The Estimation of Postwar Regional Consumption Functions in Canada

The Estimation of Postwar Regional Consumption Functions in CanadaAuthor(s): W. J. Gillen and A. GuccioneSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 3, No. 2(May, 1970), pp. 276-290Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/133677 .

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THE ESTIMATION OF POSTWAR REGIONAL CONSUMPTION FUNCTIONS IN CANADA*

W. J. GILLEN and A. GUCCIONE University of Windsor

L'estimation defonctions de consommation regionales pour la periode d'apres-guerre au Canada. Les auteurs presentent dans leur article des estimations de la fonction de consommation au Canada, au niveau intermediaire des regions. Les regions retenues sont les suivantes: les Maritimes, le Quebec, l'Ontario, les Prairies, et la Colombie-britannique, y compris le Yukon et les Territoires du Nord-ouest. On trouve dans les Comptes nationaux (edition r6vis6e) des statistiques sur la consommation totale au Canada, et sur le revenu disponible par province, pour la periode de vingt ans qui va de 1947 a 1966. Les auteurs ont obtenu des estimations de la consommation par region pour sept annees, utilisant huit releves des depenses des consom- mateurs faits durant l'apres-guerre. Les prix sont obtenus a partir des indices de prix des principales villes publi6s par le Bureau federal de la statistique. Les auteurs ont estim6 les fonctions de consommation a l'aide d'une technique de regression proposee par Theil et Goldberger.

Il y a un certain nombre de fagons differentes de proc6der 'a l'estimation de la fonction de consommation. Les auteurs en ont fait l'examen. Les solutions les plus satisfaisantes sont les suivantes. La consommation comme fonction lineaire du revenu disponible en dollars courants. La consommation comme fonction lin6aire d'une estimation du revenu permanent. Ces deux solutions procurent des propensions marginales a consommer 'a long terme pour chaque region. Elles sont, de fagon g6n6rale, plausibles prima facie et compatibles l'une par rapport a l'autre. Ces valeurs s'6chelonnent de 0.90 en Ontario et en Colombie-britannique 'a un peu plus que un aux Maritimes. De plus, l'estimation faite sur la base du revenu permanent conduit 'a des estimations des propensions marginales a consommer a court terme. Ces valeurs s'6chelonnent de 0.49 pour l'Ontario et la Colombie-britannique a 0.56 pour les Maritimes et les Prairies. Les auteurs montrent aussi que, dans un cadre d'hypothbses acceptable, les estimations pour les regions sont compatibles avec les estimations de la fonction globale pour l'ensemble du Canada.

Les auteurs font etat de recherches futures dans ce domaine des fonctions de consommation regionales. Ils examinent la possibilite de faire des estimations par region des autres variables des mod&les de d6termination du revenu.

I / Introduction

This study of regional consumption functions has two purposes: first, to add to the present knowledge of Canadian consumption patterns;' and second, to begin an investigation of the feasibility of estimating a model of interregional income determination in Canada. Regarding the first purpose, this appears to be the first time that stress has been placed on the regional disaggregation of total Canadian consumption. The second purpose requires more elaboration. *We wish to thank two referees of this JOURNAL for some very helpful comments on the first draft of this paper. 'See especially, A. Asimakopulos, "Analysis of Canadian Consumer Expenditure Surveys," Canadian Journal of Economics and Political Science, XXXI, no. 2 (May 1965), 222-41; T. M. Brown, "Habit Persistence and Lags in Consumer Behaviour," Econometrica XX, no. 3 (July 1952), 335-71; R. D. Millican, "A Factor Analysis of Canadian Urban Family Expendi- tures," Canadian Journal of Economics and Political Science, XXX, no. 2, (May 1964), 241-7; K. S. Palda, "A Comparison of Consumer Expenditures in Quebec and Ontario," ibid., XXXIII, no. 1 (February 1967), 16-26.

Canadian Journal of Economics/Revue canadienne d'Economique, III, no. 2 May/mai 1970. Printed in Canada/Imprime au Canada.



Postwar Regional Consumption Functions 277

The properties of Keynesian models of interregional income determination have been known for many years.2 Given the increasing concern with regional income differences in Canada, it is notable that no attempt to fit such a model to Canadian data has been published.3 One reason for this is the difficulty entailed in obtaining suitable regional information. Some of the required data are available only for selected years. Other necessary data are simply not available, and ad hoc assumptions must be made for the use of proxy variables. In addition, the high levels of multicollinearity between many regional time series make it difficult to obtain parameter estimates with acceptable standard errors. The difficulties encountered in the estimation of regional consumption functions are typical of the difficulties described above. Hence, the statistical techniques used here should prove useful in estimating a model of interregional income determination for Canada.4

For the purpose of estimating postwar regional consumption functions in Canada, the years 1947 to 1966 inclusive were chosen. The regions considered were the Maritimes, Quebec, Ontario, the Prairies, and British Columbia, including the Yukon and Northwest Territories. A description of the data and estimation methods used is contained in section II. In section III we investi- gate a number of alternative specifications of the regional consumption functions. Two of these give satisfactory results. Section IV summarizes the findings and indicates some of their implications.

