Modelling Panel Count Data With Excess Zeros: A Hurdle Approach

Modelling Panel Count Data With Excess Zeros: A Hurdle Approach José M. R. Murteira Faculdade de Economia, Universidade de Coimbra, and CEMAPRE March, 2007 Abstract This paper presents a hurdle-type regression model for panel count data. Several speci c features of the data generating process require a modelling approach that di ers from some commonly used count data models. The suggested model is intended for count data with excess zeros, relative to simple Poisson generated data, and bounded support. This kind of data may occur, e.g., in the context of consumer credit or behavioural scoring, with reference to the repayment of loans in a pre-determined number of xed installments. At each repayment date a debtor faces a known number of missed installments which depends on his previous repayment decisions and is bounded by the age of the contract. In any case, as most clients repay their debts on time, the number of missed installments at each date is expected to display more zero values than would be the case if this variable were to follow, e.g., the Poisson distribution. Throughout the paper a random sample of observations on time series of counts and some set of covariates is supposed to be available to the researcher. The dependence structure within each time series is addressed through the use of the "binomial thinning" operator (Steutel and Van- Harn, 979). As detailed in the text, this operator provides a exible way I am grateful to João Santos Silva for helpful comments. Support from Fundação para a Ciência e Tecnologia, program FEDER/POCI 200, is also gratefully acknowledged. The usual disclaimer applies. Address: José M. R. Murteira, Faculdade de Economia, Universidade de Coimbra, Av. Dias da Silva, 65, 3004-52 Coimbra, Portugal. e-mail: jmurt@fe.uc.pt.

to model the dependence between consecutive counts, within a Markov chain-type modelling approach. Estimation of the model through conditional least squares and maximum likelihood is illustrated on the basis of di erent simulated data sets of independent individual time sequences of counts. The performance of both methods appears somewhat similar, with slight expectable e ciency advantages of maximum likelihood in cases of correct model speci cation. The proposed speci cation can be used, for instance, as a model of repayment behaviour, to be employed in a context of credit or behavioural scoring. In this sense it may provide better estimates of default probability than, e.g., simple cross-section models of the total number of missed installments. Meanwhile, with some adjustments, the approach is rich enough to encompass such situations as early repayment or redemption of loans, as well as loan contracts with variable installments. JEL classi cation: C23, C25 Key Words: panel count data; hurdle model; conditional least squares; maximum likelihood.

Introduction This paper introduces a hurdle-type regression model for panel count data. Several speci c features of the data generating process require a modelling approach that di ers from some commonly used count data models. The proposed speci- cation is intended for count data with excess zeros, relative to simple Poisson generated data, and bounded support. Such data may occur, for instance, in the context of consumer credit or behavioural scoring, with reference to the repayment of loans in a pre-determined number of xed installments: at each repayment date a debtor faces a known number of missed installments, which depends on his previous repayment decisions and is bounded by the age of the contract. In any case, as most clients repay their debts on time, the number of missed installments at each date is expected to display more zero values than would be the case if this variable were to follow, e.g., the Poisson distribution. Throughout the text a random sample of observations on time series of counts and some set of covariates is supposed to be available to the researcher. For each statistical unit at each observation period the possibility of excess zeros, relative to the simple Poisson, is dealt with through a hurdle-type speci cation (Cragg, 97, Mullahy, 986). In addition, the inherently bounded support of the dependent count variable also excludes the Poisson or negative binomial, among others, from the set of candidate models to be selected. In the present context the binomial distribution or a binomial-based mixture seem clearly more appropriate than functions with unbounded support. The speci cation proposed here is designed to model time series of counts, one for each statistical unit. The literature proposes many possible ways to model time series of discrete variables. Integer-valued autoregressive, moving average (INARMA) models constitute a prominent, well established example, including INAR models as a special case. The latter specify the realized value of the variable of interest at period t, y t, as the sum of an integer function of past outcomes and the realization of an independent integer random variable. These models have the same serial correlation structure as linear ARMA models for continuous data, a clearly attractive feature. Di erent choices for the distribution of the innovation term (e.g. Poisson) lead to di erent marginal p.f: s for y t. Within the pure time series case, the Poisson INAR() was rst proposed

