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Statistical inference in cross-country demand systems
Economics Letters 5 (1980) 383-387 0 North-Holland Publishing Company
STATISTICAL
INFERENCE IN CROSS-COUNTRY
DEMAND SYSTEMS *
Henri THEIL, Frederick E. SUHM and James F. MEISNER University of Chicago, Chicago, IL 60637, USA Received
3 October
1980
Consumption data on eight goods in 15 countries are used for testing and estimating a system of demand equations across countries. The estimated income elasticities are compared with those of Clements et al. (1979) and Houthakker (1957).
1. Introduction Clements and Theil(1979) applied Working’s (1943) model to data on four goods in 15 countries compiled by Kravis et al. (1978) for the construction of a demand equation system. They proceeded in two steps, first estimating,income effects while ignoring the relative price differences of each good between countries, and then estimating substitution effects using the estimated income effects obtained in the first step. Suhm (1979) extended this procedure to an eight-good classification. In this paper we handle income and substitution effects simultaneously rather than successively.
2. Methodology Working’s model describes budget shares as linear functions of the logarithm of income. In a cross-country context we must specify the price vector at which this model applies. We select geometric mean prices, log pi = (l/ 15) C:,I1 log pit, where pit is the price of good i in country c. We write Working’s model as tii, = Eli + /3iS, + eic, where qc = log of per capita income of country c in Kravis dollars, iQic = budget share of good i at prices F1, .... jTn and the observed qc, and Eic = random disturbance. Therefore, Wit = Cyi+ /3jqc + (Wit
* Research
supported
-
tiic)
+ fit
,
in part by NSF Grant
(1)
SOC76-82718.
383
H. Theil et al. / Statistical inference in cross-country demand systems
384
where wiC is the observed budget share of i in c. The difference Wit ~ tiii, results from the difference between the price vectors lpic] and Fi]. We apply the differential specification of Theil(l975, p. 3 l), ’
Wit
-
Wit
= Wit
c
log~L.$
log $)
Wit
I
+ 5 I
j=1
where [nii] is the Slutsky matrix, and substitute
nii log F
,
(2)
I
(2) into (l),
n Jlic
=
oli t
/3iqc tC j=1
Tij
log
‘ic +
Eic
2
Pi
where yic = wit [ 1 - log(piJpi) + Zjwjc log(piJpj)J . The specification (3) is convenient because of its linearity in the parameters. It enables testing and imposing homogeneity and Slutsky symmetry, but the results are unattractive because of large standard errors of the Slutsky coefficients. ’ Therefore, we impose preference independence also, which implies nij = -@OiOj for i #j and Tiii = $Oi( 1 - Oi), where 4 is the income flexibility and Bi is the marginal share of good i. Under Working’s model 6’i equals wi + pi. 3 We shall therefore specify Bi of country c as Wit + /3i, and we shall allow @ to depend on real income by specifying @Jof country c as GOt @rqc. Substitution of these specifications into the equations for the nii’s shows that (3) may be written as 4 n-1
Yic = @i +
Piqc + ($0 + $1 4c)Cwic
wherextc
=
+ Pi> [xic
-
E
(Wjc + Pj> xjc] +
fit 2
lodpicipi) - log(pdPn).
We estimate CI= [CY~. . . c+_r]‘, p = [fir .. . &_r]‘, @oand $r under the assumption that [eIc .. . e,_r , ,] are independent normal vectors with zero mean and non-singular covariance matrix a. The estimation procedure is similar to that of Theil and 1 See eq. (5.24) on that page. The expression
on the first line vanishes because real income does not change (both Wit and Gic correspond to the observed qC). 2 Standard errors are large for both n = 4 and n = 8 (see section 3). Also, testing for homogeneity based on sample moments of least-squares residuals is impossible for n = 8 when the sample covers only 15 countries; see Laitinen (1978). For similar problems with symmetry testing see Meisner (1979). 3 Let M = ~ipi~i be total expenditure. Working’s model specifies Wi = “i + pi log M or PiQi = aiM+ DiMlog M. This yields Bi = LYE + pi(l + log M) = Wi + pi after differentiation with respect to M. (All logs in this paper are natural logs.) 4 In eq. (4) on is eliminated by means of Zipi = 0; there are only n - 1 unconstrained Pi’S. There are also n - 1 unconstrained 0li.s (because ~i’yi = 1). The total number of unconstrained parameters, including $0 and el, is thus 2n. The total number of degrees of freedom prior to the estimation of these parameters is 15(n - 1). Since (4) is an allocation model, we may confine ourselves to considering it for i = 1, . . . . n - 1 and c = 1, . . . . 15.
