Time series models in the Frequency domain. The power spectrum, Spectral analysis

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1 ime series models in the Frequency domain he power spectrum, Spectral analysis

2 Relationship between the periodogram and the autocorrelations = + = ( ) ( ˆ α ˆ ) β I Yt cos t + Yt sin t t= t= ( ( ) ) cosλ ( ) ( ) sinλ = yt Y t + Yt Y t t= t= λ = / ; ( λt t= t= cos = sinλ t= 0) = + t, s= = ( Yt Y)( Ys Y) cos λ ( s t) t, s= = ( Yt Y) + ( Yt Y ) ( Ys Y) cos λ ( s t ) t= ( Yt Y)( Ys Y)( cosλt cosλ s sinλt sinλ s) = 0+ p= t< s= C C cosλ p = [ C + C cos λ p] ; p=s-t 0 p= p p = C [+ r cosλ p]. 0 p= p

3 Periodogram and autocorrelations (cont.) ( ) 0 p= p. Ι () = C + C cosλ p A weighted average of the autocorrelations but the two statistics play a different role. he periodogram represents cyclical effects, which are deterministic. hus, if we have high correlations at lags, 4, we should see them in the periodogram. However, if we analyse, for example, a seasonal MA() process, it is no longer deterministic, the seasonal effects change over time and we only have r 0 with the other empirical correlations close to zero (not lie in a deterministic process). he periodogram may no longer indicate this phenomenon because the correlation is not the outcome of a single or few cycles with discrete frequencies. his suggests considering all the frequencies in the range [ 0, ], which is what Spectral analysis does. By considering the spectrum we should observe a hump around the seasonal frequencies. 3

4 he power spectrum he power spectrum of a stationary process is defined by the continuous function, f( λ ) = γ 0+ cos τ= τ γ λτ, 0 λ. Compare with the periodogram, ( ) p= f( λ ) = C0+ Cp cosλ p. f ( λ ) is defined for all λ and not just for λ / If t =. he summation is from to. Y is white noise, ( t) 0 ( ) γ = for all t 0 and f ( λ) = γ 0 / =const. hus, the spectrum is flat or the process may be regarded as consisting of an infinite number of cyclical components, all of which have equal weight. (his is what defines a white noise). 4

5 Notice that he power spectrum (cont.) γ 0 ( 0) 5 ( 0) dλ= γ (area under the spectrum). In fact, it is true in general that, 0 0 f ( λ) dλ= γ 0 ( ). (his is because ( λt) sin cos ( λt) dλ= 0 = 0). he function t f ( λ) / γ (0) is called the Spectral Density (uses correlations instead of covariances). Example MA() process- Yt = εt + θε t. ( ) ( ) ( ) ( ) γ 0 = σ + θ,, γ = θσ, γ = 0 ; ε ( + ) σ ε θ σε f ( λ) = + θ cos( λ) σε For θ = 0.5, f ( λ) = [5+ 4cos ( λ) ] 4 σε For θ = 0.5, f ( λ) = [5 4cos ( λ) ]. 4 ε

6 f σ = [5+ 4cos ]; θ = 0.5, σ ε = 4 ε ( λ) ( λ) 6

7 f σ = [5 4cos ]; θ = 0.5, σ ε = 4 ε ( λ) ( λ) 7

8 Exercises Exercise 5: Compute the power spectrum of the MA() model, Y = ε 0.5ε 0.6ε ; Var( ε ) = t t t t t Exercise 6: Let X t, Y t be two independent stationary series. 6.. Show that the series Zt = X t + Yt is also stationary. 6.. Let f ( λ), f ( λ ) denote the power X Y spectrum of the series X, Y. Show that, f ( λ) = f ( λ) + f ( λ) Z X Y 6.3. Compute f ( λ ) for the case where, X = ε 0.5 ε ; Var( ε ) = t t t t t t t t Z Y = e e ; Var( e ) = ; Cov(, e ) 0 t t ε = for all. t t 8

9 Properties of the periodogram as an estimator of the spectrum Power Spectrum: f ( λ ) = [ γ + γ cos λτ ] 0 τ= τ Periodogram: Ι () = [C + C cosλ p] 0 p= p It is natural to thin of I( λ ) / evaluated at any λ as an estimate of the spectrum. his is so because we would usually expect the covariances γ ( τ ) and hence C( τ ) to decay to zero as τ. However things are not so simple and to illustrate the problem consider the case where noise. K Y,..., Y are normal white 9

10 Poperties of the periodogram as an estimator of the power spectrum (cont.) For the white noise case, ( ˆ α ˆ ) K + βk I( λ ) σ ~ = = f ( λk ) σ K χ() E for λ = /. However, Var K 4 K σ σ f I( λ ) = 4 = = ( λk ). 4 hus, I( λk ) / is unbiased for f ( λ ), but the variance does not decay to zero with, so that the estimator is not consistent. his happens because we actually estimate parameters (the covariances appearing in the expression for the periodogram) by observations. Each covariance estimate is O(/), but the cumulative effect is O(). If U = Vi and Var(V i ) = VarU = 0.

