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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 2S 
be a linear or curvilinear trend, a sinusoid or a sum of periodic 
components. Now assuming that the model (76) is valid both 
in the observation range and the forecasting range we have, 
as a first approximation, 
(77 b) 
E(J)m1 
valid if the forecast range is short relative to the observation 
range. 
The present approach further includes forecasting by 
exogenous variables. Considering the linear case, model M is 
(78) 
Ve=B80+B x; +... +B, x, +, 
with 
(79) 
E(y,lx,, > Hing) =03, +8, Xgut... +3, XL} 
where the coefficients 3; of the exogenous variables x; usually 
are estimated from the observations. Assuming stable model 
structure we have in this case 
(80) 
E(J%)~1 
provided the forecast range is short. 
The criteria (75) and (76) can be developed in the direction 
of significance tests. The following result is due to S. MARTI- 
NELLE (*). As applied to a model (77 a) with k linear compo- 
(*) Ref. 45. I am indebted to Mr. MARTINELLE for kindly placing his 
unpublished results at disposal for the present report. 
2] Wold - pag. 42