A COMPANION TO Theoretical Econometrics

Exponential regression model

Recall that the exponential duration model depends on the parameter X, which is the constant hazard rate. We now assume an exponential distribution for each individual duration, with a rate Xi depending on the observable characteristics of this individual represented by explanatory variables. The positive sign of X is ensured by assuming that:

Xi = exp(xi0),

where 0 is a vector of unknown parameters. The survivor function is given by:

Si(y | xy 0) = exp[-(expx;0)y],

whereas the conditional pdf of the duration variable given the covariates is:

f (yi | x; 0) = X;exp(-Xiyi)

= exp(xi0)exp[-yi exp(x;0)]. (21.9)

The parameter 0 can be estimated by the maximum likelihood from a random sample of N observations on (x„ y;), i = 1,..., N. The conditional loglikelihood function is:

N

log l(y x; 0) = X log f (Уі x; 9)

і =1 N

= X [x0 - Уі ехр(х-0)]'

і=1

and the maximum likelihood estimator 0 = Argmax0 log l(y x; 0) solves the opti­mization problem. The first order conditions are:

Э!°g l(y x, 0) = 0 Э0

N

« X [! - Уі exp(Xi§)]x; = 0

і =1 N

« Xexp^-G^yi - exp(-x!-0)]x - = 0. (21.10)

і =1

Since the conditional expectation of the duration variable is E[Yixi] = exppx^), the first-order equations are equivalent to orthogonality conditions between the explanatory variables and the residuals: йі = yt - exp^x^), with weights exp^P) due to the individual heteroskedasticity.