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GENERALIZED INTEGER-VALUED AUTOREGRESSION a
Kurt Brännäs & Jörgen Hellström
a
a
Department of Economics, Umeå University, Umeå, SE-90187, Sweden Version of record first published: 06 Feb 2007.
To cite this article: Kurt Brännäs & Jörgen Hellström (2001): GENERALIZED INTEGER-VALUED AUTOREGRESSION, Econometric Reviews, 20:4, 425-443 To link to this article: http://dx.doi.org/10.1081/ETC-100106998
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ECONOMETRIC REVIEWS, 20(4), 425–443 (2001)
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GENERALIZED INTEGER-VALUED AUTOREGRESSION Kurt Bra¨nna¨s* and Jo¨rgen Hellstro¨m{ Department of Economics, Umea˚ University, SE-90187 Umea˚, Sweden
ABSTRACT The integer-valued AR(1) model is generalized to encompass some of the more likely features of economic time series of count data. The generalizations come at the price of loosing exact distributional properties. For most specifications the first and second order both conditional and unconditional moments can be obtained. Hence estimation, testing and forecasting are feasible and can be based on least squares or GMM techniques. An illustration based on the number of plants within an industrial sector is considered. Key Words: Characterization; Dependence; Time series model; Estimation; Forecasting; Entry and exit JEL Classification: C12, C13, C22, C25, C51.
1. INTRODUCTION Applied micro-economic interest in count data models has been steadily increasing in recent years, and introductory treatises can now be found in econometric textbooks (Greene, 1997) or in specialized monographs (e.g., Cameron and Trivedi, 1998; Winkelmann, 1997). While many applied studies are based on cross-sectional data some studies are based on panel data. Time series characteristics are then introduced by correlated unobserved heterogeneity (the Zeger (1988) approach) and more seldomly by an explicit lag structure in the endogenous count variable. In this paper we focus on a model with an explicit lag structure which should be of interest also for economic time series at *E-mail:
[email protected] { E-mail:
[email protected] 425 Copyright # 2001 by Marcel Dekker, Inc.
www.dekker.com
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semi-aggregate levels. The considered model class is the integer-valued autoregression (INAR). The INAR model is one useful model for non-negative sequences of dependent count variables. The first order INAR [INAR(1)] model is particularly attractive partly thanks to its interpretational appeal. It explains the present number of, say, individuals in some situation as the sum of those that remain (or survive) from the previous period, and those that enter (or are born) in the intervening period. The INAR(1) model was introduced by McKenzie (1985) and has been elaborated on in subsequent papers by McKenzie (e.g., 1988), Al-Osh and Alzaid (e.g., 1987) and others. Al-Osh and Alzaid (1987) considered Yule-Walker, conditional least squares and maximum likelihood estimation in the Poisson case. Bra¨nna¨s (1994, 1995) considered estimation by generalized method of moments for Poisson and generalized Poisson models and the inclusion of explanatory variables. Empirical economic applications of the model are still few, though it has been used in studies of, e.g., the entry and exit of plants (Berglund and Bra¨nna¨s, 1996). The related integer-valued moving average model was recently studied and employed for a financial application by Bra¨nna¨s and Hall (1998). In this paper we relax some of the independence assumptions underlying the basic INAR(1) model to make the model more readily available for economic applications. In the treatments of the original model distributional properties have been stressed, while we look for more flexibility by relaxing assumptions and by only considering the first and second order moments of the model. The main focus in the paper is on the INAR(1) model, but extensions to general INAR( p) as well as multivariate INAR(1) are also briefly considered. Beyond model properties we also focus on aspects of estimation, testing and forecasting. In Section 2 the basic model is introduced. The implications of relaxing some of the basic assumptions are presented in Section 3. To give some intuition to the modelling the discussion is in terms of the entry and exit decisions of firms. Section 4 covers estimation and testing, while forecasting is dealt with in Section 5. For two of the extended model specifications we provide Monte Carlo evidence of estimator and test statistic performance for time series of finite length in Section 6. An illustration based on the number of Swedish mechanical paper and pulp mills is presented in Section 7. Some concluding remarks close the paper.
2. BASIC MODEL The paper is concerned with extensions to the integer-valued autoregressive model of order one [INAR(1)] given by yt ¼ a ytÿ1 þ et ;
ð1Þ
where yt is a non-negative integer-valued random variable and t is the time index. The scalar multiplication of the Gaussian AR model is in the integer-valued case
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P replaced by the binomial thinning operator, defined as a y ¼ yi¼1 ui ; where fui g is a sequence of independent and identically distributed 0–1 random variables (Steutel and van Harn, 1979). The fui g sequence is independent of ytÿ1 and et ; and Prðui ¼ 1Þ ¼ 1 ÿ Prðui ¼ 0Þ ¼ a; a 2 ½0; 1: Further, ytÿ1 is assumed independent of et : The fet g sequence of non-negative, integer-valued random variables has mean l, finite variance d, and Covðet ; es Þ ¼ 0; for all t 6¼ s: This et is usually assumed to possess some specified distribution. To be able to generalize the basic model we, however, abstain from making a full distributional assumption. Instead, we only offer results for the first and second order moments, since useful results in terms, e.g., of the probability generating function are difficult to obtain for the generalizations that we consider. Under the assumptions of the basic model the thinning operator has the properties Eða yj yÞ ¼ ay; Eða yÞ ¼ aEð yÞ; V ða yj yÞ ¼ að1 ÿ aÞy; and 2 V ða yÞ ¼ a V ð yÞ þ að1 ÿ aÞEð yÞ: The first and second order conditional and unconditional moments for the base case INAR(1) model are then Eð yt jytÿ1 Þ ¼ aytÿ1 þ l Eð yt Þ ¼ l=ð1 ÿ aÞ V ð yt jytÿ1 Þ ¼ að1 ÿ aÞytÿ1 þ d V ð yt Þ ¼ ½að1 ÿ aÞEð ytÿ1 Þ þ d=ð1 ÿ a2 Þ: We note that the model embodies a conditional heteroskedasticity effect (cf. Engle, 1982). Since this does not exactly match a conventional ARCH model effect we use the label INARCH for the property. The INARCH effect is larger the larger is ytÿ1 : The autocovariance function at lag k, gk ¼ ak V ð ytÿk Þ; and the autocorrelation function is rk ¼ ak : Obviously, both functions are positive. As an example of an integer-valued process, the number of firms in a region at a certain time ( yt) is the number of firms surviving from the previous period ða ytÿ1 Þ plus the entering new firms ðet Þ: Since one would expect the survival of an individual firm to depend on the survival of other firms we incorporate this feature into the model. There is also reason to believe that survival may depend on the number of existing firms, and that the entry process may be correlated with the survival mechanism. These and other model extensions are considered in the next section.
