APPENDIX I: The Variance of the Modified Baek And Brock Test
To describe the variance of the modified Baek and Brock test and a consistent estimator for it, we begin by defining the joint probabilities
Using (A.l) and the delta method (see Serfling (1980), pp. 122-125), under the assumption that the underlying series are strictly stationary, weakly dependent, and satisfy the mixing conditions of Denker and Keller (1983), an expression for the variance of the Baek and Brock test in (7) is given by
and where E(-) denotes expected value and the Ci(-) terms are defined in (4).
and where the Ci(-,n) correlation integrals are defined in (6) and the I(.) indicators are described in Section III.2. The Ci(.,n) correlation integrals provide a consistent estimator of d in equation (A3), namely,
A consistent estimator for σ2(m,Lx,Ly,e) in (7) can then be expressed as
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Hiemstra, C., and J.D. Jones, and J.D. Jones, “Testing for Linear and Nonlinear Granger Causality in the Stock-Volume Relation,” Forthcoming, Journal of Finance, (December 1994).
Hiemstra, C., and J.D. Jones, and J.D. Jones, and C.F. Kramer (1993), “Accounting for Stock-Return Dynamics with a Macrofactor APT/Factor-ARCH Model,” unpublished manuscript, Department of Accounting and Finance, University of Strathclyde, Glasgow, Scotland, and Washington D.C.: International Monetary Fund, 1993.
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)| false ( Hiemstra, C., and J.D. Jones, and J.D. Jones, and C.F. Kramer 1993), “ Accounting for Stock-Return Dynamics with a Macrofactor APT/Factor-ARCH Model,” unpublished manuscript, Department of Accounting and Finance, University of Strathclyde, Glasgow, Scotland, and Washington D.C.: International Monetary Fund, 1993.
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Mr. Hiemstra is with the Department of Accounting and Finance, University of Strathclyde, Glasgow G4 OLN, Scotland.
Versions of this paper were presented for the Economics Department at Southern Methodist University, the Time Series Group at the Santa Fe Institute, the Chaos and Nonlinear Dynamics Study Group at the U.S. Bureau of Labor Statistics, and the 1994 North American Summer Meetings of the Econometric Society. We wish to thank the participants at these presentations for helpful comments. We also wish to thank Pedro de Lima, Robert Flood, Ted Jaditz, Jonathan Jones, Francis Longstaff, and anonymous referees for comments. We also thank Janet Shelley for her help in converting this manuscript into WordPerfect.
Examples are Chen (1991), Harvey and Ferson (1991), Chang and Pinegar (1990), and Chan, Chen and Hsieh (1985). More examples are cited in Fama (1991). Due to most readers’ familiarity with such models, our discussion of them is intentionally kept brief.
See Hsieh (1991).
For an instance of the practical use of these linear models, see Berry, Burmeister and McElroy (1988).
The literature on Granger causality testing is broad. See Geweke (1984), Geweke, Meese, and Dent (1983), and Granger (1990) for more information on the notion of Granger causality and associated statistical tests.
We calculated the Akaike criterion for every combination of a and b where each ran between 1 and 40. The combination of a and b with the smallest value of the Akaike criterion was chosen.
The Baek and Brock approach to testing for nonlinear Granger causality relies on correlation-integral estimators of certain spatial probabilities corresponding to vector time series. For certain strictly stationary and weakly dependent processes, Denker and Keller (1983) show that estimators such as these (bounded-kernel U-statistics) are consistent estimators. See Denker and Keller (1983, pp. 505-7) for the conditions under which their consistency results hold. Loosely, weakly dependent processes display short-term temporal dependence which decays at a sufficiently fast rate. Formal discussions of weakly dependent processes can be found in Denker and Keller (1983) and their references.
The maximum norm for Z≡(Z1,Z2,…,Zk)∈Rk is defined as max (Zi) i=1,2,…,K. Computational speed in implementing the test is one important reason for using the maximum norm. A more general version of the test can be devised by considering different length scales, e, corresponding to the lead and lag vectors. Also the test can easily be generalized beyond the bivariate case considered here.
By definition, the conditional probability Pr(A|B) can be expressed as the ratio Pr(AnB)/Pr(B). Note that the maximum norm allows us to write
Hiemstra and Jones (1993) modified version of the test holds for the more general assumption that the errors are weakly dependent. The fundamental differences between the two versions of the test are manifested only in estimators of (m,Lx,Ly,e) in equation (7).
Hiemstra and Jones (1993b) also find through Monte Carlo simulations that the modified Baek and Brock test is immune to the effects of contemporaneous correlation and neglected nonstationarities due to structural breaks.
See Geweke (1984). In particular, Granger causality tests can refute (but not establish) a claim of strict exogeneity. That is, finding Granger causality from Y to X implies that Y is not strictly exogenous, while a failure to find Granger causality from Y to X does not necessarily imply that Y is strictly exogenous.
Estimation of the model yielded an estimated MA parameter of -0.742 with a t-ratio of -21.52.
As can be seen in equation (7), a significant positive test statistic indicates than one series helps to predict another, while a significant negative statistic indicates that one series confounds the prediction of another. Our view is that the Brock and Baek test statistic should be evaluated using the right-tail critical value.
See Hsieh (1991) and references therein.
We used 24 autocorrelation and autocovariance terms to implement the adjusted-for-ARCH Ljung-Box test. We also used 24 autocovariance terms in implementing the Engle and McLeod and Li tests. Under the IID null these tests are asymptotically distributed x2 (24). The RESET test employed here is based on the residuals of an AR(p) model fit to the residuals series. We used the Akaike (1974) criterion using a maximum lag length of 10 periods to fit the series. The test is also based on the 2nd through 4th principal components of the raised-to-the-2nd through 6th AR(p) forecasts of the series. Under the null of no nonlinear temporal dependence in the conditional mean of the series, the test statistic is asymptotically distributed x2 (3). Our implementation of the neural net test relies on N(0,1) realizations to generate so-called hidden factors. In all other respects it conforms to Lee, White, and Granger’s NEURRALl test. Under the assumption of no nonlinear dependence in the mean of a series, the neural net test is asymptotically distributed x2 (3). In implementing the BDS test we considered length scales equal to 1.5, 1.0, and 0.5 standard deviations in the series and embedding dimensions (or maximum lag lengths) ranging from 1 to 4. Under the IID null the BDS test is asymptotically distributed N(0,1).
The test statistics corresponding to an embedding dimension of 4 are also significant at the appropriate 5 percent finite-sample level of significance for the IID N(0,1) case (see Brock, Hsieh, and LeBaron (1991)).
Factor-ARCH models of asset markets include Engle, Ng, and Rothschild (1990) and Ng, Engle and Rothschild (1992).
As previously noted, we follow the standard convention of denoting random variables in the upper case and their realizations in the lower case. These joint probabilities relate to the probability that arbitrarily selected triplet