Nearer-Normality and Some Econometric Models

NEARER-NORMALITY AND SOME ECONOMETRIC MODELS
A BIVARIATE RANDOM VARIABLE (X, Y) with characteristic function q5(S1, S2) has its
(p, q)th cumulant C,, generated by
If (X, Y) is normally distributed, then C,, =0, all p, q such that p +q >2.
DEFINITION: For the pair of random variables (XI, Y1), (X,, Y2) with each component
having zero mean and unit variance, (XI, Yl) will be said to be nearer-normal than (X2, Y2)
if
for all p, q with p +q >1,provided there is at least one strict inequality. It will be assumed
that all cumulants exist.
This is obviously a very stringent condition, and it cannot be applied to most pairs of
random variables. A number of results have recently been discussed in the univariate case
(Granger [I])where it is shown that the weighted sum of independent and identically
distributed random variables is nearer-normal than the constituent variables. This result
does not necessarily hold if the variables are not identically distributed or if they are
dependent. The corresponding bivariate theorem is as follows.