<s>
In	O
statistics	O
,	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
(	O
OLS	O
)	O
is	O
a	O
type	O
of	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
method	O
for	O
choosing	O
the	O
unknown	O
parameters	O
in	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
(	O
with	O
fixed	O
level-one	O
effects	O
of	O
a	O
linear	O
function	O
of	O
a	O
set	O
of	O
explanatory	O
variables	O
)	O
by	O
the	O
principle	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
:	O
minimizing	O
the	O
sum	O
of	O
the	O
squares	O
of	O
the	O
differences	O
between	O
the	O
observed	O
dependent	O
variable	O
(	O
values	O
of	O
the	O
variable	O
being	O
observed	O
)	O
in	O
the	O
input	O
dataset	B-General_Concept
and	O
the	O
output	O
of	O
the	O
(	O
linear	O
)	O
function	O
of	O
the	O
independent	O
variable	O
.	O
</s>
<s>
The	O
resulting	O
estimator	O
can	O
be	O
expressed	O
by	O
a	O
simple	O
formula	O
,	O
especially	O
in	O
the	O
case	O
of	O
a	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
,	O
in	O
which	O
there	O
is	O
a	O
single	O
regressor	O
on	O
the	O
right	O
side	O
of	O
the	O
regression	O
equation	O
.	O
</s>
<s>
The	O
OLS	O
estimator	O
is	O
consistent	O
for	O
the	O
level-one	O
fixed	O
effects	O
when	O
the	O
regressors	O
are	O
exogenous	O
and	O
forms	O
perfect	O
colinearity	O
(	O
rank	O
condition	O
)	O
,	O
consistent	O
for	O
the	O
variance	O
estimate	O
of	O
the	O
residuals	O
when	O
regressors	O
have	O
finite	O
fourth	O
moments	O
and	O
—	O
by	O
the	O
Gauss	O
–	O
Markov	O
theorem	O
—	O
optimal	O
in	O
the	O
class	O
of	O
linear	O
unbiased	O
estimators	O
when	O
the	O
errors	O
are	O
homoscedastic	B-General_Concept
and	O
serially	O
uncorrelated	O
.	O
</s>
<s>
In	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
,	O
the	O
response	O
variable	O
,	O
,	O
is	O
a	O
linear	O
function	O
of	O
the	O
regressors	O
:	O
</s>
<s>
where	O
and	O
are	O
vectors	O
of	O
the	O
response	O
variables	O
and	O
the	O
errors	O
of	O
the	O
observations	O
,	O
and	O
is	O
an	O
matrix	B-Architecture
of	O
regressors	O
,	O
also	O
sometimes	O
called	O
the	O
design	B-Algorithm
matrix	I-Algorithm
,	O
whose	O
row	O
is	O
and	O
contains	O
the	O
-th	O
observations	O
on	O
all	O
the	O
explanatory	O
variables	O
.	O
</s>
<s>
This	O
minimization	O
problem	O
has	O
a	O
unique	O
solution	O
,	O
provided	O
that	O
the	O
columns	O
of	O
the	O
matrix	B-Architecture
are	O
linearly	O
independent	O
,	O
given	O
by	O
solving	O
the	O
so-called	O
normal	B-Algorithm
equations	I-Algorithm
:	O
</s>
<s>
The	O
matrix	B-Architecture
is	O
known	O
as	O
the	O
normal	O
matrix	B-Architecture
or	O
Gram	B-Algorithm
matrix	I-Algorithm
and	O
the	O
matrix	B-Architecture
is	O
known	O
as	O
the	O
moment	B-Algorithm
matrix	I-Algorithm
of	O
regressand	O
by	O
regressors	O
.	O
</s>
<s>
The	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
residuals	I-Algorithm
(	O
SSR	O
)	O
(	O
also	O
called	O
the	O
error	O
sum	O
of	O
squares	O
(	O
ESS	O
)	O
or	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
RSS	O
)	O
)	O
is	O
a	O
measure	O
of	O
the	O
overall	O
model	O
fit	O
:	O
</s>
<s>
where	O
T	O
denotes	O
the	O
matrix	B-Architecture
transpose	O
,	O
and	O
the	O
rows	O
of	O
X	O
,	O
denoting	O
the	O
values	O
of	O
all	O
the	O