3. GENERALIZED MODELS The generalization of the model proceeds by considering the first and second order moments (including the autocorrelation function) for different and weaker
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assumptions about the basic model. We retain the original basic model structure in (1) but relax assumptions according to Eðui uj Þ 6¼ Eðui ÞEðuj Þ; for i 6¼ j; Eðui et Þ 6¼ Eðui ÞEðet Þ; and Eð ytÿ1 ui Þ 6¼ Eð ytÿ1 ÞEðui Þ: Further, dependence within the et process, INAR( p), Threshold INAR(1), time dependent entry and exit as well as multivariate models are also considered. We prefer in most cases to consider one extension at a time so that effects are more transparent. The number of firms in a specific region serves as a working example throughout the paper. As we wish to focus on model properties we abstain from attempts to discuss in detail the role of economic determinants to various bits of the models. Economic variables could be included by letting parameters be functions of economic variables. Other illustrative discussions based on any ‘birth-death’ type of phenomenon could equally well be used. 3.1. Dependence Between Exit Decisions It is reasonable to question the assumption that the individual firms survive or exit independently. They all operate in the same macroeconomic milieu and would, it appears, be affected in much the same way. To account for this, we modify the model by letting Eðui uj Þ ¼ ys 6¼ Eðui ÞEðuj Þ ¼ a2 ; for i 6¼ j: At this stage we maintain that fui g is independent of the past stock ytÿ1 and of the number of entrants et : The correlation between survival indicators ui and uj is ks ¼ Corrðui ; uj Þ ¼ ðys ÿ a2 Þ=að1 ÿ aÞ; i 6¼ j: For the basic model of the previous section ys ¼ a2 so that ks ¼ 0: When ys < a2 there is a negative correlation between exit decisions and when ys > a2 there is positive correlation. It is straightforward to show that both the conditional and unconditional first order moments remain unchanged from the basic model. Note that this holds under even less restrictive dependence specifications. The dependence has an effect only on the second or higher order moments. We obtain the conditional and unconditional variances as V ð yt jytÿ1 Þ ¼ ða ÿ ys Þytÿ1 þ ðys ÿ a2 Þy2tÿ1 þ d V ð yt Þ ¼ ½ða ÿ ys ÞEð ytÿ1 Þ þ ðys ÿ a2 ÞEð y2tÿ1 Þ þ d=ð1 ÿ a2 Þ: Note that the INARCH effect is in this case larger than in the basic model when ys > a2 (positively correlated ui and uj) for ytÿ1 > 1; while for ytÿ1 ¼ 0 or ytÿ1 ¼ 1 there is no INARCH effect and the conditional variance is then constant as it is in the base case model. The lag one autocorrelation coefficient can be shown to remain unchanged from the basic model, i.e. r1 ¼ a:
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3.2. Dependence Among Entrants
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To allow for dependence between entry decisions, we may write the model zt ; where b is the probability of entry among zt on the form yt ¼ a ytÿ1 þ b P zt potential entrants. Let b zt ¼ i¼1 vi ; where vi represents the independent 0–1 decision to start a firm ðvi ¼ 1Þ or not ðvi ¼ 0Þ: The zt is assumed to be independent of the fvi g sequence, and zt and fvi g are assumed to be independent of a ytÿ1 : Let Eðvi vj Þ ¼ yp ; for i 6¼ j; then it follows that the first order moments are unchanged, but with l now corresponding to bEðzt Þ: The correlation between entry decisions is kp ¼ ðyp ÿ b2 Þ=ðbð1 ÿ bÞÞ: For the second order moments we get V ð yt jytÿ1 Þ ¼ að1 ÿ aÞytÿ1 þ ðb ÿ yp ÞEðzt Þ þ yp Eðz2t Þ ÿ b2 E 2 ðzt Þ V ð yt Þ ¼ ½að1 ÿ aÞEð ytÿ1 Þ þ ðb ÿ yp ÞEðzt Þ þ yp Eðz2t Þ ÿ b2 E2 ðzt Þ=ð1 ÿ a2 Þ: The lag one autocorrelation is r1 ¼ a: 3.3. Dependence Between Entry and Exit