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Extra info for Applied Time Series Analysis.Modelling,Forecasting,Unobserved Components Analysis & the Wiener-Kolmogorov Filter.(172p)

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So, computing the Q-statistics for both the residuals and the squared residuals, an increase in the Q-value for squared residuals is an indication of nonlinearity. Similarly, a test for nonlinearity at seasonal lags may be performed by comparing Qs (at ) and Qs (a2t ). Locating the nonlinearities present in the series at some particular lags may be useful to improve the modelling of the series under analysis (see for example Fiorentini and Maravall (1996)). Yet another hypothesis to test is the residual normality.

3) as a ^t ( ) = wt 0 1 wt01 0 1 1 1 0 p wt0p 0 1 ^at01 0 1 1 1 0 q ^at0q : (3:5) The parameter  2 may be concentrated out of the likelihood function using ^ 2 = n01 S ( ), so that the function to maximize becomes: logL( ;  2 ) = 0n log S( ) n 0 n2 : (3:6) Maximizing the conditional likelihood function or minimizing the sum of squares is then equivalent. , w10p and a0; :::; a10q in order to have available the rst a^i 's. A simple solution to this problem would consist in considering the unconditional expectations of the innovations and of the wt's.

The preferred model is the one which minimizes the Bic accross the range of the di erent models. 5 Diagnostic Checking At this stage, an evaluation of the temptative model based on the estimated residual properties is to be performed. For the model to provide a good description of the series, no underlying structure might be left in the residuals. So rst of all it could be useful to examine the residual plot. For example, changes in the variance over time would suggest to modify the initial transformation of the data.

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