<s>
In	O
statistics	O
,	O
linear	B-General_Concept
regression	I-General_Concept
is	O
a	O
linear	O
approach	O
for	O
modelling	O
the	O
relationship	O
between	O
a	O
scalar	O
response	O
and	O
one	O
or	O
more	O
explanatory	O
variables	O
(	O
also	O
known	O
as	O
dependent	O
and	O
independent	O
variables	O
)	O
.	O
</s>
<s>
The	O
case	O
of	O
one	O
explanatory	O
variable	O
is	O
called	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
;	O
for	O
more	O
than	O
one	O
,	O
the	O
process	O
is	O
called	O
multiple	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
This	O
term	O
is	O
distinct	O
from	O
multivariate	O
linear	B-General_Concept
regression	I-General_Concept
,	O
where	O
multiple	O
correlated	O
dependent	O
variables	O
are	O
predicted	O
,	O
rather	O
than	O
a	O
single	O
scalar	O
variable	O
.	O
</s>
<s>
In	O
linear	B-General_Concept
regression	I-General_Concept
,	O
the	O
relationships	O
are	O
modeled	O
using	O
linear	B-General_Concept
predictor	I-General_Concept
functions	I-General_Concept
whose	O
unknown	O
model	O
parameters	O
are	O
estimated	O
from	O
the	O
data	O
.	O
</s>
<s>
Such	O
models	O
are	O
called	O
linear	B-Algorithm
models	I-Algorithm
.	O
</s>
<s>
Most	O
commonly	O
,	O
the	O
conditional	O
mean	O
of	O
the	O
response	O
given	O
the	O
values	O
of	O
the	O
explanatory	O
variables	O
(	O
or	O
predictors	O
)	O
is	O
assumed	O
to	O
be	O
an	O
affine	B-Algorithm
function	I-Algorithm
of	O
those	O
values	O
;	O
less	O
commonly	O
,	O
the	O
conditional	O
median	O
or	O
some	O
other	O
quantile	O
is	O
used	O
.	O
</s>
<s>
Like	O
all	O
forms	O
of	O
regression	O
analysis	O
,	O
linear	B-General_Concept
regression	I-General_Concept
focuses	O
on	O
the	O
conditional	O
probability	O
distribution	O
of	O
the	O
response	O
given	O
the	O
values	O
of	O
the	O
predictors	O
,	O
rather	O
than	O
on	O
the	O
joint	O
probability	O
distribution	O
of	O
all	O
of	O
these	O
variables	O
,	O
which	O
is	O
the	O
domain	O
of	O
multivariate	O
analysis	O
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
was	O
the	O
first	O
type	O
of	O
regression	O
analysis	O
to	O
be	O
studied	O
rigorously	O
,	O
and	O
to	O
be	O
used	O
extensively	O
in	O
practical	O
applications	O
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
has	O
many	O
practical	O
uses	O
.	O
</s>
<s>
If	O
the	O
goal	O
is	O
error	O
reduction	O
in	O
prediction	O
or	O
forecasting	O
,	O
linear	B-General_Concept
regression	I-General_Concept
can	O
be	O
used	O
to	O
fit	O
a	O
predictive	O
model	O
to	O
an	O
observed	O
data	B-General_Concept
set	I-General_Concept
of	O
values	O
of	O
the	O
response	O
and	O
explanatory	O
variables	O
.	O
</s>
<s>
If	O
the	O
goal	O
is	O
to	O
explain	O
variation	O
in	O
the	O
response	O
variable	O
that	O
can	O
be	O
attributed	O
to	O
variation	O
in	O
the	O
explanatory	O
variables	O
,	O
linear	B-General_Concept
regression	I-General_Concept
analysis	O
can	O
be	O
applied	O
to	O
quantify	O
the	O
strength	O
of	O
the	O
relationship	O
between	O
the	O
response	O
and	O
the	O
explanatory	O
variables	O
,	O
and	O
in	O
particular	O
to	O
determine	O
whether	O
some	O
explanatory	O
variables	O
may	O
have	O
no	O
linear	O
relationship	O
with	O
the	O
response	O
at	O
all	O
,	O
or	O
to	O
identify	O
which	O
subsets	O
of	O
explanatory	O
variables	O
may	O
contain	O
redundant	O
information	O
about	O
the	O
response	O
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
are	O
often	O
fitted	O
using	O
the	O
least	B-Algorithm
squares	I-Algorithm
approach	O
,	O
but	O
they	O
may	O
also	O
be	O
fitted	O
in	O
other	O
ways	O
,	O
such	O
as	O
by	O
minimizing	O
the	O
"	O
lack	O
of	O
fit	O
"	O
in	O
some	O
other	O
norm	O
(	O
as	O
with	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
regression	O
)	O
,	O
or	O
by	O
minimizing	O
a	O
penalized	O
version	O
of	O
the	O
least	B-Algorithm
squares	I-Algorithm
cost	O
function	O
as	O
in	O
ridge	O
regression	O
(	O
L2-norm	O
penalty	O
)	O
and	O
lasso	B-Algorithm
(	O
L1-norm	O
penalty	O
)	O
.	O
</s>
<s>
Conversely	O
,	O
the	O
least	B-Algorithm
squares	I-Algorithm
approach	O
can	O
be	O
used	O
to	O
fit	O
models	O
that	O
are	O
not	O
linear	B-Algorithm
models	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
although	O
the	O
terms	O
