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Unformatted text preview: Notes 6 Model Specification : Overall strategy - Box Jenkins Approach: 1. To decide a reasonable - but tentative - values for p,d,q . 2. Estimate φ,θ and σ 2 a , for that model. 3. Check the model’s adequacy. 4. If the model appears inadequate, consider the nature of the inadequacy to select another model. 5. Estimate the new model and check it for accuracy. To obtain a tentative order d (based on the graphical approach) A basic rule: The first (and most important) step in fitting an ARIMA model is to determine the value of d (i.e. the order of differencing). Normally, the correct amount of differencing is the lowest order of differencing that yields a time series which fluctuates around a well- defined mean value and whose autocorrelation function (ACF) plot decays fairly rapidly to zero. If the series still exhibits a long-term trend, or otherwise lacks a tendency to return to its mean value, or if its autocorrelations are positive out to a high number of lags, say 10 or more, then a higher order of differencing is needed. Properties of the sample autocorrelation function Theorem : Suppose that Z t = μ + ∞ X j =0 ψ j a t- j where a t iid ∼ (0 ,σ 2 a ) , < σ 2 a < ∞ . Assume ∑ ∞ j =0 | ψ j | < ∞ and ∑ ∞ j =0 ψ 2 j < ∞ . (This will be satisfied by any stationary ARMA model). Then, for any fixed m , the joint distribution of √ n ( r 1- ρ 1 ) , √ n ( r 2- ρ 2 ) ,..., √ n ( r m- ρ m ) approaches, as n → ∞ , a joint normal distribution with zero means, variances c ii and covariances...
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- Spring '08