In Example 6.3, estimate an MA(1) model in the differences \(z_{t}\) of the wholesale prices index using both methods applied in that example.

Data from Example 6.3

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Consider data from Enders (2004, p. 87) on the US Wholesale Price Index (WPI), specifically quarterly data from 1960Q1, with y = 30.7, through to 1990Q4, with y224 = 116.2. Consider an ARMA(1, 1) model in the first differences Z = y+1 yr. The model may be written u~ N(0,). The errors at may be represented recursively as u-c-0-1- Ou-1 uc zo - Ouo. starting with To set the recursion going, one may generate latent pre-series values {zo, uo} using a dif- fuse prior on zo and assuming u N(0, 2). The log likelihood values for quarter t are then ~ log(L) = -0.5 log(2o) - 0.5u/(20), and one may use the dloglik option in OpenBUGS. Alternatively (method 2) one may represent the lagged residual as u-1=-1-Mt-1 (e.g. Marriott et al., 1996) so that Z~N(c+2-1+0 (z1-1 - M-1), 0). Zt Flat priors are adopted on c and zo, a uniform prior on , and priors on and are constrained to stationarity and invertibility. The second halves of two chain runs of 10000 iterations provide estimates (95% CRI) for c, d, and as in Table 6.2.