"	O
least	B-Algorithm
squares	I-Algorithm
"	O
and	O
"	O
linear	B-Algorithm
model	I-Algorithm
"	O
are	O
closely	O
linked	O
,	O
they	O
are	O
not	O
synonymous	O
.	O
</s>
<s>
Given	O
a	O
data	B-General_Concept
set	I-General_Concept
of	O
n	O
statistical	O
units	O
,	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
assumes	O
that	O
the	O
relationship	O
between	O
the	O
dependent	O
variable	O
y	O
and	O
the	O
vector	O
of	O
regressors	O
x	O
is	O
linear	O
.	O
</s>
<s>
This	O
relationship	O
is	O
modeled	O
through	O
a	O
disturbance	B-General_Concept
term	I-General_Concept
or	O
error	B-General_Concept
variable	I-General_Concept
ε	O
—	O
an	O
unobserved	O
random	O
variable	O
that	O
adds	O
"	O
noise	O
"	O
to	O
the	O
linear	O
relationship	O
between	O
the	O
dependent	O
variable	O
and	O
regressors	O
.	O
</s>
<s>
The	O
decision	O
as	O
to	O
which	O
variable	O
in	O
a	O
data	B-General_Concept
set	I-General_Concept
is	O
modeled	O
as	O
the	O
dependent	O
variable	O
and	O
which	O
are	O
modeled	O
as	O
the	O
independent	O
variables	O
may	O
be	O
based	O
on	O
a	O
presumption	O
that	O
the	O
value	O
of	O
one	O
of	O
the	O
variables	O
is	O
caused	O
by	O
,	O
or	O
directly	O
influenced	O
by	O
the	O
other	O
variables	O
.	O
</s>
<s>
The	O
matrix	O
is	O
sometimes	O
called	O
the	O
design	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
corresponding	O
element	O
of	O
β	O
is	O
called	O
the	O
intercept	B-Algorithm
.	O
</s>
<s>
Many	O
statistical	O
inference	O
procedures	O
for	O
linear	B-Algorithm
models	I-Algorithm
require	O
an	O
intercept	B-Algorithm
to	O
be	O
present	O
,	O
so	O
it	O
is	O
often	O
included	O
even	O
if	O
theoretical	O
considerations	O
suggest	O
that	O
its	O
value	O
should	O
be	O
zero	O
.	O
</s>
<s>
is	O
a	O
-dimensional	O
parameter	O
vector	O
,	O
where	O
is	O
the	O
intercept	B-Algorithm
term	O
(	O
if	O
one	O
is	O
included	O
in	O
the	O
model	O
—	O
otherwise	O
is	O
p-dimensional	O
)	O
.	O
</s>
<s>
Its	O
elements	O
are	O
known	O
as	O
effects	O
or	O
regression	B-General_Concept
coefficients	I-General_Concept
(	O
although	O
the	O
latter	O
term	O
is	O
sometimes	O
reserved	O
for	O
the	O
estimated	O
effects	O
)	O
.	O
</s>
<s>
In	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
,	O
p	O
=	O
1	O
,	O
and	O
the	O
coefficient	O
is	O
known	O
as	O
regression	O
slope	O
.	O
</s>
<s>
Statistical	O
estimation	O
and	O
inference	O
in	O
linear	B-General_Concept
regression	I-General_Concept
focuses	O
on	O
β	O
.	O
</s>
<s>
This	O
part	O
of	O
the	O
model	O
is	O
called	O
the	O
error	O
term	O
,	O
disturbance	B-General_Concept
term	I-General_Concept
,	O
or	O
sometimes	O
noise	O
(	O
in	O
contrast	O
with	O
the	O
"	O
signal	O
"	O
provided	O
by	O
the	O
rest	O
of	O
the	O
model	O
)	O
.	O
</s>
<s>
The	O
relationship	O
between	O
the	O
error	O
term	O
and	O
the	O
regressors	O
,	O
for	O
example	O
their	O
correlation	O
,	O
is	O
a	O
crucial	O
consideration	O
in	O
formulating	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
,	O
as	O
it	O
will	O
determine	O
the	O
appropriate	O
estimation	O
method	O
.	O
</s>
<s>
Fitting	O
a	O
linear	B-Algorithm
model	I-Algorithm
to	O
a	O
given	O
data	B-General_Concept
set	I-General_Concept
usually	O
requires	O
estimating	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
such	O
that	O
the	O
error	O
term	O
is	O
minimized	O
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
can	O
be	O
used	O
to	O
estimate	O
the	O
values	O
of	O
β1	O
and	O
β2	O
from	O
the	O
measured	O
data	O
.	O
</s>
<s>
Standard	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
with	O
standard	O
estimation	O
techniques	O
make	O
a	O
number	O
of	O
assumptions	O
about	O
the	O
predictor	O
variables	O
,	O
the	O
response	O
variables	O
and	O
their	O
relationship	O
.	O
</s>
<s>
The	O
following	O
are	O
the	O
major	O
assumptions	O
made	O
by	O
standard	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
with	O
standard	O
estimation	O
techniques	O
(	O
e.g.	O
</s>
<s>
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
)	O
:	O
</s>
<s>
Although	O
this	O
assumption	O
is	O
not	O
realistic	O
in	O
many	O
settings	O
,	O
dropping	O
it	O
leads	O
to	O
significantly	O
more	O
difficult	O
errors-in-variables	B-Algorithm
