Comparaci´on de estrategias de actualizaci´on

Under the null hypothesis of linearity, residuals of a properly specified linear model should be independent. Any violation of independence in the residuals indicates inad- equacy of the entertained model, including the linearity assumption. This is the basic

NONLINEARITY TESTS 153 idea behind various nonlinearity tests. In particular, some of the nonlinearity tests are designed to check for possible violation in quadratic forms of the underlying time series.

4.2.1.1 Q-Statistic of Squared Residuals

McLeod and Li (1983) apply the Ljung–Box statistics to the squared residuals of an ARMA(p,q)model to check for model inadequacy. The test statistic is

Q(m)=T(T+2) m i=1 ˆ ρ2 i(at2) T −i ,

where T is the sample size, m is a properly chosen number of autocorrelations used in the test, at denotes the residual series, andρˆi(a2t)is the lag-i ACF ofat2. If the entertained linear model is adequate, Q(m)is asymptotically a chi-squared random variable withm−p−q degrees of freedom. As mentioned in Chapter 3, the prior Q-statistic is useful in detecting conditional heteroscedasticity ofat and is asymptotically equivalent to the Lagrange multiplier test statistic of Engle (1982) for ARCH models; see subsection 3.3.3. The null hypothesis of the statistics is

Ho:β1= · · · =βm =0, whereβiis the coefficient ofa_t2_−i in the linear regression a_t2=β0+β1at2−1+ · · · +βma2t−m+et

fort =m+1, . . . ,T. Because the statistic is computed from residuals (not directly from the observed returns), the number of degrees of freedom ism−p−q. 4.2.1.2 Bispectral Test

This test can be used to test for linearity and Gaussianity. It depends on the result that a properly normalized bispectrum of a linear time series is constant over all frequencies and that the constant is zero under normality. The bispectrum of a time series is the Fourier transform of its third-order moments. For a stationary time series

xtin Eq. (4.1), the third-order moment is defined as

c(u, v)=g ∞

k=−∞

ψkψk+uψk+v, (4.35)

whereu andvare integers, g = E(at3),ψ0 = 1, andψk = 0 fork < 0. Taking Fourier transforms of Eq. (4.35), we have

b3(w1, w2)= g

4π2[−(w1+w2)](w1)(w2), (4.36)

where(w)=∞_u=0ψuexp(−iwu)withi = √

−1, andwiare frequencies. Yet the spectral density function ofxtis given by

p(w)= σ 2 a

2π|(w)|

2_,

wherewdenotes the frequency. Consequently, the function

b(w1, w2)= |

b3(w1, w2)|2 p(w1)p(w2)p(w1+w2)=

constant for all (w1, w2). (4.37)

The bispectrum test makes use of the property in Eq. (4.37). Basically, it estimates the functionb(w1, w2)in Eq. (4.37) over a suitably chosen grid of points and applies

a test statistic similar to Hotelling’sT2statistic to check the constancy ofb(w1, w2).

For a linear Gaussian series,E(a3

t) =g = 0 so that the bispectrum is zero for all frequencies(w1, w2). For further details of the bispectral test, see Priestley (1988),

Subba Rao and Gabr (1984), and Hinich (1982). Limited experience shows that the test has decent power when the sample size is large.

4.2.1.3 BDS Statistic

Brock, Dechert, and Scheinkman (1987) propose a test statistic, commonly referred to as theBDS test, to detect the iid assumption of a time series. The statistic is, there- fore, different from other test statistics discussed because the latter mainly focus on either the second- or third-order properties ofxt. The basic idea of the BDS test is to make use of “correlation integral” popular in chaotic time series analysis. Given a

k-dimensional time series Xt and observations{Xt}_tT₌k₁, define the correlation inte- gral as Ck(δ)= lim Tk→∞ 2 Tk(Tk−1) i<j Iδ(Xi,Xj), (4.38)

whereI_δ(u, v)is an indicator variable that equals one ifu−v< δ, and zero oth- erwise, where.is the supnorm. The correlation integral measures the fraction of data pairs of{Xt}that are within a distance ofδfrom each other. Consider next a time seriesxt. Constructk-dimensional vectorsXkt =(xt,xt+1, . . . ,xt+k−1), which are

calledk-histories. The idea of the BDS test is as follows. Treat ak-history as a point in thek-dimensional space. If{xt}T_t=1are indeed iid random variables, then thek- histories{Xt}T_t=k₁should show no pattern in thek-dimensional space. Consequently, the correlation integrals should satisfy the relationCk(δ)= [C1(δ)]k. Any departure from the prior relation suggests thatxtare not iid. As a simple, but informative exam- ple, consider a sequence of iid random variables from the uniform distribution over [0,1]. Let[a,b]be a subinterval of[0,1]and consider the “2-history”(xt,xt+1),

which represents a point in the two-dimensional space. Under the iid assumption, the expected number of 2-histories in the subspace[a,b] × [a,b]should equal the square of the expected number ofxt in[a,b]. This idea can be formally examined by using sample counterparts of correlation integrals. Define

NONLINEARITY TESTS 155 C(δ,T)= 2 Tk(Tk−1) i<j I_δ(X∗_i,X∗_j), =1,k,

whereT=T−+1 andX_i∗=xi if=1 andX∗_i =X_ikif=k. Under the null hypothesis that{xt}are iid with a nondegenerated distribution functionF(.), Brock, Dechert, and Scheinkman (1987) show that

Ck(δ,T)→ [C1(δ)]k with probability 1, as T → ∞

for any fixed k and δ. Furthermore, the statistic √T{Ck(δ,T)− [C1(δ,T)]k}is asymptotically distributed as normal with mean zero and variance

σ2 k(δ)=4 Nk+2 k−1 j=1 Nk−jC2j+(k−1)2C2k−k2N C2k−2 , whereC =[F(z+δ)−F(z−δ)]d F(z)andN =[F(z+δ)−F(z−δ)]2d F(z). Note thatC1(δ,T)is a consistent estimate ofC, andNcan be consistently estimated by N(δ,T)= 6 Tk(Tk−1)(Tk−2) t<s<u Iδ(xt,xs)Iδ(xs,xu).

The BDS test statistic is then defined as

Dk(δ,T)= √

T{Ck(δ,T)− [C1(δ,T)]k}/σk(δ,T), (4.39) whereσk(δ,T)is obtained fromσk(δ)whenCandN are replaced byC1(δ,T)and N(δ,T), respectively. This test statistic has a standard normal limiting distribution. For further discussion and examples of applying the BDS test, see Hsieh (1989) and Brock, Hsieh, and LeBaron (1991). In application, one should remove linear dependence, if any, from the data before applying the BDS test. The test may be sensitive to the choices ofδandk, especially whenkis large.