by McKenzie (985) and further discussed, along with other INARMA models, by Al-Osh and Alzaid (987) and McKenzie (988). Brännäs (995) was the rst to extend these models to the regression case. Several other approaches are discussed in the literature on discrete variables time series analysis: surveys can be found in Cameron and Trivedi (998), ch. 7, and McKenzie (2003). The dependence structure within each time series is addressed in the present case through the use of the "binomial thinning" operator (Steutel and VanHarn, 979). This operator provides a exible way to model the dependence between counts, with the observation for one period de ning the support bound of the count variable for the following period. As detailed below, this feature seems quite adapt to the type of data the proposed model is intended for. The paper is organized as follows. After speci cation of the general model in section 2, section 3 presents its estimation through conditional least squares (CLS) and maximum likelihood (ML). The performance of these methods is assessed in section 4 on the basis of di erent simulated data sets. Algebraic derivations and proofs are included in the nal appendix. 2 Model Speci cation Consider a sample of observations on n time series of counts, y it, where t = ; :::; T i, and i = ; :::; n denote, respectively, time and individual observational indices. In what follows a model for each individual time sequence of counts is presented, enabling the speci cation of a panel count data model. Estimation and inference issues are discussed in the next section. The proposed speci cation starts with the following structural assumption about the triplet (y it,y i;t,d it ) (individual index, i, omitted): y t = d t (y t + ) + ( d t ) (p y t ) ; () y t 2 f0; ; :::; tg ; t = ; :::; T; where y 0 ( y i0 ) 0, d t ( d it ) 2 f0; g denotes a Bernoulli random variable with Pr (d t = ) = p, and the parameter p is such that 0 p. The symbol "" denotes the binomial thinning operator, de ned as follows. If y represents a positive integer, then p y P y j= b j (p ), where fb j (p ), j = ; :::; yg 2

denotes a set of i.i.d: Bernoulli random variables independent of y, for which Pr (b j (p ) = ) = p. That is, conditional on y, p y is a binomial random variable, the number of successes in y independent trials in each of which the probability of success is p. Thus, conditional on y t, the support of y t is f0; ; :::; y t + g. If d t =, then y t = y t + ; on the contrary, if d t = 0, then y t conditional on y t follows a binomial p.f. with parameters y t and p. The probabilistic model for y it jy i;t is thus speci ed as a two-part, or hurdle, model: the rst part is a binary outcome model and the second part is a count model with bounded support. Unless p = 0 (corresponding to the trivial case y t 0, 8t), the sequence fy t g is not stationary. This can be seen by considering the sequence of unconditional rst moments of y t, E (y t ) = p + r + r 2 + + r t ; where r p + p ( p). ( ) The covariance function is given by COV (y t ; y t k ) = r k V AR (y t k ) ; a similar result to the expression for the covariance function in linear AR models for continuous data. It follows that the autocorrelation function, r kp V AR (y t k ) =V AR (y t ); depends, not only on lag (k), but also on t (as V AR (y j ) involves j). (2 ) For each individual the proposed model can be cast within the general framework of a rst-order Markov chain. For each individual time sequence of T i For the trivial case p i = (, r i = ), one obtains E (y it jy i;t ) = y i;t + = t; V AR (y it jy i;t ) = 0: 2 The formula for E y 2 t can be obtained as the solution to the di erence equation E yt+ 2 = p + (r p) 2 = ( p) E yt 2 + ( r) (r p) p + r + + r t = ( p) + p; t = 2; :::; l; with initial condition E y 2 = p. Obviously, the expression for V AR (yt) involves t. 3

count variables, fy it ; t = ; :::; T i g, one can formally write the following recurrence formula for Pr (y it ) (individual index and possible conditioning covariates omitted): tx Pr (y t = y) = Pr (y t = yjy t = j ) Pr (y t = j ) ; j=maxf;yg 0 y t; t T; with transition probabilities, Pr (y t = yjy t = j ), obtained from () and Pr (y 0 = 0) =. For each t, the last recurrence formula can be written in matrix form as t = M t = M t 0 ; with 0 = (T +) 2 6 4 0. 0 3 7 5 ; t = (T +) 2 6 4 Pr (y t = 0) Pr (y t = ). Pr (y t = t) 0. 0 3 ; t = ; :::; T; 7 5 and transition matrix M = (T +)(T +) 2 6 4 m m 2 m 3 m T 0 m 2 m 22 m 23 m 2T 0 0 m 32 m 33 m 3T 0 0 0 m 43 m 4T 0..... 0 0 0 m T +;T 0 3 ; 7 5 where m jk Pr (y t = j jy t = k ), k T, j k +. For each individual the present model can also be viewed as a panel data, nonstationary version of the DAR() model, proposed by Jacobs and Lewis (978). The latter can be expressed as y t = d t y t + ( d t ) z t ; 4

where fd t g are i.i.d: binary and fz t g are i.i.d: with given distribution. If y 0 is also sampled from this distribution, then this model de nes a stationary process with that same marginal distribution. In model () not only stationarity is absent (because of the term y t + ) but also the variable replacing z t (p y t ) is not i.i.d:. In addition, the assumption of independence between d t and p y t is to be relaxed below. If one thinks of the individuals in the sample as borrowers repaying their loans through xed periodic installments, with y it denoting the number of missed installments by individual i at the end of period t, then the probabilistic model for y it jy i;t can be seen as a model of repayment behaviour. d t is de ned to be zero if the client decides to pay at period t (that is, "success" refers to a nonpayment choice, with probability denoted by p). If the client chooses to pay (d t = 0), he then decides how many installments to pay, ranging from just one (p y t = y t ) to all the installments he has missed that far plus the one for the present period (p y t = 0). That is, under the present framework each borrower is supposed to make a twofold decision at every repayment date. First, he decides whether to pay or not altogether at that date. Then, if he chooses to pay and more than one installment are due, he decides how much (how many installments) to pay. The above example illustrates the possible meaning of several limiting (even if unlikely) cases concerning di erent values of p and p. First, p = 0 (d t degenerate at zero, 8t), means that the borrower always decides to pay, one installment at least. From y 0 0 it follows that y d = 0; as y t = p y t and the individual always pays, y t = 0 = y t, 8t, so p is not identi ed. On the other hand, p = means that the client always decides not to pay; then d t =, 8t, so y t = y t + = t and p is again not identi ed. Now consider the case p = 0. Then, once the client decides to pay (d t = 0) he pays all the missed payments at that time (the b j are degenerate at zero, j = ; :::; t ), in addition to that period installment. That is, at each repayment date a client either pays all the installments he missed that far, plus the one for that period, or he pays none. On the other hand, p = means that when the client decides to pay (d t = 0) he only pays one installment (b j degenerate at one, j = ; :::; t ). That is, with p = the client s balance of missed 5