models	I-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
the	O
mean	O
of	O
the	O
response	O
variable	O
is	O
a	O
linear	O
combination	O
of	O
the	O
parameters	O
(	O
regression	B-General_Concept
coefficients	I-General_Concept
)	O
and	O
the	O
predictor	O
variables	O
.	O
</s>
<s>
This	O
technique	O
is	O
used	O
,	O
for	O
example	O
,	O
in	O
polynomial	O
regression	O
,	O
which	O
uses	O
linear	B-General_Concept
regression	I-General_Concept
to	O
fit	O
the	O
response	O
variable	O
as	O
an	O
arbitrary	O
polynomial	O
function	O
(	O
up	O
to	O
a	O
given	O
degree	O
)	O
of	O
a	O
predictor	O
variable	O
.	O
</s>
<s>
With	O
this	O
much	O
flexibility	O
,	O
models	O
such	O
as	O
polynomial	O
regression	O
often	O
have	O
"	O
too	O
much	O
power	O
"	O
,	O
in	O
that	O
they	O
tend	O
to	O
overfit	B-Error_Name
the	O
data	O
.	O
</s>
<s>
Common	O
examples	O
are	O
ridge	O
regression	O
and	O
lasso	B-Algorithm
regression	I-Algorithm
.	O
</s>
<s>
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
can	O
also	O
be	O
used	O
,	O
which	O
by	O
its	O
nature	O
is	O
more	O
or	O
less	O
immune	O
to	O
the	O
problem	O
of	O
overfitting	B-Error_Name
.	O
</s>
<s>
(	O
In	O
fact	O
,	O
ridge	O
regression	O
and	O
lasso	B-Algorithm
regression	I-Algorithm
can	O
both	O
be	O
viewed	O
as	O
special	O
cases	O
of	O
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
,	O
with	O
particular	O
types	O
of	O
prior	O
distributions	O
placed	O
on	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
.	O
)	O
</s>
<s>
homoscedasticity	B-General_Concept
)	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
person	O
whose	O
income	O
is	O
predicted	O
to	O
be	O
$	O
100,000	O
may	O
easily	O
have	O
an	O
actual	O
income	O
of	O
$	O
80,000	O
or	O
$	O
120,000	O
—	O
i.e.	O
,	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
around	O
$	O
20,000	O
—	O
while	O
another	O
person	O
with	O
a	O
predicted	O
income	O
of	O
$	O
10,000	O
is	O
unlikely	O
to	O
have	O
the	O
same	O
$	O
20,000	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
since	O
that	O
would	O
imply	O
their	O
actual	O
income	O
could	O
vary	O
anywhere	O
between	O
−$	O
10,000	O
and	O
$	O
30,000	O
.	O
</s>
<s>
(	O
In	O
fact	O
,	O
as	O
this	O
shows	O
,	O
in	O
many	O
cases	O
—	O
often	O
the	O
same	O
cases	O
where	O
the	O
assumption	O
of	O
normally	O
distributed	O
errors	O
fails	O
—	O
the	O
variance	O
or	O
standard	B-General_Concept
deviation	I-General_Concept
should	O
be	O
predicted	O
to	O
be	O
proportional	O
to	O
the	O
mean	O
,	O
rather	O
than	O
constant	O
.	O
)	O
</s>
<s>
The	O
absence	O
of	O
homoscedasticity	B-General_Concept
is	O
called	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
Formal	O
tests	O
can	O
also	O
be	O
used	O
;	O
see	O
Heteroscedasticity	B-General_Concept
.	O
</s>
<s>
The	O
presence	O
of	O
heteroscedasticity	B-General_Concept
will	O
result	O
in	O
an	O
overall	O
"	O
average	O
"	O
estimate	O
of	O
variance	O
being	O
used	O
instead	O
of	O
one	O
that	O
takes	O
into	O
account	O
the	O
true	O
variance	O
structure	O
.	O
</s>
<s>
This	O
leads	O
to	O
less	O
precise	O
(	O
but	O
in	O
the	O
case	O
of	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
,	O
not	O
biased	O
)	O
parameter	O
estimates	O
and	O
biased	O
standard	O
errors	O
,	O
resulting	O
in	O
misleading	O
tests	O
and	O
interval	O
estimates	O
.	O
</s>
<s>
The	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
for	O
the	O
model	O
will	O
also	O
be	O
wrong	O
.	O
</s>
<s>
Various	O
estimation	O
techniques	O
including	O
weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
and	O
the	O
use	O
of	O
heteroscedasticity-consistent	O
standard	O
errors	O
can	O
handle	O
heteroscedasticity	B-General_Concept
in	O
a	O
quite	O
general	O
way	O
.	O
</s>
<s>
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
techniques	O
can	O
also	O
be	O
used	O
when	O
the	O
variance	O
is	O
assumed	O
to	O
be	O
a	O
function	O
of	O
the	O
mean	O
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
in	O
some	O
cases	O
to	O
fix	O
the	O
problem	O
by	O
applying	O
a	O
transformation	O
to	O
the	O
response	O
variable	O
(	O
e.g.	O
,	O
fitting	O
the	O
logarithm	O
of	O
the	O
response	O
variable	O
using	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
