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Rank estimation of monotone hazard models
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Economics Letters 100 (2008) 80 – 82 www.elsevier.com/locate/econbase
Rank estimation of monotone hazard models ☆ Youngki Shin ⁎ Department of Economics, University of Western Ontario London, ON, Canada N6A 5C2 Received 3 April 2006; received in revised form 15 October 2007; accepted 14 November 2007 Available online 22 November 2007
Abstract I consider a class of hazard models that satisfy a flexible monotone restriction. A rank estimation procedure can be applied to this class. The result sheds light on the extension of rank estimation methods to hazard models with time-varying covariates. © 2007 Elsevier B.V. All rights reserved. Keywords: Monotone hazard function; Duration analysis; Rank correlation; Time-varying covariate; Covariate dependent censoring JEL classification: C13; C41
1. Introduction There are many economic questions related to the duration of events. For instance, labor economists may be interested in the relationship between unemployment spell length and other variables (see Kiefer (1988) for applications). Duration analysis usually starts with specifying a hazard function of an interesting variable. Cox (1972, 1975) proposed the proportional hazard (PH) model that does not require the specification of a baseline hazard function. Lancaster (1979) relaxed the restriction of proportionality and proposed the mixed proportional hazard (MPH) model by introducing unobserved heterogeneity into the PH model. Assuming additivity of the baseline hazard function leads to the additive hazard (AH) model (see Cox and Oakes (1984)). In this note I consider a monotone hazard (MH) model that is flexible enough to include all of the above models as a special case. Specifically, I assume the following hazard function: kðtjxi Þ ¼ /ðt; xiVb0 Þ
ð1Þ
where xi is a k-dimensional covariate and β0 is its corresponding parameter of interest. A non-negative function ϕ(·,·) is unknown but monotone in its second argument. Note that the baseline ☆
I would like to thank a referee and Shakeeb Khan for helpful comments. All errors are my own. ⁎ Tel.: +1 519 661 2111x85232; fax: +1 519 661 3666. E-mail address: [email protected]. 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.11.007
hazard can be defined as λ0(t) ≡ ϕ(t,0) by substituting x′i β0 = 0. Since ϕ can be any function that satisfies monotonicity in x′i β0, the MH model specification allows arbitrary link functions as well as an unspecified baseline hazard. This model is therefore more general than other semiparametric hazard models that only allow the latter. For instance, the PH model only allows arbitrary baseline hazard λ0(t) but its hazard function is required to have the multiplicative form of λ0(t) exp (xiβ) It is not difficult to check that all models mentioned above are nested within the MH model. The unknown ϕ(·) induces a broad model specification. Due to this fact, the usual counting process approach such as the partial maximum likelihood estimation (PMLE) in Cox (1972) or its extension using the pseudoscore function in Lin and Ying (1994) cannot be applied to estimate the MH model. Instead, I consider a rank estimation procedure to estimate β0 up to scale. Han (1987) proposed the maximum rank correlation (MRC) estimator and Cavanagh and Sherman (1998) did the monotone rank estimator (MRE) respectively. Recently, Khan and Tamer (2007) proposed the partial rank estimation (PRE) for the generalized accelerate failure time (GAFT) model in Ridder (1990) with general forms of random censoring. Rank estimations are more robust to different model specifications than the counting process approach. But it is more difficult to combine them with a model involving a time-varying covariate than the counting process method. This is because a hazard model with a time-varying covariate is not easily transformed into a regression model and most rank estimations require specific regression models.
