’(f*‘y*
~&+i)}yrl*~r?*
+ C;‘Ala)},
(z”*rz”*)-‘z”*ry*]‘(~*‘z”*)[jj
Q2 = [jj -
+
c;‘)
(A.lO)
- (z”*‘z*)-‘z”*‘y*],
(A.11)
Q3 = {a - (A;[C2 + (ris*‘x*)-‘]-‘z41)-1 x A;C;‘(X*‘T*
+ CT’)_‘(X*‘y*)}’
x {A; [C, + (r?*‘r?*)-‘]-‘A,} x {a - (A;[& x Q4
+[I
+ (Jz*‘8*)-‘]-541)-’
A;C;‘(2*‘X”* + c;‘)-‘(d*ry*)}, _
X”*($Wd*
+
_ Z*$*rZ*
,;l)x”*’
_
(A.12)
2*(2*+)-‘2”W
+ c;‘)-lc;‘A,
+
(r?*9z*)-‘]-‘L41}-’
x
{A;[C,
x
A;C;‘(2*‘2*
+ c;yr7*qy*.
As far as the distribution of a, 7, and d is concerned, Conditional on 6, Q3 is a constant. Therefore, conditional distribution of a and y”is proportional to exp{ - f(Ql + posterior distribution of a and p given a is a multivariate (3.11) and jj* = (z”*~z”*)-‘z”*Y*,
(A. 13) Q4 is a constant. on E, the posterior Q2)}. That is, the normal with mean
(A. 14)
C. Hsiao et al.. General framework for panel data models
respectively, and variance-covariance
85
matrix (A.15)
where &
=
(z*‘z”*)-
l.
(A.16)
Substituting (AS) into (A.14) and (A.16), we have (iii). Substituting (A.5) and (3.11) into (A.14)-(A.16), we have (i) and (iv). To obtain the unconditional distribution of a and 7 given y, we rewrite the formulation of (A.9)-(A.13) as (A.17)
Q = QT + Q2 + Q: + Q4, with
Q; =
{a- [x*7?* + C;' -
x
[xI*‘x”* + c;’
x
{a - [8*9*
C;',41(~;C;'~l)-'
- c,‘ll,@l;
c;‘A,)-‘A;
C,‘]
+ CT’ - c;‘A,(A;C;‘A,)-’
xA;C;‘]-‘(X”*‘y*)),
(A.18)
Qf = [a - (A;C;'A,)-'A;C;'a]'(A;Cz'Al) x [6 - (A;C;‘A,)-‘A’,C;‘a], Q4
=/{I
_
-
2”*(+2*)-‘2W
(A.19) _
d*[zWd*
+
C;’
c;‘Al(A;C;‘Al)-‘A;C;‘]-‘~*‘}y*.
(A.20)
Integrating exp { - $ Q} with respect to ti and 7, we have (A.21) hence (ii). Substituting (3.15) and (3.16) into (A.5) we have (v).
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C. Hsiao et al., General framework for panel data models
References Appelbe, T.W., C.R. Dineen, D.L. Solvason, and C. Hsiao, 1992, Econometric modelling of Canadian long distance calling: A comparison of aggregate time series versus point-to-point panel data approaches, Empirical Economics 17, 125-140. Balestra, P. and M. Nerlove, 1966, Pooling cross-section and time series data in the estimation of a dynamic model: The demand for natural gas, Econometrica 34, 585-612. DeFinetti, B., 1964, Foresight: Its logical laws, its subjective sources, in: H.E. Kyburg, Jr. and H.E. Smokler, eds., Studies in subjective probability (Wiley, New York, NY) 93-158. DeFontenay, A., M. Shugard, and D. Sibley, 1990, Telecommunications demand modelling: An integrated view (North-Holland, Amsterdam). Ghali, K., 1988, Econometric modelling of the United States regional electricity demand under peak-load pricing, Unpublished Ph.D. dissertation (University of Southern California, Los Angeles, CA). Hsiao, C., 1974, Statistical inference for a mode1 with both random cross-sectional and time effect, International Economic Review 15, 12-30. Hsiao, C., 1975, Some estimation methods for a random coefficients model, Econometrica 43, 305-325. Hsiao, C., 1985, Benefits and limitations of pane1 data, Econometric Reviews 4, 121-174. Hsiao, C., 1986, Analy is of panel data (Cambridge U iyersity Press, New York, NY). Hsiao, C., 1990, A m’ ed fixed and random oefficien s framewor f r pooling cross-section and time series data, Paper pre L &&I at the hird Conference o I-!&#lecommunications Demand c\ t Analysis with Dynamic Regulatio (Hilt ! n Head, SC). Hsiao, C., 1991, Panel analysis for meI ric data, in: Cl. Arminger, C.C. Clogg and M.E. Sobel, eds., Handbook of statistical modelling in the social and behavioral sciences (Plenum Press, New York, NY) forthcoming. Hsiao, C., D.C. Mountain, K.Y. Tsui, and M.W. Luke Chan, 1989, Modeling Ontario regional electricity system demand using a mixed fixed and random coefficients approach, Regional Science and Urban Economics 19, 567-587. Klein, L.R., 1988, The statistical approach to economics, Journal of Econometrics 37, 7-26. Kuh, E., 1963, Capital stock growth: A micro-econometric approach (North-Holland, Amsterdam). Larson, A.C., D.E. Lehman, and D.L. Weisman, 1990, A genera1 theory of point-to-point long distance demand, in: de Fontenay, A Shugard, and D. Sibley, eds., Telecommunications demand modelling (North-Holland, Amsterdam). Lindley, D.V. and A.F.M. Smith, 1972, Bayes estimates for the linear model, Journal of the Royal Statistical Society B 34, 1-41. Nerlove, M., 1965, Estimation and identification of Cobb-Douglas production functions (Rand McNally, Chicago, IL). Swamy, P.A.V.B., 1970, Efficient inference in a random coefficient regression model, Econometrica 38, 31 l-323. Swamy, P.A.V.B., 1971, Statistical inferences in random coefficient regression models (SpringerVerlag, Berlin). Swamy, P.A.V.B., 1974, Linear models with random coefficients, in: P. Zarembka, ed., Frontiers in econometrics (Academic Press, New York, NY) 143-168. Tavlor. L.. 1980. Telecommunications demand: A survev and critique (Ballinaer. Cambridge, MA). dels of demand,fq custom&-dialed Tefecom Canada Demand Canada, Hull). Canada-Canada and CanadaWallace, T.D. and A. Hussain, 1969, The use of error components models in combining cross-section with time series data, Econometrica 37, 55-72. Zellner, A., 1962, An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias, Journal of the American Statistical Association 57, 348-368. Zellner, A., 1971, An introduction to Bayesian inference in econometrics (Wiley, New York, NY). Zellner, A., 1988, Bayesian analysis in econometrics, Journal of Econometrics 37, 27-50.
