R,{0) = (1 +b? + b3) Re(0) (5.190) 
R,{1) = R,(- 1) = (b1b2 + by) Re (0) (5.191) 
R,{2) = R,(— 2) = bz - R-(0) (5.192) 
Rr) =0 rs & (5.193) 
because 
2 ee 
Ele; €-,]=06,-O2= Wee ee 6, is Kronecker’s delta function. 
r 
? 
R,{r) = 0 for r = 3. This is an interesting characteristic of MA(2). Generally, for 
MA(n),R,(r) = 0 for r = n+1. In the same way as for ARMA(2.1) 
a2 . . p 
s(@) = 1+ bye + bye?” | . (5.194) 
Estimation of bj, b2, and ag is possible by solving the quadratic Eqs. 5.190, 5.191, and 
5.192 for by, bz, andae. 
5.2.5.3 Example of MA(2), MA(1), and ARMA(2.2). As was done for AR(O), AR(1), 
AR(2), and ARMA(2.1) in the preceding sections, an MA(2) model was generated over 
t= 110600 by X,=€,+0.2€,)+ 0.8€,2 as in Fig. 5.32. Fore,, the same pure random 
process N[0, 1] as was generated in Fig. 5.3 in Section 5.2.1 for AR(0) is used. Its read- 
ings are listed in Appendix A1 as Table A1.6; pp. 251, 256, and 257. Figure 5.33a shows 
the theoretical R(r) using the design values b; = + 0.2, b2 = + 0.8, and o2=1.0, and 
Fig. 5.33b the estimated sample autocorrelation R(r). It is interesting to note that theo— 
retically R(r) = 0 forr = 3. 
Figure 5.34a shows the theoretical spectrum given by Eq. 5.194 where b, = + 0.2, 
by = + 0.8, ando2= 1.0, which are all design values, and Fig. 5.34b the spectrum of the 
fitted model given by Eq. 5.194 where b; = 0.18662, bz = 0.78000, and a2 = 1.0816 calcu- 
lated by the method described in Section 5.3.3. Of course again in this example, order 
determination by AIC, to be explained in Section 5.5, gave the proper value of m = 2. 
Figure 5.34c is the spectrum calculated by the nonparametric method. Here again the 
90% level confidence interval is shown by vertical lines for reference. This was originally 
generated as MA(2). However, as will be discussed in Section 5.4, often an AR model is 
fitted generally to the processes taking advantage of the fact that the determination of 
parameters is much easier than for ARMA. 
Figure 5.34d shows the spectrum of the AR model fitted to this process. The order 
of this AR process is n = 13 and is naturally higher than 2 for m as MA(m). This result 
shows that, if this MA(2) is to be approximated by AR(n), AR(13) is the closest model to 
be adopted. Again, it is interesting to note that the spectrum Fig. 5.34d of AR(13) looks 
very similar to the spectrum Fig. 5.34c estimated by the nonparametric method. 
162 
