Does a non-linear mean reverting process characterize real GDP movements?

DOI 10.1007/s00181-005-0034-5

ORIGINAL PAPER

Dimitris K. Christopoulos

Does a non-linear mean reverting processcharacterize real GDP movements?

Received: 15 June 2004 / Accepted: 11 July 2005 / Published online: 20 January 2006© Springer-Verlag 2006

Abstract This paper uses non-linear models to investigate non-stationarity of realGDP per capita for seven OECD countries over the period 1900–2000. Unit roottests based on non-linear models are more powerful than traditional ADF statisticsin rejecting the null unit root hypothesis. Empirical results show that, contrary towhat the linear ADF statistics suggest, stationarity characterizes five out of theseven countries. This finding stands at variance with other recent studies whichconclude that movements in real GDP per capita can be characterized as a non-stationary process.

Keywords Unit root tests . Non-linear models . Real GDP

JEL Classification C22 . E1

1 Introduction

An important feature of business cycle theory is the trend stationarity of real output.This means that shocks have only a transitory impact on real output movementsleading the economy towards an equilibrium value. However the stationarity of realoutput is an unlikely possibility. A substantial number of empirical studies givesupport to the contention that real output levels are non-stationary: see, forexample, Nelson and Plosser (1982), Wasserfallen (1986), Cheung and Chinn(1996), Rapach (2002), David et al. (2003) and Thangavelu and James (2004). Thisfinding has important implications for business cycle theory as it suggests that realfactors such as technology shocks play an important role in economic fluctuations.Within this context, business cycle theory no longer implies stationary fluctuationsaround a deterministic trend. Finally, the presence of a unit root in outputmovements has consequences for the way we forecast economic activity andevaluate the role and importance of macroeconomic stabilization programs.

D. K. Christopoulos (*)Department of Economic and Regional Development, Panteion University, Leof. Syngrou 136,Athens 17671, GreeceE-mail: [email protected]

Empirical Economics 31:601–611 (2006)

The empirical literature cited above reached the conclusion that real GDP levelsare non-stationary by using either univariate unit root statistics (Cheung and Chinn,1996 and Thangavelu and James, 2004) or panel unit root tests (Rapach, 2002)along the lines of the augmented Dickey–Fuller (ADF) statistic. The key feature ofall these tests is that they work upon the hypothesis that a symmetric adjustmentprocess exists. However, a very recent and expanding empirical literature allowsfor non-linear dynamics for unit root testing procedures: see for example Endersand Ludlow (2002), Caner and Hansen (2001), Shin and Lee (2001), He andSandberg (2003) and Kapetanios et al. (2003). According to Enders and Granger(1998) all standard linear unit root tests have lower power in the presence ofmisspecified dynamics. They reviewed many important examples of asymmetricadjustment of economic variables.

The aim of the present paper is to find evidence for a non-linear mean reversionfor real GDP using a battery of unit root methods with non-linear dynamics. Thesemethods are more suitable for detecting non-stationarity in the levels of real GDPper capita.

2 The Caner and Hansen threshold autoregressive (TAR) model

Caner and Hansen (2001) consider a threshold autoregressive (TAR) model toprovide statistical tests for the joint null hypothesis of linearity and stationarity.This model is given by:

�yt ¼ �ðLÞ�yt þ �01xt�1IfZt�1<�g þ �02xt�1IfZt�1>�g þ et t ¼ 1; 2; ::: ::T (1)

where δ(L) is a lag polynomial of order p, xt�1 ¼ ð1; t; yt�1Þ0, Zt � yt � yt�m forsome m ≥ 1 (m is the delay parameter which in our application is set equal to one),I(.) is the indicator function, λ is the threshold parameter and et is an i.i.d. error. Isuppose that the threshold variable Zt−1 is predetermined and strictly stationary,while λ is unknown and takes values in the interval � 2 � ¼ ½�1; �2� where λ1 andλ2 are chosen so that Pr obðZt � �1Þ ¼ �1 > 0 and Pr obðZt � �2Þ ¼ �2 < 1:1 Inso far as the description of the empirical methodology is concerned, it is convenientto consider the vectors �1 ¼ ð�1; �1; �1Þ0 and �2 ¼ ð�2; �2; �2Þ0 where α1 and α2

are the intercepts, β1 and β2 are the trend slopes and γ1 and γ2 are the slopecoefficients on the lagged levels.

