estimating consumption from expenditure data

Journal of Public Economics 23 (1984) 169-181. North-Holland

ESTIMATING CONSUMPTION FROM EXPENDITURE DATA

J.A. KAY, M.J. KEEN and C.N. MORRIS*

Institute for Fiscal Studies. II2 Castle Lane, London SWIE 6DR, U.K

Received August 1982, revised version received March 1983

Most studies of distributional issues have focused on the distribution of income. But the income information usually available suffers from well-known deficiencies. This paper explores an alternative approach to distributional analysis, based instead on expenditure data. In many countries, including the United Kingdom, available household budget data provides information on expenditures only over a short period. The problems that consequently arise in interpreting data of this sort are emphasised. A general specification for the stochastic structure of expenditure data is presented, and a procedure for estimating participant households’ underlying consumption from their observed expenditures developed and discussed.

1. Introduction

In many countries, including the United Kingdom, the principal source of information on household living standards is a large cross-section budget survey. This paper is concerned with the use of such surveys in the analysis of distributional issues. Most empirical studies of these issues have focused on the distribution of income. But the income information available is commonly only short-term in nature (as well as being subject to serious misreporting), and therefore unsatisfactory as a guide to underlying consumption. The direct information on expenditures provided by household budget surveys may be of more value, since a household’s chosen level of expenditure presumably reflects its own evaluation of its long-term economic position. Our primary purpose here is to explore this alternative approach to the analysis of living standards, emphasising expenditure rather than income. In doing so, we shall draw attention to a basic and possibly severe problem of measurement error to be faced in using survey expenditure data for any of a variety of purposes.

The main source of household budget data for the United Kingdom is the annual Family Expenditure Survey (FES). This is a cross-section survey based on diary records of expenditures over a period of two weeks.’ But in

*This paper is a product of the IFS project on the distributional implications of fiscal policy, supported by the Gatsby Foundation. We are very grateful to Nick Stern and Douglas Todd, the discussants of our conference paper, to other conference participants and to two anonymous referees for their helpful comments. Any errors are our own.

‘See Kemsley et al. (1980) for a full description of the FES.

0047-2727/84/$3.00 0 1984 Elsevier Science Publishers B.V. (North-Holland)

170 J.A. Kay et al., Estimating consumption

any particular fortnight a household either buys (for instance) a refrigerator or it does not, and in either case its observed expenditure on refrigerators is a misleading indicator of its consumption or refrigeration services. For most of the purposes for which one wishes to use FES data, however, it is the latter and not the former that is of interest. The survey therefore makes direct enquiry about the ownership of major durables, such as refrigerators and cars.’ Yet it is clear that almost all commodities are ‘durable’ relative to such a short interview period: clothes, books and cosmetics all last beyond a fortnight, and restaurant meals or haircuts can contribute to a household’s well-being without being purchased continuously. As a result, much of the variation in reported expenditures may reflect stochastic elements in purchasing behaviour rather than differences in underlying consumption: most visibly, households may simply not purchase over the interview period commodities that they actually consume. It therefore seems that recorded expenditures are most appropriately interpreted as unobservable consumptions measured with error. This simple idea has serious implications.

Three issues are of particular importance. Conventional estimators of Engel curve parameters based on expenditure data will generally be biased and inconsistent. Secondly, household expenditure may be an unreliable estimator of household consumption, with extreme values resulting from either the presence in or absence from current expenditure of infrequently purchased items. Thirdly, distributional measures based on observed expenditures are liable to give a seriously distorted picture of the true incidence of poverty or inequality as indicated by the unobservable consumption data. This paper is concerned with only the second of these issues - the relationship between the recorded expenditure and underlying consumption of individual households. [The first and third issues are considered elsewhere; see Kay, Keen and Morris (1982) for an introduction.] In particular, it considers the possibility of constructing superior estimates of household aggregate consumption levels from disaggregated components of

expenditure. The first part of the paper develops a general specification for the

stochastic structure of expenditure data (section 2). The estimation of household consumption is considered in section 3, and examples illustrating the procedure are presented in section 4. There is a short concluding section.

