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Testing Competing Models for Non-negative Data with Many Zeros

  • João M. C. Santos Silva EMAIL logo , Silvana Tenreyro and Frank Windmeijer

Abstract

In economic applications it is often the case that the variate of interest is non-negative and its distribution has a mass-point at zero. Many regression strategies have been proposed to deal with data of this type but, although there has been a long debate in the literature on the appropriateness of different models, formal statistical tests to choose between the competing specifications are not often used in practice. We use the non-nested hypothesis testing framework of Davidson and MacKinnon (Davidson and MacKinnon 1981. “Several Tests for Model Specification in the Presence of Alternative Hypotheses.” Econometrica 49: 781–793.) to develop a novel and simple regression-based specification test that can be used to discriminate between these models.

JEL Codes: C12; C52

Corresponding author: João M. C. Santos Silva, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK and CEMAPRE, Rua do Quelhas 6, 1200-781 Lisboa, Portugal, E-mail:

Acknowledgement

We are grateful to the editor Jason Abrevaya and to an anonymous referee for many helpful and constructive suggestions. We also thank Holger Breinlich, Francesco Caselli, Daniel Dias, Esmeralda Ramalho, Joaquim Ramalho, and Rainer Winkelmann for helpful comments, and John Mullahy for providing one of the datasets used in Section 4. Santos Silva acknowledges partial financial support from Fundação para a Ciência e Tecnologia (Programme PEst-OE/EGE/UI0491/2013). Tenreyro acknowledges financial support from the European Research Council under the European Community’s ERC starting grant agreement 240852, “Research on Economic Fluctuations and Globalization.” Windmeijer acknowledges financial support from ERC grant 269874 – DEVHEALTH.

Appendix

A1. Asymptotic Distribution and Adjusted Covariance Matrix

The proposed test is based on the OLS estimation of an artificial model of the form

yiy^iAy^iA=gA(xiβ^A)FA(xiγ^A)xiy^iAδ+αy^iBy^iAy^iA+νi.

The easiest way of obtaining the asymptotic distribution of the OLS estimates of θ=(δ, α), say θ^=(δ^,α^), is to consider the joint estimation of θ and ϕA=(βA, γA) by system GMM as in Newey (1984). The results in this Appendix are presented for the case in which βA and γA are jointly estimated by maximum likelihood, as in Heckman’s (1979) selection model. For cases such as the two-part model in which γA can be estimated independently of βA, the same results are valid if one considers only the moment conditions for the joint estimation of γA and θ.

Let S1 and S2 denote the vector of moment conditions for the model under the null and for the test equation, respectively, and define S(λ)=(S1,S2), with λ=(βA,γA,δ,α). The just-identified system-GMM estimator of λ is defined as the solution of

n1i=1nS(λ^)=0,

where λ^=(β^A,γ^A,δ^,α^), and we assume the following standard regularity conditions (see, e.g., theorems 2.6 and 3.4 in Newey and McFadden 1994).

A1 E(S(λ))=0 only if λ=λ0, where λ0 denotes the true value of λ.

A2λ0∈ interior of Λ, which is compact.

A3S(λ) is continuous at each λ∈Λ with probability one.

A4 With probability approaching one S(λ) is continuously differentiable in a neighborhood ς of λ0.

A5 E(supλΛS(λ)‖)<∞, E(‖S(λ0)‖2)<∞, and E(supλςλS(λ))<.

A6 The matrix M is non-singular, where M=E(λS(λ0)).

Then, the results in Newey and McFadden (1994) imply that

n(λ^λ)dN(0,M1ΣM1'),

where

Σ=E[S1S1S1S2S2S1S2S2].

Noting that

M1=[H10H21H1H1H21],

where H denotes the expectation of the matrix of derivatives of S1 with respect to ϕA and H1 and H2 denote the expectation of the derivatives of S2 with respect to ϕA and θ, respectively, the variance of θ^ can then be written as

V(θ^)=H21E(S2S2)H21H21E(S2S1)H1H1H21H21H1H1E(S1S2)H21+H21H1H1E(S1S1)H1H1H21,

or

V(θ^)=Vθ^+H21{H1V(ϕ^A)H1E(S2S1)H1H1H1H1E(S1S2)}H21,

where V(ϕ^A) is the estimated variance of ϕ^A=(β^A,γ^A) and Vθ^ is the uncorrected estimated variance of θ^.