A general framework for panel data models with an application to Canadian customer-dialed long distance telephone service * Cheng Hsiao University of California. Irvine, CA 92717, USA University of Southern California. Los Angeles, CA 90089, USA
Trent W. Appelbe and Christopher
R. Dineen
Telecom Canada, Hull, Qu.4.. Canada
We provide a general framework that covers many panel data models as special cases. A Bayes solution to this general framework is derived, and its relation to many widely used estimation and prediction formulae is discussed. Canadian long distance telephone service data are used to illustrate the effects of different ways of controlling unobserved heterogeneity across routes on estimation and prediction.
1. Introduction It is well known that blending characteristics of both cross-sectional and time series data allows us to specify more complicated behavioral hypotheses than could be done by using time series or cross-sectional data alone. Furthermore, panel data offer many more degrees of freedom, and also allow us to control for omitted variable bias and reduce the problem of multicollinearity, hence improving the accuracy of parameter estimates [e.g., Hsiao (1985, 1986, 1991)]. However, an important issue in panel data analysis is how best to control for unobserved individual heterogeneity to avoid biasing the coefficient estimates of
Correspondence to: Cheng Hsiao, Department of Economics, University of Southern California, Los Angeles, CA 90089, USA. * Cheng Hsiao’s work is supported in part by National Science Foundation Grants SES 88-21205 and SES 91-22481 and in part by Irvine Faculty Research Fellowship. We wish to thank Frank Peracchi and a referee for helpful comments and Dennis Bruce and Dave Robinson for empirical work.
0304-4076/93/$06.00 0
1993-Elsevier
Science Publishers B.V. All rights reserved
C. Hsiao et al., General framework for panel data models
64
included explanatory variables and improve the efficiency of parameter estimates. In a linear regression framework, several models have been proposed. These include: (i) a common model for all temporal cross-sectional observations, (ii) different models for different cross-sectional units, (iii) variable intercept models [e.g., Kuh (1963), Nerlove (1965)], (’tv ) error components models [e.g., Balestra and Nerlove (1966), Wallace and Hussain (1969)], (v) random coefficients models [e.g., Hsiao (1974, 1975), Swamy (1970, 1971, 1974)], and (vi) mixed fixed and random coefficients models [e.g., Ghali (1988), Hsiao et al. (1989)], etc. In this paper we provide a basic framework that unifies these models. We also combine route-specific temporal data to estimate the price and income elasticities of demand for Canadian customer-dialed long distance telephone service. In section 2 we set up a unified framework. In section 3 we provide a Bayes solution to the general model which encompasses many of the well-known estimators for each of the special cases. We specify a Canadian customer-dialed long distance telephone service demand model that takes advantage of the information provided by panel data in section 4. Empirical estimates are reported in section 5 and prediction comparisons are provided in section 6. Conclusions are in section 7.
2. A general framework Suppose there are observations of 1 + ki + k2 variables (yit, &, L&) of N cross-sectional units over T time periods, where i = 1, . . . , N and t = 1, . . . , T. Let
z NTx NkZ
=
C. Hsiao et al., General framework for panel data models
65
where
i=l
, . . . . N.
We assume that
(2-l)
y=Xa+Zy+u,
where a and y are Nkl x 1 and Nkz x 1 vectors, respectively,
(2.2) and uf = (Uil, . . . , UiT), We let the NT x NTcovariance matrix C1 be unrestricted to allow for correlations across cross-sectional units and over time. We further assume that the Nkl x 1 vector a satisfies
(2.3)
where ai is a k, x 1 vector, i = 1, . . . , N, A, is an Nkl x m matrix with known elements, 2 is an m x 1 vector of constants, and E -
N(0, C,).
(2.4)
The variance-covariance matrix C2 is assumed to be nonsingular. The Nk2 x 1 vector of y is assumed to satisfy 71 Y=
:
0
YN
=
A,?,
(2.5)
66
C. Hsiao et al.. General framework
for panel data models
where each yi is k2 x 1, A2 is an Nkz x n matrix with known elements, and r is an n x 1 vector of constants. Because A2 is known, (2.1) is formally identical to y=Xa+@+u,
(2.1’)
where 2 = ZAz. However, by postulating (2.5) we allow for various possible fixed parameter configurations. For instance, if we wish to allow for yi to be different across cross-sectional units, we can let A2 = IN @ Ik2, where I, denotes a p x p identity matrix. On the other hand, if we wish to constrain yi = yj, we can let A2 = eN @ Ik2, where eN is an N x 1 vector with each element equal to one. Many of the widely used panel data models that assume that the unobserved heterogeneity is individual-specific but time-invariant, can be treated as special cases of the model (2.1)-(2.5). These include: (i) A common model for all cross-sectional units If there is no interindividual difference in behavioral patterns, we may assume that the regression coefficients are identical for all cross-sectional units at all times. Thus, we have Yit
=
Z1* 7+
i=
Uit,
l,...,
N,
T.
t=l,...,
(2.6)
This model can be obtained from (2.1’) by letting X = 0 and A2 = (Zk,, . . . , Zk2)‘, where Ik, is a k2 row identity matrix. (ii) Diflerent models for diflerent cross-sectional units When each individual unit is viewed as different, we can write the regression model in the form Yit
=
ZftYi
+
i = 1, . . . . N,
Uitr
t = 1, . . . , T,
(2.7)
with yi different for different individuals. Model (2.7) can be obtained from (2.1’) by letting X = 0 and A2 be an Nkz x Nkz identity matrix. (iii) Variable intercept models [e.g., Kuh (1963)] The variable intercept models assume that, conditional on the observed explanatory variables, there are interindividual differences that stay constant through time. These interindividual differences can be captured by allowing the intercept terms of a regression model to vary with i; thus, we have yiz =
yli
+
y2zic2
+
“’
+
Yk,Zirk,
+
uit,
i=
l,...,N.
