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98 Chapter 3 ARMA Models
is entirely data driven. For the time series of overshorts, the data, through the graph
of the ACF, lead us to the MA(1) model. Alternatively, we can attempt to model the
mechanism generating the time series of overshorts using a structural model.Aswe
will see, the structural model formulation leads us again to the MA(1) model. In the
structural model setup, write Yt, the observed amount of fuel in the tank at time t,as
Yt y ∗
t + Ut, (3.2.17)
where y∗ t
with yt above) and Ut is the resulting measurement error. The variable y∗ t
is the true (or actual) amount of fuel in the tank at time t (not to be confused
is an idealized
quantity that in principle cannot be observed even with the most sophisticated
measurement devices. Similarly, we assume that
where a ∗ t
At a ∗
t + Vt, (3.2.18)
is the actual amount of fuel sold minus the actual amount delivered during
day t, and Vt is the associated measurement error. We further assume that {Ut} ∼
WN 0,σ2
U , {Vt} ∼WN 0,σ2
V , and that the two sequences {Ut} and {Vt} are uncorrelated
with one another (E(UtVs) 0 for all s and t). If the change of level per day
due to leakage is µ gallons (µ