Y. Shin / Economics Letters 100 (2008) 80–82
This note shows that the MH model which is not included in the GAFT model1 can be consistently estimated by the PRE. Since the hazard function in (1) is difficult to be transformed into a regression model, I derive the direct relationship between the hazard function and a key condition for identification. This sheds light on the extension of rank estimation methods to hazard models with time-varying covariates.2 In fact, if time-varying covariates satisfy a certain monotonicity assumption, then we can apply the PRE to the MH model with time-varying covariates. 2. Model and identification result In this section I introduce the model and PRE in detail and show that the key identification condition of the PRE holds. Let y⁎i be a latent duration variable following the monotone hazard function in Eq. (1): Pðt V yi⁎ V t þ hjyi⁎zt; xi Þ u/ðt; xiVb0 Þ: kð yi⁎¼ tjxi Þ¼ lim hA0 h
ð2Þ
Let ci be a random censoring variable that may depend on xi in an arbitrary way. Suppose the random sample (vi, xi′, di ) follows a right censoring model as follows: vi ¼ min yj⁎; ci di ¼ I ðyi⁎V ci Þ:
baB
X I y1i z y0j I xiVb z xjVb
ð5Þ
ip j
where y0i = vi and y1i = divi + (1 − di) · (+ ∞). The estimator maximizes rank correlation between indices and transformed duration variables. Note that if there is no censoring, that is di = 1 for all i and y0i = y1i = vi, then the PRE is reduced to the maximum rank correlation (MRC) estimation in Han (1987). The main result is stated in the next proposition whose proof and regularity conditions are in the Appendix. Proposition 2.1. Under the Assumptions I1–I3 (stated in the Appendix), the following condition holds for all i, j: Pr y1j z y0i z Pr y1i z y0j () xiVb0 z xjVb0 : ð6Þ 1
0
Thus, the following relationship holds between conditional probabilities and the hazard functions: Pr vj zvi jxi ; xj Pr vi zvj jxi ; xj ð8Þ Z þl Z þl ¼ Sj ðt ÞdFi ðt Þ Si ðt ÞdFj ðt Þ 0
Z ¼
ð3Þ
An observed duration vi may be censored and the binary variable di indicates whether it is censored or not. Note that I do not impose any regression model on y⁎i explicitly. This distinguishes the MH model from the GAFT model in Khan and Tamer (2007). Now I consider the estimation procedure. Khan and Tamer (2007) proposed the PRE defined as below for the GAFT model with covariate dependent censoring: b ¼ arg max
The condition (6) is crucial for identification of the PRE. If this condition holds, point identification, consistency and asymptotic normality follow directly from Theorems 2.2 and 2.3 in Khan and Tamer (2007) with additional regularity conditions thereof. However, the proof of Eq. (6) differs from Khan and Tamer (2007) since the model only depends on a class of hazard functions that cannot be transformed into a regression model. To help illustrate the proposition, I consider a simple case of no censoring. Let Fi(t) be a cdf and Si(t) be a survivor function of the duration yi⁎(= vi in this example) conditional on xi. Then the conditional probability that vj is greater than vi is: Z þl Pr vj zvi jxi ; xj ¼ Sj ðvi ÞdFi ðvi Þ: ð7Þ
þl
0
A simple counter example is the AH model that has a monotone hazard function but is not in accordance with the GAFT model specification. 2 Woutersen and Hausman (2005) suggest a rank estimation of the MPH model with time-varying covariates. Their model, however, is restricted to the MPH and also assumes the time-varying covariates are piecewise constant.
Z ¼
dSj ðt Þ dSi ðt Þ Sj ð t Þ Si ð t Þ dt dt dt
ð9Þ
/ðt; xiVb0 Þ / t; xjVb0 exp ðGðt ÞÞdt
ð10Þ
0
ð4Þ
81
þl
0
Rt where Gðt Þ ¼ 0 /ðs; xiVb0 Þ þ / s; xjVb0 ds. Therefore, the monotone hazard function implies Pr vj zvi jxi ; xj z Pr vi zvj jxi ; xj () xiVb0 z xjVb0 : ð11Þ This result suggests a general class of hazard models for which the rank estimation procedure may apply. It reveals the direct relationship between the hazard function and the key condition for identification, and saves us the effort of transforming the hazard function into a regression model. Such transformation may not even be possible in certain cases, such as a hazard model with an unspecified link function or time-varying covariates. It suffices to check the hazard function is monotone in xi′β0. An interesting application of this result is the MH model with time-varying covariates that satisfy certain restrictions. Consider a hazard model represented as ϕ(t,xi,(t)β0) where ϕ is monotone in the second term.3 The simplest example is where xi(t) is multiplicative separable such as xi(t) = g(t)zi for an unspecified function g(·) N 0 and a random variable zi In general, any timevarying covariate model can be included in this class provided that random processes {xi(t)β0} for i = 1,2…,n are monotone in observations over all time period t.4 The hazard function is again monotone in xi( ¯t )′β0 for any fixed time point ¯t . Thus, the proof in the appendix still holds in this case with slight modification 3 Most of current hazard models with a time-varying covariate fall in this class. It can be easily checked that the PH, MPH and AD models with a timevarying covariate satisfy this representation. 4 If we restrict our attention to the MPH model with the piecewise constant time domain, the estimator proposed by Woutersen and Hausman (2005) has an advantage that it is still consistent when { xi (t)′β0} is not monotone in i.
