COMMENTS ON: GOODNESS OF FIT: AN ECONOMIC APPROACH COWELL, FLACHAIRE & BANYOPADHAY

1. COMMENTS ON: GOODNESS OF FIT: AN ECONOMIC APPROACH COWELL, FLACHAIRE & BANYOPADHAY Lars Osberg Department of Economics Dalhousie University OPHI Workshop on Robustness Methods for Multidimensional Welfare Analysis, 5-6 May 2009

2. Improving on the “Eyeball Test” – how do we know when a predicted distribution fits “well”? “an economic approach to the problem should involve a measure of the information loss from using a badly fitting model.” So – how do we know what the cost of “bad fit” of f1(y) {theoretical} is at different points in the distribution of f2(y) {real}?

3. Ratio or Absolute Difference? Ratio: “Let the divergence between two densities f2 and f1 be λ:= f1 = f2; clearly the (page 5) Also page 17

4. Absolute Difference? (Axiomatic Foundation) Page 14

5. Why might this matter? Practical Measurement / Explanation Issue: Canada 1981-2006: all the action is in the tails Remarkable constancy in real incomes of ‘middle 90%+” Bottom 5% worse off – significant rise in poverty gap Top 1% - essentially all the gains from growth Larger percentage increases for top 0.1%, top 0.01% Models of tails of f(y) – socially important BUT – measurement error in tails + small densities Might this mean that ratio measures of divergence are potentially explosive? General Issue: the distribution of f2(y) {real} is measured with differing reliability at different y

6. Approaches: Welfare loss, Inequality Change and Distributional Change Welfare loss criterion rejected because W(x) not specific Inequality Index rejected because unchanging index consistent with multiple underlying divergence patterns Distributional Change approach preferred

7. Specifying α or specifying W(x) ? “Of course this would require the choice of a specific value or values for the parameter αin (45) according to the judgment that one wants to make about the relative importance of discrepancies in different parts of the income distribution: choosing a large positive value for α would put a lot of weight on discrepancies in the upper tail; choosing a substantial negative value would put a lot of weight on lower-tail discrepancies” If I am willing to specify α, don’t I also specify W(x) ? Atkinson(1970)

8. What is the problem to which this article is the solution? (1) Hypothesis Testing Hypothesis about specific functional form f1(y) is needed, if we are to test goodness of fit of f1(y) to actual f2(y) Historically, alternative specifications of stochastic process models generated specific hypotheses re f1(y) E.g. Champernowne (1953) – ergodic distribution is Paretian See also Solow (1951), Rutherford (1955) Not consistent with data -- ‘eyeball test’ Implausible micro-foundations Not so common in recent years Hypotheses about Income Distribution most often now concern relative size of moments of f2(y) E.g. “skill-biased technical change” hypothesis

9. What is the problem to which this article is the solution? (2) Model Fitting f1(y) = “perhaps the outcome of income or wealth simulations” “How can I show it fits “well”/”better”? a HUGE problem for micro-simulation model builders Specification of loss weighting (α) very useful ISSUE – dimensionality > 1 Micro-simulation models produce vector of outcomes for each individual in each period – how to generalize J index? Relative weight each dimension ? choice of units ? – non-transparent weighting No obvious reason why α same for all dimensions