# Estimating ratios of quantile means for sorted data

In studying the relationship between income and consumption, it is common to sort the observations by income and observe that high-income households have lower levels of consumption per dollar of income than low-income households, i.e., they have higher savings rates. This conforms to one's naive intuitions about the relationship, whether you think the higher income is the cause or consequence of the higher savings rate.

I recently took a commonly used consumption data set (the U.S. Survey of Consumer Expenditure), and sorted it into deciles based on aggregate consumption. After this sorting, the decile with the highest consumption had the lowest savings rate. In fact, they dissaved by a considerable margin.

I am looking at some real reasons why this might be true in the period in question (e.g., drawing down fictitious wealth from the housing bubble), but it seems to me that I might get the same result if income and consumption were both observed with error, that is, if the errors were not too highly correlated.

Suppose that this is true, and that the errors (or the percentage errors) of household income and consumption are uncorrelated with the true errors. How might I estimate that relationship correctly? And more specifically, can I estimate it correctly groupwise, for, e.g., income deciles or consumption deciles or quintiles? (Here I use "deciles" to refer to the groups of sample observations divided by the decile break-points, rather than by the breakpoints themselves).

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Is there a reason you can't simply correlate the two variables w/o categorizing them into quantiles first? –  gung Jul 28 '12 at 2:04
Well, the savings and consumption behavior of the rich, poor, and middle class are each of independent interest, for a variety of policy reasons. In addition, the true relationship may be nonlinear, so a simple correlation or linear regression could be unrevealing. Thirdly, business cycle effects on wealth and unemployment that affect saving impact the poor and rich quite differently, and are quantitatively important. Finally, there are special sources of measurement error which affect only the rich, and others that affect only the poor. We may want to keep these errors compartmentalized. –  andrewH Jul 28 '12 at 18:39

Some of your phrasing is slightly strange such as "Suppose ... that the errors ... of household income and consumption are uncorrelated with the true errors" but your underlying point is sensible. Here are some numbers artificially simulated so "income" and "expenditure" have similar distributions and are highly correlated ($r \approx 0.91$):

For those in the top income decile (to the right of the vertical red line) average income is greater than average income while for those in the top expenditure decile (above the horizontal blue line) average income is below average expenditure.

This need not have anything to do with errors in measuring income and expenditure, but could just be because income and expenditure are not perfectly correlated (e.g. when people have children, their expenditure can increase even if their income does not). If you fitted a least squares line of expenditure against income then expenditure would appear to rise less quickly than income but if you reflect the graph in the diagonal and fitted income against expenditure on the data in the graph then income would appear to rise less quickly than expenditure. It gets more complicated if the underlying relationship is non-linear.

In real-life there is also a timing issue. People may have lumpy expenditure, and some have lumpy incomes. Individuals whose large expenditures fall in the survey period will appear to have negative savings, while those who do not may appear to have positive savings, even if both tend to average to zero over time. There can also be a wealth effect: in the UK, it known that those with almost no reported income tend to spend more in absolute terms than those with incomes close to levels of welfare benefits: the former group tend to be wealthier and are using wealth rather than income to fund their expenditure.

One approach to address some of these issue could be to define the "richest 10%" by some combination of income and expenditure, which might catch both both the high-income miserly individuals and the high-standard-of-living profligates. Some of the points discussed in or linked from the Wikipedia article on regression dilution though once a model becomes non-linear then it can become very difficult.

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+1, there's a lot of good info here. Another possibility might be to fit a spline to the data. I discuss splines here. –  gung Jul 28 '12 at 19:53
Henry, that's wonderfully clear. Thanks so much. So, I would get more accurate results if I categorized households in terms of, say, (income + consumption)/2 - equivalent to taking diagonal slices normal to the 45 degree line. Suppose the relationship is linear over some range. If the true relationship does not have slope 1, my intuition is that it should be normal to the true line, rather than the 45 degree line. Does that seem right? –  andrewH Jul 30 '12 at 2:46
andrewH: what you say sounds sensible, until you realise that the "the true line" is a slippery concept - if you do simple linear regression then you could get two lines depending on which variable you regress against the other. One approach might be to look at the principal component –  Henry Jul 30 '12 at 9:57