GTVD : A RATS procedure to estimate Smooth Transition Regressions with Panel Data

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2 GTVD : A RATS procedure to estimate Smooth Transition Regressions with Panel Data G. Colletaz Laboratoire d Economie d Orléans, Université d Orléans, France gilbert.colletaz@univ-orleans.fr 1 Thanks to Christophe Hurlin for numerous discussions. Interested Matlab users may ask him for a practically equivalent program which runs under this software.

3 Abstract GTVD.SRC can be used with RATS software to estimate Smooth Transition Regressions with Panel Data following the recommendations given by Gonzáles, Teräsvirta and van Dick (2004). This paper exposes the model, the estimation and testing schemes and documents the various options actually available.

4 Chapter 1 The Panel Smooth Transition Regression model 1.1 The estimated structure The basic PSTR model is of the form: y it = µ i + β 0x it + β 1x it g(q it ; γ; c) + u it (1.1) for i = 1,..., N and t = 1,..., T. y it is a scalar, µ i is an unobservable timeinvariant regressor, x it is a k-dimensional vector of time-varying exogenous variables, q it is an observable transition variable and u it are the residuals. g() is a continuous bounded function of q it defined by: g(q it ; γ; c) = (1 + exp( γ m (q it c j ))) 1, γ > 0, c 1... c m (1.2) j=1 where c = (c 1,..., c m ) is a m-dimensional vector of location parameters, γ is the slope of the transition function, and γ > 0, c 1... c m are identification restrictions. The version that can be estimated via the proc PSTR is a generalization of this basic structure. It allows for r transition functions based on eventually different transition variables each with their owned vector of location parameters. Moreover, it is also possible to consider exogenous variables with constant coefficients α 0. This additive generalization has the following form: y it = µ i + β 0x it + r j=1 β jx it g j (q (j) it ; γ j; c j ) + α 0z it + u it (1.3) 1

5 and, m g j (q j it ; γ j j; c j ) = (1 + exp( γ j k=1 (q (j) it c j,k ))) 1 (1.4) 1.2 Estimation of parameters Details can be found in the paper by Gonzáles, Teräsvirta and van Dick. Here we keep the description of the estimation phase to a minimum that is sufficient to run the proc. However, reading the cited paper is a necessary condition to understand exactly what you are doing (and in particular to have a knowledge of the maintained hypothesis under which this estimation is supposed to be valid). Estimation is conducted in 2 steps : elimination of the individual effects and successive applications of OLS and NLLS to the transformed variables. 1. removing the fixed effects suppression of the µ i is done classically by centering the variables on their individual means. 2. estimation of remaining parameters with NLLS If the slopes of the transition functions, γ j, and the location parameters, c j, enter non linearly in the model this is not the case for the coefficients β and α. Thus for given values of γ j and c j an OLS regression gives estimates of β and α. With these estimates we can then search for the values of γ j and c j that minimize the residual sum of squares using a NLLS algorithm (here we use the bfgs algorithm available with the RATS instruction find). The process is repeated until convergence for the whole vector of parameters. However the convergence issue is greatly dependant upon the chosen starting values of β and α. 1.3 Choosing the starting values This is normally done by mean of a grid search, i.e. a selection of initial values for the slopes γ j and the location parameters c j, j = 1,..., r. 1. initial values for γ j User simply provides a list of possible positive values. 2. initial values for c j User selects a number of values to fill the vector c j such that min(c j,k ) > min(q (j) it ) and max(c j,k) < max(q (j) it ) for k = 1,..., m j and j = 1,..., r. To ensure that a sufficient number of observations of the transition variable are available for estimation in each side of a given location parameter, the values of min(c j,k ) and max(c j,k ) are constrained by a trimming over the observations of q (j) it, i.e. the range for the initial c j,k leaves aside a certain number of smaller and greater observations of the transition variable. 2

6 Given these grids, OLS regressions are performed for all combinations of the initial values to estimate the corresponding β and α. The vector for which the residual sum of squares is minimal is then passed as starting value for the realization of the second step of the estimation process described at the preceding point. In case of difficulties it may be a good idea to increase the number of grid values. In any case it is certainly a good idea to try different starting values choosing different specifications for the construction of these grid. Note also that if the convergence is not obtained at the end of this process, then the proc PSTR switch automatically to a simplex optimization in order to obtain new starting values for the NLLS algorithm. 1.4 Tests of linearity against PSTR 1. Before estimation of any nonlinear form If we start with the basic structure of the PSTR model given by (1.1) then a linear model is obtained by imposing γ = 0 or β 1 = 0. Whatever the chosen solution, the test will then be nonstandard due to the presence of unidentified parameters under the null. One solution adopted by Gonzáles, Teräsvirta and van Dick (2004) is to replace g() by a first-order Taylor expansion around γ = 0. This leads to an auxiliary regression: y it = µ i + β 0 x it + β 1 x it q it + + β mx it q m it + u it (1.5) Testing γ = 0 in (1.1 is then equivalent to testing β 1 = beta m = 0 in (1.5). If we compute the residual sum of squares of (1.5) estimated by OLS (after the individual-specific means have been remove ), SSR 0, and the residual sum of squares given by estimation of (1.1) then two tests can be computed : a LM one and its F-version: LM = T N(SSR 0 SSR 1 )/SSR 0 (1.6) F = (SSR 0 SSR 1 )/mk/ssr 1 /(T N N mk) (1.7) which under the null have respectively χ 2 mk and F (mk, T N N mk) distributions. In addition to these LM and F tests, the proc PSTR compute a pseudo-lrt given by : pseudo LRT = 2log(SSR 1 /SSR 0 ) (1.8) having a χ 2 mk distribution under the null hypothesis. 2. After estimation of at least one transition function The question is then the existence of remaining nonlinearity. We follow GTvD and illustrate the construction of the test with an example. Suppose 3

