Gilbert Colletaz
email: gilbert.colletaz@univ-orleans.fr.

Christophe Hurlin
email: christophe.hurlin@univ-orleans.fr.

Threshold E¤ects in the Public Capital Productivity: An International Panel Smooth Transition Approach

Keywords: Public Capital, Panel Smooth Threshold Regression Models. J C82, E22, E62

Using a nonlinear panel data model, we examine the threshold e¤ects in the productivity of the public capital stocks for a panel of 21 OECD countries observed over 1965-2001. Using the so-called "augmented production function" approach, we estimate various speci…cations of a Panel Smooth Threshold Regression (PSTR) model recently developed by Gonzalez, Teräsvirta and Van Dijk (2004). One of our main results is the existence of strong threshold e¤ects in the relationship between output and private and public inputs: whatever the transition mechanism speci…ed, tests strongly reject the linearity assumption. Moreover, this model allows crosscountry heterogeneity and time instability of the productivity without speci…cation of an ex-ante classi…cation over individuals. Consequently, it is possible to give estimates of productivity coe¢cients for both private and public capital stocks at any time and for all the countries. Finally we proposed estimates of individual time varying elasticities that are much more reasonable than those previously published.

Introduction

This paper provides an international comparison of the patterns of productivity of public capital in OECD countries. Our methodology is based on an augmented production function where public capital is an additional input, in addition to private capital and labor, following [START_REF] Aschauer | Is Public Expenditure Productive?[END_REF]. This so-called "production function approach" authorizes the derivation of the elasticity of output with respect to public capital but is not exempt from critics. At an empirical level, it is well known that studies based on national time series data and a Cobb-Douglas production function …nd very large output elasticities1 . Many reasons have been suggested to explain such results (see [START_REF] Sturm | Public Capital Expenditure in OECD Countries: the Causes and Impact of the Decline in Public Capital Spending[END_REF] for a survey).

One of them, speci…c to the panel data models, concerns the cross-section heterogeneity. It is well known that biases appear when parameter heterogeneities among cross-sectional units are ignored (see [START_REF] Hsiao | Analysis of Panel Data[END_REF][START_REF] Pesaran | Estimating Long-Run Relationships from Dynamic Heterogenous Panels[END_REF]. In a production function approach, the assumption of a common elasticity of output with respect to public and private factors is doubtful, regardless of the level of aggregation (international, regional, etc.). However, studies based on a production function approach generally specify heterogeneity only using …xed or random individual e¤ects [START_REF] Evans | Is Government Capital Productive? Evidence from a Panel of Seven Countries[END_REF][START_REF] Holtz-Eakin | Public-Sector Capital and the Productivity Puzzle[END_REF]. In this regard, there is no reason to expect the cross-section homogeneity of the other production function parameters and particularly of the public capital elasticity. For example, based on an analysis of the long run relationship between infrastructure stock and per capita income, Canning and Pedroni (1999, page 8), found evidence for considerable heterogeneity among the key parameter estimates across countries, which suggests that directly pooling certain parameters across countries may be misleading. [START_REF] Canning | Infrastructure's Contribution to Aggregate Output[END_REF] or [START_REF] Canning | The Social Rate of Return on Infrastructure Investments[END_REF] authorize a particular form of elasticity heterogeneity by splitting their sample into two groups of countries according to the observed levels of income per worker in a given year. Assuming a Cobb-Douglas production function, they show that infrastructure elasticities of poorer countries are small and statistically insigni…cant, while they remain large and signi…cant for richer countries. However, this solution implies that sub-samples, (in this case, poor and rich countries) are speci…ed ex-ante and exogenously determined. Moreover, an individual is not allowed to switch between groups across periods.

For these reasons, this kind of heterogeneity can be advantageously speci…ed in terms of the Panel Threshold Regression (PTR) model. This model, proposed by [START_REF] Hansen | Threshold E¤ects in Non-Dynamic panels: Estimation, Testing and Inference[END_REF], implies that individual observations can be divided into homogeneous classes based on the value of an observed variable. More precisely, it assumes a transition from one regime to another according to the value of a threshold variable (the income per worker 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 equation, and it is de…ned by another equation if the threshold variable exceeds the threshold parameter. However, this is not entirely satisfying, and one of the main drawbacks of this PTR model is that it allows only for a small number of classes, i.e., a small number of productivity regimes. It is highly unlikely that international or regional time varying rates of returns on public stocks can be identi…ed in a small set of constants.

A solution adopted in this paper is to use a Panel Smooth Threshold Regression (PSTR) model recently developed by Gonzalez, Teräsvirta and Van Dijk (2005) and [START_REF] Fok | A Multi-Level Panel STAR Model for US Manufacturing Sectors[END_REF]. Two interpretations of these models are possible. On one hand, the PSTR model can be thought of as a regime-switching model that allows for a small number of extreme regimes associated with the extreme value of a transition function and where the transition from one regime to the other is smooth. On the other hand, the PSTR model can be said to allow for a continuum of regimes, each one being characterized by a di¤erent value of the transition function. The logic is then similar to that developed in the standard univariate time series STAR 2 (Smooth Transition AutoRegressive) models except for the fact that the PSTR uses panel speci…cations (individual e¤ects) and is non-dynamic, i.e., without lagged endogenous variables in the explanatory variables set. In our context, the PSTR allows cross-country heterogeneity and time instability of the elasticities without speci…cation of an ex-ante classi…cation over individuals. Consequently, it is also possible to compute estimates of productivity coe¢cients for both private and public capital stocks at any time and for each country in the sample.

The paper is organized as follows. In the …rst section, we discuss the threshold speci…cation of the augmented production function. We consider a PSTR model that allows cross-country heterogeneous and time varying elasticities of the output with respect to the private and public inputs. The choice of the threshold variable, the linearity tests and the estimation of the parameters are successively presented. In the second section, we present the data and the estimates of panel linear speci…cations of the augmented production function. In the third section, we present the results of the linearity tests and the estimates obtained with various panel threshold models.

Finally, based on these PSTR estimates, we compute individual time varying estimates of the elasticities of output with respect to the public capital stocks. The last section concludes.

Threshold E¤ects in the Productivity of Public Capital Stocks

The basis of our empirical approach is exactly the same as that used by many authors since the seminal paper of [START_REF] Aschauer | Is Public Expenditure Productive?[END_REF], and more recently by [START_REF] Canning | Infrastructure's Contribution to Aggregate Output[END_REF], [START_REF] Canning | The Social Rate of Return on Infrastructure Investments[END_REF] or [START_REF] Calderon | The Output Cost of Latin America's Infrastructure Gap[END_REF] for developing countries.