,	O
which	O
implies	O
that	O
the	O
response	O
variable	O
itself	O
has	O
a	O
log-normal	O
distribution	O
rather	O
than	O
a	O
normal	O
distribution	O
)	O
.	O
</s>
<s>
Some	O
methods	O
such	O
as	O
generalized	O
least	B-Algorithm
squares	I-Algorithm
are	O
capable	O
of	O
handling	O
correlated	O
errors	O
,	O
although	O
they	O
typically	O
require	O
significantly	O
more	O
data	O
unless	O
some	O
sort	O
of	O
regularization	O
is	O
used	O
to	O
bias	O
the	O
model	O
towards	O
assuming	O
uncorrelated	O
errors	O
.	O
</s>
<s>
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
is	O
a	O
general	O
way	O
of	O
handling	O
this	O
issue	O
.	O
</s>
<s>
For	O
standard	O
least	B-Algorithm
squares	I-Algorithm
estimation	O
methods	O
,	O
the	O
design	B-Algorithm
matrix	I-Algorithm
X	O
must	O
have	O
full	O
column	O
rank	O
p	O
;	O
otherwise	O
perfect	O
multicollinearity	O
exists	O
in	O
the	O
predictor	O
variables	O
,	O
meaning	O
a	O
linear	O
relationship	O
exists	O
between	O
two	O
or	O
more	O
predictor	O
variables	O
.	O
</s>
<s>
It	O
can	O
also	O
happen	O
if	O
there	O
is	O
too	O
little	O
data	O
available	O
compared	O
to	O
the	O
number	O
of	O
parameters	O
to	O
be	O
estimated	O
(	O
e.g.	O
,	O
fewer	O
data	O
points	O
than	O
regression	B-General_Concept
coefficients	I-General_Concept
)	O
.	O
</s>
<s>
See	O
partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
regression	I-Algorithm
.	O
</s>
<s>
Methods	O
for	O
fitting	O
linear	B-Algorithm
models	I-Algorithm
with	O
multicollinearity	O
have	O
been	O
developed	O
,	O
some	O
of	O
which	O
require	O
additional	O
assumptions	O
such	O
as	O
"	O
effect	O
sparsity	O
"	O
—	O
that	O
a	O
large	O
fraction	O
of	O
the	O
effects	O
are	O
exactly	O
zero	O
.	O
</s>
<s>
Note	O
that	O
the	O
more	O
computationally	O
expensive	O
iterated	O
algorithms	O
for	O
parameter	O
estimation	O
,	O
such	O
as	O
those	O
used	O
in	O
generalized	O
linear	B-Algorithm
models	I-Algorithm
,	O
do	O
not	O
suffer	O
from	O
this	O
problem	O
.	O
</s>
<s>
A	O
fitted	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
can	O
be	O
used	O
to	O
identify	O
the	O
relationship	O
between	O
a	O
single	O
predictor	O
variable	O
xj	O
and	O
the	O
response	O
variable	O
y	O
when	O
all	O
the	O
other	O
predictor	O
variables	O
in	O
the	O
model	O
are	O
"	O
held	O
fixed	O
"	O
.	O
</s>
<s>
In	O
contrast	O
,	O
the	O
marginal	O
effect	O
of	O
xj	O
on	O
y	O
can	O
be	O
assessed	O
using	O
a	O
correlation	O
coefficient	O
or	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
model	O
relating	O
only	O
xj	O
to	O
y	O
;	O
this	O
effect	O
is	O
the	O
total	O
derivative	O
of	O
y	O
with	O
respect	O
to	O
xj	O
.	O
</s>
<s>
Care	O
must	O
be	O
taken	O
when	O
interpreting	O
regression	O
results	O
,	O
as	O
some	O
of	O
the	O
regressors	O
may	O
not	O
allow	O
for	O
marginal	O
changes	O
(	O
such	O
as	O
dummy	O
variables	O
,	O
or	O
the	O
intercept	B-Algorithm
term	O
)	O
,	O
while	O
others	O
cannot	O
be	O
held	O
fixed	O
(	O
recall	O
the	O
example	O
from	O
the	O
introduction	O
:	O
it	O
would	O
be	O
impossible	O
to	O
"	O
hold	O
ti	O
fixed	O
"	O
and	O
at	O
the	O
same	O
time	O
change	O
the	O
value	O
of	O
ti2	O
)	O
.	O
</s>
<s>
Group	O
effects	O
provide	O
a	O
means	O
to	O
study	O
the	O
collective	O
impact	O
of	O
strongly	O
correlated	O
predictor	O
variables	O
in	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
.	O
</s>
<s>
Furthermore	O
,	O
when	O
the	O
sample	O
size	O
is	O
not	O
large	O
,	O
none	O
of	O
their	O
parameters	O
can	O
be	O
accurately	O
estimated	O
by	O
the	O
least	B-Algorithm
squares	I-Algorithm
regression	I-Algorithm
due	O
to	O
the	O
multicollinearity	O
problem	O
.	O
</s>
<s>
Nevertheless	O
,	O
there	O
are	O
meaningful	O
group	O
effects	O
that	O
have	O
good	O
interpretations	O
and	O
can	O
be	O
accurately	O
estimated	O
by	O
the	O
least	B-Algorithm
squares	I-Algorithm
regression	I-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
least	B-Algorithm
squares	I-Algorithm
estimator	O
of	O
.	O
</s>
<s>
Applications	O
of	O
the	O
group	O
effects	O
include	O
(	O
1	O
)	O
estimation	O
and	O
inference	O
for	O
meaningful	O
group	O
effects	O
on	O
the	O
response	O
variable	O
,	O
(	O
2	O
)	O
testing	O
for	O
"	O
group	O
significance	O
