Estimating a cointegrating demand system

European Economic Review 41 Ž1997. 61–73

Estimating a cointegrating demand system C.L.F. Attfield Department of Economics, UniÕersity of Bristol, 8 Woodland Road, Bristol, BS8 1TN, UK Received 15 April 1994; accepted 15 September 1995

Abstract The set of variables in a time series demand model in the form of the Almost Ideal Demand System are found to be IŽ1. with the demand equations forming a cointegrating system. The parameters of the cointegrating equations are estimated and tested using a ‘triangular error correction’ procedure. The null hypothesis of homogeneity with respect to prices and nominal income in the system cannot be rejected. JEL classification: C32; D12 Keywords: Conintegration; Demand system; Homogeneity

1. Introduction A good deal of the empirical work on demand systems assumes either implicitly or explicitly that the time series data used in the analysis are stationary Že.g. Deaton and Muellbauer, 1980; Attfield, 1991.. In this paper we show that in a set of demand equations which relate budget shares, wi , to a set of prices, p 1 , p 2 , . . . , pm , and a measure of per capita real income, y, there is evidence that each of the variables is integrated of order one, IŽ1.. Of course, by construction, budget shares are bounded so, in the very long run, we would expect them to be stationary. For this sample of data however, the wi display all the characteristics of IŽ1. variates and so we treat them accordingly. 1 1

Sample autocorrelation coefficients for the wi t are virtually the same for lag correlations up to ten. Ng Ž1992. obtains similar results for budget shares on data for the USA. 0014-2921r97r$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved. SSDI 0 0 1 4 - 2 9 2 1 Ž 9 5 . 0 0 0 6 2 - 3

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C.L.F. Attfieldr European Economic ReÕiew 41 (1997) 61–73

Following the work by Engle and Granger Ž1987. it is well known that if a set of variables are IŽ1. and cointegrated then the application of ordinary least squares to the levels of the variables results in parameter estimators which are consistent but have non standard distributions. On the other hand, first differencing the variables to obtain stationarity and then modelling these first differences as a vector autoregression leads to a serious misspecification. The correct procedure, for a system of IŽ1. variates which are cointegrated, is to model them in a set of relationships which incorporate the dynamic short run movements in the variables – the first differences – with the longer run, equilibrium, cointegrating relationships in levels. Such a specification is to be found in the estimation and testing methods for cointegrating systems developed by Phillips Ž1991, 1994. where the relationships are in the form of a triangular error correction model ŽTECM.. In the next section, Section 2, we briefly present the time series properties of the variables in the model. In Section 3, within the cointegration framework, we give estimates of the parameters of the demand system and present the results of tests on individual parameters and for the homogeneity restrictions. Section 4 concludes the paper.

2. Time series properties The data analysed were constructed from the quarterly series on consumers’ expenditure and real consumers’ expenditure, real income and per capita income for the UK 1963Ži. to 1992Žii. from the Central Statistical Office Data Bank. All the groups given by the CSO were included except for durable consumer goods. The groups are: Ž1. food; Ž2. alcoholic drink and tobacco; Ž3. clothing and footwear; Ž4. energy products; Ž5. other goods; Ž6. rent, rates and water charges and Ž7. other services. The variables constructed were: w1 t , w 2 t , . . . , w 7t , p 1 t , p 2 t , . . . , p 7t , yt , where wi t is the budget share of commodity group i at time t, defined as the ratio of nominal expenditure on commodity i, say Q i t , to total nominal expenditure on all commodities, say X t s Ýis7 is1 Q i t ; pi t is the natural log of the price of the ith commodity. Real per capita income, yt , is defined as yt s lnŽ X trNt . y ln Pt) Ž . where Nt is the population at time t and ln Pt) s Ýis7 is1 wi t pi t is the Stone 1953 price index Žcf. Deaton and Muellbauer, 1980.. Each variable was deseasonalised prior to analysis using a set of centred seasonal dummies. Table 1 reports augmented Dickey–Fuller ŽADF. test statistics for the null hypothesis that the processes generating the variables in the system contain unit roots. There is evidence that the application of a linear filter to deseasonalise time series data results in a bias in the ADF test statistic ŽGhysels and Perron, 1993., and, a more serious bias in the Phillips and Perron Ž1988. test statistic. To reduce the extent of the bias in the ADF statistic Ghysels and Perron Ž1993. suggest using

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Table 1 Unit root tests a,b Test

No. of w1 lags Ž k .

w2

w3

w4

w5

w6

ADFŽ t .

