Correlations Between Alternative Welfaure Indicators

A method for judging a wider array of crude welfare indicators vis-a-vis the equivalent income welfare metric is to construct

correlation coefficients. An advantage of this approach is that these statistics are independent of the units of measurement. Both Pearson correlation coefficients and Spearman rank correlation coefficients are used. The latter indicate how closely the rankings of the observations accord for a pair of variables. The rank correlations are of special interest here as we are concerned with the ranking of households into deciles in the original position. That is, we are concerned with the ordering of households from poorest to richest. Many policy questions only require a welfare indicator which allows ordinal welfare comparisons to be made between households. The extent of agreement in the ordering with the benchmark measure thus provides a central criterion with which to judge alternative welfare indices.

Tables 4.11 and 4.12 present Pearson correlation coefficients for the loglinear and AIDS equivalent incomes respectively, and a series of existing welfare indicators. Rank correlation coefficients for the same

TABLE 4.11; Pearson Correlation Coefficients Between Loglinear YEO and Various Common Welfare Indicators

WELFARE INDICATOR FULL SAMPLE URBAN RURAL

TEX .9899 .9917 .9797 PC TEX .7573 .7076 .7537 INCOME .4412 .7830 .2524 PC INCOME .4026 .6088 .2210 FOOD EXP. .7333 .7231 .6347 PC FOOD EXP. .5289 .4588 .4415 CALORIES .3139 .3748 .3866 PC CALORIES .0944 .1860 .1969 PROTEIN .3956 .4176 .4543 PC PROTEIN .1830 .2361 .2539

TABLE 4.12; Pearson Correlation Coefficients Between AIDS YEO and Various Common Welfare Indicators

WELFARE INDICATOR FULL SAMPLE URBAN RURAL

TEX .9964 .9957 .9965 PC TEX .7417 .6964 .7177 INCOME .4442 .7864 .2567 PC INCOME .3926 .5968 .2087 FOOD EXP. .7558 .7372 .6919 PC FOOD EXP. .5102 .4418 .4164 CALORIES .3527 .4025 .4526 PC CALORIES .0802 .1771 .1648 PROTEIN .4304 .4440 .5129 PC PROTEIN .1664 .2272 .2177

TABLE 4.13; Spearman Rank Correlation Coefficients Between Loglinear YEO and Various Common Welfare Indicators

WELFARE INDICATOR FULL SAMPLE URBAN RURAL

TEX .9668 .9843 .9458 PC TEX .7761 .7276 .6968 INCOME .7912 .8449 .7171 PC INCOME .6310 .6450 .5005 FOOD EXP. .8002 .8100 .7657 PC FOOD EXP. .6300 .5782 .5392 CALORIES .4229 .4689 .5240 PC CALORIES .1703 .2534 .3035 PROTEIN .5064 .5085 .5929 PC PROTEIN .2796 .3091 .3672

TABLE 4.14; Spearman Rank Correlation Coefficients Between the AIDS YEO and Various Common Welfare Indicators

WELFARE INDICATOR FULL SAMPLE URBAN RURAL

TEX .9965 .9971 .9954 PC TEX .7087 .6733 .6057 INCOME .8190 .8608 .7580 PC INCOME .5784 .5988 .4274 FOOD EXP. .8503 .8389 .8373 PC FOOD EXP. .5691 .5267 .4602 CALORIES .4981 .5229 .6093 PC CALORIES .1225 .2276 .2163 PROTEIN .5684 .5582 .6583 PC PROTEIN .2237 .2836 .2700

T A B L E 4.15; P e a r s o n C o r r e l a t i o n C o e f f i c i e n t s B e t w e e n L o g l i n e a r YEO C a l c u l a t e d w i t h Error and V a r i o u s C o m m o n W e l f a r e Indicators

W E L F A R E I N D I C A T O R FULL SAMPLE U R B A N R U R A L T E X .9802 .9870 .9489 P C T E X .7552 .7101 .7384 INCOME .4366 .7799 .2438 P C INCOME .4004 .6105 .2144 FOOD E X P . .6961 .7054 .5366 P C FOOD E X P . .5069 .4588 .3622 C A L O R I E S .2446 .3118 .2797 PC C A L O R I E S .0143 .1043 .0788 P R O T E I N .3274 .3631 .3399 PC P R O T E I N .1075 .1670 .1376

T A B L E 4.16; P e a r s o n C o r r e l a t i o n Coefficients Between AIDS YEO