(2.8)
C. Hsiao et al., General framework for panel data models
67
Model (2.8) can be obtained from (2.1)-(2.5) by letting X = 0, the first column of Zi be a T x 1 vector of ones, eT,
P= (Yll ,
. . . . “tlN, 72,
a.3 3 ?k2),
where
I,:_, = k, x (k2 - 1)
(iv) Error components models [e.g., Balestra and Nerlove (1966), Wallace and Hussain (1969)] When the effects of the omitted variables that reflect individual time-invariant differences are treated as random variables (just like the assumption on other components of the random disturbance term of a regression model), we have the error components model Yit = Gt? +
clli
+
(2.9)
42,
with (2.10) if Zit does not include the common intercept term for all cross-sectional units, and (2.11)
EBri = 0 if Zit includes the common intercept term, and
Etali - Eali)talj - Ealj) =
i=j 2’ ifotherwiie
We can obtain (2.9) from (2.1)-(2.4) by letting Xi = eT, a’ = (all, Al
=
(2.12)
. . . , aIN),
eN?
c2 = E&E’= Cr,Z, IN,
(2.13)
68
C. Hsiao et al.. General framework for panel data models
and either rZi = 0 if zit contains the intercept term, or cl1 being an unknown constant if zic does not contain the intercept term. (v) Random coeficients models [Swamy (1970)]
When the heterogeneity among cross-sectional units cannot be captured completely by the time-invariant individual varying constants ali, while the specification of the relationships among variables appears proper given the available data, a natural generalization would be to let the parameters vary across cross-sectional units. In model (2.7) we allow the individual coefficients to be treated as fixed and different. We can also allow the regression coefficients to be treated as random variables with a probability distribution. The random coefficient specification reduces the number of parameters to be estimated substantially, while still allowing the coefficients to differ from unit to unit. Under the assumption that the explanatory variables and coefficients are independent, the Swamy (1970) type random coefficients formulation assumes that yi( =
Xi,(xi +
(2.14)
Uify
(2.15)
Eui = 6,
E(ai- 4(aj - a)’ =
A
if
o
otherwise
i=j,
,
(2.16) (2.17)
Exa(ai - a)’ = 0.
The Swamy type random coefficients model assumes that individual differences satisfy de Finetti’s (1964) exchangeability criterion. That is, the likelihood for the ith individual to have a particular realized value ui is the same as for the jth individual. The individual ai all have the same expected value. The subscript i is purely a labelling device with no substantive content. The observed difference between ai and aj is the work of chance mechanisms only. We can obtain model (2.14)-(2.17) from (2.1’) by letting 2 = 0, Al = eN @ Ik,, a=(a;
,...,
a;V),d=(al-Q
,...,
aN-ti),CZ=ZN@A.
(vi) Mixed jixed and random coejicients models [e.g., Hsiao (1990), Hsiao et al.
(19891 When some of the coefficients are treated as fixed constants and some are treated as random variables, we have the mixed fixed and random coefficients version of model (2.1’). In this case we assume that responses to changes in certain conditions are similar, but responses to changes in other conditions may be individual-specific.
C. Hsiao et al., General framework for panel data models
69
3. A Bayes solution To derive a general solution to the model (2.1)-(2.4), we shall take a Bayesian approach. We take model (2.1’) PS the basic regression model. We assume that the prior distributions of a and 7 are independent. We have no information on 7, hence a diffuse prior is assumed. But assumption (2.3) can be viewed as providing an informative prior on a, namely, a is normally distributed with mean Al 6 and varianceecovariance matrix Cz, although we may not have knowledge on 6. If there is no information on 6, the prior for 6 is assumed to be diffuse and is independent of the prior of 7. To put it formally, we assume: A.l.
The prior distributions of a and 7 are independent, that is, p(a, 7) = p(a) * P(
A.2. A.3.
7).
(3.1)
p(a) - N(A1& G).
(3.2)
There is no information about 5 and 7, that is, p(5) and p( 7) are independent and p(6) cc constant,
(3.3)
p( 7) cc constant.
(3.4)
Theorem 3.1.
Suppose that
y N N(XAlti Under A.I-A.3,
+ .@, Cl + X&X’).
(3.5)
the posterior distribution of ci and ji given y is
(3.6) where
(Cl + X&X’)-‘(XA,,
-I
.?)
1 (C, + X&X’)_ly,
v-1
=
(Cl
+X&X’)-‘(XA,,
(3.7) z”) .
1
(34
C. Hsiao et al., General framework for panel data models
70
The proof of this theorem follows straightforwardly of a linear regression model of
from the Bayes solution
y = XAla + z”y + u*,
(3.9)
with diffuse prior for a and 7, where (3.10)
u* = u + XE [e.g., Zellner (1971)]. Theorem 3.2.
Suppose that given a and f,
y - N(Xa + gf, Cl),
(3.11)
and given 6, a w N(A, a, C,). Then, under A.l-A.3, where
(3.12)
(i) the posterior distribution of a given 6 and y is N(a*, DI),
a* = {X’[C;’
- C~12(zlC~‘Z)-‘z”fC~‘]X
x (X’[C;’
- C;l~(z,C;lz”)-‘.?C;‘]y
Dl = {X’[C;’
- C;‘~(z”rC;‘z”)-‘z,C;l]X
+ CT;‘}-’ + C;‘AIE},
+ Cl’}+;
(3.13) (3.14)
(ii) the (unconditional) posterior distribution of a given y is N(a**, D:), where a** = DF*{X’[C;’
- c;1z”(zlc;‘Z)-‘~c;‘]y},
D; = {X’[C;’
- C;1z(z”‘C;‘2)-‘Z’C;‘]X
+ C;’
- C;‘Al(A;C;‘Al)-‘A;C;‘}-‘;
(3.15)
(3.16)
(iii) the posterior distribution of 7 given y and a is N(E DJ), where f=
(.?C;‘z)-‘Z’C;‘(y
6, = (pC;‘z)-1;
- Xa),
(3.17) (3.18)
C. Hsiao et al., General framework for panel data models
(iv) the posterior
distribution
f* = D:z”f([C;l
of 7 given y and ti is N( f*, OS), where
- C;‘X(X’C;‘X
- c;‘x(x’c;‘x 0;
= {z’[C;’
+ CT’)-‘X’C;‘]y
+ C;‘)-‘C;‘Ala),
- C;‘X(X’C;‘X
(v) the (unconditional)
posterior
-** = D;* S?{c;’ Y
(3.19)
+ C;‘)-‘X’C;‘]z)-‘;
distribution
off
- c;‘x[x’c;‘x
(3.20)
given y is N( 1)**, Dz*), where + c;’ (3.21)
- C;‘A1(A;C;‘Al)A;C;‘]-‘X’C;‘)-‘y, D:* = [_@C;’
11
- C;‘X[X’C;‘X
+ C;’ (3.22)
- C;‘Al(A;C;‘A~)-‘A;C;‘I-‘X’C;‘)ZI-’.