independent	O
variables	O
associated	O
with	O
a	O
particular	O
value	O
of	O
the	O
dependent	O
variable	O
,	O
are	O
Xi	O
=	O
xiT	O
.	O
</s>
<s>
The	O
product	O
N	O
=	O
XT	O
X	O
is	O
a	O
Gram	B-Algorithm
matrix	I-Algorithm
and	O
its	O
inverse	O
,	O
Q	O
=	O
N	O
–	O
1	O
,	O
is	O
the	O
cofactor	O
matrix	B-Architecture
of	O
β	O
,	O
closely	O
related	O
to	O
its	O
covariance	O
matrix	B-Architecture
,	O
Cβ	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
(	O
XT	O
X	O
)	O
–	O
1	O
XT	O
=	O
Q	O
XT	O
is	O
called	O
the	O
Moore	O
–	O
Penrose	O
pseudoinverse	O
matrix	B-Architecture
of	O
X	O
.	O
</s>
<s>
This	O
formulation	O
highlights	O
the	O
point	O
that	O
estimation	O
can	O
be	O
carried	O
out	O
if	O
,	O
and	O
only	O
if	O
,	O
there	O
is	O
no	O
perfect	O
multicollinearity	O
between	O
the	O
explanatory	O
variables	O
(	O
which	O
would	O
cause	O
the	O
gram	B-Algorithm
matrix	I-Algorithm
to	O
have	O
no	O
inverse	O
)	O
.	O
</s>
<s>
where	O
P	O
=	O
X(XTX )	O
−1XT	O
is	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
onto	O
the	O
space	O
V	O
spanned	O
by	O
the	O
columns	O
of	O
X	O
.	O
</s>
<s>
This	O
matrix	B-Architecture
P	O
is	O
also	O
sometimes	O
called	O
the	O
hat	B-Algorithm
matrix	I-Algorithm
because	O
it	O
"	O
puts	O
a	O
hat	O
"	O
onto	O
the	O
variable	O
y	O
.	O
</s>
<s>
Another	O
matrix	B-Architecture
,	O
closely	O
related	O
to	O
P	O
is	O
the	O
annihilator	O
matrix	B-Architecture
;	O
this	O
is	O
a	O
projection	B-Algorithm
matrix	I-Algorithm
onto	O
the	O
space	O
orthogonal	O
to	O
V	O
.	O
Both	O
matrices	O
P	O
and	O
M	O
are	O
symmetric	B-Algorithm
and	O
idempotent	O
(	O
meaning	O
that	O
and	O
)	O
,	O
and	O
relate	O
to	O
the	O
data	B-Algorithm
matrix	I-Algorithm
X	O
via	O
identities	O
and	O
.	O
</s>
<s>
Matrix	B-Architecture
M	O
creates	O
the	O
residuals	O
from	O
the	O
regression	O
:	O
</s>
<s>
Using	O
these	O
residuals	O
we	O
can	O
estimate	O
the	O
value	O
of	O
σ	O
2	O
using	O
the	O
reduced	B-General_Concept
chi-squared	I-General_Concept
statistic	I-General_Concept
:	O
</s>
<s>
The	O
two	O
estimators	O
are	O
quite	O
similar	O
in	O
large	O
samples	O
;	O
the	O
first	O
estimator	O
is	O
always	O
unbiased	O
,	O
while	O
the	O
second	O
estimator	O
is	O
biased	O
but	O
has	O
a	O
smaller	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
.	O
</s>
<s>
The	O
square	O
root	O
of	O
s2	O
is	O
called	O
the	O
regression	B-General_Concept
standard	I-General_Concept
error	I-General_Concept
,	O
standard	B-General_Concept
error	I-General_Concept
of	I-General_Concept
the	I-General_Concept
regression	I-General_Concept
,	O
or	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
equation	O
.	O
</s>
<s>
It	O
is	O
common	O
to	O
assess	O
the	O
goodness-of-fit	O
of	O
the	O
OLS	B-General_Concept
regression	I-General_Concept
by	O
comparing	O
how	O
much	O
the	O
initial	O
variation	O
in	O
the	O
sample	O
can	O
be	O
reduced	O
by	O
regressing	O
onto	O
X	O
.	O
</s>
<s>
The	O
coefficient	O
of	O
determination	O
R2	O
is	O
defined	O
as	O
a	O
ratio	O
of	O
"	O
explained	O
"	O
variance	O
to	O
the	O
"	O
total	O
"	O
variance	O
of	O
the	O
dependent	O
variable	O
y	O
,	O
in	O
the	O
cases	O
where	O
the	O
regression	O
sum	O
of	O
squares	O
equals	O
the	O
sum	B-Algorithm