It consists of estimating the parameters of an augmented production function where public capital appears as an explanatory variable. We also follow most studies in adopting a Cobb-Douglas speci…cation of the production function and in assuming that the public capital services are proportional to the public capital stock. For a country i = 1; ::; N at a time t = 1; :; T; we consider two speci…cations of this augmented production function:

y it k it = i + (n it k it ) + g it + v it ;
(1)

y it k it = i + (n it k it ) + (g it k it ) + v it ; (2)
where y it is the aggregate added value, k it is private capital stock, g it is public capital stock and n it is employment. All variables are expressed in logarithm. In these log-linear models, the parameter denotes the elasticity of the output with respect to the labor factor, whereas the parameter denotes the elasticity of the output with respect to the public capital stock. In both speci…cations, the endogenous variable is the productivity of private capital stock, i.e., y t k t : This normalization, used by [START_REF] Aschauer | Is Public Expenditure Productive?[END_REF], makes it possible to consider various assumptions on the nature of the scale returns. The …rst equation ( 1) corresponds to the assumption of private factors' constant returns to scale (PFCRS). The second equation ( 2) corresponds to the assumption of overall constant returns to scale (OCRS). This speci…cation corresponds to one of Aschauers' speci…cations in which he obtained a public capital elasticity of 39%, i.e., higher than the estimated private capital elasticity (26%). Finally, many studies since the …rst work done by [START_REF] Evans | Is Government Capital Productive? Evidence from a Panel of Seven Countries[END_REF] have highlighted the importance of individual e¤ects in this kind of speci…cation when panel data are considered. Consequently, we introduce …xed individual e¤ects i in order to capture all the timeless components of the productivity of the private capital stock. Thus, the speci…cations given by the equations ( 1) and (2) correspond to the general speci…cation used in the literature devoted to the so-called "production function approach" and based on panel data models [START_REF] Evans | Is Government Capital Productive? Evidence from a Panel of Seven Countries[END_REF][START_REF] Holtz-Eakin | Public-Sector Capital and the Productivity Puzzle[END_REF]Munnel, 1990;[START_REF] Pinnoi | Public Infrastructure and Private Production. Measuring Relative Contributions[END_REF][START_REF] Baltagi | Public Capital Stock and State Productivity Growth: Further Evidence from an Error Components Model[END_REF]Garcia-Mila and Mc Guire, 1996;[START_REF] Canning | Infrastructure's Contribution to Aggregate Output[END_REF][START_REF] Canning | Infrastructure's Contribution to Aggregate Output[END_REF]Bennathan, 2000 or Calderon andServen, 2004 etc..).

As was previously mentioned, in this study we propose exactly the same framework as that studied in this literature, except for the fact that we introduce non-linearity. One justi…cation for that can be found in the network dimension of most public investments (in roads and highways, in sanitation and sewer systems, etc.). This network dimension implies a non-linearity of the marginal productivity of public capital stocks as suggested, for example, by [START_REF] Fernald | Roads to Prosperity? Assessing the Link Between Public Capital and Productivity[END_REF] for the road public capital stock in the United States.

In order to take into account this speci…city, a solution consists in adopting a Panel Smooth Threshold Regression (PSTR) model as proposed by Gonzalez, Teräsvirta and Van Dijk (2005) and [START_REF] Fok | A Multi-Level Panel STAR Model for US Manufacturing Sectors[END_REF]. Let us consider the simplest case with two extreme regimes and a single transition function. The corresponding PSTR model for the PFCRS speci…cation is de…ned as:

y it k it = i + 0 (n it k it ) + 0 g it + [ 1 (n it k it ) + 1 g it ] h (q it ; ; c) + " it ; (3)
where q it denotes a threshold variable and where the error " it is assumed to be i:i:d: 0; 2 :

The transition function h (q it ; ; c) is a continuous and bounded function of the threshold variable q it . As do Gonzalez, Teräsvirta and Van Dijk (2005), we will also consider a logistic transition function de…ned as follows:

h (q it ; ; c) = f1 + exp [ (q it c)]g 1 ; > 0; (4)
where c denotes a location parameter and where determines the slope of the transition function. This model can be rewritten as:

y it k it = i + 0 0 W it + 0 1 W it h (q it ; ; c) + " it ; (5)
where j = j j 0 for j = (0; 1), W it = [(n it k it ) g it ] 0 in the case of the PFCRS speci…cation and W it = [(n it k it ) (g it k it )] 0 in the case of the OCRS speci…cation.

In our context, the PSTR model has three main advantages. The …rst one is that it allows the elasticities (in particular the public capital elasticity) to vary between countries (heterogeneity issue) but also with time (stability issue). It provides a parametric approach to the cross-country heterogeneity and the time variability of the slope coef-…cients of the production function. More precisely, this model allows the parameters of the production function to change smoothly as a function of the threshold variable q it :

For instance, if the threshold variable q it is di¤erent from g it , the elasticity of output with respect to the public capital stock g it for the i th country at time t is de…ned by the weighted average of the parameters 0 and 1 obtained in the extreme regimes:

e g it =
@y it @g it = 0 + 1 h (q it ; ; c) ; 8i; 8t;

where, by de…nition of the transition function, 0 e g it 0 + 1 ; if 1 > 0 or 0 + 1 e g it 0 if 1 < 0; since 0 h (q it ; ; c) 1; 8q it . The same conclusions are valid for the elasticities of labor and private capital inputs.

The second advantage of the PSTR model is that the value of the elasticity of public capital, for a given country, at a given date, can be di¤erent from the estimated parameters for the extreme regimes, i.e., parameters 0 and 1 : As illustrated by the equation ( 6), these parameters do not directly correspond to the public capital elasticity.

The parameter 0 corresponds to the public capital elasticity only if the transition function h (q it ; ; c) tends to 0. The sum of the parameters 0 and 1 corresponds to the public capital elasticity only if the transition function h (q it ; ; c) tends to 1. Between these two extremes, the public capital elasticity e g it is de…ned as a weighted average of the parameters 0 and 1 . Therefore, it is important to note that it is generally di¢cult to directly interpret the values of these parameters that correspond to extreme situations. It is generally preferable to interpret (i) the sign of these parameters, which indicates an increase or a decrease of the elasticity with the value of the threshold variable and (ii) the time varying and individual elasticity of the output with respect to the public capital stock (or other factor) given the equation ( 6).