payments never decreases at best it remains at the level it rose to, the last time the client chose not to pay. Naturally, some of the above limiting cases may be unlikely namely the case with p =. Also, even for those (presumably many) clients who regularly meet their obligations, p can be positive due to the possible occurrence of random unexpected (and undesirable) events, a ecting their nancial capacity and hindering their plans to pay one installment after the other during the life of the contract. In any case, a well known advantage of the hurdle speci cation is that it enables di erentiated treatment of zero and positive values of the dependent variable. In the present example, with most individuals expected to regularly meet their nancial obligations, this feature seems clearly attractive. (3 ) It may be more appropriate to consider p and p individual-speci c, rather than constant across di erent individuals (p i and p i in (), instead of p and p ). Also, a pure time series model may not lead to very useful conclusions. (4 ). In order to account for this individual heterogeneity one obvious possibility is to introduce regressors in the model (e.g: in p and p ). Nevertheless, even within a regression approach unobserved individual and/or time e ects may still be at play here, requiring a more sophisticated approach concerning model estimation. If this is the case, the obvious decision regards the choice between a xed e ects approach or a random e ects approach, marginal to unobserved individual or time e ects. As usual, the appropriate choice depends on the objective of the analysis. For instance, a random e ects approach seems more adapt for credit scoring, usually involving a large number of individual loan contracts. The assumption of independence about the joint probabilistic behavior of (d t ; b ; :::; b t ; y t ) may be inappropriate or unrealistic. On the other hand, analysis and estimation of the model is considerably eased with such an assumption. A simplifying way out of this dilemma is to assume that d t and y t are independent conditionally on p. Also, b j, j = ; :::; t and y t can be assumed conditionally independent, given p. Then, the independence assumption of these variables can be relaxed with the introduction of regressors (possibly 3 An extended survey on credit and behavioural scoring techniques and applications can be found, for instance, in Thomas, Edelman and Crook (2002). 4 For instance, credit or behavioural scoring purposes may require the classi cation of debtors on the basis of contracts and customers characteristics. 6

including lagged values of the dependent variable) and/or unobservable e ects in the model, through p and p. For instance, p and p can be speci ed as logits: with time-invariant regressor vectors x i and z i, p i and p i can be expressed as p i ( + exp ( x 0 i)) ; (2) p i ( + exp ( zi)) 0 ; where and denote regressor vectors. Obviously, time-varying regressors can also be introduced then one would naturally use p it and p it, instead. Unobserved heterogeneity can also be accounted for but, as expected, this concern can actually lead to increased di culty in what regards model estimation. 3 Estimation and Inference The present section discusses estimation and testing of the panel count data model introduced above. Two estimation methods are considered in what follows: CLS and ML. Use of both methods in the context of the stationary Poisson INAR is studied by Al-Osh and Alzaid (987), Jin-Guan and Yuan (99) and Ronning and Jung (992). Brännäs (994) introduces GMM and extends its use to a panel data generalized Poisson INAR(). Brännas (995) proposes a Poisson INAR() regression model and studies its estimation through CLS and GMM. A very recent account of estimating methods for panel count data, namely GMM, can be found in Windmeijer (2006). 3. Conditional Least Squares Under the assumptions of the model and in view of the de nition of the binomial thinning operator, the conditional mean of y it can be written as E (y it jy i;t ) = E (d it (y i;t + ) + ( d it ) (p i y i;t )) = p i + r i y i;t ; t = ; :::; T i ; (3) where y i0 0 and r i p i + p i ( p i ). Consider, for now, a pure time series framework, with p and p assumed constant and a panel of n independent individual time series, fy i ; :::; y iti g, i = ; :::; n, (y it independent of y ju, 8i 6= 7

