PDF Ebook Bayesian Inference in Threshold Stochastic Frontier Models
Recently, threshold and smooth transition models have received much attention in the literature. Threshold models have a wide variety of application in economics. Some of the applications include models of separating and multiple equilibria, empirical sample splitting when the sample split is based on a continuously-distributed variable such as firm size. Another potential application of the threshold and smooth transition models is in the study of efficiency measurement.
Our motivations of extending the threshold model to a stochastic frontier framework are as follow. First, the idea is to introduce heterogeneity and to allow for non-linearities in the production function without over-parameterization. The second motivation comes from some practical problems. For example, in analyzing the relationship between firms’ sizes and efficiency, a common approach is to split the sample of firms into sub-group based on some measures that related to the size of the firms (e.g., total asset to debt ratio). However, some decision must be made concerning what is the appropriate threshold (i.e., how big must a firm be to be categorized as “large”) at which to split the sample. When this value is unknown, some method must be employed in its selection (see for example Tran and Tsionas 2006)). Another example comes from the analysis of cross-country pattern of economics growth and factor accumulation (e.g., Koop, Osiewalski and Steel (1999), Koop et al. (2000), Limam and Miller (2004)). Advocate of endogenous growth theory (e.g., Romer (1986) and Lucas (1988)) suggest that physical capital growth alone cannot explain per capita output growth. They argue that the large Solow residual responds to variables such as technology and human capital accumulation, which are endogenous to the model. Thus by incorporating technological change, those models allows for the diffusion of technology between countries and ability of developing countries to adopt and implement foreign technology. Benhabib and Spiegel (1994) suggest that human capital affects total factor productivity (TFP) growth through the adoption and implementation of new technologies. They present a model where they decompose TFP growth into two separate components: a catch-up term and a technological component. Instead of including human capital as an input in the production, they suggest that human capital affects income through its effect on TFP.
Koop, Osiewaski and Steel (1999) use a stochastic frontier model that allow for technological change to affect the marginal product of capital and labour for a group of OECD countries. Their specification of technological change represents a deviation from the standard cross-country and panel-growth literature, where technological change often disembodies and assumes to be the same for all countries. Liman and Miller (2004) also use a stochastic frontier model to examine the decomposition of TFP. In their model, they allow explicitly for human capital and the age of capital stock (which represent the input quality) to affect the marginal product of labour and capital. However, both Koop, Osiewaski and Steel (1999) and Limam and Miller (2004) did not allow for transition, adoption and implementation of new technologies.
Motivated by the above discussion, in this paper we propose a class of threshold stochastic frontier models that allow for transition, adoption and implementation of new technologies based on the class of threshold models. In particular, we model the transition to the different technology using another perspective. We allow the transition to depend on certain exogenous variables such as human capital and the age of capital stock that represent input quality, the time trend that allows modeling structural change, and latent technical inefficiency. The transition is smooth but the threshold effect can be either discontinuous (in which case we obtain threshold or regime switching models, see Hansen, 2000) or continuous (in which case we obtain smooth transition models that allow modeling learning and gradual adjustment to the new technology). The models proposed here allow for single or multiple covariates in the transition process, and also allow inefficiency to be a determinant of the switching process. One of the models is precisely the latent class model when group probabilities depend on covariates (Orea and Kumbhakar, 2004). Certain models are more general than the Markov switching model in Tsionas and Kumbhakar (2004) in the sense that we allow for persistent use of technology and endogenous switching or adjustment to the new technology. We allow for freely time varying technical inefficiency and latent cross firm heterogeneity to be correlated with the included variables. In that sense, the purpose of the models proposed here is to model endogenous gradual adjustment in technology, a feature that is absent from the current literature on efficiency and productivity measurement.
We propose Bayesian procedures organized around Gibbs sampling with data augmentation for the analysis of all new models proposed here. The new techniques are applied to a panel of world production functions using as switching or transition variables, human capital, the age of the capital stock (representing input quality), and a time trend to capture structural switching or structural smooth transition.
The paper is organized as follows. Section 2 briefly reviews the standard stochastic frontier model. Section 3 proposes and discusses five different threshold stochastic frontier models. Bayesian inferences for the five models are detailed in Section 4. Section 5 discusses model comparisons and hypothesis testing. Section 6 extends the models discussed in Section 3 to the multiple threshold case. An empirical application is presented in Section 7. Section 8 concludes the paper. Details on the numerical methods for Bayesian inference and marginal likelihood considerations are given in the Appendices.
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PDF Ebook Bayesian Inference in Threshold Stochastic Frontier Models
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