2. The stochastic structure of expenditure data

The importance of the idea that whereas commodities are purchased only at isolated points in time consumption is essentially a continuous process is obvious from the most casual inspection of expenditure data, which is

‘These enquiries are severely limited in both coverage and kind: the FES simply records which of six major durables a household possesses.

J.A. Kay et al., Estimating consumption 171

characterized by a high incidence of zeros at even substantial levels of aggregation over commodities: see, for instance, the first column of table 1 below, which shows the proportions of non-zero expenditures for one subsample of the 1977 FES3 This striking feature of expenditure data - which is even more marked in the budget surveys of other countries - cannot be attributed entirely to variation in preferences and/or circumstances across the sample: one cannot believe that 18 percent of the sample had no clothing. An immediate corollary is that strictly positive expenditure entries are typically upward biased estimates of consumption over the interview period.

To proceed further it is necessary to consider the stochastic structure of expenditure data in more detail. For this purpose, consider the behaviour of household h in respect of some commodity k over an interview period I. Suppose that qk: = [t,$,, tf] II is such that the total consumption of commodity k by household h over Thk is equal to its total real purchases over the same interval, these purchases being made at a sequence of V points vi E Thk. Thus, in obvious notation,

(1)

The problem is to construct a plausible distribution of expenditure on k over the interview period I, which we may take to be of length unity. In the circumstances just described the number of purchases per unit interval is

V/A,,> where A,,: = tjfk - &, and so on a relative frequency argument the probability that h will make some purchase of k over the interview period is min{V/A,,, l}. Now suppose we are given the information that h makes some purchase of k during the interview period. The expected value of that

purchase is X:=1 e,,k(Ui)/l/: If we take it that the expected number of purchases over a unit subinterval of Thk conditional on at least one such purchase being made is 1 if V-C A,, and V/A,, otherwise, then the expected value of purchases over the interview period conditional on some purchase being made is:

(4

where c,,~: = JThr chk(t)dt/A,, denotes average consumption of k by h per unit interval. Denoting by ehk the relevant expenditure over the interview period we therefore have:

phk: = Prob [ehk > 0] = {min V/A,,, l}, (3)

3These proportions are in a number of cases (particularly that of Drink) rather lower than they are for most other household groups.


(4)

A simple example may clarify these ideas. Suppose that over a period of two years h purchases and consumes footwear to the value of &SO, making four purchases in all. Then (3) says that the probability of observing h purchasing footwear over an interview period of two weeks is 4/52= l/13, and (4) that

the expected value of such a purchase is (13/1)(&80/52) =&20. Denoting by H (respectively K) the set of households (commodities) for

which observations are available, the preceding arguments suggest a general specification of the form:

(5)

where ch: = Ck chk denotes the total consumption of household h, yf and oi are, respectively, vectors of observable household characteristics and of parameters to be estimated, i = 1,2, whk is distributed Bernouilli with

Prob[wh,=l]=pk(ch,$&O,), V~EH, VkEK, (6)

and ph: = (phk)4 is a vector random variable such that:

where 0, denotes the N-vector of zeros, and

Prob[~,,<-f,(ch,y:,e,)]=O, V~EH, VkEK, (8)

corresponding to the restriction that expenditures cannot be negative. Note

that if

Prob[~hk=--fk(ch,Y:,e,)]=O, VkkEK, (9)

then w,, is observable. In addition, there is the adding-up condition on the Engel curves:

~fk(Ch,Y;,&)=Ch, V’hEH.

4Throughout, x,, denotes the N-vector (x,,~).


The further assumptions will be made in what follows that:

Al. w,,~ is distributed independently of w,,,,, Vh E H, Vk E K, Vm E K -k.

A2. ph is distributed independently of w,,, Vh E H. This implies that there are

no cash flow or other constraints on total expenditure; it is therefore a natural assumption to make in the present context, since our essential concern is with the implications for the interpretation of expenditure data of the durability problem described above.

Note that this formulation emphasises two conceptually distinct elements

in the stochastic relationship between expenditure and consumption: one is the simple buy/do not buy dichotomy; the other is a more familiar disturbance of strictly positive expenditures about their conditional means. For convenience, these two sources of error will be referred to collectively as ‘lumpiness’.