H. Theil et al. / Statistical inference
in cross-country
demand systems
385
Laitinen (1979). We write (4) for i = 1, .... n - 1 asy, = X,0 + E,, where y, = bi,], e, = [eic], and X, and 8 are partitioned matrices consisting of I, qJ, zc = [zic], qczc and (Y,f3,Go and @r, respectively, with n-1
zic
(Wit+ ai>xic -
=
[
The asymptotic K=
_
5 c=l
with
C
Cwjc
j=1
f
Pj>Xjc
.
1
covariance matrix of the ML estimator
acw’ ae
n_,
of 9 is -K-l,
ww ae”
where
(6)
a(x,eyae'= [I,q,z+ ~(az,/ap'),~,,~,z,]. 5
3. Empirical results We applied (4) to the Clements-Theil n = 4 goods and to Suhm’s n = 8 goods with qc = 0 for c = U.S. (i.e., the U.S. per capita income is selected as income unit). The ML estimates of Go and $r for n = 4 are -0.735 (0.214) and -0.323 (0.146) suggesting that the income flexibility takes larger negative values at a higher level of affluence. 6 The corresponding estimates for n = 8 are -0.705 (0.075) and -0.099 (0.059) which allows constancy of 4. lmposing constancy for n = 8 yields a 4 estimate of -0.618 (0.05 l), which is in good agreement with earlier estimates, and (Yi and pi estimates shown in the first two lines of table 1. ’ For comparison, Suhm’s (1979) & estimates are shown in the third line. The present estimates are mostly closer to zero, which implies income elasticities closer to 1. The last 15 lines contain the income elasticities of the eight goods at the income levels of the I5 countries and geometric mean prices, followed in the last column by the Divisia variance of these elasticities. 8 A comparison with table 1 of Clements et al. (1979) which is based on Suhm’s (1979) estimates, confirms that the income elasticities are mostly
5 azC/a@ has off-diagonal elements azi,/aQ = -(Wit + pi) Xjc and diagonal elements a+/a& = (1 - Wit - Pi) Xic - ZFl'(Wjc + flj)Xjc. 6 This effect agrees with Frisch’s (1959) conjecture. However, the point estimates imply a positive value of the income flexibility for India (the poorest of the 15 countries), which is unacceptable. The four goods are those of the first two columns of table 1, the sum of the next two, and the sum of the last four. 7 The ai estimates are estimated budget shares at the U.S. income level and geometric mean prices; see (4) and recall that qC = 0 for c = U.S. 8 The budget share of good i at the income level of country c and geometric mean prices equals CY~ + pjqC and the income elasticity is 1 + &/(cY~+ PiQc). The income elasticities in table 1 are obtained by substituting the estimates of the first two lines. Their Divisia mean and variance are 1 and X:ip:/(oli + piqc), respectively.
elasticities
estimates
1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05
893 (61) 42 (46) 60
2130 (130) -1526 (96) -1711
0.75 0.71 0.70 0.68 0.67 0.67 0.56 0.53 0.52 0.46 0.46 0.45 0.45 0.43 0.28
Clothing and footwear
1.31 1.27 1.26 1.24 1.24 1.24 1.20 1.20 1.20 1.19 1.19 1.19 1.19 1.18 1.17 by 10,000.
1.90 1.59 1.57 1.49 1.48 1.47 1.35 1.33 1.33 1.31 1.31 1.30 1.30 1.30 1.27
2.28 1.74 1.70 1.58 1.57 1.56 1.39 1.37 1.37 1.34 1.34 1.34 1.33 1.33 1.29
881 (45) 258 (35) 260
995 (79) 267 (60) 266
1514 (112) 261 (83) 270
elasticities. Medical care
income
Home furnishings, operations
and implied
Gross rent and fuel
system
all be divided
demand
Food, beverages, tobacco
of a cross-country
a The entires in the first three lines should
India Korea Philippines Malaysia Colombia Iran Hungary Italy Japan Netherlands United Kingdom West Germany France Belgium United States
Income
Suhm’s pi
Table 1 Parameter a
1.42 1.34 1.33 1.30 1.30 1.30 1.24 1.23 1.23 1.22 1.22 1.22 1.22 1.22 1.20
1050 (125) 210 (93) 228
Transport, communication
1.35 1.29 1.29 1.26 1.26 1.26 1.22 1.21 1.21 1.20 1.20 1.20 1.20 1.20 1.18
1327 (76) 243 (64) 361
Recreation and education
1.44 1.35 1.34 1.31 1.31 1.30 1.25 1.24 1.24 1.22 1.22 1.22 1.22 1.22 1.20
1209 (88) 246 (67) 266
Other consumption expenditures
0.1317 0.1092 0.1079 0.1045 0.1043 0.1041 0.1087 0.1119 0.1123 0.1191 0.1192 0.1203 0.1212 0.1231 0.1424
Divisia variance
H. Theil et al. / Statistical inference
in cross-country
demand systems
387
closer to 1 and that the Divisia variances are smaller. The U.S. food elasticity has increased from 0.10 to 0.28 relative to the Clements et al. (1979) result; this is in closer agreement with Houthakker’s (1957) double-log Engel curve result.