Finally, this model can be analyzed as a generalization of the Panel Threshold Regression (PTR) model proposed by [START_REF] Hansen | Threshold E¤ects in Non-Dynamic panels: Estimation, Testing and Inference[END_REF] and the panel linear model with individual e¤ects. When the parameter tends to in…nity, the transition function h (q it ; ; c) tends to the indicator function I (q it c) : Thus, when tends to in…nity, the PSTR model gives the PTR model:

y it k it = i + 0 0 W it + 0 1 W it I (q it c) + " it ; (7)
I (q it c) = ( 1 if q it c 0 if q it < c ; (8)
In this case, the public capital elasticity for the country i at time t, denoted e g it ; switches between two extreme values given the level of the threshold variable: e g it = 0 if q it < c and e g it = 0 + 1 if q it c: When tends to zero, the transition function h (q it ; ; c) is constant. Then, the model corresponds to the standard linear model with individual e¤ects (with constant and homogenous elasticities). In this case, the public capital elasticity is simply de…ned by e g it = 0 ; 8i = 1; ::; N and 8t = 1; ::; T: In other cases, the choice of the logistic transition function implies a monotonic change of the coe¢cient of public capital from 0 to 0 + 1 as q it increases.

A more complicated relationship between the transition variable q it and the public capital elasticity can be obtained in a model with r + 1 extreme regimes. Such a generalization can be de…ned as:

y it k it = i + 0 (n it k it ) + 0 g it + r X j=1 j (n it k it ) + j g it h j q it ; j ; c j + " it ; (9)
or equivalently

y it k it = i + 0 0 W it + r X j=1 0 j W it h j q it ; j ; c j + " it ; (10)
where the transition function h j q it ; j ; c j ; 8j = 1; ::; r; depends on the slope parameters j and on the location parameters c j : Even if it is possible to consider a threshold variable di¤erent for each transition function, in our application, we consider that all transition functions share the same threshold variable. In this generalization, if the threshold variable q it is di¤erent from g it , the elasticity of public capital for the i th country at time t is de…ned by the weighted average of the r + 1 elasticities j obtained in the r + 1 extreme regimes:

e g it = @y it @g it = 0 + r X j=1 j h j q it ; j ; c j ; 8i; 8t; (11)
Such an expression authorizes a variety of con…gurations for the relationships between the time varying public capital elasticity and the level of the threshold variable (the public capital stock for instance) as we will discuss in the next part.

Estimation and Speci…cation Tests

The estimation of the parameters of the PSTR model3 is relatively straightforward and consists of eliminating the individual e¤ects i by removing individual-speci…c means and then applying nonlinear least squares to the transformed model (for more details, see Appendix A). However, in this threshold model, there are two main speci…cation issues. The …rst one consists of choosing the threshold variable. The second concern involves testing the number of regimes or testing the threshold speci…cation.

Choice of the Threshold Variable

No technical constraint is imposed on the choice of the threshold variable. Therefore, this choice is mainly an economic issue. If we want to asses the idea that public investments have a network character, the threshold variable can be de…ned as an indicator of the completion of the main networks. It is precisely the idea raised by [START_REF] Gramlich | Infrastructure Investment: a Review Essay[END_REF] or [START_REF] Fernald | Roads to Prosperity? Assessing the Link Between Public Capital and Productivity[END_REF]: the construction of the network substantially boosts the productivity and the output, but, when the construction of the network is completed, the public capital is not exceptionally productive at the margin. In this perspective, a natural candidate for the threshold variable is the existing level of available public capital stock, i.e., q it = g it : However, this speci…cation obviously captures the threshold e¤ects due to a simple country size e¤ect. For instance, if we consider Luxembourg and the United States in the same panel, the threshold e¤ects associated with the variable g it will mainly re ‡ect the di¤erences of size and not the potential network e¤ects in public investment productivity. So, in order to avoid these size e¤ects, we propose here a model in which the marginal productivity of public capital depends on the ratio of public capital to private capital. Finally, we consider a lagged value of this ratio in order to avoid a simultaneity issue since the private capital stock is a part of the endogenous variable of our regressions. Thus, our …rst speci…cation of the transition function is based on the following threshold variable:

Model A :

q it = g i;t 1 k i;t 1 ; (12)
Another choice for the threshold variable consists of using the lagged level of private capital per worker, i.e., q it = k i;t 1 n i;t 1 : This second speci…cation is related to the issue of the heterogeneity of the production function between "rich" countries and "poor" countries as suggested by [START_REF] Canning | The Social Rate of Return on Infrastructure Investments[END_REF]. However, in contrast with Canning and Bennathan, the heterogeneity in our PSTR model is endogenous in the sense that it is the threshold variable, which determines the di¤erent regimes of productivity. Moreover, a country with low productivity in the beginning of the period can have a medium or high productivity at the end of the sample period. So, a second speci…cation, denoted Model B, is given by: Model B:

q it = k i;t 1 n i;t 1 ; (13)
The impact of this threshold variable on the public capital productivity is a priori unclear. Based on a World sample, Canning and Bennathan conclude that "infrastructure in the poorer countries appears to have the same e¤ectiveness in raising output as other types of physical capital while having a greater e¤ectiveness than other types of capital in richer countries" (Canning and Bennathan, 2000, page 13). The issue is to verify if this conclusion is robust in our panel of OECD countries, in which the heterogeneity between rich countries and poor countries is less important.

Estimation and Speci…cation Tests

In order (i) to test the linearity against the PSTR model and (ii) to determine the number, r; of transition functions, i.e., the number of extreme regimes that is equal to r + 1; we use here the procedure proposed by Gonzalez, Teräsvirta and Van Dijk.

Testing the linearity in the augmented production function under PFCRS (equation 3) can be done by testing H 0 : = 0 or H 0 : 1 = 1 = 0: But in both cases, the test will be non-standard since under H 0 the PSTR model contains unidenti…ed nuisance parameters as it was the case in the Hansen's PTR model. This issue is well known in the literature devoted to the time series threshold models [START_REF] Hansen | Inference when a Nuisance Parameter is not identi…ed under the Null Hypothesis[END_REF]. A possible solution consists of replacing the transition function h (q it ; ; c) by its …rst-order

Taylor expansion around = 0 and to test an equivalent hypothesis in an auxiliary regression. If we consider the augmented production function under PFCRS (equation 1), we obtain:

y it k it = i + 0 0 W it + 0 1 W it q it + " it ; (14)
where 0 = ( 0 0 ) 0 , W it = [(n it k it ) g it ] 0 and the parameter vectors 0 i are a multiple of the slope parameter (see Appendix B). Thus, testing the linearity against the PSTR model simply consists of testing H 0 : 1 = 0 in this linear panel model. If we de…ne SSR 0 as the panel sum of squared residuals under H 0 (linear panel model with individual e¤ects) and SSR 1 as the panel sum of squared residuals under H 1 (PSTR model with two regimes), the corresponding F-statistic4 is then de…ned by:

