Estimation of Panel Smooth Transition Regression Models - A RATS Procedure PSTR.SRC

Transcription

1 Estimation of Panel Smooth Transition Regression Models - A RATS Procedure PSTR.SRC Gilbert Colletaz February 7, 2018 Abstract This document only describes the referenced RATS program. A prerequisite for its comprehension is the reading of [1]. 1 The Model The base PSTR Model discussed in González, Teräsvirta, van Dijk, and Yang is the following: n z n x n x y it = μ i + δ k z kit + β 0k x kit + β 1k x kit g(q it ; γ, c) + u it (1) where g(q it ; θ) is a transition function having a logistic specification given by : 1 m g(q it ; γ, c) = 1 + exp γ (q it c j ), γ > 0 (2) Note that actually only one transition function g() is allowed in the above specification within the RATS procedure. It contains only one or two threshold parameters, i.e. m = 1 or m = 2 in (2). Thereafter, the explanatory variables, z 1,..., z nz will have constant coefficients over time and individuals, while the variables x 1, x 2,..., x nx enter in both the linear and nonlinear part of the model, their coefficients depending on the value of the transition function. The equation to be estimated is written more compactly as: j=1 y it = μ i + δ z it + β 0 x it + β 1 x itg(q it ; γ, c) + u it (3) 1

2 2 The Data The following structure is required for the data : each variable, i.e. y, z, x and q, has to be grouped by individuals, and each individual to be observed over T periods of time. With N individuals observed over T periods of time we have : Variable individual 1, time 1 individual 1, time 2.. individual 1, time T individual 2, time 1 individual 2, time 2. individual 2, time T individual 3, time 1 individual 3, time 2.. individual 3, time T. individual N, time 1 individual N, time 2.. individual N, time T The number of time entries, T, is fixed. The RATS code for missing values (.) must be used to pad ranges if the data are not aligned correctly (so that, even if some entries are missing, the number of lines for any individual is always T). In other words, if nindiv is the number of individuals, ntime the periods by individual, begin and end the first and last dates in the sample, the main program must be: calendar(panelobs=ntime,period) begin allocate nindiv//end where period is the periodicity of the data. 3 Parameter Estimation A within transformation is done to remove the fixed effects Initial values for cˆ 1 and cˆ 2 (if m = 2), are searched among the centiles of the variable q;more precisely, between centile(t 0 ) and centile(1 t 0 ), where t 0 is given by the option trimming=t 0 and 0 t 0 < 0.5. The 2

3 default value is t 0 = 0.0 and in this case the initials values are searched between the minimum and the maximum values of the transition variable q. Noting that the coefficient of an x k variable in (1) is β 0k + β 1k g(q it ; γ, c), i.e. a combination of its coefficients in two extreme regimes (it is β 0k when g() = 0, or β 0k + β 1k when g() = 1). In order to identify these extreme regimes, we have to constrained γ 0.0, in this case obtaining the so-called low regime when g() = 0, and the high one if g() = 1. The constraint γ 0.0 is imposed in the proc. If g() is always zero or one, then we only have one regime, i.e. a usual panel model, and of course the two extreme regimes cannot be identified. The initial default values for the slope γ are taken among the list : {0.1, 0.2, 0.3, 0.4, 0.5, 0.6,..., 3.8, 3.9, 4.0}. To input their own preferences, the user can use the option gamma=v where v is a vector of (positive) values. The optimal initial values for γ and c are those for which the tentative {γ, c} leads to the lowest RSS when estimating β 0 and β 1 in (3) with a fixed effect panel regression. These optimal initial values are then used as initial values in an iterative process which alternates a BFGS algorithm for {γ, c} and a panel data regression for {β 0, β 1 }. If m = 2 : a model with m = 1 is estimated and the estimates obtained for γ and c 1 are used to find the initial value of c 2. These three values are then passed to the iterative (BFGS/panel regression) optimization. Before switching to BFGS, some simplex iterations are done in order to improve the initial values of the parameters. The estimation of c 1 and c 2 during the BFGS optimization can be constrained depending on the value of the option cfree=. if cfree=yes, then c 1 and c 2 are freely estimated, if cfree=no, then ĉ 1 and ĉ 2 must lie between c min and c max which are the lower and upper bounds of the segment over which their initial values are searched according to the option trimming=. The standard errors reported for ˆγ, ĉ 1 and ĉ 2 are those given by the BFGS algorithm. However, if ĉ 1 or ĉ 2 is equal to c min or c max, we report a missing value for its standard error. 3