II / The data

Data on personal disposable income are available annually, by province, from the National Accounts.5 However, the National Accounts do not contain data on provincial constant dollar incomes6 or on regional consumption. In this section estimates of annual constant dollar income are computed for each region by using the consumer price indexes for major Canadian cities. Also,

2See J. Airov, "Fiscal-Policy Theory in an Interregional Economy: General Interregional Multipliers and their Application," The Regional Science Association Papers, 19 (1967), 83-108; J. S. Chipman, "The Multi-Sector Multiplier," Econometrica 18, no. 4 (October 1950), 355-74; J. S. Chipman, The Theory of Inter-Sectoral Money Flows and Income Formation (Baltimore, 1950); R. M. Goodwin, "The Multiplier as a Matrix," Economic Journal, 59, (December 1949), 537-55; L. A. Metzler, "A Multiple-Region Theory of Income and Trade," Econometrica 18, no. 4 (October 1950), 329-54. 3Indeed, there have been few attempts to apply the model anywhere. R. E. Bolton estimated a simplified version for the United States. R. E. Bolton, Defense Purchases and Regional Growth (Washington, 1966). 4We are currently working on the estimation of a Metzler-type model for Canada. Preliminary investigation of some of the functions in the model suggests that the technique used in this work is promising. 5The Canadian National Income and Expenditure Accounts are currently being revised. Some of the revised data are published in: Dominion Bureau of Statistics, National Income and Expenditure Accounts 1926-1968 (Ottawa, 1969). A revised table on "Personal Disposable Income, Geographical Distribution, 1926-68" was kindly made available, prior to its publica- tion, by the Dominion Bureau of Statistics. We are indebted to one of the referees of this JOURNAL for informing us of the availability of these revisions. 6Hereinafter the term "income" refers to "personal disposable income."



278 W. J. GILLEN and A. GUCCIONE

estimates of constant dollar regional consumption are obtained for seven years, by using the postwar surveys of consumer expenditures.7

The Dominion Bureau of Statistics reports consumer price indexes for at least one major city in each Province except Prince Edward Island.8 A first approximation to constant dollar income in each province was obtained by deflating the province's current dollar income with the price index of the city located in that province. For Prince Edward Island the index for Halifax was used.9 Constant dollar income (1949 dollars) for Canada was calculated by dividing the current dollar figures by the Canadian Consumer Price Index. The first approximations to provincial constant dollar income were then pro-rated to obtain final estimates of provincial income in 1949 dollars. The pro-rating was done in order to force the total of the provincial estimates in each year to equal the Canadian (constant dollar) income. The provincial estimates were then aggregated to obtain regional data.

The Canadian Consumer Price Index was used to deflate total Canadian consumption. The data on consumption were obtained from the National A ccounts.

An unsuccessful attempt was made to estimate the regional marginal propensities to consume, for a simple form of the consumption function, using only the data described above. Assume that the regional consumption functions are of the form

(1) Crt Bro + Brlyrt + Urt (r = 1, ..., 5)

where Cr = total consumption in region r at time t; Yrt = total personal disposable income in region r at time t; Ur t = a stochastic term for region r at time t; and Bro and Br, are constants depending on r. Equation 1 implies

(2) Ct = Bo + B11Yit + B21Y2t + ... + B51Y5t + Ut

5 5 5

where Ct =Z Crt, Bo = E BrO, Uz = E Urt. r=1 r=1 r=1 Equation 2 was fitted by ordinary least squares to the data for 1947-66. Unfortunately the large size of the standard errors prevented any reasonable inference from the point estimates.'0 It appears that the high correlations

7See the following publications of the Dominion Bureau of Statistics: Canadian Non-Farm Family Expenditure, 1947-1948, Reference Paper no. 42 (Ottawa, 1953); City Family Expendi- ture, 1953, Reference Paper no. 64 (Ottawa, 1956); City Family Expenditure, 1955, Reference Paper no. 83 (Ottawa, 1957); City Family Expenditure, 1957, Occasional no. 62-517 (Ottawa, 1961); Farm Family Living Expenditure, 1958, Occasional no. 62-523 (Ottawa, 1966); Urban Family Expenditure, 1959, Occasional no. 62-521 (Ottawa, 1963); Urban Family Expenditure, 1962, Occasional no. 62-525 (Ottawa, 1967); Urban Family Expenditure, 1964, Occasional no. 62-527 (Ottawa, 1968). 8The Consumer Price Indexes for Canada and "regional cities" are reported in: Dominion Bureau of Statistics, Prices and Price Indexes (Ottawa, monthly). 9The published index for St. John's, Newfoundland, uses 1951 as the base year, and data for 1949 and 1950 are not available. For these years, the Halifax index was substituted and the index for St. John's was converted to a 1949 base. Also, the Toronto index was chosen from the two indexes available for Ontario. '0The results of the regression were:

