outcome
dependent
Two-variable ↑
regression Specify
.
:
ETQIQ
-
= L+ BL in
equinately :
2
·ndepome/ual Y L + P + negremio
=
.
or
er
Basic notel to
we observe (22 3) .
as
having come
P .
dist and E =
y -
E(y)M) =
y -
x
-
BM
-
lawe Passical Lineor Regression Mode)
May
causal interpretation . .
>
- can be u re a
for forecasting
Conditional enfectation >
-
Data : n observations ,
(2 , 4, 1
, .... (Ru ,
In)
of of Y Yi Ei for
-
·
It y conditional th = 2 + XiB +
each
given ze-average on i = 1 m .
, ....
being
I
a
particular value ·
on : - (II ElEl + ] = 0
Ei has mean 8
for anyth
Assumptions
·
ie
. Condition (restrict) the data set for fined # of education (V(3i(X) =
V(Ei) =
02 >
-
navance-constant .
I look of A r.
Earnings for that restricted group .
B Cou(Ei ,
3 ; (2) =
0 for ifj
·
E(9(2) -'conditional enfectation 14) Eil X ~ Normas (0 5
,
-(no beteroscedasticity)
*
↳ ↳ has
function
a constant variance .
regression
.
Explaining
Conditions :
Correlation & Causality (video 2) .
(1) E
: = 0
for any
22 .
Do condition violated when :
↳
hiB represent out else than conditional enfectation -
check
Statistical Correlation
*
· - In 84 have systematic relations .
when E(3/mi) 70 when 22 depends on 3 .
coontry)
=if at l
>
-
of
regresion error hasindegenevariance
(2) i s
regoe
Reminder
·
- Could 41 ,
=
-(x-Mse) (4 My) - L . education example
:
nagex Education years
-
↳ low
where
g
E(Xh
My E14) (a) Cor(x A for
= =
vor
=
HS education
; ; , von uage
among
Tron
for mage
among Uni education -
>
- constant variance inplier ,
a
plot of re s i d u a l vs (2) har a
pattern
that forms a holitontal and
palle n
B) information .
abst Ith person has new
infor alt .
Ith person
Correlation
·
# Correlation .
holdswhenwahaueassectionaldaa radorDateene
O
(4) Assumption for nathematical Convenience
.
When can we view correlation as a causal relationship ? Ordinary Least Square Estimation : (0()) (Video 4)
↳ Medical & Economic Example . ·
Estimate <, B from (m , y 1 . ,
...
(W , Un) .
↳ (a b) (x B)
wite , for estimates
of ,
.
Classical Linear Regression Hokel. (video 3) idea :
Yi-a-bhi-B-bluEi ;
+
close 100
& b B.
diff
by minimizing ,
a x =
Conditional Expectation (lined
.
Least square estimation :
regression) -
choose a b such that
↓
- ,
minimited
nb) This in
b) (Yn function
=1412S-aimB'where B # from aa (4 + + a
-
a
-
a, some not m,
minay
-
are entiate
-
he ,
...
4
(close to 07 .
-
↑
linor function .
How computer ?
is a & b
- stateevation the
calculus
.
"observations
*
t
, Hof) 0:
%a (Yi-a-Mib) ,
= -
2 (4 ; - a - Mib) () Interpreting Regression Coefficient moder 2
(video 21
o
=
4/b * (4 ,
-
a -
Mib)" = -
z (4 , -
a -
zib)x; Ch ↳
Y; = x +
Blog(xi) +
Ei
Population:) Fac a o (4 , - a
-
-i b) = 8 : 3 (4 , hib)
- E if 2 4 by small amount $ :
p[log(n b) togle
+
-
- (4 , - ib E
[Y14I = C +
Blogb)
E(y(x Al
na =
,
(4 :
-
2; b) + = 2 +
Blog(n A) +
a:
Y ,
(4 , -Rib) 6 Y, Y
is e I compute the
change in
average
to
Blog (I +
n
I
embl
Substitute into other foc . (2) ↳2)
By approximation rule :
if $/ in small (cose to 01
0
: (4 :- Jn)ze;
-us l el i s e. >
-
log
(1 + 9/2) =
&ke
-Simplify algebraically mone a
cane
- b
- derot
Benital On
Wil
Therefore
:
5(4( -x + 11 -
E(414-N =
BPk
/ 100 =
/ is % change inse ,
so when 2 4
by 1).,
regression
re s i d u a l the
average of Y
changes by P/100
L
I
.