11 Poperties of the periodogram as an estimator of the power spectrum (cont.) Notice also that since the periodograms are uncorrelated, Cov[ fˆ ( λ ), fˆ ( λ )] = 0 for j, so that the periodogram has a very irregular appearance. It can be shown that for any stationary fˆ ( λ) process, χ() for all 0 λ f ( λ) ( ˆ E f ( λ)) f ( λ) ; Var( fˆ ( λ)) f ( λ) j K. Also, the ordinates at different frequencies are asymptotically independent, giving rise to the erratic form of the estimates. In the white noise case, fˆ ( λ) ( ˆ α ˆ ) ˆ ˆ K + βk γ 0 ( α K + βk ) = / =. f ( λ) σ

12 Estimation of the power spectrum We are interested in estimating a continuous function and hence we lie our estimate to be a smooth function. Spectral analysis: How to estimate the power spectrum consistently. Assumption: he series is stationary, i.e., no trend or deterministic seasonality. Also needed from a practical point of view; we may not be able to detect frequencies other than those corresponding to deterministic trends and seasonals).

13 More on the stationarity requirement he stationarity requirement seems to rule out the use of the model, [ n / ] Y = α + ( u cosλ t+ v sin λ t) + ε. t 0 t = However, if we assume that u and v are random variables satisfying, E( u ) = E( v ) = 0 ; Var( u ) = Var( v ) =σ E( u u ) = E( v v ) = 0 i j i j i j E( u v ) = 0 for all i and j, i E( Y t ) [ n / ] = [ n / ] = j α =, γ ( τ ) = E( Y Y ) = = 0 Var( Y ) t [ n / ] ε σ = = σ +, and σ {cos( λ t)cos[ λ ( t τ )] + sin( λ t)sin[ λ ( t τ )]} t t τ σ cos( λτ ). 3

14 Stationarity requirement (cont.) he signal is stationary and yet deterministic (fixed) for any given realization, so that it is more a matter of interpretation. 4

15 Estimation of the power spectrum Method : Use estimators of the form, M fˆ( λ ) = [ w C + w C cos( λ )], 0 0 = where the { w } are decreasing weights called the lag window and M << is the truncation point. Basic Idea: we only consider some of the covariances and since the precision of the estimates C decreases as increases, the weights decrease in. Under normality assumptions, the periodogram estimates Ι ( ˆ λ K ) are sufficient statistics for the periodograms Ι ( λ K ) (considered as parameters). his means that if we are looing for smoothed estimates of the power spectrum, all that we need to do is to smooth the periodogram estimators. Indeed, the estimator f ˆ( λ ) defined above can be shown to be a smoothed function of the periodograms. 5

16 Lag windows in common use ) Parzen window M / M M 3 M wk = M M 0 > M ) uey window w K + cos = 0,..., M = M 0 > M he two windows are similar, but uey s weights are somewhat larger. 3) Daniel (rectangular) window w K sin( / M ) = =,... ; w 0=. / M 6

17 Sampling properties of spectral estimates w = for = 0,,,M, Suppose first that M such that, fˆ( λ ) = [ C0+ C cos( λ )]. = E[ ( Y )( ) t µ y Y t t µ y = γ = +, - C obtained by substituting Y for µ y and dividing by instead of -, is consistent for γ. We find that for M,, ( M / ) 0, M ˆ E[ f ( λ)] [ γ + ( ) γ cos( λ)] f ( λ) and 0 =, Var[ fˆ ( λ)] 0, since we only estimate M covariances and M<<. 7

18 Properties of spectral estimates (cont.) fˆ( λ ) = [ w0c0 + wc cos( λ )]. = he following results can be shown to hold: lim E fˆ( λ ) = f ( λ ) for all λ, ˆ Var [ f ( λ)] ( + δλ,0, ) f ( λ) w K= ( ) λ= 0, w = w ; δλ,0, =. 0 otherwise Also, f ˆ( λ ) and f ˆ( λ ) are correlated positively when λ λ 0. he correlation increases as λ λ decreases f ˆ( λ ) is a smooth function. he asymptotic results assume certain behaviour of the weights w. For consistency it is sufficient that M / 0 as. 8

19 Bias and variance of window estimates Window Bias var ˆ { f ( λ)}/ Mf ( λ) 6 Parzen M f ''( λ ) 0.54 uey Daniel 4M 6M f ( λ) 0.75 f ( λ).00 he smaller the bias, the larger the variance. he values in the table assume λ 0,. For λ= 0,, the variances should be multiplied by. 9

20 Example MA() Yt = εt 0.5ε t 0.6ε t ; Var( ε t ) = γ (0) = =.6 γ () = = 0.; γ () = 0.6 f ( λ) = [ γ (0) + γ ( τ )cos( λτ )] τ= = [.6 + ( 0.cosλ 0.6cos λ)] = [.6 0.4cosλ.cos λ]. f '( λ) = [0.4sinλ+.4sin λ] f ''( λ) = [0.4cosλ+ 4.8cos λ] 0

21 f ( λ ) [.6 0.4cosλ.cos λ] = ; σ ε =

22 Window Example (cont.) Bias 6 Parzen M uey Daniel 4M 6M f var ˆ ''( λ ) 0.54 f ( λ) 0.75 f ( λ).00 Suppose that =00, M=0 { f ( λ)}/ Mf ( λ) = =. ( M / ) f ( ) 0.6 ; ( M / ) f ( ) 0.03 Window Bias Variance MSE λ / / / Parzen uey Daniel f '( ) =.3, f '( ) = 0 ; f ''( ) =.53, f ''( ) =.4.

