A Note on Binary Choice Duration Models

Transcription

1 A Note on Binary Choice Duration Models Deepankar Basu Robert de Jong October 1, 2008 Abstract We demonstrate that standard methods of asymptotic inference break down for a binary choice duration model in a time series setting. This is because the dependent variable has a degenerate limit distribution, which makes the asymptotic variancecovariance matrix singular. JEL Codes: C22, C25 Keywords: binary choice; duration models. 1 Introduction This note points out a problem of internal consistency in a binary choice duration model in a time series framework. The problem emerges whenever duration in the current state is included as a regressor in a binary choice model. In such a situation, the dependent variable converges in probability to unity and standard methods of asymptotic inference breaks down. Recently, Frederiksen, Honoré and Hu (2007) have studied a discrete choice duration model with group level heterogeneity in a panel data setting with large N and small T. The problem that we point out will be relevant for any attempt to extend this model to the case where both N and T are large. To fix ideas consider the following binary choice probit model y t = I(β 0 z t + ε t > 0) (1) Corresponding author. Department of Economics, Colorado State University, 1771 Campus Delivery, Fort Collins, CO Ph: (970) ; Fax: (970) ; deepankar.basu@colostate.edu Department of Economics, Ohio State University, 429 Arps Hall, 1945 N High Street, Columbus, OH 43210, dejong@econ.ohio-state.edu. 1

2 where I(.) is the indicator function, β 0 > 0 is the parameter of interest, the process starts at time t = 0 at an arbitrary starting value y 0 and z t is defined as the number of consecutive ones of the y t sequence leading up to the current period, i.e., for t = 1, 2,..., T z t = t j y t i. j=1 i=1 (2) In (1) and (2), y t is a binary variable taking values in {0, 1}, z t is the duration in the current state (looking back from the current time period) and the error term ε t is distributed as i.i.d. N(0, 1). For instance, y t could capture whether a person is unemployed or not (y t = 1 if unemployed, and y t = 0 otherwise) with z t measuring the current spell of unemployment; we show below that the structure of the model in (1) and (2) implies that eventually every individual will be unemployed. Our focus on the case β 0 > 0 is motivated by the assumption that long spells of unemployment increase the probability of further unemployment. The joint density of the observed sample conditional on y 0 is f(y 1,..., y T y 0 ; β) = f(y T y T 1,..., y 1, y 0 ; β)f(y T 1 y T 2,..., y 1, y 0 ; β)... f(y 2 y 1, y 0 ; β)f(y 1 y 0 ; β). From (1) and (2), we see that f(y t y t 1,..., y 1, y 0 ; β) = [Φ(βz t )] yt [1 Φ(βz t )] 1 yt (t = 1, 2,..., T ). Hence, the conditional log-likelihood function for the sample is log L(β) = y t log Φ(βz t ) + (1 y t ) log[1 Φ(βz t )]. (3) The question that we want to pose and answer is the following: can we conduct asymptotically valid inference on β 0 using standard procedures? 2 Main Result and Discussion Theorem 1 Consider the model in (1) and (2) for y t. Let F ε ( ) denote the distribution function of ε t. If (i) ε t is i.i.d., (ii) y R, F ε (y) > 0 and (iii) E[ ε t I(ε t < 0)] < then p 1 as t. y t Note that since Theorem 1 does not impose a centering assumption on ε t, the result will remain true if an intercept is added to the specification of Equation (1). The intuition behind this result is the following. Since in the probit setting P (y t = 1) = E[y t ] (1/2), the y t sequence will eventually hit unity. But once it hits unity, the probability of y t attaining unity in the next period increases because z t increases by one. Thus, in a sense, the y t sequence gradually drifts away from zero and towards unity, eventually getting stuck at unity. 2

