Journal of Econometrics 135 (2006) 311347

Large shocks vs. small shocks.

the stock market, and to US GNP to investigate the asymmetric behavior of its shocks.

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www.elsevier.com/locate/jeconom

Corresponding author.0304-4076/$ - see front matter r 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.jeconom.2005.07.022

E-mail address: jesus.gonzalo@uc3m.es (J. Gonzalo).

URL: http://www.eco.uc3m.es/english/staff/contact_info/jgonzalo.html.r 2005 Elsevier B.V. All rights reserved.

JEL classification: C22; C51

Keywords: Asymmetries; Nonlinear permanenttransitory decomposition; Persistence; Shocks; Threshold

models(Or does size matter? May be so.)

Jesus Gonzalo, Oscar Martnez

Department of Economics, U. Carlos III de Madrid, 28903 Getafe, Madrid, Spain

Available online 2 November 2005

Abstract

What are the shocks that drive economic uctuations? The answer to this question requires as

a rst step solving the shock identication issue. This paper proposes a new identication

scheme based on two aspects: the long-run effect of the shock (permanent or transitory), and the

size of the shock (Large or small). This is done by using a threshold integrated moving average

model (TIMA) previously introduced in the literature by the authors. Based on this model we

develop a testing strategy to determine whether Large and small shocks have different long-run

effects, as well as whether one of them is purely transitory. The paper analyzes the impulse

response function of both types of shocks, and provides the asymptotic results sufcient to

implement the above testing strategy. Based on these results we develop a new nonlinear

permanenttransitory decomposition, that is applied to US stock prices to analyze the quality of

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 3113473121. Introduction

What are the shocks that drive the economy? Where does the persistence present inmost of the economic time series come from? Is the persistence of output shockssymmetric or asymmetric? What fraction of output variation is due to supply ordemand shocks? All the attempts to answer these questions pass through the criticaldecision of shock identication.Traditionally this has been done by decomposing the analyzed variable into

unobserved permanent and transitory components (see Watson, 1986). Some examplesare: (1) Beveridge and Nelson (1981) decomposition (BN hereafter), where thepermanent component is a random walk and there is perfect correlation betweenpermanent and transitory shocks due to the fact that there is only one shock, thepermanent one; (2) Unobserved component models with uncorrelated components(UC0) (see Harvey, 1985; Clark, 1987), where the permanent component is alsoassumed to be a random walk but its innovation is uncorrelated with the one in thetransitory term. Note that not all the ARIMA models admit an UC0 decomposition.This problem can be solved eliminating the random walk constraint on the permanentcomponent. In this case we have an extra identication problem that is overcome byimposing an ad hoc smooth condition on the permanent term (see the canonicaldecomposition in Pierce, 1979). All these decompositions present two basic problemsthat are not solved in the literature: (i) at every time period t there is always apermanent shock, and (ii) none of the assumptions on permanent and transitorycomponents are testable. The approach proposed in this paper tries to solve theidentication issue without encountering these problems.Our proposal is based on two pillars. First, as in the above mentioned

identication attempts we accept that there are permanent and transitory shocks,and secondly, shocks behave differently in the long-run according to their owncharacteristics (sign, size, etc.) or some characteristics of the economy. The rstpremise, as is well known (see Quah, 1992) is not enough to identify the permanentand transitory shocks of a univariate time series. This paper shows that with oursecond assumption we cannot only identify the permanent and transitory shocks of asingle economic variable, but we can test whether in fact these shocks are transitoryor permanent. In order to implement our proposal we use a class of thresholdmodels, the threshold integrated moving average models (TIMA) previouslyintroduced in the literature by Gonzalo and Martnez (2004). These are modelswith a unit root in the autoregressive part to capture persistence, and a thresholddesign in the moving average side to allow for asymmetries. The threshold variablecan represent any characteristic of the shocks (large or small, positive or negative,etc.) or of the economy (expansion or recession, ination or deation, etc.) we areinterested on. This threshold design is able to capture the possibility that any of thesecharacteristics may have different long run effects. By allowing the existence of a unitroot in some of the moving average regime, we permit that the characteristicstriggering those regimes have only a transitory effect.In principle, TIMAmodels can deal with different shock characteristics, but in thispaper we will focus on shock size as in the StopBreak model of Engle and Smith

shock than after a small one. For example in the stock market, investors may

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 313perceive large shocks as containing informational content and small shocks as merenoise. In macroeconomics, an unsolved issue is whether economic uctuations aredue to an accumulation of small shocks or instead mostly to infrequent large shocks.A positive aspect of our framework is that we are able to test our maintainedhypothesis.The identication of permanent and transitory shocks with our procedure, allows

us to construct an orthogonal nonlinear permanenttransitory decomposition of theoriginal variable. We also obtain a nonlinear BN decomposition that while sharingpart of the spirit of both, the standard linear BN and the UC0 decompositions, itsbehavior lies in between.Threshold moving average (TMA) models have already been considered in

Wecker (1981) and in De Gooijer (1998). Both works are centered on presenting thenew TMA model and on analyzing some of the moment properties in detail. Theyboth assume normality, the threshold parameter to be known and equal to zero, andthey do not present asymptotic results. These can be found in Guay and Scaillet(2003) using indirect inference. One part of our paper can be consideredcomplementary to this one in the sense of also developing asymptotic results; butin our case using Hansen (1996), Gonzalez and Gonzalo (1998), and Caner andHansen (2001) approach to threshold models.The paper nishes with two applications of our TIMA Shock-Size model: rst, to

stock prices, where following Hasbrouck (1993) the size of the transitory componentcan be considered a measure of the quality or efciency of the stock market; andsecond to GNP, where the majority of the research that investigates the size of itspermanent component has implicitly imposed symmetry, and the minority thatallows for asymmetries driven by the sign of the shocks has not reached any nalconclusion yet (see Elwood, 1998).The rest of the paper is structured as follows. Section 2 introduces the TIMA

Shock-Size model and analyzes in detail its impulse response function (IRF). Section3 presents the asymptotic theory results needed for testing the hypothesis of interests.In Section 4 we dene two new nonlinear permanenttransitory decompositions andcompare them with the existing linear decompositions. Sections 5 and 6 present twoempirical applications of our TIMA Shock-Size model, one to measure the quality ofthe stock market and the other to analyze whether or not the persistence of shocks toGNP is asymmetric. Finally, Section 7 draws some concluding remarks. Theappendix contains technical derivations and proofs of the results in the main text.

2. TIMA Shock-Size models

In this section we lay out the basis of the TIMA Shock-Size model. In order to(1999). Our maintained hypothesis is that large shocks will have permanent effectswhile small shocks will produce only transitory effects. This hypothesis is based onthe assumption that certain times series are less likely to mean revert after a largebetter understand this model, we start with a more general structure, the

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347314autoregressive threshold integrated moving average ARTIMA (p;m; 1; 1) modelintroduced in Gonzalo and Martnez (2004),

FpL1 Lyt mXm1j1

1 yjLt1rj1oztprj, (1)

where L is the lag operator, FpL 1 f1L fpLp with all its roots outsidethe unit circle, 1: denotes the indicator function, and zt the threshold variable thattriggers the regime switches. The random error term t is a real i:i:d sequence withEjtj2go1 for some g42. The threshold parameters denoted r1; . . . ; rm withr0 1, rm1 1 are such that ri 2 Rm 8i 1; . . . ;m with Rm fr1; . . . ; rm :1o ror1o ormoro1g. Thus we require all threshold parameters to lie inthe bounded subset r; r of the threshold variable sample space.Some examples of threshold variables zt are: (i) t, for instance when the sign is the

shock characteristic that triggers the regime switches (as in Wecker, 1981; Elwood,1998; Guay and Scaillet, 2003); (ii) 1 Lyt or any other economic variable that isstrictly stationary and ergodic (as in De Gooijer, 1998); and (iii) jtj, the shocks size.In this paper we are only concerned with the asymmetry produced by the size of a

shock (zt jtj). To present the main results of ARTIMA models producing suchasymmetry (ARTIMA Shock-Size), and without loss of generality, we will use asimpler version of model (1). A version with no autoregressive part (FpL 1) andonly two regimes (the case of more regimes can be handled as in Gonzalo andPitarakis, 2002) that is denoted TIMA Shock-Size model:

1 Lyt mt y1t1 if jt1j4r;t y2t1 if jt1jpr;

((2)

or in a more compact way

1 Lyt m t yt1t1, (3)with yt1 y1 if jt1j4r, and equal to y2 otherwise. This TIMA model is a timevarying moving average model, and in that sense can be seen as the threshold versionor approximation of the StopBreak model of Engle and Smith (1999). The maindifference is that in the latter all the shocks are permanent (a shock is transitory if itis equal to zero), while in the former the existence of both permanent and transitoryshocks is allowed.

