"Frontmatter". In: Analysis of Financial Time Series

NONLINEARITY TESTS 153idea behind various nonlinearity tests. In particular, some of the nonlinearity testsare designed to check for possible violation in quadratic forms of the underlyingtime series.4.2.1.1 Q-Statistic of Squared ResidualsMcLeod and Li (1983) apply the Ljung–Box statistics to the squared residuals of anARMA(p, q) model to check for model inadequacy. The test statistic isQ(m) = T (T + 2)m∑i=1ˆρ 2 i (a2 t )T − i ,where T is the sample size, m is a properly chosen number of autocorrelationsused in the test, a t denotes the residual series, and ˆρ i (at 2) is the lag-i ACF of a2 t .If the entertained linear model is adequate, Q(m) is asymptotically a chi-squaredrandom variable with m − p − q degrees of freedom. As mentioned in Chapter 3,the prior Q-statistic is useful in detecting conditional heteroscedasticity of a t and isasymptotically equivalent to the Lagrange multiplier test statistic of Engle (1982)for ARCH models; see subsection 3.3.3. The null hypothesis of the statistics isH o : β 1 =···=β m = 0, where β i is the coefficient of at−i 2 in the linear regressiona 2 t = β 0 + β 1 a 2 t−1 +···+β ma 2 t−m + e tfor t = m + 1,...,T . Because the statistic is computed from residuals (not directlyfrom the observed returns), the number of degrees of freedom is m − p − q.4.2.1.2 Bispectral TestThis test can be used to test for linearity and Gaussianity. It depends on the resultthat a properly normalized bispectrum of a linear time series is constant over allfrequencies and that the constant is zero under normality. The bispectrum of a timeseries is the Fourier transform of its third-order moments. For a stationary time seriesx t in Eq. (4.1), the third-order moment is defined asc(u,v)= g∞∑k=−∞ψ k ψ k+u ψ k+v , (4.35)where u and v are integers, g = E(a 3 t ), ψ 0 = 1, and ψ k = 0fork < 0. TakingFourier transforms of Eq. (4.35), we haveb 3 (w 1 ,w 2 ) =g4π 2 Ɣ[−(w 1 + w 2 )]Ɣ(w 1 )Ɣ(w 2 ), (4.36)where Ɣ(w) = ∑ ∞u=0 ψ u exp(−iwu) with i = √ −1, and w i are frequencies. Yet thespectral density function of x t is given by