686  -both -
  members  of  the  subspace  of  functions  that  is  orthogonal
to  the  function
olnf(yx,e.
h(x,)  -g(x)d(y) -

(2.13)
90
with  regard  to  the  inner  product
h(x,o),p(x)  »  =    h(x,p(x)d(x).
The  difference  between  the  unweighted  estimator  6,  for  6,  and
the  weighted  estimator  6  for  0,,  forms  the  basis  of  the  Hausman
type  test  (Hausman  1978,  White  1982).
3.  Testing  against  misspecification
3.1  White's  information  matrix  test
If  the  model  is  not  misspecified,  that  is  g(u)  -  f(u,e)  the
matrix  A(9)  =  -B(0)  or  alternatively  A(9)  +  B(0)  -  O.  To
construct  a  test  of  the  Wald  type,  the  following  notation  is
used.  The  symbol  lowtri  symbolizes  the  operation  of  stacking
the  lower  triangular  of  a  symmetric  matrix  in  a  column  vector.
If  A  -  (a,)  i,j-1,...,p,  then  lowtri  A  -  (a,,a1.a9.
a  will  have      p(p1)/2  elements.