The derivation of these results are in the appendix. The Corollaq 3.1. **, Dz), where N(a Ii** =
(unconditional)
posterior
distribution
B2{A;C;1(X’[C;1 - c;‘z”(z”lc;‘2)-‘z’c;1]x - c;~z”(zlc;‘~)-‘z”,c;‘lu),
x x’[c;’ D2 = {A; cc;’ x
z’c,‘x
= {A;(& x
- c;‘(x’c;‘x
of ii given y is
+ CT’)_’ (3.23)
- xC;‘.z(~c;‘z”)-’
+ C;‘)-‘C;‘]Al}-l
+ [x’(c;’
-
2c;1)x]-‘)-‘Al}-‘.
c;1.2(.Fc;‘2)-’ (3.24)
The derivation of (3.23) and (3.24) follows straightforwardly from the joint distribution of 6 and jJ [(3.6)] is equal to the product of the conditional distribution of 7 given ti C(3.19) and (3.20)] and the marginal distribution of 6. Remark
1. The Bayes estimators of a, a, and 7 are derived conditional on Cl and C2. Of course, unconditional Bayes estimators can be derived by introducing the prior distributions of Cl and Cz. However, with unrestricted Cl and Cz, the computation can be burdensome with little gain in insight.
72
C. Hsiao et al., General framework
for panel data models
Therefore, in the event that Ci and Cz can be consistently estimated, we shall confine our analysis with the assumption that Ci and C2 are known. Remark 2.
In a classical framework, it makes no sense to predict the independently drawn random variable ai. However, in panel data, we actually operate with two dimensions - a cross-sectional dimension and a time series dimension. Even though ai is an independently distributed random variable across i, once a particular ai is drawn, it stays constant over time. Therefore, it makes sense to predict ai. The classical predictor of ai is the generalized least squares estimator of the model (2.1’). The Bayes predictor (3.13) or (3.15) is the weighted average between the generalized least squares estimator of a for the model (2.1’) and the overall mean A 1ti if 6 is known [ (3.13)] or A 1ti ** if ti is unknown, ,**
= D1(x’[c;’
-
c;‘z(z”lc;l~)-lz”~c~l]x~
+ C;‘A,a**}, (3.25)
where B = {x’[c;’ x {x’[c;’
- c;‘z(z”,c;‘2)-‘z”lc;‘]x}~ -
c;‘z”(z”‘c;‘z)-‘z”lc;‘ly).
(3.26)
[For details, see Hsiao (1991).] Remark 3. Many of the widely used panel data model estimators are special cases of the Bayes estimators. For instance, the generalized least squares estimator for the error components model is the Bayes estimator (3.21) after substituting Xi = eT, Ai = eN and Cl = crzINT, Cz = ozlZN into (3.21). The Swamy (1971) random coefficients model estimator or Lindley-Smith (1972) linear hierarchy model estimator can be obtained from (3.23) by letting Zi = 0, A1=eN@Ik,,a=(a; ,..., &),&‘=(ai-ti )..., aN-6),Cz=IN@d. Remark 4. Given that most panel data involves a large number of individuals, the Bayes estimators for the completely stacked model (3.9) and (2.1’) may involve the inversion of matrices of very high order. Computation can be simplified by operating with lower-dimension matrices. Eqs. (3.13) to (3.22) and (3.23) and (3.24) provide formulae to convert operations involving matrices of high dimensions to operations with matrices of much smaller dimensions. Remark 5. There appears no straightforward generalization of the above results for the single equation models to the simultaneous equations models. However, there are many ways the above results can still be applied to simultaneous
73
C. Hsiao et al., General framework for panel data models
equations models. For instance, consider a simultaneous equations model of the form KBi = XiRi + Zi& + Ui,
i=
l,...,N,
(3.27)
where x is the T x G matrix of endogenous variables, Xi and Zi are T x kl and T x k2 matrices of the exogenous variables of the ith cross-sectional unit, respectively; Bi, Ri, and fi are G x G, kl x G, and k2 x G matrices of coefficients; and Ui is a T x G matrix of errors. We assume that Bi and 4 are matrices of fixed constants, and Ri is random with Ri =
R+
(3.28)
Cip
where J? and &iare kl x G matrices of fixed constants and random variables, respectively, with E(ei) = 0, and the variancecovariance matrix of Eidefined by COV(&i)= E [(vet ei)(vec ei)‘] = CZ,
vi.
(3.29)
The reduced form of (3.27) is of the form Yj = XiIZil + Zis?7i, + 6,
(3.30)
where
iii,
= ly;‘.
(3.31)
Then, nia is fixed and nil is random with mean I?ii and variance-covariance matrix of the form COV($)
=
E[vec($)vec($)‘]
= Ci*,.
(3.32)
Ignoring the possible restrictions on pi,, ITizp and Cz, (3.30) is in the form of a multivariate regression model analogous to (2.1’). Considering one equation of (3.30) at a time, the results of this section can be applied straightforwardly. Of course, one can also derive the analogous results for the multivariate regression model (3.30) directly by following the procedures outlined here. If one wishes to take into account the prior restrictions on r’iil, Z7iz, and Cg explicitly, restricted reduced form estimators will have to be derived. The computation, though, can be very laborious.
74
C. Hsiao et al., General framework for panel data models
If the interest is in the structural form parameters, a computationally manageable approach appears to take the limited information method conditioning on the estimated reduced form parameters. More specifically, consider the first equation in the model (3.27) Yil
=
FlBil
+
Xilail
+
Zil
yil
+
Uil)
i=
l,...,N,
(3.33)
where yil and x denote the TX 1 and T x Cl matrices of the included endogenous variables with the coefficient Of yii set equal to one; Xii and Zil are T x kr and T x kt included exogenous variables, respectively; Uil denotes the TX 1 vector of disturbance term; Bil, ail, and yil are G, x 1, k: x 1, and k: x 1 vectors of coefficients with pii and yii fixed and ail random. Let the reduced form of x1 be expressed as
Substituting (3.34) into (3.33), we have Yil =
Xilail
+
zilYil
+
El/&l
+
Vi13
(3.35)
where vii = uil + Kl Bii. Conditional on the estimated fli: and I?i*,we can treat ZI, and 6, as the set of exogenous variables with fixed coefficients and Xii as the set of exogenous variables with random coefficients, and utilize the results of this section. Of course, one should keep in mind that the resulting Bayes estimator is a conditional posterior mean given the estimated fiz and ni*,.