This general specification is closely related to the frequency purchase

model considered by Deaton and Irish (1982), which it includes as a special case. Nevertheless the motivations of the two papers are very different. In particular, Deaton and Irish are especially concerned with the possibility of misreporting, and this leads them to a sophisticated treatment of expenditure on a single commodity; the emphasis here, in contrast, is on the information on living standards contained in the full set of commodity expenditures, and this will lead us to a rather crude treatment of a complete system of demand equations.

3. Estimating consumption from expenditure data

For all the reasons discussed above, the expenditures reported by households over the relatively brief interview periods characteristic of most household surveys are potentially misleading as indicators of underlying consumption patterns. But such surveys may nevertheless also be potentially the richest source of information on household consumption and living standards available - this is certainly true of the FES in the case of the United Kingdom. The question is how to make the fullest use of the imperfect data available. It is to this problem that we now turn.

Note first from (5)47) and A2 that:

E[eh]=ch, VhEH, (11)

where eh: = xk ehk denotes the total expenditure of household h. Thus, under the rather general specification arrived at above, total expenditure turns out to be an unbiased estimator of total consumption. But it may also be verified

174

that:

J.A. Kay et al., Estimating consumption

Var b”] =c it1 -Phk) dk + dk>/phk +c 2 drnr k k mfk

(12)

where oh km: =Cov[phk, ,u~,,J and the summation is restricted to commodities for which Phk > 0, from which it is clear that the variance of total expenditure may be substantial, particularly at higher consumption levels. In practice, the high variance of recorded expenditures is abundantly clear from the data. Because of this variability, expenditure data may give a seriously distorted picture of the distribution of consumption. This in turn has implications for policy formation: one is likely to wish to act differently, for instance,

according to whether a few households are very poor or a larger number moderately so. .Thus, although total expenditure possesses the desirable property of unbiasedness, the potential value of a more efficient estimator of household consumption is considerable.

To construct such an estimator it is necessary to add more structure to the general specification of section 2. We make the following assumptions (to be thought of as applying to subsamples of the full survey defined by demographic or other characteristics).

A3. fk(ch, 7:) 0,) = uk + bkch, V k E K, V h E H. Thus Engel curves are assumed

to be both linear and identical across households. Although unattractive as a maintained hypothesis, linearity has the great merit of simplicity; we hope to consider in future work the extension to alternative functional forms of the general approach adopted here. At a more general level, note that this assumption precludes the sort of tobit structure considered by Deaton and Irish (1982): it implies (in effect) that all households consume all commodities, so that zero expenditures occur only as a result of infrequent purchase. There can be little doubt that both preference variation and the existence of an extensive margin of choice make this implausible. Indeed, the extent to which the observed variability of expenditures is attributable to factors such as these rather than to simple lumpiness is an important question in its own right; the work of Deaton and Irish (1982) suggests that there are unlikely to be any easy answers. These issues are not, however, considered here.

A4. pk(ch, yi, 0,) =pk(y!, O,), Vk E K, Vh E H. This says that the probability of purchase over the interview period is independent of total consumption (though it may depend upon other factors), so that the act of purchase does not of itself convey information on underlying consumption. The implication is that if, for instance, household h has twice the clothing consumption of household h’, then this will be reflected not in the former’s buying clothes


twice as often as the latter, but rather in its spending twice as much on clothing when it does. One can certainly tell plausible stories that violate this assumption: high consumption households may simply go to the cinema more often. But the direction of any dependency of purchase probabilities is not a priori obvious: high consumption households may also be households that tend to place a relatively high value on their members’ time, leading them to make purchases less frequently. The determination of purchase probabilities is likely to be a complex matter, and the independence assumption is no more than the obvious first approximation to explore.

A5. ,uh- N(O,, C), Vlh, C: = [c~,,,] being positive definite. Note the implied homoscedasticity of the disturbance term relating positive expenditures to consumption grossed-up by the probability of purchase. Note too that normality vi,olates (8); however, truncation adds considerably to the complexity of the analysis, and the assumption may be a reasonable one at moderate consumption levels.