Assumptions.

A.0. t iid 0;s, with a uniformly continuous density function 0of o1, andE2gt o1 with g42:

A.1. jy01 y02j h040:A.2. E Ejyt1j rhf mo1 and jy1jojy2j, where

h jy1 y2j, and f m maxef r e f r e with e dened in the

support of t.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 315A.3. The vector of the true parameter values W0 m0; y01; y02; r0 2 Y m; m 1 d; 1 d 1 d; 1 r; r s.t.E Ey1; y2; r rhf mo1 and jy1jojy2j 8W 2 Y, where

Ey1; y2; r jy1j1 pr jy2jpr, prXsupkPjt kjor,f m 4maxef e, d40, and 0o roro1.

A.4. Et=It1 0 and Etj1jtjj4r=It1 0 for jX0, with sigma-fieldIt deffj; jptg.

A.5. yi 1 for some i 1; 2.

Assumptions A.0A.1 are standard identication assumptions in threshold modelsand empirical processes (see Bai, 1994). Assumption A.2 is a sufcient condition forinvertibility. We can distinguish two different parts in it. In the rst one, Ejyt1jmust be less than 1. This takes care of the case of overdifferencing, especially whenwe allow a unit root in the MA part. Non-invertibility is not only a problem ofoverdifferencing, but a problem of nonlinearity too. The second part, rhf m; takescare of the degree of nonlinearity, measured as a product of the gap (rh) and itsupper limit probability f m. A.3 describes the parametric space of TIMA Shock-Sizemodels (partially related to the invertibility condition) and constitutes a sufcientrequirement for the asymptotic results obtained in the paper. Finally, assumptionsA.4 and A.5 are helpful assumptions to interpret the outcomes of our model, i.e. toprove the existence of an orthogonal permanenttransitory decomposition in termsof the IRF. A.4 is a symmetry condition of the conditional distribution of the shocksthat can be relaxed. It allows the identication of permanent and transitory shocksbased on the IRF. Otherwise, the IRF is unknown and has to be estimated (see forexample, Clarida and Taylor, 2003). A.5 (together with A.4) guarantees the existenceof pure transitory shocks.

2.1. Impulse response function

To analyze in detail the asymmetric persistence of the shocks present in a TIMAmodel we have to study the behavior of its IRF. This function measures the effect ofa perturbation at time t in the sample path fytkg1k0. If this effect on ytk does notvanish when k !1 we say that the shock is persistent. With linear models there is ageneral consensus about the relation of the IRF and the persistence of the shock.However with nonlinear models three main aspects of the time series come up todetermine the denition of the IRF and its relationship with persistence. Theseaspects are the history of the series at time t 1; the shock t and future shocks. Tocapture all these three new aspects we use the Generalized impulse response function(GIRF) introduced and dened in Koop et al. (1996) and in Potter (2000) as

GIRF k; t;wt1 Eytk=t;wt1 Eytk=wt1 for k 0; 1; 2; . . . , (4)where wt is the sample history of the process until time t. According to this denition,a shock is persistent if the effect of knowing it on the conditional expectation of ytk

(given the past) does not vanish when k !1:

In order to show that TIMA Shock-Size models can admit both type of shocks,permanent as well as transitory, we use the following result.

Result 1. Let yt follow the TIMA process (2). Under assumption A.4, the GIRF of yt isgiven by

GIRF k; t;wt1 1 ytt.

From this result is clear that the shock t is transitory if yt 1 (assumption A.5). Itis worth noting that it is the size of t, that determines the persistent or transitoryeffect of t. It is also important to notice that when A.4 does not hold the GIRF maybe different from 1 ytt. This is the case, for instance, when the threshold variableis the sign of the shock. In this situation, TIMA model will not have any transitoryshock.To better understand the behavior of the GIRF and the main differences between

our TIMA model and other linear and nonlinear models, we consider four examplesfor which we calculate their theoretical GIRF. In all these examples, the shocks t are

X1

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k

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347316Fig. 1. Mean of the estimated GIRF (divided by t). The mean of the GIRF is obtained conditional on theshock size (jtjp0:5 and jtj40:5). The mean is calculated over 1000 Monte Carlo replications for 200yt j0

yjtj,

GI

0 10 20 300.0

0.2

0.4

0.6

0.8

1.0

TIMA Big ShockTIMA Small ShockTAR Big ShockTAR Small ShockRW Big ShockRW Small Shockassumed to be iid. A simulation exercise where the different GIRFs are compared isshown in Fig. 1.

Example 1 (Linear model). In this example we consider the standard general linearmodel,values of t N0; 1. We select a random history wt1 with t 201.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 317with y0 1. Then

ytk Xk1j0

yjtkj ykt X1j1

ykjtj.

Therefore

GIRF k; t;wt1 ykt.There are two possibilities: limk!1yk 0 or limk!1yka0: In the rst case, all theshocks ftg11, are transitory. In the second one, all of them are permanent. The longrun properties of the shock, t, only depend on the parameter yk, they do not dependeither on t or on the past. There is no asymmetric persistence. For the simulationand comparison exercise of Fig. 1, we use as an example of linear model a randomwalk (RW)

yt yt1 t.This model produces the following GIRF:

GIRF k; t;wt1 t,where clearly all the shocks are permanent.

Example 2 (TAR model). In this example we consider the following TAR model

yt f1yt11jt1j4r f2yt11jt1jpr t.Rewriting this model at time t k

ytk X1j0

ytk;jtkj

Xk1j0

ytk;jtkj ytk;kt X1jk1

ytk;jtkj,

with yt;j Qj

i1fti, fti f11jtij4r f21jtijpr for jX1, and yt;0 1.Therefore

GIRF k; t;wt1 Xk1j0

Eytk;jtkj=wt1; t Eytk;jtkj=wt1

Eytk;k=t;wt1t Eytk;kt=wt1. 5For TAR models the GIRF is nonlinear and it depends on the past of t. In

general, the problem is to obtain the expectations involved in the right-hand side of(5). This problem can be solved by simulation (see Koop et al., 1996; Potter, 2000).For the particular TAR model of this example, it can be proved that underassumption A.4,

GIRF k; t;wt1 ftYtk1

Efj=t;wt1t.

jt1

ARTICLE IN PRESSAssuming the parameters (f1, f2) satisfy some stationarity and ergodicityconditions (see Petrucelli and Woodford, 1984), the above GIRF tends to 0 whenk !1. Therefore although we can obtain asymmetric behavior in the effect of theshocks, all of them are transitory.For the simulation and comparison study, we use the following TARUR model

(introduced by Gonzalez and Gonzalo, 1998) with a unit root in the regimecorresponding to large shocks,

yt yt1 t if jt1j40:5;0:6yt1 t if jt1jp0:5:

(Its GIRF is

GIRF k; t;wt1 Eftk1t if jtj40:5;0:6Eftk1t if jtjp0:5;

(where Efto1. Note that in this case all shocks are transitory, although the effectdepends on the shocks size.