OLS can be used to estimate model (1) for a fixed λ ∈Λ. Letting b�2ð�Þ ¼ T�1PT1betð�Þ2 the computed residual variance, the LS point estimates of the threshold λ

and the corresponding vectors ρ1 and ρ2 are found by minimizing b�2ð�Þ:b� ¼ argmin

�2Λb�2ð�Þ

λ

λλ2

λ1

λ

λ

1 Since only the magnitude of the change in the deviation matters and not the sign, I consider theabsolute value of change as the switching variable. Therefore I treat π1 and π2 symmetrically sothat �2 ¼ 1� �1 �1 ¼ ��2 ¼ �. This makes my TAR a two-regime symmetric model.

602 D. K. Christopoulos

To test for linearity ðH0 ¼ �1 ¼ �2Þ in a TAR model against the alternative of athreshold effect I can use a Wald statistic, WT ( λ)

WT ð�Þ ¼ Tðb�20=b�2ð�Þ � 1Þ

where b�2ð�Þ is the generated residual variance from model (1) for a fixed thresholdλ and b�2

0 is the residual variance from OLS estimation of the null linear model(ρ1=ρ2). Since WT (λ) is a decreasing function of b�2ð�Þ the Wald statistic is

WT ¼ WT ðb�Þ ¼ sup�2�

WT ð�Þ

However, since the distribution of the WT is not identified under the null Canerand Hansen (2001) prove that although an asymptotic distribution can be generatedthey suggest the use of a model-based bootstrap approach.

In model (1) the parameters γ1 and γ2 control the stationarity of the process yt.A leading case is when yt is a unit root process. To test for unit roots, I use the one-sided formulation of Caner and Hansen (2001), namely H0: �1 ¼ �2 ¼ 0 versusthe alternative H1: �1 < 0 or γ2 < 0. The test statistic is a two sided Wald test of theform RT ¼ t21 þ t22 where ti signifies the t-ratio for b�i from OLS regression in theTAR model. Exact probabilities values for this test can be computed using abootstrap approach.

3 A general test of non-linear mean reversion

Enders and Ludlow (2002) and Ludlow and Enders (2000) follow a differentstrategy from Caner and Hansen (2001) to construct unit root tests based on non-linear models. In particular, they adopt the following extension to the standardlinear AR (1) model:

yt ¼ �ðtÞyt�1 þ et t ¼ 1; 2; :::::T (2)

where yt is a time series variable, et is a white noise disturbance term and α(t) is adeterministic but unknown function of time. α(t) can also be a p-th order differenceequation, a threshold function or a switching function. They showed that, althougha sufficiently long Fourier series represents the function α(t) exactly, equivalentresults can be obtained by considering only a single frequency so that:

�ðtÞ ¼ ’o þ ’1 sin2�k

T� t þ ’2 cos

2�k

T� t (3)

where k is an integer in the interval 1 to T/2.If ϕ1=ϕ2=0 then a sufficient condition for stationarity (and in this case for linear

mean reversion) is ’oj j < 1: However, with ’1 6¼ 0 and ’2 6¼ 0, ’oj j < 1 is neithera necessary nor sufficient condition for mean reversion of the yt series.

The main advantage in using Eq. (3) to detect non-stationarity in comparison toCaner and Hansen’s (2001) unit root test is that I do not need to specify the preciseadjustment mechanism or the nature of the asymmetry. Thus, I am not forced a

Does a non-linear mean reverting process characterize real GDP movements? 603

priori to impose a particular dynamic structure on the adjustment coefficient.2 All Ineed to find are the most appropriate values of the coefficients ϕ1 (i=0,1,2) and k.In this context Enders and Ludlow (2002) proved that a necessary and sufficientcondition for mean reversion is that ’oj j > 0 and ’o < 1þ r2

4 where r< 2, r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi’21 þ ’22

p. Finally, for various values of ϕo I can derive the adjustment paths of the

yt series.In Fig. 1, I present the various adjustment paths of the {yt}sequence for positive

values of ϕ0. The arc eb is constructed so that ’0 ¼ 1þ r2

4 shows the boundary lineof decay. Any (r, ϕ0) combination above this locus implies a divergent sequence.The restriction r < 2 limits the region of decay to the area οebf. On the other hand, astandard Dickey–Fuller test restricting the value of r equal to zero, considers onlythe line segment οe. Inside region οebf I identify four separate areas of interestscorresponding to four different types of decay.