of	I-Algorithm
squares	I-Algorithm
of	I-Algorithm
residuals	I-Algorithm
:	O
</s>
<s>
where	O
TSS	O
is	O
the	O
total	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
for	O
the	O
dependent	O
variable	O
,	O
,	O
and	O
is	O
an	O
n×n	O
matrix	B-Architecture
of	O
ones	O
.	O
</s>
<s>
(	O
is	O
a	O
centering	B-Algorithm
matrix	I-Algorithm
which	O
is	O
equivalent	O
to	O
regression	O
on	O
a	O
constant	O
;	O
it	O
simply	O
subtracts	O
the	O
mean	O
from	O
a	O
variable	O
.	O
)	O
</s>
<s>
In	O
order	O
for	O
R2	O
to	O
be	O
meaningful	O
,	O
the	O
matrix	B-Architecture
X	O
of	O
data	O
on	O
regressors	O
must	O
contain	O
a	O
column	O
vector	O
of	O
ones	O
to	O
represent	O
the	O
constant	O
whose	O
coefficient	O
is	O
the	O
regression	O
intercept	O
.	O
</s>
<s>
The	O
variance	O
in	O
the	O
prediction	O
of	O
the	O
independent	O
variable	O
as	O
a	O
function	O
of	O
the	O
dependent	O
variable	O
is	O
given	O
in	O
the	O
article	O
Polynomial	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
If	O
the	O
data	B-Algorithm
matrix	I-Algorithm
X	O
contains	O
only	O
two	O
variables	O
,	O
a	O
constant	O
and	O
a	O
scalar	O
regressor	O
xi	O
,	O
then	O
this	O
is	O
called	O
the	O
"	O
simple	B-General_Concept
regression	I-General_Concept
model	O
"	O
.	O
</s>
<s>
In	O
the	O
previous	O
section	O
the	O
least	B-Algorithm
squares	I-Algorithm
estimator	O
was	O
obtained	O
as	O
a	O
value	O
that	O
minimizes	O
the	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
residuals	I-Algorithm
of	O
the	O
model	O
.	O
</s>
<s>
Thus	O
,	O
the	O
residual	O
vector	O
will	O
have	O
the	O
smallest	O
length	O
when	O
y	O
is	O
projected	B-Algorithm
orthogonally	I-Algorithm
onto	O
the	O
linear	O
subspace	O
spanned	O
by	O
the	O
columns	O
of	O
X	O
.	O
</s>
<s>
Introducing	O
and	O
a	O
matrix	B-Architecture
K	O
with	O
the	O
assumption	O
that	O
a	O
matrix	B-Architecture
is	O
non-singular	O
and	O
KT	O
X	O
=	O
0	O
(	O
cf	O
.	O
</s>
<s>
Orthogonal	O
projections	B-Algorithm
)	O
,	O
the	O
residual	O
vector	O
should	O
satisfy	O
the	O
following	O
equation	O
:	O
</s>
<s>
The	O
equation	O
and	O
solution	O
of	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
are	O
thus	O
described	O
as	O
follows	O
:	O
</s>
<s>
Another	O
way	O
of	O
looking	O
at	O
it	O
is	O
to	O
consider	O
the	O
regression	B-General_Concept
line	I-General_Concept
to	O
be	O
a	O
weighted	O
average	O
of	O
the	O
lines	O
passing	O
through	O
the	O
combination	O
of	O
any	O
two	O
points	O
in	O
the	O
dataset	B-General_Concept
.	O
</s>
<s>
 [ proof ] 	O
This	O
normality	O
assumption	O
has	O
historical	O
importance	O
,	O
as	O
it	O
provided	O
the	O
basis	O
for	O
the	O
early	O
work	O
in	O
linear	B-General_Concept
regression	I-General_Concept
analysis	O
by	O
Yule	O
and	O
Pearson	O
.	O
</s>
<s>
This	O
is	O
the	O
so-called	O
classical	O
GMM	O
case	O
,	O
when	O
the	O
estimator	O
does	O
not	O
depend	O
on	O
the	O
choice	O
of	O
the	O
weighting	O
matrix	B-Architecture
.	O
</s>
<s>
There	O
are	O
several	O
different	O
frameworks	O
in	O
which	O
the	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
can	O
be	O
cast	O
in	O
order	O
to	O
make	O
the	O
OLS	O
technique	O
applicable	O
.	O
</s>
<s>
Mathematically	O
,	O
this	O
means	O
that	O
the	O
