<s>
In	O
statistics	O
,	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
is	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
with	O
a	O
single	O
explanatory	O
variable	O
.	O
</s>
<s>
It	O
is	O
common	O
to	O
make	O
the	O
additional	O
stipulation	O
that	O
the	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
(	O
OLS	B-General_Concept
)	O
method	O
should	O
be	O
used	O
:	O
the	O
accuracy	O
of	O
each	O
predicted	O
value	O
is	O
measured	O
by	O
its	O
squared	O
residual	O
(	O
vertical	O
distance	O
between	O
the	O
point	O
of	O
the	O
data	O
set	O
and	O
the	O
fitted	O
line	O
)	O
,	O
and	O
the	O
goal	O
is	O
to	O
make	O
the	O
sum	O
of	O
these	O
squared	O
deviations	O
as	O
small	O
as	O
possible	O
.	O
</s>
<s>
Other	O
regression	O
methods	O
that	O
can	O
be	O
used	O
in	O
place	O
of	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
include	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
(	O
minimizing	O
the	O
sum	O
of	O
absolute	O
values	O
of	O
residuals	O
)	O
and	O
the	O
Theil	O
–	O
Sen	O
estimator	O
(	O
which	O
chooses	O
a	O
line	O
whose	O
slope	O
is	O
the	O
median	O
of	O
the	O
slopes	O
determined	O
by	O
pairs	O
of	O
sample	O
points	O
)	O
.	O
</s>
<s>
Deming	B-Algorithm
regression	I-Algorithm
(	O
total	O
least	O
squares	O
)	O
also	O
finds	O
a	O
line	O
that	O
fits	O
a	O
set	O
of	O
two-dimensional	O
sample	O
points	O
,	O
but	O
(	O
unlike	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
,	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
,	O
and	O
median	O
slope	O
regression	O
)	O
it	O
is	O
not	O
really	O
an	O
instance	O
of	O
,	O
because	O
it	O
does	O
not	O
separate	O
the	O
coordinates	O
into	O
one	O
dependent	O
and	O
one	O
independent	O
variable	O
and	O
could	O
potentially	O
return	O
a	O
vertical	O
line	O
as	O
its	O
fit	O
.	O
</s>
<s>
The	O
remainder	O
of	O
the	O
article	O
assumes	O
an	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
This	O
relationship	O
between	O
the	O
true	O
(	O
but	O
unobserved	O
)	O
underlying	O
parameters	O
and	O
and	O
the	O
data	O
points	O
is	O
called	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
.	O
</s>
<s>
As	O
mentioned	O
in	O
the	O
introduction	O
,	O
in	O
this	O
article	O
the	O
"	O
best	O
"	O
fit	O
will	O
be	O
understood	O
as	O
in	O
the	O
least-squares	B-General_Concept
approach	O
:	O
a	O
line	O
that	O
minimizes	O
the	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
residuals	I-Algorithm
(	O
see	O
also	O
Errors	O
and	O
residuals	O
)	O
(	O
differences	O
between	O
actual	O
and	O
predicted	O
values	O
of	O
the	O
dependent	O
variable	O
y	O
)	O
,	O
each	O
of	O
which	O
is	O
given	O
by	O
,	O
for	O
any	O
candidate	O
parameter	O
values	O
and	O
,	O
</s>
<s>
This	O
shows	O
that	O
is	O
the	O
slope	O
of	O
the	O
regression	B-General_Concept
line	I-General_Concept
of	O
the	O
standardized	O
data	O
points	O
(	O
and	O
that	O
this	O
line	O
passes	O
through	O
the	O
origin	O
)	O
.	O
</s>
<s>
We	O
can	O
see	O
that	O
the	O
slope	O
(	O
tangent	O
of	O
angle	O
)	O
of	O
the	O
regression	B-General_Concept
line	I-General_Concept
is	O
the	O
weighted	O
average	O
of	O
that	O
is	O
the	O
slope	O
(	O
tangent	O
of	O
angle	O
)	O
of	O
the	O
line	O
that	O
connects	O
the	O
i-th	O
point	O
to	O
the	O
average	O
of	O
all	O
points	O
,	O
weighted	O
by	O
because	O
the	O
further	O
the	O
point	O
is	O
the	O
more	O
"	O
important	O
"	O
it	O
is	O
,	O
since	O
small	O
errors	O
in	O
