Introduction to Spectral and Time-Spectral Analysis with some Applications

Transcription

1 Introduction to Spectral and Time-Spectral Analysis with some Applications A.INTRODUCTION Consider the following process X t = U cos[(=3)t] + V sin[(=3)t] Where *E[U] = E[V ] = 0 *E[UV ] = 0 *V ar(u) = V ar(v ) = : 1

2 QUESTIONS: 1. Simulated values of X t?.is it possible to detect visually the frequency =3 from the simulated data? 3. Show that X t is covariance stationary. 4. Does the graph of estimated values of (k) allow to easily identify the particular frequency =3? 5. Give the graphic of the fourier Transform of X t and the one of the fourier Transform of (k). Is the frequency, =3, can now be easily identied from the graph of the fourier Transform of (k)?

3 B.GENERALISATION Any covariance-stationary process X t, with E(X t ) = 0, can be written as follows: X t = Z e it! dz(!) *dz(!) is an orthogonal process. *The spectral density of X t is dened as: f(!) = 1 * ) (k) = X e ik! (k), k Z kz Z e ik! f(!)d! *) f(!) gives information about X t in the frequency domain (![0; ]) *(k) gives the same informations but in the time domain (k Z) 3

4 C. SPECTRAL DENSITY OF ARMA PROCESS * It can be shown that if X t is an ARMA(p; q) stationary process as (B)X t = (B)" t where " t IID(0; ), then the spectral density of X t is given by: f(!) = (e i! ) (e i! ) *=) For a white Noise process, " t IID(0; ): f(!) = is a constant. *Example (see exercice) 4

5 C. How to estimate f(!)? Let us consider the time series,fx t g T t=1, coming from the covariance stationary process X t with E(X t ) = 0 and f the true spectral density of X. I. Periodogram: I(!) = 1 T *We have: TX X t e i!t t=1 * E[ I(!)] = f(!) + O(T 1 ) *V ar(i(!)) = f(!) +O(T 1 ) () I(!) is not a consistant estimator) *Cov[I(!)(! 1 ); I(!)(! )] = O(T 1 ) *I(!) = b f P (!) bf P (!) = 1 XT 1 k= (T 1) e ik! b(k) Where b(k) = 1 T 5 P T jkj t=1 x t x t+k

6 II.Estimate using a window bf K (!) = 1 XT 1 k= (T 1) K M (k)e ik! b(k) Where K M (k) = K M ( k), the max of K M (k) is obtained for k = 0 and K M (0) = 1. * The Statistical properties of b f K (!) are given by the ones of K M (k) and the ones of the true spectral density f. * The parameter M of K M (:) allows to deal betwen the Bias and Variance of b f K (!) (Large value of M gives low bias but high variance. vsv) *Example: K M (k) = 1 jkj =M if jkj < M, and K M (k) = 0 if jkj M (Bartlett window) 6

7 BIBLIOGRAPHY [1] Priestley,M.B.: Spectral analysis and Time Series. Academic Press, London,1981. [] Brockwell,P.J and Davis, R.A: Time series, Theory and methods. Second Edition, Springer Series in statistics [3] Hamilton J.D., Time Series Analysis,Princeton University Press

8 D. Extension to the concept of Time- Spectral Density. I. Introduction * For a stationary process, f(!) is supposed to be time-invariant a) ) (k) is also time-invariant for a stationary process b)) (0) =variance, is also timeinvariant for a stationary process * a) et b) are strong assumptions for some time series (see the SP500 returns). *Thus, a lot of theory have tryed to propose a time varying spectral density: -Priestley theory (1965), -Dahlhauss theory (1986), -Wavelets theory, etc.. 8

9 [1] Priestley,M.B.: Spectral analysis and Time Series. Academic Press, London,1981 [] Dahlhaus,R. Fitting time series models to nonstationary processes:ann.statist,1997,5,p1-37. [3] Dahlhaus,R. On the Kullback-Leiber information divergence of locally stationary processes. Stochastic.process.appl.(1996),6,p

10 II.Estimate of the time-spectral density from Priestley *Consider time series fx t g T t=1 from oscillatory process(see Priestley), where E(X t ) = 0. * Denote by h t (!), the value of the spectral density at the frequency! and for the time t. b ht (!) = P vz W v ju t v (!)j Where: a) U t (!) = P uz g ux t u e i!(t u) b) {W v g et g u are two chosen windows. * b h t (!) can be approximatelly an unbiased estimator of h t (!) with low variance(see priestley) 10

11 D. EXERCICES Ex.1. Give the spectral density and the estimators when X t R: *Ex. Find the spectral density for a MA(1) process, an AR(1) process an for a ARMA(1,1)((Theoretical and from simulated data, How to detect unit root from the spectral density?). * Ex.3 Find an AR() process with high spectral density in the frequencies band of Business cycl.(simulation) 11

12 E. APPLICATION 1. US Business cycle using the spectral density?.. Performance of the BK lter and HP lter to detect business cycle?. 3. Bartlett White Noise test 4.Covariance stationnarity tests. 5. Detecting break in the covariance 4. Stability of the SP500 returns. 1