23 Exercise 7: Compute the bias, variance and MSE of the windows of Parzen, uey and Daniel at λ = 0., 0.5, 0.9 for the model, Y = ε ε σ =. Assume =00, M=0., t t t ε Choice of M (truncation point) Can be chosen in an optimal way if the underlying process is nown. Compromise between bias and variance; he smaller M, the smaller is the variance but the larger is the bias. Small Values of M possibly too smooth but may give an idea of the large peas (but may hide the small peas). Large values of M possibly too many peas (lie with the periodogram). 3

24 Choice of M (cont.) A possible procedure for determining M use the estimated autocorrelations, set M such that ˆ ρ j 0 for j> M Not very efficient because the ˆ ρ j are autocorrelated, so that they tend to decay slower than the theoretical autocorrelations. Another procedure is window closing where we try several values of M, say 3, such that ( M 3 / M) 4 and study the results. (Low values will highlight the large peas, high values will show other important peas but also spurious peas; compromise achieved by studying the results for the intermediate values of M. 4

25 Choice of M (cont.) It is sometimes recommended in the literature to choose M as a fixed proportion of, say M = (implies ( M / ) 0 as ) but there is no theoretical justification for this strategy. For example, in the white noise case the best choice is M=0. In principle, we can estimate the spectrum at any value of λ but often it is evaluated at frequencies j/q, where Q is chosen to be sufficiently large to show all the important cycles (common choice Q=M). 5

26 Estimation of the power spectrum, an alternative method So far we considered estimators of the form, M fˆ( λ ) = [ w C + w C cos( λ )]. 0 0 = Method : An alternative method is to use weighted averages of the periodograms, i.e., l ( j+ l) l= m l= m * * fˆ( λj ) = wl I ( λj+ ) = wl I ( ) l= m l= m j ; λ = j /, and l varies over m+ successive integers such that the λ l are symmetric around the λ for which the estimator is computed. j Example: l= m l fˆ( λj ) = I ( λj+ ) m +. l= m See later for other examples of weights. 6

27 Alternative method (cont.) General expression of estimators: l ; l= m * fˆ( λ j ) = w l m lι ( λ = j+ ) l * * * * * * l l 0 m l = l w = w, w w... w > 0 w = At the end points, m * fˆ(0) wl I ( l / ) l = m * f I wl I l= =, l ˆ( ) = [ ( ) + ( )]. ;. Assumes symmetry of the periodogram around 0 and and I (0) = 0. 7

28 Rationale of Method he periodogram estimators are asymptotically unbiased and since neighbouring estimators are asymptotically uncorrelated, the variance of f ˆ( λ ) obtained this way is of order [/(m+)]. However, f ˆ( λ ) is generally biased; ˆ ( j+ l) ( j+ l) E f E w w f [ ; l= m * l= m * [ ( λj)] = [ lι ( )] = l ( ) l= m l= m E[ ( ) f ( ) Ι ]. However, if f ( λ ) is approximately linear around λ, the bias will be small. 8

29 Method (cont.) l l= m * fˆ( λ j ) = w l m lι ( λ = j + ). In general, the bias is small if m is small and ( ) f λ is a reasonably smooth function of λ. Choice of m same problem as with M under the previous procedure. he larger is m, the smaller is the variance but the larger is the bias. Advantage of Method over method Method is more efficient computationally, particularly with the use of the Fast Fourier ransform, and this is now the standard method used in computer software. 9

30 Confidence Intervals under Method It can be shown that for the estimator, M fˆ( λ ) = ( w C + w C cos( λ ) (Method ) 0 0 = Asym.. q fˆ( λ) / f ( λ) ~ χ ; M 0 q q = /[ w + w ]. = A 00( -α)% confidence interval for f ( λ ) is, q fˆ ( λ) q fˆ ( λ) [, ] χ χ q, α / q, α / For Parzen window, q = 3.7 / M. For uey window, q =.67 / M. 30

Prof. Dr. Roland Füss Lecture Series in Applied Econometrics Summer Term Introduction to Time Series Analysis

Prof. Dr. Roland Füss Lecture Series in Applied Econometrics Summer Term Introduction to Time Series Analysis Introduction to Time Series Analysis 1 Contents: I. Basics of Time Series Analysis... 4 I.1 Stationarity... 5 I.2 Autocorrelation Function... 9 I.3 Partial Autocorrelation Function (PACF)... 14 I.4 Transformation