6 8 10

ADFŽ F .

6 8 10

Test

No. of lags Ž k .

p1

ADFŽ t .

6 8 10

y2.563 y2.465 y1.705 y1.799 y1.770 y0.929 y2.226 y1.591 y3.334 y2.401 y2.561 y1.976 y2.281 y1.078 y2.379 y2.386 y3.062 y2.618 y3.112 y1.724 y1.859 y0.475 y2.238 y1.453

ADFŽ F .

6 8 10

y1.121 y1.876 y3.734 y1.262 y3.087 y1.753 y1.885 y3.491 y0.970 y2.617 y2.213 y2.219 y3.784 y1.520 y3.216 0.896 1.716 2.723

2.809 2.624 3.254 p2

3.394 5.892 4.791

7.019 6.105 7.163 p3

3.141 2.957 3.547

1.848 1.626 1.807 p4

1.940 3.525 5.419

4.764 3.433 5.174 p5

1.800 2.040 1.668

w7 0.238 0.174 0.508

1.913 2.056 1.288

2.560 1.634 2.219

7.901 5.956 3.463

p6

1.653 2.677 2.025

p7

1.360 1.380 0.958

2.588 3.084 2.546

y

2.018 3.319 1.953

a ADFŽ t . is the augmented Dickey–Fuller test statistic based on the Student’s ratio to test the hypotheses H 0 : r s 0 in the model D x t s a q b t q r x ty1 q u 1 D x ty1 q . . . q u k D x tyk q u t . ADFŽ F . is the augmented Dickey–Fuller test statistic based on the F-ratio to test the hypotheses H 0 : b s 0 and r s 0 in the model D x t s a q b t q r x ty1 q u 1 D x ty1 q . . . q u k D x tyk q u t . b Critical values for the statistics are Žfor a sample of size ns100.: ADFŽ t .: y4.04 Ž1%., y3.45 Ž5%.; ADFŽ F .: 8.73 Ž1%., 6.49 Ž5%..

more lagged differences than the length of the seasonal adjustment. In Table 1 therefore the ADF statistics are computed with lag lengths of six, eight and ten. The ADFŽ t . statistic tests the null of a unit root with both an intercept and a time trend in the model while the ADFŽ F . statistic tests the null of a unit root and a zero coefficient on the time trend with an intercept in the model. For both tests the only variable that comes close to having the unit root hypothesis rejected is w 3 and then the null cannot be rejected at the 1% level. We can conclude therefore that all variables are generated by a unit root process without a time trend. Having established that the variables in the system display all the characteristics of IŽ1. variables we next turn to testing the demand equations for cointegration. The budget share equations, in the form of the AIDS model, are jsmq1

wi t s b 0 q

Ý

b j pjt q g i yt q u i t ,

i s 1, . . . ,m; t s 1, . . . ,n,

Ž 1.

js1

where, to avoid the singularity implicit in the system, the equation for the commodity group, ‘other services’, is omitted and so m s 6. Two stage budgeting is assumed so that consumers allocate expenditure among commodity groups given an exogenously determined total expenditure. We tested for cointegration in

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64

each of the equations in Ž1. by obtaining the OLS residuals and testing them for stationarity using the Phillips and Ouliaris Ž1990. Za statistic which Haug Ž1995., in a Monte Carlo study, finds satisfactory. Table 2 gives the test statistics for the case of the model in Ž1. excluding an intercept, the ‘standard’ statistic; including an intercept the ‘demeaned’ statistic and with both an intercept and a time trend the ‘demeaned and detrended’ statistic. For a sample of size 100 with five explanatory variables Haug Ž1992. gives 5% critical values for Za of y34.77, y37.94 and y41.91 for the standard, demeaned and demeaned and detrended statistics respectively. Critical values for more than five explanatory variables are not computed but since the increment for each additional explanatory variable is less than five, in Haug’s tables, with eight explanatory variables, critical values would be around y50, y53 and y57 respectively. The null hypothesis being tested is ‘no cointegration’ and, except for the w6 equation in Ž1., the test statistic is in the rejection region for all forms of Za , for all lag windows between one and ten, and so the null hypothesis of no cointegration can be rejected for the w 1 , . . . , w5 equations in Ž1.. In the case of w6 the result where both a constant term and a time trend are included is highly significant, the Za test statistic rejecting the null of no cointegration. 2 Eq. Ž1. then represent a cointegrating demand system in the ‘almost ideal’ form of Deaton and Muellbauer Ž1980.. The parameters of these equations are appropriately estimated and tested in the next section.

