Introduction to Time Series Analysis. Lecture 21. 1. Review: The periodogram, the smoothed periodogram. 2. Other smoothed spectral estimators.

3. Consistency. 4. Asymptotic distribution.

1 Review: Periodogram

The periodogram is defined as

2 2 I(ν)= Xc (ν)+ Xs (ν).

1 n X (ν)= cos(2πtν)x , c √n t t=1 1 n X (ν)= sin(2πtν )x . s √n j t t=1

Under general conditions, Xc(νj ), Xs(νj ) are asymptotically independent and N(0, f(ν )/2). Thus, EI(ˆν(n)) f(ν), but Var(I(ˆν(n))) f(ν)2. j → →

2 Review: smoothed periodogram

If f(ν) is approximately constant in a band of frequencies [ν L/(2n),ν + L/(2n)], we can average the periodogram over this k − k band:

(L−1)/2 1 fˆ(ν )= I(ν l/n) k L k − l=−(L−1)/2 (L−1)/2 1 2 2 = X (νk l/n)+ X (νk l/n) . L c − s − l=−(L−1)/2

3 Review: smoothed periodogram

Under general conditions, the X (ν l/n) and X (ν l/n) are c k − s k − asymptotically independent and N(0, f(ν l/n)/2). Thus, k − Efˆ(ν(n)) f(ν) and Varfˆ(ν(n)) f 2(ν)/L. → → Notice the bias-variance trade off: 1. Our assumption that f is approximately constant on [ν L/(2n),ν + L/(2n)] becomes worse as L increases, so the difference − between fˆ(ˆν(n)) and f(ν) (the bias) will increase with L. 2. The variance of our estimate, Varfˆ(ˆν(n)) decreases with L.

4 Other smoothed spectral estimators

Instead of computing an unweighted average of the periodogram at all nearby frequencies, it is common to consider other weighted averages, typically with a smoother weighting function. Consider the weighted average

fˆ(ν)= W (j)I(ˆν(n) j/n), n − |j|≤Ln where the bandwidth Ln is allowed to vary with n, and Wn is called the spectral window function.

5 Other smoothed spectral estimators

For example, if 1  L if j < L/2, Wn(j)=  | | 0 otherwise,  then we have the smoothed spectral estimator we consider earlier,

fˆ(ν)= W (j)I(ˆν(n) j/n) n − j 1 = I(ˆν(n) j/n). L − |j|

6 Consistency of nonparametric spectral estimation

Suppose Ln and Wn satisfy L L , n 0, n → ∞ n → W (j) 0,W (j)= W ( j), W (j) = 1, n ≥ n n − n |j|≤Ln W 2(j) 0 n → |j|≤Ln as n , then... → ∞

7 Consistency of nonparametric spectral estimation

... for a large class of stationary processes, fˆ(ν) f(ν) in the mean → square sense. In particular, Efˆ(ν) f(ν) and → −1 f 2(ν ) if ν = ν (0, 1/2)  2  ˆ ˆ  1 1 2 Wn (j) Cov(f(ν1), f(ν2))  ∈ → 0 if ν = ν . |j|≤Ln  1 2  The conditions on the bandwidth parameter Ln ensure that, as the sample size grows, the window width goes to zero, but includes an infinite number of terms. The conditions on the spectral window function Wn ensure that the expectation of fˆ(ν) converges to f(ν) and its variance converges to zero.

8 Consistency of nonparametric spectral estimation

E fˆ(ν) = W (j)E I(ˆν(n) j/n) n − |j|≤Ln W (j)f(ν)= f(ν). ≈ n |j|≤Ln

Var fˆ(ν) = W (j)W (k)Cov I(ˆν(n) j/n),I(ˆν(n) k/n) n n − − j,k W 2(j)Var I(ˆν(n) j/n) ≈ n − j f 2(ν) W 2(j) 0. ≈ n → j

9 Introduction to Time Series Analysis. Lecture 21. 1. Review: The periodogram, the smoothed periodogram. 2. Other smoothed spectral estimators.

3. Consistency. 4. Asymptotic distribution.

10 Nonparametric spectral estimation: asymptotics

Recall that for Daniell’s estimator we have

(L−1)/2 1 2 2 fˆ(νk)= X (νk l/n)+ X (νk l/n) , L c − s − l=−(L−1)/2

2 which is (asymptotically) a sum of 2L independent χ1 random variables, so χ2 fˆ(ν ) f(ν ) 2L . k ∼ k 2L 2 But for a non-uniform weighting, we form a weighted sum of these χ1 random variables, so we cannot count up the degrees of freedom in the same way. But we can still approximate a general smoothed spectrum by fˆ(ν ) c χ2 for some c and d. k ∼ k d k

11 Nonparametric spectral estimation: asymptotics

Suppose that fˆ(ν ) c χ2. What values should c and d take? We have, k ∼ k d k for a suitable spectral window Wn, f(ν ) Efˆ(ν )= c d, k ≈ k k f 2(ν ) W 2(j) Varfˆ(ν ) = 2c2d. k n ≈ k k |j|≤Ln Thus, we get f(ν ) c = k , k d 2c = f(ν ) W 2(j), k k n |j|≤Ln 2 d = . W 2(j) |j|≤Ln n

12 Nonparametric spectral estimation: asymptotics

f(ν ) c = k , k d 2 d = . W 2(j) |j|≤Ln n This d is often referred to as the equivalent degrees of freedom for a smoothed spectrum. Under suitable conditions (and for a slightly different definition of d), it can be shown that, asymptotically, χ2 fˆ(ν(n)) f(ν) d . ∼ d

13 Nonparametric spectral estimation: the lag window

We can also view smoothing the spectrum in the frequency domain as smoothing in the time domain, via

fˆ(ν)= w (j)ˆγ(j)e−2πiνj, n |j|≤Ln where wn is the inverse Fourier transform of the spectral window. This is known as the lag window.

Tapering techniques are also popular: Define yt = htxt for some weighting function ht. Then the tapered estimator is the smoothed spectral estimator for the tapered series yt. For a weighting function ht that smoothly diminishes values near the ends of the time series, we see less leakage.

14 Introduction to Time Series Analysis. Lecture 21. 1. Review: The periodogram, the smoothed periodogram. 2. Other smoothed spectral estimators.

3. Consistency. 4. Asymptotic distribution.

15