References Clements, K.W. and H. Theil, 1979, A cross-country analysis of consumption patterns, Report 7924 of the Center for Mathematical Studies in Business and Economics (The University of Chicago, Chicago, IL). Clements, K.W., F.E. Suhm and II. Theil, 1979, A cross-country tabulation of income elasticities of demand, Economics Letters 3, 1999202. Frisch, R., 1959, A complete scheme for computing all direct and cross demand elasticities in a model with many sectors, Econometrica 27, 177-196. Houthakker, H.S., 1957, An international comparison of household expenditure patterns, Commemorating the Centenary of Engel’s Law, Econometrica 25, 532-551. Kravis, l.B., A.W. Heston and R. Summers, 1978, International comparisons of real product and purchasing power (Johns Hopkins University Press, Baltimore, MD). Laitinen, K., 1978, Why is demand homogeneity so often rejected? Economics Letters 1, 1877191. Meisner, J.F., 1979, The sad fate of the asymptotic Slutsky symmetry test for large systems, Economics Letters 2, 231-233. Suhm, F.E., 1979, A cross-country analysis of Divisia covariances of prices and quantities consumed, Economics Letters 3, 2877291. Theil, H., 1975, Theory and measurement of consumer demand, Vol. 1 (North-Holland, Amsterdam). Theil, H. and K: Laitinen, 1979, Maximum likelihood estimation of the Rotterdam model under two different conditions, Economics Letters 2, 2399244. Working, II., 1943, Statistical laws of family expenditure, Journal of American Statistical Association 38, 43-56.
STATISTICAL
INFERENCE IN CROSS-COUNTRY
DEMAND SYSTEMS *
Henri THEIL, Frederick E. SUHM and James F. MEISNER University of Chicago, Chicago, IL 60637, USA Received
3 October
1980
Consumption data on eight goods in 15 countries are used for testing and estimating a system of demand equations across countries. The estimated income elasticities are compared with those of Clements et al. (1979) and Houthakker (1957).
1. Introduction Clements and Theil(1979) applied Working’s (1943) model to data on four goods in 15 countries compiled by Kravis et al. (1978) for the construction of a demand equation system. They proceeded in two steps, first estimating,income effects while ignoring the relative price differences of each good between countries, and then estimating substitution effects using the estimated income effects obtained in the first step. Suhm (1979) extended this procedure to an eight-good classification. In this paper we handle income and substitution effects simultaneously rather than successively.
2. Methodology Working’s model describes budget shares as linear functions of the logarithm of income. In a cross-country context we must specify the price vector at which this model applies. We select geometric mean prices, log pi = (l/ 15) C:,I1 log pit, where pit is the price of good i in country c. We write Working’s model as tii, = Eli + /3iS, + eic, where qc = log of per capita income of country c in Kravis dollars, iQic = budget share of good i at prices F1, .... jTn and the observed qc, and Eic = random disturbance. Therefore, Wit = Cyi+ /3jqc + (Wit
* Research
supported
-
tiic)
+ fit
,
in part by NSF Grant
(1)
SOC76-82718.