3. System estimates and tests Phillips parameters framework

Ž1991, 1994. shows that maximum likelihood ŽML. estimates of in cointegrating relationships can be obtained within a triangular ECM ŽTECM.. In essence this amounts to writing the system

y 1 t s By 2 t q u1 t ,

D y 2 t s C1 D y 2 ty1 q . . . qCk D y 2 tyk q u 2 t , where y 1 t is the Ž r = 1. vector of left-hand side variables in the r-equation cointegrating system and y 2 t is the Ž q = 1. vector of right-hand side variables. For the demand system in Ž1., r s m s 6 and q s 8 and we have y 1 t s Ž w1 t , w 2 t , . . . , w6 t ., and y 2 t s Ž p 1 t , p 2 t , . . . , p 7t , y . where all variables are de2

The results reported in the next section and in Tables 3 and 4 are for the case where a time trend was not included in the cointegrating equations. Because there is a suggestion from the Za test statistic that the w6 equation requires a time trend, to be cointegrated, the whole analysis was repeated with a time trend included. The results, however, are so similar to those in Tables 3 and 4 that they are not presented here. In particular, the conclusions drawn from the hypothesis tests discussed in Section 3, are not altered by the inclusion of a time trend.

y110.704 y121.587 y130.363 y133.559 y134.154

y111.815 y125.474 y134.630 y137.719 y137.494

Za demeaned

y87.031 y94.544 y99.038 y103.377 y103.273

Za demeaned

y113.678 y125.283 y131.828 y131.807 y128.729

Z a demeaned and detrended

y99.100 y101.990 y101.345 y102.227 y99.292

Z a demeaned and detrended

y39.196 y43.067 y45.139 y46.394 y44.646

Za standard

w6

y79.098 y94.854 y108.418 y115.895 y119.922

Za standard

w3

y22.767 y24.205 y26.200 y26.731 y27.323

Za demeaned

y75.408 y90.018 y102.560 y109.517 y113.238

Za demeaned

y104.727 y113.367 y116.545 y114.340 y111.115

Z a demeaned and detrended

y76.261 y91.070 y103.513 y110.218 y113.664

Z a demeaned and detrended

The Z a test for cointegration is defined in Phillips and Ouliaris Ž1990.. The standard statistic is based on the residual from the regression: wi t sÝ b j p jt qg i yt q u i t . The demeaned statistic is based on the residuals from the above regression including an intercept and the demeaned and detrended statistic includes both a time trend and an intercept in the regression. b 5% critical values for the Z a statistic are: for 1 explanatory variable: y14.71 Ž Z a standard., y19.29 Ž Z a demeaned., y25.06 Ž Z a demeaned and detrended.; for 2 explanatory variables: y20.80 Ž Z a standard., y24.66 Ž Z a demeaned., y29.89 Ž Z a demeaned and detrended.; for 3 explanatory variables: y25.72 Ž Z a standard., y29.36 Ž Z a demeaned., y34.10 Ž Z a demeaned and detrended.; for 4 explanatory variables: y30.53 Ž Z a standard., y33.86 Ž Z a demeaned., y38.25 Ž Z a demeaned and detrended.; for 5 explanatory variables: y34.77 Ž Z a standard., y37.94 Ž Z a demeaned., y41.91 Ž Z a demeaned and detrended.. Source: Haug Ž1992, pp. 477–478, Tables 1–3..

a

y87.817 y90.520 y94.240 y92.725 y90.846

y71.811 y79.377 y87.506 y90.217 y90.923

2 4 6 8 10 y75.556 y83.804 y92.191 y95.550 y97.074

Za standard

Z a demeaned and detrended

Za standard

No. of lags

Za demeaned

w5

y76.148 y85.251 y89.299 y92.389 y90.838

w4

y97.179 y97.932 y98.576 y98.777 y98.721

y68.864 y74.565 y77.805 y77.998 y76.104

2 4 6 8 10 y88.321 y92.124 y94.950 y96.081 y96.141

Za standard

Z a demeaned and detrended

Za standard

No. of lags

Za demeaned

w2

w1

Table 2 Cointegration tests a,b

C.L.F. Attfieldr European Economic ReÕiew 41 (1997) 61–73 65

0.0554 Ž1.01. 0.1195 Ž2.15. y0.0384 Žy0.72. y0.0365 Žy0.62. y0.0201 Žy0.48. y0.0055 Žy0.22.