LM F = [(SSR 0 SSR 1 ) =2] = [SSR 0 = (T N N 2)] ; (15)
Under the null hypothesis, the F-statistic is distributed as a 2 (2) ; and the F statistic has an approximate F (2; T N N 2) distribution. The logic is similar when it comes to test the number of transition functions in the model or equivalently the number of extreme regimes. The idea is as follows: we use a sequential approach by testing the null hypothesis of no remaining nonlinearity in the transition function. For instance, let us assume that we have rejected the linearity hypothesis. The issue is then to test whether there is one transition function (H 0 : r = 1) versus at least two transition functions (H 1 : r = 2). Let us assume that the model with r = 2 is de…ned as:

y it k it = i + 0 0 W it + 0 1 W it h 1 (q it ; 1 ; c 1 ) + 0 2 W it h 2 (q it ; 2 ; c 2 ) + " it ; (16)
The logic of the test consists of replacing the second transition function by its …rstorder Taylor expansion around 2 = 0 and then in testing linear constraints on the parameters. If we use the …rst-order Taylor approximation of h 2 (q it ; 2 ; c 2 ), the model becomes:

y it k it = i + 0 0 W it + 0 1 W it h 1 (q it ; 1 ; c 1 ) + 0 1 W it q it + " it ; (17)
and the test of no remaining nonlinearity is simply de…ned by H 0 : 1 = 0. Let us de…ne SSR 0 as the panel sum of squared residuals under H 0 ; i.e., in a PSTR model with one transition function. Let us de…ne SSR 1 as the sum of squared residuals of the transformed model (equation 17). As in the previous cases, the statistic LM F can be computed according to the same de…nition by adjusting the number of degrees of freedom. The testing procedure is then the following. Given a PSTR model with r = r , we will test the null H 0 : r = r against H 1 : r = r + 1. If H 0 is not rejected, the procedure ends. Otherwise, the null hypothesis H 0 : r = r + 1 is tested against

H 1 : r = r + 2.
The testing procedure continues until the …rst acceptance of H 0 .

Given the sequential aspect of this testing procedure, at each step of the procedure the signi…cance level must be reduced by a constant factor 2 ]0; 1[ in order to avoid excessively large models. We postulate = 0:5 as suggested by Gonzalez, Teräsvirta and Van Dijk (2005).

Data and Panel Linear Models

In this study, we consider a panel of 21 OECD countries over the period 1965-2001. Included countries are: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Iceland, Ireland, Italy, Japan, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, United Kingdom and United States. As suggested by [START_REF] Hansen | Threshold E¤ects in Non-Dynamic panels: Estimation, Testing and Inference[END_REF], we consider a balanced panel since it is unknown if the results of estimation and testing procedures presented below extend to unbalanced panels. The data on private and public net capital stocks are drawn from [START_REF] Kamps | The Dynamic Macroeconomic E¤ects of Public Capital: Theory and Evidence for OECD Countries[END_REF]. Both series have been estimated according to the perpetual inventory method from the OECD series of In order to asses the comparability of our data sets to the data sets used in previous studies, we …rst estimate the augmented production function in linear panel models.

In Table 1, we report various estimates of the parameters of the production functions

(1) and ( 2) in three speci…cations: (i) with …xed individual e¤ects, (ii) with random individual e¤ects or (iii) with …xed individual e¤ects and time dummies.

Insert Table 1. Linear Panel Models with Fixed E¤ects and Year Dummies

Our results are globally similar to those reported in the literature based on a production function approach. First, the elasticity of output with respect to the public capital stock is largely signi…cant and positive. Whatever the considered assumption on the scale returns, the estimated elasticity is particularly high and even superior to 0.30.

Hence, our estimated elasticities are close to those generally reported in time series models. Indeed, since the seminal article of [START_REF] Aschauer | Is Public Expenditure Productive?[END_REF], many empirical studies based on this approach have yielded very high estimated elasticities on American data as well as on OECD data sets (see [START_REF] Gramlich | Infrastructure Investment: a Review Essay[END_REF][START_REF] Sturm | Public Capital Expenditure in OECD Countries: the Causes and Impact of the Decline in Public Capital Spending[END_REF] for a survey). The emblematic illustration of these "stratospheric" [START_REF] Gramlich | Infrastructure Investment: a Review Essay[END_REF] estimated elasticities

for the public capital input is the value of 0.39 obtained by Aschauer for the United

States under the OCRS hypothesis. Second, the productive contributions of private factors are generally lower than the share of their respective remuneration in added value. As in [START_REF] Aschauer | Is Public Expenditure Productive?[END_REF], [START_REF] Ram | Government Capital and Private Output in the United States[END_REF], [START_REF] Eisner | Real Government Saving and the Future[END_REF], [START_REF] Vijverberg | Public Capital and Private Productivity[END_REF] or [START_REF] Sturm | Is Public Expenditure Really Productive?[END_REF], the elasticity of private capital is lower than that of public capital in our OCRS speci…cation. Third, our results are also similar to the results obtained in the literature based on panel data models. For instance, [START_REF] Kamps | The Dynamic Macroeconomic E¤ects of Public Capital: Theory and Evidence for OECD Countries[END_REF] obtained an estimated elasticity of 0.31 for the public capital input and 0.72 for the labor input on the same sample when he used a group mean fully modi…ed OLS estimator [START_REF] Pedroni | Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors[END_REF]. Based on a smaller sample of seven OECD countries, Evans and Karras (2004) showed the importance of the introduction and the speci…cation of the individual e¤ects in order to capture a part of the heterogeneity of the production function. They found that, when individual and time e¤ects are introduced, the estimated elasticities for the public capital input are less important and even not signi…cant. This is not the case in our sample. The introduction of …xed or random individual e¤ects does not reduce the "stratospheric" estimated value of the public capital elasticity. However, we can observe that the introduction of time e¤ects with …xed individual e¤ects leads to a slight reduction of the estimated elasticity.

As observed in the literature, the implied annual marginal yields of public capital are then extremely high. [START_REF] Tatom | Public Capital and Private Sector Performance[END_REF] and [START_REF] Gramlich | Infrastructure Investment: a Review Essay[END_REF] calculated (starting from the elasticities estimated by [START_REF] Aschauer | Is Public Expenditure Productive?[END_REF]) that the annual marginal productivity of public infrastructures in the United States would range between 75% in 1970 and more than 100% in 1991. Thus, these results "mean that one unit of government capital pays for itself in terms of higher output in a year or less, which does strike one as implausible" [START_REF] Gramlich | Infrastructure Investment: a Review Essay[END_REF](Gramlich , page 1186). The issue is then to determine if the introduction of threshold e¤ects in the productivity of the public capital stock would allow estimating more reasonable rates of return on these stocks.

