FOrmUlaS:
HANdOUt I
since
Its mean is also a random
E(X) Mx
105):Var(x)
0
If the observation
sample
COV(d +
is an
+
b284
bx
+
MY
+
cV,X) b.COV(X,X) c.COV(V,X)
+
=
E(XX) COV(X, y) E(X) E(Y) Oxy+MxMy
distribution of the
aid, then the
bMx cuy
+
+
n
=
>
a
+
=
=
>
+
(ax+by):aevar (x) b2 Var(y) 2abCOv(X,y)
E(y2) 8z
(X) ox
bEX CEY
+
=
var
error
a
=
(a + by) b2Var(X)
var
mean
107):standard
=
+
+
variable, thus there is also
point estimate for the sample
a
E(a bX cY)
the observed data are random variables,
=
+
=
X-NCMn,of)
- EX(2
Var(X) E(X
mean:
=
EX - (EX)2
=
approximationofdistribution (asymptotic approximation)
M
relies on the value
sample mean (x):n
of a being large in -> as
law oflarge number (LIN):asncs, the sample mean
populationmean
theorem
Central limit
(CLT):when nas, then the distribution
·
probability to the
sample variance
(SR).n
·
Sample
correlation
-
x)2
"(Xi y)(yi y)2
.
-
-
1
-
i1
=
SxSy
sample standard deviation (5x):sq
average ECD) Mx
=
population
"
(E(X)orMus:
mean
yip;
i1
=
efficient:var(x)<var (xi), then x
is more
population
eff.
d, N10,1),
W-E(W)
normal:If
(xi
i1
=
(rxy):SxY
a
consistent:true on value, as nc, the distribution will be more clustered
asymptotically
M
-
n
around the true value
·
1
sample covariance (Sxx):
is a normal
AMRas re
estimators
·unblased:true on
in
(xP>Ma)
distribution NCO, 1)
properties of
converges
xi
=
wisthe estimator
variance
(8):
n
population cov(Oxx)
var(W)
(xi
-
x)(xi y
h,y(x,
=
x)2
-
care more about
consistency than blas
HandOUT 2
OLS assumptions
regression
univariate
population regression line Elyx) p0+B1X when ElUilxi) 0
unbiased, consistent, asymptoticallynormal
=
=
>
the
actual observation
Ui
+
·
5 B" xi
sample regression line:
where the
is scattered around vi=Bo B,Xi
+
B and B is estimated using the
a minimum squared
OLS
cordinary
prediction
leastsquares)
error
min
·no extreme
u
Total sum of squares (TSS):(yi -7)
var (xi) <c
BY
2
(Yi-T)"
Estimated sum of squares (ESS):
Residual sum of squares (SSR)
R
outliers OE(Xi)<0, 0E1Ui) <a
OLS estimator for
Goodness of fit
2
=
(xi,yi) are iid (SRS)
+
=
ECUi1xi) 0 (correct specifications)
(yi-y)"
=
estimated sum of squares
=
or
total sum of squares
explained
variations
unexplained
variations
·
R
COV(X,Y) SAY
=
2
=
and
B TN
=
rxy2
=
1. Residual sum of squares
total sum of squares
HandOUT3
A symptoticallynormal
HomOSkedasticity
U(B-B1)
"tendency
>
N
0,002
BY B,d>N(0, 1)
-
Ov
mOn3
SV var(V)
=
-Mx).U
V (X
SECBY)
NOTE:
the standard error approaches of
more
33
where
=
V
variation
in reduces the
Heteroskedasticity
scatter"
·homoskedasticity:varcuiixi):ou
8n4
equivalent to
to
Us
constant)
If it is homoskedastic,
OLSIS OBLUE]
·heteroskedastic:Var(uilxi):f(xi)
as n
<
variation
in
B
homoskedasticity-only formula
asymptotic
variance of
B
estimated standard error
2
Op
BY
of
Orp
=
8B
S
=
2
nSn
population regression
B1FD
EXAMPle: HO: B1 = b, H1 :
E-values
&
and t-test.
·after
finding urt-value, you
If the It Evalue of (x(), where X: significance level
generalidea, but
then
we use z-value when
Ifurt-value
In
BY-b
SE(BY
'I
I
BO
+
B1X1,+...
+
ElyiXi, X2
BkXki
Y i = BO
BiX1i+...
+
+
BkXki + Ui
sample regression line
lies In
the reasnaded
=
line:
Population regression model
2 bc. It's a two talled test
hull
Visualization:
t-test when n<30
t-statIStIc:
you reject
Y
would want to compare it
when to use which? both has the same
n > 30 and we use
regression
multivariate
Hypothesis testing guide
Yi B0 +BYX1i+... BYXki
area
+
=
rejecthulihypothesis
Eluilxii...., xki)
0
=
z.VAIUC
of
& confidence
&
Acceptance
intervals
estimator
Confidence level. SE
tested value
BY = 1.96.SE(BY
Reject
b I
null
&
p-value
region
confidence (v1.SE. after getting the +-value, find the probability that
1.96.SE(BY
of the shaded
+ 951. confidence
(a
95% confidence level)
& at
4/2
hypothesis if bisoutside
-
the confidence interval
level)
by 2bc. two-talled
PY is outside the
Reject null If
(using the E-table) and multiply
area
test.
then
r
,1
acceptance region
Compare with
x.
Ifp(X,
rejectnull
Handout 4
·conditions for omitted variable bids:
&
the
variables (included and omitted) must be correlated
②
the
omitted variable should have
the omitted
regressors
effect on the dependent
variable
(B270)
OVB,
such that when there is an
·
an
[Ox,Xz70]
are absorbed
by the error term
and the assumption that
E(UiIXi) tO.
·
OLS estimator
↑
&I
is
based and Inconsistent, no matter the size of the
8x12
-
inconsistent
If
blasterm
OX1X2
because it depends
0x1x2
on the value
highly
If high
Is
the
of BC
negative then
a
No
or
under
perfect collinearity
regressors
Rescaling
vice versed
collinear but not
perfectly
collinear
it
Inigh R-value,
How to solve for
&
impose
is
(rxxk 0.8,
will be difficult to
then check for
effect but not individual effect, then
=
BO
+
Y i = BO
+
B1X1,+...
line:
+
If
BiX1i+...
+
BkXki + Ui
Yi B0 +BYX1i+... BYXki
+
=
...,
=
a
constant, then slope
+
E
+(()(kX)
cx
+
C
ushifty, they-intercept shifts
Goodness of fit pt.2
Adjusted R2
Xki) 0
=
=
factor adjustment
1
=
-
n
n
ElyiXi, X2i,Xki)
BkXki
ou
If Uscale Y,
(Y
R
regression
sample regression line
x1i,
by
m
population regression model
E(Ui
multicollinearity
·
nothing!
Y
·
multicollinearity
population regression
=
no.
-
-
1.SSR 1=
k -1
each other
shifting
(,)(kX)
a+
restrictions
multivariate
X (i)
maintaining equality
② Or if you only wish to find out the overall
do
.
=
few significant t-ratios
·If sample correlation
. . .
of
y x + BX+
(Imperfect multicollenearity)
separate the Individual effects
·
and
If you scale
multicollinearity, then
degree of imperfect
are not linear combinations
Homoskedasticity Var(Uilxi,
estimated the impact of BI
8x,2
>If
either the value
Additional OLS assumption for multivariate
-
=
Ox
OLS estimator
Is
cx B1 + B2Ox142,
O,B1 + B20x1x2
w
n
TSS
of regressors
2
S4
2
Sy
=k