383
H. Theil et al. / Statistical inference in cross-country demand systems
384
where wiC is the observed budget share of i in c. The difference Wit ~ tiii, results from the difference between the price vectors lpic] and Fi]. We apply the differential specification of Theil(l975, p. 3 l), ’
Wit
-
Wit
= Wit
c
log~L.$
log $)
Wit
I
+ 5 I
j=1
where [nii] is the Slutsky matrix, and substitute
nii log F
,
(2)
I
(2) into (l),
n Jlic
=
oli t
/3iqc tC j=1
Tij
log
‘ic +
Eic
2
Pi
where yic = wit [ 1 - log(piJpi) + Zjwjc log(piJpj)J . The specification (3) is convenient because of its linearity in the parameters. It enables testing and imposing homogeneity and Slutsky symmetry, but the results are unattractive because of large standard errors of the Slutsky coefficients. ’ Therefore, we impose preference independence also, which implies nij = -@OiOj for i #j and Tiii = $Oi( 1 - Oi), where 4 is the income flexibility and Bi is the marginal share of good i. Under Working’s model 6’i equals wi + pi. 3 We shall therefore specify Bi of country c as Wit + /3i, and we shall allow @ to depend on real income by specifying @Jof country c as GOt @rqc. Substitution of these specifications into the equations for the nii’s shows that (3) may be written as 4 n-1
Yic = @i +
Piqc + ($0 + $1 4c)Cwic
wherextc
=
+ Pi> [xic
-
E
(Wjc + Pj> xjc] +
fit 2
lodpicipi) - log(pdPn).
We estimate CI= [CY~. . . c+_r]‘, p = [fir .. . &_r]‘, @oand $r under the assumption that [eIc .. . e,_r , ,] are independent normal vectors with zero mean and non-singular covariance matrix a. The estimation procedure is similar to that of Theil and 1 See eq. (5.24) on that page. The expression
on the first line vanishes because real income does not change (both Wit and Gic correspond to the observed qC). 2 Standard errors are large for both n = 4 and n = 8 (see section 3). Also, testing for homogeneity based on sample moments of least-squares residuals is impossible for n = 8 when the sample covers only 15 countries; see Laitinen (1978). For similar problems with symmetry testing see Meisner (1979). 3 Let M = ~ipi~i be total expenditure. Working’s model specifies Wi = “i + pi log M or PiQi = aiM+ DiMlog M. This yields Bi = LYE + pi(l + log M) = Wi + pi after differentiation with respect to M. (All logs in this paper are natural logs.) 4 In eq. (4) on is eliminated by means of Zipi = 0; there are only n - 1 unconstrained Pi’S. There are also n - 1 unconstrained 0li.s (because ~i’yi = 1). The total number of unconstrained parameters, including $0 and el, is thus 2n. The total number of degrees of freedom prior to the estimation of these parameters is 15(n - 1). Since (4) is an allocation model, we may confine ourselves to considering it for i = 1, . . . . n - 1 and c = 1, . . . . 15.
H. Theil et al. / Statistical inference
in cross-country
demand systems
385
Laitinen (1979). We write (4) for i = 1, .... n - 1 asy, = X,0 + E,, where y, = bi,], e, = [eic], and X, and 8 are partitioned matrices consisting of I, qJ, zc = [zic], qczc and (Y,f3,Go and @r, respectively, with n-1
zic
(Wit+ ai>xic -
=
[
The asymptotic K=
_
5 c=l
with
C
Cwjc
j=1
f
Pj>Xjc
.
1
covariance matrix of the ML estimator
acw’ ae
n_,
of 9 is -K-l,
ww ae”
where
(6)
a(x,eyae'= [I,q,z+ ~(az,/ap'),~,,~,z,]. 5
3. Empirical results We applied (4) to the Clements-Theil n = 4 goods and to Suhm’s n = 8 goods with qc = 0 for c = U.S. (i.e., the U.S. per capita income is selected as income unit). The ML estimates of Go and $r for n = 4 are -0.735 (0.214) and -0.323 (0.146) suggesting that the income flexibility takes larger negative values at a higher level of affluence. 6 The corresponding estimates for n = 8 are -0.705 (0.075) and -0.099 (0.059) which allows constancy of 4. lmposing constancy for n = 8 yields a 4 estimate of -0.618 (0.05 l), which is in good agreement with earlier estimates, and (Yi and pi estimates shown in the first two lines of table 1. ’ For comparison, Suhm’s (1979) & estimates are shown in the third line. The present estimates are mostly closer to zero, which implies income elasticities closer to 1. The last 15 lines contain the income elasticities of the eight goods at the income levels of the I5 countries and geometric mean prices, followed in the last column by the Divisia variance of these elasticities. 8 A comparison with table 1 of Clements et al. (1979) which is based on Suhm’s (1979) estimates, confirms that the income elasticities are mostly
5 azC/a@ has off-diagonal elements azi,/aQ = -(Wit + pi) Xjc and diagonal elements a+/a& = (1 - Wit - Pi) Xic - ZFl'(Wjc + flj)Xjc. 6 This effect agrees with Frisch’s (1959) conjecture. However, the point estimates imply a positive value of the income flexibility for India (the poorest of the 15 countries), which is unacceptable. The four goods are those of the first two columns of table 1, the sum of the next two, and the sum of the last four. 7 The ai estimates are estimated budget shares at the U.S. income level and geometric mean prices; see (4) and recall that qC = 0 for c = U.S. 8 The budget share of good i at the income level of country c and geometric mean prices equals CY~ + pjqC and the income elasticity is 1 + &/(cY~+ PiQc). The income elasticities in table 1 are obtained by substituting the estimates of the first two lines. Their Divisia mean and variance are 1 and X:ip:/(oli + piqc), respectively.
