1308 Testing Econometric Software
of squares. In the case of CLS, the relevant quantity is:
S∗(φ,θ)=∑nt= 1a^2 t(φ,θ|w∗,a∗,w), (28.5)where the subscript asterisks emphasize conditioning on the choice of starting
values. The problem, obviously, is in the choice of pre-sample values forwanda,
i.e., how to definew∗anda∗.
BJ (1976, p. 211) consider two approaches. The first involves settingw∗anda∗
equal to their unconditional expectations, which areμand 0, respectively. In the
event thatμ =0,w ̄ can be substituted for each element ofw∗. This preserves
the full sample. However, due to potential instabilities when the roots ofφ(B)are
near the unit circle, this method is not much used. A more reliable method is to
discard observations so that actual values ofware used for all calculations. This
implies that the sum of squares can be taken only over observationsp+1 through
n. (Clearly, whenp=0 this method is equivalent to the previous method.) Thus,
the second approach setsat=0 fort=1,...,pand calculates thea’s fromap+ 1
onward; observations must be dropped from the sample. It is the second approach
that BJ adopt for CLS, and which is considered here.
Setting to zero values that might vary substantially from zero may induce a tran-
sient effect that can adversely affect the quality of the final estimates. BJ (1976, p.
211) notes that CLS is “not satisfactory” for seasonal models.
To derive the ULS method, BJ introduce the unconditional sum of squares:
S(φ,θ)=∑nt=−Q[at|(φ,θ,w]^2 , (28.6)where the at,t ≤ 0 are computed recursively by taking expectations of
equation (28.4). The values necessary to compute theseat,t≤0, are the[wt],t≤0,
which can be calculated via the procedure known asbackcasting.
In the usual fashion, the backward and forward representations are given by:
et=(wt−μ)−φ(wt+ 1 −μ)+θet+ 1 (28.7)
at=(wt−μ)−φ(wt− 1 −μ)+θat− 1. (28.8)Given initial estimates of the parameters,μ 0 ,φ 0 andθ 0 , settingen+ 1 =0 allows
equation (28.7) to be executed fromt=ntot=1. Att=0 the valueet=0, and
so the equation can be rewritten as an expression forwt, from which[wt],t≤ 0
can be calculated back to somet=−Q, the quantity (wt−μ) being neglible for
t<−Q. These backcasted values ofwtcan then be used to start the recursion
equation (28.8) from−Qupon settinga−Q− 1 =0.
Exactly when the quantitywt−μbecomes neglible is not explicitly stated in
BJ. Their example in their Table 7.4 (1976, p. 218) suggests thatwt−μ<0.01
is an appropriate stopping rule. Other stopping rules for backcasting have been
described in the literature. For example, Granger and Newbold (1977, p. 88) suggest
stopping when the magnitude of three successive values ofEc(Yt)is less than 1%
of the standard deviation ofyt.