Ž1. y0.0025 Žy0.19. 0.0156 Ž1.18. 0.0141 Ž1.12. y0.0522 Žy3.71. y0.0124 Žy1.24. y0.0519 Žy8.51.

P2 0.0597 Ž1.39. y0.1049 Žy2.40. 0.0733 Ž1.76. y0.0803 Žy1.73. 0.0209 Ž0.64. y0.0995 Žy4.95.

P3 y0.0758 Žy4.57. 0.0235 Ž1.39. 0.0076 Ž0.47. 0.0693 Ž3.85. 0.0050 Ž0.40. y0.0104 Žy1.33.

P4 y0.0142 Žy0.73. 0.0002 Ž0.01. 0.0048 Ž0.25. y0.0291 Žy1.38. 0.0248 Ž1.66. y0.0025 Žy0.27.

P5 y0.0104 Ž0.45. y0.0052 Žy0.22. y0.0327 Žy1.45. y0.0239 Žy0.95. y0.0239 Žy1.34. 0.1261 Ž11.57.

P6 0.0101 Ž0.17. y0.0741 Žy1.21. y0.0214 Žy0.37. 0.1378 Ž2.12. 0.0053 Ž0.11. 0.0379 Ž1.34.

P7 y0.1835 Žy5.75. y0.0329 Žy1.01. 0.0153 Ž0.49. y0.0817 Žy2.36. 0.0668 Ž2.73. y0.1368 Žy9.13.

Expenditure

0.99

0.94

0.95

0.97

0.98

0.99

R2

0.74

1.80

1.42

1.31

1.60

1.72

DW

wy2.93x

Ž0.009.

Ž0.009. w2.61x

y0.0254

0.0221

Ž2. 0.0073

w0.89x

Ž0.008.

Ž3.

wy1.62x

Ž0.009.

y0.0149

Ž4.

wy0.04x

Ž0.007.

y0.0003

Ž5.

wy1.44x

Ž0.004.

y0.0057

Ž6.

Notes: t-ratio is for a test of the null hypothesis that the sum of the coefficients on the price variables is zero.

Sum of price coefficients Estimated standard error t-ratio

Ž1.

(b) Single equation tests for homogeneity

Notes: t-ratios in parentheses, calculated as described in Section 3. Commodity groups: Ž1. Food; Ž2. Alcoholic drink and tobacco; Ž3. Clothing and footwear; Ž4. Energy products; Ž5. Other goods; Ž6. Rent rates and water charges; Ž7. Other services.

Ž6.

Ž5.

Ž4.

Ž3.

Ž2.

P1

Commodity

Table 3 Phillip’s triangular ECM estimates assuming all variables are IŽ1.. (a) Estimates of cointegrating Õectors from the model y1 t s By 2 t qC0 D y 2 t qC1 D y 2 ty1 q . . . q D 3 D y 2 ty3 q e t where the y1 vector contains the budget shares and the y2 vector the price and real per capita expenditure variables

66 C.L.F. Attfieldr European Economic ReÕiew 41 (1997) 61–73

1.611

y0.531 0.052

y0.840 0.730

y0.132 1.184

Ž3.

3.12yFŽ6,61.

16.80y x 2 Ž6.

1% Critical value

Notes: All elasticities are computed at the point of the sample means.

Own price Expenditure

Ž1.

Ž2.

2.25yFŽ6,61.

16.799

(d) Estimated elasticities

Wald test statistic F-test statistic

12.60y x 6.

2Ž

5% Critical value

(c) System tests for homogeneity

y0.075 0.006

Ž4. y0.845 1.595

Ž5. 0.150 y0.099

Ž6.