elasticities
estimates
1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05
893 (61) 42 (46) 60
2130 (130) -1526 (96) -1711
0.75 0.71 0.70 0.68 0.67 0.67 0.56 0.53 0.52 0.46 0.46 0.45 0.45 0.43 0.28
Clothing and footwear
1.31 1.27 1.26 1.24 1.24 1.24 1.20 1.20 1.20 1.19 1.19 1.19 1.19 1.18 1.17 by 10,000.
1.90 1.59 1.57 1.49 1.48 1.47 1.35 1.33 1.33 1.31 1.31 1.30 1.30 1.30 1.27
2.28 1.74 1.70 1.58 1.57 1.56 1.39 1.37 1.37 1.34 1.34 1.34 1.33 1.33 1.29
881 (45) 258 (35) 260
995 (79) 267 (60) 266
1514 (112) 261 (83) 270
elasticities. Medical care
income
Home furnishings, operations
and implied
Gross rent and fuel
system
all be divided
demand
Food, beverages, tobacco
of a cross-country
a The entires in the first three lines should
India Korea Philippines Malaysia Colombia Iran Hungary Italy Japan Netherlands United Kingdom West Germany France Belgium United States
Income
Suhm’s pi
Table 1 Parameter a
1.42 1.34 1.33 1.30 1.30 1.30 1.24 1.23 1.23 1.22 1.22 1.22 1.22 1.22 1.20
1050 (125) 210 (93) 228
Transport, communication
1.35 1.29 1.29 1.26 1.26 1.26 1.22 1.21 1.21 1.20 1.20 1.20 1.20 1.20 1.18
1327 (76) 243 (64) 361
Recreation and education
1.44 1.35 1.34 1.31 1.31 1.30 1.25 1.24 1.24 1.22 1.22 1.22 1.22 1.22 1.20
1209 (88) 246 (67) 266
Other consumption expenditures
0.1317 0.1092 0.1079 0.1045 0.1043 0.1041 0.1087 0.1119 0.1123 0.1191 0.1192 0.1203 0.1212 0.1231 0.1424
Divisia variance
H. Theil et al. / Statistical inference
in cross-country
demand systems
387
closer to 1 and that the Divisia variances are smaller. The U.S. food elasticity has increased from 0.10 to 0.28 relative to the Clements et al. (1979) result; this is in closer agreement with Houthakker’s (1957) double-log Engel curve result.
References Clements, K.W. and H. Theil, 1979, A cross-country analysis of consumption patterns, Report 7924 of the Center for Mathematical Studies in Business and Economics (The University of Chicago, Chicago, IL). Clements, K.W., F.E. Suhm and II. Theil, 1979, A cross-country tabulation of income elasticities of demand, Economics Letters 3, 1999202. Frisch, R., 1959, A complete scheme for computing all direct and cross demand elasticities in a model with many sectors, Econometrica 27, 177-196. Houthakker, H.S., 1957, An international comparison of household expenditure patterns, Commemorating the Centenary of Engel’s Law, Econometrica 25, 532-551. Kravis, l.B., A.W. Heston and R. Summers, 1978, International comparisons of real product and purchasing power (Johns Hopkins University Press, Baltimore, MD). Laitinen, K., 1978, Why is demand homogeneity so often rejected? Economics Letters 1, 1877191. Meisner, J.F., 1979, The sad fate of the asymptotic Slutsky symmetry test for large systems, Economics Letters 2, 231-233. Suhm, F.E., 1979, A cross-country analysis of Divisia covariances of prices and quantities consumed, Economics Letters 3, 2877291. Theil, H., 1975, Theory and measurement of consumer demand, Vol. 1 (North-Holland, Amsterdam). Theil, H. and K: Laitinen, 1979, Maximum likelihood estimation of the Rotterdam model under two different conditions, Economics Letters 2, 2399244. Working, II., 1943, Statistical laws of family expenditure, Journal of American Statistical Association 38, 43-56.