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Published byElissa Warford Modified over 3 years ago

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Definitions Periodic Function: f(t +T) = f(t)t, (Period T)(1) Ex: f(t) = A sin(2Πωt + )(2) has period T = (1/ω) and ω is said to be the frequency (angular), A is amplitude, is the phase (2) can also be written as f(t) = C sin(2Πωt) + D cos (2Πωt)(3) with A = C² + D², = tan -1 (D/C)

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Fourier Series Theorem:If y(t) is periodic function of period 2Π then its behavior is captured by studying it on (-Π, Π) satisfying 1. 2.y(t) has a finite number of discontinuities of the first kind on (-Π, Π), then (4) Where, for j=0,1,2… (5) for j=1,2,… (6) The R.H.S. of (4) is called the Fourier series of y(t)

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Fourier Transforms (FT) Let y(t) be a continuous function of t (not necessarily periodic ), then the FT of y(t) is defined as (7) The inverse FT of Y( ) is simply y(t) (8) Parsevals Identity: (9) (conservation of energy principle)

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Discrete Fourier Transforms (DFT) In case y t is a discrete series defined only at discrete time points …-2,-1,0,1,2,… then the DFT of y t is (10) and (11) In applications, the series is usually observed only for N periods say 0,1,…,(N-1) and the finite Fourier transforms is then defined as (12)

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Spectrum Let X t be a discrete covariance stationary process, with =cor{X t X t+ } the autocorrelation at lag, then spectrum h( ) of X t may be defined as (13) h( ) is periodic (of period 2 ) and hence is usually studied on (-, ) We also have (14) In case the series X t is real we may simplify (13) to (15)

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Examples 1. If X t is a WNP,(flat spectrum) 2. If X t = t - 1 t-1 (MA(1)) then 3. X t =a X t-1 + t with |a| < 1 then

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Spectrum Estimation Data: X(1), X(2),…,X(N)in mean deviation form (N is even) Periodogram of X(t) defined as Є (-, )(16) Fourier frequencies p=0,1,…,(N/2) Define I p =I N ( p ) Plot of I p against p (or p ) is called Periodogram

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Theorem: Є (-, ) Where Modified Periodogram: is an asymptotically unbiased estimator of the spectrum but is inconsistent (variance does not vanish asymptotically)

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Source of the problem All the autocorrelation possible from s=-(N-1), to (N-1) are included in but for high values of s, is an unreliable estimate of the true K(s). Solution: Sacrifice some of the information available in the later, thereby deliberately introducing bias but reducing unreliability of the estimates.

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Two Choices Truncation: Ignore all for |s|>S 0 (some fixed no. (N-1)) Tail / Tapering: Give lower weight to for increasing s, through a properly chosen weighting scheme. This leads to the window selection problem, and a window estimate of the spectrum. with, lag window generator

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Continued.. We get different spectral estimates by choosing different forms of 1. Truncated Periodogram Window: 2. Bartlett Window:

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Continued.. 3. Daniell Window: 4. Parzen Window:

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Theorem: Spectral (window ) estimates with lag windowcan be viewed as a weighted average of the modified periodogram in the frequency range with (continuous) weighting function Note: is called as the spectral window generator

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Comparison of Windows: Bartlett window can lead to negative spectral estimates, a problem not present with the Daniell and Parzen windows. The Parzen window has lower variance but higher bias than the Daniell window. Bandwidth Concepts: Resolvability: Suppose the true spectrum has twin peaks at 1 and 2, then we should like the estimated spectrum also to have peaks at 1 and 2 Good Resolution implies that the two peaks are not merged together and shown as a single intermediate peak.This requires that the width of the spectral window generator should not exceed | 1- 2 |, the distance between the 2 peaks.Thus as an informal rule,we should try to choose as narrow a window as possible.

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Theorem (17) The window estimates are asymptotically unbiased, consistent and normal provided that 1. 2.

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Bandwidth Half-power Bandwidth: Parzen Bandwidth: Jenkins Bandwidth: M truncation parameter of window,,

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Greander Uncertainty Principle: Choice of M: High values of Mlow bias, small bandwidth, high variance Low value of Mlow variance, wide bandwidth, high bias

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Filter Suppose X(t) and Y(t) are two series s.t. is a sequence of constants Y(t) is filtered series of X(t) is parameter function of the filter

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Slutskis Theorem: Pre-whitening: Choose or g(u) such that Y(t) is white noise. Fit a sufficiently high order AR model to X(t) and take Y(t) as the residual. Estimate the spectrum of Y(t) by choosing a low value of M, since good resolution is automatic for WNP.

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Dynamic Models, Autocorrelation and Forecasting ECON 6002 Econometrics Memorial University of Newfoundland Adapted from Vera Tabakova’s notes.

Dynamic Models, Autocorrelation and Forecasting ECON 6002 Econometrics Memorial University of Newfoundland Adapted from Vera Tabakova’s notes.

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