<s>
In	O
statistics	O
,	O
the	O
Wald	B-General_Concept
test	I-General_Concept
(	O
named	O
after	O
Abraham	O
Wald	O
)	O
assesses	O
constraints	B-Application
on	O
statistical	O
parameters	O
based	O
on	O
the	O
weighted	O
distance	O
between	O
the	O
unrestricted	O
estimate	O
and	O
its	O
hypothesized	O
value	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
where	O
the	O
weight	O
is	O
the	O
precision	B-General_Concept
of	O
the	O
estimate	O
.	O
</s>
<s>
Intuitively	O
,	O
the	O
larger	O
this	O
weighted	O
distance	O
,	O
the	O
less	O
likely	O
it	O
is	O
that	O
the	O
constraint	B-Application
is	O
true	O
.	O
</s>
<s>
While	O
the	O
finite	O
sample	O
distributions	O
of	O
Wald	B-General_Concept
tests	I-General_Concept
are	O
generally	O
unknown	O
,	O
it	O
has	O
an	O
asymptotic	O
χ2-distribution	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
a	O
fact	O
that	O
can	O
be	O
used	O
to	O
determine	O
statistical	B-General_Concept
significance	I-General_Concept
.	O
</s>
<s>
Together	O
with	O
the	O
Lagrange	B-General_Concept
multiplier	I-General_Concept
test	I-General_Concept
and	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
,	O
the	O
Wald	B-General_Concept
test	I-General_Concept
is	O
one	O
of	O
three	O
classical	O
approaches	O
to	O
hypothesis	O
testing	O
.	O
</s>
<s>
An	O
advantage	O
of	O
the	O
Wald	B-General_Concept
test	I-General_Concept
over	O
the	O
other	O
two	O
is	O
that	O
it	O
only	O
requires	O
the	O
estimation	O
of	O
the	O
unrestricted	O
model	O
,	O
which	O
lowers	O
the	O
computational	O
burden	O
as	O
compared	O
to	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
However	O
,	O
a	O
major	O
disadvantage	O
is	O
that	O
(	O
in	O
finite	O
samples	O
)	O
it	O
is	O
not	O
invariant	O
to	O
changes	O
in	O
the	O
representation	O
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
;	O
in	O
other	O
words	O
,	O
algebraically	O
equivalent	O
expressions	O
of	O
non-linear	O
parameter	O
restriction	O
can	O
lead	O
to	O
different	O
values	O
of	O
the	O
test	O
statistic	O
.	O
</s>
<s>
That	O
is	O
because	O
the	O
Wald	B-General_Concept
statistic	I-General_Concept
is	O
derived	O
from	O
a	O
Taylor	O
expansion	O
,	O
and	O
different	O
ways	O
of	O
writing	O
equivalent	O
nonlinear	O
expressions	O
lead	O
to	O
nontrivial	O
differences	O
in	O
the	O
corresponding	O
Taylor	O
coefficients	O
.	O
</s>
<s>
Another	O
aberration	O
,	O
known	O
as	O
the	O
Hauck	B-General_Concept
–	I-General_Concept
Donner	I-General_Concept
effect	I-General_Concept
,	O
can	O
occur	O
in	O
binomial	O
models	O
when	O
the	O
estimated	O
(	O
unconstrained	O
)	O
parameter	O
is	O
close	O
to	O
the	O
boundary	O
of	O
the	O
parameter	O
space	O
—	O
for	O
instance	O
a	O
fitted	O
probability	O
being	O
extremely	O
close	O
to	O
zero	O
or	O
one	O
—	O
which	O
results	O
in	O
the	O
Wald	B-General_Concept
test	I-General_Concept
no	O
longer	O
monotonically	O
increasing	O
in	O
the	O
distance	O
between	O
the	O
unconstrained	O
and	O
constrained	O