82
Y. Shin / Economics Letters 100 (2008) 80–82
such that xi′β0 = ( ¯t )′β0.5 To give some illustrations of the model with time-varying covariates, I may consider the following applications. One is an analysis of treatment effects for a disease. I may think of the case that the treatment intensity changes over time but it heavily depends on the initial status of each patient. Another is a duration analysis of firms with the cash flow that changes over time. If the time-varying cash flow depends on interest rates and the initial condition of each firm, then the MH model can be applied in the analysis.
Then, the first and the second terms are respectively, ð15Þ Pr y0i z y1i ; di ¼ 1; dj ¼ 1 ¼ Pr yi⁎ V yj⁎ V cj ; yi⁎ V cj Z ¼
Z
cj y⁎i
0
dFj yj⁎ dFi ð yi⁎Þ
Z ¼ Fj cj Fi cj
cj 0
ð16Þ
Fj ðyi⁎ÞdFi yi⁎ ; ð17Þ
3. Conclusion I have introduced a general class of duration models by assuming a monotone hazard function. I showed that the key condition for identification holds so that the PRE can be applied without any change. This result makes it possible to apply the rank estimation procedure to the whole class by simply checking the monotonicity condition. Certain time-varying covariates can be combined with the model but it is still an open question if I can estimate it with (general) time-varying covariates. Neither existing counting process approaches nor semiparametric approaches can resolve the problem yet. I leave this for future research.
Pr y0j z y1i ; di ¼ 1; dj ¼ 0 ¼ Pr
Appendix A. Regularity conditions The following regularity conditions come from Khan and Tamer (2007). We slightly change the condition I2 since the error term ɛi is not explicitly defined in the MH model. It still maintains the main idea of covariate dependent censoring. Let SX denote the support of xi Then Xuc defined as follow has positive measure: ð12Þ
The random variable ci is independent of yi⁎ conditional on xi. The first component of xi is continuously distributed on the real line with positive Lebesgue measure conditional on the remaining (k − 1) components of xi. Proof of Proposition 2.1 Let Fi(t) and Si(t) be defined as before. I suppress the notation but all probabilities are conditional on xi and xj I will prove the following condition that is equivalent to Eq. (6): Pr y0j z y1i z Pr y0i z y1j iff xiVb0 z xjVb0 : ð13Þ I partition the censoring variable into two cases: ci N cj and ci ≤cj. I start with the first case. Since di = 0 implies Pr(y0i ≥y1j) = 0, Pr y0j z y1i ¼ Pr y0j z y1i ; di ¼ 1; dj ¼ 1 ð14Þ þ Pr y0j z y1i ; di ¼ 1; dj ¼ 0 : 5 As pointed out by a referee, the censoring variable ci; however, requires a further restriction in the time-varying covariate case. Since ci itself is a random variable whose support is on the time domain, it cannot depend on xi (t) in an arbitrary way. However, it may depend on xi(t) in a restrictive way. For example, ci can be any arbitrary function of zi in the case of multiplicative separable xi (t) =g(t) zi.