matrix	B-Architecture
X	O
must	O
have	O
full	O
column	O
rank	O
almost	O
surely	O
:	O
</s>
<s>
Then	O
the	O
matrix	B-Architecture
is	O
finite	O
and	O
positive	O
semi-definite	O
.	O
</s>
<s>
In	O
such	O
case	O
the	O
value	O
of	O
the	O
regression	B-General_Concept
coefficient	I-General_Concept
β	O
cannot	O
be	O
learned	O
,	O
although	O
prediction	O
of	O
y	O
values	O
is	O
still	O
possible	O
for	O
new	O
values	O
of	O
the	O
regressors	O
that	O
lie	O
in	O
the	O
same	O
linearly	O
dependent	O
subspace	O
.	O
</s>
<s>
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
in	O
dimension	O
n	O
,	O
and	O
σ2	O
is	O
a	O
parameter	O
which	O
determines	O
the	O
variance	O
of	O
each	O
observation	O
.	O
</s>
<s>
Homoscedasticity	B-General_Concept
:	O
,	O
which	O
means	O
that	O
the	O
error	O
term	O
has	O
the	O
same	O
variance	O
σ2	O
in	O
each	O
observation	O
.	O
</s>
<s>
When	O
this	O
requirement	O
is	O
violated	O
this	O
is	O
called	O
heteroscedasticity	B-General_Concept
,	O
in	O
such	O
case	O
a	O
more	O
efficient	O
estimator	O
would	O
be	O
weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
In	O
such	O
cases	O
generalized	O
least	B-Algorithm
squares	I-Algorithm
provides	O
a	O
better	O
alternative	O
than	O
the	O
OLS	O
.	O
</s>
<s>
In	O
some	O
applications	O
,	O
especially	O
with	O
cross-sectional	B-General_Concept
data	I-General_Concept
,	O
an	O
additional	O
assumption	O
is	O
imposed	O
—	O
that	O
all	O
observations	O
are	O
independent	O
and	O
identically	O
distributed	O
.	O
</s>
<s>
no	O
perfect	O
multicollinearity	O
:	O
is	O
a	O
positive-definite	B-Algorithm
matrix	I-Algorithm
;	O
</s>
<s>
homoscedasticity	B-General_Concept
:	O
.	O
</s>
<s>
The	O
stochastic	O
process	O
{	O
xi	O
,	O
yi}	O
is	O
stationary	B-Algorithm
and	O
ergodic	B-Algorithm
;	O
if	O
{	O
xi	O
,	O
yi}	O
is	O
nonstationary	O
,	O
OLS	O
results	O
are	O
often	O
spurious	O
unless	O
{	O
xi	O
,	O
yi}	O
is	O
co-integrating	O
.	O
</s>
<s>
The	O
p×p	O
matrix	B-Architecture
is	O
of	O
full	O
rank	O
,	O
and	O
hence	O
positive-definite	B-Algorithm
;	O
</s>
<s>
 { xiεi } 	O
is	O
a	O
martingale	O
difference	O
sequence	O
,	O
with	O
a	O
finite	O
matrix	B-Architecture
of	O
second	O
moments	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
each	O
coefficient	O
is	O
equal	O
to	O
square	O
root	O
of	O
the	O
j-th	O
diagonal	O
element	O
of	O
this	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
estimate	O
of	O
this	O
standard	B-General_Concept
error	I-General_Concept
is	O
obtained	O
by	O
replacing	O
the	O
unknown	O
quantity	O
σ2	O
with	O
its	O
estimate	O
s2	O
.	O
</s>
<s>
The	O
Gauss	O
–	O
Markov	O
theorem	O
states	O
that	O
under	O
the	O
spherical	O
errors	O
assumption	O
(	O
that	O
is	O
,	O
the	O
errors	O
should	O
be	O
uncorrelated	O
and	O
homoscedastic	B-General_Concept
)	O
the	O
estimator	O
is	O
efficient	O
in	O
the	O
class	O
of	O
linear	O
unbiased	O
estimators	O
.	O
</s>
<s>
in	O
the	O
sense	O
that	O
this	O
is	O
a	O
nonnegative-definite	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
we	O
are	O
willing	O
to	O
allow	O
biased	O
estimators	O
,	O
and	O
consider	O
the	O
class	O
of	O
estimators	O
that	O
are	O
proportional	O
to	O
the	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