"	O
of	O
the	O
variables	O
via	O
testing	O
versus	O
,	O
and	O
(	O
3	O
)	O
characterizing	O
the	O
region	O
of	O
the	O
predictor	O
variable	O
space	O
over	O
which	O
predictions	O
by	O
the	O
least	B-Algorithm
squares	I-Algorithm
estimated	O
model	O
are	O
accurate	O
.	O
</s>
<s>
Numerous	O
extensions	O
of	O
linear	B-General_Concept
regression	I-General_Concept
have	O
been	O
developed	O
,	O
which	O
allow	O
some	O
or	O
all	O
of	O
the	O
assumptions	O
underlying	O
the	O
basic	O
model	O
to	O
be	O
relaxed	O
.	O
</s>
<s>
The	O
very	O
simplest	O
case	O
of	O
a	O
single	O
scalar	O
predictor	O
variable	O
x	O
and	O
a	O
single	O
scalar	O
response	O
variable	O
y	O
is	O
known	O
as	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
The	O
extension	O
to	O
multiple	O
and/or	O
vector-valued	O
predictor	O
variables	O
(	O
denoted	O
with	O
a	O
capital	O
X	O
)	O
is	O
known	O
as	O
multiple	O
linear	B-General_Concept
regression	I-General_Concept
,	O
also	O
known	O
as	O
multivariable	O
linear	B-General_Concept
regression	I-General_Concept
(	O
not	O
to	O
be	O
confused	O
with	O
multivariate	O
linear	B-General_Concept
regression	I-General_Concept
)	O
.	O
</s>
<s>
Multiple	O
linear	B-General_Concept
regression	I-General_Concept
is	O
a	O
generalization	O
of	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
to	O
the	O
case	O
of	O
more	O
than	O
one	O
independent	O
variable	O
,	O
and	O
a	O
special	O
case	O
of	O
general	O
linear	B-Algorithm
models	I-Algorithm
,	O
restricted	O
to	O
one	O
dependent	O
variable	O
.	O
</s>
<s>
In	O
the	O
more	O
general	O
multivariate	O
linear	B-General_Concept
regression	I-General_Concept
,	O
there	O
is	O
one	O
equation	O
of	O
the	O
above	O
form	O
for	O
each	O
of	O
m	O
>	O
1	O
dependent	O
variables	O
that	O
share	O
the	O
same	O
set	O
of	O
explanatory	O
variables	O
and	O
hence	O
are	O
estimated	O
simultaneously	O
with	O
each	O
other	O
:	O
</s>
<s>
Nearly	O
all	O
real-world	O
regression	O
models	O
involve	O
multiple	O
predictors	O
,	O
and	O
basic	O
descriptions	O
of	O
linear	B-General_Concept
regression	I-General_Concept
are	O
often	O
phrased	O
in	O
terms	O
of	O
the	O
multiple	O
regression	O
model	O
.	O
</s>
<s>
Another	O
term	O
,	O
multivariate	O
linear	B-General_Concept
regression	I-General_Concept
,	O
refers	O
to	O
cases	O
where	O
y	O
is	O
a	O
vector	O
,	O
i.e.	O
,	O
the	O
same	O
as	O
general	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
The	O
general	O
linear	B-Algorithm
model	I-Algorithm
considers	O
the	O
situation	O
when	O
the	O
response	O
variable	O
is	O
not	O
a	O
scalar	O
(	O
for	O
each	O
observation	O
)	O
but	O
a	O
vector	O
,	O
yi	O
.	O
</s>
<s>
Conditional	O
linearity	O
of	O
is	O
still	O
assumed	O
,	O
with	O
a	O
matrix	O
B	O
replacing	O
the	O
vector	O
β	O
of	O
the	O
classical	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
.	O
</s>
<s>
Multivariate	O
analogues	O
of	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
(	O
OLS	B-General_Concept
)	O
and	O
generalized	O
least	B-Algorithm
squares	I-Algorithm
(	O
GLS	O
)	O
have	O
been	O
developed	O
.	O
</s>
<s>
"	O
General	O
linear	B-Algorithm
models	I-Algorithm
"	O
are	O
also	O
called	O
"	O
multivariate	O
linear	B-Algorithm
models	I-Algorithm
"	O
.	O
</s>
<s>
These	O
are	O
not	O
the	O
same	O
as	O
multivariable	O
linear	B-Algorithm
models	I-Algorithm
(	O
also	O
called	O
"	O
multiple	O
linear	B-Algorithm
models	I-Algorithm
"	O
)	O
.	O
</s>
<s>
Various	O
models	O
have	O
been	O
created	O
that	O
allow	O
for	O
heteroscedasticity	B-General_Concept
,	O
i.e.	O
</s>
<s>
For	O
example	O
,	O
weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
is	O
a	O
method	O
for	O
estimating	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
when	O
the	O
response	O
variables	O
may	O
have	O
different	O
error	O
variances	O
,	O
possibly	O
with	O
correlated	O
errors	O
.	O
</s>
<s>
(	O
See	O
also	O
Weighted	B-Algorithm
linear	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
,	O
and	O
Generalized	O
least	B-Algorithm
squares	I-Algorithm
.	O
)	O
</s>
<s>
Heteroscedasticity-consistent	O
standard	O
errors	O
is	O
an	O
improved	O
method	O
for	O
use	O
with	O
uncorrelated	O
but	O
potentially	O
heteroscedastic	B-General_Concept