Mechanisms It may be reasonable, e.g., for management and technology reasons, to expect some dependence between the entry and exit mechanisms. We consider the case where exit and entry are correlated in the following sense, Eðui et Þ ¼ ye 6¼ Eðui ÞEðet Þ; for any i.1 We obtain ke ¼ Corrðui ; et Þ ¼ ðye ÿ alÞ=½ða ÿ a2 Þd1=2 : Since the model in (1) is additive we immediately see that the first order moments remain unchanged. The situation is more involved when it comes to the second order moments, since we now have a covariance term in the variance expression: V ð yt Þ ¼ V ða ytÿ1 Þ þ V ðet Þ þ 2Covða ytÿ1 ; et Þ: We have that Covða ytÿ1 ; et Þ ¼ ðye ÿ alÞEð ytÿ1 Þ: Note that under independence ye ¼ al; so that the covariance term is equal to zero. The conditional and unconditional variances are given by V ð yt jytÿ1 Þ ¼ ½að1 ÿ aÞ þ 2ðye ÿ alÞ ytÿ1 þ d V ð yt Þ ¼ f½að1 ÿ aÞ þ 2ðye ÿ alÞEð ytÿ1 Þ þ dg=ð1 ÿ a2 Þ 1 We could Pzt also consider a more basic dependence with the number of entrants. Let, as in Section 3.2, et ¼ i¼1 vi ; where zt may represent the number of potential entrepreneurs and vi represents the independent 0–1 decision to start a firm ðvi ¼ 1Þ or not ðvi ¼ 0Þ: Let Eðvi Þ ¼ b and Eðui vi Þ ¼ c: From this follows that Eðui et Þ ¼ cEðzt Þ ¼ ye :
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and the lag one autocorrelation coefficient is again r1 ¼ a: The variance expressions exceed those of the basic model only when ye > al:
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3.4. Stock Dependent Survival Mechanism In a market one may expect dependence between the exit mechanism and the number of operating firms. This would imply that Eðui jytÿ1 Þ ¼ Prðui ¼ 1jytÿ1 Þ ¼ ay;t is a function of ytÿ1 and hence varying over time. This dependence makes it difficult to obtain general and explicit expressions for the unconditional moments except for under some very P simplified and then probably artificial specifications. Since Eðay;t yÞ ¼ Ey ½ yi¼1 Eðui jyÞ ¼ Ey ðay;t yÞ we get the moments Eð yt jytÿ1 Þ ¼ ay;t ytÿ1 þ l Eð yt Þ ¼ Eðay;t ytÿ1 Þ þ l V ð yt jytÿ1 Þ ¼ ay;t ð1 ÿ ay;t Þytÿ1 þ d V ð yt Þ ¼ V ðay;t ytÿ1 Þ þ E½ay;t ð1 ÿ ay;t Þytÿ1 þ d: Not surprisingly there is no explicit expression for r1. The ay;t probability may, for instance, be modelled by a logistic distribution function. In a related way we could also introduce stock dependent entry behavior, e.g., through the conditional representation Eð yt j ytÿ1 Þ ¼ ay;t ytÿ1 þ ly;t ; where ly;t is a function of ytÿ1 and then time-varying. 3.5. INAR( p) While the INAR(1) models have features that makes for easy interpretations higher order autoregressions are not equally easy to interpret. They may be viewed as duals to INMA(q) models for which Al-Osh and Alzaid (1988) and Bra¨nna¨s and Hall (1998) have offered some interpretations. Let the INAR( p) process be defined as yt ¼ a1 ytÿ1 þ þ ap ytÿp þ et with ai 2 ½0; 1; i ¼ 1; . . . ; p ÿ 1; and ap 2 ð0; 1: To simplify we present results for the p ¼ 2 case, but generalize previous model suggestions to account for dependence between exit decisions and dependence between entry and exit decisions. We get Eð yt jy1 ; . . . ; ytÿ1 Þ ¼ a1 ytÿ1 þ a2 ytÿ2 þ l Eð yt Þ ¼ l=ð1 ÿ a1 ÿ a2 Þ V ð yt jy1 ; . . . ; ytÿ1 Þ ¼ ½a1 ÿ ys þ 2ðye1 ÿ a1 lÞ ytÿ1 þ ðys ÿ a21 Þy2tÿ1 þ ½a2 ÿ ys þ 2ðye2 ÿ a2 lÞ ytÿ2 þ ðys ÿ a22 Þy2tÿ2 þ 2ðy12 s ÿ a1 a2 Þ ytÿ1 ytÿ2 þ d:
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Here, ykls ¼ Eðui;tÿk uj;tÿl Þ is a measure of survival dependence at times t ÿ k and t ÿ l; with ys for k ¼ l ¼ 0; fej ¼ Eðet uij Þ is the dependence measure between the entry mechanism at time t and the survival decision at time t ÿ j; j ¼ 1; 2: The variance can be obtained from the given moments.