C a l c u l a t e d w i t h Error and V a r i o u s C o m m o n W e l f a r e Indicators

W E L F A R E INDICATOR FULL SAMPLE U R B A N R U R A L

TEX .9966 .9977 .9917 PC TEX .7261 .6741 .7117 INCOME .4464 .7924 .2565 PC INCOME .3864 .5829 .2069 FOOD EXP. .7584 .7506 .6694 PC FOOD EXP. .5012 .4358 .3927 C A L O R I E S .3416 .3987 .4242 PC C A L O R I E S .0495 .1423 .1257 P R O T E I N .4216 .4441 .4840 PC P R O T E I N .1390 .1982 .1809

T A B L E 4.17; S p e a r m a n R a n k C o r r e l a t i o n C o e f f i c i e n t s B e t w e e n L o g l i n e a r YEO C a l c u l a t e d w i t h Error and V a r i o u s C o m m o n W e l f a r e Indicators

W E L F A R E I N D I C A T O R FULL SAMPLE U R B A N R U R A L T E X .8883 .9525 .8038 P C TEX .7496 .7723 .6031 INCOME .7164 .8049 .5945 PC INCOME .6013 .6752 .4156 FOOD E X P . .6646 .7399 .5498 PC FOOD E X P . .5387 .5893 .3481 C A L O R I E S .2735 .3297 .3354 PC C A L O R I E S .0088 .1117 .0873 P R O T E I N .3470 .3813 .3786 PC P R O T E I N .1185 .1898 .1434

T A B L E 4.18; S p e a r m a n R a n k C o r r e l a t i o n Coefficients Between the AIDS YEO C a l c u l a t e d w i t h Error and V a r i o u s C o m m o n W e l f a r e Indicators

W E L F A R E INDICATOR FULL SAMPLE U R B A N R U R A L

TEX .9855 .9928 .9763 PC T E X .7128 .6924 .5947 INCOME .8051 .8527 .7376 PC INCOME .5743 .6125 .4126 FOOD E X P . .8161 .8210 .7868 PC FOOD EXP. .5488 .5338 .4122 C A L O R I E S .4548 .4763 .5638 PC C A L O R I E S .0727 .1736 .1583 P R O T E I N .5233 .5165 .6069 PC P R O T E I N .1756 .2386 .2113

measures are given in Tables 4.13 and 4.14. Tables 4.15, 4.16, 4.17, and 4.18 repeat the above for the YEOs with error. In each Table, the

correlations are given for the sample as a whole, and for the sample subdivided into rural and urban households.

Inspection of the Tables suggests the following observations: 1. All the coefficients are positive. Under the null hypothesis that the Pearson correlation equals zero, the 5 percent cut off point is

.1946 and the 1 percent level is .2540. Thus the alternative hypothesis that the observations are not orthogonal is accepted at the 1 percent

level for most measures. Per capita protein and calorie intakes, rural per capita income and rural income for the loglinear are the exceptions.

(There are more exceptions for the YEOs with error although the same indicators do poorly.) In terms of the rank correlations, every

coefficient is concluded to significantly differ from zero. (Again, there are exceptions for YEOs with error.) For the whole sample and the two subsamples, the rank correlations are lowest between the AIDS

equivalent income and per capita calories (Table 4.14). However, the full sample correlation coefficient of .1225, the urban coefficient of .2276, and the rural coefficient of .2163 are respectively 8.8, 9.5 and 12.7 standard deviations from zero.^

2. In all cases, total expenditure is the best performing indicator of equivalent income. This is as expected since the equivalent incomes are derived from TEX. Per capita TEX and food expenditures compete for the second highest Pearson correlations with their rankings differing according to the sample and functional form. In contrast to evidence of its relatively low Pearson correlation coefficient with YEO (for the sample as a whole), the rank ordering of money income accords closely

with that of YEO. The contrast is particularly pronounced for rural households. The data also suggest a close ranking relationship with food expenditures. The rank correlation coefficients for both measures

surpass that for per capita TEX. (This is not true for the loglinear YEO with error.)