Panel Threshold Models

The …rst step consists of testing the log-linear speci…cation of the production function against a speci…cation with threshold e¤ects. If the linearity hypothesis is rejected, it will be necessary to determine the number of transition functions required to capture the nonlinearity of the augmented production function, or equivalently, all the heterogeneity of the parameters of the production function. The results of these linearity tests and speci…cation tests of no remaining nonlinearity are reported in Table 2. Given the de…nition of the threshold variable q it (models A or B) and the speci…cation of the returns to scale, four cases are considered. For each speci…cation, we compute the LM F statistics for the linearity tests (H 0 : r = 0 versus H 1 : r = 1) and for the tests of no remaining nonlinearity (H 0 : r = a versus H 1 : r = a + 1). The values of the statistics are reported until the …rst acceptance of H 0 . For computational tractability, we limit our analysis to PSTR models with at most four transition functions.

Insert Table 2. LM F Tests for Remaining Nonlinearity

The linearity tests clearly lead to the rejection of the null hypothesis of linearity of the relationships between the output and public and private inputs. The only exception is model A under the assumption of PFRCS. In this case, the linearity assumption is not rejected. In all other cases, whatever the choice made for the threshold variable,

the assumption on the returns to scale, the LM F statistics lead to strongly reject the null H 0 : r = 0. The results (not reported) are similar when we consider the other LM or pseudo-LRT statistics. This …rst result implies that there is strong evidence that the relationship between output and the considered inputs (in particular public capital stocks) is nonlinear. Our results are then compatible with the results obtained by [START_REF] Fernald | Roads to Prosperity? Assessing the Link Between Public Capital and Productivity[END_REF] for the road sector. For a variety of reasons, perhaps due to the network aspect of public investments in infrastructure, there is a strong nonlinearity in the productivity of these equipment and structures.

Thus, what could be the consequences of using a linear panel model in order to estimate the elasticity of public capital? Let us assume that one comes to estimate the augmented production function with a homogenous linear model, i.e., a model in which the parameters of the production are assumed (wrongly) to be common to all the countries. This approach leads to ignoring the heterogeneity of the productivity of the public capital and consequently leads to presenting an estimate that is a nonsensical average of heterogeneous rates of return. It is perhaps a reason why, for the same countries and for comparable linear speci…cations, the estimated elasticities obtained in linear panel models with …xed e¤ects are generally lower than the ones reported in studies based on time series [START_REF] Evans | Is Government Capital Productive? Evidence from a Panel of Seven Countries[END_REF]. Besides, a linear approach leads to ignoring the potential changes of productivity regimes and proposes an estimate that could be analyzed, for a given country, as a nonsensical average of the di¤erent historical values of the productivity. This second source of misevaluation could have drastic consequences especially for industrialized countries in which the main networks of infrastructure are globally completed.

The speci…cation tests of no remaining nonlinearity (see Table 2) lead to speci…cations with generally one, two or three transition functions. In other words, in a PSTR model, a small number of extreme regimes is su¢cient to capture the non-linearity of the technological relationships, or equivalently the cross-country heterogeneity and the time variability of the public and private factors productivity. Recall that a smooth transition model, even with two extreme regimes (r = 1), can be viewed as a model with an in…nite number of intermediate regimes of productivity. The elasticities of inputs are de…ned at each date and for each country as weighted averages of the values obtained in the two extreme regimes. The weights depend on the value of the transition function (equation 4). So, even if r = 1, this model allows a "continuum" of elasticities (or regimes), with each one associated with a di¤erent value of the transition function h(:) between 0 and 1. In Table 3, for each model we report the corresponding optimal number of transition functions deduced from the LM F tests of remaining nonlinearity, the number of parameters, the residual sum of squares, the Akaike and the Schwarz information criteria.

Insert Table 3. Summary Statistics

Finally, for each assumption on the returns to scale, we choose the appropriate variable between the two "candidate" threshold variables (model A or model B). For that we consider the variable that gives rise to the strongest rejection of linearity whatever is the value of m (LM F statistic for H 0 : r = 0, see Table 2). Obviously, the best transition variable for the OCRS speci…cation is given by the lagged ratio of public capital stock to private capital stock g i;t 1 k i;t 1 (model A), while it is the lagged stock of private capital per worker k i;t 1 n i;t 1 (model B) when the assumption of PFCRS is considered.

Table 4 contains the parameter estimates of the …nal PSTR models. Recall that the estimated parameters j = j ; j 0 cannot be directly interpreted as elasticities.

As in logit or probit models, the value of the estimated parameters is not directly interpretable, but their signs can be interpreted. For instance, let us consider the production function estimated under OCRS (third column of Table 4). In this case, the threshold variable is de…ned as the lagged ratio of public capital stock to private capital stock, and there is only one transition function. A negative (respectively positive) parameter 1 in the vector 1 only signi…es that, when the threshold variable increases, the elasticity of the public capital decreases (respectively increases). In other words, if 1 is positive, it implies that an increase in the ratio of public capital to private capital stocks induces an increase of the public capital elasticity. This observation can be generalized in a model with more than one transition function (r > 1) even if things are slightly more complicated. In a model with two transition functions, if the parameter 1 is positive and the parameter 2 is negative, this implies that an increase of the threshold variable has two opposite e¤ects on the elasticity. The results of these two opposite e¤ects will depend on the value of the (i) slope parameters j and (ii) the location parameters c j . No general result can be deduced. It is precisely the case for our PFCRS model B. Thus, when the ratio of private capital per worker increases in a country, it induces one positive e¤ect and one negative e¤ect on the marginal productivity of the public capital stocks. Similar opposite e¤ects are observed on the private capital productivity.

Insert Table 4. Parameter Estimates for the Final PSTR Models

For all the models, we observe that the transition functions are not quite sharp.

Recall that, when the slope parameter tends to in…nity, the transition function tends to an indicator function as in the threshold model without a smooth transition. On the contrary, in our application, the maximum of the estimated slope parameters is equal to 7.4 in the PFCRS speci…cation and 4.863 in the OCRS speci…cation. The corresponding estimated transition function (see Figure 1) is relatively smooth. This point is particularly important, since it implies that the non-linearity of the augmented production function can not be reduced to a limited number of regimes with di¤erent elasticities. Indeed, it is important to recall that, contrary to a PTR model, a PSTR model with a smooth transition function can be interpreted as a model that allows a "continuum" of regimes. This "continuum" of regimes is clearly required when it comes to measure the threshold e¤ects of the public capital productivity. This result also points out the fact that the solution which consists in grouping some countries in a panel and in estimating a linear relationship between the public capital and the productivity of private inputs, or other measure of the activity, may be unsatisfactory.

It is well known that this approach neglects the heterogeneity of the relationships between the countries. But for a given country, using a linear speci…cation also leads to neglect the continuum of di¤erent productivity regimes which could be observed over the time periods.

