Measurement error in recall surveys and the relationship between household size and food demand.

National Accounts (NA) estimates of household food consumption are also not a plausible source of validation data, at least in developing countries. Comparisons between survey and NA estimates of food consumption have been hotly debated in India where both the survey and national account statisticians have concluded that discrepancies more likely reflect errors in the national accounts (Minhas 1988; Kulshreshtha and Kar 2005). For example, some foods that are also ingredients in restaurant meals get counted twice in NA estimates because their use by the food-away-from-home (FAFH) sector is not deducted when household consumption is derived from aggregate production and net exports. The rising importance of FAFH with economic growth induces a trend error in the national accounts (Deaton and Kozel 2005). Moreover, expenditure in restaurants is classified as nonfood consumer services in the NA estimates but as part of the food group in the household surveys (Minhas 1988).

While validation data for directly studying measurement errors in household expenditure surveys are hard to find, the literature on cognitive processes gives plausible reasons for why variations in survey design may create correlated measurement errors. First, information appears to be encoded, and eventually retrieved, differently when reporting for oneself rather than others (Eisenhower, Mathiowetz, and Morganstein 1991). This may help explain why results for diary surveys (with self-reporting) differ from recall surveys (with proxy reporting). A special case of this proxy reporting is "composite households" (those comprising individuals other than either a single person, a couple or a couple or their children) who have much greater item non-response for consumption questions (Browning, Crossley, and Weber 2003). Larger households are more likely to be composite, (4) so one reason why reported per capita expenditures may fall in larger households is that item nonresponse wrongly gets treated as zero spending.

Second, the cognitive strategies used by respondents depend on the length of the recall period and the number of events in that period. Respondents tend to give an actual count for infrequent events ("episodic enumeration") but for higher frequency events they switch to an estimation strategy (Blair and Burton 1987). This matters because enumeration and estimation are not equally reliable. According to Eisenhower, Mathiowetz, and Morganstein (1991, p. 140) "when the number of events are large or closely spaced [...] the direction of response error would be predicted to be an [...] underestimation of events." Hence, if a questionnaire uses a shorter, less detailed, recall list, there will be more purchases in each category in a given time period, especially for larger households. Thus, a respondent from a larger household, when given a less detailed recall questionnaire, might tend to use an estimation strategy that is likely to understate frequent, closely spaced purchases. Food is typically purchased more frequently than nonfood, and purchase frequency is more in proportion to household size, so this understatement for larger households should especially affect measured food expenditure. (5)

Third, the greater the length of a recall period over which a respondent is required to remember information, the greater the expected bias (Eisenhower, Mathiowetz, and Morganstein 1991). The errors related to recall period are due either to telescoping, which is a mis-dating of events, or recall decay, which is a forgetting of events. Telescoping is most relevant to nonroutine events, and can bias survey reports either upwards or downwards. But for routine events, like buying food, recall decay is the most likely source of error. This decay could explain why recall surveys often have lower expenditures than diary surveys because most diary-keepers record on the day of their purchase so there should be less memory loss. (6)

Two Motivating Examples for Studying Correlated Measurement Errors

Two motivations for studying correlated errors in food expenditure data are that (1) they may cause empirical fragility in Engel method estimates of household scale economies, adding to the other problem besetting this method, which is its atheoretical nature, (7) and (2) they may also at least partially cause the puzzle about food demand reported by Deaton and Paxson. These two examples are in fact in conflict with each other because the puzzle that Deaton and Paxson report was identified during an attempt to develop an alternative to the Engel method of measuring scale economies. An alternative was needed because even though the Engel method is atheoretical it continues to be used. The aim here is not to resolve that conflict, but rather, to show how correlated measurement errors might affect the empirical results reported in each area.