yi⁎V cj ; yj⁎ Ncj
¼ Fi cj 1 Fj cj : Thus, the conditional probability is Z cj Fj ð yi⁎ÞdFi yi⁎ Pr y0j zy1i ¼ Fi cj
ð18Þ ð19Þ
ð20Þ
0
Z
cj
¼ 0
Xuc ¼ fxaSX jPðdi ¼ 1jxi ¼ xÞ N 0g:
cj
Sj yi⁎ dFi y⁎i :
I next evaluate P(y0i ≥ y1j) in the same way: Z cj Si yj⁎ dFj yi⁎ : Pr y0i zy1j ¼
ð21Þ
ð22Þ
0
Therefore, the desired result follows from the equation: Pr y0j zy1i Pr y0i zy1j ð23Þ Z cj dSj ðt Þ dSi ðt Þ ¼ S j ðt Þ Si ð t Þ dt dt dt 0 Z ¼
cj
/ðt; xiVb0 Þ / t; xjVb0 exp ðGðt ÞÞdt:
ð24Þ
0
The case of ci ≤ cj is analogous.
□
References Cavanagh, C., Sherman, R., 1998. Rank estimation for monotonic index models. Journal of Econometrics 84, 351–381. Cox, D., 1972. Regression models and life tables. Journal of the Royal Statistical Societry Series B 34, 187–220. Cox, D., 1975. Partial likelihood. Biometrika 62, 269–276. Cox, D., Oakes, D., 1984. Analysis of Survival Data. Chapman and Hall, London. Han, A., 1987. Non-parametric analysis of a generalized regression model. Journal of Econometrics 35, 303–316. Khan, S., Tamer, E., 2007. Partial rank estimation of transformation models with general forms of censoring. Journal of Econometrics 136, 251–280. Kiefer, N., 1988. Economic duration data and hazard functions. Journal of Economic Literature 26, 646–679. Lancaster, T., 1979. Econometric methods for the duration of unemployment. Econometrica 47, 939–956. Lin, D., Ying, Z., 1994. Semiparametric analysis of the additive risk model. Biometrika 81, 61–71. Ridder, G., 1990. The non-parametric identification of generalized accelerated failure-time models. Review of Economic Studies 57, 167–182. Woutersen, T., Hausman, J., 2005. Estimating a semi-parametric duration model without specifying heterogeneity. Johns Hopkins University, Working Paper.
Economics Letters 100 (2008) 80 – 82 www.elsevier.com/locate/econbase
Rank estimation of monotone hazard models ☆ Youngki Shin ⁎ Department of Economics, University of Western Ontario London, ON, Canada N6A 5C2 Received 3 April 2006; received in revised form 15 October 2007; accepted 14 November 2007 Available online 22 November 2007
Abstract I consider a class of hazard models that satisfy a flexible monotone restriction. A rank estimation procedure can be applied to this class. The result sheds light on the extension of rank estimation methods to hazard models with time-varying covariates. © 2007 Elsevier B.V. All rights reserved. Keywords: Monotone hazard function; Duration analysis; Rank correlation; Time-varying covariate; Covariate dependent censoring JEL classification: C13; C41
1. Introduction There are many economic questions related to the duration of events. For instance, labor economists may be interested in the relationship between unemployment spell length and other variables (see Kiefer (1988) for applications). Duration analysis usually starts with specifying a hazard function of an interesting variable. Cox (1972, 1975) proposed the proportional hazard (PH) model that does not require the specification of a baseline hazard function. Lancaster (1979) relaxed the restriction of proportionality and proposed the mixed proportional hazard (MPH) model by introducing unobserved heterogeneity into the PH model. Assuming additivity of the baseline hazard function leads to the additive hazard (AH) model (see Cox and Oakes (1984)). In this note I consider a monotone hazard (MH) model that is flexible enough to include all of the above models as a special case. Specifically, I assume the following hazard function: kðtjxi Þ ¼ /ðt; xiVb0 Þ
ð1Þ
where xi is a k-dimensional covariate and β0 is its corresponding parameter of interest. A non-negative function ϕ(·,·) is unknown but monotone in its second argument. Note that the baseline ☆
I would like to thank a referee and Shakeeb Khan for helpful comments. All errors are my own. ⁎ Tel.: +1 519 661 2111x85232; fax: +1 519 661 3666. E-mail address: [email protected]. 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.11.007
