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## The Feldstein-Horioka Puzzle: a Panel Smooth Transition Regression Approach

### BibTeX

@MISC{Tesar_thefeldstein-horioka,

author = {; Tesar and Baxter and ; Crucini and Coakley},

title = {The Feldstein-Horioka Puzzle: a Panel Smooth Transition Regression Approach},

year = {}

}

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### Abstract

Abstract This paper proposes an original framework to determine the relative influence of five factors on Feldstein and Horioka outcome of a high saving-investment association among OECD countries. Based on panel threshold regression models, we derive country-specific and time-specific saving retention coefficients for 24 OECD countries over 1960-2000. These coefficients are assumed to change smoothly as a function of five threshold variables considered as the most important in the literature devoted to the Feldstein and Horioka puzzle. The results show that degrees of openness, country sizes and ratios of current account to GDP have the greatest influence on investment and saving relationship. based on a two steps approach, using five-year averages over the period 1965the period to 1989the period , Taylor (1994 shows that when the FH cross-section regression of the ratio of domestic investment to GDP on the ratio of domestic savings to GDP is controlled for domestic relative prices, age structure of the population and the interaction between the dependency rate and the rate of growth, the standard correlation of saving and investment disappears. Regarding size of countries, A third method consists in using time-series data (or panel data) instead of cross-section averages. Thus, with annual data, OECD countries over the period 1962 to 1990. In this context, the main issue is the cross-country heterogeneity of the FH coefficients, even for OECD countries as highlighted by The purpose of this paper is to propose an original and unified framework to determine the relative influence of five factors on the saving retention coef-3 Feldstein and Bachetta (1991) and 3 ficient, while explaining cross-country heterogeneity and time variability. For that, we propose to test the existence of threshold effects in the relationships between the ratio of national investment to GDP and the ratio of national savings to GDP on a panel of OECD countries. Indeed, the idea that capital mobility depends on other exogenous variable (openness, size, demography etc.) clearly corresponds to the definition of a threshold regression model: "threshold regression models specify that individual observations can be divided into classes based on the value of an observed variable" (Hansen, 1999, page 346). (2005). This threshold regression model authorises a smooth transition mechanism between regimes. Our approach has two main advantages. Firstly, based on the PSTR estimates, we derive FH parameters (i.e. the saving retention coefficients) that vary between countries but also with time. So, our approach provides a simple parametric approach to capture both cross-country heterogeneity and time instability of the saving retention coefficients 4 . Secondly, our approach allows the country-specific FH saving retention coefficients to change smoothly as a function of the threshold variable. Consequently, in such model it is possible (i) to test the existence of thresholds effects and (ii) to evaluate the influence of the five potential threshold variables previously mentioned on saving retention coefficient by comparing the estimated FH parameters in the different regimes. The rest of the paper is organized as follows. In the next section, we discuss the threshold specification of FH regression and particularly, the crosscountry heterogeneity and the time variability of saving retention coefficients. The Feldstein-Horioka puzzle: Toward a Threshold Specification The basis of our empirical approach is exactly the same as that used by many authors since the seminal paper of where I it is the ratio of domestic investment to GDP observed for the i th country at time t, S it is the ratio of domestic savings to GDP and α i denotes an individual fixed effect. The residual ε it is assumed to be i.i.d. (0, σ 2 ε ). Used in particular by Besides, when one comes to include these factors in the regression (1) as additional explanatory variables, it does not solve the problem: the conditional relationship between investment and saving is assumed to be homogeneous as long as β i is common for all i. Secondly, the equation where q it denotes threshold variables, c a threshold parameter and where the 5 For a presentation of Random Coefficients Models, see In these two extreme regimes model, the FH coefficient is equal to β 0 if the threshold variable is smaller than c and is equal to β 0 + β 1 if the threshold variable is larger than c. However, this PTR model imposes that the value of saving retention coefficient can be divided into a small number of classes. Such an assumption may be unrealistic even for OECD countries and, at least, must be tested. Besides, the transition mechanism between the regimes is too simple to allow interesting non linear effects of capital mobility. As usual in the literature, the solution to these problems consists in using In this case, the transition function is a continuous and bounded function of the threshold variable. Gonzàlez et al. (2005), following the work of Granger and Teräsvirta (1993) for the time series STAR models, consider the following logistic transition function: The vector c with by definition of the transition function, The PSTR model can be generalized to r + 1 extreme regimes as following: where the r transition functions g j (q it , γ j , c j ) depend on the slope parameters γ j and on m location parameters c j . In this generalization, if the threshold variable q it is different from S it , the coefficient for the i th country at time t is defined by the weighted average of the r + 1 parameters β j associated to the r + 1 extreme regimes: The expression of the coefficient is slightly different if the threshold variable q it is a function of the ratio of domestic savings to GDP. For instance, if we assume that the threshold variable corresponds to the ratio of domestic savings to GDP, i.e. q it = S it , the expression of the FH coefficient is then defined as: Such an expression authorizes a variety of configurations for the relationships between the ratio of domestic investment to GDP and the ratio of domestic savings to GDP as we will discuss in the next part. 3 Estimation and Specification Tests The estimation of the parameters of the PSTR model consists in eliminating the individual effects α i by removing individual-specific means and then in applying non linear least squares to the transformed model (see Gonzàlez et al., 2005) or Colletaz and Hurlin In this first-order Taylor expansion, the parameters θ i are proportional to the slope parameter γ of the transition function. Under the null hypothesis, the F-statistic has an approximate versus there is at least two transition functions (H 0 : r = 2). Let us assume that the model with r = 2 is defined as: