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Figure 8-5
The one-step predictors ˆYt
for the airline passenger
data (solid line) and the
actual data (square boxes).
8.6 State-Space Models with Missing Observations 283
(thousands)
100 200 300 400 500 600
1949 1951 1953 1955 1957 1959 1961
must be nearly constant and equal to ˆX12 2.171. The first three components of the
predictors ˆXt are plotted in Figure 8.4. Notice that the first component varies like a
random walk around a straight line, while the second component is nearly constant as
a result of ˆσ 2 2 ≈ 0. The third component, corresponding to the seasonal component,
exhibits a clear seasonal cycle that repeats roughly the same pattern throughout the
12 years of data. The one-step predictors ˆXt1 + ˆXt3 of Yt are plotted in Figure 8.5
(solid line) together with the actual data (square boxes). For this model the predictors
follow the movement of the data quite well.
8.6 State-Space Models with Missing Observations
State-space representations and the associated Kalman recursions are ideally suited
to the analysis of data with missing values, as was pointed out by Jones (1980) in the
context of maximum likelihood estimation for ARMA processes. In this section we
shall deal with two missing-value problems for state-space models. The first is the
evaluation of the (Gaussian) likelihood based on {Yi1 ,...,Yir }, where i1,i2,...,ir
are positive integers such that 1 ≤ i1 < i2 < ··· < ir ≤ n. (This allows for
observation of the process {Yt} at irregular intervals, or equivalently for the possibility
that (n−r)observations are missing from the sequence {Y1,...,Yn}.) The solution of
this problem will, in particular, enable us to carry out maximum likelihood estimation
for ARMA and ARIMA processes with missing values. The second problem to be
considered is the minimum mean squared error estimation of the missing values
themselves.