Diagnostics in time series analysis
The portmanteau diagnostic test for goodness of model fit is studied. It is found that the true variances of the estimated residual autocorrelation function are potentially deflated considerably below their asymptotic level, and exhibit high correlations with each other. This suggests a new portmanteau test, ignoring the first p + q residual autocorrelation terms and hence approximating the asymptotic chi-squared distribution more closely. Simulations show that this alternative portmanteau test produces greater accuracy in its estimated significance levels, especially in small samples. Theory and discussions follow, pertaining to both the Dynamic Linear Model and the Bayesian method of forecasting. The concept of long-term equivalence is defined. The difficulties with the discounting approach in the DLM are then illustrated through an example, before deriving equations for the step-ahead forecast distribution which could, instead, be used to estimate the evolution variance matrix W(_t). Non-uniqueness of W in the constant time series DLM is the principal drawback with this idea; however, it is proven that in any class of long-term equivalent models only p degrees of freedom can be fixed in W, leading to a potentially diagonal form for this matrix. The bias in the k(^th) step-ahead forecast error produced by any TSDLM variance (mis)specification is calculated. This yields the variances and covariances of the forecast error distribution; given sample estimates of these, it proves possible to solve equations arising from these calculations both for V and p elements of W. Simulations, and a "head-to-head" comparison, for the frequently-applied steady model illustrate the accuracy of the predictive calculations, both in the convergence properties of the sample (co)variances, and the estimates Ṽ and Ŵ. The method is then applied to a 2-dimensional constant TSDLM. Further simulations illustrate the success of the approach in producing accurate on-line estimates for the true variance specifications within this widely-used model.