Direct decay Within region οec, ϕ0 > r and ’0 þ r < 1. The value of a(t) is nevernegative and never greater than unity. Decay towards the attractor is direct althoughthe speed of adjustment changes over time.

Explosive decay Within region ebc, ϕ0 > r so that a(t) is never negative. Since’0 þ r > 1 and cos(2πkt) can equal unity, there will be values of t such that a(t) isgreater than unity. Although the {yt} sequence exhibits periods of explodingbehavior, the overall process ultimately reverts towards the attractor.

Non-explosive decay with oscillations Inside region οcg, ’0 þ r < 1 so that thereare no explosive periods. Given that cos(2πkt) takes the value −1, there will beperiods such that a(t) is less than zero. During these periods, {yt} will exhibitoscillating decay towards the attractor.

Oscillations and explosive decay Within region bfgc, ϕ0 < r and ’0 þ r > 1. Therewill be periods such that a(t) < 0 and others such that a(t) > 0. The {yt} sequencewill exhibit periods of oscillating reversion in conjunction with periods ofexplosive behavior.

Given that Eq. (3) is not known to the investigator I need to derive parameterestimates for ϕi, i=0,1,2 and k as well as their statistical levels of significance byperforming the following steps.

To find the most appropriate value for the integer k I estimate the followingregression in first differences using each integer value of k in the interval 1 to T/2.

�yt ¼ þ ’1 sin2�k

T� t þ ’2 cos

2�k

T� t

� �yt�1 þ et (4)

Parameter μ in Eq. (4) corresponds to the value (ϕο−1) in Eq. (3). Reversionrequires j j < r2

4 and r < 2.

2 In a recent paper Blake and Kapetanios (2003) consider a unit root test based on a neuralnetwork pure significance test, thus avoiding the specification of an alternative hypothesis. Ananonymous referee has suggested this to me.


The optimal level k* is chosen so as to minimize the sum of squared errors. Thecoefficients that correspond to the optimal k* are denoted by μ*, ϕ1* and ϕ2*. Basedon the estimated coefficients of Eq. (4) I can now test the following hypotheses:

* ¼ 0 (5)

’*1 ¼ ’*2 ¼ 0 (6)

* ¼ ’*1 ¼ ’*2 ¼ 0 (7)

* ¼ ðr *Þ24

(8)

where ðr *Þ2 ¼ ð’ *1Þ2 þ ð’ *2Þ2The null hypothesis (5) corresponds closely to the Dickey-Fuller unit root test.

Within this context, rejection of the null hypothesis (5) in favor of the alternativeμ*< 0 is a necessary condition for decay only when ϕ1 =ϕ2 = 0. However,rejecting the null hypothesis (5) in favor of the alternative μ*< 0 with ϕ1 ≠ 0 andϕ2 ≠ 0, is not a necessary condition to guarantee decay, see Enders and Ludlow

ϕ 0

“Oscillations” with Explosive

Decay

“Explosive” Reversion

Direct

Oscillationsr

2.0

f g

c=(0.5, 0.5)

b

e

2.0

1.0

1.0

0

Divergent

Fig. 1 The range of reversion


(2002). Rejection of the null hypothesis (6) indicates that any adjustment towardsan attractor is non-linear. Within this context the implied value of r * ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið’ *1Þ2 þ ð’ *2Þ2

qyields the ordinate on line of in Fig. 1; as such it can be

considered as a measure of the degree of non-linearity in the data. Non-rejection ofthe null hypothesis (7) implies that the series yt contains a unit root. Finally,rejection of the null hypothesis (8) guarantees decay. In other words rejecting the

null of the * ¼ ðr * Þ24 in favor of the alternative * < ðr *Þ2

4 is a necessary andsufficient condition for decay of the {yt}sequence. Further rejection of the nullhypothesis (8) implies that the larger region oebf of Fig. 1 describes better the areaof decay than the space οe considered in the standard linear Dickey-Fuller.3 A t-statistic can be used to test the null hypothesis (5) while an F- statistic is necessaryfor the null hypotheses (6)–(8). Monte Carlo simulations approximate all theseempirical distributions and are tabulated in Enders and Ludlow (2002).