Example 3 (TIMA model). In this example we analyze our TIMA Shock-Size model(2) with GIRF described in Result 1. For the simulation and comparison study ofFig. 1, and in order to compare closely to the TAR model of Example 2, we use thefollowing ARTIMA (1,1,1,1) model:

1 0:6L1 Lyt t 0:6t1 if jt1j40:5;t t1 if jt1jp0:5:

(From Result 1 it is obvious that the GIRF of this model is

GIRF k; t;wt1 t if jtj40:5;0:6k1t if jtjp0:5:

(If jtjp0:5 (small shock), t is transitory. When jtj40:5 (large shock), t ispermanent. Recall that although the threshold in t depends on t1; it is the size of twhich determines the type of persistence effect.Examples 2 and 3 are very similar. In both examples, yt behaves like a random

walk in the big shock regime, and like a stationary process in the small shock regime.Nevertheless the GIRF is totally different. Comparing the three theoretical GIRFs itis observed that: (i) in the RW case, there is no asymmetric behavior, GIRF is thesame for all shocks; (ii) in the TAR case, the size of the shock affects its persistence(asymmetric behavior), but all of them are transitory; and (iii) for the ARTIMAShock-Size model, GIRF is equal to the one produced by a random walk whenshocks are large, and closer to the GIRF produced by the TAR model when shocksare small.Fig. 1 shows the results of a simulation exercise, where the GIRFs (divided by t) of

Examples 13 have been estimated. To estimate the GIRF we generate 1000 future30

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347318paths of ftjgj1 for each 200 realizations of t, and only one history wt1. The GIRF

is obtained as the sample mean of these 1000 future paths conditional on the shocksize. Fig. 1 summarizes graphically the results obtained in Examples 13.Finally we compare the TIMA Shock-Size model (2) with the StopBreak model of

Engle and Smith (1999).

Example 4 (StopBreak model). In its simplest form, the StopBreak process is

yt mt t,

where mt is a time-varying conditional mean which is updated via

mt mt1 qt1t1,

with qt qt. In particular, Engle and Smith (1999) consider the following q:function

qgt 2t

g 2 ; g40.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 319t

The GIRF function for the StopBreak model (assuming Eqtt 0) is

GIRF k; t;wt1 qtt.

Fig. 2 compares graphically the GIRF (divided by t) of a StopBreak process (g 5),with the one of a TIMA Shock-Size model (y1 0, y2 1 and r 1). From thisgure, it is clear that in the StopBreak process all the shocks are permanent, while inthe TIMA Shock-Size model both type of shocks (permanent and transitory) are ableto co-exist.

-4 -2 0 2 40.0

0.2

0.4

0.6

0.8

1.0

1.2

StopBreakTIMA

shock

Fig. 2. GIRF of StopBreak (g 5) and TIMA (y1 0; y2 1; r 1 models (divided by t). StopBreakmodel: yt mt t, with mt mt1 qt1gt1 and qt 2t =g 2t . TIMA Shock-Size:

1 Lyt t y1t11jt1j4r y2t11jt1jpr.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 3113473203. Estimation and inference of TIMA Shock-Size models

In TIMA, as well as in TAR models, it is necessary to distinguish between thecases of observable and unobservable threshold variable. When the thresholdvariable is observable (for instance, zt 1 Lyt), it can be proved (see Gonzaloand Martnez (2004)) that the least squares (LS) estimators of the TIMA modelparameters behave asymptotically like the corresponding LS estimators of thediscontinuous TAR model parameters: T1=2 consistent for bm;by1;by2 and T for br. Thelatter speed of convergence is due to the kind of discontinuity in r, present in bothmodels via the indicator function.In the TIMA Shock-Size model, the situation becomes different because the

threshold variable t is unobservable, and therefore has to be estimated. The model isdiscontinuous in all the parameters, again via the indicator function. This is the case,even if the threshold parameter r is known. The discontinuity implies that the rate ofconvergence for all the parameters becomes now T1. In more detail, the (joint)estimation of the parameter vector W m; y1; y2; r is carried out by conditional leastsquares (CLS) (see Chan, 1993). For simplicity, without lack of generality we assumem 0. First we minimize

QT W XTt1

e2t W,

with

etW yt1et1W xt; e0 0,

yt1 y1 if jet1j4r;y2 if jet1jpr;

(and xt 1 Lyt. For a given r, the LS solution is by1r and by2r. Second, weminimize QT by1r;by2r; r, obtaining br. The CLS estimator of W y1; y2; r isbW by1;by2;br by1br;by2br;br.The rate of convergence of bW is provided in the next result.Result 2. Let yt follow the TIMA process (2). Under A.0, A.1 and A.3,

byi;T y0i OpT1 for i 1; 2, and brT r0 OpT1:In TIMA Shock-Size models, all the parameters enter into the indicator functionthrough etW, and therefore there is a discontinuity in r, as well as in the slopeparameters yi (i 1; 2).When the threshold variable is observable the asymptotic distribution of by1;by2 is

obtained in Gonzalo and Martnez (2004). For the TIMA Shock-Size case we havenot been able yet to obtain the asymptotic distribution directly. This is a verydifcult task and we recognize that it is beyond the scope of this paper. Therefore wepropose an alternative way of obtaining the asymptotic distribution of by by1;by2.

This alternative procedure consists on two steps. In a rst step, the TIMA

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 321Shock-Size model is estimated by CLS as previously described. In a second step, weconsider t and r0 to be known and equal to the CLS estimates of the rst step.Basically, this can be done because in the rst step everything is Tconsistent. Inmore detail, in the second step we obtain bybW by minimizing

QT y; bW XTt1

e2t y;bW,with

ety; bW y11jbet1j4brbet1 y21jbet1jpbrbet1 xt.The asymptotic normality of bybW, is derived from the fact that bybW y0

OpT1=2; together with Result 2. This is established in the next result.Result 3. Let yt follow the TIMA process (2). Under A.0, A.1 and A.3,

T1=2bybW y0 !d N0;O,where

O E2t E2t11jt1j4r0 0

0 E2t11jt1jpr0

" #1.

From this result it is easy to test whether a shock is transitory or not. This is doneby testing if one of the y parameters is equal to one, for instant, via a standardt test.

3.1. A testing strategy

In order to be able to detect whether the persistence of shocks is asymmetric ornot, and to discover the existence of pure transitory shocks, we propose a simpletesting strategy inside the framework of TIMA models. This testing strategy followsthe lines of Gonzalez and Gonzalo (1998), Caner and Hansen (2001), and Gonzaloand Montesinos (2002) for TAR models with unit roots. It consists on testing rstfor the existence of a threshold effect

H0 : y1 y2Ha : y1ay2, 6

and second, on testing for a unit root in one of the moving average regimes (e.g. thesecond regime)

H0 : y2 1Ha : y2a1. 7

There are many ways of carrying out a test for the rst hypothesis. In this paper,we test the null of linearity versus the alternative of a threshold effect, in a similar

fashion as Engle and Smith (1999) test linearity versus a StopBreak process. Under

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347322the null hypothesis of linearity we t an ARIMA(p; 1; q) model to yt

FpL1 Lyt mYqLt, (8)

and use the residuals, bt, to construct the following auxiliary regressionbt abt11jbt1jor ut. (9)Under the null, a 0, and under the alternative, aa0. This can be tested by at test. The only problem is that r is not identied under the null, so we proceed asin Davis (1977, 1987) taking the sup of the absolute value of the t-statistic over allvalues of r 2 r; r. The asymptotic distribution is provided in the next result, where) denote weak convergence with respect to the Skorohod metric. FollowingWooldridge (1994), let bl bm; bf1; . . . ; bfp;by1; . . . ;byq be the LS estimator of l0 m0;f01; . . . ;f0p; y01; . . . ; y0q from model (8), dened as

bl arg minl

XTt1

e2t l arg minlXTt1

qtwt; l,

with etl the error term from model (8) for a given value of the parameter l. Then,from Theorem 4.4 in Wooldridge (1994), it is easy to prove that under A.0,

T1=2bli l0i T1=2H1i: l0XTt1

stl0 op1; for i 0; . . . ; p q,

where stl is the score vector, and Hi:l0 the i-row of Hl0 limT!1 T1

PTt1 Ehtl0 with htl the Hessian matrix.