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3.6. A Bivariate Model We consider a bivariate process and note that results are easily generalized to a general multivariate context. We write yit ¼ ai yi;tÿ1 þ ei;t and allow for a general dependence structure. Specifically, we let fkls ¼ Eðuik ujl Þ; i; j ¼ 1; 2; reflect the dependence between survival=exit decisions in equations k and l, fkle ¼ Eðuik elt Þ; for k, l ¼ 1, 2, reflects the dependence between entry and exit mechanisms, and f ¼ Eðe1t e2s Þ; for t ¼ s; and f ¼ 0 otherwise. Given this setup we find no changes from previous results for the first order moments. For the second order moments we get among other results that V ð yit Þ ¼ f½ai ÿ fiis þ 2ðfiie ÿ li ai ÞEð yi;tÿ1 Þ þ ðfiis ÿ a2i ÞEð y2i;tÿ1 Þ þ di g=ð1 ÿ a2i Þ; i ¼ 1; 2 Covð y1t ; y2t Þ ¼ f12 s Eð y1;tÿ1 y2;tÿ1 Þ ÿ a1 a2 Eð y1;tÿ1 ÞEð y2;tÿ1 Þ 21 þ ðf12 e ÿ a1 l2 ÞEð y1;tÿ1 Þ þ ðfe ÿ a2 l1 ÞEð y2;tÿ1 Þ þ ðf ÿ l1 l2 Þ:
Note that these expressions simplify under stronger assumptions. In particular, given the basic independence assumptions on single equations as well as between equations the covariance between y1t and y2t reduces to zero. Other results such as the full cross-covariance function can be obtained, but is not given. Empirical applications of related multivariate models have been reported by Blundell et al. (1999), Berglund and Bra¨nna¨s (1999) and Bra¨nna¨s and Bra¨nna¨s (1998). 3.7. Time Dependent Entry and Exit Following Bra¨nna¨s (1995), we may introduce explanatory variables through the parameters of the model, keeping in mind that the restrictions at 2 ½0; 1 and lt 0 should be respected. Two convenient and in other situations widely adopted specifications are the logistic distribution function, i.e. at ¼ 1=½1 þ expðxt bÞ, and the exponential function, i.e. lt ¼ expðzt pÞ: The explanatory variable vectors xt and zt are treated as fixed and measured at the beginning of the period starting at time t ÿ 1: The b and p are the corresponding vectors of parameters. Note the relationship between this specification and the stock dependent model of Section 3.4. The full time varying model is written yt ¼ at ytÿ1 þ et :
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We have the following unconditional moment relations Eðyt Þ ¼ at Eðytÿ1 Þ þ lt V ðyt Þ ¼ a2t V ðytÿ1 Þ þ at ð1 ÿ at ÞEðytÿ1 Þ þ dt " # kÿ1 Y gk;t ¼ atÿi V ðytÿk Þ; k ¼ 1; 2; . . . i¼0
rk;t ¼ gk;t =½V ðyt ÞV ðytÿk Þ
1=2
¼
" k ÿ1 Y
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i¼0
# atÿi
V ðytÿk Þ V ðyt Þ
1=2 ; k ¼ 1; 2; . . . ;
where gk;t and rk;t are the autocovariance and the autocorrelation functions at time t and lag k. Hence, both functions are time dependent. With variances approximately equal we expect that rk;t > rkþ1;t ; for any t and k ¼ 1; 2; . . . : The conditional moments are of the form given for the basic model, albeit with time dependent parameters.
3.8. Threshold INAR(1) Models We consider two types of threshold models, in one the switching between regimes is governed by a random and non-observable process, while in the other switching occurs with respect to a threshold level and the past stock ytÿ1 : For the first case, let the previous stock ytÿ1 be split randomly into two parts wtÿ1 and ytÿ1 ÿ wtÿ1 ; with corresponding survival probabilities a1 and a2 : We assume that wtÿ1 cannot be observed and that wtÿ1 given ytÿ1 follows a binomial distribution such that Eðwtÿ1 jytÿ1 Þ ¼ pytÿ1 and V ðwtÿ1 jytÿ1 Þ ¼ pð1 ÿ pÞytÿ1 ¼ pp ytÿ1 : With this we write the model as yt ¼ ða1 wtÿ1 Þ þ ða2 ðytÿ1 ÿ wtÿ1 ÞÞ þ et : Assume that the assumptions about the basic model are otherwise satisfied and that wtÿ1 is independent of et : After some tedious but straightforward algebra we can prove the following results: Eðyt jytÿ1 Þ ¼ ½a1 p þ a2 p ytÿ1 þ l ¼ a~ ytÿ1 þ l Eðyt Þ ¼ l=ð1 ÿ a~ Þ V ðyt jytÿ1 Þ ¼ ½a1 pð1 ÿ a1 pÞ þ a2 p ð1 ÿ a2 p Þytÿ1 þ d V ðyt Þ ¼ ½1 ÿ ða21 p2 þ a22 p 2 Þÿ1 f½a1 pð1 ÿ a1 pÞ þ a2 p ð1 ÿ a2 p ÞEðytÿ1 Þ þ dg r1 ¼ a~ : Bo¨ckenholt (1999) studies a mixture Poisson INAR(1) model in which mixing is related to the et -part of the model. For the second type of threshold INAR(1) model we assume that there are two mean functions governed by the lagged level ytÿ1 of the process:
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ytÿ1 y0 : ytÿ1 > y0
This model reduces to an INAR(1) if a1 ¼ a2 and e1t ¼ e2t and can have the low order moments of INAR(1) if a1 ¼ a2 and the low order moments of the eit are equivalent. The conditional first order moment is a1 ytÿ1 þ l1 ; ytÿ1 y0 : Eðyt jytÿ1 Þ ¼ a2 ytÿ1 þ l2 ; ytÿ1 > y0
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In this case the unconditional and conditional variances are formidable and not very illuminating.
3.9. Remarks As could be anticipated, changes in the dependence structure of the basic INAR model will generally not change the first order moments, but will change higher order moments. There are a number of other generalizations that could have been considered. For instance, we could let the thinning operations be dependent over time in the INAR(1). There will be no effect of this on moments of the type considered here, but there may be effects on other moments. Obviously, we could also consider combinations of the studied extensions and let some of the dependence parameters be functions of time varying economic variables. We find that the INARCH effect as well as the variance properties vary substantially with model type, and for this reason we argue that empirical discrimination between the model types should be possible. Note that as dependence is introduced obtaining distributional properties, e.g., using probability generating functions, will become exceedingly complicated for most specifications and hence too complicated for empirical use.