Insert Figure 1. Estimated Transition Function

Individual Estimates of the Public Capital Productivity

Given the parameter estimates of the …nal PSTR models, it is now possible to compute, for each country of the sample and for each date the time varying elasticity of output with respect to the public capital stock g it , denoted e g it (see equation 11 ). The individual averages of these smoothed elasticities for public capital stock and labor, as well as their variances, are reported in Tables 5 and6 6 . In the second column, the individual OLS estimated elasticity is reported with the corresponding t-statistic. In the third column, the panel Within estimated elasticity obtained in a linear speci…cation is reported for the 21 countries of the sample. In this case, the estimated elasticity is common to all the countries. In the fourth column, the averages of the estimated individual smoothed elasticities are reported for the optimal PSTR model. These values correspond to the mean of the individual estimates:

e g i = 1 T T X t=1 e g it = 1 T T X t=1 @y it @g it ; 8i = 1; ::; N; (18)
e n i = 1 T T X t=1 e n it = 1 T T X t=1 @y it @n it ; 8i = 1; ::; N; (19)
where the individual time varying elasticities are given by equations (11).

Insert Table 5. Output Elasticities of the Public Capital Stocks.

Insert Table 6. Output Elasticities of the Labor Input

When OLS is used to estimate the parameters of the augmented production function country by country, we …nd exactly the same nonsensical values as those reported in the literature; that is, when the estimated elasticity of the public capital stock is positive, its value is so important that it can not be considered as reasonable. The estimates are greater than 0:30 for Australia, Denmark, Finland, France, Germany, Norway and Switzerland. The estimated elasticity is even greater than that, for Greece.

For instance, if we assume that for most of these countries the public capital to output ratios range from 0:40 to 0:70 [START_REF] Kamps | The Dynamic Macroeconomic E¤ects of Public Capital: Theory and Evidence for OECD Countries[END_REF], these results imply estimates of the marginal product of public capital that range from 57% to 99% per year for France or from 85% to 150% for Germany. For the United States, our estimated elasticity (0:39) is equal to that obtained by [START_REF] Aschauer | Is Public Expenditure Productive?[END_REF]. As noticed by Gramlich, these original OLS estimates imply estimated rates of return on public capital that exceed 100 percent. While these "stratospheric" [START_REF] Gramlich | Infrastructure Investment: a Review Essay[END_REF] rates of return on public capital are implausibly high, such a result is a common …nding in the literature. On the contrary, the estimated coe¢cient on the public capital input is negative in four countries (Iceland, Ireland, Portugal and New Zealand), and the elasticity of the labor input is also negative in two other countries (Greece and Italy). Such a …nding is not uncommon in the literature. This was the case in some linear speci…cations used by [START_REF] Sturm | Is Public Expenditure Really Productive?[END_REF] or [START_REF] Vijverberg | Public Capital and Private Productivity[END_REF] for the United States.

As usual in the litterature (see [START_REF] Romp | Public Capital and Economic Growth: A critical Survey[END_REF] for a survey), the use of a linear panel model with individual …xed e¤ects (Holtz-Eakin 1994; Evans and

Karras, 1994) leads to more reasonable estimates. The estimated elasticity of output with respect to public capital (Table 5, second column) is then equal to 31%. The estimated elasticity of the labor capital, i.e., 48:9%; is slightly less important than what is generally considered as valid for the industrialized countries. However, these results are obtained in a panel speci…cation that ignores the heterogeneity of the production technology. The parameters of the augmented production function are assumed to be homogeneous. Given the individual-country regression results, it is evident that this homogeneity assumption is fallacious. Consequently, these within estimates correspond to a weighted average of individual heterogeneous elasticities.

In contrast, the results derived from the various PSTR models presented in this paper are economically reasonable and robust to the changes in the composition of the sample. The individual estimated elasticities of output with respect to the public capital stocks and private inputs and the corresponding rates of returns are largely more reasonable than those obtained in linear OLS models. For instance, in the United States, we found an average estimated elasticity of 6:66% under the OCRS assumption (Table 5, last column). Recall that, under the same assumption, Aschauer found an elasticity of 39%, and that, in a simple optimal growth model, the optimal public investment ratio must be equal to this elasticity. Since the historical public investment ratio over the post war period is roughly equal to 5% in the United States, our estimate implies only a slight sub-optimality of the public investment policy in this country compared to the requirement implied by the standard OLS results. In our case, for an average ratio of government capital stock to GDP equal to 59; 52%, the implied marginal rate of return on the public capital stock is equal to 11:19% per year. Similar results are obtained for the other OECD countries. The average estimated public capital elasticity ranges from 6:38% in New Zealand to 38:33% in Portugal. Except for Finland, Portugal, Belgium and Norway, the estimated elasticity is always smaller than 15%. For most of the major OECD countries, including Germany, France, Canada, United Kingdom, the estimated elasticities are roughly comparable to the historical public investment to GDP ratios. Besides, the estimated elasticities for the labor input are quite reasonable even if they are smaller than those obtained in linear panel models.

Insert Figure 2. Output Elasticities of the Public Capital Stocks

Figure 2 displays the estimated elasticities e g it of the output with respect to the public capital stock over the period 1966-2001 for the 21 countries of our sample. For most of the countries, these elasticities are quite stable over the time period. The main exceptions are obtained for Finland, Belgium, Iceland, Japan, Spain, Norway, Portugal, Sweden and Switzerland. In these countries (except for Spain), the elasticity is decreasing over time. This decrease is clearly related to a fall in the ratio of public to private capital stocks in these countries. Indeed, it has been previously mentioned that, given the negative estimated parameter 1 associated with the transition function (see Table 4), an increase of the threshold variable induces a decrease of the public capital stock elasticity. On the contrary, in the case of the United Kingdom, the recent increase in the public investments has induced a slight increase in the public capital elasticity.

Conclusion

In this paper, we propose an empirical evaluation of the threshold e¤ects in the productivity of public capital stocks in OECD countries. Our assessment is based on the estimation of various threshold panel speci…cations of public capital augmented production functions. Our main results can be summarized in two main points. First, the relationship between the output and public capital stocks is nonlinear. More precisely, strong threshold e¤ects can be identi…ed in these relationships. This conclusion is robust to changes in the speci…cations of the production function and in the threshold variable. In addition, it seems that the productivity of the public capital cannot be reduced to a small number of regimes and must be studied through a model allowing a continuum of regimes. This result reveals the importance of the heterogeneity of the economic situations of the OECD countries even in a nonlinear environment. Second, we propose individual time varying estimates of the public capital elasticities for 21 OECD countries. These estimates, issued from a variety of PSTR models, take into account both the cross-sectional heterogeneity and the threshold e¤ects in the production technology. These estimates con…rm the in ‡uence of the public capital stocks on the productivity of other factors. However, the "stratospheric" estimates of the productivity sometimes reported in the empirical literature disappear when these threshold e¤ects are controlled. However, our approach may su¤er from two main limits. The …rst one is the non stationarity of the data used in the augmented production function. In a linear context, several empirical studies, mainly conducted on American time series [START_REF] Tatom | Public Capital and Private Sector Performance[END_REF][START_REF] Sturm | Is Public Expenditure Really Productive?[END_REF][START_REF] Crowder | Balanced Growth and Public Capital: an Empirical Analysis[END_REF] have highlighted the fact that these are not stationary and not cointegrated, i.e. that the total factor productivity is a non stationary process. Thus, the issue is to transpose these conclusions in a non linear panel context: it is possible to test the non stationarity using panel unit root tests. But, to the best of our knowledge none panel cointegration test has been yet

proposed against an alternative of non linear model. Such a testing procedure is largely beyond the scope of this paper.