Result 4. Let tar be the t-ratio of the parameter a in the auxiliary regression (9) for agiven r. Under the null of linearity plus assumption A.0,

tar ) Dr in r; r,with

Dr D1r Xpqi0

VliSr" #

Srs2 1=2,

where D1r is a continuous gaussian process with covariance functionM1r1; r2 E2t E2t11jt1jpminr1; r2,

V li a random variable with distribution given by the asymptotic normal distribution of

T1=2bli l0i , and Sr E2t 1jtjpr. Finally, to complete the definition of Dr,CovD1r;V li H1i: l0limT!1 T1

PTt1 Ett11jt1jprstl0.

Then, by the continuous mapping theorem,

sup jtarj !d sup jDrj.

r2r;r r2r;r

parameters.Therefore to obtain the p-values of the test we propose a simple bootstrap method.

Bl1 r2r;r

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 323If the null of linearity is rejected, then we proceed to test the existence of transitoryshocks by constructing the t-statistic of y2 1 in model (2). According to Result 3,this statistic follows a standard normal distribution.

4. Permanenttransitory decomposition

For both, theoretical and empirical reasons, it is frequently desirable todecompose an economic time series into the sum of unobservable permanent andtemporary components that generate the series. Most of the literature has focused ondecomposing linear models, see Watson (1986) and more recently Mortely et al.(2001). As the latter authors note, this implicitly imposes the restriction that thetransitory component (related to business cycle) is symmetric, whereas recentresearch suggests the opposite (see for example, Neftci, 1984; Hamilton, 1989;Beaudry and Koop, 1993). In this section, we propose two new nonlinear permanentand transitory (PT hereafter) decompositions based on the ARTIMA Shock-Sizemodel.First we present a modied version of Quah (1992)s denition of a PT

decomposition.

Denition 1. Let Y be an integrated sequence. A permanenttransitory (PT)decomposition for Y is a pair YP;YT such that:

(i) YP is integrated and YT is covariance stationary;PThis is a model-based bootstrap (see Davison and Hinkley, 1997; Section 8.2.2), thatis next described in a computer algorithm format.Algorithm 1 generates bootstrap samples fy%t gTt1 from model (8) and calculates the

bootstrap approximation of the distribution of supr2r;r jtarj.Algorithm 1 (Model-based Bootstrap procedure).

1. l 1.2. Generate f%t gTt1 resampling from fbt bgTt1, with b T1PTt1bt.3. Generate fy%t gTt1 from bFpL1 Ly%t bm bYqL%t .4. From model (8), with bootstrap sample fy%t gTt1, obtain fb%t gTt1 by LS.5. From model (9), with fb%t gTt1, estimate a%r by LS and compute

tla supr2r;r jta%rj.

6. l l 1. Go to step 2 while lpB.7. Estimate the p-value, pv, from the bootstrap approximation,

pv 1 XB

1 tla4 sup jtarj

!.Th(ii)e asymptotic distribution of supr2r;rjDrj depends on nuisanceY does not have any transitory shock;

y lim Ey jy ; y ; . . . kEy y ,

t t t

, we use the

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347324wher

welle CLL F1L1 y1L and CsL F1L1 y2L. Second

yt F11m yt1 CLLLt CsLst ,To apply this to the ARTIMA Shock-Size model (10), we rst rewrite it astk!1 tk t t1 t t1

yT y yP.(iii) VarnYP and VarnYT are strictly positive, with n 1 L;(iv) Yt YPt YTt .

Further, if

(v) nYP is uncorrelated with YT at all leads and lags,

then the PT decomposition is said to be orthogonal.

The new condition (ii) guarantees that the permanent component is formed onlyby permanent shocks. It is worth mentioning that the Beveridge and Nelsondecomposition does not satisfy condition (v), because its permanent and transitoryshocks are perfectly correlated.Given the following ARTIMA Shock-Size model,

FL1 Lyt m 1 y1LLt 1 y2Lst , (10)with y2 1, the rst new PT decomposition proposed is

yt yPt yTt ,yPt F11m yPt1 FL11 y1LLt , 11

yTt FL1st , (12)where the permanent component is formed only by permanent shocks(Lt t1jtj4r), and the transitory component, only by transitory shocks(st t1jtjpr). It is obvious that this nonlinear decomposition is an orthogonaldecomposition according to Denition 1. An important characteristic of this newPT decomposition is that its existence can be tested, by testing y2 1. Besides, itallows the possibility of a richer dynamic structure in the permanent component thanthe one produced by a pure random walk.There are situations where economic theory suggests the permanent component be

a random walk (see the application to stock prices). For these cases we propose toapply BN directly to our ARTIMA Shock-Size model. The method developed byBeveridge and Nelson (1981) consists on dening the permanent component as thelong-horizon level forecast of the original series, or the part that remains after alltransitory dynamics have worked themselves out. More precisely,

yt yPt yTt ,P-known algebraic result (see Phillips and Solo, 1992) by which a lag polynomial

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 325CL can be written as CL C1 1 L eCL. Then the corresponding BNdecomposition can easily be obtained as follows:

yPt F11m yPt1 CL1Lt Cs1st , (13)

yTt eCLLLt eCsLst . (14)This decomposition constitutes our second proposal of a nonlinear PT

decomposition. For the simple TIMA model (2),

yPt m yPt1 1 y1Lt 1 y2st ,yTt y1Lt y2st .

When the small shocks are transitory (y2 1) the nonlinear BN becomesyPt m yPt1 1 y1Lt , (15)

yTt y1Lt st . (16)The big advantage of this nonlinear BN decomposition, as the following result

shows, does not come from the permanent component but from the transitory one.The above examples show that neither the shocks of the permanent and transitorycomponents are perfectly correlated nor that all the shocks at time t are permanent.In this way we overcome the two main drawbacks of the linear BN decomposition(more examples of nonlinear BN decompositions can be found in Clarida andTaylor, 2003).The next result establishes the relationship between the BN and UC0

decompositions applied to the Wold (1938) representation of a TIMA Shock-Sizeprocess, and the nonlinear BN decomposition (15)(16). Note that, in order forUC0 to exist, the parameter y1 must satisfy 0py1p1.Result 5. Let yt be the TIMA model (2), then

VBNDyPt VTIMADyPt VUC20DyPt ,VBNyTt pVTIMAyTt pVUC20yTt ,

where VBN and VUC20 are the variances of the components of the BN and UC0decompositions applied to the Wolds representation of 1 Lyt, and VTIMA is thevariance of the components forming the nonlinear BN decomposition of the TIMAmodel. Strict inequality is obtained if 0oy1o1.As it is well known, (see Quah, 1992), the variance of the innovations of the

random walk component is always the same (equal to the height of the spectraldensity of 1 Lyt at frequency zero), independently of the used decomposition.However, for 0oy1o1, the variance of the transitory component of eachdecomposition is different. In particular, the variance of the transitory componentof the nonlinear BN decomposition for the TIMA model (2) lies always between thecorresponding variances of the UC0 and the linear BN decompositions. This fact

has important consequences for the applications analyzed in the next two sections.

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0.6 UC-0

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 3113473260.0 0.2 0.4 0.6 0.8 1.00.5

TIMA Shock Size0.7

0.8

0.9

1.0

B-N

2,L/2=0.2In Fig. 3 we illustrate graphically the inequalities of Result 5 for y2 1, anddifferent values of y1 2 0; 1 and of the ratio E2t 1jtj4r=E2t .

5. Application to stock prices: a measure of the stock market quality

In this section we apply ARTIMA Shock-Size models to measure the deviationsbetween actual transaction and implicit efcient prices, following the methodologydeveloped by Hasbrouck (1993). Measurement of this difference arises in nancialmarket analysis in two important contexts. First, with the purpose of evaluating the

2,L/2=0.5

2,L/2=0.8

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

1

0.0 0.2 0.4 0.6 0.8 1.01

0.0 0.2 0.4 0.6 0.8 1.01

Fig. 3. Variance of the transitory component of different PT decompositions of a TIMA Shock-Size

Process. s2;L E2t 1jtj4r and s2t E2t . y2 1 and t N0; 1.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 327brokers performance, and second, for a comparative analysis of the marketregulatory structures. Hasbroucks approach is based on decomposing thetransaction price pt, into two components, the efcient price mt, and the pricingerror st, and on measuring the relative size of st. In more detail Hasbroucks modeltakes the logarithm of the observed transaction price at time t, as the sum of twocomponents:

pt mt st.The element mt (efcient price) is assumed to follow a random walk,

mt mt1 t,and st (pricing error) is a zero-mean covariance-stationary process that is modelledas

st bt Zt,where Zt is uncorrelated with t. Given that the permanent component is identiedwith a random walk, the correct PT decomposition depends only on theassumption about the correlation structure of both components. If the shocks ofboth components are perfectly correlated (Zt 0), we are in the BN scenario. Ifboth shocks are uncorrelated (b 0), UC0 is the decomposition to be used.Unfortunately, the two identication restrictions are obviously not testable, and as itis shown in the previous section they imply different dispersion of the pricing error.Besides, economic theory does not argue conclusively in favor of any of theidentication restrictions.Roll (1984) bid-ask spread model corresponds to b 0 and Zt spread=2.