C.L.F. Attfieldr European Economic ReÕiew 41 (1997) 61–73 67

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68

meaned and deseasonalised with the set of centred seasonal dummies. Assuming normality of the errors u1 t and u 2 t ML estimates are then obtained from the model y 1 t s By 2 t q C0 D y 2 t q C1 D y 2 ty1 q . . . qCk D y 2 tyk q Õt

Ž 2.

where the rq elements in B are the coefficients of the cointegrating demand system in Ž1.. The coefficients in the C matrices are treated as nuisance parameters. Let Zt s Ž D y 2 t , D y 2 ty1 , . . . , D y 2 t y k .X and C s Ž C0 , C1 , . . . , C k . and rewrite Ž2. in terms of all observations and we have Y s XBX q ZCX q V

Ž 3.

where Y contains all observations on y 1 t – that is the wi t – and X all observations on y 2 t – the pi t ’s and real per capita income, yt . Estimates of the coefficients of interest are then obtained from B˜X s Ž X X Q z X .

y1

X X QzY

where Q z s I y ZŽ ZX Z .y1 ZX . Phillips Ž1994. shows that a Wald test is valid for testing a hypothesis on the elements of B of the form Ho: R 1 Br 2 s r 3 , where R 1 is Ž g = r ., with rank g, r 2 is Ž q = 1. and r 3 is Ž g = 1. and all elements of R 1 , r 2 and r 3 are known. For the model in Ž1. g is the number of restrictions being tested. The form of the Wald test statistic is Ws

X Ž R1 Br 2 y r 3 . Ž R1 S˜u u RX1 .

r 2X

X

Ž X Qz X .

y1

y1

Ž R1 Br 2 y r 3 . r2

where S˜ u u s ny1 V˜ X V˜ and W has the exact finite sample distribution: ng Ws F Ž g , n y Ž 2 q k . q y g q 1. . ny Ž2qk . qygq1

Ž 4.

Eq. Ž4. is very similar to the relation between the F and Wald statistics derived by Phillips Ž1986. for the case of the multivariate linear model with fixed regressors and stationary dependent variables. The term Ž2 q k . q in Ž4. gives the total number of variables in the matrices X and Z. As an example of the use of Ž4. consider a test of the null hypothesis that a single coefficient in B, say bi j , is zero. In this case g s 1 and so R 1 s iXi , r 2 s i j and r 3 s 0 where i i is an Ž r = 1. vector with one in position i and zeros elsewhere and i j is a Ž q = 1. vector with one in position j and zeros elsewhere. The square root of the F-ratio obtained from Ž4. is then the t-ratio given in Table 3. To test the homogeneity hypothesis for the ith demand equation requires R 1 s iXi , r 2 s Ž1,1, 1, 1, 1, 1, 1, 0.X , and r 3 s 0. To test homogeneity in the complete system requires R 1 s Ig , r 2 s Ž1, 1, 1, 1, 1, 1, 1, 0.X , and r 3 the Ž g = 1. null vector Žwith g s 6..

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Table 3, panel Ža., gives the set of estimates obtained in the manner described above. Panels Žb. and Žc. give single equation and system test results for homogeneity and panel Žd. gives estimated own price and income elasticities of demand for each of the commodity groups analysed. For comparison purposes, multivariate least squares estimates and test statistics, which assume stationarity of all variables, are given in Table 4. Since the variables in the system are IŽ1., least squares estimators are consistent but the conventional t-ratios reported are incorrect. Assuming stationarity Attfield Ž1991. argues that, because equation errors are likely to be correlated with the expenditure variable, least squares estimators will be biased and inconsistent. When the variables are IŽ1. and cointegrated, however, Stock Ž1987. has shown that because least squares estimators are ‘super consistent’ any finite sample bias vanishes asymptotically whether the bias is due to an ‘errors in variables’ or simultaneity type problem. For the non stationary case, the TECM estimates in Table 3, a lag length of k s 3 was selected, although with k s 4 the values of ‘significant coefficients’ are not very different. 3 Also, because there is evidence of serial correlation in the equation errors, the covariance matrix S˜ u u was replaced by a Newey and West Ž1987. type formulation: jsp

Sˆ u u s Vˆ 0 q Ý f Ž j, p . Vˆ j q Vˆ jX

Ž 5.

js1 X Ž . w Ž .x where Vˆ j s ny1 Ýtsn tsjq1Õt Õtyj and f j, p s 1 y jr p q 1 . The results for the homogeneity test statistics for this case wnot reported herex were very similar to those quoted in Table 3 panels Žb. and Žc. with values for p, the lag window, of 3 and 4. For example with k s 4 and p s 4 the F-ratio is 2.437 which lies between 0.05 0.01 the 5% critical value of F, F6,52 s 2.278, and the 1% critical value, F6,52 s 3.170. Strictly, the F-test is no longer valid with the long-run covariance matrix given in Ž5. because the Newey–West result is asymptotic, but some finite sample adjustment to the Wald statistic is in order and that given by Ž4. is reasonable as an approximation. Since the homogeneity restrictions cannot be rejected for the whole system it follows that the model in Ž1. with the restricted estimates in place will still be a cointegrated system. Comparing the least squares estimates in Table 4, which assume stationarity, with the TECM estimates in Table 3, point estimates are different in magnitude and sometimes sign although the signs and magnitude of those coefficients with high t-ratios are generally very similar. After a finite sample adjustment, homogeneity for the complete demand system cannot be rejected for the multivariate linear estimates, while tests for homogeneity equation by equation suggest that the

