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PROBABILITYANDMATHEMATICAL STATISTICSVol. 31, Fasc. 1 (2011), pp. 300–000NONLINEARITY OF ARCH AND STOCHASTIC VOLATILITY MODELSAND BARTLETT’S FORMULABYPIOTR S . K O K O S Z K A∗(LOGAN) AND DIMITRIS N . P O L I T I S∗∗(SAN DIEGO)Abstract. We review some notions of linearity of time series and showthat ARCH or stochastic volatility (SV) processes are not only non-linear:they are not even weakly linear, i.e., they do not even have a martingalerepresentation. Consequently, the use of Bartlett’s formula is unwarrantedin the context of data typically modeled as ARCH or SV processes such asfinancial returns. More surprisingly, we show that even the squares of anARCH or SV process are not weakly linear. Finally, we discuss an alterna-tive estimator for the variance of sample autocorrelations that is applicable(and consistent) in the context of financial returns data.2000 AMS Mathematics Subject Classification: Primary: 62M10;Secondary: 60G42.Key words and phrases: ARCH processes, GARCH processes, lineartime series, stochastic volatility.1. INTRODUCTIONIn the theory and practice of time series analysis, an often used assumption isthat a time series {Xt, t ∈ Z} of interest is linear [17], i.e., that(1.1) Xt= a +∞∑i=−∞αiξt−i, where ξt∼ i.i.d. (0, 1),which means the ξts are independent and identically distributed with mean zeroand variance one.1Recall that a linear time series {Xt} is called causal if αk= 0 for k < 0, thatis, if(1.2) Xt= a +∞∑i=0αiξt−i, where ξt∼ i.i.d. (0, 1).∗Partially supported by NSF grants DMS-0804165 and DMS-0931948.∗∗Partially supported by NSF grant DMS-0706732.1When writing an infinite sum as in (1.1), it will be tacitly assumed throughout the paper thatthe coefficients αiare (at least) square-summable, i.e., that∑iα2i< ∞.2 P. S. Ko koszka and D . N. P o lit i sEquation (1.2) should not be confused with the Wold decomposition that all purelynondeterministic time series have [5]. In the Wold decomposition the ‘error’ series{ξt} is only assumed to be a white noise, i.e., uncorrelated, and not i.i.d. A weakerform of (1.2) amounts to relaxing the i.i.d. assumption on the errors to the assump-tion of a martingale difference, i.e., to assume that(1.3) Xt= a +∞∑i=0αiνt−i,where {νt} is a stationary martingale difference adapted to Ft, the σ-field gener-ated by {Xs, s ¬ t}, i.e., that(1.4) E[νt|Ft−1] = 0 and E[ν2t|Ft−1] = 1 for all t.For conciseness, we propose to use the term weakly linear for a time series{Xt, t ∈ Z} that satisfies (1.3) and (1.4).Many asymptotic theorems in the literature have been proven under the as-sumption of linearity or weak linearity [5]. In the last thirty years, however, therehas been a surge of research activity on nonlinear time series models. One of thefirst such examples is the family of bilinear models [16], that is a subclass of thefamily of ARCH/GARCH models introduced in the 1980s (see [3] and [10]) tomodel financial returns. A popular alternative to ARCH/GARCH models is thefamily of stochastic volatility models introduced by Taylor [24] around the sametime.An important result shown to hold under weak linearity, [12], [17], is the cel-ebrated Bartlett’s formula for the asymptotic variance of the sample autocorrela-tions, which has recently been modified in [11] for processes (1.3) whose innova-tions satisfy Eνt1νt2νt3νt4= 0 if t1= t2, t1= t3, t1= t4. Very early on, Grangerand Andersen [16] warned against the use of Bartlett’s formula in the context ofbilinear time series; Diebold [9] repeated the same warning for ARCH data. De-spite additional such warnings ([2], [19], [20], [22]), even to this day practitionersoften give undue credence to the Bartlett ±1.96/√n bands – that many softwareprograms automatically overlay on the correlogram – in the context of financialreturns data. A possible reason for such misuse of Bartlett’s formula is its abovementioned validity for weakly linear series. However, as will be apparent from themain developments of this paper, ARCH/GARCH processes and their squares arenot only non-linear: they are not even weakly linear. The intuition that these mod-els are somehow not linear is not new, but precise results stating assumptions underwhich it holds have not been formulated. One of the goals of this note is to formu-late such general results. It is however a fairly common intuition that the squaresof ARCH models can be treated as linear, for example, the popular ARMA repre-sentation of the squares of GARCH processes is often taken to imply that resultsproven for weakly linear time series extend to such models. We show that this isNonlinearity of ARCH 3not the case and, in particular, that Bartlett’s cannot be used. We first illustrate theseissues with the following motivating example which is continued in Section 3.EXAMPLE 1. Consider a simple ARCH(1) process, i.e., let(1.5) Xt= εt√β0+ β1X2t−1,where εt∼ i.i.d. (0, 1). If β1< 1, then EX2t= β0/(1 − β1). Write Yt= X2tand let Ft−1be the σ-field generated by Yt−1, Yt−2, . . . Let σ2t= β0+ β1LYtbethe volatility function, where L denotes the lag-operator, i.e., LYt= Yt−1. Sinceβ1LYt= σ2t− β0, it follows that(1 −β1L)(Yt− EYt) = Yt− β1LYt− (1 − β1)EYt= σ2tε2t− (σ2t− β0) −(1 − β1)β01 −β1= σ2tε2t− σ2t.Setting νt= σ2t(ε2t− 1), by the above calculation we obtain(1.6) (1 −β1L)(Yt− EYt) = νt,and hence(1.7) Yt=β01 −β1+∞∑i=0βi1νt−i.In view of the fact that the innovations νtconsitute a white noise [15], equa-tion (1.6) is simply the recursive equation of an AR(1) model with nonzero mean.In this light, equation (1.7) is the usual MA representation of an AR(1) process,thereby giving an allusion toward linearity.Nevertheless, this allusion is false: linearity does not hold true here, not evenin its weak form; this is a consequence of the fact that the innovations νtdo nothave a constant conditional variance as required in the martingale representations(1.3) and (1.4). To see this, just note thatE[ν2t|Ft−1] = E[σ4t(ε2t− 1)2|Ft−1] = σ4tE[(ε2t− 1)2] = σ4tVar[ε2t].The above simple example shows that the common intuition that the squares ofan ARCH process are weakly linear is inaccurate. We will show in Section 2 thatneither the general ARCH(∞) nor stochastic volatility (SV) models are weaklylinear; more surprisingly, we show that this negative result also extends to