Roll proves that the bid-ask spread generates a rst order autocorrelation differentfrom 0 in the increments of the transaction prices, and this correlation explains theexistence of the pricing error. The perfect correlation restriction is a particular caseof Glosten (1987) model. In this model, the spread is partially due to the asymmetricinformation revealed in the trade. With no nontrade public information, it isobtained b40 and Zt 0.Starting from these two models, we develop a more complete model which takes

into account several issues not considered in any of the previous ones. We assume themarket description of Glosten (1987), that is, two kinds of agents, uninformed andinformed, with a market maker. The rst aspect to be considered is that when there isa transaction cost, not all the new information will be translated into the transactionprices. Only the new information which implies a prot greater than the transactioncost will be translated into the transaction price. In other words, the shocks thatdrive the efcient price component must be big shocks to the transaction price.The second aspect is that the transactions of the uninformed agents cannotgenerate big inefcient changes in the transaction prices, because the informedtraders will arbitrate the situation. This implies that the shocks driving the pricingerror component must be small shocks to the transaction price. Based on these

two aspects and taking Glosten model as starting point we propose the following

pricing specication

pt mt st,mt mt1 1jtj4rt,st 1jtjprt.

The parameter r is identied as the spread or transaction cost, and only the shocksgreater than this cost will affect the efcient price mt. From this specication thefollowing representation for the transaction prices is obtained

1 Lpt t if jtj4r;t t1 if jtjpr:

((17)

Model (17) is a TIMA Shock-Size model. At this point, it is worth mentioningagain that although we only can observe one shock, t, in the structural model there

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347328are two type of shocks. One related to new information and therefore driving theefcient price process, and another coming from the uninformed traders andtherefore of transitory nature. Using the reasoning described above, these two typesof structural shocks translate into the transaction price process in terms of big andsmall shocks. Within this TIMA model we are able to test the existence of thesetwo type of structural shocks.Fig. 4 represents the logarithm of the S&P500 daily series. Table 1 shows the Least

Squares estimates of the linear ARIMA and nonlinear ARTIMA model tted toS&P500 daily series. Parameter estimates of the TIMA Shock-Size model wereobtained by numerically minimizing the squared estimated error using theFORTRAM optimization procedure DBCPOL. The moving average parameterestimates of the TIMA model are clearly different. Following the testing strategydescribed in Section 3.1, we test the validity of this model in two steps. In the rstone, the null hypothesis of linearity is tested against the alternative of threshold

6.8

7.0

7.2

1998:01:02 1999:08:04 2001:03:06 2002:10:10Fig. 4. Logarithm of S&P500, 2 January 199829 July 2003.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 329Table 1

Least squares estimates of ARIMA(0,1,1) and ARTIMA(0,1,1,1) models for stock prices

Parameters y y1 y2 r s AIC

ARIMA0; 1; 1 0:01760:0267

0.0136 5.7521ARTIMA(0,1,1,1) 0:0131

0:02680:97170:3632

0.0035 0.0135 5.7564

Note: In brackets are the corresponding standard errors. The ARTIMA parameter estimates correspond

to the second step procedure described in Section 3.effect. If we reject this null, in a second step we test the null hypothesis of a unit rootin the MA regime driven by small shocks.To test linearity, we t an ARIMA(0,1,1) to the logarithm of the series, construct

the following auxiliary regression with the estimated residuals,bt a0 a1bt1 a2bt11jbt1jpr ut,and test the signicance of a2r. Given that under the null hypothesis the parameterr is unidentied, the test proposed is suprjta2rj, where ta2r is the t-ratio for the nullof a2r 0. The results of this test are shown in Table 2.Linearity is rejected at a 10% signicant level (p-value 0:067). From Table 1 the

hypothesis of a unit root in the MA side corresponding to the regime of small shockscannot be rejected at the usual signicance levels. With these two results we conclude

Table 2

Testing linearity versus threshold Shock-Size effect

Auxiliary regressionbt a0 a1bt1 a2bt11jbt1jprsupr jta2 j p-value ba0r ba1r ba2r r2.213 0.067 0:000

0:00030:0040:0268

0:8060:3645

0.0035

Note: bt are the residuals of the null linear model. Bootstrap p-value is obtained from 1000 replications. Inbrackets are the corresponding standard errors.

Table 3

Variance decomposition of different PT decompositions for stock prices

BN ARIMA(0,1,1) BN ARTIMA(0,1,1,1) UC0 ARIMA(0,1,1)

sDmt 102 1.338 1.370 1.339sst 102 0.024 0.074 0.176

Note: pt mt st, with mt a random walk and st the transitory component. sDmt and sst are the standarddeviation of Dmt and st, respectively.

shocks are responsible for economic uctuations. This question has been analyzed in

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347330detail by Blanchard and Quah (1989), Cochrane (1994), and many other authors.Cochrane (1994) remarks two key issues: (i) shock identication and (ii) endogenousshocks. About the latter, he demands new theoretical models, probably nonlinear, thatare able to explain the uctuation of the business cycle in terms of endogenous shocks.TIMA Shock-Size models are perfect candidates to handle both issues simultaneously.Fig. 5 represents the logarithm of quarterly real GNP for the period 1947:01-

2003:03. Table 4 shows the least square estimates of several linear ARIMA modelsand a nonlinear ARTIMA model tted to GNP. Following Campbell and Mankiw(1987), we have considered all ARIMA models with p and q less than 3. For spaceconsiderations, Table 4 only presents the results for the ARIMA(1,1,0), ARI-MA(0,1,1) and ARIMA(1,1,1). The ARTIMA Shock-Size tted model is

1 fL1 Lyt m 1 y1LLt 1 y2Lst .Table 4 shows that all linear models t the data very similarly. According to the

AIC criteria the best linear model is the ARIMA(1,1,0), however to eliminate a fewthat the analyzed S&P500 data do not reject the proposed TIMA Shock-Size model.Therefore in order to measure the size of the pricing error component st, we obtainthe nonlinear BN decomposition of the estimated TIMA model. To do that weimpose y2 1. The results are in Table 3. Notice that, as the theory suggests, thecontribution of the random walk component is practically the same in all the PTdecompositions, and the sizes of the transitory components satisfy the inequalityestablished in Result 5.Besides this measure of the quality of the stock market, the TIMA Shock-Size

model provides some extra information. For example, the estimated value of thethreshold parameter br 0:00283 (imposing y2 1) implies that the estimatedtransaction cost is 0.283 percent of the stock price. This gure is very similar tothat obtained by Hasbrouck (1993) for intra-day data. Also, the percentage of puretransitory shocks can be estimated (23:65%). The periods where these shocks occurredcan be identied in this context with periods of non-outstanding information.