4. ESTIMATION Maximum likelihood (ML) estimation of a and l in the basic INAR(1) model is more complicated than ML estimation in the Gaussian AR(1). This is due to more complicated distributional forms that complicate numerical calculations. Estimation with conditional least squares (CLS), conditional ML as well as exact ML and the Yule-Walker estimators were studied by Al-Osh and Alzaid (1987) in the Poisson case. In a recent study Park and Oh (1997) show asymptotic normality for the Yule-Walker type estimator for a slightly different parameterization and also show that the Yule-Walker asymptotically is more efficient than the CLS estimator. Generalized method of moments (GMM) estimation was considered by Bra¨nna¨s (1994) for the Poisson and the generalized Poisson model. In this section we will consider estimation of the generalized INAR(1) models of Section 3.
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Method of moments or Yule-Walker based on unconditional moments, CLS (weighted and unweighted) and GMM are the considered estimators.
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4.1. Yule-Walker Estimation It is simple to get estimates of a, l and d in the basic INAR(1) model. The method of moments is related to the Yule-Walker estimator and yields a^ ¼ r1 ; l^ ¼ ð1 ÿ r1 Þy and d^ ¼ ð1 ÿ r12 Þs2 ÿ r1 ð1 ÿ r1 Þy; where r1 is the sample autocorrelation coefficient at lag one, y is the sample mean, and s2 is the sample variance. Obtaining estimates for the generalized models is not straightforward, since we then have, at least, four unknown parameters, but only have ready access to the mean, variance and autocorrelations. Since we have alternative estimators there is no strong reason for pursuing a search for additional unconditional moments of higher orders to make this approach feasible.
4.2. Conditional Least Squares Estimators Weighted or unweighted conditional least squares (WCLS or CLS) estimators are simple to use and have been found to perform well for univariate models and short time series (Bra¨nna¨s, 1995). The conditional mean or the one-step-ahead prediction error can be used to obtain the estimates. The conditional mean is for most of the specifications considered in Section 3: Eðyt jytÿ1 Þ ¼ aytÿ1 þ l; where a and l are the unknown parameters to be estimated. The CLS estimators of a and l minimize the criterion function Q¼
T X
½yt ÿ aytÿ1 ÿ l2 :
t¼2
For both the basic and generalized models V ðyt jytÿ1 Þ vary with both a and l as well as other parameters. Depending on which model type is considered we may apply OLS to estimate any remaining parameters (y and d) from the empirical conditional variance expression e^2t ¼ gðy; d; a^ ; l^ ; ytÿ1 Þ þ xt ; where e^t is the residual from the CLS estimation phase and xt is a disturbance term. The WCLS estimator of a and l minimize a criterion function in which the conditional variance is taken as given (i.e. evaluated at estimates)
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QW ¼
T X ðyt ÿ aytÿ1 ÿ lÞ2
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gðy^ ; d^ ; a^ ; l^ ; ytÿ1 Þ
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:
In this case the estimators of a and l have the shape of the CLS estimator expressions, but where each sum contains a term 1=gð:; ytÿ1 Þ and T ÿ 1 is replaced P by Tt¼2 1=gð:; ytÿ1 Þ: For both the CLS and WCLS estimators, estimated covariance matrices for parameters based on the Gauss-Newton algorithm was studied by Bra¨nna¨s (1995). He also considered Eicker-White type covariance matrices for the CLS estimator and found that the WCLS estimator has better bias and mean square error (MSE) properties and that its associated test statistics had the best power properties. For the stock dependence specification as well as when explanatory variables are present estimation by the Gauss-Newton algorithm with or without weighting is straightforward (cf. Bra¨nna¨s, 1995).
4.3. Generalized Method of Moments In this subsection we consider GMM (Hansen, 1982) estimation. Two approaches to GMM estimation can be considered. One approach employs unconditional moment restrictions and the other, considered here, is based on conditional moment restrictions (e.g., Newey, 1985; Tauchen, 1986). Note that estimation based on unconditional moment restrictions is related to the Yule-Walker estimator. The conditional GMM estimator can be seen as an extension of the CLS estimator and minimizes the quadratic form ^ ÿ1 mðcÞ; q ¼ mðcÞ0 W where mðcÞ is a vector of moment restrictions and c is the vector of unknown parameters. The estimator is consistent and asymptotically normal subject to mild regularity conditions (e.g., Davidson and MacKinnon, 1993, ch.17) for any ^ : The estimator is efficient symmetric and positive definite weight matrix W ^ ^ ; q can in a when W is the asymptotic covariance matrix of mðcÞ: To obtain W ^ : In a first stage be minimized using, for instance, the identity matrix I for W ^ ^ second stage the consistent estimates c from stage one can be used to form W based on the consistent Newey and West (1987) estimator. PT ÿ1 To estimate a and l the empirical moment restrictions T t¼2 et ¼ 0; PT ÿ1 for l, and T t¼2 ytÿ1 et ¼ 0; for a, with corresponding theoretical moments Eðet Þ ¼ 0 and Eðytÿ1 et Þ ¼ 0 can be used. Here, et ¼ yt ÿ aytÿ1 ÿ l is the onestep-ahead forecast error. The restrictions match the normal equations of the CLS estimator. For d and y (depending on which extension of the base case is considered) the GMM estimators may, e.g., be formed by letting the moment restrictions be the difference between sample and theoretical moments
