Lecture 11

Lecture 11 - USC, Econ 513, Fall 2005 Lecture 11: Maximum...

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USC, Econ 513, Fall 2005 Lecture 11: Maximum Likelihood Estimation: Testing After estimating the exponential model for the unemployment durations, Lancaster con- siders an extension. Consider the hazard function or escape rate h ( y | x,θ ) = lim Δ 0 Pr ( y y < y + Δ | y y,X ) / Δ = f ( y | x,θ ) 1 - F ( y | x,θ ) . The hazard function is just another way of characterizing a distribution, like the density function, the distribution function, the survivor function, the moment generating function or the characteristic function. It is just a particularly convenient and interpretable way of describing a distribution or durations. Given the hazard you can calculate the distribution function as F ( y | x,θ ) = 1 - exp ± - Z y 0 h ( s | x,θ ) ds ² , and hence the density function. The exponential model implies that the hazard function stays constant over the duration of the spell, equal to exp( x 0 β ) in our previous specification. To see what this means, take a person and look at their chances of finding a job on the first day of being unemployed. These chances are the same as the chances that this same person would find a job on the fifthieth day given that he has been unsuccessful in finding work in the first forty–nine days. This may be reasonable, but it might also be something you do not wish to impose from the outset. Lancaster therefore considers an extension allowing the hazard function to either increase, stay constant, or decrease over time. This extension is known as the Weibull distribution : h ( y | x,β,α ) = ( α + 1) · y α exp( x 0 β ) . Note that this reduces to the exponential distribution if α = 0. The implied density function for the Weibull distribution is f ( y | x,β,α ) = ( α + 1) · y α exp( x 0 β ) exp ± - y α +1 exp( x 0 β ) ² . The moments of this distribution are E [ y k | X ] = exp ± - k · x 0 β . ( α + 1) ² · Γ ³ k + 1 α + 1 ´ . 1
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(Note that for the case with α = 0 this reduces to the exponential case with E [ y k | X ] = exp( - k · x 0 β ) · Γ(1 + k ) , and thus with k = 1 the mean of the exponential distribution is E [ y | X ] = exp( - x 0 β ).) The log likelihood function for this model is L ( α,β ) = N X i =1 ± ln( α + 1) + α ln y i + x 0 i β - y ( i α + 1) · exp( x 0 i β ) ² . One can estimate this model using any of the numerical methods described before (Newton- Raphson, Davidon-Fletcher-Powell). The one (minor) complication is that numerical algo- rithms have to take account of the restriction that α > - 1; with α = - 1 the density is degenerate and all probability mass piles up at y = 0. Table 1 presents the maximum likelihood results for both the exponential and Weibull models. The standard errors are based on the second derivatives, evaluated at the maximum likelihood estimates (see discussion below).
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Lecture 11 - USC, Econ 513, Fall 2005 Lecture 11: Maximum...

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