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THE BISPECTRUM AND TESTS FOR NONLINEARITY The Bispectrum hereas the spectrum provides a decomposition of the second moment into contributions from individual component frequencies, the bispectrum provides a similar decomposition of the thir oment. First, consider the spectral density. This is the function ) such that ef (1 ir As long as the autocovariances xx ] do not go to zero too slowly, we can invert this to give hhh )exp( ir ), so that the spectral density is the Fourier transform of the autocovariance function. Using the spectral representation, we ﬁnd that xx EedZ edZ eeE dZ dZ )] 11 λ= λ= λ+ ince the lefthand side does not depend on , neither does the righthand side. Thus, we must have dZ dZ )] 0 unless λ+λ= 0. Therefore, 11 λ= eE dZ dZ )] Comparing this with (1), we see that λ= dZ dZ )]. Thus, is the joint secon oment of the components of from two frequencies which sum to zero, one of which is . Since from (1) we have (0) we can also think of as the contribution to the variance of from frequency The bispectrum ) is the contribution to the third moment of { } from the pair of fre- 12 quencies . Consider the third order moment function

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-2- xx x ], tt rt assumed not to depend on . The bispectrum is deﬁned b eef (2) 12 12 = λ= ir is 12 12 This can be inverted (under suitable conditions) to giv (2 hhhh ee rs 12 so that the bispectrum is the (two dimensional) Fourier transform of ). Using the spectral represen ation, we have xx x Ee dZ edZ edZ rt 12 12 1 2 λ+λ+ ti ri 12 eeeE dZ dZ dZ )] ince the lefthand side does not depend on , neither does the righthand side, so dZ dZ dZ )] 12 12 nless λ+λ+λ= 0, and therefore eeE dZ λ+ )) dZ dZ )] 12 12 ri 12 1 2 Comparison with (2) reveals that λ= dZ λ+ )) dZ dZ )] , 12 12 12 1 2 212 so that is the joint third moment of the components of from three frequencie hich sum to zero, two of which are . Since from (2) we have 12 12 12 12 we can also think of as the contribution to ] from frequencies ( ). If { }i 12 12 12 Gaussian, then 0 for all , and the bispectrum will be identically equal to zero.

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-3- Subba Rao’s Test for Nonlinearity and Non-Gaussianit lthough many time series observed in practice may be nonlinear, it would be unrealistic to assume that the data were generated exactly by one of the known parametric models. These model Bilinear, TAR, GARCH, etc.) are merely approximations which will sometimes be useful. The question of whether the series is nonlinear, however, seems more basic. By considering the bispectrum, an erhaps higher order spectra, it is possible to construct tests for nonlinearity without specifying any ﬁnite-parameter models, either under the null hypothesis or the alternative hypothesis. The nul ypothesis (of linearity) is that ge , where the { } are iid . It can be shown (homework) that ut ut f{ } is linear, then the quantity λ+ hhhhhhhhhhhhhhhhhh 12 12 121 23 hhhhh , where = ]. Furthermore, if { } is Gaussian, then it i hhh will be constant, inear with = 0, so ) (and )) will be identically equal to zero. 12 12 he ﬁrst stage of Subba Rao’s test examines whether the series is Gaussian, by testing whether the estimated bispectrum over a grid of frequencies is signiﬁcantly different from zero. If it is (as meas red by a test statistic related to Hotelling’s ), then the series is declared to be non Gaussian, and we 12 can ask whether the process is linear or not. If { } is linear, then ) will be constant. So th econd stage of Subba Rao’s test examines whether the sample version of ) is signiﬁcantly non- 12 constant, over a grid of frequencies. If it is, we declare the series to be nonlinear The advantage of Subba Rao’s test is that it is nonparametric. A drawback of the test is that, depending on exactly how we estimate the spectrum and bispectrum, our conclusions about Gaussianity nd linearity of a given observed series may change. A more serious problem is that not all nonlinear processes will have a nonzero and nonconstant bispectrum. For example, the bispectrum of an ARCH rocess is identically zero. Thus Subba Rao’s test will have no chance of detecting nonlinearity (or even non Gaussianity) in this case. On the other hand, we are not likely to ever see a time series which s exactly ARCH. Furthermore, it is possible to generalize Subba Rao’s test to higher order spectra. For

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-4- xample, the third-order spectrum ("trispectrum") of an ARCH process is nonzero (see Milhoj), while the third-order spectrum of a Gaussian process is identically zero. Subba Rao’s test reveals that two of the data sets which traditionally have been analyzed in the time series literature using ARMA models (the sunspot series and the Canadian Lynx series) actually eem to be nonlinear.