Ct = 499.41 + 0.99 Ylt + 0.08 Y2t + 2.29 Y3 - 0.55 Y4t + 0.27 Y5u (R2 = 0.99). (422.93) (1.49) (0.63) (0.56) (0.30) (1.03)




among the regional incomes preclude the use of this technique. Therefore alternative methods had to be tried. These methods required the introduction of additional information which was obtained from the postwar surveys of consumer expenditures.

Eight postwar surveys of consumer expenditures were used to estimate regional consumption for the years 1948, 1953, 1955, 1957, 1959, 1962, and 1964.11 The surveys are very heterogeneous, and none used a sample represen- tative of the total regional population.12 For this reason a number of ad hoc assumptions were required to derive regional consumption estimates from the survey data. With one exception,'3 estimates of regional consumption were obtained by a four-step procedure.

Step 1 The following areas were distinguished in each province. Area A: cities of 30,000 or more persons; area B: other urban areas; area C: farms; area D: other rural areas. Plausible proxy variables for per capita consumption in many of these areas are available from the surveys. These proxy variables were used as a first approximation to per capita consumption. Table I shows the areas and the survey data used.

Step 2 Ad hoc assumptions were used to obtain a first approximation to per capita consumption in areas for which the survey data did not provide proxy variables. These assumptions are: (i) Within each province, the ratio of per capita consumption in one area to per capita consumption in another area was constant between 1947 and 1964. This assumption allows the 1959 estimates to be used as a benchmark. (ii) Between 1958 and 1959 per capita consumption on farms in each province increased at the same rate as per capita consumption in Canada. This assumption allows the 1958 survey of farm family living expenditures to be up-dated to 1959 and integrated with the 1959 survey of urban family expenditures. (iii) The ratio of per capita consumption in each area in Newfoundland to per capita consumption in each area of Nova Scotia was constant between 1953 and 1959. This assumption was used because data are not available for Newfoundland in 1953, 1955, 1958 and 1959. (iv) Between 1953 and 1955, per capita consumption in area A in Alberta increased at the same rate as per capita consumption in area A in Saskat- chewan. This assumption was used because data are not available for Edmon- ton, Alberta, in 1953.

Step 3 The estimates of per capita consumption obtained in steps one and two were deflated by regional price indexes,'4 and then multiplied by popula-

"Data from the survey for August 1947 to August 1948 were taken to refer to the year 1948. Some experimentation with the data showed that this simplification had no effect on the estimates of the regional consumption functions. 12See I. McWhinney "Surveys of Family Expenditure," Special reprint from The Canadian Statistical Review (June 1967) (Ottawa, 1967). '3As a first approximation it was assumed that consumption in the Yukon and Northwest Territories equalled personal disposable income. This was done because of the lack of any data directly related to consumption in the Yukon and Northwest Territories. Since personal disposable income in these regions is relatively small, little bias is introduced by the assumption. 14The regional price indexes used were those implicit in the estimates of regional constant dollar personal disposable income calculated above.




TABLE I

FIRST APPROXIMATIONS BASED ON SURVEY DATAa

First approximations per capita consumption in: Survey data

Provinces Areas" Years Per capita constumption in

Newfoundland A 1957,62,64 St. John's Prince Edward Island A 1953,55,57,59,62,64 Halifaxc Nova Scotia A 1953,55,57, 59, 62, 64 Halifaxc New Brunswick A 1953,55,57,59,62,64 Halifaxc Quebec A 1953,55,57,59,62,64 Montreal Ontario A 1953,55,57,59,62,64 Toronto Manitoba A 1953,55,57,59, 62, 64 Winnipeg Sasketchewan A 1953,55,57,59,62,64 Winnipeg Alberta A 1955,57,59,62,64 Edmonton British Columbia A 1953,55,57,59,62,64 Vancouver All provinces A, B, D 1948 Non-farm areas in

each regiond except C, D 1958 Farms in each

province Newfoundland B 1959 Urban areas in each

regiond excluding the above cities

aThe sources for the survey data are shown in n. 7. Consumption was calculated as "all current consumption" plus "gifts and contributions" reported in the surveys. 5See text for area definitions. e1959 data for Halifax were provided by Dominion Bureau of Statistics in correspondence. dThe 1947-8 and 1959 survey data are published by region; the regions are those listed on pp. 1-2 above, except that Newfoundland is not included in the Maritimes in 1947-8.

tion estimates'5 for the areas to produce a first approximation to total constant dollar consumption in each area.