residuals	I-Algorithm
(	O
SSR	O
)	O
of	O
the	O
model	O
,	O
then	O
the	O
best	O
(	O
in	O
the	O
sense	O
of	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
)	O
estimator	O
in	O
this	O
class	O
will	O
be	O
,	O
which	O
even	O
beats	O
the	O
Cramér	O
–	O
Rao	O
bound	O
in	O
case	O
when	O
there	O
is	O
only	O
one	O
regressor	O
(	O
)	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
estimators	O
and	O
s2	O
are	O
independent	O
,	O
the	O
fact	O
which	O
comes	O
in	O
useful	O
when	O
constructing	O
the	O
t	O
-	O
and	O
F-tests	B-General_Concept
for	O
the	O
regression	O
.	O
</s>
<s>
To	O
analyze	O
which	O
observations	O
are	O
influential	O
we	O
remove	O
a	O
specific	O
j-th	O
observation	O
and	O
consider	O
how	O
much	O
the	O
estimated	O
quantities	O
are	O
going	O
to	O
change	O
(	O
similarly	O
to	O
the	O
jackknife	B-Algorithm
method	I-Algorithm
)	O
.	O
</s>
<s>
where	O
is	O
the	O
j-th	O
diagonal	O
element	O
of	O
the	O
hat	B-Algorithm
matrix	I-Algorithm
P	O
,	O
and	O
xj	O
is	O
the	O
vector	O
of	O
regressors	O
corresponding	O
to	O
the	O
j-th	O
observation	O
.	O
</s>
<s>
From	O
the	O
properties	O
of	O
the	O
hat	B-Algorithm
matrix	I-Algorithm
,	O
,	O
and	O
they	O
sum	O
up	O
to	O
p	O
,	O
so	O
that	O
on	O
average	O
.	O
</s>
<s>
Usually	O
the	O
observations	O
with	O
high	O
leverage	O
ought	O
to	O
be	O
scrutinized	O
more	O
carefully	O
,	O
in	O
case	O
they	O
are	O
erroneous	O
,	O
or	O
outliers	O
,	O
or	O
in	O
some	O
other	O
way	O
atypical	O
of	O
the	O
rest	O
of	O
the	O
dataset	B-General_Concept
.	O
</s>
<s>
where	O
M1	O
is	O
the	O
annihilator	O
matrix	B-Architecture
for	O
regressors	O
X1	O
.	O
</s>
<s>
where	O
Q	O
is	O
a	O
p×q	O
matrix	B-Architecture
of	O
full	O
rank	O
,	O
and	O
c	O
is	O
a	O
q×1	O
vector	O
of	O
known	O
constants	O
,	O
where	O
.	O
</s>
<s>
In	O
this	O
case	O
least	B-Algorithm
squares	I-Algorithm
estimation	O
is	O
equivalent	O
to	O
minimizing	O
the	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
residuals	I-Algorithm
of	O
the	O
model	O
subject	O
to	O
the	O
constraint	O
A	O
.	O
</s>
<s>
The	O
constrained	O
least	B-Algorithm
squares	I-Algorithm
(	O
CLS	O
)	O
estimator	O
can	O
be	O
given	O
by	O
an	O
explicit	O
formula	O
:	O
</s>
<s>
This	O
expression	O
for	O
the	O
constrained	O
estimator	O
is	O
valid	O
as	O
long	O
as	O
the	O
matrix	B-Architecture
XTX	O
is	O
invertible	O
.	O
</s>
<s>
It	O
was	O
assumed	O
from	O
the	O
beginning	O
of	O
this	O
article	O
that	O
this	O
matrix	B-Architecture
is	O
of	O
full	O
rank	O
,	O
and	O
it	O
was	O
noted	O
that	O
when	O
the	O
rank	O
condition	O
fails	O
,	O
β	O
will	O
not	O
be	O
identifiable	O
.	O
</s>
<s>
where	O
R	O
is	O
a	O
p×( p−q	O
)	O
matrix	B-Architecture
such	O
that	O
the	O
matrix	B-Architecture
is	O
non-singular	O
,	O
and	O
.	O
</s>
<s>
Such	O
a	O
matrix	B-Architecture
can	O
always	O
be	O
found	O
,	O
although	O
generally	O
it	O
is	O
not	O
unique	O
.	O
</s>
<s>
The	O
least	B-Algorithm
squares	I-Algorithm
estimators	O
are	O
point	O
estimates	O
of	O
the	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
parameters	O
β	O
.	O
</s>
<s>
where	O
q	O
denotes	O
the	O
quantile	O
function	O
of	O
standard	O
normal	O
distribution	O
,	O
and	O
[·]jj	O
is	O
the	O
j-th	O
diagonal	O
element	O
of	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
no	O
explanatory	O
value	O
of	O
the	O
estimated	O
regression	O
is	O
tested	O
using	O
an	O
F-test	B-General_Concept
.	O
</s>
<s>
If	O
the	O
calculated	O
F-value	O
is	O
found	O
to	O
be	O
large	O
enough	O
to	O
exceed	O
its	O
critical	O
value	O
for	O
the	O
pre-chosen	O
level	O
of	O
significance	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
rejected	O
and	O
the	O
alternative	B-General_Concept
hypothesis	I-General_Concept
,	O
that	O
the	O
regression	O
has	O
explanatory	O
power	O
,	O
is	O
accepted	O
.	O
</s>
<s>
Otherwise	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
no	O
explanatory	O
power	O
is	O
accepted	O
.	O
</s>
<s>
Here	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
the	O
true	O
coefficient	O
is	O
zero	O
.	O
</s>
<s>
This	O
hypothesis	O
is	O
tested	O
by	O
computing	O
the	O
coefficient	O
's	O
t-statistic	O
,	O
as	O
the	O
ratio	O
of	O
the	O
coefficient	O
estimate	O
to	O
its	O
standard	B-General_Concept
error	I-General_Concept
.	O
</s>
<s>
If	O
the	O
t-statistic	O
is	O
larger	O
than	O
a	O
predetermined	O
value	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
rejected	O
and	O
the	O
variable	O
is	O
found	O
to	O
have	O
explanatory	O
power	O
,	O
with	O
its	O
coefficient	O
significantly	O
different	O
from	O
zero	O
.	O
</s>
<s>
Otherwise	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
a	O
zero	O
value	O
of	O
the	O
true	O
coefficient	O
is	O
accepted	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
Chow	B-General_Concept
test	I-General_Concept
is	O
used	O
to	O
test	O
whether	O
two	O
subsamples	O
both	O
have	O
the	O
same	O
underlying	O
true	O
coefficient	O
values	O
.	O
</s>
<s>
The	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
residuals	I-Algorithm
of	O
regressions	O
on	O
each	O
of	O
the	O
subsets	O
and	O
on	O
the	O
combined	O
data	B-General_Concept
set	I-General_Concept
are	O
compared	O
by	O
computing	O
an	O
F-statistic	O
;	O
if	O
this	O
exceeds	O
a	O
critical	O
value	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
no	O
difference	O
between	O
the	O
two	O
subsets	O
is	O
rejected	O
;	O
otherwise	O
,	O
it	O
is	O
accepted	O
.	O
</s>
<s>
The	O
following	O
data	B-General_Concept
set	I-General_Concept
gives	O
average	O
heights	O
and	O
weights	O
for	O
American	O
women	O
aged	O
30	O
–	O
39	O
(	O
source	O
:	O
The	O
World	O
Almanac	O
and	O
Book	O
of	O
Facts	O
,	O
1975	O
)	O
.	O
</s>
<s>
When	O
only	O
one	O
dependent	O
variable	O
is	O
being	O
modeled	O
,	O
a	O
scatterplot	B-Application
will	O
suggest	O
the	O
form	O
and	O
strength	O
of	O
the	O
relationship	O
between	O
the	O
dependent	O
variable	O
and	O
regressors	O
.	O
</s>
<s>
It	O
might	O
also	O
reveal	O
outliers	O
,	O
heteroscedasticity	B-General_Concept
,	O
and	O
other	O
aspects	O
of	O
the	O
data	O
that	O
may	O
complicate	O
the	O
interpretation	O
of	O
a	O
fitted	O
regression	O
model	O
.	O
</s>
<s>
The	O
scatterplot	B-Application
suggests	O
that	O
the	O
relationship	O
is	O
strong	O
and	O
can	O
be	O
approximated	O
as	O
a	O
quadratic	O
function	O
.	O
</s>
<s>
The	O
output	O
from	O
most	O
popular	O
statistical	B-Algorithm
packages	I-Algorithm
will	O
look	O
similar	O
