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

Sample text

Notice however that signi cance tests may only help in distinguishing between nested models. For example, they are useful in evaluating the relevance of an Ar(1) against an Arma(1,1). Distinguishing between non-nested models cannot be drawn in that way: this procedure cannot be applied to test for an Ar(1) against a Ma(1). The Bayesian Information Criterion (Bic), proposed by Akaike and discussed for example in Hannan (1980) (see also Sneek, 1984) is more useful in this respect. Denoting k the number of estimated parameter (for an Arima(p; d; q); k = p + q + 1), the Bic is obtained as: BIC = 02 log L( j ;  2 w ) + k + k log n (3:9) The Bic can be used to compare di erent models tted to a time series.

4 Hypothesis Testing and Con dence Interval Test of signi cance of the parameters estimated may be carried out as t-tests. The variance-covariance matrix of the parameters estimates may be obtained from the 55 second derivative of the likelihood function according to: ' [ 0ddLd( 0) j =^ ]01 (3:8) In practice, the matrix of variance-covariance is directly given by the algorithm used to maximized the likelihood function. The square root of the i-th element of the matrix gives the standard error of the i-th parameter estimator.

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