4. A point-to-point
telephone service demand model
When aggregate data are used to estimate the demand for long distance telephone service, a common specification is to regress quantity on price, income, and other socio-demographic variables [e.g., Appelbe et al. (1991), de Fontenay et al. (1990), Taylor (1980)]. When panel data are used, because they contain information on specific routes, we can model a unique feature of telephone service. That is, a long distance call requires the interaction of two economic agents, both of whom derive utility from it, but only one, the originator, bears the cost. However, an implicit repayment agreement may exist in the form of an obligation on the part of the receiver of the call to make a return call at some future time.’ The joint consumption nature of calling is further illustrated by the observation that it is not primarily from the call itself that agents derive utility, but ‘See Larson, Lehman, and Weisman (1990) for details on the theoretical underpinnings of this type of model.
C. Hsiao et al., General framework
for panel data models
15
rather from the exchange of information facilitated by the call. Thus, for a route with endpoint areas A and B, an agent in A derives utility from calls originating in both A and B, and similarly an agent in B derives utility from calls originating in both areas. As a result of these demand characteristics, it is assumed that calling in one direction affects return calling through a reciprocal calling effect. Hence, the demand function of an individual in A for calls from A to B includes calls from B to A as an argument along with the usual determinants of demand such as price and income. Analogously, the agent in B has calls from A to B in his demand function for calls from B to A. Therefore, a pair of demand equations from area A to B and area B to A are postulated: 1%
QA
=
uOA
+
log
QB
=
BOB
+
+
&Al%
pA
a,A1og~A
+
JIB
+
log
+
a2A
USADA
PB
+
b2s
1%
+
log
XA
+
a3A
1%
Qr, (4.1)
#A,
XB
hi 1%ZB + ~SB DB + %,,
+
b3B
log
QA (4.2)
where QA denotes the demand from A to B, PA denotes real price from A to B, XA denotes real income in A, ZA denotes other socio-economic factors such as market size, postal strikes, etc. in A, QB denotes the demand from B to A, PB denotes the real price from B to A, Xa denotes real income in B, Za denotes other factors in B (including market size, postal strikes, etc.), DA and DB denote the seasonal dummies, and UAand aa are the unobservable error terms in eqs. (4.1) and (4.2), respectively. The point-to-point panel data model (4.1) and (4.2) allows us to derive much more information than aggregate models and open up new possibilities for the evaluation of the impacts of price changes on demand and revenues. For example, it provides unique information on uni-directional price elasticities (the elasticity of demand for company A with respect to a price change for traffic originating in company A), cross-price elasticities between companies (the elasticity of demand for company A with respect to a price change for traffic originating in company B), bi-directional price elasticities (the elasticity of demand for company A with respect to simultaneous equal percentage price changes for traffic originating in both company A and company B), and the reciprocal calling coefficient (the responsiveness of calling from company A to company B due to a change in calling from company B to company A).2 Elasticities may be similarly defined with respect to other determinants of demand such as income and market size. With this additional information, the ‘For
detailed
derivation,
see Appelbe
et al. (1992)
76
C. Hsiao et al., General framework for panel dara models
separate effects of a price change for traffic originating in company A on traffic from A to B and from B to A may be calculated using the appropriate uni-directional and cross-company elasticities. Or, in the case of an across the board price change such as the recent Canadian federal goods and services tax (GST), the bi-directional elasticity may be employed. To fit this basic model to panel data, we consider several different ways of modeling unobserved heterogeneity. These include: Model 1.
Fully Constrained Model.
responses
across
%A
a4A
=
bm
Model 2.
=
We assume there are no differences in Thus, we have aOA= bOB, aIA = bIs, aZA = bzB, and asA = bsB, VA, B.
rOUteS. b4B,
Fully Unconstrained Model.
We allow the response coefficients to
be different across routes. Model 3. Generalized Variable Intercept Model (or Partially Constrained Model). We assume there are route-specific effects that do not vary over time.
In other words, we let the coefficients of the intercept and seasonal dummies vary across rOUteS, but we COILStrain alA = blB, aZA = bzB, asA = baB, VA, B. In some cases, we let the coefficients of some of the variables (e.g., market size) differ across cross-sections to take into account regional effects such as different company sizes, geographic characteristics, and socio-economic factors. Model 4. Mixed Fixed and Random CoefJicients Model. We invoke the representative consumer argument in assuming that consumers respond in more or less the same way to price and income changes. The coefficients of these variables across routes are considered random draws from a common population with constant mean and variance-covariance matrix. However, we also assume that there are route-specific effects, and these effects are more appropriately captured by fixed and different coefficients for route-specific seasonal dummy variables and, occasionally, the market size variable.
5. Estimates of the demand for Canadian long distance service We use data on the demand for customer-dialed long distance service (DDD) between the nine major telephone companies in Canada. These are British Columbia Telephone Company (BCT), Alberta Government Telephones (AGT), Saskatchewan Telecommunications (SASK), Manitoba Telephone System (MTS), Bell Canada (BELL), New Brunswick Telephone Company (NBT), Maritime Telegraph and Telephone Company (MTT), The Island Telephone Company (ITC), and Newfoundland Telephone Company (NTC). Bell is split into Ontario region and Quebec region.