④ =
Y; - a-Tib in an estimate
of Ei =
Yi-x-miB ⑳
⑱@
diffaction caluim conclusion I
#wrough
-
we can estimate - =
V(2) by same .
si =
yn Ee? -
> sample analogue
- O
↑
i =
1
You ( <+ Blogu) =
P
gn = B'
dof correction .
let the [CUIU) (2
the BY be
charge in when we
change
- -
by Dm .
- For male bu
,
we have embY/ Bu
the
We-Two variable Cont -B/x
regression =
4 B
. : =
Interpreting Regression crefflient (video 1) .
same
definition -
L what does the
reg conffient represent
.
?
Interpreting Regression Coefficient (Model 3) (Video 31
Yi = L+
BR, +
Er =
y wage ,
-causation
,
compare (212/25
E(Y(2 13) =
= d +
B13 .
instead , we can
transform Y to
log(4) leg earnings)
.
-
E(4(x =
12) = > + B12 log(Yi) =
a +
Bxi +
Ei
- o
x+px
differentiate
=
difference = E(4(x =
B) -
+
(4(k 1) = =
(c +
BB) -
( + BR) =
B art .
in o N =
B
sx
The 1 unit then Y
by B
to interpret B by
How =
changes
-
Ar
.
.
,
take the
definition of
literal derivative
T-xm to
differential Calculu La (rate a
of charge of
&
By in the 12
same
laying
: as
ofhange E(log Y ! )
=
4/jn(k + pm) =
By
same a n swe r relative As
to rate is I
~log (4 -Log (10) ,
, . . .
LetYo be the
original
va l u e
of Y and Y, be the s
ummary
:
4 c a re
(i)
·mine( it zen byjwheree4
i
ades a nde
Y
of by
Dann
I
changes
new va l u e DD i Yo
when . e
, AY =
Y,
log(4,) -Ag(y) 34ly B .
↓
=
p
=
0
log
(41) -
lig (10) =
BAM
/ An
sever
emplonator and
log
(4 1
, -log (10) =
log (" /y) =
log (1 +
** ) *
2/
4
# = tyo.
hi
is
Box
/When Hypothesis Testing
AY
by 1 unit (video 5)
Mo
Co = 24 24
,
by 10
. .
↳
Aft computing estimators what do e do ? - It .
Methods (1 At[0 1)
Note 1) = D
only good for
:
approximation
log
on + is .
I Eei
↳ Y
Bo
Another Method : Ho to I, and
Yo to , ↳
selm -
~
Tro
red
by
Review HT Procedure -
(1)
Specify Ho :
B =
Po
Specify
12-sdal
12) H: BFBo
table
.
(1 Choose
significance
Level (21 ; Find correspond ,
value from t-dist
Bob
14 Compute T :
(video 4)
Interpreting Regression Coefficients
Model 4 15) Decide
. whether to accept on
reject Ho
(indep ) .
(Video 6)
Forecasting
↑
Both depended & explanator variable are
transformed .
.
loglin
log (Yi) L
Blog (i) Ei deviative
= + +
Im
L :
g(1/2)
Ja
=
0(x +
Ploglill =
Blog en
day I
of
nation
Because log litlog
10)
in
sm
log
log (70)
= P/p = big (n) -Mogl in
Bl Forming Confidence Internal
provde
cur tai
a
range of values on which the actual values u
probability -
-BW/m T0)
Y
log (3 /) log) *
=
0 1+
B
=
=
prediction
Lin el tele9
,
M
In
,
Unt
_
log (1 + 4 M) :
hi
Yo
.
6
open / Mobability 1-2 .
er ror
-
#x
/sli In the
↳ 4 4
When in 4
my 1 . by B% >
-
+
Clarial linear regression Model Assumption 4
.