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P P ðT ÿ 1Þÿ1 Tt¼2 e2t ÿ V ðyt jytÿ1 Þ ¼ 0 and ðT ÿ 1Þÿ1 Tt¼2 ½e2t ÿ V ðyt jytÿ1 Þ ytÿ1 ¼ 0: When the numbers of unknown parameters and moment restrictions are equal the estimated asymptotic covariance matrix of the GMM estimator is ^ 0W ^ ÿ1 ; ^ Þ ¼ T ÿ1 ½G ^ ÿ1 G Covðc ^: ^ matrix with rows G j ¼ @mj =@c0 and W ^ are both evaluated at c where the G
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4.4. Remarks on Specification Testing For some of the discussed INAR specifications the changes arise in the first order moment, while for others tests need to focus on second order moments. The former problem is of a more conventional type and standard procedures can be expected to perform relatively well. To test if we have dependence between the survival probability and the existing stock as well as on a vector of explanatory variables we may specify a functional form for ay,t, e.g., of the logistic distribution function type: ay;t ¼ 1=ð1 þ expðxt b þ yytÿ1 ÞÞ: A Wald test for y ¼ 0 is therefore an immediate test of stock dependence, while a test for b ¼ 0 tests for the presence of explanatory variables. A joint test of y ¼ 0 and b ¼ 0 is a test for constant survival probability. The Wald tests can be based on CLS, WCLS or GMM estimators. For the other specifications, testing must be based on the second order moments of the form V ðyt jytÿ1 Þ ¼ gðy; d; a; l; ytÿ1 Þ: The GMM estimator provides a unified framework for doing this and testing can be related to LR, LM or Wald testing ideas. Testing the hypothesis of no dependence in the exit mechanism corresponds to testing RðcÞ ¼ ys ÿ a2 ¼ 0: The simple Wald test in ^ Þ0 ½hðc ^ Þ0 W ^ Þ ¼ @RðcÞ=@c0 ; ^ ÿ1 hðc^ Þÿ1 Rðc^ Þ with hðc the GMM framework, W ¼ Rðc 2 is distributed w ð1Þ: Note that when CLS or WCLS techniques are used, it will not be possible to account for the covariances between estimators of parameters contained in the first and second order moments, so that the resulting test statistics are at most approximative.
5. FORECASTING We consider the forecasting of future values yT þh of the INAR(1) process given past observations up through time T. Since the first order moments of the basic and extended models in most cases are the same this leaves their forecasts unaltered. By repeated substitution we can write the future values of the process as
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Then the h-step-ahead forecast is obtained as
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y^ TþhjT ¼ EðyT þh jy1 ; . . . ; yT Þ ¼ ah yT þ lð1 þ a þ . . . þ ahÿ1 Þ l l h ¼ a yT ÿ þ ; 1ÿa 1ÿa where the equality ð1 þ a þ . . . þ ahÿ1 Þ ¼ ð1 ÿ ah Þ=ð1 ÿ aÞ has been used. The term in brackets measures the deviation of the process from the mean of the process. As h goes to infinity and with a < 1, the first part of the expression goes to zero and hence the forecast approaches the mean of the process. As a ! 1 the forecast approaches yT , which is to be expected on comparison with a random walk model. Bra¨nna¨s (1995) gives corresponding results for the time-varying parameter model. The forecast error is eT þh ¼ yT þh ÿ y^ TþhjT ; so that the forecast is unbiased. The one-step-ahead forecast error variance is in the basic model V ðeT þ1 Þ ¼ að1 ÿ aÞEðyTþ1 Þ þ d ¼ al þ dð1 ÿ a2 ÞV ðyT Þ: It can be shown that the forecast error variances are affected by the generalizations of Section 3 only through changes in the variance term V(yT). The h-step-ahead forecast error variance is in the basic model V ðeT þh Þ ¼ að1 ÿ aÞEðyT Þ þ a2 ð1 ÿ ah ÞV ðyT Þ þ d ¼ ð1 ÿ a2h ÞV ðyT Þ; where the error variance increases with the forecast horizon, h, for 0 < a < 1: For the extended models we can show that the forecast error variance h steps ahead will again change only due to the changes in the variance term V(yT). 6. FINITE SAMPLE PROPERTIES To give an indication of the small sample performance of estimators and tests we conduct two Monte Carlo experiments. One is for the case of correlated survival=exit decisions (cf. Section 3.1) and the other for the stock dependent case (cf. Sections 3.4 and 4.4). The factors to be varied in the first experiment are a ¼ 0:5; 0:7; 0:9; l ¼ 5; 10; and to obtain a positive correlation ks, we set ys ¼ a2 þ ði ÿ 1Þ 0:02; i ¼ 1; . . . ; 5: Positively correlated binary data are generated by specializing a result of Lunn and Davies (1998).2 The time series length is set at 2 To obtain a constant correlation between binary random variables we modify the algorithm of Lunn and Davies (1998, Section 2.1), such that Ui ¼ ð1 ÿ Wi ÞVi þ Wi Z; i ¼ 1; . . . ; ytÿ1 : In Ui all variables are independently Bernoulli distributed with probabilities a for Vi and Z, and jk1=2 s j for Wi. This gives a constant correlation ks 0 between any pair Ui, Uj.
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T ¼ 50; 100; 200; and the distribution for et is throughout Poisson. For the second experiment we specify ay;t ¼ 1=½1 þ expðÿ2:2 þ yytÿ1 Þ with y ¼ 0ð0:02Þ0:14; so that for y ¼ 0 ay;t ¼ 0:9 and smaller for larger y. The other parts of the model are as in the basic model, i.e. with ys ¼ a2 so that ks ¼ 0: The other values are set as in the first experiment. In each cell 1000 replications are generated, and to avoid start-up transients a first set of 150 observations is dropped in each replication. For the first experiment the CLS estimator is based on the explicit expressions of Section 4.2 and a LS estimator is used for the conditional variance expression with a and l set at CLS estimates. The obtained estimates are used to initialize a GMM estimator with W ¼ I. In the second experiment a Gauss-Newton algorithm is used for the nonlinear CLS estimator.