j; 8t; u). Then, the CLS estimator of the parameters p and p minimizes the criterion function nx XT i (y it p ry i;t ) 2. i= t= This objective function is the one associated with nonlinear least squares (NLS) estimation of the pooled regression model y it = p + ry i;t + u it ; (4) t = ; :::; T i, i = ; :::; n; so the corresponding estimators are easily obtained with common econometric packages. For n!, these estimators are consistent, p n-asymptotically normal with the usual covariance matrix associated with (heteroskedastic) NLS estimation. The CLS procedure assumes homoskedastic errors, which is not so by construction. Under correct speci cation of V AR (y it jy i;t ) an alternative estimator, more e cient than CLS is conditional weighted least squares (CWLS), with the weighting functions obtained from V AR (u it jy i;t ) = V AR (y it jy i;t ) = y 2 i;t p ( p) ( p ) 2 + y i;t ( p) ( p ) (2p + p ) + p ( p) ; (5) replacing p and p with its CLS estimates. As the true form of heteroskedasticity may be unknown, the robust "sandwich" covariance estimator should also be used for inference purposes (either with CLS or CWLS). As previously mentioned, the assumption of constant parameters, p and p, 8i, can be rather restrictive. This assumption can be relaxed, even within a pure time series context. (5 ) Within a regression framework one can specify p i 5 The assumption can be relaxed within a pure time series context by adopting individualspeci c parameters, p i and p i. Then, y it = p i + r i y i;t + u it ; u it = (d it p i ) (y i;t + ) + ( d it ) (p i y i;t ) + ( p i ) (p i y i;t ) ; where E (d it jp i ) = p i, r i p i + p i ( p i ) and p i y i;t P y i;t j= b ij, with E (b ij jp i ) = p i. The model parameters can be estimated through separate CLS regressions (one for each individual with T i > 2), or a single regression with individual dummies the resulting estimators are consistent with T i!. The precision of individual estimates of this xed 8

and p i, for instance, as in (2). Accordingly, the CLS estimator of 0 ; 0 0 can be de ned as ^CLS = arg min nx XT i (y it p i r i y i;t ) 2 : i= t= This estimator is also relatively straightforward to implement, given access to a software package including NLS estimation, although reported standard errors may wrongly assume homoskedastic errors. CWLS is a more e cient method, for which the weighting functions can be obtained from (3) and (5), with expectations now conditional on x i and y i;t. Again, if the true form of heteroskedasticity is unknown, the robust "sandwich" covariance matrix estimator should also be used for inference purposes. Accounting for unobserved individual heterogeneity can be di cult in a regression context if the form of this heterogeneity is left unspeci ed. A feasible approach introduces unobservables in the model only through p i p i. One presumably useful simpli cation considers individual-speci c, timeinvariant random intercepts in both p i and p i, leading to the formal de nitions (6 ) and p i ( + exp ( x 0 i " i )) ; (6) p i ( + exp ( z 0 i " i )) : Then, for each individual the model for the conditional rst moment of y it becomes E (y it jy i;t ; x i ) = q i + r i y t Z ; (7) q i p i f " (") d"; Z r i (p i + p i ( p i )) f " (") d" where " " i, i = ; :::; n, denote i.i.d: random variables with zero mean and some speci ed density f ". It is noted that, conditional on observed regressors, e ects approach can be poor for individuals with low T i. Still, it can be useful as a rst step in CWLS estimation or as a means to test the hypothesis of parameter constancy with, e.g., a standard F test of model (4) against the latter speci cation. Alternatively, a random e ects approach can be adopted, with estimation of the relevant parameters of the distributions of the random variables p i and p i over di erent individuals. 6 Obviously, unobserved individual heterogeneity can be dealt with in this same way within a pure time series approach the only di erence being the absence of observable regressors. 9

p i and p i are no longer independent. (7 ) The parameters of the model (, and parameters in f " ) can be estimated through nonlinear CLS, provided intervening integrals are conveniently computed (with quadrature or simulation approximation methods, if necessary). (8 ) Accounting for unobservables in this panel data regression model has now led to full speci cation of the conditional distribution of the sequence fy t ; t = ; :::; T g given x. Though easy to implement, the CLS approach does not make use of all the information this full speci cation provides. One obvious alternative, possibly more e cient, is provided by the ML method, to be addressed in the next section. 3.2 Maximum Likelihood Under the assumption of independent individual time sequences ML estimation requires full speci cation of the joint probabilistic model for the Markov chain fy t ; t = ; :::; T g. This model is already implicit in previous assumptions () 7 In general, accounting for unobserved individual heterogeneity a ects the Markov nature of each individual sequence fy t; t = ; :::; T g. For simplicity, let " it = " i, 8t. If, conditional on x and ", the sequence constitutes a Markov chain, that is, f (y ; :::; y T j") = f (y j") TY f (y tjy t ; ") (conditioning on x, and function indices omitted), the same is not true when " is integrated out from this joint density. Formally, f (y ; :::; y T ) = 6= Z Z = f (y ) f (y j") t=2 TY f (y tjy t ; ") f " (") d" t=2 TY Z f (y j") f " (") d" f (y tjy t ; ") f " (") d" t=2 TY f (y tjy t ) : t=2 Nevertheless, the formal inclusion of " in the model (through p and p ) causes y t to be dependent on other elements of the chain only through y t and ". This leads to consideration of the conditional rst moment of y t as in (7), thereby enabling CLS estimation analogously as before. 8 Other schemes could be adopted for the inclusion of unobservables, such as random and/or coe cients (other than the intercepts), time-varying individual e ects and/or di erent (correlated) disturbances in p and p. Presumably, the usefulness of such sophistications should be weighted against the added estimation di culty they are bound to represent. 0