7 that you estimate a model with one transition function (r=1 in equation (1.3)): y it = µ i + β 0x it + β 1x it g 1 (q (1) it ; γ 1; c 1 ) + α 0z it + u it (1.9) and that you now think about the addition of a second one. The model to be considered is then: y it = µ i + β 0x it + β 1x it g 1 (q (1) it ; γ 1; c 1 ) + β 2x it g 2 (q (2) it ; γ 2; c 2 ) + α 0z it + u it (1.10) where eventually q (1) it = q (2) it. As before, the test consider the slope of the second transition function and the null hypothesis is then H0 : γ 2 = 0. Again a first-order Taylor expansion around γ 2 = 0 leads to an auxiliary regression: y it = µ i +β 0 x it +β 1x it g 1 (q (1) it ; γ 1; c 1 )+β 21x it q it + +β 2mx it q m it +α 0z it +u it (1.11) With this approximation the test is now H0 : β 21 =... = β 2m = 0 Based on the residual sum of squares given by the estimations of (1.11) and (1.9) one can construct LM, F tests as explained before. Note however that in the paper by Gonzáles, Teräsvirta and van Dick the estimation of (1.11) is made conditionally upon the values of γ 1 and c 1 given by the adjustment of (1.9). The procedure PSTR gives the results so obtained, but it also gives those obtained after a unconstrained re-estimation of (1.11) allowing in particular new optimal estimates for γ 1 and c 1. It also prints a pseudo-lrt statistics. 4

8 Chapter 2 Instructions This part lists the various options that can be specified and the syntax of the call to the proc. It also gives some calling examples. 2.1 Calling the procedure Data must be described using instructions allocate and calendar according to the panel data structure of RATS. Call to the procedure is then done with a standard syntax : proc gtvd depvar #x1 x2 #q1 q2 #z1 z2 Where depvar is the name of the dependent variable, x1 x2 are the name of the exogenous variables upon which the transition functions are active, q1 q2 are the name of the transition variables and z1 z2 are exogenous variables with constant coefficients. 2.2 Specifying the transitions The number of transition functions, r, is equal to the number of transition variables. Hence #q1 q1 defines two functions each having q1 as argument transition. m = m is a vector having r components giving the number of restrictions to 5

9 be put on the transition functions. For example m= 3, 2 applies to two transition functions, the first one having 3 restrictions and the second having 2 restrictions. In this case we must have two transition variables in the call to the proc. Note that it is obligatory to specify the contents of this vector. rc = rc specifies the number of transition functions that must be incorporated in one unique estimated model. If rc is not specified then the proc implements a sequential treatment. For example #q1 q2 asks for the estimation of a model having q1 as transition variable and then for the estimation of a model having q1 and q2 as transition variables. In this case the number of transition variables must be equal to the dimension of the vector m. If rc is specified then only the model with rc transition variables is estimated and the dimension of vector m must be at least rc. If the number of transition variables specified in the calling is greater than rc then linearity tests versus PSTR are realized within the estimated model and this is done successively for each variable in excess in the list (these tests are realized only conditionally to the value of the option mlin, see below). If rc is used then its value must be strictly positive. 2.3 Specific exogenous variables exoz = If exogenous variables of type z are present, i.e. if some independent variables having constant coefficients are used then the user must indicate exoz = 1. In this case their names are read on the third supplementary card. If exoz = 0 or if exoz is not specified then this third card must not be present in the call to the proc. 2.4 estimation : initial values of parameters and trimming Given the slopes of the transition functions and the location parameters, others coefficients are estimated by minimizing the sum of squared residuals. The values that minimize the overall sum of squared residuals are used as startingvalues of the estimation algorithm based on BFGS. Following options are used to specify array of numbers for slopes and location parameters upon which the search of these starting values is undertaken. methodini = simplex / [genetic] / grid This option specifies the way the proc find optimal initial values of various parameters before switching the a bfgs algorithm. With methodini=grid initial values for the slopes and the restriction parameters are taken in 6