3. According to their Pearson correlation coefficients, money income per household and per capita, perform considerably better as

indicators of equivalent income for urban households than for their rural counterparts. An explanation for this is that in rural areas, income in any one period can be a poor proxy for welfare. Rural workers and

peasants are less likely to be employed in permanent work or to receive a steady stream of income from own production due in large part to the seasonality of work and yearly weather uncertainties in a sector which is largely agricultural. Storage, savings, and credit can often be used to buffer consumption from income variability. In addition, rural

households are more likely to benefit from gifts, payments in kind and other village level support structures (Ravallion and Dearden, 1988). Consequently, because of (i) the highly irregular nature of conventional income and (ii) consumption stabilization practices, information on current incomes often tends to be a less accurate indicator of

consumption for rural dwellers than for urban ones. In the absence of data on permanent incomes, income in the survey month is likely to remain a poorer indicator of a rural household's physical well being than

consumption expenditures and equivalent incomes based on the latter. This is clearly less a problem in urban areas.

Interestingly, although the urban coefficient is also greater in the case of the Spearman rank correlation, the disparity is dramatically

diminished. This suggests the possibility that the rural data may be characterized by more extreme values of income. Pearson correlations can be dramatically affected by the latter through the product moment

effects. Rank correlations are less sensitive to this. At any rate, money income appears to provide a more reliable basis than most other indicators on which to rank household welfare levels in both rural and urban areas. Thus, although income can be a poor cardinal welfare measure, it may still be a satisfactory indicator of whether one household is better off than another. For urban households, income appears to be a good indicator on both counts, at least in comparison to the other welfare measures examined.

4. Food expenditures per household and per capita also exhibit higher correlation coefficients with equivalent incomes for urban than rural households, though the effect is less marked than for money

income. In contrast, the calorie and protein consumption indices (per household and per capita) in general perform better for the rural

sample. This observation is reversed for per capita intakes under the AIDS function with and without regression errors (in accordance with the arguments of 5. below).

5. On the whole, the per capita measures are associated with lower Pearson and Spearman correlation coefficients than their per household equivalents. In addition, the per capita indicators consistently show a lesser degree of covariability with the AIDS than with the loglinear equivalent incomes. (Exactly the opposite is true for the YEOs with error.) This result follows from the discussion in Section 4.4 which establishes that referencing household characteristics exerts a greater impact on the loglinear equivalent income function. It follows that the

loglinear YEO will tend to be more closely related to per capita measures then the AIDS YEO.

6. As in most examples in this Chapter, loglinear equivalent incomes incorporating demand model residuals provide the exceptions to otherwise quite general patterns and features of equivalent incomes.

In many situations, it is not practical to compute a behaviorally consistent money metric of welfare. We may need to rely on a more easily available indicator. So, which do we choose as the most suitable proxy for welfare? On the basis of the present data saimple, a few general conclusions can be drawn. Firstly, as a juxtaposition of the Pearson and Spearman correlation coefficients shows, certain indicators perform

better as cardinal measures of welfare, and others as ordinal measures. TEX and per capita TEX provide the most reliable measures of an

individual household's welfare as measured by equivalent income. Food expenditures may not do as well due to the fact that households do not face homogeneous prices. The extent of reranking in the welfare ordering is at a minimum when using TEX, followed by income and food

expenditures. On the whole, expenditure based indicators in their per household form (rather than per capita) are better proxies for equivalent income.

Notes: Chapter 4

1. The importance of regional price variations on purchasing power has been explicitly recognized through the widespread use of cost of living indices. Atkinson (1975) reviews the empirical evidence. On relative price changes over time see Muellbauer's investigations of the effect of variable inflation on inequality in the U.K. (1974b, 1974c) and Murty, G.V.S.N., and Murty, K.N. (1977) and Murty, G.V.S.N. (1984) on the same question in an application to India.

2. For a general review of research in this area see Deaton and

Muellbauer (1980, Chapter 8). Muellbauer (1974a, 1974b, 1974c) looks at how household composition effects together with relative price changes affect inequality. In a fast growing literature on the costs of

children, Deaton and Muellbauer's (1985) paper is notable for its application to Indonesia (and Sri Lanka).

3. A separate issue involves the introduction of "optimization errors" which arise when households fail to reach their desired positions

(Zabalza and Arrufat, 1983, 1986). However, this practice precludes using the indirect utility function (and thus the equivalent income function approach) as households will no longer be at their utility maximum. This issue is not considered here.

4. On choosing a suitable welfare indicator see Anand and Harris (1988) who analyze and compare per capita income, total expenditures, and food

expenditures, and the food share of expenditure using Sri Lankan survey data. Anand and Harris suggest that (among these options) food

expenditure per capita is the better welfare measure.

5. With sample sizes greater than 20, the sampling distribution for the rank correlation approaches normality where the test statistic is the correlation coefficient divided by the standard error (the variance being