and (2) (or (7) with speci cation of f " ). Here, this full speci cation is made explicit in order to implement ML. In the present context y 0 0, so y d and Pr (y ) = p y ( p) y, with y 2 f0; g. Assumption () leads to the following model of transition probabilities (individual indices omitted, so T T i ): (yty t ) yt Pr (y t jy t ) = p ( (yt=yt +) p) p yt ( p ) yt yt ; where () denotes the usual indicator function. Then, the joint conditional density of each individual sequence fy t ; t = ; :::; T g can be written as y t = = / f y (y ; :::; y T jp; p ) TY p ( (yt=yt +) p) t= TY p p t= yt y t p p p p yt y t p yt ( p ) yt yt (y ty t )! (yt=y t +) ( p) (8) p p yt ( p ) yt (yty t )! P T t= (yt=yt +) ( p) T P T! (T 2) t=2 (ytyt )yt P T t=2 ( p ) (ytyt )yt ; exhibiting, as expected, the usual split of hurdle models into two separate components: the rst, involving p, refers to the binary process that splits individual sequences into success (periods for which y t = y t for which y t y t model for periods with y t y t. + ), and failure (periods ); the second, involving p, refers to the binomial part of the ML estimates can be obtained on the basis of an individual contribution to the log-likelihood of the form, XT i p LL i = const: + (y it = y i;t + ) log p + T i log ( p) t= + (T i 2) XT i t=2 (y it y i;t ) y it log! p + y i;t log ( p ) : p It is readily seen that estimation of the rst component of the hurdle (p estimation) uses all observations in the sample. Estimation of the second component (involving p ) disregards data on the rst period for every individual

(consequently disregarding individuals with only one observation), as well as observations for periods with d it =, so y it = y i;t +. If one wants to analyze a regression model, covariates can be introduced through parameters p and p, speci ed, for instance, as logits. (9 ) Unobserved individual heterogeneity can also be dealt with, analogously as in the previous section. However, if such heterogeneity is present in both p and p, ML estimation is bound to become more di cult because, in general, the likelihood no longer factors into two functionally independent terms. (0 ) In addition, as the likelihood may involve complex integrals with no known analytical expressions, simulation or quadrature approximation techniques may be required to estimate the model. 4 Simulation Results 4. Monte Carlo Design This section illustrates the performance of the CLS and ML estimators of the panel data hurdle model on the basis of a simulated panel data set. This panel contains information on each individual time series of counts up to the sampling date, as well as on a set of covariates. As before, the value of the dependent variable for individual i at period t is denoted by y it, where t T i, and T i represents the length of the i-th series up to the observation date (for instance, T i can be measured in months). Two samples were generated: the rst with n = 360 individual time sequences and the second with n = 5400 sequences. In each sample T i 36, with the same number of individuals for each di erent T i value. That is, the 9 The above expressions hold with time-invariant regressors. If time-varying covariates are introduced, notation must be altered accordingly. 0 This is in line with the approach of Winkelmann (2004). In this case the individual contribution to the likelihood can be written as Z Z L = A (p) B (p ) df (p; p ) ; where A and B denote, respectively, the rst and second parts of the hurdle likelihood, and F (p; p ) denotes the joint distribution of p and p (possible conditioning on observed regressors omitted). Only with independent p and p can this expression be written as a product of two functionally independent terms. 2

smaller sample contains 0 (= 360=36) individuals per each T i, whereas the larger sample contains 50 (= 5400=36) individuals per each T i. Individual time sequences are drawn independently (independence across i); each individual sequence fy it ; t = ; :::; T i g is drawn from the joint (conditional) p.f: de ned in (8) with p and p speci ed as logits. These include the following regressors from the corresponding marginals: x (intercept term), x 2 Bernoulli (:25), x 3 N (0; 2:25), and x 4 a rank variable generated by 8 ; 0 < w < 2; 2; 2 w < 4; >< x 4 = 3; 4 w < 6; 4; 6 w < 8; >: 5; w 8; with w a chi-squared variable with three degrees of freedom. Covariates are assumed time-invariant (x kit x ki, k = 2; 3; 4) with p and p speci ed as p it ( + exp ( x 0 i y i;t " i )) ; p i ( + exp ( x 0 i " i )) ; with x i (; x 2i ; x 3i ; x 4i ) 0, (:85; :75; :2; :5) 0, (; :5; :5; :5) 0, is a parameter and " i N 0; " 2, 8i, denotes an unobserved individual e ect. For each of the two sample sizes various DGP s are considered, according to di erent values of and 2 ". Respectively, 2 f0; :3g and 2 " 2 f0; g. For instance, ; " 2 = (0; 0) means that the sample is generated without unobserved individual heterogeneity and with the assumption that, at each date, individual decisions are independent of the previous value of the count variable (y i;t ). Alternatively a negative value is chosen for, under which p it is lower the higher y i;t. The following table labels these DGP s: Data Generating Processes DGP DGP2 DGP3 DGP4 0 :3 0 :3 2 " 0 0 3