The logic of the test consists in replacing the second transition function by its first-order Taylor expansion around γ 2 and then in testing linear constraints on the parameters. If we use the first-order Taylor approximation of g 2 (q it , γ 2 , c 2 ), the model becomes: and the test of no remaining nonlinearity is simply defined by But, while ageing decreases investment, the younger a population, the more investment it makes. Thus, raising youth dependency is expected to decrease the saving retention coefficient, whereas it will increase with ageing. Finally, we consider (Model F) the ratio of current account to GDP as the threshold to choose a particular model among the six models proposed, the "optimal" model would correspond to this model. Indeed, as suggested by Gonzàlez et al. (2005), the "optimal" threshold variable corresponds to the variable which leads to the strongest rejection of the linearity hypothesis. The specification tests of no remaining non-linearity (see Finally, in the PSTR model, it is necessary to choose the number of location parameters used in the transition functions, that is the value of m. In We estimate the PSTR models for each potential specification m, r(m), and report the number of parameters and the residual sum of squares. We suggest here the use of a Schwarz information criteria in order to choose a benchmark specification for each specification of the saving retention coefficient. Consequently, we consider the specification with m = 1 and r = 1 as optimal for the model A, B, D, E, and the specification with m = 2 and r = 2 as optimal for the model C. growth, the larger the country, the younger the population, the higher is the saving retention coefficient (models A, C and D) and thus the less capital is mobile. On the contrary, we confirm that openness, ageing and current account balances tend to decrease FH coefficient (models B, E and F) and thus to signal greater international mobility of capital. However, the main difference with the previous studies is that our model makes it possible to asses the relative quantitative importance of these various variables on capital mobility. of the model C (threshold equal to size) are more surprising, but they confirm Ho's results. When the relative size of the country is inferior to 4%, the estimated FH coefficient is found to be roughly equal to 0.15, but when the size exceeds this threshold the FH is roughly equal to 0.70 (as in homogenous models in panel). As for dependency ratios, our outcomes confirm the intuition. The more important the share of young in population, the higher the FH coefficient, as Brooks PSTR Estimates of Capital Mobility 17 The average estimated FH parameters 16 are reported in It is important to note that when the threshold variable is not well chosen, it implies associating countries according to fallacious criteria. Consequently, at each date the countries are split into a small number of randomly constituted groups and associated with different slope parameters, according to the value of the fallacious threshold variable. Therefore, the estimated slope parameters obtained in this context on random groups shouldn't be different from those estimated for the whole sample. However, here we notice that this interpretation is not justified. The individual estimated parameters obtained are different from the within coefficient in linear panel. In addition, this conclusion is reinforced by the fact that the linearity tests lead to a strong rejection of the linearity. These threshold variables make it possible to take into account the unstability of the FH coefficient but not of the cross section heterogeneity. This result is confirmed by the high individual standard errors. We obtained similar conclusion for the ratio of current account used in threshold variable (Model F). PSTR Estimates and Endogeneity In order to assess the robustness of our PSTR estimates to potential endogeneity biases, we propose to consider an instrumental variable (IV) extension of the estimation method generally used in this context. Recall that the parameters of a PSTR model are estimated with non linear least squares. For a given threshold parameter and a given value of the threshold variable, the model is linear and the IV estimator can be adapted in order to take into account the potential endogeneity of savings. Let us consider a simple PSTR model with one transition function (r = 1) : The estimation of the parameters is carried out in two steps. In the first step, Consequently, the matrix of transformed explanatory variables x * it (γ, c) = S it :w it (γ, c) and the matrix of instrumental variables η * it (γ, c) = [z itζit (γ, c)] depend on the parameters of the transition function. So, it has to be recomputed at each iteration. More precisely, given a couple (γ, c), the FH parameter can be estimated by IV, which yields: . In a second step, conditionally toβ IV (γ, c), the parameters of the transition function γ and c are estimated by NLS according to the program: Given γ and c, it is then possible to estimate the elasticities of the production function in the extreme regimes as follows:β IV =β IV (γ,ĉ) . In Conclusion In this paper we propose an empirical evaluation of the influence of various threshold variables on the saving-retention coefficient. This assessment is based on a Panel Smooth Transition Regression Model. Our main results can be summarized in three main points. Firstly the relationship between domestic investment and saving is non linear and strong threshold effect can be identified. This conclusion is robust to changes in the threshold variable and to potential endogeneity biases. More precisely, we found that three variables have the greatest influence on capital mobility: the degree of openness, the size of the country and the ratio of current account to GDP. In addition, for five out of six models, it seems that the saving retention coefficient 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 strong heterogeneity in the degree of mobility of OECD countries. Secondly in line with the literature, we can observe that the estimated FH parameters are decreasing between 1960 and 2000 for most of the countries of our sample. Thirdly, we propose an original method for specifying the heterogeneity and the time variability of the FH coefficient. Notes: For each model (each specification), the optimal number of locations parameters used in the transitions functions can be determined as follows. For each value of m, the corresponding optimal number of thresholds, denoted r * (m), is determined according to a sequential procedure based on the LMF statistics of the hypothesis of non remaining nonlinearity. Thus, for each couple (m, r * ), the value the RSS of the model is reported. The total number of parameters is determined by the formula K(r * + 1) + r * (m + 1) , where K denotes the number of explicative variables, i.e. K = 1 in our specifications. Notes: The standard errors in parentheses are corrected for heteroskedasticity. For each model and each value of m the number of transition functions r is determined by a sequential testing procedure (see 21 23 24