A6. ok,,, =O, V k E N, Vm e N - k. This diagonality assumption does not seem particularly objectionable in itself. It may be useful to extend the present analysis by relaxing diagonality and A3 to incorporate a limited degree of preference variation, allowing the Engel curve intercepts ak to vary across households.

Suppose now that the parameters a, b, P and C are known, and denote by 0 the parameter vector constructed from them. (For simplicity, we ignore the possible variation of p across households.) Using Al-A6, it is routine to show that the maximum likelihood estimator of ch (conditional on 0) can be written as:’

zh(e) =eh +c ehk -tak + bkeh)hk

b,fP,

d hk?

k

where

.

(13)

(14)

Written in this form, the estimator can be given a straightforward intuitive rationale. Total consumption is estimated by adding to observed total expenditure a correction factor which is simply a weighted average of the discrepancies between recorded expenditure on each commodity and the

‘To ensure that the estimator is always well-defined and that its moments exist, it is assumed throughout that there is some commodity j for which pj = 1 and that b, #O, Vk EK.


expenditure that would be predicted (conditional on some purchase being made) under the null hypothesis that total consumption is equal to total expenditure, each discrepancy being grossed-up by the marginal propensity to consume to give the implied error in commodity consumption. The weights are zero for those commodities not purchased by the household over the interview period and largest for those characterized by a high value of

%/a,, relative to the others purchased: for the information on the total consumption of a particular household h implicit in its consumption of k

increases with 1 b, 1, whilst the efficiency with which that information is incorporated into observed expenditure on k when some purchase is made decreases with okk. It is important to note the rather indirect way in which c”“(O) gives h credit for the commodities which it happens not to have purchased over the interview period. In so far as h has not purchased various commodities, its total expenditure will be low and the corresponding terms in the sum in (13) will vanish. At first sight it might therefore seem that Fh(0) will be too small. However, the individual commodity expenditures are likely to appear rather large relative to those that would be expected if h’s true consumption were in fact as low as its total expenditure (suppose that h merely buys a &l train ticket over the interview period: given total consumption of El, El is a surprisingly large amount to spend on travel);

those discrepancies ehk -(uk + b,eh)/p, which have positive weight are therefore likely to be substantially positive, pulling total expenditure upwards in arriving at an estimate of consumption. It is easily verified, for instance, that if h purchases only a single commodity with ehk = c,,/p,,, then Eh(H) = ch.

Consider now the statistical properties of Eh(f)). Note first that using (5)

and A3:

c”“(O) = Ch + ih, (15)

where

(16)

It is apparent from this that Eh(0) is not consistent for ch. This reflects the non-standard nature of the likelihood maximization problem considered here. Intuitively, of course, the point is that with t3 known, an increase in the number of households participating in the survey does not itself bring any additional information on the error with which the expenditure on (say) shirts by any particular household over the interview period measures its true shirt consumption. More formally, the familiar asymptotic properties of maximum likelihood (ML) estimators are typically predicated on various compactness conditions on the parameter space that are not satisfied in the present case. We have not calculated the exact distribution of Eh(G). But A2


and A4 imply that E[ch 1 w,J =O, and hence that Zh(0) is an unbiased estimator of total consumption:

E[Zh(B)] = ch. (17)

It may similarly be shown that:

(18)

(19)

where

(20)

is the Cramer-Rao bound, which exists in the usual way. Thus, Fh(B) is both unbiased and in an approximate sense, efficient.6 Comparison of (20) with (12), on the other hand, confirms the suspicions that expenditure is an inefficient estimator of total consumption and that the extent of this inefficiency may be substantial. Note further that both (17) and (18) continue to hold if the normality assumption fails (as, strictly, it must); the substantive loss is the approximate efficiency result. Thus, the estimator may be of value

even if normality is rejected. It has been assumed until now that the parameter vector 8 is known. In

practice, of course, this is not the case. However, if gH is a consistent estimator of 0 then the estimator Z”(g*) defined as in (13) converges to c”(0) in probability limit, and in that sense inherits the properties of c”“(0) as the sample size tends to infinity. Bearing in mind that household surveys such as the FES typically enable one to work with very large sample sizes, the proven optimality properties of c”“(e) - although limited - seem sufftcient for such an estimator Fh(i?“) to be of potential value as an additional tool in the analysis of household budget data.