The second limit is due to the potential endogeneity and reverse causality. But, in order to properly test for weak exogeneity, it would be necessary to consider a multivariate nonlinear framework as in Jansen and Teräsvirta (1996). While a single equation non-linear model may lack e¢ciency if weak exogeneity does not hold, it has the great advantage of avoiding the speci…cation of additional non-linear equations and possible misspeci…cations that would a¤ect estimation of all equations in the system. But, instrumental variable methods are not actually available in a non linear panel context. We only can note that our estimates of elasticities are much more reasonable than those previously published (in a linear context with potential endogeneity bias) with for example an average elasticity of 6.6% for the United States where the public investment ratio is roughly equal to 5%.

A Appendix

A.1 Estimation of Parameters

Let us consider a production function under PFCRS with one threshold function:

y it k it = i + 0 0 W it + 0 1 W it h (q it ; ; c) + " it ; (20)
where j = j j 0 for j = f0; 1g and W it = [(n it k it ) g it ] 0 : The estimation of the parameters is carried out in two steps. In the …rst step, the individual e¤ects i are eliminated by removing individual-speci…c means to the variables of the model. This …rst step is standard in linear models (within transformation), but it requires more careful treatment in the context of a threshold model. Let us denote e y it = y it k it y i k i and e " it = " it " i : The explanatory variables must be transformed as follows. The vector W it is simply transformed as f

W it = W it W i where W i = n i k i g i 0 :
But the transformed explanatory variables in the second regime depends on the parameters and c of the transition function since:

e Z it ( ; c) = W it h (q it ; ; c) Z i ( ; c) ; (21)
Z i ( ; c) = 1 T T X t=1 W it h (q it ; ; c) ; (22)
Consequently, the matrix of transformed explanatory variables denoted x it ( ; c) = h f W 0

it : e Z 0 it ( ; c) i 0 depends on the parameters of the transition function. So, it has to be recomputed at each iteration. More precisely, given a couple ( ; c), the elasticities of the production function in the extreme regimes can be estimated by ordinary least squares, which yields:

b ( ; c) = " N X i=1 T X t=1 x it ( ; c) x it ( ; c) 0 # 1 " N X i=1 T X t=1 x it ( ; c) e y it # ; where b ( ; c) = h b 0 0 ( ; c) b 0 1 ( ; c)
However, the convergence issue of this estimation procedure is greatly dependant upon the chosen starting values of and c. This is normally done by mean of a grid search, i.e. a selection of initial values for the slopes j and the location parameters c j , j = 1; : : : ; r. Given these grids, OLS regressions are performed for all combinations of the initial values to estimate the corresponding and . The vector for which the residual sum of squares is minimum is then passed as a starting value for the realization of the second step of the estimation process described at the preceding point. Details on the choice of initial conditions can be found in Gonzalez, Teräsvirta and Van Dijk (2005).

A.2 First-Order Taylor Expansion of the Transition Function

Let us consider the …rst-order Taylor expansion around = 0 of the function y it k it = i + 0 0 W it + 0 1 W it h (q it ; ; c) + " it in the case m = 1: For simplicity, we consider the case in which the threshold variable q it is di¤erent from the explanatory variables.

y it k it = i + 0 0 W it + 0 1 W it 1 2 c 1 4 + 0 1 W it 4 q it + " it (25)
So, the …rst-order Taylor expansion depends only on q it since m = 1 and the parameter associated to q it is a multiple of the slope parameter : When m = 2; this …rst-order Taylor expansion is de…ned as:

y it k it = i + 0 0 + 1 2 0 1 + c 1 c 2 4 0 1 W it 4 0 1 W it (c 1 + c 2 ) q it + 4 0 1 W it q 2 it + " it (26)
This expression depends on q it and q 2 it ; and the corresponding parameters vectors 1 and 2 depend on the slope parameter : Notes: For each speci…cation, the corresponding optimal number of thresholds, denoted r , is determined according to a sequential procedure based on the LMF statistics of the hypothesis of non remaining nonlinearity. The total number of parameters is determined by the formula 2 (r + 1) + 2 r . Notes: The Model A corresponds to the threshold variable gi;t 1 ki;t 1 and the Model B to the threshold variable ki;t 1 ni;t 1: The standard errors in parentheses are corrected for heteroskedasticity. For each model, the number of transition functions r is determined by a sequential testing procedure (see Table 2). For the j th transition function, with j = 1; ::r;, the estimated location parameter cj and the corresponding estimated slope parameter j are reported. The PSTR parameters can not be directly interpreted as elasticities.

B References

  private and public investments (Codes: IBV and IGV, OECD AnalyticalDatabase, 2002). One of the main advantages of these data is the international comparability of the de…nition used for the public sector. Indeed, in panel studies based on national sources, the de…nition of the public sector underlying the investments or stocks series generally varies across countries. This lack of international comparability may have strong consequences when considering the threshold e¤ects based on the level of the public capital stocks. On the contrary, the estimated net stocks proposed by Kamps are based on public investments series issued from a homogenous de…nition for all the countries of the panel 5 . So, in our study, the stocks correspond to the general government sector including the central, local and state government subsectors. Therefore, the estimated series of public capital stock corresponds to the government net capital stock in volume. Similarly, the estimated private capital stock corresponds to the private non-residential stock in volume. Finally, the data for the real GDP are taken from the OECD Analytical Database (Code: GDPV), and the data for the total employment are issued from the OECD Economic Outlook (Code: ET). All the data, except employment, are expressed in 1995 prices and in billions of national currency units.

  the values ( ; c) : In a second step, conditionally to b ( ; c) ; the parameters of the transition function and c are estimated by NLS according to the program: (b ; b c) = ArgM in b and b c; it is possible to estimate the elasticities of the production function in the extreme regimes: b j b j 0 = b 0 j (b ; b c) ; j = 0; 1;

Figure 1 :

 1 Figure 1: Estimated transition function (c=0) for the OCRS speci…cation

Figures

  Figures 2. Output Elasticities of the Public Capital Stocks.Individual PSTR Estimates under OCRS

  dependent variable is log GDP per unit of private capital stock. All variables are expressed in logs. The t-statistics are in parenthesis. The parameter and respectively denote the elasticities of output with respect to the labor and the public capital stock. The PFCRS speci…cation corresponds to an assumption of Private Factors Constant Returns to Scale. The OCRS speci…cation corresponds to the assumption of Overall Constant Returns to Scale.