These simulated data sets can be thought of as samples of loan repayment histories in xed installments. Each individual represents a di erent contract aged T i months at the sampling date. The assumption of time-invariant regressors may re ect the frequent fact that contracts and clients characteristics are recorded at the time of loan applications and remain subsequently unchanged over the repayment period. A negative value for (p it decreasing in y i;t ) is naturally interpreted as a decrease in the probability not to pay, in response to an increase in the number of previously missed installments. In the simulation experiment model () is denoted, respectively, as PHL and PH, according to whether it is speci ed with or without lagged dependent variable in p. The performance of CLS and ML estimation methods is assessed on the basis of 2000 replications of the described samples with regressors newly drawn at each replica. All computations were performed using TSP 5.0 (Hall and Cummins, 2005). 4.2 Monte Carlo Results Estimation results for models PH and PHL are displayed in tables. through 4.2, corresponding to the four DGP s and two di erent sample sizes. The tables contain estimates averages and standard errors for parameters in p ( through 4, and for model PHL) and in p ( through 4 ). ML estimates and standard errors for the parameters are the same under models PH and PHL because the second part of the log-likelihood (involving p ) is identical in both models (see (8)). Obviously, this is not the case for the CLS estimates of. The estimation results suggest some remarks. First, under DGP both models are correctly speci ed. Accordingly, estimates from both models are relatively similar. Both CLS and ML seem to work slightly better (with lower standard errors) under model PH than with PHL, which is expectable because the former assumes the true value for, while it is estimated in the latter. For CLS this nding extends to p parameters as well. Understandably, the advantage of model PH over PHL is reduced as the sample size increases. Meanwhile, under DGP ML standard errors are invariably lower than CLS, a consequence of the former method full usage of the available information. Under DGP2 only model PHL is correctly speci ed. Nevertheless, model PH seems to produce reasonable CLS estimates with both sample sizes, quite 4

better than ML and close to PHL estimates. Actually this pattern can be deceiveing as it appears to be associated with a relatively low absolute value of, bringing models PH and PHL close to one another. Unreported simulation results for model PH do indicate a strong sensitivity of both and CLS estimates to the true value of. Meanwhile, for the reason already mentioned, ML estimates of coincide with those from model PHL and are, therefore, obtained from consistent estimators. Given the functional separability of the likelihood function, ML estimates are immune to the misspeci cation of p. Under DGP3 and DGP4 no model is correctly speci ed, as none of them allows for unobserved individual heterogeneity. The consequences of such misspeci cation appear somewhat widespread, a ecting both p and p parameters estimates (tables 3. through 4.2). These e ects are particularly severe for the CLS estimates of the intercept term in p but, overall, results are not trustworthy under neglected individual heterogeneity. 5 Conclusion This paper presents a hurdle model for panel count data with excess zeros and bounded support, suggesting its estimation through CLS and ML. The use and behaviour of both methods is illustrated with simulated data sets of independent individual time sequences of counts. The performance of CLS and ML appears somewhat similar under the considered DGP s, with a slight and expectable advantage of the latter in cases of correct model speci cation. Naturally, e ciency advantages of ML should be weighted against the considerably easier coding of CLS estimation. The proposed speci cation can be used, for instance, as a model of repayment behaviour to be employed in a context of credit or behavioural scoring. In this sense it may provide better estimates of default probability than, e.g., simple cross-section models of the total number of missed installments. Meanwhile, with some adjustments, the approach is rich enough to encompass such situations as early repayment or redemption of loans, as well as loan contracts with variable installments. Naturally, these examples can provide the ground for future work. 5

Estimation Results - DGP Table. - Sample: 360 individuals Model PH PHL Est. Meth. CLS ML CLS ML est. s.d. est. s.d. est. s.d. est. s.d..876.84.85.096.902.80.790.2 2 -.753.50 -.752.093 -.764.5 -.732.094 3.207.047.20.024.23.049.90.026 4 -.53.5 -.502.067 -.53.7 -.477.070 -.05.03.028.026.949.338.004.29.6.555.004.29 2 -.59.29 -.506.28 -.578.322 -.506.28 3.497.098.50.034.523.3.50.034 4 -.488.9 -.507.096 -.575.262 -.507.096 Table.2 - Sample: 5400 individuals Model PH PHL Est. Meth. CLS ML CLS ML est. s.d. est. s.d. est. s.d. est. s.d..852.047.849.025.854.045.779.029 2 -.75.038 -.75.024 -.752.038 -.728.024 3.200.02.200.006.20.02.88.007 4 -.50.029 -.499.08 -.502.029 -.47.08 -.003.02.032.006.995.085.000.033.0.42.000.033 2 -.50.076 -.500.032 -.505.082 -.500.032 3.50.024.500.009.503.029.500.009 4 -.499.048 -.500.025 -.505.067 -.500.025 6