6.1. Results Starting with the dependence-between-exits case the bias results for the CLS estimators of a and ys are illustrated in Figure 1. The bias is smaller the larger is T and there is a weak tendency for increasing bias for the ys parameter with larger correlation ks. The biases (and MSEs) are practically the same for the GMM estimator based on the four moment restrictions mentioned in Section 4.3. This similarity is anticipated in view of the Ahn and Schmidt (1995) asymptotic argument. With moment conditions of the present sequential nature no efficiency gain can be expected for the parameters contained in the conditional mean function. For the other parameters, i.e. l and d we find biases that are quite large for large a and ys for both estimators. One explanation is that these instances correspond to much larger variances, V(yt), of the series. For the largest a and ys the variance is 136.3, while for a ¼ 0:5 and ys ¼ a2 the variance of the series is only 10.
Figure 1. Biases of CLS estimators of a (right) and ys (left) against ks for the dependence between exits model with true a ¼ 0:9 and l ¼ 5: Solid line and white symbol ðT ¼ 50Þ; dot-dashed line and grey symbol (T ¼ 100) and dotted line and black symbol (T ¼ 200).
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The power properties of the GMM based Wald test of H0 : ys ¼ a2 ; i.e. of zero-correlation between exit decisions, are illustrated in Figure 2. The test is more powerful for a ¼ 0:5 than for a ¼ 0:9 and moreover the size properties are better in the former case. Again this is likely to be due to the very different variances in the two cases. For the second experiment with stock dependence we give the bias and MSE properties for the CLS estimator of y in Figure 3. Both measures appear smaller for larger T and the bias is smaller for larger y. A larger y corresponds to a smaller ay,t (and smaller V(yt)), so that there appears to be a general bias improvement as a gets smaller. In terms of the power of a Wald test of y ¼ 0, using a White-type of covariance matrix estimator, we find sizes that increase with sample size and are
Figure 2. Power functions for Wald test statistic of the hypothesis ys ¼ a2 plotted against ks for T ¼ 50, 100, 200 at true values l ¼ d ¼ 5; a ¼ 0:5 (left) and a ¼ 0:9 (right). Lines and symbols are defined in Figure 1.
Figure 3. Biases (left) and MSEs (right) for the CLS estimator of y in the stock dependence case with l ¼ 5. Lines and symbols are defined in Figure 1.
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significantly too large for T ¼ 200. Numerically, the Gauss-Newton algorithm diverged in a large number of replications for small y, i.e. when ay,t is large.
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7. ILLUSTRATION Consider as an illustration (cf. Bra¨nna¨s, 1995) the number of Swedish mechanical paper and pulp mills 1921–1981, Figure 4. This industrial production technology is obviously on its way out and new production capacity is created in plants of a more recent technology and larger scale. From this follows that 1 ÿ a may reflect exits that are entries in other production technologies. Table 1 gives parameter estimates for a simple model with industrial gross profit margin and GNP used as explanatory variables. The fit of the model is exhibited in Figure 4 and is quite good. Note that the fit for a model without explanatory variables is almost equally good (R2 ¼ 0.95 instead of 0.96). Testing for stock dependence
Figure 4. The number of Swedish mechanical paper and pulp mills (solid line) and fitted values (dashed line), and estimated conditional variances (right) for base case model (dotted line), models for exit dependence (dash-dotted line) and dependence between entry and exit (solid line).
Table 1.
Estimation Results (CLS With S.E. in Parentheses) and Variable Definitions
Variable Gross Profit Margin (1950–72 ¼ 100) GNP (1900 ¼ 100) Constant
Survival Probability
Mean Entry
ÿ 0.055 (0.009) –
ÿ 0.038 (0.006) ÿ 0.001 (0.000) 5.051 (0.607)
3.605 (0.673)
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indicates a nonsignificant effect. For these two reasons we only present additional results for models without explanatory variables. We present CLS and GMM estimates for all parameters in the first and second order moment specifications for dependent exits and dependent entry and exit in Table 2. The former specification suggests that d is much larger than l, and that ys > a2 : It follows that there is a positive correlation of ks ¼ 0:09 between the survival=exit decisions. For the specification with dependence between entry and exit decision d is also larger than l. Also, since ye > al there is a positive correlation of ke ¼ 0:54 between the entry and exit decisions. The hypothesis of no correlation was tested using a Wald test based on GMM estimates. The correlation between exits was not significant, while the entry=exit correlation was significant at the 0.05 level. The parameters a and ys are throughout precisely estimated, while l and in particular d are imprecisely estimated. Figure 4 also reports graphs for conditional variances for the basic as well as for the two models with dependence. The latter two have rather similar paths, while the former is quite flat. The differences arise from the weights given to ytÿ1 in the conditional variances. Note also the closeness (except for the level) between the right and left panels of Figure 4.