4 Empirical results

Table 1 reports single country ADF linear tests using annual data on real GDP percapita in logarithmic form for seven OECD countries, namely, Denmark, Finland,France, Italy, the Netherlands, the USA and the UK over the period 1900–2000.Data description and sources are given in Maddison (1995).4 The unit root nullhypothesis is not rejected at conventional levels of significance for any country,except for the USA at the 5% level. This finding is consistent with the real GDPunit root literature.

I now apply the potentially more powerful unit root statistic based on a TARmodel of Caner and Hansen. Before applying this test I tested whether the data setis trended. I concluded in favor of the presence of a time trend in the fitted

Table 1 Linear ADF unit root tests

Country ADF statistic L

Denmark −2.18 1Finland −2.55 4France −2.38 1Italy −2.00 1Netherlands −2.62 1UK −2.58 1USA −3.90** 1

ADF is the augmented Dickey-Fuller t-test for a unit with a constant and trend. The number oflags (L) was selected optimally using the Schwarz criterion. Critical values for the ADF test are−4.05, −3.45 and −3.15 at the 1, 5 and 10% levels of statistical significance, respectively.Boldface values denote sampling evidence in favour of unit roots. A (**) indicates statisticalsignificance at 5% statistical level

3

3 In fact the null hypothesis (8) tests whether μ* and r* lie in area oebf of Fig. 1.4 The data can be downloaded from the Maddison web site. http://www.eco.rug.nl/˜Maddison/


http://www.eco.rug.nl/�Maddison/

regression (1). In the following Table 2 I present the threshold test WT (column 2).I see that the threshold statistic WT rejects the null hypothesis of no threshold foronly three countries out of seven, namely, Denmark, Finland and the Netherlands.For the remaining four countries, that is, France, Italy, the UK and the USA, athreshold effect cannot be established. This means that for these four countries Icannot reject the linear AR model in favor of the TAR model (1). Thus, for France,Italy, the UK and the USA a linear representation is sufficient to describe theadjustment of the real GDP towards the equilibrium level. Therefore, for these fourcountries I do not proceed with the estimation of the TAR model (1).

In Table 2 (column 3) I also display the unit root tests RT for the three countrieswhere the TAR model (1) remains a valid specification. Parameter estimates of thismodel are presented in Table 3.

The RT test indicates that the null hypothesis of a unit root can be rejected forDenmark and the Netherlands, but not in Finland where a unit process seems tocharacterize real GDP movements. Therefore I can conclude that for Denmark andthe Netherlands the RT statistics reject for a globally stationary TAR process.However, in both countries the individual ti–ratios (i=1,2) identify the presence ofa unit root only in one of the two regimes. Further, the OLS parameter estimatesshown in Table 3 indicate that the dynamics of the real GDP movements aredifferent depending on whether the change in real GDP is above or below theestimated threshold value.

Table 2 The Wald tests for a threshold - WT and unit root - RT

Country WT RT t1 t2

Denmark 22.2 (0.02) 18.6 (0.06) 3.80 (0.05) 2.05 (0.39)Finland 17.5 (0.07) 5.36 (0.81) 1.27 (0.60) 1.94 (0.40)France 13.4 (0.16)Italy 4.13 (0.94)Netherlands 68.0 (0.004) 71.2 (0.004) −5.02 (0.99) 6.78 (0.006)UK 6.71 (0.74)USA 8.93 (0.52)

Bootstrap p values are reported in parentheses. For bootstrapping 10,000 replications have beenused. t1 is the one-sided Wald test for the null of a unit root H0:γ1=γ2=0 versus the alternative ofstationarity only in the first regime (Zt−1< λ), that is, H1:γ1<0 and γ2=0 while t2 tests the null of aunit root against the alternative of stationarity only in the second regime (Zt−1< λ), that is,H1:γ1=0and γ2<0

Table 3 Parameter estimates for the TAR unit root model (1)

Zt−1< λ Zt−1> λ

Constant Trend yt−1 Δyt−1 Constant Trend yt−1 Δyt−1

Denmark 4.47[1.15]

0.01[0.004]

−0.49[0.13]

1.76[0.38]

0.94[0.45]

0.003[0.001]

−0.10[0.05]

−0.03[0.15]

Netherlands −4.99[0.99]

−0.02[0.003]

0.52[0.10]

−1.26[0.28]

2.94[0.43]

0.009[0.001]

−0.30[0.04]

−0.06[0.11]