First consider the spectral density This is the function such that ef 1 ir As long as the autocovariances xx do not go to zero too slowly we can invert this to give hhh exp ir so that the spectral density is the Fourier transform of the autocovar ID: 22555

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Page 1

THE BISPECTRUM AND TESTS FOR NONLINEARITY The Bispectrum hereas the spectrum provides a decomposition of the second moment into contributions from individual component frequencies, the bispectrum provides a similar decomposition of the thir oment. First, consider the spectral density. This is the function ) such that ef (1 ir As long as the autocovariances xx ] do not go to zero too slowly, we can invert this to give hhh )exp( ir ), so that the spectral density is the Fourier transform of the autocovariance function. Using the spectral representation, we ﬁnd that xx EedZ edZ eeE dZ dZ )] 11 λ= λ= λ+ ince the lefthand side does not depend on , neither does the righthand side. Thus, we must have dZ dZ )] 0 unless λ+λ= 0. Therefore, 11 λ= eE dZ dZ )] Comparing this with (1), we see that λ= dZ dZ )]. Thus, is the joint secon oment of the components of from two frequencies which sum to zero, one of which is . Since from (1) we have (0) we can also think of as the contribution to the variance of from frequency The bispectrum ) is the contribution to the third moment of { } from the pair of fre- 12 quencies . Consider the third order moment function

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-2- xx x ], tt rt assumed not to depend on . The bispectrum is deﬁned b eef (2) 12 12 = λ= ir is 12 12 This can be inverted (under suitable conditions) to giv (2 hhhh ee rs 12 so that the bispectrum is the (two dimensional) Fourier transform of ). Using the spectral represen ation, we have xx x Ee dZ edZ edZ rt 12 12 1 2 λ+λ+ ti ri 12 eeeE dZ dZ dZ )] ince the lefthand side does not depend on , neither does the righthand side, so dZ dZ dZ )] 12 12 nless λ+λ+λ= 0, and therefore eeE dZ λ+ )) dZ dZ )] 12 12 ri 12 1 2 Comparison with (2) reveals that λ= dZ λ+ )) dZ dZ )] , 12 12 12 1 2 212 so that is the joint third moment of the components of from three frequencie hich sum to zero, two of which are . Since from (2) we have 12 12 12 12 we can also think of as the contribution to ] from frequencies ( ). If { }i 12 12 12 Gaussian, then 0 for all , and the bispectrum will be identically equal to zero.

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-3- Subba Rao’s Test for Nonlinearity and Non-Gaussianit lthough many time series observed in practice may be nonlinear, it would be unrealistic to assume that the data were generated exactly by one of the known parametric models. These model Bilinear, TAR, GARCH, etc.) are merely approximations which will sometimes be useful. The question of whether the series is nonlinear, however, seems more basic. By considering the bispectrum, an erhaps higher order spectra, it is possible to construct tests for nonlinearity without specifying any ﬁnite-parameter models, either under the null hypothesis or the alternative hypothesis. The nul ypothesis (of linearity) is that ge , where the { } are iid . It can be shown (homework) that ut ut f{ } is linear, then the quantity λ+ hhhhhhhhhhhhhhhhhh 12 12 121 23 hhhhh , where = ]. Furthermore, if { } is Gaussian, then it i hhh will be constant, inear with = 0, so ) (and )) will be identically equal to zero. 12 12 he ﬁrst stage of Subba Rao’s test examines whether the series is Gaussian, by testing whether the estimated bispectrum over a grid of frequencies is signiﬁcantly different from zero. If it is (as meas red by a test statistic related to Hotelling’s ), then the series is declared to be non Gaussian, and we 12 can ask whether the process is linear or not. If { } is linear, then ) will be constant. So th econd stage of Subba Rao’s test examines whether the sample version of ) is signiﬁcantly non- 12 constant, over a grid of frequencies. If it is, we declare the series to be nonlinear The advantage of Subba Rao’s test is that it is nonparametric. A drawback of the test is that, depending on exactly how we estimate the spectrum and bispectrum, our conclusions about Gaussianity nd linearity of a given observed series may change. A more serious problem is that not all nonlinear processes will have a nonzero and nonconstant bispectrum. For example, the bispectrum of an ARCH rocess is identically zero. Thus Subba Rao’s test will have no chance of detecting nonlinearity (or even non Gaussianity) in this case. On the other hand, we are not likely to ever see a time series which s exactly ARCH. Furthermore, it is possible to generalize Subba Rao’s test to higher order spectra. For

Page 4

-4- xample, the third-order spectrum ("trispectrum") of an ARCH process is nonzero (see Milhoj), while the third-order spectrum of a Gaussian process is identically zero. Subba Rao’s test reveals that two of the data sets which traditionally have been analyzed in the time series literature using ARMA models (the sunspot series and the Canadian Lynx series) actually eem to be nonlinear.

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