A unit root test based on smooth transitions and nonlinear adjustment

Transcription

1 MPRA Munich Personal RePEc Archive A uni roo es based on smooh ransiions and nonlinear adjusmen Aycan Hepsag Isanbul Universiy 5 Ocober 2017 Online a hps://mpra.ub.uni-muenchen.de/81788/ MPRA Paper No , posed 6 Ocober :57 UTC

2 A uni roo es based on smooh ransiions and nonlinear adjusmen Aycan Hepsag * Absrac In his paper, we develop a new uni roo esing procedure which considers joinly for srucural breaks and nonlinear adjusmen. The srucural breaks are modeled by means of a logisic smooh ransiion funcion and nonlinear adjusmen is modeled by means of an ESTAR model. The empirical size of es is quie close o he nominal one and in erms of power, he new uni roo es is generally superior o he alernaive es. Keywords: Smooh Transiion, nonlineariy, uni roo, ESTAR JEL Classificaion: C12, C22 * Isanbul Universiy, Faculy of Economics, Deparmen of Economerics, Tel.: , Beyazı, Isanbul, Turkey, hepsag@isanbul.edu.r 1

3 1. Inroducion In he las fory years, he ime series analysis of models wih uni roos has increasingly become one of he major opics for he invesigaors and praciioners o undersand he response of economic sysems o shocks. The firs ess for uni roo were proposed by Fuller (1976) and Dickey and Fuller (1979). However, i is well known ha he presence of srucural breaks and nonlineariies in ime series migh affec he power of he radiional uni roo ess. Accordingly, he Dickey-Fuller es fails o rejec he null hypohesis of uni roo and hese ypes of ess would be powerless o separae he behaviour of a uni process from he behaviour of a saionary process wih srucural breaks. Perron (1989) proposed a uni roo es which akes ino accoun srucural breaks exogenously in he deerminisic componens and displayed ha he radiional uni roos ess deec incorrecly ha he series have a uni roo when in fac hey are saionary wih srucural breaks. Apar from Perron (1989), many auhors have developed uni roo ess in order o ake ino accoun srucural breaks (Zivo and Andrews (1992); Lumsdaine and Papell (1997); Lee and Srazicich (2003). The main feaure of hese uni roo ess is ha he deerminisic srucural changes are assumed o occur insananeously, only in cerain poins of ime. Noneheless, individual agens can reac simulaneously o a given economic simulus; while some may be able o reac insananeously and so will adjus wih differen ime lags. Thus, when considering aggregae behavior, he ime pah of srucural changes in economic series is likely o be beer capured by a model whose deerminisic componen permis gradual raher han insananeous adjusmen beween differen values (Leybourne e. al., 1998). From his poin of view, some auhors proposed differen uni roo ess ha consider smooh raher 2

4 han a sudden change. The main idea behind of hese ess is ha nonlineariies can be presen in ime series as an asymmeric speed of mean reversion and auoregressive parameer varies depending upon he values of a variable. This nonlinear behavior implies ha here is a cenral regime where he series behave as a uni roo whereas for values ouside he cenral regime, he variable ends o rever o he equilibrium (Cuesas and Ordóñez, 2014). The nonlinear dynamics for uni roo esing procedures and he join analysis of nonlineariy and nonsaionariy have been popularised since abou he las weny years. Kapeanios e. al. (2003) proposed a uni roo es wihin an exponenial smooh ransiion auoregressive (ESTAR) model. Apar from Kapeanios e. al. (2003), Sollis (2009), Kruse (2011) presen invaluable conribuions o he esing of uni roos considering nonlineariy. Alhough hese sudies consider asymmeric speed of mean reversion, hey do no ake ino accoun nonlineariies in he deerminisic componens. On he oher hand, Chrisopoulos and León-Ledesma (2010) developed ess for uni roos ha accoun joinly for srucural breaks and nonlinear adjusmen. The prominen conribuion of uni roo es of Chrisopoulos and León-Ledesma (2010) is ha his es akes ino accoun asymmeric speed of mean reversion, as well as srucural changes in he inercep, approximaed by means of a Fourier funcion. Cuesas and Ordóñez (2014) also proposed a uni roo es which exends he uni roo es of Leybourne e. al. (1998) and akes ino accoun boh sources of nonlineariies, i.e. in he deerminisic componens, approximaed by a logisic smooh ransiion funcion no only in he inercep, bu also in he slope, an asymmeric adjusmen of mean reversion. 3

