Note on “Equilibirum-correction vs. di¤erencing
in macroeconometric forecasting”
Øyvind Eitrheim and Ragnar Nymoen
July 18, 2000
Eitrheim et al. (1999) notes that the error term ²y;t in equation (2.9) is autocorrelated. However it also has a non-zero mean, since the omitted equilibrium
correction term has a mean which is di¤erent from zero. Thus, the formulation
implies that the “dVAR forecaster”picks up the wrong growth rate in yt . This is a
consequence of the assumption that both the intercept and derivative parameters of
E[¢yt jdVAR] are identical to the corresponding parameters of the EqCM.
It appears to be more relevant to assume that the dVAR forecast model has
a zero mean disturbance. Formally, a mean zero forecast error for the dVAR can
formally be obtained by adding E[²T +1 ] = ¯' to the dVAR forecast (implicitly
de…ned in (2.12)). However, this seems to imply that the dVAR-forecaster is able
to use the mathematical expectation of ²t given the true DGP, which we assume is
unknown to the forecaster.
However, consider a dVAR de…ned by
¢yt = ° d + ¼ d ¢xt + ²dt
and E[²dt ] = 0; since ° d and ¼ d are the plims of estimated coe¢cients (just as the
coe¢cients in the EqCM given by (2.5) and (2.6) can be viewed as plims of their
respective estimators). Then from OLS-properties (and (2.5)-(2.6))
° d = ¯' ¡ ¼ d ' = ¯' ¡ ¼' + ®b'
¼ d = ¼ ¡ ®b
where b is the plim of the regression coe¢cient of the omitted variable EqCM t with
respect to ¢xt . Then
y^T +1;dV AR = ¯' + yT
and
¯'
] ¡ ¯'
®
= ¡®[(yT ¡ yT0 ) ¡ ¯(xT ¡ x0T )] + biast+1;EqCM ;
biasT +1;dV AR = ¡®[(yT ¡ yT0 ) ¡ ¯(xT ¡ x0T ) + (³ ¡ ³ ¤ ) ¡
so that ¯'=® vanishes from (2.16). Likewise, in the subsequent dVAR forecast
biases, the terms ¯' (1.step) and ¯'h (multi-step) disappear. So “assuming parameters to be known” seems equivalent to the assumption that E[²T +1 ] = ¯' is known
to the dVAR forecaster.
References
Eitrheim, Ø., T. A. Husebø and R. Nymoen (1999). Equilibrium-Correction versus
Di¤erencing in Macroeconomic Forecasting. Economic Modelling, 16 , 515–544.
1