Question

(b) Jan has collected a monthly time series consisting of 167 observations on the differenced log Yen/SAU exchange rate and plotted the sample ACF for the series (see below). Use the QLB(2) statistic to test whether or not the series is a white-noise process at the 5% level of significance. Write down your hypotheses, decision rule and conclusion. [3 marks] Lag-1.00 -0.60-0.20 0.20 0.60 ACF 1.00 I 0.001073 0.121080 1 -0.084039 1 -0.052353 I -0.054732 1 -0.095489 0 . 044049 1 -0.057184 I 0.062974 1 0.009568 R I R I R I 9 10 Here 'R' represents the magnitudes of ACF values 100 (c) Suppose that we have a time series (y)9. How would you compute the sample partial autocorrelation function p(r) for τ 1.237 [3 marks]

Answer 1

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(c) In the following please replace X_t with y_t .

Alternatively, we can fit AR models of increasing orders; then the estimate of the last coefficient in each model is the sample PACF. Specifically, the estimates Pnj,j=1, hin can be updated recursively from the estimates h-1 at step h 1 corresponding to The updating recursive equations (Durbin-Levinson Algorithm) are given by These recursive equations avoid the inversion of h × h mlatrices. Example. Assume that sample ACF ph were calculated previously. φ22 = (p(2) _ φιιρ(1))/(1-φιιρ(1)) = (p(2)-武1))/(1-p?(1)) 112211

To calculate sample ACF:

For observations r1,.. . , Vn of a time series, i=-Σ he sample autocovariance function is Ct. the sample mean is for -n <h< n. t=1 The sample autocorrelation function is 5(h) ρ(h) =-(0)