In this paper, we propose to use these threshold regression models in order to take into account this original dimension in the estimation of the rates of return on public capital stocks, i.e. the potential presence of threshold e¤ects. Applying the so-called production function approach introduced by Aschauer (1989), we consider various speci...cations of public capital stock augmented production functions. At this stage, a remark has to be made. So far, we indi¤erently used the terms of public capital and infrastructure. However, the notion of infrastructure does not fully correspond to the concept of public sector investment expenditure. This distinction is particularly important when it comes to interpret the threshold e¤ects. Indeed, it would be naïve to directly interpret the threshold e¤ects in the productivity of public capital as network e¤ects: the public investments are so heterogeneous, that the level of public investments or the public capital stocks can not be used to reveal information on the completion of the main infrastructure networks. For this reason, in this paper, we only consider the threshold e¤ects of the productivity of the infrastructure investments.

More precisely, we propose to re-evaluate the marginal productivity of infrastructure stocks using threshold regression models, in which the threshold variable may be de...ned as the level of the infrastructure stock actually available. The main concern with these models is that they require an important number of observations to be estimated since they depend on an important number of parameters. This concern is particularly relevant in our context, since the time dimension of the series of infrastructure stocks is generally not su¢ cient to estimate such models, even if we consider only two regimes<sup>1</sup> . One solution is to use panel data as it is generally done in the literature devoted to linear representations of the productivity of infrastructure. In a seminal paper, Hansen (1999) proposed the ...rst procedure to estimate and to test the threshold e¤ects in nondynamic panels. His Panel Threshold Regression (PTR) model allows for dividing the individual observations into classes according to an observable variable. In this case, the time series and cross sections are used in order to identify the regimes. Based on his procedure, it is then possible to test and to estimate the threshold e¤ects in the marginal productivity of infrastructures in a panel of countries, without assuming the homogeneity of the aggregated production function. This model assumes a transition from one regime to another based on the value of a threshold variable, the infrastructure stock for instance: in a model with two regimes, if the threshold variable is below a certain value, called the threshold parameter, the productivity is de...ned by one model, and it is de...ned by another model if the threshold variable exceeds the threshold parameter.

It is important to note that the existence of threshold e¤ects in the productivity of infrastructure is compatible with the three major criticisms addressed to the linear speci...cations of the production function in the huge literature devoted to this topic (see Gramlich, 1994; Sturm, 1998 or Romp and De Haan, 2005 for a survey). Indeed, when it comes to estimate the rates of return on infrastructure (or public capital) with an augmented production function in a panel, three major problems are generally raised. The ...rst one is the potential reverse causation. If public investments depend on income,

> 1 This is the main reason why these models have not been used even in the case of OECD countries. To the best of our knowledge no study devoted to the productivity of public capital and based on threshold models has yet been proposed.

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