5 In his paper, we develop a new uni roo esing procedure which considers joinly for srucural breaks and nonlinear adjusmen. In our proposed es, srucural breaks are modeled by means of a logisic smooh ransiion funcion ha allows in he inercep, in he inercep under a fixed slope and in he inercep and slope erms. Nonlinear adjusmen is modeled by means of an ESTAR model as suggesed by Kruse (2011). The res of he paper is organized as follows: Secion 2 describes he proposed es saisics and provides asympoic criical values. Secion 3 presens he resuls of power and size of our proposed es via Mone Carlo simulaion experimens. The las secion concludes he paper. 2. The Uni Roo Tes In his secion, we propose a uni roo es which accouns joinly for srucural breaks and nonlinear adjusmen. The es, which is considered as an alernaive o Leybourne e. al. (1998) and Kruse (2011), aemps o model srucural change as a smooh ransiion beween differen regimes over ime and also model he nonlineariies by means of ESTAR model. In order o develop he new uni roo esing sraegy, we consider he following hree logisic smooh ransiion models by following Leybourne e. al. (1998): y = + S + v (1) Model A: α α ( λ τ ) 1 2, y = + + S + v (2) Model B: α β α ( λ τ ) 1 1 2, Model C: α β α ( λ, τ ) β ( λ, τ ) y = + + S + S + v (3)

6 where v is error erm which is normally disribued wih zero mean and uni variance and (, ) S λ τ is he logisic smooh ransiion funcion, based on a sample of size T : ( λ τ ) = + λ ( τ ) S, 1 exp{ } 1 T λ > 0 (4) The parameer τ deermines he iming of he ransiion midpoin and he speed of ransiion is deermined by he parameer λ. If we assume v is a zero-mean ( 0) I process, he in model A y is saionary around a mean which changes from he iniial value α 1 o he final value α1 + α2. Model B is similar o Model A, wih he inercep changing from α 1 o α1 + α2, bu i allows for a fixed slope erm. Finally, in Model C, in addiion o he change in inercep from α 1 o α1 α2 +, he slope also changes conemporaneously, and wih he same speed of ransiion β 1 o β1 + β2. The null of uni roo hypohesis may be saed as follows: H : y = µ, µ = µ + ε (5) 0 1 where ε is assumed o be an ( 0) I process wih zero mean. The es saisics are calculaed via a wo sep procedure. In he firs sep, we use nonlinear leas squares (NLS) algorihm for esimaing only deerminisic componen in model A, B and C, hen we compue he NLS residuals, Model A: vˆ ˆ ˆ 1 2 ( ˆ, ˆ y α α S λ τ ) = (6) Model B: vˆ ˆ ˆ ˆ ( ˆ, ˆ y α β α S λ τ ) = (7) 5

7 Model C: vˆ ˆ ˆ ˆ ( ˆ, ˆ) ˆ 2 ( ˆ, ˆ y α β α S λ τ β S λ τ ) = (8) In he second sep, we apply he uni roo es of Kruse (2011) o he residuals obained in he firs sep: p i i i= 1 vˆ = δ vˆ + δ vˆ + ψ vˆ + ε (9) Kruse (2011) ess he null of uni roo agains he alernaive of globally saionary ESTAR process, i.e. ( 1 exp{ ( ) }) 2 vˆ = γ vˆ θ vˆ c + ε (10) 1 1 In order o es of he null of uni roo, Kruse (2011) propose a firs order Taylor approximaion for equaion (10) and obain he auxiliary regression shown a equaion (9). The es saisics of our new procedure are compued as a modified Wald ype es saisic by following Kruse (2011) (For deails see Kruse (2011)). We denoe he value of es saisics as τ SNLα if Model A is used o consruc he v ˆ, τ if Model B is used and τ SNLαβ if Model C is used. Thus, he criical values of τ SNLα, τ and τ SNLαβ es saisics have been obained via sochasic simulaions a 1%, 5% and 10% significance levels based on 50,000 replicaions for T = 50,100, 250, 500. The criical values are repored in Table 1. 6