Figures in brackets are standard errors


Next, I apply the Enders and Ludlow (2002) unit root test which does not placea priori restrictions on the exact form of the adjustment mechanism or the nature ofthe asymmetry. Given that my series yt contains a trend I regress the yt series on thetrend attractor z+ z1t and save the residuals, thus generating a new variable which isde-meaned and de-trended. Next I replace yt by the corresponding residual seriesand estimate Eq. (4) for each integer value of k in the interval 1 to (101/2). Havingidentified the optimal k* I present in Table 4 the values of the test statistics for thehypotheses (5)–(8) while in Table 5 I report parameter estimates for Eq. (4) alongwith their diagnostic statistics. To make the test more robust to a wider class oferror terms I introduced additional terms �yt�L on the right hand side of (4). Thenumber of lags (L) was selected optimally using Schwarz’s Bayesian criterion.

The results in Table 4 indicate that, firstly, μ* is not significantly different fromzero since the null hypothesis μ*= 0 is not rejected in any of the cases. This findingis consistent with the ADF results in Table 1 which show acceptance of the unit roothypothesis. It is easy to conclude that real GDP per capita does not enter the modellinearly. However, as discussed in Section 3, μ*= 0 is neither a necessary norsufficient condition for stationarity. A second finding from Table 4 is that the nullhypotheses μ*=ϕ1* =ϕ2* = 0 and ϕ1*=ϕ2*= 0 are rejected at conventional levels ofstatistical significance for all countries apart from the UK, thus one can concludethat rejection is most likely due to the part ϕ1*=ϕ2*= 0. Thirdly, the non-linear

restriction * ¼ ð’*1 Þ24 þ ð’*2 Þ2

4 is strongly rejected for four of the seven countries butnot for Italy, the Netherlands and the UK. This means that for all countries exceptthese three, a non-linear mean reversion process characterizes real GDP per capitamovements. Other things being equal, all shocks have temporary effects on outputlevels, which subsequently revert to their equilibrium values. This finding isconsistent with what real business cycle theory predicts. For the three countries inwhich the non-linear restriction was not rejected, I examined the estimated value ofμ* and r*. According to Enders and Ludlow (2002) who used power functions,

Table 4 Non-linear unit root tests

Country Hypothesis testing

μ*= 0 t-statistic ϕ1* =ϕ2

* = 0 F-statistic μ* =ϕ1* =ϕ2

* = 0F-statistic

* ¼ ð’ *1 Þ24 þ ð’ *

2 Þ24

F-statistic

Denmark −1.59 13.98*** 11.37*** 11.60*Finland −2.59 8.50** 8.83** 15.64**France −1.89 12.09*** 10.85*** 12.35*Italy −1.16 13.27*** 10.54*** 8.80Netherlands −0.87 21.46*** 17.63*** 9.72UK −2.25 2.49 3.98 9.21USA −3.09 9.46** 11.33** 20.50***Critical Values10% −3.21 7.14 6.53 11.595% −3.58 8.03 7.33 14.251% −4.27 9.95 9.30 19.55

(***), (**) and (*) denote rejection of the null hypothesis at the 1, 5 and 10% levels of statisticalsignificance, respectively. The source of the critical values is Enders and Ludlow (2002), Table 1


when μ* is near −0.1 and r* is near 0.27 the power of the Dickey-Fuller, μ*= 0 and

* ¼ ð’*1 Þ24 þ ð’*2 Þ2

4 tests are all very low. In these circumstances the F-tests forμ* =ϕ1*=ϕ2* = 0 and ϕ1* =ϕ2*= 0 are both quite powerful. In my case the μ* valuesare −0.03 for Italy, −0.03 for the Netherlands and −0.07 for the UK. The cor-responding r* values are 0.17, 0.36, and 0.09 respectively. It is obvious that theestimates of μ* and r* are close to the values proposed by Enders and Ludlow(2002) only for the Netherlands. This means that for this country my evidence furtherfavors the existence of a non-linear reverting process. For the remaining two countries,that is, Italy and the UK, results based on ADF regression are more powerful comparedto those derived from a non-linear unit root model. However, given that r*>0 and thatthe μ* test indicates that I can set μ*=0 it is possible to argue that real GDP is re-verting. Nevertheless, the inability to reject the more general restriction * ¼ ðr*Þ2=4leaves the question somewhat open.