Step 4 Final estimates of total consumption in each area were obtained by pro-rating first approximations. The pro-rating was done in such a wNay as to force the sum of the final estimates in each year to equal total Canadian constant dollar consumption. In each year the number used for pro-rating was the ratio of total consumption to the sum of the first approximations.

The final estimates were aggregated into the regional estimates slhown in Table II.

The data in Table II could be used directly to estimate some specifications of the regional consumption functions. However the limitation of the statistical analysis to these data would involve a considerable loss of information. In particular, the information contained in the National Accounts for postwar

'6Total population for each Province in all years was obtained from Dominion Bureau of Statistics, National Accounts Income and Expenditure (Ottawa, annual). Population data for each area in the Province in 1951, 1956, and 1961 were taken from Dominion Bureau of Statistics, 1961 Census of Canada, General Review, Rural and Urban Population, Series 7.1 (Ottawa, 1963), 7, 22-9. Area population estimates were then obtained by assu-ming a linear trend in the proportion of total population in each area in each Province from 1947 to 1956 and from 1956 to 1964.




TABLE I I

REGIONAL CONSUMPTION ESTIMATES (millions of 1949 dollars)a

British Year Maritilmiesb Quebec Ontario Prairies Columbia

1948 744 2606 4059 1881 1071 1953 1114 3402 5250 2518 1323 1955 1423 3825 5762 2810 1559 1957 1596 4389 6431 3103 1614 1959 1364 4930 7041 3451 1886 1962 1661 5532 7325 3780 2081 1964 1678 5813 9042 3799 2303

aThe estimates for each year total to the National Accounts figure for Canadian constumptioni (after deflation). bExcludes Newfoundland for 1948.

years other than the seven years in Table II would be neglected. A preferable alternative is to combine the National Accounts' data with the regional consumption figures by means of an appropriate statistical model. A technique capable of conmbining data from different samples has been suggested by Theil and Goldberger."6 The specific application of this model to the estimation of regional consumnption functions is described in sectioi III.

III / Emppirical estimates of alternative models

In this section we specify and estimate a number of alternative consumption functions. We begin by writing current consumption as a linear function of current incomle only, and then introduce additional variables in an attempt to capture the effect of past income and/or consumption.'7 It turns out that some specifications, appealing on a priori grounds, are not useful because of statistical difficulties. In fact, out of the seven forms described below, only two are satisfactory.

The first form considered is that shown in equation 1 above. It states the very simple hypothesis that current consumption is a liinear function of current income. To apply the Theil-Goldberger estimation technique to this form, equations 1 and 2 are rewritten in the following way. For the seven years for which regional consumption data were obtained in section II, write equation 1 as

(3) [C]r = [A]rr[B]r + [U]r (r = 1, ... , 5)

"H. Theil and A. S. Goldberger, "On Pure and Mixed Statistical Estimation in Economics," International Economic Review, 2, no. 1 (January 1961), 65-78. The reader is referred to this paper for a discussion of the statistical properties of this technique. 17All variables are defined in constanit dollar terms. Sorne experimentation suggests that estimates using current dollar data, while clifficult to justify theoretically, are not substantially different from those obtained using constant dollar data.




where

FCrl1 1 Yrl1 [Url] Cr121 1 Ir2 Ur2 II [ 1 [~~~~Bro] .

[Chr= 0. ,[A]rr= j ,[B]r LI Lr

Lr7J 1 YrJ Lu77i

Note that the subscripts denoting the year have been rearranged so that the first seven observations refer to the years 1948, 53, 55, 57, 59, 62, and 64, respectively. The last thirteen observations will now refer to the years 1947 to 1966 excluding the above seven years. For the last thirteen observations write equation 2 as

(4) [C]6 = [A]61[B]1 + [A]62[B]2 + + [A ]65[B]5 + [U]6 where

[C8 1 Yr8 U8 C9 [1 Yr9 I U9I

[C16= ,A6r ,[U]6=

LC20_ 1 Yr2oj LU20J

Combining equations 3 and 4 we obtain

[C]1 r[A ll _ [B], I U]i [C]2 [A ]22 [B]2 [ U]2

[C]5 [A]55 L_[B]5_J [ U]5 _[C]6 L[A ]61 [A ]62 [A]65 [U]6

or, in more compact notation,

(5) C=AB+ U.