Estimation Results - DGP2 Table 2. - Sample: 360 individuals Model PH PHL Est. Meth. CLS ML CLS ML est. s.d. est. s.d. est. s.d. est. s.d..838.52.433.087.899.56.742.4 2 -.74.33 -.629.090 -.766.40 -.72.099 3.99.036.33.023.208.040.83.027 4 -.468.098 -.34.062 -.52.04 -.459.07 -.298.84 -.24.042 -.238.299.005.37.842.588.005.37 2 -.99.259 -.506.34 -.476.273 -.506.34 3.43.074.503.036.495.085.503.036 4 -.024.63 -.507.02 -.444.257 -.507.02 Table 2.2 - Sample: 5400 individuals Model PH PHL Est. Meth. CLS ML CLS ML est. s.d. est. s.d. est. s.d. est. s.d..87.037.43.022.853.04.733.028 2 -.734.033 -.624.022 -.750.035 -.74.023 3.95.009.32.006.200.00.8.007 4 -.456.024 -.340.06 -.50.027 -.454.08 -.30.06 -.209.00 -.90.073.00.035.989.76.00.035 2 -.88.066 -.500.034 -.498.074 -.500.034 3.42.08.500.009.500.022.500.009 4 -.037.040 -.500.026 -.496.073 -.500.026 7

Estimation Results - DGP3 Table 3. - Sample: 360 individuals Model PH PHL Est. Meth. CLS ML CLS ML est. s.d. est. s.d. est. s.d. est. s.d. -.359.362.798.70 -.344.335.49.47 2 -.546.363 -.658.77 -.45.306 -.499.33 3.22.24.74.048.064.00.09.035 4 -.89.264 -.330.0 -.009.26 -.08.083.79.90.24.024 7.0 5.059.34.209 2.306.999.34.209 2 -.730 4.297 -.282.29 -.350.939 -.282.29 3.84.996.386.059.56.468.386.059 4 -.546.57 -.44.42 -.202.832 -.44.42 Table 3.2 - Sample: 5400 individuals Model PH PHL Est. Meth. CLS ML CLS ML est. s.d. est. s.d. est. s.d. est. s.d. -.443.02.790.044 -.407.08.37.038 2 -.635.7 -.645.046 -.4.074 -.487.034 3.263.032.72.02.056.022.089.009 4 -.84.074 -.323.026 -.904.046 -.0.02.235.04.25.006 6.84.544.23.054.464.257.23.054 2 -.070.529 -.260.056 -.094.55 -.260.056 3.803.43.38.05.299.05.38.05 4 -.444.247 -.30.036.06.30 -.30.036 8

Estimation Results - DGP4 Table 4. - Sample: 360 individuals Model PH PHL Est. Meth. CLS ML CLS ML est. s.d. est. s.d. est. s.d. est. s.d. -.45.280.266.25 -.64.259.079.42 2 -.424.28 -.534.39 -.463.246 -.487.3 3.059.090.02.036.047.080.078.035 4 -.07.72 -.70.080 -.04.59 -.07.08.20.436.095.030 2.536.698.309.23.930.23.309.23 2 -.506.659 -.347.240 -.25.549 -.347.240 3.504.232.406.065.35.26.406.065 4 -.45.47 -.228.48.09.478 -.228.48 Table 4.2 - Sample: 5400 individuals Model PH PHL Est. Meth. CLS ML CLS ML est. s.d. est. s.d. est. s.d. est. s.d. -.52.075.255.03 -.262.06.062.036 2 -.434.079 -.528.036 -.42.057 -.480.033 3.079.023.02.009.04.07.076.009 4 -.963.045 -.60.020 -.958.037 -.096.020.25.03.097.008 2.598.67.300.060.557.66.300.060 2 -.424.7 -.322.06.006.08 -.322.06 3.426.054.404.07.270.035.404.07 4 -.465.02 -.26.039.246.09 -.26.039 9

Appendix This appendix presents algebraic derivations of expressions for relevant moments of the marginal, joint and conditional distributions involved in the sequence fy t ; t = ; :::; T g. Also included is a brief statement of asymptotic properties of the CLS estimator, with n!. Section 2 E (y t ) = p + r + ::: + r t. Proof. Under the de nition of the model in () and subsequent assumptions, E (y ) = E (d ) = p. That is, the proposed formula is valid for t =. Supposing that E (y t ) = p + r + ::: + r t, it follows that E (y t+ ) = E (E (y t+ jy t )) = E (p + ry t ) = p + r p + r + ::: + r t = p + r + ::: + r t ; which con rms the generality of the proposed result, for any positive integer t. Section 2 COV (y t ; y t k ) = r k V AR (y t k ). Proof. i. The formula is derived on the basis of the following preliminary result for the conditional rst moment of y t, given y t k : E (y t jy t k ) = p + r + ::: + r k + r k y t k : (9) This can be shown by mathematical induction as well. It is rst noted that, given the Markov nature of the sequence fy t g, one can write E (y t jy t k ) = E (E (y t jy t k ) jy t k ). (0) Now, for k =, (9) is veri ed, because E (y t jy t ) = p + ry t. Then, (9) and (0) lead to the required result E (y t jy t k ) = E (E (y t jy t k ) jy t k ) = p + r + ::: + r k + r k E (y t k jy t k ) = p + r + ::: + r k + r k (p + ry t k ) = p + r + ::: + r k + r k+ y t (k+) : 20