I can derive the various adjustment paths towards the equilibrium of the realGDP per capita based on the values of μ* and r*. Given that *j j < r* and

Table 5 Estimates of the Fourier model (4)

Country Estimated parameters k

ϕ0 ϕ1 ϕ2 Δyt−1 Δyt−2 Δyt−3

Denmark −0.05[1.59]

−0.09* [1.62] 0.24***[4.92]

0.08 [0.83] 29

Finland −0.08[2.59]

0.02 [0.49] 0.19***[4.06]

0.36***[3.98]

34

France −0.06[1.89]

−0.16***[3.07]

−0.14***[3.17]

0.22**[2.31]

−0.19**[2.11]

0.28***[2.92]

11

Italy −0.03[1.17]

−0.11***[3.08]

0.13***[3.58]

0.29***[3.02]

18

Netherlan. −0.03[0.87]

0.26***[4.09]

0.25***[4.98]

0.02 [0.25] 12

UK −0.07[2.25]

−0.06 [1.53] 0.07* [1.64] 0.32***[3.33]

19

USA −0.16[3.09]

0.24***[3.68]

0.16***[2.38]

0.28***[2.93]

0.02[0.17]

0.11***[1.90]

19

Diagnosticstatistics

adjR2 DW

Denmark 0.24 2.06Finland 0.25 1.89France 0.32 2.02Italy 0.29 1.99Netherlands 0.36 1.94UK 0.20 2.03USA 0.33 2.05

The optimal number of lags yt−L was selected using Schwarz’s Bayesian criterion. k stands for theoptimal frequency in the interval 1 to T/2. Figures in brackets indicate absolute t-ratios (***), (**)and (*) indicate statistical significance at the 1, 5 and 10% levels of statistical significance,respectively. D-W is the Durbin and Watson statistic for first-order autocorrelation


*j j þ r* < 1 for all countries examined, the real GDP per capita in every countryexhibits periods of exploding behavior although the overall process reverts towardsthe equilibrium level. This means that I have periods with â(t) > 1 and the numberof explosive periods equal k*, and the rest of the periods are non-explosive suchthat 0 < â(t) < 1 holds. This behavior is described in Fig. 1 within the region ebc.Therefore I can conclude that in the majority of cases I examine I can overturn thenon-stationary conclusion of the linear ADF test by applying unit root statisticsbased on the non-linear model (4).

Finally, comparing the findings shown in Table 4 with those reported in Table 3,I observe that: (a) the Enders and Ludlow procedure correctly tells us that Finland,France and the USA can be characterized by a non-linear mean-reversion process;(b) for two countries, France and the USA, where the TARmodel failed to establisha threshold the Enders and Ludlow unit root test produced strong evidence in favorof a non-linear mean reverting process; (c) the Enders and Ludlow statisticprovides stronger evidence in favor of the null unit root hypothesis than a TAR testin the case of the Netherlands; (d) for one country, Denmark, both tests producestrong evidence in favor of a non-linear mean reversion process; (e) the Enders andLudlow test gives strong support for non-linear mean-reversion in three countries(Finland, France and the USA), and weak support for one country (theNetherlands), whereas the Caner and Hansen procedure gives support for onlytwo countries (Denmark and the Netherlands); and (f) the econometric methodol-ogy of Enders and Ludlow used in rejecting the null unit root hypothesis in thepresent study should be emphasized, since important differences emerge when Icompare standard linear ADF statistics or unit root tests based on TAR modelsagainst general tests of non-linear mean reversion.

5 Concluding remarks

Using models that do not assume a linear adjustment towards the equilibriumvalue, this paper investigates non-linear mean-reversion of real GDP per capita forseven OECD countries over the period 1900–2000. Standard linear ADF statisticsshow that the data are basically non-stationary for all these countries apart from theUSA. In contrast, when I adopt a non-linear model which has higher power than astandard univariate unit root statistic to reject a false null hypothesis of unit rootbehavior, the empirical evidence suggests that real GDP per capita is wellcharacterized by a non-linear mean reverting process which exhibits periods ofexploding behavior. This might offer an alternative explanation for the difficultyresearchers have encountered in rejecting the unit root hypothesis for real GDP percapita.

Acknowledgements Thanks go to three anonymous referees for extremely detailed and usefulcomments on a previous version of this paper. Thanks are also due to M. León-Ledesma forproviding helpful computational assistance with Gauss code. The usual disclaimer applies to anyremaining errors or omissions.


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Does a non-linear mean reverting process characterize real GDP movements?