6. Application to GNP: asymmetric persistence of shocks to output

The goal of this section is to answer empirically an important question aboutoutput uctuations: Is the persistence of shocks to output asymmetric? This questionhas been analyzed by Beaudry and Koop (1993), Hess and Iwata (1997) and Elwood(1998) among others. All these authors deal with asymmetries produced by the signof the shocks, and there is not a clear conclusion. The rst authors give anafrmative answer about the existence of asymmetries, while the others contradictthis result. We will use our TIMA Shock-Size model to conclude that shocks withdifferent sizes have different long run effects on output. Small shocks will betransitory, while large shocks will be permanent.In this section, indirectly we will also try to answer the question of which type ofcorrelations in the residuals, we select the ARIMA(1,1,1) model for our testing

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 3311947Q1 1956Q4 1966Q4 1976Q4 1986Q4 1996Q4 2003Q4

7.5

8.0

8.5

9.0strategy. This testing strategy is the one suggested in Section 3.1 and used in theprevious application. To test the null of linearity versus the alternative of a thresholdeffect, we construct the following auxiliary regression with the residuals of the linearARIMA (1,1,1)

bt a0 a1bt1 a2bt11jbt1jpr ut.In this regression we test the signicance of a2r with the suprjta2rj, where ta2r is the

t-ratio for the null of a2r 0. The results of this test are shown in Table 5. Clearly thenull of linearity is rejected with a p-value smaller than 0:05. This implies that the size ofthe shocks is able to generate asymmetries in the persistence of shocks to GNP.In order to test whether or not small shocks are transitory, we test y2 1. From

Result 3, this is done directly from last column of Table 4, concluding that the null ofa unit root in the MA side cannot be rejected. Therefore the GNP data do not rejectthat large shocks are persistent while small shocks are only transitory.

Table 4

Least square estimates of ARIMA and ARTIMA models for real GNP

Parameter ARIMA(1,1,0) ARIMA(0,1,1) ARIMA(1,1,1) ARTIMA(1,1,1,1)

f 0.335 (0.062) 0.453 (0.171) 0.647 (0.182)y 0.260 (0.064) 0.131 (0.191)y1 0.320 (0.196)y2 1.210 (0.396)r 102 0.477s 102 0.956 0.969 0.950 0.941AIC 6.452 6.424 6.447 6.475

Note: In brackets are the corresponding standard errors. The ARTIMA parameter estimates correspond

to the second step procedure described in Section 3.

Fig. 5. Log of real GNP, 1947:01-2003:03.

Imposing y2 1, we calculate the nonlinear orthogonal PT decomposition((11)(12)) proposed in Section 4. This becomes

1 LyPt m

1 f1 y1L1 fL

Lt ,

yTt 1

1 fL st ,

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Table 5

Testing linearity versus threshold Shock-Size effect

Auxiliary regressionbt a0 a1bt1 a2bt11jbt1jprsuprjta2 j p-value ba0r ba1r ba2r r2.362 0.045 0:000

0:00060:0210:0676

0:7890:3363

0.004

Note: bt are the residuals of the null linear model. Bootstrap p-value is obtained from 1000 replications. Inbrackets are the corresponding standard errors.

1947Q1 1956Q4 1966Q4 1976Q4 1986Q4 1996Q4 2003Q47.35

8.31

9.26B-N Permanent Component of GNP and NBER Reference Points, 1947:01-2003:03

0.01

0.02B-N Transitory Component of GNP and NBER Reference Points, 1947:01-2003:03

(a)

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 3113473321947Q1 1956Q4 1966Q4 1976Q4 1986Q4 1996Q4 2003Q4-0.02

-0.01

0

(b)

Fig. 6. BN decomposition of real GNP based on an ARIMA(1,1,1): (a) BN Permanent component of

GNP and NBER reference points, 1947:01-2003:03; (b) BN Transitory Component of GNP and NBERReference Points, 1947:01-2003:03.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 3337.35

-6.40

1947Q1 1956Q4 1966Q4 1976Q4 1986Q4 1996Q4 2003Q4

Non-Linear Transitory Component of GNP and NBER Reference Points, 1947:01-2003:03

-0.014

-0.007

0.0

0.008

0.016

(a)Non-Linear Permanent Component of GNP and NBER Reference Points, 1947:01-2003:039.26where st corresponds to shocks smaller than br 0:0046. This implies that shockssmaller than approximately s=2 are pure transitory. They amount to 51% of theGNP shocks (the remain 49% are permanent).Figs. 68 represent graphically the PT decompositions discussed in Section 4,

together with the NBER reference points. Numerically this comparison issummarized in Table 6. Both linear and nonlinear BN decompositions producevery similar results (the differences are due to estimation issues). However, theorthogonal nonlinear PT decomposition is clearly different. First, the transitorycomponent is less volatile, and second, the permanent component is smoother.To nish, it is worthwhile to mention the close link between the large shocks from

the ARTIMA (1,1,1,1) model and the NBER turning points. From Fig. 9 it can beseen that peaks and troughs are related to large shocks of opposite signs. Situationswhere output leaves a trough are generated by a positive large shock, and theopposite occurs when the output abandons a peak.

7. Conclusion

In this paper we introduce a new class of simple nonlinear models (TIMA) withthe aim of being able to identify the shocks of a dynamic system. This identication

1947Q1 1956Q4 1966Q4 1976Q4 1986Q4 1996Q4 2003Q4(b)

Fig. 7. BN decomposition of real GNP based on an ARTIMA(1,1,1,1): (a) Nonlinear permanent

component of GNP and NBER reference points, 1947:01-2003:03; (b) Nonlinear transitory component of

GNP and NBER reference points, 1947:01-2003:03.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 3113473348.31

9.26Non-Linear Permanent Component of GNP and NBER Reference Points, 1947:01-2003:03is based on the long run effects of the shocks (permanent or transitory) and on theirsizes (Large or small). The latter is a special case rather than a necessity. It could beany other shock or economic characteristic. TIMA models provide a nice frameworkto test if a shock with a given characteristic (in this paper its size) has a permanent ora transitory effect. Once shocks are identied we construct two new nonlinearpermanenttransitory decomposition, that are compared with the standard linearBN and UC0 permanenttransitory decompositions available in the literature.

Table 6

Variance decomposition of different PT decomposition for real GNP

s2DyPt

103 s2yTt 103

BN ARIMA(1,1,1) 0.2288 0.0398

BN ARTIMA(1,1,1,1) 0.1825 0.0229

Orthogonal ARTIMA(1,1,1,1) 0.0982 0.0032

Note: yt yPt yTt . In the BN decomposition yPt is a random walk. In the orthogonal ARTIMAdecomposition yPt is formed only by permanent shocks, s.t EDyPt ; yTt 0 (see (11) and (12)). s2DyPt and s

2yTt

are the variance of DyPt and yTt , respectively.

1947Q1 1956Q4 1966Q4 1976Q4 1986Q4 1996Q4 2003Q47.36

1947Q1 1956Q4 1966Q4 1976Q4 1986Q4 1996Q4 2003Q4-0.005

-0.002

0

0.002

0.004Non-Linear Transitory Component of GNP and NBER Reference Points, 1947:01-2003:03

(a)

(b)

Fig. 8. Orthogonal decomposition of real GNP based on an ARTIMA(1,1,1,1): (a) Nonlinear permanent

component of GNP and NBER reference points, 1947:01-2003:03; (b) Nonlinear transitory component of

GNP and NBER reference points, 1947:01-2003:03.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 3351947Q1 1956Q4 1966Q4 1976Q4 1986Q4 1996Q4 2003Q4

-0.03

-0.01

Fig. 9. Large shocks and the NBER reference points, 1947:01-2003:03.0

0.01

0.03Two applications of these new nonlinear PT decompositions are shown in thepaper. The rst one studies the quality of the stock market by measuring theimportance of the transitory component (pricing error) in the total transaction price.The second one studies the asymmetries in the persistence of the GNP shocks. Whilethe literature has been inconclusive to this respect when the asymmetry comes fromthe shock sign, we nd a clearly afrmative response when the asymmetry isgenerated by the shock size.Extensions to a multivariate framework is under current research by the authors.

Acknowledgements

This work was partially supported by the Spanish Ministry of Science andTechnology under DGCYT Grant SEC01-0890 and SEJ2004-04101. Helpfulcomments from Juan Jose Dolado, Juan Carlos Escanciano, Bruce Hansen,Francesc Marmol, Jean-Yves Pitarakis, Barbara Rossi, Carlos Velasco, and seminarparticipants at Bilkent University, ECARES, NBER Time Series Conference 2004,Sabanci University, Southampton University, UC Riverside, UCSD Conference inhonor of Clive Granger, Winter Meetings of the North American EconometricSociety 2005, Zaragoza University, the editor and anonymous referees are gratefullyacknowledged. This research is part of Oscar Martnez Ph.D. dissertation.