10 grids defined by options grilleg and nbvalinic. In this case all combinations of these values are tried in order to find those associated with the minimal Residuals Sum of Squares. Even if the proc takes into account the ordered structure of the restriction parameters, this can lead to an enormous number of initial regressions and can be very much time consuming. When methodini=simplex or methodini=genetic, only one value of the slope parameter is considered for each of the transition functions, values of the restrictions parameters being always given by the option nbvalinic: Given the slope of the functions, the proc does initial regressions in order to find optimal initial restrictions parameters, but the number of the initial regressions is gretly reduces, and then switch to a simplex or a genetic algorithm to find initial values of parameters passed to the bfgs algorithm. grilleg = Vector specifying an array of positive values for the slopes of the transition functions. Use of these values depends on the method specified for finding optimal initial values before switching to the bfgs optimisation, i.e., on the choice made for the option methodini. When methodini=grid the same vector is used for each of these functions and all its components are successively tried. Default values are, without any particular justification: grilleg = The number of grid values is determined by the dimension of the vector and thus can be arbitrarily fixed by the user. For example, grilleg = search for a starting value in a list of only three potential numbers before going to a bfgs optimization. When methodini=simplex or methodini=genetic the dimension of grilleg is simply equal to the number of transition functions and its ith component is the initial value of the slope parameter for the ith transition function. By default, grilleg is taken in this case, again without any particular justification, as a random realization of pseudo-uniform numbers over [0.2,3.5]. nbvalinic = Integer that gives the number initial values to be considered in the grid for each restriction parameters. Values themselves are taken at regular intervals among the observations of the corresponding transition variable. Observations corresponding to trimc and (1 trimc) are selected so that nbvalinic must be greater or equal to 2. The default value corresponds to nbvalinic =10. trimc = It gives the percentage of observations used for the trimming applied to the transition variables. The default value is trimc=.15. For example, this default value consider initial values for the restriction parameter from the 15%-tile (included) to the 85%-tile (included) of the corresponding transition variable, and in this range the proc takes nbvalinic 2 other values with equal spacing. 7

11 piters = This integer indicates the number of iterations to be realized with the simplex or the genetic algorithm chosen according to the option methodini. By default, piters = 6. If the simplex or the genetic method converges then its solutions are taken as initial values for a bfgs optimization. 2.5 linearity tests and tests of parameter constancy mlin = Integer which gives the maximum power to be considered in the auxiliary regression derived from a Taylor expansion of the initial representation. mlin is the value of the parameter m in equations (1.5) or (1.11). This Taylor development is made for the last transition variable of the adjusted equation. Fox example, if rc is not specified, then at the first step it corresponds to the first variable in the list q1 q2. If mlin = 0 (which is the default value) then the test of linearity against PSTR is not realized. mh = Integer which gives the maximum power to be considered in the auxiliary regression used to test the parameter constancy against the hypothesis of a smooth change from one regime to another (corresponding to the socalled TV-PSTR model). If mh = 0 (which is the default value) then this test is not done. graphd = Yes / [No] If graphd = Yes, the proc evaluates the derivatives relative to variables x1 x2 and plots the results. Actually formulas suppose that none of these variables is used as transition variables, i.e. x i q j i, j. Results are given for each individual at any point of time. Moreover the proc prints, for each individual, the mean of these derivatives over all time periods. name = Specifies the name of every individual in the construction of the preceeding graphs. The order of these names must match the order of the entry in the data file. If not specified, individuals are named by their order of apparition in this file, i.e; (1,2,3,...). 2.6 = 2 ) y #x #q 8

12 requests estimation of the model: y it = µ i + β 0 x it + β 1 x it (1 + exp( γ all options are taken their default = 1, 1, nbvalinic = 20) y #x #q q 2 (q it c j ))) 1 + u it (2.1) j=1 requests successive estimations of models: and, y it = µ i + β 0 x it + β 1 x it (1 + exp( γ(q it c 1 ))) 1 + u it (2.2) y it = µ i +β 0 x it +β 1 x it (1+exp( γ 1 (q it c 1 ))) 1 +β 2 x it (1+exp( γ 2 (q it c 2 ))) 1 +u it (2.3) Linearity tests with m = 1, 2 are realized before estimation of the first model (here redundant results are given as q is repeated) and after. Note that for the second model the grid search imposes 40, 000 OLS adjustments in order to find initial = 1, nbvalinic = 20, mh = 0, mlin = 4, exoz = 1, rc = 1) y #x #q #z requests estimation of the model: y it = µ i + β 0 x it + β 1 x it (1 + exp( γ(q it c 1 ))) 1 + α 0 z it + u it (2.4) before estimation it does tests of linearity successively for q and x. After estimation it does tests of remaining linearity for x. These tests are realized for m = 1, 2, 3, 4 in equations (1.5) or (1.11). Tests of parameters constancy are not made (the same result if we omit the option mh=0). The grid search for c 1 is made over 20 = 1, 1, nbvalinic = 5, methodini = genetic, grilleg =.8.3, rc = 2) y #x #q q requests estimation of the model: y it = µ i +β 0 x it +β 1 x it (1+exp( γ 1 (q it c 1 ))) 1 +β 2 x it (1+exp( γ 2 (q it c 2 ))) 1 +u it (2.5) 9

13 before doing a bfgs optimization, this calls for a genetic optimization for which initial values are respectively 0.8 and 0.3 for the slope parameters of the two transition functions and initial values of the restriction parameters are found among five numbers selected by means of OLS regressions. 10