hazard can be defined as λ0(t) ≡ ϕ(t,0) by substituting x′i β0 = 0. Since ϕ can be any function that satisfies monotonicity in x′i β0, the MH model specification allows arbitrary link functions as well as an unspecified baseline hazard. This model is therefore more general than other semiparametric hazard models that only allow the latter. For instance, the PH model only allows arbitrary baseline hazard λ0(t) but its hazard function is required to have the multiplicative form of λ0(t) exp (xiβ) It is not difficult to check that all models mentioned above are nested within the MH model. The unknown ϕ(·) induces a broad model specification. Due to this fact, the usual counting process approach such as the partial maximum likelihood estimation (PMLE) in Cox (1972) or its extension using the pseudoscore function in Lin and Ying (1994) cannot be applied to estimate the MH model. Instead, I consider a rank estimation procedure to estimate β0 up to scale. Han (1987) proposed the maximum rank correlation (MRC) estimator and Cavanagh and Sherman (1998) did the monotone rank estimator (MRE) respectively. Recently, Khan and Tamer (2007) proposed the partial rank estimation (PRE) for the generalized accelerate failure time (GAFT) model in Ridder (1990) with general forms of random censoring. Rank estimations are more robust to different model specifications than the counting process approach. But it is more difficult to combine them with a model involving a time-varying covariate than the counting process method. This is because a hazard model with a time-varying covariate is not easily transformed into a regression model and most rank estimations require specific regression models.
Y. Shin / Economics Letters 100 (2008) 80–82
This note shows that the MH model which is not included in the GAFT model1 can be consistently estimated by the PRE. Since the hazard function in (1) is difficult to be transformed into a regression model, I derive the direct relationship between the hazard function and a key condition for identification. This sheds light on the extension of rank estimation methods to hazard models with time-varying covariates.2 In fact, if time-varying covariates satisfy a certain monotonicity assumption, then we can apply the PRE to the MH model with time-varying covariates. 2. Model and identification result In this section I introduce the model and PRE in detail and show that the key identification condition of the PRE holds. Let y⁎i be a latent duration variable following the monotone hazard function in Eq. (1): Pðt V yi⁎ V t þ hjyi⁎zt; xi Þ u/ðt; xiVb0 Þ: kð yi⁎¼ tjxi Þ¼ lim hA0 h
ð2Þ
Let ci be a random censoring variable that may depend on xi in an arbitrary way. Suppose the random sample (vi, xi′, di ) follows a right censoring model as follows: vi ¼ min yj⁎; ci di ¼ I ðyi⁎V ci Þ:
baB
X I y1i z y0j I xiVb z xjVb
ð5Þ
ip j
where y0i = vi and y1i = divi + (1 − di) · (+ ∞). The estimator maximizes rank correlation between indices and transformed duration variables. Note that if there is no censoring, that is di = 1 for all i and y0i = y1i = vi, then the PRE is reduced to the maximum rank correlation (MRC) estimation in Han (1987). The main result is stated in the next proposition whose proof and regularity conditions are in the Appendix. Proposition 2.1. Under the Assumptions I1–I3 (stated in the Appendix), the following condition holds for all i, j: Pr y1j z y0i z Pr y1i z y0j () xiVb0 z xjVb0 : ð6Þ 1
0
Thus, the following relationship holds between conditional probabilities and the hazard functions: Pr vj zvi jxi ; xj Pr vi zvj jxi ; xj ð8Þ Z þl Z þl ¼ Sj ðt ÞdFi ðt Þ Si ðt ÞdFj ðt Þ 0
Z ¼
ð3Þ
An observed duration vi may be censored and the binary variable di indicates whether it is censored or not. Note that I do not impose any regression model on y⁎i explicitly. This distinguishes the MH model from the GAFT model in Khan and Tamer (2007). Now I consider the estimation procedure. Khan and Tamer (2007) proposed the PRE defined as below for the GAFT model with covariate dependent censoring: b ¼ arg max
The condition (6) is crucial for identification of the PRE. If this condition holds, point identification, consistency and asymptotic normality follow directly from Theorems 2.2 and 2.3 in Khan and Tamer (2007) with additional regularity conditions thereof. However, the proof of Eq. (6) differs from Khan and Tamer (2007) since the model only depends on a class of hazard functions that cannot be transformed into a regression model. To help illustrate the proposition, I consider a simple case of no censoring. Let Fi(t) be a cdf and Si(t) be a survivor function of the duration yi⁎(= vi in this example) conditional on xi. Then the conditional probability that vj is greater than vi is: Z þl Pr vj zvi jxi ; xj ¼ Sj ðvi ÞdFi ðvi Þ: ð7Þ
þl
0
A simple counter example is the AH model that has a monotone hazard function but is not in accordance with the GAFT model specification. 2 Woutersen and Hausman (2005) suggest a rank estimation of the MPH model with time-varying covariates. Their model, however, is restricted to the MPH and also assumes the time-varying covariates are piecewise constant.