3

With k s 4 there are 288 parameters estimated. With larger values of k estimates display some instability.

P2 0.0065 Ž0.70. y0.0042 y0.47. 0.0082 Ž1.13. y0.0566 Žy5.18. y0.0116 Žy2.11. y0.0575 y13.14.

P1 0.0419 Ž1.56. 0.0415 Ž1.63. y0.0026 Žy0.12. 0.0216 Ž0.69. 0.0114 Ž0.72. y0.0070 Žy0.56.

0.0801 Ž3.08. y0.0371 Žy1.51. 0.0380 Ž1.89. y0.0658 Žy2.16. 0.0121 Ž0.79. y0.0662 Žy5.45.

P3

P4 y0.0260 Žy2.62. 0.0107 Ž1.14. y0.0267 Žy3.47. 0.0920 Ž7.92. 0.0005 Ž0.08. y0.0037 Žy0.79.

P5 y0.0268 Žy1.61. 0.0030 Ž0.19. 0.0396 Ž3.07. 0.0063 Ž0.32. 0.0548 Ž5.59. y0.0010 Žy0.13.

P6 y0.0707 Žy5.32. y0.0051 Žy0.41. 0.0068 Ž0.66. 0.0026 Ž0.17. y0.0026 Žy0.34. 0.1269 Ž20.41. 0.0134 Ž0.39. y0.0209 Žy0.65. y0.0579 Žy2.20. y0.176 Žy0.44. y0.0648 Žy3.23. 0.0080 Ž0.50.

P7

‘Expenditure y0.1146 Žy6.29. y0.0428 Žy2.48. y0.0141 Žy1.00. y0.0069 Žy0.33. 0.0744 Ž6.95. y0.1104 Žy12.96.

0.99

0.89

0.80

0.94

0.93

R2 0.99

0.47

1.89

1.45

1.22

1.55

DW 1.62

wy2.00x

Ž0.006.

Ž0.006. w2.84x

y0.0122

0.0183

Ž2. 0.0054

w1.09x

Ž0.005.

Ž3.

wy2.31x

Ž0.008.

y0.0174

Ž4.

wy0.05x

Ž0.004.

y0.0002

Ž5.

wy0.19x

Ž0.003.

y0.0006

Ž6.

Notes: t-ratio is for a test of the null hypothesis that the sum of the coefficients on the price variables is zero.

Sum of price coefficients Estimated standard error t-ratio

Ž1.

(b) Single equation tests for homogeneity

Notes: t-ratios in parentheses. Commodity groups: Ž1. Food; Ž2. Alcoholic drink and tobacco; Ž3. Clothing and footwear; Ž4. Energy products; Ž5. Other goods; Ž6. Rent rates and water charges; Ž7. Other services.

Ž6.

Ž5.

Ž4.

Ž3.

Ž2.

Commodity Ž1.

Table 4 Unrestricted mulivariate least squares estimates assuming variables are stationary. (a) Parameter estimates

70 C.L.F. Attfieldr European Economic ReÕiew 41 (1997) 61–73

14.696

16.166

17.414 12.6

5% Critical valuey x 2 Ž6.

16.8

1% Critical valuey x 2 Ž6.

y0.672 0.415

y0.991 0.651

Ž2. y0.532 0.831

Ž3. 0.132 0.915

Ž4.

Notes: All elasticities are computed at the point of the sample means.

Own price Expenditure

Ž1.

(d) Estimated Elasticities

y0.584 1.665

Ž5. 0.139 0.105

Ž6.

Notes: Bartlett adjustment for the multivariate linear model calculated from the formula given by Attfield Ž1995.