C. Hsiao et al., General framework
77
for panel data mo&ls
Thirty-two of the possible point-to-point intercompany routes are modeled. The various modeled routes were divided into three groupings. These pointto-point groupings were formed by considering the average mileage of the calls made over a specific route. Group 1 (short haul) has the majority of its calls in the O-680 miles range and has an average haul length of approximately 340 miles. Group 2 (medium haul) has the majority of its calls in the 291-1200 miles range and has an average haul length of approximately 775 miles. Group 3 (long haul) has the majority of its calls over 920 miles and has an average haul length of approximately 1990 miles. The routes forming the three groups are: Group 1: BCT-AGT, AGT-BCT, AGT-SASK, SASK-AGT, SASK-MTS, MTS-SASK, QUE-NBT, NBTQUE, NBT-MTT, MTT-NBT, MTT-ITC, ITC-MTT; Group 2: BCT-SASK, SASK-BCT, AGT-MTS, MTS-AGT, MTS-ONT, ONT-MTS, ONT-NBT, NBT-ONT, ONT-MTT, MTT-ONT; Group 3: BCT-QUE, QUE-BCT, BCT-MTS, MTS-BCT, BCT-ONT, ONT-BCT, SASK-ONT, ONT-SASK, ONT-NTC, NTC-ONT. The traffic is further divided according to full-rate (peak) and discount-rate (off-peak). The full-rate category consists of calls placed between 8 a.m. and 6 p.m. Monday to Saturday. The discount-rate category consists of all other calls. The quantity demanded is represented by price deflated revenues (PDR). The explanatory variables include own price, the level of income of economic activity, the size of the market, seasonality, special or atypical events (i.e., postal strikes and data problems), and traffic in the reverse direction for each route. The price variable is a chained Laspeyres index deflated by the relevant Consumer Price Index (CPI). Income is represented by Total Wages and Salaries or Retail Sales deflated by CPI or Employment. Market size is represented by the number of telephone lines in service (NAS) in the originating region only. The reciprocal calling variable is represented by the price deflated revenues for the traffic in the reverse direction of a specific route. We use quarterly data from 1980.1 to 1989.IV to estimate the various versions of the basic panel data model C(4.1)and (4.2)]. However, because (4.1) and (4.2) is a pair of simultaneous equations, the full information approach of deriving the route-specific coefficients estimates for the mixed fixed and random coefficients model (model 4) can be computationally unwieldy. Therefore, we take a limited information approach as outlined in remark 5 of section 3. That is, we substitute the reduced form estimates of log QB and log QA into the righthand side of (4.1) and (4.2), respectively, and treat the resulting system as a seemingly unrelated regression model. While this procedure may not be necessary for other models, it is also followed to facilitate comparison. The error terms UAitand aair are assumed to follow the pattern
UAit
=
PAUAi,t-
1 +
&Air,
UBir
=
PBUBi,t-1
+
&Bit,
(5.1)
78
C. Hsiao et al., General framework for panel data models
where &Aitand &Bitare assumed to be independently distributed over time, but are COntemporaneOuSly correlated, that is, E(&Air&Bjt) = Iterations between the regression coefficients and error term parameters are applied until the estimates converge. Results for the price and income coefficients of the off-peak models are reported in tables l-3. For confidentiality reasons, the routes reported in these tables are not in the same order as listed in paragraph 3 of this section. We note that the estimates of price and income coefficients are indeed sensitive to the way the unobserved heterogeneity is being modelled. For instance, the point estimates of the short-, medium-, and long-haul off-peak price coefficients are - 0.01, - 1.1, - 0.38, respectively, for fully constrained DABije
Table 1 Short-haul regression coefficients.” Fully constrained
Partially constrained
Unconstrained
Mixed coefficients
Price 1 2 3 4 5 6 7 8 9 10 11 12
-
0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102 0.0102
Average
- 0.0102
(- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28) (- 0.28)
-
0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38) 0.3091 ( - 8.38)
-
0.1203 ( 0.34) 0.2915 ( 0.98) 0.4910 ( - 3.87) 0.0069 ( - 0.03) 0.0634 ( - 0.43) 1.1448 ( - 4.72) 0.3697 ( - 3.48) 0.8745 ( - 4.40) 0.2372 ( - 2.14) 0.1728 ( - 2.05) 0.3670 ( - 6.33) 0.6379 ( - 4.69) N/A
- 0.3091
- 0.0385 (N/A) 0.0659 (N/A) - 0.4684 (N/A) - 0.2452 (N/A) - 0.0993 (N/A) - 0.4479 (N/A) - 0.2926 (N/A) - 0.2618 (N/A) - 0.2421 (N/A) - 0.2447 (N/A) - 0.4294 (N/A) - 0.4308 (N/A) - 0.2612
Income 1 2 3 4 5 6 7 8 9 10 11 12
0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906 0.0906
Average
0.0906
( ( ( ( ( ( ( ( ( (
( (
3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40) 3.40)
at-statistics are in parentheses.
0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028 ( 0.4028
24.81) 24.81) 24.8 1) 24.8 1) 24.81) 24.81) 24.81) 24.8 1) 24.81) 24.81) 24.81) 24.81)
0.65) 0.2168 ( 0.38) 0.0693 ( 5.18) 0.6144 ( 0.16) 0.0313 ( 2.52) 0.3188 ( 4.59) 1.3680 ( 2.52) 0.4308 ( - 0.1451 ( - 1.08) 0.4383 ( 6.63) 2.25) 0.2413 ( 0.3691 ( 12.13) 0.7948 ( 13.30)
WA
0.2895 0.1457 0.5406 0.2482 0.3390 0.4156 0.3064 0.2709 0.3947 0.3492 0.4105 0.3858 0.3413
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
C. Hsiao et al.. General framework for panel dara models
79
Table 2 Medium-haul regression coefficients. Fully constrained
Partially constrained
Mixed coefficients
Unconstrained
Price 1 2 3 4 5 6 7 8 9 10
-
l.lOOO(l.lOOO(l.lOOO(l.lOOO(1.1000 (1.1000 (l.lOOO(l.lOOO(l.lOOO(l.lOOO(-
Average
- 1.1000
32.17) 32.17) 32.17) 32.17) 32.17) 32.17) 32.17) 32.17) 32.17) 32.17)
- 0.2147 - 0.2147 - 0.2147 - 0.2147 - 0.2147 - 0.2147 -L 0.2147 - 0.2147 - 0.2147 - 0.2147
( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30) ( - 5.30)
- 0.2147
0.0534 - 0.0758 0.0376 - 0.9261 - 0.3328 0.1713 - 0.9495 - 0.8326 - 0.1955 - 0.4106
( ( ( ( ( ( ( (
0.35) - 0.34) 0.13) - 3.21) - 3.23) 1.02) - 4.47) - 3.71) ( - 2.23) ( - 3.70)
-
0.0113 0.1827 0.0391 0.4698 0.1888 0.0498 0.7456 0.6994 0.2014 0.4345
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
- 0.3000
N/A Income
1 2 3
Average
0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244 0.3244
( ( ( (
( ( ( ( (
(
6.71) 6.71) 6.71) 6.71) 6.71) 6.71) 6.71) 6.71) 6.71) 6.71)
0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779 0.3779
( ( ( ( (
( ( ( ( (
14.47) 14.47) 14.47) 14.47) 14.47) 14.47) 14.47) 14.47) 14.47) 14.47)
1.1559 ( 0.2382 ( 0.7090 ( 1.1264 ( 0.3084 ( 0.2810 ( 0.7673 ( 0.1274 ( 0.3566 ( 0.1475 ( N/A
5.47) 1.43) 2.84) 3.46) 3.49) 1.98) 4.98) 1.10) 10.51) 0.95)
0.4956 0.4081 0.6557 0.6244 0.3722 0.4346 0.5803 0.1118 0.3620 0.2560
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
0.4301
model (model 1) and - 0.309, - 0.215, - 0.235, respectively, for partially constrained model (model 3). This suggests that perhaps there are route-specific effects which should not be ignored. In fact, tests for the homogeneity of the seasonal dummy coefficients across the routes are rejected by the conventional F-statistics. A further test of the homogeneity of the price and income coefficients across routes was also rejected by the data. Therefore, a fully unconstrained model (model 2) is estimated. Unfortunately, because of the multicollinearity among the variables and the shortage of degrees of freedom, the individual route estimates are quite diverse. Many of them are either not statistically significant and/or have the wrong signs. Column 4 of tables 1-3 reports the alternative estimates when the price and income coefficients are assumed to be randomly distributed with common mean and constant varianceecovariance matrix (model 4). It is apparent from the comparison of the estimates between models 2 and 4 that the estimates of model 4 are much more concentrated around the
80
C. Hsiao et al., General framework for panel data models
Table 3 Long-haul Fully constrained
regression
coefficients.