Thus it remains to derive a consistent estimator of the parameter vector 8. This is straightforward for the purchase probabilities, since the sample proportion of non-zero expenditures on commodity k is easily shown to be consistent for pk (although not ML for the present specification). Consistent estimators of both the Engel curve parameters and the covariance matrix C are developed in Kay, Keen and Morris (1982).

6The approximation result is derived in Kay, Keen and Morris (1982). Note that the Cram&r- Rao bound is not in general attained unless pk = 1, Vk E K.


4. An illustration

Preliminary empirical results from the procedure described above are discussed in Kay, Keen and Morris (1982) to which the reader is also referred for details of the data and methods employed. The purpose of this section is simply to illustrate how the consumption estimator derived above acts to correct raw expenditure data for the implications of lumpiness.

The following discussion revolves around table 1. The first column shows the estimated purchase probabilities (i.e. the proportion of non-zero expenditures) for the subsample of respondents to the 1977 FES consisting of one-parent families with less than two working members. (This subsample is chosen for concreteness and convenience only.) The second and third

columns give the estimated Engel slope parameters for the same subsample. The fourth column shows the commodity weighting factors &ik2/ekk which are in turn used to construct the weights d,, in (14); the latter, it will be recalled, are household-specific, depending on the pattern of purchases w,,. Clearly, considerable weight is attached in estimating consumption to expenditures on Services and on Transport and Vehicles, and very little to expenditures on either Drink or Durables. A similar pattern seems to emerge for most other subsamples, reflecting the relatively high marginal propensities to consume the former commodities. Rather little weight, on the other hand, is attached to Food expenditure. This too has been a general feature of our results. and

Probability Engel curve Weighting Fictitious households of purchase parameters fdCtOrS

Commodity group fix ti, 6, (h:,‘ci,,) IO’ A B C D

Fuel, light and power 0.995 2.788 0.046 0.356 3.32 8.42 3.21 1.05

Food 1.000 9.745 0.101 0.312 9.94 22.59 3.24 14.85 Drink 0.42 1 ~ 0.02 1 0.028 0.052 0.00 2.80 0.00 0.00 Clothing 0.815 -3.281 0.195 1.374 4.93 3.79 0.33 0.29 Durables 0.718 1.613 0.028 0.041 62.18 0.00 0.00 5.35 Others and

miscellaneous 1.000 0.215 0.088 0.845 4.27 6.36 3.51 1.72 Transport and

Vehicles 0.872 - 5.945 0.268 1.92 2.60 18.47 0.00 0.42 Services 0.974 -5.114 0.246 2.46 4.64 39.92 5.00 0.00

Table 1

Estimating consumption.

Total expenditure 91.88 99.35 15.29 23.68 Estimated consumption Eh(GH) 42.75 101.72 26.48 18.29

(a) Parameter estimates are those for the subsamples of the 1977 Family Expenditure Survey consisting of one-parent families with less than two working members. (Sample size H = 195.)

(b) Asymptotic standard error of ?‘(o^,) z 11.99.


is perhaps rather counterintuitive - one might have expected expenditure on food to be one of the most reliable indicators of underlying consumption. However, the point is that although Food expenditure is characteristically rather well-determined (in the sense, for instance, that 8ii2 is small relative to non-zero mean expenditure) this does not of itself imply that Food is a good indicator commodity: this depends, inter alia, on the magnitude of the marginal propensity to consume which, in the case of Food, typically proves to be rather low.