4 4 Selection of a Model Users can impose a model by choosing select=no. In this case it is mandatory to specify a value for m (m=1 or m=2). They can also let the program try finding an optimal value for the m parameter, m {1, 2}. When select=yes, a sequence of tests is done as described in [1] : 1. Starting with m=3, the following auxiliary regression is estimated : y it = μ i + δ z it + β 0 x it + β 1 x itq it β m x it q m it + u it (4) 2. An homogeneity test is done, H 0 : β 1 = β 2 = β = 0. As it is often 3 recommended for this type of test, we use a F-version of the χ 2 statistics. In fact, two F-stats are evaluated, a standard one and a robust F, based on a consistent covariance matrix estimator allowing for heteroscedasticity. If H is not rejected, this means that the linear 0 model is as good as a PSTR with m=3, and logically the PSTR story has to stop here. However, no matter what the preceding conclusion is, three other F tests are computed : 3. (a) H 03 : β 3 = 0, (b) H 02 : β 2 = 0 β 3 = 0, (c) H 01 : β 1 = 0 β 2 = β 3 = 0. We used the Wald form for all these tests. According to [1], the program selects m = 2 if the rejection of H is the strongest one, 02 and m = 1 otherwise. The significance level is that of a standard F-test or of a robust Chi-Square, depending on the choice test=f or test=chi, the second choice, activating the option robusterror, being the default one. 4. Using an auxiliary regression based on a first-order Taylor expansion of (3) with the selected m, an homogeneity test is done, using one the following specifications and tests : if m = 1 : if m = 2 : The transition variable q: y it = μ i + δ i z it + β 0 x it + β 1 x itq it + u it (5) H 0 : β 1 = 0 (6) y it = μ i + δ i z it + β 0 x it + β 1 x itq it + β 2 x itq 2 it + u it (7) H 0 : β 1 = β 2 = 0. (8) 4

5 1. Users can impose a particular transition variable simply by specifying the desired variable as candidate for explaining the transition. 2. More than one variable can be given as candidate: (a) If select=yes, the program considers successively each of these variables, and retains for the final model the one that gives the strongest rejection of the homogeneity test (6 ) or (8) depending on m. (b) If select=no, then the first variable in this list is included in the model as argument of the transition function, the other variables in the list being considered only during the test of no remaining heterogeneity (see Section 5). Note also that any explanatory variable x i, i = 1,..., n x or z i, i = 1,..., n z can be used as transition variable. Finally, with the value of m and the transition variable q imposed by the user or selected by the program, the final model (3) is estimated as described in section 3. 5 Model Evaluation 5.1 Testing no remaining heterogeneity Equation (3) with its logistic transition function (2) is a two-regime PSTR. An immediate extension is to consider the three regimes version : y it = μ i + δ z it + β 0 x it + β 1 x itg(q it ; γ 1, c 1 ) + β 2 x itg( q it ; γ 2, c 2 ) + u it where q it can be q it or another variable. Nullity of γ 2 can be tested by considering a first-order Taylor expansion of this last model around γ 2 = 0, leading to the following equation and test : y it = μ i + δ z it + β 0 x it + β 1 x itg(q it ; γ 1, c 1 ) + and, H 0 : β 1 =... = β m = 0 m j=1 β j x it q j it + u it m being the number of elements of c 2. In the program we consider m = (1, 2). If only one transition variable has been specified, then q = q. When a list of transition variables has been entered, the program will do the above test with q being successively each of these variables, including the one selected as optimal when select=yes, i.e. q. 5