Z ¼
dSj ðt Þ dSi ðt Þ Sj ð t Þ Si ð t Þ dt dt dt
ð9Þ
/ðt; xiVb0 Þ / t; xjVb0 exp ðGðt ÞÞdt
ð10Þ
0
ð4Þ
81
þl
0
Rt where Gðt Þ ¼ 0 /ðs; xiVb0 Þ þ / s; xjVb0 ds. Therefore, the monotone hazard function implies Pr vj zvi jxi ; xj z Pr vi zvj jxi ; xj () xiVb0 z xjVb0 : ð11Þ This result suggests a general class of hazard models for which the rank estimation procedure may apply. It reveals the direct relationship between the hazard function and the key condition for identification, and saves us the effort of transforming the hazard function into a regression model. Such transformation may not even be possible in certain cases, such as a hazard model with an unspecified link function or time-varying covariates. It suffices to check the hazard function is monotone in xi′β0. An interesting application of this result is the MH model with time-varying covariates that satisfy certain restrictions. Consider a hazard model represented as ϕ(t,xi,(t)β0) where ϕ is monotone in the second term.3 The simplest example is where xi(t) is multiplicative separable such as xi(t) = g(t)zi for an unspecified function g(·) N 0 and a random variable zi In general, any timevarying covariate model can be included in this class provided that random processes {xi(t)β0} for i = 1,2…,n are monotone in observations over all time period t.4 The hazard function is again monotone in xi( ¯t )′β0 for any fixed time point ¯t . Thus, the proof in the appendix still holds in this case with slight modification 3 Most of current hazard models with a time-varying covariate fall in this class. It can be easily checked that the PH, MPH and AD models with a timevarying covariate satisfy this representation. 4 If we restrict our attention to the MPH model with the piecewise constant time domain, the estimator proposed by Woutersen and Hausman (2005) has an advantage that it is still consistent when { xi (t)′β0} is not monotone in i.
82
Y. Shin / Economics Letters 100 (2008) 80–82
such that xi′β0 = ( ¯t )′β0.5 To give some illustrations of the model with time-varying covariates, I may consider the following applications. One is an analysis of treatment effects for a disease. I may think of the case that the treatment intensity changes over time but it heavily depends on the initial status of each patient. Another is a duration analysis of firms with the cash flow that changes over time. If the time-varying cash flow depends on interest rates and the initial condition of each firm, then the MH model can be applied in the analysis.
Then, the first and the second terms are respectively, ð15Þ Pr y0i z y1i ; di ¼ 1; dj ¼ 1 ¼ Pr yi⁎ V yj⁎ V cj ; yi⁎ V cj Z ¼
Z
cj y⁎i
0
dFj yj⁎ dFi ð yi⁎Þ
Z ¼ Fj cj Fi cj
cj 0
ð16Þ
Fj ðyi⁎ÞdFi yi⁎ ; ð17Þ
3. Conclusion I have introduced a general class of duration models by assuming a monotone hazard function. I showed that the key condition for identification holds so that the PRE can be applied without any change. This result makes it possible to apply the rank estimation procedure to the whole class by simply checking the monotonicity condition. Certain time-varying covariates can be combined with the model but it is still an open question if I can estimate it with (general) time-varying covariates. Neither existing counting process approaches nor semiparametric approaches can resolve the problem yet. I leave this for future research.
