time series final project _ sample

# time series final project _ sample - TIME SERIES FINAL...

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TIME SERIES FINAL PROJECT MARC-ANDRE ROUSSEAU 1. Introduction The topic of this project is Gegenbauer models for long memory processes. In this report, we will discuss the paper by P.M. Lapsa and ﬁt this model to the Farallon data set. There will also be some examples of diﬀerent techniques one can use to estimate the parameters in these sorts of models. 1.1. Long Memory Models. A long memory model is usually deﬁned as a time series process who’s autocorrelation function is not absolutely summable. One particular example of such a process is the fractionally integrated ARMA process (ARIMA). Such a process can be represented as: (1 - B ) d φ ( B ) X t = θ ( B ) Z t for 0 < | d | < 1 2 . One property of these models is that the autocorrelation function ρ ( h ) has the property that ρ ( h ) h 1 - 2 d c as h → ∞ [1]. This demonstrates a very slow convergence to zero and that the correlation structure is such that the random events continue to be inﬂuenced by earlier values for quite some time. 1.2. Gegenbauer Models. A particular type of long-memory models have a periodic behaviour at some frequency ω . To model these processes, we use the following model involving three parameters ρ,ω,d rep- resenting the underlying noise variance, periodic frequency and the magnitude of the diﬀerencing. The ﬁrst two parameters control the variance and the periodicity of the correlation structure while the third inﬂuences the rate at which the correlations decrease to zero. The representation for the Gegenbauer process is: Y t (1 - (2cos ω ) B + B 2 ) d = ρW t where W t is white noise. This process can also be written as: Y t = ρ X k =0 g k B k W t where the g k are Gegenbauer coeﬃcients [2]. In the paper by Lapsa, several recursive properties of these models are discussed, the following two results mentioned in the paper are useful for estimation of the parameters. 1.2.1. Auto-covariance Function. A formula for the ACVF using generalized hypergeometric functions F ( a ; b ; z ) is given. Letting a ( k ) = (1 - 2 d + k, 1 - 2 d - k ), b = 3 / 2 - 2 d : γ ( k ) = ρ 2 π Γ(1 - 2 d ) Γ(3 / 2 - 2 d ) sin 1 - 4 d ω £ F ( a ( k ) ,b, cos 2 ( ω/ 2)) + ( - 1) k F ( a ( k ) ,b, sin 2 ( ω/ 2)) / (1) During the maximization process for the Farallon data, problems arose resulting in an incorrect estimation of the ρ parameter. it is unclear whether this was as a result of a coding problem or if there is a slight typo in the ACFV equation. Time constraints did not permit a full inquiry into the source of the problem. 1

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2 MARC-ANDRE ROUSSEAU 1.2.2. Recursive Hypergeometric Functions. While modern computational tools permit the evaluation of hypergeometric functions directly, it is sometimes preferable to use recursive deﬁnitions to save on computing time. The paper by Lapsa provides a recursive deﬁnition which contains some small mistakes which are corrected here. We let a ( k ) = ( δ + k,δ - k ), using the Gauss contiguity formulas for the hypergeometric
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## This note was uploaded on 05/26/2010 for the course STAT 443 taught by Professor Yuliagel during the Winter '09 term at Waterloo.

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time series final project _ sample - TIME SERIES FINAL...

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