errors	O
.	O
</s>
<s>
Generalized	O
linear	B-Algorithm
models	I-Algorithm
(	O
GLMs	O
)	O
are	O
a	O
framework	O
for	O
modeling	O
response	O
variables	O
that	O
are	O
bounded	O
or	O
discrete	O
.	O
</s>
<s>
prices	O
or	O
populations	O
)	O
that	O
vary	O
over	O
a	O
large	O
scale	O
—	O
which	O
are	O
better	O
described	O
using	O
a	O
skewed	B-General_Concept
distribution	I-General_Concept
such	O
as	O
the	O
log-normal	O
distribution	O
or	O
Poisson	O
distribution	O
(	O
although	O
GLMs	O
are	O
not	O
used	O
for	O
log-normal	O
data	O
,	O
instead	O
the	O
response	O
variable	O
is	O
simply	O
transformed	O
using	O
the	O
logarithm	O
function	O
)	O
;	O
</s>
<s>
when	O
modeling	O
ordinal	B-General_Concept
data	I-General_Concept
,	O
e.g.	O
</s>
<s>
Generalized	O
linear	B-Algorithm
models	I-Algorithm
allow	O
for	O
an	O
arbitrary	O
link	O
function	O
,	O
g	O
,	O
that	O
relates	O
the	O
mean	O
of	O
the	O
response	O
variable(s )	O
to	O
the	O
predictors	O
:	O
.	O
</s>
<s>
Logistic	O
regression	O
and	O
probit	B-Architecture
regression	I-Architecture
for	O
binary	O
data	O
.	O
</s>
<s>
Multinomial	O
logistic	O
regression	O
and	O
multinomial	B-General_Concept
probit	I-General_Concept
regression	O
for	O
categorical	O
data	O
.	O
</s>
<s>
Ordered	O
logit	O
and	O
ordered	O
probit	B-Architecture
regression	I-Architecture
for	O
ordinal	B-General_Concept
data	I-General_Concept
.	O
</s>
<s>
Single	O
index	O
models	O
allow	O
some	O
degree	O
of	O
nonlinearity	O
in	O
the	O
relationship	O
between	O
x	O
and	O
y	O
,	O
while	O
preserving	O
the	O
central	O
role	O
of	O
the	O
linear	O
predictor	O
β′x	O
as	O
in	O
the	O
classical	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
.	O
</s>
<s>
Under	O
certain	O
conditions	O
,	O
simply	O
applying	O
OLS	B-General_Concept
to	O
data	O
from	O
a	O
single-index	O
model	O
will	O
consistently	O
estimate	O
β	O
up	O
to	O
a	O
proportionality	O
constant	O
.	O
</s>
<s>
Hierarchical	B-General_Concept
linear	I-General_Concept
models	I-General_Concept
(	O
or	O
multilevel	O
regression	O
)	O
organizes	O
the	O
data	O
into	O
a	O
hierarchy	O
of	O
regressions	O
,	O
for	O
example	O
where	O
A	B-Application
is	I-Application
regressed	O
on	O
B	O
,	O
and	O
B	O
is	O
regressed	O
on	O
C	O
.	O
It	O
is	O
often	O
used	O
where	O
the	O
variables	O
of	O
interest	O
have	O
a	O
natural	O
hierarchical	O
structure	O
such	O
as	O
in	O
educational	O
statistics	O
,	O
where	O
students	O
are	O
nested	O
in	O
classrooms	O
,	O
classrooms	O
are	O
nested	O
in	O
schools	O
,	O
and	O
schools	O
are	O
nested	O
in	O
some	O
administrative	O
grouping	O
,	O
such	O
as	O
a	O
school	O
district	O
.	O
</s>
<s>
Errors-in-variables	B-Algorithm
models	I-Algorithm
(	O
or	O
"	O
measurement	B-Algorithm
error	I-Algorithm
models	I-Algorithm
"	O
)	O
extend	O
the	O
traditional	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
to	O
allow	O
the	O
predictor	O
variables	O
X	O
to	O
be	O
observed	O
with	O
error	O
.	O
</s>
<s>
In	O
Dempster	O
–	O
Shafer	O
theory	O
,	O
or	O
a	O
linear	O
belief	O
function	O
in	O
particular	O
,	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
may	O
be	O
represented	O
as	O
a	O
partially	O
swept	O
matrix	O
,	O
which	O
can	O
be	O
combined	O
with	O
similar	O
matrices	O
representing	O
observations	O
and	O
other	O
assumed	O
normal	O
distributions	O
and	O
state	O
equations	O
.	O
</s>
<s>
The	O
combination	O
of	O
swept	O
or	O
unswept	O
matrices	O
provides	O
an	O
alternative	O
method	O
for	O
estimating	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
.	O
</s>
<s>
A	O
large	O
number	O
of	O
procedures	O
have	O
been	O
developed	O
for	O
parameter	O
estimation	O
and	O
inference	O
in	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
Some	O
of	O
the	O
more	O
common	O
estimation	O
techniques	O
for	O
linear	B-General_Concept
regression	I-General_Concept
are	O
summarized	O
below	O
.	O
</s>
<s>
The	O
"	O
locus	O
of	O
horizontal	O
tangential	O
points	O
"	O
passing	O
through	O
the	O
leftmost	O
and	O
rightmost	O
points	O
on	O
the	O
ellipse	O
(	O
which	O
is	O
a	O
level	O
curve	O
of	O
the	O
bivariate	O
normal	O
distribution	O
estimated	O
from	O
the	O
data	O
)	O
is	O
the	O
OLS	B-General_Concept
estimate	O
of	O
the	O