The generalized least squares estimator of B is

(6) B = [A'Q-'A]-'A'Q-'C

where Q is the 48 by 48 variance-covariance matrix of the stochastic terms. In applying this model, the stochastic terms were assumed to be serially

independent, independent across regions, and homoscedastic with each region. These assumptions imply that Q is of the form

-2 I7,L7

0217

05211 q2'1




where ot2 is the variance from equation 3 for region r, (J2 iS the variance from equation 4, and In, is the n by n identity matrix.'8 Since the diagonal elements are unknown, an iterative procedure was used to obtain B. The seven observations for each region's consumption and the twenty observations on national consumption were treated as separate samples to obtain initial estimates of the diagonal elements. These were used to form a first approximation to U- and, using equation 6, to B. A second round of approximation was obtained by inserting f3 into equation 5, computing the resulting estimate of Q-1, and re-estimating B from equation 6 with the new values of '-1.19

The regional consumption functions obtained by this procedure are: 20

Maritimes: Clt- 72.87 + 0.994Y,t (R2 = 0.85) (231.82) (0.174)

Quebec: C2 t -120.36 + 0.974Y2t (R2 = 0.99) (219.85) (0.047)

Ontario: C3G = 60.67 + 0.909Y35 (R?2 = 0.97) (396.43) (0.056)

Prairies: C4t= 95.29 + 0.917Y41 (R 2 0.89) (451.12) (0.139)

British Columbia: C5t = -12.93 + 0.901 Y5t (RJ2 0.96) (153.81) (0.080)

A priori one would expect all intercepts to be non-negative. The estimated intercepts are negative for Quebec and British Columbia, but none of the regional intercepts is significantly different from zero. Indeed, in all cases the standard error of the intercept is larger than the point estimate. All the marginal propensities to consume lie between 0.90 and 0.99, a result consistent

18More generally one could assume that ro fort s

Urt prUp,t_l + Vrt with E(VrtVke) =

Or,k for t = s and-1 < Pr < 1. It can be shown that, for t _ s,

E(Urt Uks) prtS5rk

E(Urt U.) = Z5k-1 Ptark

E(U,8Ut) =,5r_1 prtark

E(Ut U.) =6r=i L,k-1 Prt a6rk where (ry = cTrk/l PrPk) = b5kr.

These equations determine the elements of Q in terms of the 20 parameters 5rk and Pr (r, k = 1, ... 5; k > r). It would be possible to estimate these parameters on the basis of the 48 observations available here. It is our judgement, however, that the resulting loss of degrees of freedom is not justified by the increased generality of the assumption. 191n the second round of approximations an unbiased estimate of rl (r = 1, ..., 5) could be obtained from the formula (1/nr)[ U]r'[f U]r, where [ U]r is the first approximation to [ U]r, and nrNr - trace[A/JAr(A'A)<11 with Nr defined as the number of observation on CGt (i.e., seven) and Ar defined as the rows of A containing [A]rr. It follows that 5 < nr _ 7. We used the approximation nr = 6. Similarly to estimate a2 in the second round the number of degrees of freedom was approximated as 17.5. 20The R2 here and in the following regressions is defined as the proportion of total variation explained using the regression coefficients obtained from the iterative procedure described in the text. The years used in this computation were the seven years for which regional consumption estimates are available.




with theory and with the literature on time series estimation of the consumption function. There are no, significant differences among the regional marginal propensities. However, it may be noted that the ,point estimates for the Maritimes and Quebec are relatively high. There is in fact a, tendency,for the estimates of the regional marginal propensities to consume to rank in the same order as regional per capita income.21

These regional consumption functions are, consistent with: the. results obtained by fitting an aggregate consumption function for.Canada. Define the aggregate consumption function as

(7) Ct = ao + alYt + Ut

where U, is stochastic and

ct = LCr t, Yt L YT t-. rl r==1 The parameters of equation 7 can be estimated directly by ordinary least squares. They can also be estimated indirectly from the regional consumption functions if it is assumed that the regional distribution of income has been .approximately constant,22 i.e.,

(Yrt/Yt) = Kr + (Vrt/Yt) (r 1, ..., 5) where V,, is stochastic and Kr is a constant depending o-nly on r. If Kr iS estimated as:

19l66 F. R1r = SF -(r = 1..., 5) 20 t=1947 yt

and ao and &i,, are defined as:23 5 5

aAo - rO; &i -E riKr r=1 r=1

then the regional estimates imply the following function for Canada:

Ct = 95.53 + 0.932Yt This is virtually identical with the direct estimate of equation 7:

ct= -27.55 + 0.940Yt (R2 = 0.99) (285.59) (0.016) (d = 0.98).

