<s>
Structural	O
stability−	O
i.e.	O
,	O
the	O
time-invariance	O
of	O
regression	O
coefficients−	O
is	O
a	O
central	O
issue	O
in	O
all	O
applications	O
of	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
.	O
</s>
<s>
For	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
,	O
the	O
Chow	B-General_Concept
test	I-General_Concept
is	O
often	O
used	O
to	O
test	O
for	O
a	O
single	O
break	O
in	O
mean	O
at	O
a	O
known	O
time	O
period	O
for	O
.	O
</s>
<s>
The	O
Chow	B-General_Concept
test	I-General_Concept
is	O
not	O
applicable	O
in	O
these	O
situations	O
,	O
since	O
it	O
only	O
applies	O
to	O
models	O
with	O
a	O
known	O
breakpoint	O
and	O
where	O
the	O
error	O
variance	O
remains	O
constant	O
before	O
and	O
after	O
the	O
break	O
.	O
</s>
<s>
For	O
cases	O
1	O
and	O
2	O
,	O
the	O
sup-Wald	O
(	O
i.e.	O
,	O
the	O
supremum	O
of	O
a	O
set	O
of	O
Wald	B-General_Concept
statistics	I-General_Concept
)	O
,	O
sup-LM	O
(	O
i.e.	O
,	O
the	O
supremum	O
of	O
a	O
set	O
of	O
Lagrange	B-General_Concept
multiplier	I-General_Concept
statistics	I-General_Concept
)	O
,	O
and	O
sup-LR	O
(	O
i.e.	O
,	O
the	O
supremum	O
of	O
a	O
set	O
of	O
likelihood	B-General_Concept
ratio	I-General_Concept
statistics	I-General_Concept
)	O
tests	O
developed	O
by	O
Andrews	O
(	O
1993	O
,	O
2003	O
)	O
may	O
be	O
used	O
to	O
test	O
for	O
parameter	O
instability	O
when	O
the	O
number	O
and	O
location	O
of	O
structural	O
breaks	O
are	O
unknown	O
.	O
</s>
<s>
These	O
tests	O
were	O
shown	O
to	O
be	O
superior	O
to	O
the	O
CUSUM	O
test	O
in	O
terms	O
of	O
statistical	B-General_Concept
power	I-General_Concept
,	O
and	O
are	O
the	O
most	O
commonly	O
used	O
tests	O
for	O
the	O
detection	O
of	O
structural	O
change	O
involving	O
an	O
unknown	O
number	O
of	O
breaks	O
in	O
mean	O
with	O
unknown	O
break	O
points	O
.	O
</s>
<s>
The	O
sup-Wald	O
,	O
sup-LM	O
,	O
and	O
sup-LR	O
tests	O
are	O
asymptotic	O
in	O
general	O
(	O
i.e.	O
,	O
the	O
asymptotic	O
critical	O
values	O
for	O
these	O
tests	O
are	O
applicable	O
for	O
sample	O
size	O
as	O
)	O
,	O
and	O
involve	O
the	O
assumption	O
of	O
homoskedasticity	B-General_Concept
across	O
break	O
points	O
for	O
finite	O
samples	O
;	O
however	O
,	O
an	O
exact	B-General_Concept
test	I-General_Concept
with	O
the	O
sup-Wald	O
statistic	O
may	O
be	O
obtained	O
for	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
with	O
a	O
fixed	O
number	O
of	O
regressors	O
and	O
independent	O
and	O
identically	O
distributed	O
(	O
IID	O
)	O
normal	O
errors	O
.	O
</s>
<s>
There	O
are	O
several	O
statistical	B-Algorithm
packages	I-Algorithm
that	O
can	O
be	O
used	O
to	O
find	O
structural	O
breaks	O
,	O
including	O
R	B-Language
,	O
GAUSS	B-Algorithm
,	O
and	O
Stata	B-Algorithm
,	O
among	O
others	O
.	O
</s>
