Testing Competing Models for Non-negative Data with Many Zeros

João M. C. Santos Silva; Silvana Tenreyro; Frank Windmeijer

doi:10.1515/jem-2013-0005

Published by De Gruyter March 29, 2014

Testing Competing Models for Non-negative Data with Many Zeros

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

From the journal Journal of Econometric Methods

https://doi.org/10.1515/jem-2013-0005

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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.

Keywords: health economics; international trade; non-nested hypotheses; C test; P test

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: jmcss@essex.ac.uk

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

yi−y^iAy^iA=∇gA(x′iβ^A)FA(x′iγ^A)x′iy^iAδ+αy^iB−y^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 ϕ_A=(β_A, γ_A) by system GMM as in Newey (1984). The results in this Appendix are presented for the case in which β_A and γ_A are jointly estimated by maximum likelihood, as in Heckman’s (1979) selection model. For cases such as the two-part model in which γ_A can be estimated independently of β_A, the same results are valid if one considers only the moment conditions for the joint estimation of γ_A 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)=(S′1, S′2)′, with λ=(β′A, γ′A, δ, α)′. The just-identified system-GMM estimator of λ is defined as the solution of

n−1∑i=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.

A2 λ₀∈ interior of Λ, which is compact.

A3 S(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(sup_λ_∈_Λ‖S(λ)‖)<∞, E(‖S(λ₀)‖²)<∞, and E(supλ∈ς‖∂∂λ′S(λ)‖)<∞.

A6 The matrix M is non-singular, where M=E(∂∂λ′S(λ0)).

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

n(λ^−λ)→dN(0, M−1ΣM−1'),

where

Σ=E[S1S′1S1S′2S2S′1S2S′2].

Noting that

M−1=[H−10−H2−1H1H−1H2−1],

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

V(θ^)=H2−1E(S2S′2)H2−1′−H2−1E(S2S′1)H−1′H′1H2−1′−H2−1H1H−1E(S1S′2)H2−1′+H2−1H1H−1E(S1S′1)H−1′H′1H2−1′,

V(θ^)=Vθ^+H2−1{H1V(ϕ^A)H′1−E(S2S′1)H−1′H′1−H1H−1E(S1S′2)}H2−1′,

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(S2S′1)=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 γ_A is estimated by maximum likelihood. In this case, V(ϕ^A)=−H−1 and E(S2S′1)=−H1, and therefore

V(θ^)=Vθ^−H2−1H1(−H−1)H′1H2−1′,

implying that V(θ^) is smaller than Vθ^ (see Pierce 1982; Lee 2010, pp. 104–105). Therefore, when γ_A 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 M_A 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=x′iδ1+η1i, η1i|xi~N(0,1)ln(yi)=x′iδ2+η2i(η2i|Ii>0, xi)~f(0, σ2)(η2i|Ii≤0, xi)≡−∞(yi≡0),

where f is a continuous distribution with mean zero and variance σ². Hence, (ln(yi)|Ii>0, xi)~f(x′iδ2, σ2).

To show that correlation between η₁ and η₂ is possible Duan et al. (1984) constructed the following example (pp. 285–286): Let Z₁_i and Z₂_i follow a standard bivariate normal distribution with correlation coefficient ρ. Let G_i be the left- and H_i be the right-truncated standard normal cdf, with −x′iδ1 as truncation point:

Gi(u)=∫−x′iδ1uϕ(z)dz/Φ(x′iδ1),−x′iδ1≤u,Hi(v)=∫−∞vϕ(z)dz/Φ(x′iδ1),v≤−x′iδ1,

where ϕ denotes the standard normal pdf.

Construct (η₁_i, η₂_i) as follows: with probability Φ(x′iδ1), define

η1i=Gi−1(Φ(Z1i));η2i=f−1(Φ(Z2i)).

With probability (1−Φ(x′iδ1)) define

η1i=Hi−1(Φ(Z1i));η2i=−∞.

Then the two-part model assumptions are satisfied and there is correlation between η₁_i and η₂_i. 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>−x′iδ1,E(η2i|η1i)≡−∞,η1i≤−x′iδ1

and consequently η₁_i and η₂_i are stochastically dependent and positively associated.

The problem with this argument lies in the fact that with probability Φ(x′iδ1) we draw an η₁_i such that η₁_i is larger than −x′iδ1. This essentially introduces a new uniformly distributed random variable, say ζ_i, and changes the model to

ζi|xi~U(0, 1)Ii=x′iδ1+η1iIi>0 if ζi<Φ(x′iδ1),

so ζ_i determines the outcome I_i>0 and is independent of η₂_i. Therefore, there is no selection problem, as

E(ln(yi)|Ii>0, xi)=x′iδ2+E(η2i|Ii>0, xi)=x′iδ2+E(η2i|ζi<Φ(x′iδ1))=x′iδ2.

Clearly, the model of the example can be specified as

ζi|xi~U(0, 1)Ii∗=1(ζi<Φ(x′iδ1))ln(yi)=x′iδ2+η2i(η2i|Ii∗=1, xi)~f(0, σ2)(η2i|Ii∗=0, xi)≡∞ (yi≡0),

with ζ_i independently distributed of η₂_i and the value of η₁_i is immaterial. Therefore, this example does not show that the errors η₁_i and η₂_i 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 η₂_i is homoskedastic but non-normal, the two-part model can be used to consistently estimate the conditional expectation of y_i, while that is not possible with the sample selection model. In that sense, the two models are indeed not nested.

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Supplemental Material

The online version of this article (DOI: 10.1515/jem-2013-0005) offers supplementary material, available to authorized users.

Published Online: 2014-3-29

Published in Print: 2015-1-1

Testing Competing Models for Non-negative Data with Many Zeros

Abstract

Acknowledgement

Appendix

A1. Asymptotic Distribution and Adjusted Covariance Matrix

A2. Correlation in the Two-Part Model

References

Supplemental Material

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