2iAn attempt was made to introduce explicitly the effect of per capita income on the, -marginal propensities to consume. If

=;t -B,0 + Brt Y,0 + Ur- and Br,5 czo + ari(Yrt/Prt) + Vrt

where P,t is regional population, then Crt = IIAO + 1171 Y7t + IEI2( Yr2/Pr) + W7t

where W7t is stochastic. This function was estimated but the results were inconclusive. 22During 1947 to 1966, the largest change in the per cent of Canadian income in a.region occurred in the Prairies, where income in 1948 was 20.5 per cent and in 1961 was 16.4 per cent of Canadian income. In the other regions the extremes were: 26.5 and 24.2 per cent in Quebec; 39.4 and 37.7 per cent in Ontario; 11.2 and 9.9. per cent in British Columbia; 7.8 and 6.8 per cent in the Maritimes. 231t is easily shown that the regional consumption functions are related to equation 7 by

ceo = 5r1rOg a1 = Zr=371K7, Ut = Zr=i(7V7t + o-rt)




The regional estimates of the simple function discussed above appear reasonable when considered individually, and when compared with the results obtained from the nationally aggregated function. Nevertheless, there is some reason to believe that a more complex consumption function is needed. It is well known, especially from the experience in the United States, that the simple form used above can fail to provide accurate forecasts of consumption, especially in periods of rapid change.24 From a slightly different point of view, the assumption of no serial correlation has frequently been rejected when this function was estimated.25 These difficulties have led to attempts to introduce lagged variables into the consumption function. In particular, T. M. Brown has experimented with the introduction of several forms of lagged income and lagged consumption in his analysis of the Canadian data.

It seemed desirable, therefore, to introduce into the regional functions some measure of consumers' dependence on past patterns of behaviour. The following forms were tried without success:

(8) .Crt = Bro + BrlYrt + BT2Yr,t-1

(9) Cr = BTO + BriYrt + Br2Yrtyr,t.l

(10) Crt = Bro + Brlyrt + BT2YTt2

(11 ) Cr t-BrO + Brlyrt + Br2Ct-.

Equations 8, 9, and 10 are taken directly from T. M. Brown's equations 2, 7, and 7a. Equation 11 is also derived from Brown,26 but the additional assumption is made that consumption in each region depends on total Cana- dian consumption of the previous year rather than on lagged regional consumption. While this assumption was suggested by the lack of a complete time series for regional consumption, it can be interpreted in terms of a demonstra- tion effect operating across regions. Equations 8 to 11 were estimated by applying the Theil-Goldberger technique. The results were poor. In all cases the majority of the coefficients was not statistically significant, and none of the equations provided a significantly better fit to the data than did equation 1.27 The size of the standard errors suggests that the cause of the poor results is mainly the high correlation between current and lagged income and between current income and lagged consumption.

As frequently suggested when multicollinearity is present, additional information or assumptions may overcome the problem. One possibility in the present case is to make use of the relationship between Brown's equations

24See G. Ackley, Macroeconomic Theory (New York, 1961), 246-9. 25See, for example, Brown, "Habit Persistence and Lags in Consumer Behaviour," 356. There are difficulties in testing for serial correlation in regressions estimated by the technique used in the present paper. However, the Durbin-Watson test can be applied to equation 7. The Durbin-Watson statistic for the estimate of equation 7 is 0.98, which is significant at the five per cent level. 26For a detailed discussion of these equations see Brown, "Habit Persistence and Lags in Consumer Behaviour," 358-60. 27On the other hand, we should note Brown's statement: "In the statistical testing of the above. . . hypotheses (i.e. Brown's equations 2, 4, 7, 7a) it was found that all of them ... gave rather good explanations of consumer behaviour in the period studied." "Habit Persistence and Lags in Consumer Behaviour," 360.




and Friedman's theory of the consumption function.28 Using this notion, a model which avoids multicollinearity while still capturing historical influences on consumption, is as follows. Assume

(12) Crt = Bro + Br1Zrt + Urt (r = 1, ..., 5)

where

(13) Zrt = a Ej XAYr,t-s (0 < X < 1). s=O

a and X are constants and Ur, is a stochastic term. Assume also that the regional distribution of income is approximately constant:

(14) (Yrt/Yt) = Kr + (Vrt/Yt) (r = 1, ..., 5)

where Vr7 is stochastic and independent of (Bria) * Zrt may be interpreted either as an attempt to estimate Friedman's "permanent income"29 by region, or more simply, as a statistical device to replace a series of lagged regional incomes by a single variable. In either case, the substitution of Z for a number of lagged incomes provides a device to avoid the effects of the intercorrelation between the incomes. It may be noted that the weight X is assumed constant across regions.30 Equations 12 and 13 imply

(15) Crt = BrO + (Bria) E XsYr,t-s + Urt s=O

which expresses current consumption as a linear function of present and past income, and which can be estimated with the available data. The assumptions of the model make it possible to derive an estimate of X from the National Accounts data. Given the estimate of X,

co

E: XS Yr, t-s s=O

can be approximated and, in turn, can be used to estimate Bro and (Bria). Specifically, equations 14 and 15 imply

