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Deterministic impulse response in a nonlinear model. An analytical expression
Economics Letters 95 (2007) 315 – 319 www.elsevier.com/locate/econbase
Deterministic impulse response in a nonlinear model. An analytical expression Ioannis A. Venetis a,⁎, Ivan Paya b,1 , David A. Peel b a
University of Patras, Department of Economics, University Campus Rio, 26504, Greece b Lancaster University Management School, Lancaster, LA1 4YX, UK
Received 24 November 2005; received in revised form 25 August 2006; accepted 11 October 2006 Available online 26 March 2007
Abstract This paper derives an analytical expression of the “impulse response function” for the skeleton of a restricted version of the typical ESTAR model reported in the real exchange rate literature. The expression involves the Lambert function and provides further insights to the complexities that nonlinear models introduce to impulse response analysis. © 2007 Published by Elsevier B.V. Keywords: Impulse response; ESTAR; Lambert W-function JEL classification: C22; C53
1. Introduction One class of nonlinear models that is often considered in applied econometric analysis is the exponential smooth transition autoregressive (ESTAR) models (see Granger and Terasvirta, 1993; van Dijk et al., 2002, for extensive analysis and review). This model assumes the presence of two or more ⁎ Corresponding author. Tel.: +30 2610 969964. E-mail address: [email protected] (I.A. Venetis). 1 Ivan Paya acknowledges financial support from the Spanish Ministerio de Educacion y Ciencia Research Project SEJ200502829/ECON. 0165-1765/$ - see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.econlet.2006.10.007
316
I.A. Venetis et al. / Economics Letters 95 (2007) 315–319
regimes that display a smooth transition from one regime to another. The typical univariate exponential STAR model is set as follows yt ¼ /1V xt ½1−Gð yt−d ; g; cÞ þ /2V xt Gð yt−d ; g; cÞ þ ut
ð1Þ
where xt ¼ ð 1 yt−1 yt−2 ddd yt−p Þ V /i ¼ /i;0 /i;1 /i;2 ddd /i; p V for i = 1, 2 and ut is an error term with zero conditional mean, constant conditional variance (to emphasize modelling of the conditional mean) and is uncorrelated. Function G(.) takes the form Gð yt−d ; g; cÞ ¼ ½1−expf−gð yt−d −cÞ2 g where the specification of the (transition) function is such that regimes are associated with small and large absolute values of yt and the model is characterized by symmetric adjustment. In this note, we consider a restricted version of (1) where p = 1, d = 1, ϕ2′ = 0 and series yt is viewed as deviations from the mean. This form is often reported in empirical work on real exchange rate deviations (see, e.g., Taylor et al., 2001). Then yt ¼
/yt−1 expf−gy2t−1 g þ ut
ð2Þ
In Eq. (2), when yt −1 is close to zero, yt behaves as an AR(1) process yt = ϕyt−1 + ut (which takes the form of a random walk if ϕ = 1) while as yt−1 deviates from zero yt becomes increasingly less persistent. In the limit, yt = ut. A number of authors have written on the subject of nonlinear models impulse response functions, see Gallant et al., 1993; Koop, 1995; Koop et al., 1996; Potter, 2000; van Dijk et al., 2006-in press. Their research shows that impulse response analysis is considerably more complex for nonlinear models when compared to linear (e.g. AR(p)) models. The impulse response functions of nonlinear models are not invariant to the series history, size of shocks, past shocks, and future shocks. In addition monotonic behavior with respect to parameters of interest is not granted. Finally, in computation terms Monte Carlo or bootstrap integration methods are required to overcome the issue of future shocks intrinsically incorporated in the model. In Section 2, we provide an analytical expression for the “naive” impulse response function of Eq. (2). The term naive is borrowed from the forecasting literature where naive forecasts are produced when future shocks entering the optimal forecast equation are ignored. Even so, the analytical expression provides insights on the complexity that nonlinear models introduce to impulse analysis. Section 3 presents a brief conclusion. 2. Deterministic impulse response in a non-linear model The “half-life” of a shock δ is the periods h necessary for the impulse response function (IRFh) to satisfy IRFh < 0.5δ where the impulse response is defined as the difference between two expected paths of