6 5.2 Testing parameter constancy The program implements the test of constancy of parameters described in [1]. The idea is that under the alternative, the coefficients change smoothly over time so that we may augment (3) with a transition function depending on the time, g(t/t; γ 2, c 2 ). y it = μ i + δ z it + β 0 x it + β 1 x itg(q it ; γ 1, c 1 ) + ( β 2 x it + β 3 x itg(q it ; γ 1, c 1 ) ) g(t/t; γ 2, c 2 ) + u it In order to test H 0 : γ 2 = 0, a first-order Taylor expansion is done around γ 2 = 0 leading to the following auxiliary regression: y it = μ i + δ z it + β 0 x it + β 1 x itg + β 1 x it( t T ) + β 2 x it( t T ) β m x it( t m )t T + β m+1 x itg(q it ; γ 1, c 1 )( t T ) + β m+2 x itg(q it ; γ 1, c 1 )( t T ) β 2 m x itg(q it ; γ 1, c 1 )( t T ) m + u it H0 becomes : H 0 : β i = 0, i = 1,..., 2 m where m is the number of elements of c 2. As above, the program will consider m = (1, 2). Note also that the coefficients of the variables z 1,..., z nz that are a priori constant over time are not considered in this test. 6 Derivatives The derivatives of y with respect to the explanatory variables are given by the following expressions : 6

7 = δ(k) if q z k, k = 1,..., n z δz k = δz k δq = δ(k) + γ g(q it; γ, c) [ 1 g(q it ; γ, c) ] nx β 1 (k)x kit if q = z k = δz k δq = δ(k) + γ g(q it; γ, c) [ 1 g(q it ; γ, c) ] [ ] nx 2q it c 1 c 2 β 1 (k)x kit and m = 1 if q = z k and m = 2 δx k = β 0 (k) + β 1 (k) g(q it ; γ, c) m = (1, 2) if x k q δx k δx k = δq = β 0(k) + β 1 (k) g(q it ; γ, c) + γ g(q it ; γ, c) [ 1 g(q it ; γ, c) ] n x β 1 (k)x kit if x k = q and m = 1 = δq = β 0(k) + β 1 (k) g(q it ; γ, c) + γ g(q it ; γ, c) [ 1 g(q it ; γ, c) ] [ ] n x 2q it c 1 c 2 β 1 (k)x kit if x k = q and m = 2 δq = γ g(q it; γ, c) [ 1 g(q it ; γ, c) ] nx β 1 (k)x kit δq = γ g(q it; γ, c) [ 1 g(q it ; γ, c) ] [ ] nx 2q it c 1 c 2 β 1 (k)x kit if q z j and q x k and m = 1 if q z j and q x k and m = 2 In order to evaluate these expressions, unknown parameters are replaced by their estimated values. An Excel file, "Derives.xls", is created. It contains δy/δz k, k = 1,..., n z (if n z > 0), δy/δxk, k = 1,..., n x, and, eventually, δy/δq if q {z 1, z 2..., z nz, x 1, x 2..., x nx }. 7 Calling the proc After the command source pstr.scr has been processed successfully, users can do something like : 7

8 @pstr(list of options) #explained variable #list of x variables #list of q variables #list of z variables The last list must be present only if the option varz=yes is specified when calling the procedure. The list of variables obey to the standard rules of RATS. For exemple, use name{lag} to introduce the variable name delayed by lag periods. Finally, note that an option smpl=series is available, where series is a zeros/nonzeros variable. As usual, a zero indicates an observation that must not be considered in the estimation of the model. Some examples of calls are given in the file "exemples.prg". References [1] Andrés González, Timo Teräsvirta, Dick van Dijk, and Yukai Yang, Panel Smooth Transition Regression Models,, October 2017 (revised and updated version of the Working Paper No. 604 (2005) in the Working Paper Series of Economics and Finance, Stockholm School of Economics). 8