Partially constrained
Mixed coefficients
Unconstrained
Price 1 2 3 4 5 6 7 8 9 10
-
Average
- 0.3848
0.3848 0.3848 0.3848 0.3848 0.3848 0.3848 0.3848 0.3848 0.3848 0.3848
1 2 3 4 5 6 7 8 9 10
0.5433 0.5433 0.5433 0.5433 0.5433 0.5433 0.5433 0.5433 0.5433 0.5433
Average
0.5433
( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78) ( - 9.78)
-
0.2360 0.2360 0.2360 0.2360 0.2360 0.2360 0.2360 0.2360 0.2360 0.2360
( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84) ( - 3.84)
- 0.2360
( ( ( ( ( ( ( ( ( (
22.59) 22.59) 22.59) 22.59) 22.59) 22.59) 22.59) 22.59) 22.59) 22.59)
0.4076 0.4076 0.4076 0.4076 0.4076 0.4076 0.4076 0.4076 0.4076 0.4076
- 0.0712(0.1694 ( - 1.0142 (- 0.4874 (- 0.3190 (0.0365 ( - 0.3996 (- 0.1033 (- 0.3965 (- 0.6187(-
0.15) 0.44) 5.22) 2.29) 2.71) 0.20) 3.92) 0.95) 4.22) 4.82)
11.31) 11.31) 11.31) 11.31) 11.31) 11.31) 11.31) 11.31) 11.31) 11.31)
0.4076
1.4301 - 0.0348 0.3698 0.2497 0.5556 0.1119( 0.9197 0.3886 0.6688 0.1928
N/A
0.2875 0.0220 0.7743 0.1686 0.2925 0.0568 0.3881 0.2504 0.2821 0.5934
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
- 0.3116
N/A
( ( ( ( ( ( ( ( ( (
-
( (( ( ( ( ( ( (
3.07) 0.09) 1.95) 0.70) 2.71) 0.95) 8.10) 3.88) 6.16) 2.39)
0.4740 0.2679 0.3394 0.3145 0.3501 0.1344 0.5342 0.5255 0.5648 0.2574
(N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A) (N/A)
0.3762
mean, and many of the routes that have unreasonable unconstrained estimates now have the correct sign.3 6. A prediction comparison We have seen in the last section that the parameter estimates are very sensitive to the way the unobserved heterogeneity is modelled. To select a representative formulation for the demand for Canadian long distance telephone service we note that ‘a severe test for an economic theory, the only test and the ultimate test is its ability to predict’ [Klein (1988, p. 21)] and ‘the prediction principle that predictive performance is central in evaluating hypothesis is of key importance
3This is because, as pointed model is the weighted average
out in remark 2 of section between the unconstrained
3, the individual estimates of the mixed estimates and the overall mean.
81
C. Hsiao et al., General framework for panel data models
in econometrics’ [Zellner (1988, p. 31)]. We therefore provide a predictive performance comparison of the four models in this section. We use the data from 1988.1 to 1990.111 to evaluate the root mean square percentage prediction error,
of the four models for each route. The predicted value is generated from the reduced form of (4.1) and (4.2) by treating the estimated structural form parameters as if they were true and explicitly taking account of the first-order autoregressive structure of the error terms. The results are reported in tables 4-6. In general the fully constrained model performs very badly. That is, the homogeneity across routes assumption is not supported by the data. As for the prediction performance of the other three models, the results vary. However, it does appear that on average the mixed model (model 4) has a slight edge. In addition, we note: (i) The unconstrained coefficients for seasonal dummies are much more spread out than the price and income coefficient. Models 2, 3, and 4 all allow the coefficients of seasonal dummies to vary across routes. (ii) Model 4 estimates of price and income coefficients for each route appear to be more reasonable on a priori grounds than fully unconstrained estimates (model 2). (iii) Similarity of the constrained price and income coefficient estimates of model 3 and their respective mean estimates from model 4. (iv) Under a mixed fixed and random coefficients formulation (model 4), in principle, we
Table 4 Short-haul
root mean square
percentage
prediction
error.
Fully constrained
Partially constrained
1 2 3 4 5 6 7 8 9 10 11 12
0.10826 0.07539 0.09788 0.07222 0.10707 0.10357 0.07358 0.08434 0.06095 0.09667 0.11890 0.07325
0.10280 0.03288 0.03565 0.07217 0.06121 0.04126 0.05825 0.05572 0.10561 0.10209 0.03322 0.06256
0.04371 0.04085 0.02508 0.02772 0.05982 0.04862 0.04576 0.03640 0.08456 0.08598 0.04524 0.10759
0.06808 0.03282 0.02749 0.05285 0.09564 0.05355 0.05766 0.05118 0.10481 0.10321 0.06457 0.03958
Average
0.08934
0.06362
0.05428
0.06262
0.08748
0.06590
0.07272
0.06540
Revenue-weighted
average
Unconstrained
Mixed coefficients
82
C. Hsiao et al., General framework for panel data models
can generate predictions using the overall mean 6 rather than using individual ai. A priori, one would expect that these predictions on average should perform reasonably well. In this sense, we may view the prediction generated by model 3 as using ti instead of ai, except that under the mixed coefficients framework the h is an inefficient estimate. Tables 4-6 show that the difference in predictive performance between models 3 and 4 is on average not that large, except for the long-haul off-peak model. In view of these observations, we would be in favor of selecting a mixed fixed and random coefficients formulation in which the coefficients of price and income variables are assumed to be random with
Table 5 Medium-haul
root mean square
percentage
prediction
error.
