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# Lecture16 - Lecture 16 ARIMA GARCH Models Steven Skiena Department of Computer Science State University of New York Stony Brook NY 117944400

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Lecture 16: ARIMA / GARCH Models Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794–4400 http://www.cs.sunysb.edu/ skiena

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Moving Average Models A time series is said to be a moving average process of order q if it is a weighted linear sum of the last q random shocks / errors. In general, an MA ( q ) model has form r t = c 0 + i = q - 1 s i =0 θ i a t - i The history of the model dictates how long the effects of the random shocks last. For an MA (2) model, r t = c 0 + θ 0 a t + θ 1 a t - 1 The order of such a model can be determined by analysis of the autocorrelation function, which is zero for all lags greater than q .
Fitting Moving Average Models The parameters of this model cannot be set using least squares, because (1) the random shocks themselves are not given, and (2) solving for the error terms a t from r t depends upon the parameters being known. A numerical fitting method tries all coefficient sets on a finite grid/space of parameter values, and selects the one which minimizes the sum of the random shocks, i.e. a t .

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Transformations Linear time series methods do not work well on non-linear functions. Defining the transform y t = ln x t converts an exponential function to a linear one. Any trend f n observed in the time series (say, through curve fitting, regression analysis or first principles) can be subtracted out to leave what remaines to be modeled: y t = x t - f n
Differencing Models Differencing transforms a time series X into another series Y where y t = x t - x t - 1 , trying to find a better fitting model. Differencing does not require estimating a parameter, al- though it costs us one series point per difference. Differencing is a better way to remove locally varying trends

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## This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

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Lecture16 - Lecture 16 ARIMA GARCH Models Steven Skiena Department of Computer Science State University of New York Stony Brook NY 117944400

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