parameter	O
.	O
</s>
<s>
Under	O
the	O
Wald	B-General_Concept
test	I-General_Concept
,	O
the	O
estimated	O
that	O
was	O
found	O
as	O
the	O
maximizing	O
argument	O
of	O
the	O
unconstrained	O
likelihood	O
function	O
is	O
compared	O
with	O
a	O
hypothesized	O
value	O
.	O
</s>
<s>
If	O
the	O
hypothesis	O
involves	O
only	O
a	O
single	O
parameter	O
restriction	O
,	O
then	O
the	O
Wald	B-General_Concept
statistic	I-General_Concept
takes	O
the	O
following	O
form	O
:	O
</s>
<s>
which	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
follows	O
an	O
asymptotic	O
χ2-distribution	O
with	O
one	O
degree	O
of	O
freedom	O
.	O
</s>
<s>
The	O
square	O
root	O
of	O
the	O
single-restriction	O
Wald	B-General_Concept
statistic	I-General_Concept
can	O
be	O
understood	O
as	O
a	O
(	O
pseudo	O
)	O
t-ratio	O
that	O
is	O
,	O
however	O
,	O
not	O
actually	O
t-distributed	O
except	O
for	O
the	O
special	O
case	O
of	O
linear	O
regression	O
with	O
normally	O
distributed	O
errors	O
.	O
</s>
<s>
where	O
is	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
maximum	O
likelihood	O
estimate	O
(	O
MLE	O
)	O
,	O
the	O
square	O
root	O
of	O
the	O
variance	O
.	O
</s>
<s>
There	O
are	O
several	O
ways	O
to	O
consistently	O
estimate	O
the	O
variance	O
matrix	O
which	O
in	O
finite	O
samples	O
leads	O
to	O
alternative	O
estimates	O
of	O
standard	B-General_Concept
errors	I-General_Concept
and	O
associated	O
test	O
statistics	O
and	O
p-values	B-General_Concept
.	O
</s>
<s>
The	O
Wald	B-General_Concept
test	I-General_Concept
can	O
be	O
used	O
to	O
test	O
a	O
single	O
hypothesis	O
on	O
multiple	O
parameters	O
,	O
as	O
well	O
as	O
to	O
test	O
jointly	O
multiple	O
hypotheses	O
on	O
single/multiple	O
parameters	O
.	O
</s>
<s>
The	O
distribution	O
of	O
the	O
test	O
statistic	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
:	O
</s>
<s>
In	O
the	O
standard	O
form	O
,	O
the	O
Wald	B-General_Concept
test	I-General_Concept
is	O
used	O
to	O
test	O
linear	O
hypotheses	O
that	O
can	O
be	O
represented	O
by	O
a	O
single	O
matrixR	O
.	O
</s>
<s>
where	O
is	O
the	O
derivative	B-Algorithm
of	O
c	O
evaluated	O
at	O
the	O
sample	O
estimator	O
.	O
</s>
<s>
The	O
fact	O
that	O
one	O
uses	O
an	O
approximation	O
of	O
the	O
variance	O
has	O
the	O
drawback	O
that	O
the	O
Wald	B-General_Concept
statistic	I-General_Concept
is	O
not-invariant	O
to	O
a	O
non-linear	O
transformation/reparametrisation	O
of	O
the	O
hypothesis	O
:	O
it	O
can	O
give	O
different	O
answers	O
to	O
the	O
same	O
question	O
,	O
depending	O
on	O
how	O
the	O
question	O
is	O
phrased	O
.	O
</s>
<s>
For	O
example	O
,	O
asking	O
whether	O
R	O
=	O
1	O
is	O
the	O
same	O
as	O
asking	O
whether	O
logR	O
=	O
0	O
;	O
but	O
the	O
Wald	B-General_Concept
statistic	I-General_Concept
for	O
R	O
=	O
1	O
is	O
not	O
the	O
same	O
as	O
the	O
Wald	B-General_Concept
statistic	I-General_Concept
for	O
logR	O
=	O
0	O
(	O
because	O
there	O
is	O
in	O
general	O
no	O
neat	O
relationship	O
between	O
the	O