The Deaton and Paxson Puzzle

Deaton and Paxson use a version of the model first developed in Barten (1964). An egalitarian household with n members allocates consumption between food, [q.sub.f] and a nonfood good, such as housing, [q.sub.h], in order to maximize utility, u:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where x is total household expenditures, [p.sub.f] and [p.sub.h] are the price of food and nonfood, and [[phi].sub.k](n) (where k = f, h) is the scaling function that transforms the number of members, n into "effective" size. (8) The commodity-specific degree of economies of household scale is:

(2) [[sigma].sub.k] = 1 [partial derivative]ln[[phi].sub.k](n)/ [partial derivative]ln n.

The per capita food demand function is:

(3) [q.sub.f]/n = [[phi].sub.f](n)/n[g.sub.f](x/n, [p.sub.f][[phi].sub.f](n)/n, [p.sub.h][[phi].sub.h](n)/n)

where [g.sub.f](x, [p.sub.f], [p.sub.h]) is the food demand function for a single person household. Differentiating the logarithm of equation (3) with respect to Inn yields the conditions needed if per capita food consumption is to increase with household size, holding x/n constant:

(4) [partial derivative]ln([q.sub.f]/n)/[partial derivative]ln n > 0 [??] [[sigma].sub.h]([[epsilon].sub.fx] + [[epsilon].sub.ff]) - [[sigma].sub.f] (1 + [[epsilon].sub.ff]) > 0

where [[epsilon].sub.ff] and [[epsilon].sub.fx] are the own-price and income elasticities of demand for food. If nonfood contains some public goods, so that [[sigma].sub.h] [not equal to] 0, while food is a pure private good ([[sigma].sub.f] = 0), and if the (absolute) own-price elasticity is less than the income elasticity of food demand, per capita food consumption will increase with household size. This condition is most likely to hold for poor consumers, so the positive effect of household size on per capita food consumption (and hence food budget shares) is predicted to be greatest in poor countries.

To test whether the empirical evidence is consistent with this prediction, Deaton and Paxson estimate the following food share model on household survey data from seven countries:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [r.sub.ji] = [n.sub.ji]/[n.sub.i] is the proportion of persons in household i in demographic group j, z is a vector of other household characteristics, [u.sub.i] is a disturbance term, and [alpha], [beta], [gamma], [eta], and [delta] are parameters to be estimated. While [??] was expected to be positive, especially for poor countries, the empirical results showed the opposite pattern. Deaton and Paxson estimated [??] to be negative in surveys from six out of seven countries (positive only in Britain), and while it was close to zero for the rich countries (-0.008 for the United States and France) it was quite large for the poor countries (approximately -0.06 to -0.10 for Thailand, Pakistan, and Africans in South Africa).

Several unsuccessful attempts have been made to explain this puzzle. Horowitz (2002) suggests that the two-good model used to derive the predictions is too restrictive. In a three-good model, food demand rises with household size only if food and the public good are gross complements. However, a multi-good equivalent to equation (4) derived by Deaton and Paxson (2003) provides no resolution to the puzzle. Gan and Vernon (2003) suggest that there are economies of scale in food preparation but this only deepens the puzzle because a reduction in per capita preparation costs should allow an increase in food expenditures per head. Abdulai (2003) suggests that bulk discounts allow larger households to spend less on food even as they consume more. But he provides no evidence of these bulk discounts, other than a negative effect of household size on the average unit value for all food--which could just as easily reflect a tendency for larger households to buy lower quality foods (Deaton 1997). It is therefore worth seeing whether correlated errors bias [??] downwards especially because of the variation in household survey methods among the countries considered by Deaton and Paxson.

Engel Estimates of Household Scale Economies

A reparameterized version of equation (5) can provide Engel estimates of size economies, albeit with assumptions substantially different to those used by Deaton and Paxson. In the case of the Engel method, no distinction is made between private and public goods (hence, the economies of scale are not commodity-specific). Scale economies are calculated by comparing the total outlays of different-sized households with the same food shares. For example, Lanjouw and Ravallion (1995) use data from Pakistan to estimate:

(6) [w.sub.f,i] = [alpha] + [beta] ln ([x.sub.i]/[n.sup.1-[sigma].sub.i]) + [J-1.summation over.(j=1)] [[eta].sub.j] [r.sub.j,i] + [delta] x z + [u.sub.i],