ARTICLE IN PRESSAppendix

Proof of Result 1. This result is easily proved by rewriting model (2) at time t k as

ytk 1

1 L m1 y1L1 L tk11jtk1j4r

1 y2L1 L tk11jtk1jpr. 18

Conditioning on wt1, the past history of the process (included 0), and using (18),the GIRF has the following expression:

GIRF k; t;wt1 1 y1ft1jtj4r Et1jtj4rjwt1g 1 y2ft1jtjpr Et1jtjprjwt1g

Xk2i0

1 y1fEtk1i1jtk1ij4rjt;wt1

Etk1i1jtk1ij4rjwt1g

Xk2i0

1 y2fEtk1i1jtk1ijprjt;wt1

Etk1i1jtk1ijprjwt1g.

Under Assumption A.4, we obtain that

Etj1jtjj4rjwt1 Etj11jtj1j4rjt;wt1 0,

for jX0. Hence,

GIRF k; t;wt1 1 ytt 1 y1t if yt y11 y2t if yt y2:

(&

Proofs of Results 2 and 3. They can be found in Gonzalo and Martnez (2004) andare available upon request. &

Proof of Result 4. For simplicity and without loss of generality, this result is provedfor the case of p 1, q 1 and m 0. Part of the proof makes use of some of theresults in Bai (1994).Let us write the ARMA(1,1) representation of xt 1 Lyt as t xt

f0xt1 y0t1 and the corresponding residuals as bt xt bfxt1 bybt1, wherebf;by are the LS estimators of the parameters f0; y0. These estimators satisfy that1=2 1=2

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347336T bf f0 and T by y0 are Op1.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 337Subtracting the errors from the residuals, making use of b0 0, and by repeatedsubstitution we obtain

bt t 1t1byt0 bf f0Xt1j0

1jbyjxt1j by y0Xt1j0

1jbyjt1j.(19)

Denote r^ by; Tp bf f0; Tp by y0 and r u; v;w 2 R3: DeneLr;t ut0 T1=2 v

Xt1j0

ujxt1j wXt1j0

ujt1j

! ut0 T1=2xrt. (20)

From (9) and the denition of tar

tar barbsa T1=2 X

T

t1btbst1r T1XT

t1bst1r2T1XT

t1bu2t r

" #1=2, (21)

where bstr bt1jbtjpr and butr bt barbst1r.The proof basically consists on decomposing the RHS term of (21) in several

pieces, and analyzing the asymptotic behavior of these pieces. For the rst term,T1=2

PTt1btbst1r, we prove that,

T1=2XTt1btbst1r X 1;T r X 2;TX 3;T r op1,

where the op1 is uniform for all r 2 r; r and

X 1;T r ) D1r,X 2;T !d Vf V y,X 3;T r!

pSr uniformly in r,

with Vf and V y normal random variables with distribution, the asymptoticdistribution of T1=2bf f0 and T1=2by y0, respectively. For the other term in(21), we prove that

T1XTt1

bst1r2T1 XTt1

bu2t r X 4;T r op1,where

X 4;T r!pSrs2 ,uniformly in r:

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347338We start with T1=2PT

t2btbst1r. DeneUx;tby Xt1

j01jbyjxt1j,

U;tby Xt1j0

1jbyjt1j.From (19), T1=2

PTt2btbst1r can be written as

T1=2XTt1btbst1r T1=2 XT

t1tt11jbt1jpr

bf f0T1=2 XTt1

tUx;t1by1jbt1jpr by y0T1=2 XT

t1tU;t1by1jbt1jpr bf f0T1=2

XTt1

t1Ux;tby1jbt1jpr by y0T1=2XTt1

t1U;tby1jbt1jpr R1;tbf;by; r. 22The rst term of the RHS of (22) can be written as

T1=2XTt1

tt11jbt1jpr T1=2 XTt1

tt11jt1jpr T1=2

XTt1

tt11jbt1jpr 1jt1jpr. 23The Lemma 3.1 of Koul and Stute (1999) establishes that

T1=2XTt1

tt11jt1jpr ) D1r,

where D1r is a gaussian process with a covariance function given byM1r1; r2 E2t E2t11jt1jpminr1; r2.

For the last term of the RHS of (23) it sufces to prove that

supr2r;r

T1=2XTt1

tt11bt1or 1t1or

op1.Given that tt1 tt1 tt1; where t t1t40 and t t1to0, it is

enough to prove that

supr2r;r

T1=2XT

tt11bt1or 1t1or

op1, (24)t1

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 339since the case of tt1 is similar. For that, we use the proof of Theorem 1 in Bai(1994). Dene

r Et11t1or,with 1 0 and 1 K o1: Then, using (19) and (20)

T1=2XTt1

tt11bt1or 1t1or

T1=2XTt1

tt11t1or Lr;t1 1t1or

bKT r KT r.To study this process, bKT r KT r, it sufces to study the auxiliary process,

GT r; r T1=2XTt1

tt11t1or Lr;t1 1t1or.

Expression (24) is implied by the following result:

supr2Db

supr2r;r

jGT r; rj op1 for every b40, (25)

where Db y; y b; b2 with 0oyo1 and b40: Dene

ZT r;r T1=2XTt1

ftt11t1or Lr;t1 r Lr;t1

tt11t1or rg,

PT r; r T1=2XTt1

t r Lr;t1 r.

Then, jGT r;rjpjZT r;rj jPT r;rj: To show (25), it sufces to prove thefollowing two propositions.

Proposition 5. Under Assumption A.0, we have

supr2Db

supr2r;r

jZT r; rj op1 for every b40.

Proposition 6. Under Assumption A.0, we have

supr2Db

supr2r;r

jPT r;rj op1 for every b40.

The proof of both propositions follows Bai (1994) and therefore we do not show it

in full detail. We will focus only on those details that are different. Given

non-decreasing monotonicity of the indicator function used in Bai (1994).

lemma is needed.

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347340Lemma 2. Under Assumption A.0, for every given r u; v;w 2 Db and every l 2 R;we have

(a) T1=2PT

t1 t jut0j ty

t1jl0j op1;(b) T1=2max1ptpT t jxrtj jlZtj op1;(c) T1

PTt1

t jxrtj jlZtj Op1.Proof of Proposition 5. Dene Zt CPt1

j0 tjjxtj1j jtj1j for some C40 and

t 2 0; 1, and for every l 2 R;

eZT r;r; l T1=2XTt1

t ft11t1or Gt1r; l

r Gt1r; l t11t1or rg,

where Gtr; l ut0 ltyt1j0j T1=2xrt lT1=2Zt. Note that eZT r; r; 0 ZT r;r.Using the compactness of Db; this set can be partitioned into a nite number of

subsets such that the diameter of each one is less than z: Denote these subsets byn1;n2; . . .nmz. Fix k and considernk and rk uk; vk;wk 2 nk: Proceeding as Bai(1994), choose C large enough and t 2 0; 1 such that

jLr;t Lrk ;tjpztyt1j0j zT1=2Zt for all r 2 nk.

With this inequality and the monotonicity of the functions involved in ZT r;r, itcan be proved that

ZT r; rp eZT r;rk; z T1=2 XTt1

t r Gt1rk; z r Lr;t1,

for all r 2 nk, and the reverse inequality with z replaced by z: Now the followingthat t t t , it is enough to prove Proposition 5 for

ZT r;r T1=2XTt1

ft t11t1or Lr;t1 r Lr;t1

t t11t1or rg.

The case t is similar. Note that t

t11t1or Lr;t1 and t r maintain theThis lemma can be proved in a similar way as Lemma 1 in Bai (1994).