regression	O
of	O
parents	O
 '	O
heights	O
on	O
children	O
's	O
heights	O
,	O
while	O
the	O
"	O
locus	O
of	O
vertical	O
tangential	O
points	O
"	O
is	O
the	O
OLS	B-General_Concept
estimate	O
of	O
the	O
regression	O
of	O
children	O
's	O
heights	O
on	O
parent	O
's	O
heights	O
.	O
</s>
<s>
The	O
major	O
axis	O
of	O
the	O
ellipse	O
is	O
the	O
TLS	B-Algorithm
estimate	O
.	O
</s>
<s>
In	O
the	O
least-squares	B-Algorithm
setting	O
,	O
the	O
optimum	O
parameter	O
is	O
defined	O
as	O
such	O
that	O
minimizes	O
the	O
sum	O
of	O
mean	O
squared	O
loss	O
:	O
</s>
<s>
This	O
is	O
provided	O
by	O
the	O
Gauss	B-Algorithm
–	O
Markov	O
theorem	O
.	O
</s>
<s>
Linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
methods	O
include	O
mainly	O
:	O
</s>
<s>
Maximum	O
likelihood	O
estimation	O
can	O
be	O
performed	O
when	O
the	O
distribution	O
of	O
the	O
error	O
terms	O
is	O
known	O
to	O
belong	O
to	O
a	O
certain	O
parametric	B-General_Concept
family	O
ƒθ	O
of	O
probability	O
distributions	O
.	O
</s>
<s>
When	O
fθ	O
is	O
a	O
normal	O
distribution	O
with	O
zero	O
mean	O
and	O
variance	O
θ	O
,	O
the	O
resulting	O
estimate	O
is	O
identical	O
to	O
the	O
OLS	B-General_Concept
estimate	O
.	O
</s>
<s>
Ridge	O
regression	O
and	O
other	O
forms	O
of	O
penalized	O
estimation	O
,	O
such	O
as	O
Lasso	B-Algorithm
regression	I-Algorithm
,	O
deliberately	O
introduce	O
bias	O
into	O
the	O
estimation	O
of	O
β	O
in	O
order	O
to	O
reduce	O
the	O
variability	O
of	O
the	O
estimate	O
.	O
</s>
<s>
The	O
resulting	O
estimates	O
generally	O
have	O
lower	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
than	O
the	O
OLS	B-General_Concept
estimates	O
,	O
particularly	O
when	O
multicollinearity	O
is	O
present	O
or	O
when	O
overfitting	B-Error_Name
is	O
a	O
problem	O
.	O
</s>
<s>
Least	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
(	O
LAD	O
)	O
regression	O
is	O
a	O
robust	O
estimation	O
technique	O
in	O
that	O
it	O
is	O
less	O
sensitive	O
to	O
the	O
presence	O
of	O
outliers	O
than	O
OLS	B-General_Concept
(	O
but	O
is	O
less	O
efficient	O
than	O
OLS	B-General_Concept
when	O
no	O
outliers	O
are	O
present	O
)	O
.	O
</s>
<s>
Bayesian	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
applies	O
the	O
framework	O
of	O
Bayesian	O
statistics	O
to	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
(	O
See	O
also	O
Bayesian	B-General_Concept
multivariate	I-General_Concept
linear	I-General_Concept
regression	I-General_Concept
.	O
)	O
</s>
<s>
In	O
particular	O
,	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
β	O
are	O
assumed	O
to	O
be	O
random	O
variables	O
with	O
a	O
specified	O
prior	O
distribution	O
.	O
</s>
<s>
The	O
prior	O
distribution	O
can	O
bias	O
the	O
solutions	O
for	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
in	O
a	O
way	O
similar	O
to	O
(	O
but	O
more	O
general	O
than	O
)	O
ridge	O
regression	O
or	O
lasso	B-Algorithm
regression	I-Algorithm
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
Bayesian	O
estimation	O
process	O
produces	O
not	O
a	O
single	O
point	O
estimate	O
for	O
the	O
"	O
best	O
"	O
values	O
of	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
but	O
an	O
entire	O
posterior	O
distribution	O
,	O
completely	O
describing	O
the	O
uncertainty	O
surrounding	O
the	O
quantity	O
.	O
</s>
<s>
Mixed	B-General_Concept
models	I-General_Concept
are	O
widely	O
used	O
to	O
analyze	O
linear	B-General_Concept
regression	I-General_Concept
relationships	O
involving	O
dependent	O
data	O
when	O
the	O
dependencies	O
have	O
a	O
known	O
structure	O
.	O
</s>
<s>
Common	O
applications	O
of	O
mixed	B-General_Concept
models	I-General_Concept
include	O
analysis	O
of	O
data	O
involving	O
repeated	O
measurements	O
,	O
such	O
as	O
longitudinal	O
data	O
,	O
or	O
data	O
obtained	O
from	O
cluster	O
sampling	O
.	O
</s>
<s>
They	O
are	O
generally	O
fit	O
as	O
parametric	B-General_Concept
models	O
,	O
using	O
maximum	O
likelihood	O
or	O
Bayesian	O
estimation	O
.	O
</s>
<s>
In	O
the	O
case	O
where	O
the	O
errors	O
are	O
modeled	O
as	O
normal	O
random	O
variables	O
,	O
there	O
is	O
a	O
close	O
connection	O
between	O
mixed	B-General_Concept
models	I-General_Concept
and	O