Fully constrained
Partially constrained
1 2 3 4 5 6 I 8 9 10
0.05244 0.10361 0.08650 0.04167 0.09534 0.07910 0.11093 0.08284 0.09650 0.15154
0.07276 0.05566 0.03421 0.02623 0.04215 0.05730 0.10649 0.06750 0.12336 0.13097
0.04858 0.02937 0.04173 0.11346 0.04814 0.10088 0.09737 0.05463 0.11992 0.10765
0.09223 0.0603 1 0.04570 0.04755 0.04369 0.06124 0.10073 0.05348 0.12347 0.11940
Average
0.09005
0.07166
0.07617
0.07478
0.08723
0.06573
0.07562
0.07039
Revenue-weighted
average
Unconstrained
Mixed coefficients
Table 6 Long-haul
root mean square
percentage
prediction
error. Mixed coefficients
Fully constrained
Partially constrained
1 2 3 4 5 6 7 8 9 10
0.11236 0.07635 0.15800 0.06146 0.05981 0.09863 0.10154 0.13085 0.05136 0.12980
0.05764 0.00966 0.09175 0.07964 0.06273 0.10256 0.07203 0.08542 0.12341 0.09124
0.15916 0.07221 0.07625 0.03852 0.05282 0.11077 0.04639 0.02724 0.03882 0.04634
0.01744 0.08550 0.08579 0.06567 0.08538 0.13918 0.02608 0.03406 0.07537 0.07347
Average
0.09802
0.07761
0.06685
0.06879
0.10612
0.07834
0.05443
0.05220
Revenue-weighted
average
Unconstrained
C. Hsiao el al., General framework for panel data models
83
a common mean to invoke a representative consumer argument and the coefficients of seasonal dummies to be fixed and different across routes to capture the fixed route-specific effects.
7. Conclusions In this paper we provide a unified framework to link various panel data models. A Bayes solution is derived which can be specialized to many specific estimators discussed in the literature. We illustrate the application of this framework by estimating a point-to-point Canadian long distance telephone service demand model. It does appear that the approach offers wide possibilities for controlling unobserved heterogeneity. However, many issues remain unresolved. In particular, the choice between random versus fixed parameters formulations and a more manageable solution for simultaneous equations models.
Appendix
In this appendix we sketch the derivation of Theorem 3.2. Let y* = c;‘Py,
x* = c;‘Px,
z”* = c-r/zz, 1
** = c;‘/Zu,
(A-1)
and f* = [I _ ~*(~*~~*)-‘z*~]x*.
(A.2)
We can rewrite (2.1’) as y* =
_f*a+ z”*7 + u*,
(A.3)
where z”*1d* = 0,
(A.4)
jj = ji + (Z*‘Z”*)-‘Z”*‘X*a,
(A.5)
and Eu* = 0,
Eu*u*’ = 1.
(A@
By Bayes theorem,
(A-7)
84
C. Hsiao et al., General framework for panel data models
where, under A.2-A.3, P(a) cc P(a 1G)P(ii).
(A.8)
The product on the right-hand where Q is given by
side of (A.7) is proportional
to exp{ - f Q>,
Q = QI + Q2 + QS + Q4,
(A.9)
with Q1 =
{a- (z*‘x”* + C;')-'(r?*'y* x
+
{a- (r?*Tz* + c;y’(f*‘y*
~&+i)}yrl*~r?*
+ C;‘Ala)},
(z”*rz”*)-‘z”*ry*]‘(~*‘z”*)[jj
Q2 = [jj -
+
c;‘)
(A.lO)
- (z”*‘z*)-‘z”*‘y*],
(A.11)
Q3 = {a - (A;[C2 + (ris*‘x*)-‘]-‘z41)-1 x A;C;‘(X*‘T*
+ CT’)_‘(X*‘y*)}’
x {A; [C, + (r?*‘r?*)-‘]-‘A,} x {a - (A;[& x Q4
+[I
+ (Jz*‘8*)-‘]-541)-’
A;C;‘(2*‘X”* + c;‘)-‘(d*ry*)}, _
X”*($Wd*
+
_ Z*$*rZ*
,;l)x”*’
_
(A.12)
2*(2*+)-‘2”W
+ c;‘)-lc;‘A,
+
(r?*9z*)-‘]-‘L41}-’
x
{A;[C,
x
A;C;‘(2*‘2*
+ c;yr7*qy*.
As far as the distribution of a, 7, and d is concerned, Conditional on 6, Q3 is a constant. Therefore, conditional distribution of a and y”is proportional to exp{ - f(Ql + posterior distribution of a and p given a is a multivariate (3.11) and jj* = (z”*~z”*)-‘z”*Y*,
(A. 13) Q4 is a constant. on E, the posterior Q2)}. That is, the normal with mean
(A. 14)
C. Hsiao et al.. General framework for panel data models
respectively, and variance-covariance
85
matrix (A.15)
where &
=
(z*‘z”*)-
l.
(A.16)
Substituting (AS) into (A.14) and (A.16), we have (iii). Substituting (A.5) and (3.11) into (A.14)-(A.16), we have (i) and (iv). To obtain the unconditional distribution of a and 7 given y, we rewrite the formulation of (A.9)-(A.13) as (A.17)
Q = QT + Q2 + Q: + Q4, with
Q; =
{a- [x*7?* + C;' -
x
[xI*‘x”* + c;’
x
{a - [8*9*
C;',41(~;C;'~l)-'
- c,‘ll,@l;
c;‘A,)-‘A;
C,‘]
+ CT’ - c;‘A,(A;C;‘A,)-’
xA;C;‘]-‘(X”*‘y*)),
(A.18)
Qf = [a - (A;C;'A,)-'A;C;'a]'(A;Cz'Al) x [6 - (A;C;‘A,)-‘A’,C;‘a], Q4
=/{I
_
-
2”*(+2*)-‘2W
(A.19) _
d*[zWd*
+
C;’
c;‘Al(A;C;‘Al)-‘A;C;‘]-‘~*‘}y*.
(A.20)
Integrating exp { - $ Q} with respect to ti and 7, we have (A.21) hence (ii). Substituting (3.15) and (3.16) into (A.5) we have (v).
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