(16) Crt = Bro(1 - X) + (Br,aKr)Yt + XCr, t-1 + Wrt

where the stochastic term Wrt is assumed to be homoscedastic and serially independent.3' Summation over r yields

5 (5 5

(17) Ct = E Bro(1 - X) + Br7aKr} Yt + XCt-1 + E Wrt. r=1 r=1 r=1

28See L. R. Klein, "The Friedman-Becker Illusion," Journal of Political Economy, LXVI, no. 6 (December 1958), 539-45. 29For an interpretation of a and X in terms of permanent income, see M. Friedman, A Theory of the Consumption Function (Princeton, 1957), 142-52. 30A. A. Walters states that X is about 3 in the United States and about I in the United Kingdom. This variation suggests X may vary from region to region. Therefore the assumption that X is constant across Canada, an assumption imposed by the lack of data, may hold only approximately. See A. A. Walters, An Introduction to Econometrics (Toronto, 1968), 263. We have taken the liberty of correcting the obvious transposition of countries in the source. 31For the properties of ordinary least squares estimators for such a model, see J. Johnston, Econometric Methods (New York, 1960), 212.




Equation 17 was estimated from the National Accounts data for 1947 to 1966 by ordinary least squares. The results are:

C= -100.99 + 0.511Yt + 0.483Ct1 (R]2 = 0.99) (219.35) (0.116) (0.130) (d = 1.54).

The coefficient of lagged consumption here is an estimate of X necessary for the computation of permanent income.32 One may note that, although equation 17 was derived from a permanent income type of argument, it is in fact identical to T. M. Brown's equation 4. Similarly, after equation 15 has been estimated for each region, it can be transformed into the regional counterpart of Brown's specification (compare equation 16 with Brown's equation 4).

Approximations to

E: XsYr,t-s s=O

were obtained by substituting the estimate of X obtained above and dropping all terms in powers of X higher than seven.33 These approximations are denoted by Yr,*. The Theil-Goldberger technique was applied to the resulting data to estimate equation 15. The regional consumption functions obtained by this procedure are:

Maritimes: C1t = 10.67 + 0.556Y,t* (R2 = 0.83) (248.33) (0.100)

Quebec: C2 = -179.85 + 0.535Y2t* (R2 = 0.99) (194.06) (0.022)

Ontario: C3t = 74.55 + 0.489Y3t* (R2 = 0.97) (387.52) (0.029)

Prairies: C4 = -311.66 + 0.560Y4 * (R2 = 0.91) (408.58) (0.067)

British Columbia: C5= -15.22 + 0.491Y51* (R2 = 0.97) (122.36) (0.034)

We would have preferred to fit equation 16 directly for each region. Lack of regional data however prevented this, causing a number of statistical difficulties. One such difficulty requires elaboration. Since equation 16 cannot be estimated directly we have no way to analyze the behaviour of its residuals. However the Durbin-Watson statistic for equation 17 suggests that the stochastic terms in 17 are serially independent. This would lead one to suspect that the stochastic terms in equation 16 are also serially independent, since 16 is simply the regional version of equation 17. Now the assumptions that the

32There may be some downward bias in this estimate of X, resulting from the effect of transitory consumption. See Walters, An Introduction to Econometrics, 262. However we assume that the use of annual data and aggregation across Canada reduce substantially the importance of transitory consumption. 33Some experimentation suggests the regressions are not sensitive to small variations in the number of terms used. For a similar point of view see Friedman, A Theory of the Consumption Function, 146.




residuals in equation 15 and in equation 16 are serially independent are incon- sistent.34 We conclude therefore that equation 15 contains serially correlated residuals. This by itself does not impair the consistency property of the point estimates of the coefficients. However, the estimates of the standard errors are biased downward and hence the usual tests of significance are impaired.35

Turning to the point estimates of the coefficients of equation 15, two comments are appropriate. First, none of the intercepts are significantly different from zero. Second, the ranking of the regional income coefficients has changed slightly from that obtained from the estimates for equation 1. The Maritimes and Quebec remain somewhat high relative to Ontario and British Columbia, but the coefficient for the Prairies has increased relative to all other regions. This last result is what one would expect given the apparently large transitory part of personal disposable income in the Prairies.36 More detailed comparison between these two sets of results is reported in the next section.

As already noted one can transform a consumption function in which consumption depends on permanent income (as written above) into one in which consumption depends on current income and lagged consumption. Making this transformation on our regional estimates, and aggregating by the technique described above, one obtains the following function for Canada in total:

C: =-217.92 + 0.519Yt + 0.483Ct_1.

