TimeSeriesBook.pdf

4 as a reminder the theoretical autocorrelation

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4 As a reminder: the theoretical autocorrelation coefficients are ρ ( h ) = φ | h | .
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82 CHAPTER 4. ESTIMATION OF MEAN AND ACF 0 5 10 15 20 25 30 35 40 -1 -0.5 0 0.5 1 order correlation coefficient theoretical ACF estimated ACF upper bound for AR(1)-confidence interval lower bound for AR(1)-confidence interval Figure 4.3: Estimated autocorrelation function of an AR(1) process with φ = 0 . 8 and corresponding 95 percent confidence interval for T = 100 4.3 Estimation of the partial autocorrelation function According to its definition (see definition 3.2), the partial autocorrelation of order h , α ( h ), is equal to a h , the last element of the vector α h = Γ - 1 h γ h (1) = R - 1 h ρ h (1). Thus α h and consequently a h can be estimated by ˆ α h = b Γ - 1 h ˆ γ h (1) = ˆ R - 1 h ˆ ρ h (1). As ρ ( h ) can be consistently estimated and is asymptotically nor- mally distributed (see Section 4.2), the continuous mapping theorem (see Appendix C) ensures that the above estimator for α ( h ) is also consistent and asymptotically normal. In particular we have for an AR(p) process (Brock- well and Davis, 1991) T ˆ α ( h ) d ----→ N(0 , 1) for T → ∞ and h > p. This result allows to construct, as in the case of the autocorrelation coef- ficients, confidence intervals for the partial autocorrelations coefficients. The 95 percent confidence interval is given by ± 1 . 96 T . The AR(p) process is char- acterized by the fact that the partial autocorrelation coefficients are zero for h > p . ˆ α ( h ) should therefore be inside the confidence interval for h > p and outside for h p . Figure 4.4 confirms this for an AR(1) process with φ = 0 . 8.
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4.4. ESTIMATION OF THE LONG-RUN VARIANCE 83 0 2 4 6 8 10 12 14 16 18 20 -1 -0.5 0 0.5 1 order partial autocorrelation coefficient estimated PACF lower bound for confidence interval upper bound for confidence interval Figure 4.4: Estimated PACF for an AR(1) process with φ = 0 . 8 and corre- sponding 95 percent confidence interval for T = 200 Figure 4.5 shows the estimated PACF for an MA(1) process with θ = 0 . 8. In conformity with the theory, the partial autocorrelation coefficients converge to zero. They do so in an oscillating manner because θ is positive (see formula in Section 3.5) 4.4 Estimation of the long-run variance For many applications 5 it is necessary to estimate the long-run variance J which is defined according to equation (4.1) as follows 6 J = X h = -∞ γ ( h ) = γ (0) + 2 X h =1 γ ( h ) = γ (0) 1 + 2 X h =1 ρ ( h ) ! . (4.5) This can, in principle, be done in two different ways. The first one con- sists in the estimation of an ARMA model which is then used to derive the implied covariances as explained in Section 2.4 which are then inserted into 5 For example, when testing the null hypothesis H 0 : μ = μ 0 in the case of serially correlated observations (see Section 4.1); for the Phillips-Perron unit-root test explained in Section 7.3.2.
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