The sunspot numbers {X_t, t = 1, 2, ..., 100} have sample autocovariances gamma(

Question

The sunspot numbers {X_t, t = 1, 2, ..., 100} have sample autocovariances gamma(0) = 1382.2, gamma (1) = 1114.4, gamma (2) = 591.73, and gamma (3) = 96.216. Let Y_t = X_t - 46.93, t = 1, 2, ...., 100, be the mean-corrected sunspot series. (a) Use the Durbin-Levinson algorithm to compute the sample partial autocorrelations phi_11, phi_22 and phi_33 for Y_t. Is the value of phi_33 compatible with the hypothesis that the data are generated by an AR(2) process? (USe significance level 0.05). (b) Find the Yule-Walker estimates of phi_1, phi_2, sigma^2 in the model: Y_t = phi_1 Y_t - 1 + phi_2 Y_t + Z_t, {Z_t} ~ W N(0, sigma^2). Assuming that the data really are realization of an AR(2) process, find 95% confidence intervals for phi_i, i = 1, 2.

Explanation / Answer

A generic AR model
x[n]+a1x[n1]+a2x[n2]+…+aNx[nN]=w[n](1)
x[n]+a1x[n1]+a2x[n2]+…+aNx[nN]=w[n](1)
can be written in compact form as
k=0Nakx[nk]=w[n],a0=1(2)
k=0Nakx[nk]=w[n],a0=1(2)
Note that the scaling factor x[n]x[n] term is a0=1a0=1
Multiplying (2) by x[nl]x[nl]
k=0NakE{x[nk]x[nl]}=E{w[n]x[nl]},a0=1(3)
k=0NakE{x[nk]x[nl]}=E{w[n]x[nl]},a0=1(3)
One can readily identify the auto-correlation and cross-correlation terms as
Yule Walker Equations

k=0Nakrxx[lk]=rwx[l],a0=1(4)
k=0Nakrxx[lk]=rwx[l],a0=1(4)
The next step is to compute the identified cross-correlation rwx(l)rwx(l) term and relate it to the auto-correlation term rxx(lk)rxx(lk)
The term x[nl]x[nl] can also be obtained from equation (1) as
x[nl]=k=1Nakx[nkl]+w[nl](5)
x[nl]=k=1Nakx[nkl]+w[nl](5)
Note that data and noise are always uncorrelated, therefore – x[nkl]w[n]=0x[nkl]w[n]=0. Also the auto-correlation of noise is zero at all lags except at lag 0 where its value is equal to 22 (Remember the flat Power Spectral Density of white noise and its auto-correlation). These two properties are used in the following steps. Restricting the lags only to positive values and zero ,

rwx(l)=E{w[n]x[nl]}=E{w[n](k=1Nakx[nkl]+w[nl])}=E{k=1Nakx[nkl]w[n]+w[nl]w[n]}=E{0+w[nl]w[n]}=E{w[nl]w[n]}={0,l>02,l=0(6)
rwx(l)=E{w[n]x[nl]}=E{w[n](k=1Nakx[nkl]+w[nl])}=E{k=1Nakx[nkl]w[n]+w[nl]w[n]}=E{0+w[nl]w[n]}=E{w[nl]w[n]}={0,l>02,l=0(6)
Substituting (6) in (4),

k=0Nakrxx[lk]={0,l>02,l=0,a0=1(7)
k=0Nakrxx[lk]={0,l>02,l=0,a0=1(7)
Here there are two cases to be solved – l>0l>0 and l=0l=0
For l>0l>0 case, equation (7) becomes,

k=1Nakrxx[lk]=rxx[l](8)
k=1Nakrxx[lk]=rxx[l](8)
Clue: notice the lower limit of the summation which changed from k=0k=0 to k=1k=1.

Now, equation (8) can be written in Matrix form

Yule Walker Equations Matrix form

This is the Yule-Walker Equations which comprises of a set of NN linear equations and NN unknown parameters.

Representing equation (9) in a compact form
R¯a¯=r¯(10)
R¯a¯=r¯(10)
The solutions a¯a¯ can be solved by
a¯=R¯1r¯(11)
a¯=R¯1r¯(11)
Once we solve for a¯a¯ , equivalently akak, – the model parameters, the noise variance 22 can be found by applying the estimated values of akak in equation (7) by setting l=0l=0
Matlab’s “aryule” efficiently solves the “Yule-Walker” equations using “Levinson Algorithm”[4][5]

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