I.A. Venetis et al. / Economics Letters 95 (2007) 315–319
317
the series initiated by a nonzero and zero shock respectively 2 . For example, the stationary AR(1) model has, IRFhAR(1) = δϕ h , where ϕ is the autoregressive parameter. The number of periods that it takes for half of an initial shock δ to dissipate is the solution of δϕ h = 0.5δ with respect to h. That is, h ¼ logð0:5Þ logðuÞ . Obviously, the half-life is independent of the size and sign of the shock or the relevant history up to the time the shock occurs. That result does not generalize to nonlinear models and certainly it is not the case when we deal with the exponential smooth transition model. Let us provide some heuristics regarding the impulse response function of model (2). We will consider only the model skeleton in order to expose its complicated nature with respect to perturbations. Such heuristics would coincide with the impulse response function in linear models. For example, the AR(1) model. Nevertheless, in nonlinear models, such an approach would not take the error feedback mechanism (through the nonlinear function) into account. Let the skeleton of the model be the following nonlinear difference equation ytþ1 ¼ /yt expf−gyt2 g
ð3Þ
where yt is thought as deviations from the mean. Assume the typical solution yt = Abt to the difference Eq. (3). Then, we obtain b ¼ /expf−gA2 b2t g with solution
−W ð2tgA2 e2tln/ Þ þ 2tln/ b ¼ exp 2t
hence
ð4Þ
1 2 2t ln / Þ−2t ln / ⇔ yt ¼ y0 exp − ½W ð2t gy0 e 2 1 yt ¼ y0 /t exp − W ð2tgy20 /2t Þ 2
ð5Þ
ð6Þ
where W(x) is the Lambert W-function3 (see Corless et al., 1996) defined by solutions of the equation WeW = x with W(0) = 0. Clearly, Eq. (3) can be seen as the deterministic equivalent of an AR(1) process with endogenously time-varying coeffcient ϕ(t) = ϕ exp{−γyt2 } and the typical solution involves a function of time A[b(t)]t . For γ = 0, Eq. (6) reduces to the familiar yt = y0ϕt . Assume two “shocks” at time 0, namely y0 ≠ 0 and y0′ = 0. The impulse response function at horizon h given by the difference of the paths initiated with y0 and y0′ is simply (substitute t with h) 1 h 2 2h ð7Þ IRFh ¼ y0 / exp − W ð2hgy0 / Þ 2 2
The definition is not flawless. See Chortareas and Kapetanios (2004) for an alternative definition based on the cumulative impact of shocks. 3 A quick overview can be found online in http://mathworld.wolfram.com/LambertW-Function.html. W(x) is real for x ≥ −1 / e with W(− 1 / e) = −1 and W(0) = 0. The function is positive for x > 0. For approximations of the function for real x see Barry et al., 2000. Further material can be found in R.M Corless website. http://www.apmaths.uwo.ca/~corless/frames/PAPERS/ LambertW.
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I.A. Venetis et al. / Economics Letters 95 (2007) 315–319
As a result, we have the following formula expressing the x-life of shock y0 where (1 − x) ∈ [0, 1] expresses the fraction of shock (y0) absorbed. Equate xy0 with IRFh to obtain 1 2 2h xy0 ¼ y0 / exp − W ð2hgy0 / Þ 2
ð8Þ
h
Solving Eq. (8) with respect to h gives h¼
−ln x
ð9Þ
gy20 x2 −ln /
For example, the “half-life” of shocks corresponds to x = 0.5 and we obtain h¼
0:69315 0:25gy20 −ln /
ð10Þ
while h = 0 only for x = 1 (100% of the initial shock still remains). The following remarks can be made: • Eq. (9) depends on γ nonlinearly. As expected, larger transition speeds imply faster convergence. In addition for x ∈ [0, 1] and ϕ ∈ [0, 1] it is positive since the model assumes γ > 0. • Eq. (9) depends nonlinearly on the initial shock y0. Larger shocks imply increasingly faster convergence. In addition, Eq. (9) is symmetric, as expected, with respect to the sign of the initial shock since it depends on y02 . • since −lnϕ > 0 is positive for 0 < ϕ < 1, the stationary model implies always faster convergence than the asymptotically stationary models with ϕ ≥ 1 • let 0 < ϕ < 1. For simplicity treat IRFh as a function of h and write it as IRF(h). We will denote IRF(a, h) the impulse response function when arguments γ or y02 are multiplied