Table 1 .

 1 2. Output Elasticities of the Public Capital Stocks. Individual PSTR Estimates under OCRS (1966-2001) Public Capital Augmented Production Function 7 Linear Panel Models with Fixed E¤ects and Year Dummies

		0 .4									0.45
									Australia								Denmark
									Au s tr ia									Finland
		0.35							Belgium Canada	0 .4							France Germany
											0.35
		0 .3
	Elasticity	0.25								Elasticity	0 .3
	Public Capital	0 .2								Public Capital	0 .2 0.25
		0.15
											0.15
		0 .1									0 .1
		1965 0.05	1970	1975	1980	1985	1990	1995	2000	2005	1965 0.05	1970	1975	1980	1985	1990	1995	2000	2005
		0.18									0.45
									Greece								Japan
									Iceland								New Zealand
									Ireland Ita ly	0 .4							Norway Portugal
		0.16
											0.35
		0.14
	Elasticity									Elasticity	0 .3
	Capital	0.12								Capital	0.25
	Public									Public	0 .2
		0 .1
											0.15
		0.08
											0 .1
		1965 0.06	1970	1975	1980	1985	1990	1995	2000	2005	1965 0.05	1970	1975	1980	1985	1990	1995	2000	2005
		0.15
									Spain
									Sweden
		0.14							United Kingdom
									United States
		0.13
		0.12
	Elasticity	0.11
	Public Capital	0 .1
		0.09
		0.08
		0.07
		1965 0.06	1970	1975	1980	1985	1990	1995	2000	2005

Table 3 .

 3 Summary Statistics 9

	Returns: PFCRS		Model A	Model B
	Threshold variable	g i;t 1 k i;t 1 k i;t 1 n i;t 1
	Optimal Number of Transition Functions r		0	2
	Residual Sum of Squares		4:296	3:135
	Number of Parameters		2	10
	AIC Criterion		5:161	5:444
	Schwarz Criterion		5:148	5:382
	Returns: OCRS		Model A	Model B
	Threshold variable g Optimal Number of Transition Functions r	1	3
	Residual Sum of Squares		3:982	3:607
	Number of Parameters		6	14
	AIC Criterion		5:221	5:288
	Schwarz Criterion		5:184	5:202

i;t 1 k i;t 1 k i;t 1 n i;t 1

Table 5 .

 5 Output Elasticities of the Public Capital Stocks Average of Individual PSTR Estimates under OCRS 11

	Model	OLS-Linear Within	PSTR
	Threshold Variable	None	None	Model A
	Australia	0:439 (4:33)	0:314 (16:8)	0:136 (1:31)

Hence, for the US economy, Aschauer gives estimates for the return on public stock varying between 60% and 80%, and such values have been considered too large to be credible by[START_REF] Gramlich | Infrastructure Investment: a Review Essay[END_REF].

See Granger and Teräsvirta (1993), Teräsvirta (1998) orVan Dijk, Franses and Teräsvirta (2002) for a survey on the STAR models.

All the corresponding codes have been simultaneously developed under WinRats and Matlab 7.0 and are available on our websites.

Three statistics have been proposed by Gonzàlez and al.: an LM one, an F-statistic and a pseudo-LRT. Since the F-version of the test has better size properties in a small sample than the asymptotic 2 -based statistic (Dijk et al., 2002), we only report the results based on the F-version. Other results are available upon request.

Expect for Japan and the United States, for which the stocks correspond to the public sector, including the general government and non…nancial public corporations.

We only report the individual elasticities obtained under the OCRS assumption. The elasticities obtained under PFCRS are available upon request.

Notes: For each model (i.e. for each threshold variable), the testing procedure works as follows. First, test a linear model (r = 0) against a model with one threhosld (r = 1) : If the null hypothesis is rejected, test the single threshold model against a double threshold model (r = 2). The procedure is continued until the hypothesis no additional threhold is not rejected. The corresponding LMF statistic has an asymptotic F [2; T N N 2(r + 1) ] distribution under H0: The corresponding pvalues are reported in parentheses. Notes: For each country, the results of the individual-country regressions with a linear time trend (OLS-Linear) and the panel linear model (Within) are reported. The corresponding t-statistics are in parenthesis. For the PSTR models, the …gures in parenthesis correspond to the standard errors of the individual estimates expressed in percent. The Model A corresponds to the threshold variable gi;t 1 ki;t 1 and the Model B to the threshold variable ki;t 1 ni;t 1 33 Notes: For each country, the results of the individual-country regressions with a linear time trend (OLS-Linear) and the panel linear model (Within) are reported. The corresponding t-statistics are in parenthesis. For the PSTR models, the …gures in parenthesis correspond to the standard errors of the individual estimates expressed in percent. The Model A corresponds to the threshold variable gi;t 1 ki;t 1 and the Model B to the threshold variable ki;t 1 ni;t 1:


Earthquake fault heterogeneity is often studied on a set of subfaults in kinematic inversion, while it is sometimes described with spatially localized geometry. Aochi and Ide (EPS, 2011) and Ide and Aochi (submitted to Pageoph and AGU, 2012) apply a concept of multi-scale heterogeneity to simulate the dynamic rupture process of the 2011 Tohoku-oki earthquake, introducing circular patches of different dimension in fault fracture energy distribution. Previously the patches are given by the past moderate earthquakes in this region, and this seems to be consistent with the evolution process of this mega earthquake, although a few patches, in particular, the largest patch, had not been known previously. In this study, we try to identify patches by inversion. As demonstrated in several earthquakes including the 2010 Maule (M8.8) earthquake, it is possible to indentify two asperities of ellipse kinematically or dynamically (e.g. Ruiz and Madariaga, 2011, and so on). In the successful examples, different asperities are rather visible, separated in space. However the Tohoku-oki earthquake has hierarchical structure of heterogeneity.

We apply the Genetic Algorithm to inverse the model parameters from the ground motions (K-net and Kik-net from NIED) and the high sampling GPS (GSI). Starting from low frequency ranges (> 50 seconds), we obtain an ellipse corresponding to M9 event located around the hypocenter, coherent with the previous result by Madariaga et al. (pers. comm.). However it is difficult to identify the second smaller with few constraints. This is mainly because the largest covers the entire rupture area and any smaller patch improves the fitting only for the closer stations. Again, this needs to introduce the multi-scale concept in inversion procedure. Instead of finding the largest one at first, we have to start to extract rather smaller moderate patches from the beginning of the record, following the rupture process.