A NEW DIAGNOSTIC TEST OF MODEL INADEQUACY WHICH USES THE MARTINGALE DIFFERENCE CRITERION

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Abstract
Let {x(t)} denote a discrete‐time random process. Given a sample of increments e(t) = x(t) ‐ x(t ‐ 1) from the time series, we wish to test formally whether the sample is consistent with the assumption that {e(t)} is a martingale difference. It is shown that the martingale criterion is more general than the white noise criterion in analyzing fitted residuals for signs of model inadequacy. In this paper we present such a test which approximately achieves a given type 1 error probability for samples. We assume that (1) the process is strictly stationary, (2) all its kth‐order cumulant functions exist and (3) the kth‐order cumulants are absolutely summable and satisfy a mixing condition. The martingale assumption implies that most third‐order cumulants of the increment process are zero, and thus the third‐order cumulant sequence is sparse. This result is used to derive test statistics based on a modified sample bispectrum. The test can be regarded as a two‐dimensional portmanteau test of serial dependence. The large‐sample results are demonstrated through the use of artificial data. Finally, the test is applied to a daily financial series.