Given the purchase probabilities, Engel curve parameters and commodity weighting factors, the total consumption of any household can be estimated from its observed expenditures in the manner of (13). Confidentiality restrictions on the use of FES data prevent our reporting results relating to individual households in the sample. But it is possible to give some flavour of these by showing how the procedure would work if applied to various fictitious households. For this purpose, the final columns of table 1 give the expenditures and estimated total consumption of four households which,

although imaginary, would not have been out of place in the subsample. Households A and B have very similar and high total expenditures, but the high expenditure of the former is almost entirely the product of a large entry for Durables, whilst the expenditures of household B are relatively evenly distributed across commodities. The comparison between these two households is typical of those that the investigator using

household survey data to examine distributional issues will frequently find himself making, and which to a large extent prompted the present work: intuition strongly suggests that B is considerably ‘better off’ than A, but expenditure totals do not bring this out. However, the weighting scheme developed above does just that, more than halving the total expenditure of A in arriving at an estimate of consumption whilst leaving that of B virtually unchanged. The effects of the procedure are less dramatic in the cases of households C and D. The low total expenditure of C is pulled up as a result of its moderately high expenditure on Services, to which considerable weight is attached in the absence of any expenditure on Transport and Vehicles. Instinct is perhaps rather less sure here; this seems to be partly because the low weight attached to expenditure on Food - which is in C’s case very small - is not intuitively appealing, and partly because of a feeling that the zero entry for Transport and Vehicles is trying to tell us that C really is a low consumption household. The procedure pulls down household D’s expenditure somewhat as a result of its rather low expenditure on Transport and Vehicles. However, the adjustment is not substantial, which indeed proves to be true for most households in the various samples examined. This is as one would hope, since it is the outliers like our household A that provide the most spectacular demonstration of the potentially distorting nature of expenditure data as a guide to living standards.


5. Concluding comments

Inspection of the disaggregated information provided by cross-section budget surveys similar to the FES is liable to suggest a fundamental problem: recorded expenditure over the interview period may be an unreliable indicator of underlying consumption. Given the absence of stock information for ‘diary’ commodities, this measurement error can present the user of such data with serious difiiculties. The paper has concentrated on just one of these difficulties - that of inferring a household’s unobserved consumption from its observed expenditure.

We have seen that under fairly general conditions (in the absence of systematic misreporting) a household’s aggregate expenditure is an unbiased estimator of its aggregate consumption. But the variance of this estimator can be considerable. In view of the wide variety of purposes to which information on household consumption patterns might usefully be put, we believe that the construction of more efficient estimators is an important task. To make progress in this direction it is necessary to impose falsifiable restrictions on the data. Of the assumptions used here perhaps the most

unacceptable is the requirement that purchase probabilities be independent of total consumption.7 A less cavalier treatment of aggregation issues and preference variation may also be appropriate. More fundamental extensions td the method are also required. For instance, the procedure described here is unsuited to deal with the consumption of housing and leisure - both important items in any analysis of living standards - since the prices households pay for these commodities vary widely across the sample.’ It also remains to exploit the zero/one information on ownership of major durables.

Despite these and other limitations the procedure described here goes some way towards overcoming the principal difficulty in using available data on expenditure to assess household living standards. The estimated consumption data generated by such a procedure must of course be used with care, the properties of the estimator being borne in mind in analyzing distributional issues. Nevertheless, estimators of this type open the way to a new perspective on distributional questions in applied economics. Our preliminary work9 indicates that this perspective may demand a radical reassessment of views on such matters as the incidence of poverty, the distributional impact of fiscal policies and trends over time in the degree of inequality.

‘See Kay, Keen and Morris (1982). ‘See, for example, King (1980) on the prices paid for housing services in the FES ‘Again, see Kay, Keen and Morris (1982).


References

Deaton, A. and M. Irish, 1982, Statistical models for zero expenditures in household budgets, paper presented to the NBERjSSRC conference on Micro-data and Public Economics, June, Journal of Public Economics 23, 59-80, this issue.

Kay, J.A., M.J. Keen and C.N. Morris, 1982, Consumption, income and the interpretation of household expenditure data, Working Paper No. 42, Institute for Fiscal Studies, London.

Kemsley, W.F.F., R.U. Redpath and M. Holmes, 1980, Family Expenditure Survey handbook (HMSO, London).

King, M.A., 1980, An econometric model of tenure choice and demand for housing as a joint decision, Journal of Public Economics 14, 137-159.

estimating consumption from expenditure data

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