by a, for a > 0. Then IRFða;hÞ IRFðhÞ Y1 only if 0 < ϕ < 1. That is, there is no difference asymptotically between the functions. Now let ϕ = 1 (in many cases ESTAR models have been estimated with their “inner regime” corresponding to a random walk). For simplicity denote the function IRFh as IRF(γh,.) or IRF(y0h,.) to give emphasis on the elements γ (transition speed), and y0 (initial condition). Consider the ratio of two different impulse response functions differing in one of those elements with the nominator element being a positive − 1/2 as h → +∞ and a > 0 the function is multiple of the denominator element. Since IRFðahÞ IRFðhÞ → a regularly varying at infinity with index of regular variation −1/2. Thus, as horizon increases, two impulse responses whose transition speeds or initial shocks differ by a > 0 will never intersect. The larger the difference a the closer they will appear asymptotically. • it is worth noting that ϕ < 0 values are not permitted. x-life is a measure of persistence and looses meaning when applied to an antipersistent framework • the explosive inner regime case ϕ > 1 again confirms the complexities that arise in nonlinear models. For a given x there could be certain combinations of the parameters γ, ϕ and the initial condition magnitude y0 that produce a valid x-life expression. On the other hand negative values are meaningless for Eq. (9). This could be the case of ϕ > 1 being large enough relative to γ as to produce explosive behavior thus x-life is not defined. A relatively simple example can be produced for the half-life case where x = 1/2. In an attempt to “emulate” the applied econometrics literature let y0 = 1σu for Tσu > 0 where σu represents the disturbance standard deviation. Further assume the normalization g ¼ gr2 often y
I.A. Venetis et al. / Economics Letters 95 (2007) 315–319
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seen in estimated ESTAR models where σy2 represents the variance of yt. Finally, let the disturbance variance be k times thevariance of yt where k ∈ (0,1), that is σu2 = kσy2 . Then Eq. (9) is a valid expression only if γ > 1k 4 ln ϕ. 3. Concluding remarks Nonlinear econometric models may have impulse response functions that are analytically intractable. This is certainly the case for ESTAR models. Notwithstanding, in this letter we provide an analytical expression for a deterministic skeleton that ignores disturbances but can provide insight. The analysis enlightens several issues regarding the persistence profile of the particular class of nonlinear models. References Barry, D.A., Parlange, J.-Y., Li, L., Prommer, H., Cunningham, C.J., Stagnitti, F., 2000. Analytical approximations for real values of the Lambert W-function. Mathematics and Computers in Simulation 53, 95–103. Chortareas, G., Kapetanios, G., 2004. How Puzzling is the PPP Puzzle? An Alternative Half-Life Measure of Convergence to PPP. Department of Economics Working Paper, vol. 522. Queen Mary University of London. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E., 1996. On the Lambert W-function. Advances in Computational Mathematics 5, 329–359. Gallant, A.R., Rossi, P.E., Tauchen, G., 1993. Nonlinear dynamic structures. Econometrica 61, 871–908. Granger, C.W.J., Terasvirta, T., 1993. Modelling Nonlinear Economic Relationships. Oxford University Press, Oxford. Koop, G., 1995. Parameter uncertainty and analysis. Journal of Econometrics 72, 135–149. Koop, G., Pesaran, H., Potter, S., 1996. Impulse response analysis in nonlinear multivariate models. Journal of Econometrics 74, 119–148. Potter, S.M., 2000. Nonlinear impulse response functions. Journal of Economic Dynamics and Control 24 (10), 1425–1444 (September 2000). Taylor, M.P., Peel, D.A., Sarno, L., 2001. Nonlinear mean-reversion in real exchange rates: toward a solution to the purchasing power parity puzzles. International Economic Review 42 (4), 1015–1042. van Dijk, D., Terasvirta, T., Franses, P.H., 2002. Smooth transition autoregressive models — a survey of recent developments. Econometric Reviews 21, 1–47. van Dijk, D., Franses, P.H., Boswijk, H.P., 2006-in press. Absorption of shocks in nonlinear autoregressive models Computational Statistics & Data Analysis.