standard	B-General_Concept
errors	I-General_Concept
of	O
R	O
and	O
logR	O
,	O
so	O
it	O
needs	O
to	O
be	O
approximated	O
)	O
.	O
</s>
<s>
There	O
exist	O
several	O
alternatives	O
to	O
the	O
Wald	B-General_Concept
test	I-General_Concept
,	O
namely	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
and	O
the	O
Lagrange	B-General_Concept
multiplier	I-General_Concept
test	I-General_Concept
(	O
also	O
known	O
as	O
the	O
score	B-General_Concept
test	I-General_Concept
)	O
.	O
</s>
<s>
Robert	O
F	O
.	O
Engle	O
showed	O
that	O
these	O
three	O
tests	O
,	O
the	O
Wald	B-General_Concept
test	I-General_Concept
,	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
and	O
the	O
Lagrange	B-General_Concept
multiplier	I-General_Concept
test	I-General_Concept
are	O
asymptotically	O
equivalent	O
.	O
</s>
<s>
There	O
are	O
several	O
reasons	O
to	O
prefer	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
or	O
the	O
Lagrange	O
multiplier	O
to	O
the	O
Wald	B-General_Concept
test	I-General_Concept
:	O
</s>
<s>
Non-invariance	O
:	O
As	O
argued	O
above	O
,	O
the	O
Wald	B-General_Concept
test	I-General_Concept
is	O
not	O
invariant	O
under	O
reparametrization	O
,	O
while	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
tests	I-General_Concept
will	O
give	O
exactly	O
the	O
same	O
answer	O
whether	O
we	O
work	O
with	O
R	O
,	O
logR	O
or	O
any	O
other	O
monotonic	O
transformation	O
ofR	O
.	O
</s>
<s>
The	O
other	O
reason	O
is	O
that	O
the	O
Wald	B-General_Concept
test	I-General_Concept
uses	O
two	O
approximations	O
(	O
that	O
we	O
know	O
the	O
standard	B-General_Concept
error	I-General_Concept
or	O
Fisher	O
information	O
and	O
the	O
maximum	O
likelihood	O
estimate	O
)	O
,	O
whereas	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
depends	O
only	O
on	O
the	O
ratio	O
of	O
likelihood	O
functions	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
and	O
alternative	O
hypothesis	O
.	O
</s>
<s>
The	O
Wald	B-General_Concept
test	I-General_Concept
requires	O
an	O
estimate	O
using	O
the	O
maximizing	O
argument	O
,	O
corresponding	O
to	O
the	O
"	O
full	O
"	O
model	O
.	O
</s>
<s>
In	O
some	O
cases	O
,	O
the	O
model	O
is	O
simpler	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
so	O
that	O
one	O
might	O
prefer	O
to	O
use	O
the	O
score	B-General_Concept
test	I-General_Concept
(	O
also	O
called	O
Lagrange	B-General_Concept
multiplier	I-General_Concept
test	I-General_Concept
)	O
,	O
which	O
has	O
the	O
advantage	O
that	O
it	O
can	O
be	O
formulated	O
in	O
situations	O
where	O
the	O
variability	O
of	O
the	O
maximizing	O
element	O
is	O
difficult	O
to	O
estimate	O
or	O
computing	O
the	O
estimate	O
according	O
to	O
the	O
maximum	O
likelihood	O
estimator	O
is	O
difficult	O
;	O
e.g.	O
</s>
<s>
the	O
Cochran	B-General_Concept
–	I-General_Concept
Mantel	I-General_Concept
–	I-General_Concept
Haenzel	I-General_Concept
test	I-General_Concept
is	O
a	O
score	B-General_Concept
test	I-General_Concept
.	O
</s>