ii. From COV (y t ; y t k ) = E (y t (y t k E (y t k ))) = E (E (y t jy t k ) (y t k E (y t k ))) ; and from i., it follows that COV (y t ; y t k ) = E p + r + ::: + r k + r k y t k (yt k E (y t k )) = r k E (y t k (y t k E (y t k ))) = r k V AR (y t k ) : Section 3. Consistency and p n-asymptotic normality of the CLS estimator of (p; p ) 0 (bounded T i, 8i). Proof. Asymptotic properties are stated with respect to (p; r) 0 for the case n!. From these the desired properties of the CLS estimator of (p; p ) 0 follow immediately, with the asymptotic covariance matrix of the (p; p ) 0 estimator obtained through the delta method. Let (p; r) 0 and ^ denote its CLS estimator. Then, one can write ^ = + n nx XT i z it zit 0 i= t=! n! nx XT i z it u it ; i= t= with z it (; y i;t ) 0, y i0 0 and u it y it p p y i;t. Suppose, for simplicity, that p < (the case p = does not a ect asymptotic properties for xed T i, P Ti i= t= z itzit 0 is 8i). With xed T i, P T i t= z itzit 0 is necessarily bounded, so P n O p (n). Also, p lim n! n P n i= P Ti t= z itu it = 0, because of the following readily established results: E (u it ) = 0; E (u it y i;t ) = 0; and 2 P n P 3 Ti V AR lim 4 n i= t= u it n! P n P 5 Ti V AR n i= t= y i;t u it 2 P 3 nx Ti = lim 4 t= V AR (u it) n!n 2 P 5 ; Ti i= t= V AR (y i;t u it ) PTi because cross-product terms are null in V AR t= z itu it, given the fact that V AR XT i! XT i u it = V AR (u it ) t= t= 2

and V AR XT i t=! XT i y i;t u it = V AR (y i;t u it ) ; as y it and y i;t k are independent, conditionally on y i;t. Then, the last expression is equal to lim n!n 2 i= t= 2 3 nx 4 T i (p ( p) + pp ) p Ti = ( p ) 5 = 0; p ( p) E yi;t 2 + p ( p ) E yi;t 3 because E y 3 it is necessarily bounded for xed Ti. ( ) Therefore, the CLS estimator of (p; r) 0 is consistent. Asymptotic normality of p n (^ ) follows immediately from the fact that both elements of P Ti t= z itu it ( P T i t= u it and P T i t= y i;t u it ) have zero mean and bounded variance for bounded T i, enabling application of a central limit theorem to the sequence n PTi t= z itu it o. In matrix notation one can write p n (^ ) = n Z0 Z pn Z 0 u; where Z ( P T i2) 2 6 4 Z. Z n 3 7 5 ; Z i (T i2) 2 6 4 0 y i y i;ti 3 ; 7 5 and u denotes the column ( P T i ) vector of residuals, y it (p + ry i;t ), t = ; :::; T i, i = ; :::; n. Thus, p n (^ ) _N (0; V ) ; Speci cally, and E yit 2 p t = p p 2 p2t p p 2 p + p 2 t 2 ; p E yit 3 jy i;t = p 3 yi;t 3 + 3p2 p 3p3 + 3p2 y i;t + 6p p 3p 2 p 3p2 + 2p3 + p yi;t + p; as can be obtained, for instance, from the conditional moment generating function of y t, given y t, E (exp (sy t) jy t ) = ( p + pe s ) ( p + p e s ) y t ; (the expression for the unconditional moment, E yit 3 is quite cumbersome to obtain). 22

where V p lim n! n Z0 Z p lim n! n Z0 uu 0 Z p lim n! n Z0 Z : (The formal expression for this variance involves the unconditional variances of y it, t = ; :::; T i ). Section 3. CWLS estimation, expression (5): V AR (y it jy i;t ) = y 2 i;t p ( p) ( p ) 2 + y i;t ( p) ( p ) (2p + p ) + p ( p) : Proof. The model assumptions lead to E y 2 t jy t = pe y 2 t jy t ; d t = + ( p) E y 2 t jy t ; d t = 0 = p (y t + ) 2 + ( p) V (p y t ) + E 2 (p y t ) = p (y t + ) 2 + ( p) y t p ( p ) + y 2 t p 2 : Then, expression (5) follows from the equality V AR (y t jy t ) = E y 2 t jy t E 2 (y t jy t ) ; with E (y t jy t ) = p + (p + p ( p)) y t. 23

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