3 The main implication of this result is that standard maximum likelihood-based asymptotic inference on β 0 is problematic. To see this, note that the score function equals s T (β) = s t (β) = and the Hessian is H T (β) = H t (β) = φ(βz t )z t [y t Φ(βz t )] Φ(βz t )[1 Φ(βz t )] {φ(βz t )} 2 zt 2 Φ(βz t )[1 Φ(βz t )] + [y t Φ(βz t )]L(βz t ) (5) where φ(.) denotes the density and Φ(.) the distribution function of a standard normal random variable, and L( ) is a known function. Following Wooldridge (2002), it is easy to see that E[s t (β 0 ) z t ] = 0, and one might be tempted to conclude that T (β0 β) d N(0, A 1 0 ) (6) where A 0 = lim T T 1 E[H t (β 0 )]. (7) However, because z t diverges as t, E(H t (β 0 ) z t ) = {φ(β 0 z t )} 2 zt 2 Φ(β 0 z t )[1 Φ(β 0 z t )] ) p 0 as t. Because E(H t (β 0 ) z t ) is bounded uniformly in z t, it follows by the dominated convergence theorem that EH t (β 0 ) 0 as t. Therefore, it follows that A 0 = 0. This violates standard assumptions made in the theory of minimization estimators and obviously, the standard result of Equation (6) ceases to be valid. In conclusion therefore, standard methods for justifying asymptotic inference for minimization estimators will break down in the binary choice duration model when T is large. This implies that in a panel data analysis with a relatively large number of time periods standard methods of inference might be misleading. Appendix To analyze the properties of the stochastic dynamical system in (1) and (2) and to prove Theorem 1, let m N be a positive integer and define a random variable τ m as τ m = inf{k : y k m+1 = 1,..., y k 1 = 1, y k = 1}. (8) 3 (4)

4 Then, τ m is the first time period when the y t sequence gets a realization of m consecutive ones; thus, τ m takes values in {m, m + 1, m + 2,...}. Let F ε ( ) be the distribution function of ε t. Note that the event {τ m > Km} implies that K times in a row, the y t sequence did not come up with even a single m-tuple of ones (i.e., m consecutive ones) between t = 1 and t = Km; thus, letting denote the complement of an event, we have P (τ m > Km) P (τ m > (K 1)m and (y (K 1)m+1 = 1,..., y Km = 1)) P (τ m > (K 1)m and (ε (K 1)m+1 > 0,..., ε Km > 0)) (1 [F ε (0)] m ) K, because the ε t are i.i.d. Since y R, F ε (y) > 0, we have P (τ m > Km) (1 [F ε (0)] m ) K 0 as K. Thus, τ m is an a.s. finite random variable, where a.s. stands for almost surely. Proof of Theorem 1: Choose any j N such that j m. Define the index set, A m = {j N : P (τ m = j) > 0}. Letting i.o. stand for infinitely often, we have P (y t = 0 i.o.) = P (y t = 0 i.o. & j Am {τ m = j}) = P (y t = 0 i.o. & τ m = j) = P (y t = 0 i.o. τ m = j)p (τ m = j) The first equality follows because: P ( j Am {τ m = j}) = P ({τ m = j}) = 1. (9) and the second follows because the events {τ m = j} are disjoint, i.e., {τ m = j} {τ m = k} =, for j k. Note that we can legitimately condition on the event {τ m = j} in the third equality because τ m is a well-defined random variable. Now, P (y t = 0 i.o. τ m = j)p (τ m = j) 4

5 sup P (y t = 0 i.o. τ m = j) P (τ m = j) = sup P (y t = 0 i.o. τ m = j) (by Equation (9)) sup P ( t j, y t = 0 τ m = j) sup P (ε j+1 < β 0 m or ε j+2 < β 0 (m + 1) or ε j+3 < β 0 (m + 2)... τ m = j) P (ε t β 0 j) j=m where the last inequality follows by the independence of the ε t, countable subadditivity of the probability measure and because {τ m = j} σ(y 0, ε 1,..., ε j 1, ε j ) where σ(y 0, ε 1,..., ε j 1, ε j ) is the sub-sigma field generated by (y 0, ε 1,..., ε j 1, ε j ). Now, P (ε t β 0 j) j=m = P ( β 0 (k + 1) < ε t β 0 k) j=m k=j β0 k k k=m j=1 β 0 (k+1) β0 k k k=m β 0 (k+1) β0 m β 1 0 df ε (x) ( x /(β 0 k))df ε (x) x df ε (x), and the last expression converges to 0 as m if E[ ε t I(ε t < 0)] <, which completes the proof of Theorem 1. References Frederiksen, A., B. Honoré and L. Hu, 2007, Discrete Time Duration Models with Grouplevel Heterogeneity, Journal of Econometrics, forthcoming. Wooldridge, J. M, 2002, Econometric Analsysis of Cross Section and Panel Data. Press, Cambridge, Massachussets). (MIT 5