8. CONCLUSIONS The paper has demonstrated that several empirically motivated extensions to the basic integer-valued autoregressive model can be made while maintaining that models be easy to interpret as well as be estimable. Full characterizations of the models could generally be obtained in terms of the first and second order conditional and unconditional moments. Full distributional properties could not be obtained on explicit forms, so that empirically maximum likelihood estimation is not feasible, and instead least Table 2. Estimation Results by CLS and GMM for Different Dependence Structures (S.E. in Parentheses) Specification CLS-estimates Basic model Dependent exits (Section 3.1) Dependent entry=exit (Section 3.3) GMM-estimates Dependent exits (Section 3.1) Dependent entry=exit (Section 3.3)
a
l
d
ys
0.958 (0.028) 0.958 (0.028) 0.958 (0.028)
0.233 (0.698) 0.233 (0.698) 0.233 (0.698)
6.668 (3.832) 4.640 (3.167) 1.340 (2.823)
0.921 (0.003)
0.958 (0.001) 0.958 (0.001)
0.233 (0.212) 0.233 (0.212)
3.740 (5.527) 1.340 (7.551)
ye
ks
ke
0.09 0.348 (0.197)
0.922 (0.002)
0.54 0.12
0.348 (0.183)
0.54
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squares and generalized method of moments (GMM) estimators are employed. These estimators and the corresponding tests performed reasonably well in a small scale Monte Carlo experiment. No doubt there is room for further improvements, at least, in terms of GMM estimation, since additional as well as alternative moment restrictions may be preferable to the ones evaluated here. An interesting aspect of the model class is its conditional heteroskedasticity property, which we label INARCH. It is through the conditional variance that the various model extensions come through most clearly. In the reported Monte Carlo results as well as in the empirical illustration the dependence parameters were quite precisely estimated and the corresponding Wald test statistic based on GMM had reasonable properties. ACKNOWLEDGMENTS The financial support from The Swedish Research Council for the Humanities and Social Sciences is acknowledged. Thomas Aronsson, Colin Cameron, Xavier de Luna and two anonymous referees are thanked for their comments on previous versions of the paper. A previous version of the paper has been presented at the Umea˚ and Uppsala universities and at the 1998 EC2 Conference. REFERENCES Ahn, S.C.; Schmidt, P.A. Separability Result for GMM Estimation, with Applications to GLS Prediction and Conditional Moments Tests. Econometric Reviews 1995, 14, 19–34. Al-Osh, M.A.; Alzaid, A.A. First-Order Integer Valued Autoregressive (INAR(1)) Process. Journal of Time Series Analysis 1987, 8, 261–275. Al-Osh, M.A.; Alzaid, A.A. Integer-Valued Moving Average (INMA) Process. Statistical Papers 1988, 29, 281–300. Berglund, E.; Bra¨nna¨s, K. Entry and Exit of Plants: A Study Based on Swedish Panel Count Data for Municipalities. Yearbook of the Finnish Statistical Society 1996, 95–111. Berglund, E.; Bra¨nna¨s, K. Plants’ Entry and Exit in Swedish Municipalities. Umea˚ Economic Studies 497, 1997. Blundell, R.; Griffith, R.; Windmeijer, F. Individual Effects and Dynamics in Count Data Models. Working Paper W99=3, Institute of Fiscal Studies, London, 1999. Bo¨ckenholt, U. Mixed INAR(1) Poisson Regression Models: Analyzing Heterogeneity and Serial Dependencies in Longitudinal Count Data. Journal of Econometrics 1999, 89, 317–338. Bra¨nna¨s, K. Estimation and Testing in Integer-Valued AR(1) Models. Umea˚ Economic Studies 335, 1994. Bra¨nna¨s, K. Explanatory Variables in the AR(1) Count Data Model. Umea˚ Economic Studies 381, 1995. Bra¨nna¨s, E.; Bra¨nna¨s, K. A Model of Patch Visit Behaviour in Fish. Biometrical Journal 1998, 40, 717–724.
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Bra¨nna¨s, K.; Hall, A. Estimation in Integer-Valued Moving Average Models. Umea˚ Economic Studies 477, 1998. Cameron, A.C.; Trivedi, P. Regression Analysis of Count Data; Cambridge University Press: Cambridge, 1998. Davidson, R.; MacKinnon, J.G. Estimation and Inference in Econometrics; Oxford University Press: Oxford, 1993. Engle, R.F. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica 1982, 50, 987–1007. Greene, W.H. Econometric Analysis; Prentice-Hall: Upper Saddle River, 1997. Hansen, L.P. Large Sample Properties of Generalized Method of Moments Estimators. Econometrica 1982, 50, 1029–1054. Lunn, A.D.; Davies, S.J. A Note on Generating Correlated Binary Variables. Biometrika 1998, 85, 487–490. McKenzie, E. Some Simple Models for Discrete Variate Time Series. Water Resources Bulletin 1985, 21, 645–650. McKenzie, E. Some ARMA Models for Dependent Sequences of Poisson Counts. Advances in Applied Probability 1988, 20, 822–835. Newey, W.K. Generalized Method of Moment Specification Testing. Journal of Econometrics 1985, 29, 229–256. Newey, W.K.; West, K.D. A Simple, Positive Definite, Heteroskedasticity and Auto Correlation Consistent Covariance Matrix. Econometrica 1987, 55, 703–708. Park, Y.; Oh, C.W. Some Asymptotic Properties in INAR(1) Processes with Poisson Marginals. Statistical Papers 1997, 38, 287–302. Steutel, F.W.; van Harn, K. Discrete Analogues of Self-Decomposability and Stability. The Annals of Probability 1979, 7, 893–899. Tauchen, G.E. Statistical Properties of Generalized Method of Moments Estimation of Structural Parameters Obtained from Financial Data (with discussion). Journal of Business and Economic Statistics 1986, 4, 397–424. Winkelmann, R. Econometric Analysis of Count Data; Springer-Verlag: Heidelberg, 1997. Zeger, S.L. A Regression Model for Time Series of Counts. Biometrika 1988, 75, 621– 629. Zeger, S.L.; Qagish, B. Markov Regression Models for Time Series: A Quasi-Likelihood Approach. Biometrics 1988, 44, 1019–1031.
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