its	O
position	O
will	O
affect	O
the	O
slope	O
connecting	O
it	O
to	O
the	O
center	O
point	O
more	O
.	O
</s>
<s>
A	O
possible	O
interpretation	O
of	O
is	O
to	O
imagine	O
that	O
defines	O
a	O
random	O
variable	O
drawn	O
from	O
the	O
empirical	B-General_Concept
distribution	I-General_Concept
of	O
the	O
x	O
values	O
in	O
our	O
sample	O
.	O
</s>
<s>
Sometimes	O
it	O
is	O
appropriate	O
to	O
force	O
the	O
regression	B-General_Concept
line	I-General_Concept
to	O
pass	O
through	O
the	O
origin	O
,	O
because	O
and	O
are	O
assumed	O
to	O
be	O
proportional	O
.	O
</s>
<s>
Description	O
of	O
the	O
statistical	O
properties	O
of	O
estimators	O
from	O
the	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
estimates	O
requires	O
the	O
use	O
of	O
a	O
statistical	O
model	O
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
evaluate	O
the	O
properties	O
under	O
other	O
assumptions	O
,	O
such	O
as	O
inhomogeneity	B-General_Concept
,	O
but	O
this	O
is	O
discussed	O
elsewhere	O
.	O
</s>
<s>
Under	O
such	O
interpretation	O
,	O
the	O
least-squares	B-General_Concept
estimators	O
and	O
will	O
themselves	O
be	O
random	O
variables	O
whose	O
means	O
will	O
equal	O
the	O
"	O
true	O
values	O
"	O
and	O
.	O
</s>
<s>
The	O
formulas	O
given	O
in	O
the	O
previous	O
section	O
allow	O
one	O
to	O
calculate	O
the	O
point	O
estimates	O
of	O
and	O
—	O
that	O
is	O
,	O
the	O
coefficients	O
of	O
the	O
regression	B-General_Concept
line	I-General_Concept
for	O
the	O
given	O
set	O
of	O
data	O
.	O
</s>
<s>
The	O
standard	O
method	O
of	O
constructing	O
confidence	O
intervals	O
for	O
linear	B-General_Concept
regression	I-General_Concept
coefficients	O
relies	O
on	O
the	O
normality	O
assumption	O
,	O
which	O
is	O
justified	O
if	O
either	O
:	O
</s>
<s>
Under	O
the	O
first	O
assumption	O
above	O
,	O
that	O
of	O
the	O
normality	O
of	O
the	O
error	O
terms	O
,	O
the	O
estimator	O
of	O
the	O
slope	O
coefficient	O
will	O
itself	O
be	O
normally	O
distributed	O
with	O
mean	O
and	O
variance	O
where	O
is	O
the	O
variance	O
of	O
the	O
error	O
terms	O
(	O
see	O
Proofs	B-Algorithm
involving	I-Algorithm
ordinary	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
)	O
.	O
</s>
<s>
At	O
the	O
same	O
time	O
the	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
residuals	I-Algorithm
is	O
distributed	O
proportionally	O
to	O
with	O
degrees	O
of	O
freedom	O
,	O
and	O
independently	O
from	O
.	O
</s>
<s>
The	O
confidence	O
intervals	O
for	O
and	O
give	O
us	O
the	O
general	O
idea	O
where	O
these	O
regression	B-General_Concept
coefficients	I-General_Concept
are	O
most	O
likely	O
to	O
be	O
.	O
</s>
<s>
In	O
order	O
to	O
represent	O
this	O
information	O
graphically	O
,	O
in	O
the	O
form	O
of	O
the	O
confidence	O
bands	O
around	O
the	O
regression	B-General_Concept
line	I-General_Concept
,	O
one	O
has	O
to	O
proceed	O
carefully	O
and	O
account	O
for	O
the	O
joint	O
distribution	O
of	O
the	O
estimators	O
.	O
</s>
<s>
Although	O
the	O
OLS	B-General_Concept
article	O
argues	O
that	O
it	O
would	O
be	O
more	O
appropriate	O
to	O
run	O
a	O
quadratic	O
regression	O
for	O
this	O
data	O
,	O
the	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
model	O
is	O
applied	O
here	O
instead	O
.	O
</s>
<s>
These	O
quantities	O
would	O
be	O
used	O
to	O
calculate	O
the	O
estimates	O
of	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
and	O
their	O
standard	O
errors	O
.	O
</s>