Pr y0j z y1i ; di ¼ 1; dj ¼ 0 ¼ Pr
Appendix A. Regularity conditions The following regularity conditions come from Khan and Tamer (2007). We slightly change the condition I2 since the error term ɛi is not explicitly defined in the MH model. It still maintains the main idea of covariate dependent censoring. Let SX denote the support of xi Then Xuc defined as follow has positive measure: ð12Þ
The random variable ci is independent of yi⁎ conditional on xi. The first component of xi is continuously distributed on the real line with positive Lebesgue measure conditional on the remaining (k − 1) components of xi. Proof of Proposition 2.1 Let Fi(t) and Si(t) be defined as before. I suppress the notation but all probabilities are conditional on xi and xj I will prove the following condition that is equivalent to Eq. (6): Pr y0j z y1i z Pr y0i z y1j iff xiVb0 z xjVb0 : ð13Þ I partition the censoring variable into two cases: ci N cj and ci ≤cj. I start with the first case. Since di = 0 implies Pr(y0i ≥y1j) = 0, Pr y0j z y1i ¼ Pr y0j z y1i ; di ¼ 1; dj ¼ 1 ð14Þ þ Pr y0j z y1i ; di ¼ 1; dj ¼ 0 : 5 As pointed out by a referee, the censoring variable ci; however, requires a further restriction in the time-varying covariate case. Since ci itself is a random variable whose support is on the time domain, it cannot depend on xi (t) in an arbitrary way. However, it may depend on xi(t) in a restrictive way. For example, ci can be any arbitrary function of zi in the case of multiplicative separable xi (t) =g(t) zi.
yi⁎V cj ; yj⁎ Ncj
¼ Fi cj 1 Fj cj : Thus, the conditional probability is Z cj Fj ð yi⁎ÞdFi yi⁎ Pr y0j zy1i ¼ Fi cj
ð18Þ ð19Þ
ð20Þ
0
Z
cj
¼ 0
Xuc ¼ fxaSX jPðdi ¼ 1jxi ¼ xÞ N 0g:
cj
Sj yi⁎ dFi y⁎i :
I next evaluate P(y0i ≥ y1j) in the same way: Z cj Si yj⁎ dFj yi⁎ : Pr y0i zy1j ¼
ð21Þ
ð22Þ
0
Therefore, the desired result follows from the equation: Pr y0j zy1i Pr y0i zy1j ð23Þ Z cj dSj ðt Þ dSi ðt Þ ¼ S j ðt Þ Si ð t Þ dt dt dt 0 Z ¼
cj
/ðt; xiVb0 Þ / t; xjVb0 exp ðGðt ÞÞdt:
ð24Þ
0
The case of ci ≤ cj is analogous.
□
References Cavanagh, C., Sherman, R., 1998. Rank estimation for monotonic index models. Journal of Econometrics 84, 351–381. Cox, D., 1972. Regression models and life tables. Journal of the Royal Statistical Societry Series B 34, 187–220. Cox, D., 1975. Partial likelihood. Biometrika 62, 269–276. Cox, D., Oakes, D., 1984. Analysis of Survival Data. Chapman and Hall, London. Han, A., 1987. Non-parametric analysis of a generalized regression model. Journal of Econometrics 35, 303–316. Khan, S., Tamer, E., 2007. Partial rank estimation of transformation models with general forms of censoring. Journal of Econometrics 136, 251–280. Kiefer, N., 1988. Economic duration data and hazard functions. Journal of Economic Literature 26, 646–679. Lancaster, T., 1979. Econometric methods for the duration of unemployment. Econometrica 47, 939–956. Lin, D., Ying, Z., 1994. Semiparametric analysis of the additive risk model. Biometrika 81, 61–71. Ridder, G., 1990. The non-parametric identification of generalized accelerated failure-time models. Review of Economic Studies 57, 167–182. Woutersen, T., Hausman, J., 2005. Estimating a semi-parametric duration model without specifying heterogeneity. Johns Hopkins University, Working Paper.