Whether V(θ^) is smaller, larger, or equal to Vθ^, in the positive semidefinite sense, depends on the particular case being considered. For example, if H1=0, the two matrices are equal and when E(S2S1)=0,V(θ^) is larger than Vθ^ in the positive semidefinite sense.

In the context of the HPC test, it is of special interest to consider the case where γA is estimated by maximum likelihood. In this case, V(ϕ^A)=H1 and E(S2S1)=H1, and therefore

V(θ^)=Vθ^H21H1(H1)H1H21,

implying that V(θ^) is smaller than Vθ^ (see Pierce 1982; Lee 2010, pp. 104–105). Therefore, when γA is estimated by maximum-likelihood, the test-statistic constructed using the uncorrected covariance will have variance smaller than 1 and, therefore, the test will be asymptotically undersized.

Finally, we reiterate that the correction of the covariance matrix is needed only when MA is a double-index model.

A2. Correlation in the Two-Part Model

In Duan et al. (1984) an example is given that argues that there can be correlation between the two error terms in the two-part model and that therefore this model is not nested by the sample selection model, in the sense that the two-part model cannot be obtained by imposing a restriction on the selection model. Since then, numerous papers have quoted this result, for example, Leung and Yu (1996) and Norton et al. (2008). Here we argue that the example is misleading.

In the notation of Duan et al. (1984), the two-part model is given by

Ii=xiδ1+η1i,η1i|xi~N(0,1)ln(yi)=xiδ2+η2i(η2i|Ii>0,xi)~f(0,σ2)(η2i|Ii0,xi)(yi0),

where f is a continuous distribution with mean zero and variance σ2. Hence, (ln(yi)|Ii>0,xi)~f(xiδ2,σ2).

To show that correlation between η1 and η2 is possible Duan et al. (1984) constructed the following example (pp. 285–286): Let Z1i and Z2i follow a standard bivariate normal distribution with correlation coefficient ρ. Let Gi be the left- and Hi be the right-truncated standard normal cdf, with xiδ1 as truncation point:

Gi(u)=xiδ1uϕ(z)dz/Φ(xiδ1),xiδ1u,Hi(v)=vϕ(z)dz/Φ(xiδ1),vxiδ1,

where ϕ denotes the standard normal pdf.

Construct (η1i, η2i) as follows: with probability Φ(xiδ1), define

η1i=Gi1(Φ(Z1i));η2i=f1(Φ(Z2i)).

With probability (1Φ(xiδ1)) define

η1i=Hi1(Φ(Z1i));η2i=.

Then the two-part model assumptions are satisfied and there is correlation between η1i and η2i. Duan et al. (1984) show that when f is assumed to be normal then the conditional expectation is given by

E(η2i|η1i)=ρσΦ(Gi(η1i)),η1i>xiδ1,E(η2i|η1i),η1ixiδ1

and consequently η1i and η2i are stochastically dependent and positively associated.

The problem with this argument lies in the fact that with probability Φ(xiδ1) we draw an η1i such that η1i is larger than xiδ1. This essentially introduces a new uniformly distributed random variable, say ζi, and changes the model to

ζi|xi~U(0,1)Ii=xiδ1+η1iIi>0ifζi<Φ(xiδ1),

so ζi determines the outcome Ii>0 and is independent of η2i. Therefore, there is no selection problem, as

E(ln(yi)|Ii>0,xi)=xiδ2+E(η2i|Ii>0,xi)=xiδ2+E(η2i|ζi<Φ(xiδ1))=xiδ2.

Clearly, the model of the example can be specified as

ζi|xi~U(0,1)Ii=1(ζi<Φ(xiδ1))ln(yi)=xiδ2+η2i(η2i|Ii=1,xi)~f(0,σ2)(η2i|Ii=0,xi)  (yi0),

with ζi independently distributed of η2i and the value of η1i is immaterial. Therefore, this example does not show that the errors η1i and η2i in the original model can be correlated.

In summary, under the maintained assumptions, there is no evidence to support the view that the two-part model cannot be obtained by imposing a restriction on the sample selection model. However, the assumptions of the sample selection model are unlikely to hold when it is used to describe corner solutions data, and in that case there is no guaranty that the conditional expectation implied by the sample selection model will fit the data better than the conditional expectation implied by the two-part model. For example, if η2i is homoskedastic but non-normal, the two-part model can be used to consistently estimate the conditional expectation of yi, while that is not possible with the sample selection model. In that sense, the two models are indeed not nested.

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Published Online: 2014-3-29
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