8 Table 1: Criical Values τ τ SNLα τ SNLαβ T 1% 5% 10% 1% 5% 10% 1% 5% 10% Mone Carlo Sudy This secion involves he Mone Carlo invesigaion of he size properies and power performance of our new uni roo es and also he power comparison of he new es wih Kruse (2011) es. Firs, we sudy he empirical size of es for differen sample sizes i.e. T = 50,100 wih a nominal size of We generae he DGP as follows: 1 0 ( ) y = µ, µ = µ + ε, µ = 0 ε ~ NIID 0,1 (11) The resuls of empirical size of es, based on 5000 replicaions, are presened in Table 2. In general, we could conclude ha he empirical size of es is quie close o he nominal one, 5%. A significan size disorion is only deermined for T = 50 for τ es. Noneheless, he size disorion disappears for T = 100. Table 2: Size Properies of Tes T τ τ SNLα τ SNLαβ

9 Nex, we invesigae he power of τ SNLα, τ, τ SNLαβ ess based on he following models, respecively: y = v 1 + exp{ λ ( τt )} + (12) 10 y = v 1 + exp{ λ ( τt )} + (13) y = v 1+ exp{ λ ( τt )} 1+ exp{ λ ( τt )} (14) { 1 exp ( ( ) )} 2 v = γ v θ v c + ε (15) 1 1 wih λ = 1.0, τ = 0.5 and γ = 1.5. The locaion parameer c is allowed by drawing from a uniform disribuion wih lower and upper bound of 5( 10) and 5( 10 ), respecively. Analogously, he parameer θ is allowed by drawing from a uniform disribuion wih lower and upper bound of ( 0.001, 0.01 ) wih slow ransiion beween regimes ( θ l ) and ( 0.01,0.1 ) wih fas ransiion beween regimes ( θ h), respecively. The nominal size of he ess are deermined a 0.05, he number of replicaions is 5000 and he sample size is considered for T = 50,100. The resuls of power experimens and power comparison wih Kruse (2011) es are displayed in Table 3. 8

10 Table 3: Power Experimens and Comparison Model A c θl c θh,, ± 5 ± 5 c, θ ± 10 c, ± 10 τ SNLα τ τ SNLα τ τ SNLα τ τ SNLα τ T= T= l θh c l Model B c θh, θ, ± 5 ± 5 c, θ ± 10 c, ± 10 τ τ τ τ τ τ τ τ T= T= l θh c l Model C c θh, θ, ± 5 ± 5 c, θ ± 10 c, ± 10 τ SNLαβ τ τ SNLαβ τ τ SNLαβ τ τ SNLαβ τ T= T= τ, τ and τ SNLαβ ess and bold values display he cases where each es performs beer. Noes: The values are rejecion raes of Kruse es ( τ ) and SNLα l θh The resuls of he power experimens and comparison show ha he new uni roo es is generally superior o he Kruse es. Only in some cases where he uni roo es is applied for Model B, he Kruse es performs beer han τ es. 4. Conclusions In his paper, we develop a new uni roo esing procedure which considers joinly for srucural breaks and nonlinear adjusmen. The empirical size of es is quie close o he nominal one and in erms of power, he new uni roo es is generally superior o he Kruse es. 9

11 References Chrisopoulos, D., León-Ledesma, M.A., Smooh breaks and non-linear mean reversion: pos-breon-woods real exchange raes. Journal of Inernaional Money and Finance. 29, Cuesas, J.C., Ordóñez, J., Smooh ransiions, asymmeric adjusmen and uni roos. Applied Economics Leers. 21, Dickey, D.A., Fuller. W.A., Disribuion of he esimaors for auoregressive ime Series wih a uni roo. Journal of he American Saisical Associaion. 84, Fuller, W.A., Inroducion of saisical ime series, Wiley, New York. Kapeanios, G., Shin, Y., Snell, A., Tesing for a uni roo in he nonlinear STAR framework. Journal of Economerics. 112, Kruse, R., A new uni roo es agains ESTAR based on a class of modified saisics. Saisical Papers. 52, Lee, J., Srazicich, M.C., Minimum Lagrange muliplier uni roo es wih wo srucural breaks. The Review of Economics and Saisics. 85, Leybourne, S., Newbold, P., Vougas, D Uni roos and smooh ransiions. Journal of Time Series Analysis. 19,

12 Lumsdaine, R.L., Papell, D.H Muliple rend breaks and he uni-roo hypohesis. Review of Economics and Saisics. 79, Sollis, R., A simple uni roo es agains asymmeric STAR nonlineariy wih an applicaion o real exchange raes in Nordic counries. Economic modelling. 26, Perron, P., The grea crash, he oil price shock and he uni roo hypohesis. Economerica. 57, Zivo, E., Andrews, D.W. K., Furher evidence on he Grea Crash, he oil-price shock, and he uni-roo hypohesis. Journal of Business and Economic Saisics. 10,