ARTICLE IN PRESS

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 341Applying Lemma 2 to the next inequality, that has been obtained from a meanvalue theorem,

T1=2XTt1

t r Gt1rk z r Lr;t1

p2zf eT1=2

XTt1

t t 1yt2j0j T1=2Zt1; for all r and r 2 Db,

with f e maxe ef eoKfeo1, it is straightforward to show that

supr2Db

supr2r;r

T1=2XTt1

t r Gt1rk z r Lr;t1

zOp1,where the Op1 is uniform in r and r 2 Db. Therefore,

supr2Db

supr2r;r

jZT r; rjp maxkpmz

supr2r;r

j eZT r;rk; zj max

kpmzsup

r2r;rj eZT r;rk;zj zOp1. 26

Then, zOp1 can be arbitrary small in probability, so Proposition 5 follows if therst two terms of the RHS of (26) are op1: This will be true if

supr2r;r

j eZT r;r; zj op1 for everyr; z. (27)For that Lemma 3 is needed.

Lemma 3. For every 2 0; 1=2; every r u; v;w 2 Db and every l 2 R;

supr1 ;r22BT ;

T1=2XTt1

t j r2 Gt1r; l r1 Gt1r; lj op1,

where BT ; fr1; r2 2 R R; j r1 r2jpK ;1T1=2g for some K ;1o1.

Given the uniform continuity and boundedness of our r, the proof of thislemma is analogous to that of Lemma 2 of Bai (1994).Now divide the real line into NT T1=2 1 parts, with 2 0; 1=2; by

points r r0or1o orNT r with ri iNT1K ;1: Since t t11t1orand t r are non-decreasing, for rkorork1, we have

j eZT r;r; zjpj eZT rk1; r; zj T1=2 XTt1

t rk1 Gt1 r Gt1

T1=2XTt1

ft t11t1ork1 rk1

1 or rg,t t1 t1

ARTICLE IN PRESSand the reverse inequality with rk1 replaced by rk: Therefore,

supr2r;r

j eZT r;r; zjpmaxk

j eZT rk;r; zjmax

kT1=2

XTt1

t rk Gt1 r Gt1

sup

jgg0jpK NT1T1=2

XTt1

ft t11t1o 1g g

t t11t1o 1g0 g0g. 28For the third summand of the RHS of expression (28), it is needed to prove that

T1=2PT

t1 t t11t1o 1g g is tight. This process can be written as

T1=2XTt1

t t11t1oF1g g

T1=2XTt1

t Et t11t1o 1g g

T1=2XTt1

Et t11t1o 1g g.

The tightness of the rst and second summand can be proved by using Lemma 3.1of Koul and Stute (1999). Then, the op1 of both summands follows from thetightness of the process and the fact that K ;1NT1 op1. The convergence inprobability to 0 of the second summand of the RHS of (28) follows from Lemma 3.Finally, eZT rk;r; z can be expressed as

eZT rk; r; z T1=2 XTt1

t Et dt1rk; r; z T1=2Et XTt1

dt1rk;r; z,

with dt1rk;r; zt11t1ork Gt1r; z rk Gt1r; z t11t1orkrk. Then, the processes involved in both sums are martingale arrays, and we canapply directly Bai (1994) to obtain that

maxk

j eZT rk;r; zj op1: &Proof of Proposition 6. This proof is analogous to Proposition 2 of Bai (1994), and itis thus omitted. &

With this, result (24) is proved. Then, getting back into expression (23), we haveproved

T1=2XTt1

tt11jbt1jpr ) D1r op1,

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347342where the op1 term is uniform in r:

ARTICLE IN PRESSFor the rest of the terms in the RHS of (22), using that

E supu

U2x;tu

pKUxo1,

E supu

U2;tu

pKUo1,

and that t is iid, it can be established, in a similar way as (23) and (24), that

T1XTt1

tUx;t1by1jbt1jpr op1,T1

XTt1

tU;t1by1jbt1jpr op1,T1

XTt1

t1Ux;tby1jbt1jpr T1 XTt1

2t11jt1jpr op1,

T1XTt1

t1U;tby1jbt1jpr T1 XTt1

2t11jt1jpr op1,

T1XTt1

2t11jt1jpr !p

E2t11jt1jpr Sr uniformly in r 2 r; r,

supr2r;r

R1;tbf;by; r op1,where the op1 terms are uniformly in r. Now, given that T1=2bf f0; by y0converges in distribution to Vf;V y, which is a multivariate normal randomvariable with mean 0 and variance covariance matrix Mf;y, it is straightforward toprove that

T1=2XTt1btbst1r ) D1r Vf VySr, (29)

with CovD1r;Vf H11: l0limT!1 T1PT

t1 Ett11jt1jprstl0 andCovD1r;V y H12: l0limT!1 T1

PTt1 Ett11jt1jprstl0.

Finally, it is necessary to prove that

T1XTt1

bst1r2!p Sr, (30)

T1XTt1

bu2t r !p s2 , (31)2

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 343uniformly in r 2 r; r with 0oSro1 and 0oso1:

ARTICLE IN PRESS

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347344The proofs of (30) and (31) are tedious and similar to the above proofs. So, weonly present an outline of them. For (30) note that bstr2 can be written as

bstr2 t t bt21jbtjpr 2t 1jtjpr 2t 1jbtjpr 1jtjpr

t bt21jbtjpr 2tt bt1jbtjpr.About the rst term, T1

PTt1

2t 1jtjpr; given Assumption A.0, it is

straightforward to show that

T1XTt1

2t 1jtjpr !pSr,

uniformly in r 2 r; r with 0oSro1: The rest of terms can be handledas T1

PTt1

t

t11bt1or 1t1or; R1;tbf;by; r and T1PTt1 tUx;t1by1

jt1jor, respectively. Once (30) is proved, it is straightforward to show that

bar !p 0, (32)uniformly in r 2 r; r: From (30) and (32), it is easy to obtain (31).Result 4,

supr2r;r

jtarj !d supr2r;r

jDrj,

with

Dr D1r Vf V ySrSrs2 1=2,follows directly from the continuous mapping theorem and (29)(31). &

Proof of Result 5. For simplicity and without lack of generality we present the prooffor the case of y2 1.Let s2x V xt, and rx1 Covxt; xt1 with xt 1 Lyt. If yt follows the

TIMA model (2), then,

s2x 2s2 1 y21s2;L,rx1 s2 1 y1s2;L, 33

where s2;L ELt 2. From Eqs. (15) and (16),VTIMADyPt s2x 2rx1,VTIMAyTt rx1 y21 y1s2;L.

To obtain the corresponding results for linear BN decomposition, we use theWold representation of the TIMA model,1 Lyt m ut yut1, (34)

Finally, proceeding in a similar way, the UC0 decomposition of (34) is

ARTICLE IN PRESS

J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 3451 LyPt m ZPt ,yTt ZTt ,

with EZPt EZTt 0, EZPt 2 s2ZP , EZTt 2 s2

ZT, EZPt ; ZTt 0, and such that

s2ZP s2x 2rx1 and s2ZT rx1. ThereforeVUC20DyPt s2x 2rx1,VUC20yTt rx1.

Then, the VarDyPt is the same for the three decompositions. With respect to theVaryTt , using the fact that the parameters y and y1 are less than 1, it isstraightforward to show the largest one is obtained in the case of the UC0decomposition. To prove that VBNyTt is less or equal than VTIMAyTt , it is enoughto prove that s2uXs

2 , since

VBNyTt s2x s2u,VTIMAyTt s2x s2 .

From the Wold representation (34),

s2u s2x

s2x2 4r2x1

q2

.

Given that rx1 ys2u and jyjo1, the valid solution is

s2u s2x

s2x2 4r2x1

q2

.

Substituting s2x and rx1 in terms of y1, s2 and s

2;L from (33), we have

s4x 4rx12 s4;L1 y212 s2s2;L s4;L2 2y12.Then, by using that s2x 2s2 1 y21s2;L and s24s2;L it is obtained that s2uXs2

and the result follows. &

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J. Gonzalo, O. Martnez / Journal of Econometrics 135 (2006) 311347 347

Large shocks vs. small shocks. (Or does size matter? May be so.)IntroductionTIMA Shock-Size modelsImpulse response function

Estimation and inference of TIMA Shock-Size modelsA testing strategy

Permanent-transitory decompositionApplication to stock prices: a measure of the stock market qualityApplication to GNP: asymmetric persistence of shocks to outputConclusionAcknowledgementsAppendixReferences