Deterministic impulse response in a nonlinear model. An analytical expression Ioannis A. Venetis a,⁎, Ivan Paya b,1 , David A. Peel b a
University of Patras, Department of Economics, University Campus Rio, 26504, Greece b Lancaster University Management School, Lancaster, LA1 4YX, UK
Received 24 November 2005; received in revised form 25 August 2006; accepted 11 October 2006 Available online 26 March 2007
Abstract This paper derives an analytical expression of the “impulse response function” for the skeleton of a restricted version of the typical ESTAR model reported in the real exchange rate literature. The expression involves the Lambert function and provides further insights to the complexities that nonlinear models introduce to impulse response analysis. © 2007 Published by Elsevier B.V. Keywords: Impulse response; ESTAR; Lambert W-function JEL classification: C22; C53
1. Introduction One class of nonlinear models that is often considered in applied econometric analysis is the exponential smooth transition autoregressive (ESTAR) models (see Granger and Terasvirta, 1993; van Dijk et al., 2002, for extensive analysis and review). This model assumes the presence of two or more ⁎ Corresponding author. Tel.: +30 2610 969964. E-mail address: [email protected] (I.A. Venetis). 1 Ivan Paya acknowledges financial support from the Spanish Ministerio de Educacion y Ciencia Research Project SEJ200502829/ECON. 0165-1765/$ - see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.econlet.2006.10.007
316
I.A. Venetis et al. / Economics Letters 95 (2007) 315–319
regimes that display a smooth transition from one regime to another. The typical univariate exponential STAR model is set as follows yt ¼ /1V xt ½1−Gð yt−d ; g; cÞ þ /2V xt Gð yt−d ; g; cÞ þ ut
ð1Þ
where xt ¼ ð 1 yt−1 yt−2 ddd yt−p Þ V /i ¼ /i;0 /i;1 /i;2 ddd /i; p V for i = 1, 2 and ut is an error term with zero conditional mean, constant conditional variance (to emphasize modelling of the conditional mean) and is uncorrelated. Function G(.) takes the form Gð yt−d ; g; cÞ ¼ ½1−expf−gð yt−d −cÞ2 g where the specification of the (transition) function is such that regimes are associated with small and large absolute values of yt and the model is characterized by symmetric adjustment. In this note, we consider a restricted version of (1) where p = 1, d = 1, ϕ2′ = 0 and series yt is viewed as deviations from the mean. This form is often reported in empirical work on real exchange rate deviations (see, e.g., Taylor et al., 2001). Then yt ¼
/yt−1 expf−gy2t−1 g þ ut
ð2Þ
In Eq. (2), when yt −1 is close to zero, yt behaves as an AR(1) process yt = ϕyt−1 + ut (which takes the form of a random walk if ϕ = 1) while as yt−1 deviates from zero yt becomes increasingly less persistent. In the limit, yt = ut. A number of authors have written on the subject of nonlinear models impulse response functions, see Gallant et al., 1993; Koop, 1995; Koop et al., 1996; Potter, 2000; van Dijk et al., 2006-in press. Their research shows that impulse response analysis is considerably more complex for nonlinear models when compared to linear (e.g. AR(p)) models. The impulse response functions of nonlinear models are not invariant to the series history, size of shocks, past shocks, and future shocks. In addition monotonic behavior with respect to parameters of interest is not granted. Finally, in computation terms Monte Carlo or bootstrap integration methods are required to overcome the issue of future shocks intrinsically incorporated in the model. In Section 2, we provide an analytical expression for the “naive” impulse response function of Eq. (2). The term naive is borrowed from the forecasting literature where naive forecasts are produced when future shocks entering the optimal forecast equation are ignored. Even so, the analytical expression provides insights on the complexity that nonlinear models introduce to impulse analysis. Section 3 presents a brief conclusion. 2. Deterministic impulse response in a non-linear model The “half-life” of a shock δ is the periods h necessary for the impulse response function (IRFh) to satisfy IRFh < 0.5δ where the impulse response is defined as the difference between two expected paths of