to	O
this	O
:	O
</s>
<s>
|	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
stat	I-General_Concept
.	I-General_Concept
</s>
<s>
The	O
Std	B-General_Concept
error	I-General_Concept
column	O
shows	O
standard	B-General_Concept
errors	I-General_Concept
of	O
each	O
coefficient	O
estimate	O
:	O
</s>
<s>
The	O
t-statistic	O
and	O
p-value	B-General_Concept
columns	O
are	O
testing	O
whether	O
any	O
of	O
the	O
coefficients	O
might	O
be	O
equal	O
to	O
zero	O
.	O
</s>
<s>
Large	O
values	O
of	O
t	O
indicate	O
that	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
can	O
be	O
rejected	O
and	O
that	O
the	O
corresponding	O
coefficient	O
is	O
not	O
zero	O
.	O
</s>
<s>
The	O
second	O
column	O
,	O
p-value	B-General_Concept
,	O
expresses	O
the	O
results	O
of	O
the	O
hypothesis	O
test	O
as	O
a	O
significance	B-General_Concept
level	I-General_Concept
.	O
</s>
<s>
Conventionally	O
,	O
p-values	B-General_Concept
smaller	O
than	O
0.05	O
are	O
taken	O
as	O
evidence	O
that	O
the	O
population	O
coefficient	O
is	O
nonzero	O
.	O
</s>
<s>
Even	O
though	O
the	O
assumption	O
is	O
not	O
very	O
reasonable	O
,	O
this	O
statistic	O
may	O
still	O
find	O
its	O
use	O
in	O
conducting	O
LR	B-General_Concept
tests	I-General_Concept
.	O
</s>
<s>
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
statistic	I-General_Concept
tests	O
whether	O
there	O
is	O
any	O
evidence	O
of	O
serial	O
correlation	O
between	O
the	O
residuals	O
.	O
</s>
<s>
Akaike	O
information	O
criterion	O
and	O
Schwarz	B-General_Concept
criterion	I-General_Concept
are	O
both	O
used	O
for	O
model	O
selection	O
.	O
</s>
<s>
Standard	B-General_Concept
error	I-General_Concept
of	O
regression	O
is	O
an	O
estimate	O
of	O
σ	O
,	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
error	O
term	O
.	O
</s>
<s>
Total	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
,	O
model	O
sum	O
of	O
squared	O
,	O
and	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
tell	O
us	O
how	O
much	O
of	O
the	O
initial	O
variation	O
in	O
the	O
sample	O
were	O
explained	O
by	O
the	O
regression	O
.	O
</s>
<s>
This	O
statistic	O
has	O
F( p	O
–	O
1	O
,	O
n	O
–	O
p	O
)	O
distribution	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
and	O
normality	O
assumption	O
,	O
and	O
its	O
p-value	B-General_Concept
indicates	O
probability	O
that	O
the	O
hypothesis	O
is	O
indeed	O
true	O
.	O
</s>
<s>
Note	O
that	O
when	O
errors	O
are	O
not	O
normal	O
this	O
statistic	O
becomes	O
invalid	O
,	O
and	O
other	O
tests	O
such	O
as	O
Wald	B-General_Concept
test	I-General_Concept
or	O
LR	B-General_Concept
test	I-General_Concept
should	O
be	O
used	O
.	O
</s>
<s>
Ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
analysis	O
often	O
includes	O
the	O
use	O
of	O
diagnostic	O
plots	O
designed	O
to	O
detect	O
departures	O
of	O
the	O
data	O
from	O
the	O
assumed	O
form	O
of	O
the	O
model	O
.	O
</s>
<s>
Different	O
levels	O
of	O
variability	O
in	O
the	O
residuals	O
for	O
different	O
levels	O
of	O
the	O
explanatory	O
variables	O
suggests	O
possible	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
We	O
need	O
to	O
find	O
the	O
least-squares	B-Algorithm
approximation	I-Algorithm
of	O
and	O
for	O
the	O
given	O
data	O
.	O
</s>