This is very similar to the ordinary least squares estimate of equation 17.37

IV / Conclusion

In the previous section a number of specifications of the consumption function were considered. The best results were obtained from the two forms:

(1) Crt = Bro + Brlyrt + Urt and

(12) Crt = Bro + BrlZrt + Vrt

34See A. S. Goldberger, Econometric Theory (New York, 1964), 274-78. 35It is recognized that an estimate of the variance-covariance matrix for equation 15 could be obtained using the estimate of X and the assumptions of the model. Generalized least squares could then be applied. However, other statistical problems cannot be excluded - for example, errors in variables, and simultaneous bias. The state of the econometric art is such that .... the researcher may well have to decide which complicating factor is most important in his data and adopt the model and procedure that would be appropriate if the others were absent." (Gold- berger, Econometric Theory, 388). In our view, circumventing the scarcity of regional data and the effects of multicollinearity is the central problem at the present stage of regional research. 36Year-to-year fluctuations in real income in the Prairies are noticeably wider than in any other region. For example, there are six year-to-year decreases in the Prairies income, two in the Maritimes, one in Quebec and British Columbia, and none in Ontario. This suggests a relatively large transitory component in the Prairies income. Alternatively, the same conclusion can be reached by computing the variance of permanent income relative to the variance of current income. 87For comparable results see Table I in R. J. Ball and P. S. Drake, ''The Relationship Between Aggregate Consumption and Wealth," International Economic Review, 5, no. 1 (January 1964) 63-81.




7

(13) Zr- a E 2ayrtY 8=0

with the implication

(18) Crt = Bro + (Bria)yrt* + Vrt.

Algebraically the first form does not allow one to distinguish between long and short run marginal propensities to consume. However, since the equations in form 1 were estimated from annual data for twenty years, the slopes may be interpreted as longrun propensities. The fact that the estimates of the intercept term do not differ significantly from zero gives support to this interpretation. The second form allows the derivation of both shortrun and longrun marginal propensities to consume. For the shortrun it is defined as the derivative of equation 18 with respect to current income, i.e., (Brla). For the longrun it is defined as the derivative of equation 18 with respect to aYrt*. Given an estimate of a, the longrun propensity can be obtained. We define:38

7 s s-1

where g is the rate of growth. Setting g equal to 0.044 (the average annual rate of growth of personal disposable income in Canada between 1947 and 1966), estimates of the regional long run marginal propensity to consume are obtained.

Table III summarizes the estimates obtained from the two specifications. Excepting the Prairies, the longrun marginal propensities generated by the two models are similar; the estimates obtained from the second model are slightly higher than those from equation 1 but they reveal the same pattern. The two point estimates of the longrun marginal propensity for the Prairies are noticeably different. In the belief that annual income in the Prairies contains a relatively large transitory component, we expected the first form to underestimate in that region. Yet the longrun estimate given by the second model appears unacceptably high. This suggests that additional research on regional consumption functions in Canada might be directed toward a disaggregation of income, especially in the Prairies.

No attempt has been made here to relate explicitly our findings to the literature on the role of wealth in the determination of consumption. Before any empirical work can be done on this relation at a regional level, the consumption models developed for closed economies need to be adapted to open (regional)

880ur definition of a is obtained as follows. From equations 12 and 13 Crt -Bro + Bria Z78=oX8 Yr,t-. + Vrt

Assume a constant growth rate g. Then it is true that Yr,t-.s = [11(1 + g)M Yt

hence, by substitution in the above equation, we obtain Crt Bro + Bria L78=o[X/(l + g)] Yt + Vrt

The definition of a given in the text then implies that CrGt BrJ + Br1 Yrt + Vrt

and the longrun marginal propensity to consume is Bri, which is consistent with equation 12. Our definition of a and of the longrun marginal propensity to consume are equivalent to the expressions given by M. K. Evans, "The Importance of Wealth In The Consumption Function," Journal of Political Economy, 75, no. 4 (August 1967), 335-51. In particular, see Appendix B, equation 1.




TABLE III

ESTIMATED REGIONAL MARGINAL PROPENSITIES TO CONSUME

From equations 12, 13, 18 From equation 1

Region longrun Longrun Shortrun

Maritimes 0.994 1.031 0.556 Quebec 0.974 0.992 0.535 Ontario 0.909 0.907 0.489 Prairies 0.917 1.039 0.560 British Columbia 0.901 0.911 0.491

economies. In particular, the connection between regional saving and regional wealth needs not to be the same as the connection between saving and wealth in a closed economy. Nevertheless, the estimates for our second specification of the consumption function are close enough to those obtained by R. J. Ball and P. S. Drake39 to suggest that further research in this area may be fruitful.

More generally, regional consumption functions are only a part of the estimation of a model of regional income determination. There are well known limitations on the data available for this purpose. We suggest, however, that the results reported in this study warrant at least mild optimism about the feasibility of estimating such a model for Canadian regions.

39See R. J. Ball and P. S. Drake, "The Relationship Between Aggregate Consumption and Wealth."



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The Estimation of Postwar Regional Consumption Functions in Canada