generalized	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
Fixed	B-General_Concept
effects	I-General_Concept
estimation	I-General_Concept
is	O
an	O
alternative	O
approach	O
to	O
analyzing	O
this	O
type	O
of	O
data	O
.	O
</s>
<s>
Principal	B-Algorithm
component	I-Algorithm
regression	I-Algorithm
(	O
PCR	O
)	O
is	O
used	O
when	O
the	O
number	O
of	O
predictor	O
variables	O
is	O
large	O
,	O
or	O
when	O
strong	O
correlations	O
exist	O
among	O
the	O
predictor	O
variables	O
.	O
</s>
<s>
This	O
two-stage	O
procedure	O
first	O
reduces	O
the	O
predictor	O
variables	O
using	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
and	O
then	O
uses	O
the	O
reduced	O
variables	O
in	O
an	O
OLS	B-General_Concept
regression	I-General_Concept
fit	O
.	O
</s>
<s>
While	O
it	O
often	O
works	O
well	O
in	O
practice	O
,	O
there	O
is	O
no	O
general	O
theoretical	O
reason	O
that	O
the	O
most	O
informative	O
linear	O
function	O
of	O
the	O
predictor	O
variables	O
should	O
lie	O
among	O
the	O
dominant	O
principal	B-Application
components	I-Application
of	O
the	O
multivariate	O
distribution	O
of	O
the	O
predictor	O
variables	O
.	O
</s>
<s>
The	O
partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
regression	I-Algorithm
is	O
the	O
extension	O
of	O
the	O
PCR	O
method	O
which	O
does	O
not	O
suffer	O
from	O
the	O
mentioned	O
deficiency	O
.	O
</s>
<s>
Least-angle	O
regression	O
is	O
an	O
estimation	O
procedure	O
for	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
that	O
was	O
developed	O
to	O
handle	O
high-dimensional	O
covariate	O
vectors	O
,	O
potentially	O
with	O
more	O
covariates	O
than	O
observations	O
.	O
</s>
<s>
It	O
has	O
similar	O
statistical	O
efficiency	O
properties	O
to	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
but	O
is	O
much	O
less	O
sensitive	O
to	O
outliers	O
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
is	O
widely	O
used	O
in	O
biological	O
,	O
behavioral	O
and	O
social	O
sciences	O
to	O
describe	O
possible	O
relationships	O
between	O
variables	O
.	O
</s>
<s>
It	O
tells	O
whether	O
a	O
particular	O
data	B-General_Concept
set	I-General_Concept
(	O
say	O
GDP	O
,	O
oil	O
prices	O
or	O
stock	O
prices	O
)	O
have	O
increased	O
or	O
decreased	O
over	O
the	O
period	O
of	O
time	O
.	O
</s>
<s>
A	O
trend	O
line	O
could	O
simply	O
be	O
drawn	O
by	O
eye	O
through	O
a	O
set	B-General_Concept
of	I-General_Concept
data	I-General_Concept
points	O
,	O
but	O
more	O
properly	O
their	O
position	O
and	O
slope	O
is	O
calculated	O
using	O
statistical	O
techniques	O
like	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
The	O
capital	O
asset	O
pricing	O
model	O
uses	O
linear	B-General_Concept
regression	I-General_Concept
as	O
well	O
as	O
the	O
concept	O
of	O
beta	O
for	O
analyzing	O
and	O
quantifying	O
the	O
systematic	O
risk	O
of	O
an	O
investment	O
.	O
</s>
<s>
This	O
comes	O
directly	O
from	O
the	O
beta	O
coefficient	O
of	O
the	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
that	O
relates	O
the	O
return	O
on	O
the	O
investment	O
to	O
the	O
return	O
on	O
all	O
risky	O
assets	O
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
is	O
the	O
predominant	O
empirical	O
tool	O
in	O
economics	O
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
finds	O
application	O
in	O
a	O
wide	O
range	O
of	O
environmental	O
science	O
applications	O
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
plays	O
an	O
important	O
role	O
in	O
the	O
subfield	O
of	O
artificial	B-Application
intelligence	I-Application
known	O
as	O
machine	O
learning	O
.	O
</s>
<s>
The	O
linear	B-General_Concept
regression	I-General_Concept
algorithm	O
is	O
one	O
of	O
the	O
fundamental	O
supervised	B-General_Concept
machine-learning	I-General_Concept
algorithms	O
due	O
to	O
its	O
relative	O
simplicity	O
and	O
well-known	O
properties	O
.	O
</s>
<s>
Least	B-Algorithm
squares	I-Algorithm
linear	B-General_Concept
regression	I-General_Concept
,	O
as	O
a	O
means	O
of	O
finding	O
a	O
good	O
rough	O
linear	B-General_Concept
fit	I-General_Concept
to	O
a	O
set	O
of	O
points	O
was	O
performed	O
by	O
Legendre	O
(	O
1805	O
)	O
and	O
Gauss	B-Algorithm
(	O
1809	O
)	O
for	O
the	O
prediction	O
of	O
planetary	O
movement	O
.	O
</s>
