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Elements of Exploratory Time Series Analysis 2. 5) zt := 1/yt ∼ 1/ E(Yt ) = 1/flog (t), t = 1, . . , n. Then we have the linear regression model zt = a + bzt−1 + εt , where εt is the error variable. Compute the least squares estimates a ˆ, ˆb of a, b and motivate the estimates βˆ1 := − log(ˆb), βˆ3 := (1 − exp(−βˆ1 ))/ˆ a as well as n ✏n + 1 ✑✑ 1 ❳ ✏ βˆ3 βˆ2 := exp βˆ1 + −1 , log 2 n t=1 yt proposed by Tintner (1958); see also the next exercise. 3. The estimate βˆ2 defined above suffers from the drawback that all observations yt have to be strictly less than the estimate βˆ3 .

1. Autocorrelation functions of AR(1)-processes Yt = aYt−1 + εt with different values of a. 2 Moving Averages and Autoregressive Processes 57 PROC GPLOT DATA = data1 ; PLOT rho * s = a / HAXIS = AXIS1 VAXIS = AXIS2 LEGEND = LEGEND1 VREF =0; RUN ; QUIT ; ✝ ✡ The data step evaluates rho for three different values of a and the range of s from 0 to 20 using two loops. The plot is generated by the procedure GPLOT. The LABEL option in the AXIS2 statement uses, in addition to the greek font CGREEK, the font COMPLEX assuming this to be the default text font (GOPTION FTEXT=COMPLEX).

Models of Time Series In this case we obtain for a stationary process (Zt ) that almost surely Yt = au Zt−u and u bw Yt−w = Zt , t ∈ Z. 1) w The filter (bu ) is, therefore, called the inverse filter of (au ). Causal Filters An absolutely summable filter (au )u∈Z is called causal if au = 0 for u < 0. 10. Let a ∈ C. The filter (au ) with a0 = 1, a1 = −a and au = 0 elsewhere has an absolutely summable and causal inverse filter (bu )u≥0 if and only if |a| < 1. In this case we have bu = au , u ≥ 0. Proof.

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