Wald test statistic Likelihood ratio test statistic Bartlett adjusted LR statistic

(c) System tests for homogeneity

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commodity groups food, alcoholic drink and tobacco and energy products do not satisfy the homogeneity postulate. A similar conclusion holds for the tests on the TECM model in Table 3. Homogeneity cannot be rejected for the complete system but in this case only the food and tobacco and alcohol groups fail the single equation homogeneity test. 4 Ng Ž1992. using different techniques to estimate and test a cointegrating demand system, finds, for per capita data for the USA, that homogeneity cannot be rejected for food and clothing commodity groups, after allowing for polynomial trends in the demand equations. The most striking differences are in the estimated price and income elasticities. For the TECM models all own price elasticities, with the exception of group Ž6., rent, rates and water charges, have the correct sign and all display price inelasticity. Income elasticities have the correct sign, excluding group Ž6. again, but not only are food and tobacco and alcohol income inelastic, which might be expected, but energy products are highly income inelastic. Not surprisingly, clothing and footwear and other goods are income elastic. 4. Conclusion This paper has applied recent time series techniques for estimation and testing models in which variables are non-stationary in levels but stationary in first differences to the Deaton and Muellbauer Ž1980. demand model. The demand equations for the commodity groups for Ž1. food; Ž2. alcoholic drink and tobacco; Ž3. clothing and footwear; Ž4. energy products; Ž5. other goods; Ž6. rent, rates and water charges form a cointegrating system for data for the UK 1963Ži. to 1992Žii.. Appropriate estimation and testing for this situation demonstrates that homogeneity of prices and nominal income cannot be rejected for the demand system as a whole. Acknowledgements I am grateful to an Editor and two anonymous referees for constructive and helpful comments any errors or omissions are my own responsibility. References Attfield, C.L.F., 1991, Estimation and testing when explanatory variables are endogenous, Journal of Econometrics 48, 395–408.

4 There is no inconsistency in this result. It is always possible to reject an individual null hypothesis while accepting a joint hypothesis because the confidence region for the joint null can contain points which are not in the confidence interval for the individual hypothesis Žsee Goldberger, 1991, pp. 207–217; Davidson and MacKinnon, 1993, pp. 71–77.. When homogeneity within commodity groups is tested jointly across all commodities there is not enough evidence to reject the null.

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Attfield, C.L.F., 1995, A Bartlett adjustment to the likelihood ratio test for a system of equations, Journal of Econometrics 66, 207–223. Davidson, Russell and James G. MacKinnon, 1993, Estimation and inference in econometrics ŽOxford University Press, New York.. Deaton, A. and J. Muellbauer, 1980, An almost ideal demand system, American Economic Review 70, 312–326. Engle, Robert F. and C.W.J. Granger, 1987, Co-integration and error correction: Representation, estimation and testing, Econometrica 55, 251–276. Ghysels, E. and P. Perron, 1993, The effect of seasonal adjustment filters on tests for a unit root, Journal of Econometrics 55, 57–98. Goldberger, Arthur S., 1991, A course in econometrics ŽHarvard University Press, Cambridge, MA.. Haug, A.A., 1992, Critical values for the Zˆa-Phillips–Ouliaris test for cointegration, Oxford Bulletin of Economics and Statistics 54, 473–480. Haug, A.A., 1995, Tests for cointegration: A Monte Carlo comparison, Journal of Econometrics, forthcoming. Newey, W. and K.D. West, 1987, A simple positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica 55, 703–708. Ng, S., 1992, Testing for homogeneity in demand systems with non-stationary regressors, Mimeo. ŽPrinceton University, Princeton, NJ.. Phillips, P.C.B., 1986, The exact distribution of the Wald statistic, Econometrica 54, 881–895. Phillips, P.C.B., 1991, Optimal inference in cointegrated systems, Econometrica 59, 283–306. Phillips, P.C.B., 1994, Some exact distribution theory for maximum likelihood estimators of cointegrating coefficients in error correction models, Econometrica 62, 73–93. Phillips, P.C.B. and S. Ouliaris, 1990, Asymptotic properties of residual based tests for cointegration, Econometrica 58, 165–193. Phillips, P.C.B. and P. Perron, 1988, Testing for a unit root in time series regression, Biometrika 75, 335–346. Stock, James H., 1987, Asymptotic properties of least squares estimators of cointegrating vectors, Econometrica 55, 1035–1056. Stone, J.R.N., 1953, The measurement of consumers’ expenditure and behaviour in the United Kingdom, 1920–1938, Vol. 1 ŽCambridge University Press, Cambridge..