I.A. Venetis et al. / Economics Letters 95 (2007) 315–319
317
the series initiated by a nonzero and zero shock respectively 2 . For example, the stationary AR(1) model has, IRFhAR(1) = δϕ h , where ϕ is the autoregressive parameter. The number of periods that it takes for half of an initial shock δ to dissipate is the solution of δϕ h = 0.5δ with respect to h. That is, h ¼ logð0:5Þ logðuÞ . Obviously, the half-life is independent of the size and sign of the shock or the relevant history up to the time the shock occurs. That result does not generalize to nonlinear models and certainly it is not the case when we deal with the exponential smooth transition model. Let us provide some heuristics regarding the impulse response function of model (2). We will consider only the model skeleton in order to expose its complicated nature with respect to perturbations. Such heuristics would coincide with the impulse response function in linear models. For example, the AR(1) model. Nevertheless, in nonlinear models, such an approach would not take the error feedback mechanism (through the nonlinear function) into account. Let the skeleton of the model be the following nonlinear difference equation ytþ1 ¼ /yt expf−gyt2 g
ð3Þ
where yt is thought as deviations from the mean. Assume the typical solution yt = Abt to the difference Eq. (3). Then, we obtain b ¼ /expf−gA2 b2t g with solution
−W ð2tgA2 e2tln/ Þ þ 2tln/ b ¼ exp 2t
hence
ð4Þ
1 2 2t ln / Þ−2t ln / ⇔ yt ¼ y0 exp − ½W ð2t gy0 e 2 1 yt ¼ y0 /t exp − W ð2tgy20 /2t Þ 2
ð5Þ
ð6Þ
where W(x) is the Lambert W-function3 (see Corless et al., 1996) defined by solutions of the equation WeW = x with W(0) = 0. Clearly, Eq. (3) can be seen as the deterministic equivalent of an AR(1) process with endogenously time-varying coeffcient ϕ(t) = ϕ exp{−γyt2 } and the typical solution involves a function of time A[b(t)]t . For γ = 0, Eq. (6) reduces to the familiar yt = y0ϕt . Assume two “shocks” at time 0, namely y0 ≠ 0 and y0′ = 0. The impulse response function at horizon h given by the difference of the paths initiated with y0 and y0′ is simply (substitute t with h) 1 h 2 2h ð7Þ IRFh ¼ y0 / exp − W ð2hgy0 / Þ 2 2
The definition is not flawless. See Chortareas and Kapetanios (2004) for an alternative definition based on the cumulative impact of shocks. 3 A quick overview can be found online in http://mathworld.wolfram.com/LambertW-Function.html. W(x) is real for x ≥ −1 / e with W(− 1 / e) = −1 and W(0) = 0. The function is positive for x > 0. For approximations of the function for real x see Barry et al., 2000. Further material can be found in R.M Corless website. http://www.apmaths.uwo.ca/~corless/frames/PAPERS/ LambertW.
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I.A. Venetis et al. / Economics Letters 95 (2007) 315–319
As a result, we have the following formula expressing the x-life of shock y0 where (1 − x) ∈ [0, 1] expresses the fraction of shock (y0) absorbed. Equate xy0 with IRFh to obtain 1 2 2h xy0 ¼ y0 / exp − W ð2hgy0 / Þ 2
ð8Þ
h
Solving Eq. (8) with respect to h gives h¼
−ln x
ð9Þ
gy20 x2 −ln /
For example, the “half-life” of shocks corresponds to x = 0.5 and we obtain h¼
0:69315 0:25gy20 −ln /
ð10Þ
while h = 0 only for x = 1 (100% of the initial shock still remains). The following remarks can be made: • Eq. (9) depends on γ nonlinearly. As expected, larger transition speeds imply faster convergence. In addition for x ∈ [0, 1] and ϕ ∈ [0, 1] it is positive since the model assumes γ > 0. • Eq. (9) depends nonlinearly on the initial shock y0. Larger shocks imply increasingly faster convergence. In addition, Eq. (9) is symmetric, as expected, with respect to the sign of the initial shock since it depends on y02 . • since −lnϕ > 0 is positive for 0 < ϕ < 1, the stationary model implies always faster convergence than the asymptotically stationary models with ϕ ≥ 1 • let 0 < ϕ < 1. For simplicity treat IRFh as a function of h and write it as IRF(h). We will denote IRF(a, h) the impulse response function when arguments γ or y02 are multiplied by a, for a > 0. Then IRFða;hÞ IRFðhÞ Y1 only if 0 < ϕ < 1. That is, there is no difference asymptotically between the functions. Now let ϕ = 1 (in many cases ESTAR models have been estimated with their “inner regime” corresponding to a random walk). For simplicity denote the function IRFh as IRF(γh,.) or IRF(y0h,.) to give emphasis on the elements γ (transition speed), and y0 (initial condition). Consider the ratio of two different impulse response functions differing in one of those elements with the nominator element being a positive − 1/2 as h → +∞ and a > 0 the function is multiple of the denominator element. Since IRFðahÞ IRFðhÞ → a regularly varying at infinity with index of regular variation −1/2. Thus, as horizon increases, two impulse responses whose transition speeds or initial shocks differ by a > 0 will never intersect. The larger the difference a the closer they will appear asymptotically. • it is worth noting that ϕ < 0 values are not permitted. x-life is a measure of persistence and looses meaning when applied to an antipersistent framework • the explosive inner regime case ϕ > 1 again confirms the complexities that arise in nonlinear models. For a given x there could be certain combinations of the parameters γ, ϕ and the initial condition magnitude y0 that produce a valid x-life expression. On the other hand negative values are meaningless for Eq. (9). This could be the case of ϕ > 1 being large enough relative to γ as to produce explosive behavior thus x-life is not defined. A relatively simple example can be produced for the half-life case where x = 1/2. In an attempt to “emulate” the applied econometrics literature let y0 = 1σu for Tσu > 0 where σu represents the disturbance standard deviation. Further assume the normalization g ¼ gr2 often y
I.A. Venetis et al. / Economics Letters 95 (2007) 315–319
319
seen in estimated ESTAR models where σy2 represents the variance of yt. Finally, let the disturbance variance be k times thevariance of yt where k ∈ (0,1), that is σu2 = kσy2 . Then Eq. (9) is a valid expression only if γ > 1k 4 ln ϕ. 3. Concluding remarks Nonlinear econometric models may have impulse response functions that are analytically intractable. This is certainly the case for ESTAR models. Notwithstanding, in this letter we provide an analytical expression for a deterministic skeleton that ignores disturbances but can provide insight. The analysis enlightens several issues regarding the persistence profile of the particular class of nonlinear models. References Barry, D.A., Parlange, J.-Y., Li, L., Prommer, H., Cunningham, C.J., Stagnitti, F., 2000. Analytical approximations for real values of the Lambert W-function. Mathematics and Computers in Simulation 53, 95–103. Chortareas, G., Kapetanios, G., 2004. How Puzzling is the PPP Puzzle? An Alternative Half-Life Measure of Convergence to PPP. Department of Economics Working Paper, vol. 522. Queen Mary University of London. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E., 1996. On the Lambert W-function. Advances in Computational Mathematics 5, 329–359. Gallant, A.R., Rossi, P.E., Tauchen, G., 1993. Nonlinear dynamic structures. Econometrica 61, 871–908. Granger, C.W.J., Terasvirta, T., 1993. Modelling Nonlinear Economic Relationships. Oxford University Press, Oxford. Koop, G., 1995. Parameter uncertainty and analysis. Journal of Econometrics 72, 135–149. Koop, G., Pesaran, H., Potter, S., 1996. Impulse response analysis in nonlinear multivariate models. Journal of Econometrics 74, 119–148. Potter, S.M., 2000. Nonlinear impulse response functions. Journal of Economic Dynamics and Control 24 (10), 1425–1444 (September 2000). Taylor, M.P., Peel, D.A., Sarno, L., 2001. Nonlinear mean-reversion in real exchange rates: toward a solution to the purchasing power parity puzzles. International Economic Review 42 (4), 1015–1042. van Dijk, D., Terasvirta, T., Franses, P.H., 2002. Smooth transition autoregressive models — a survey of recent developments. Econometric Reviews 21, 1–47. van Dijk, D., Franses, P.H., Boswijk, H.P., 2006-in press. Absorption of shocks in nonlinear autoregressive models Computational Statistics & Data Analysis.