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Econometric Methods
Reviewing Assumptions
Assumption 1: “Correctly specified, linear model”
Dummy variables
o Case 1: Binary, model specified as: yi = μ +δ Di + ε i , (Di = 1 if female)
female male
male
Alternately: yi =μ fDi +μ fHi + ε i
o Case 2: Several Categories
summer fall i
i i spring
D D
c y D
2 3
1 where:
fall w er
sum w er
spring w er
w er
c c
c c
c c
c
int
int
int
int
3
2
1
1
Alternately: fall w er i
i i spring summer
D D
c y D D
3 3 int
1 2 where:
4
3
2
1
4
3
2
1
c
c
c
c
o Case 3: (many categories, many values, just examined) o Case 4: Threshold Effects If wanted to measure effect of increasing levels of, say, education, one couldn’t set up a variable where Ei = 1 if high school, 2 if bachelors, 3 if masters, 4 if PhD, etc. since this assumes that each ‘jump’ (1 to 2, 2 to 3) has equal ‘value.’
Instead, use dummy variables: = α +β +∑(δ )+ε
3
1
i
income age i Ei
(notice that dummy variable trap is avoided by summing through 3, not 4 (which is the total number of categories) o Case 5: Interaction Terms
Suppose have a model: income = α +β( age )+γ( gender )+ ε
- ( ) ( ) ( ) ( ) 1 E income age age Di =α +β +γ=α+γ + β =
( ) = + =
0 E income age Di
Allow for interaction:
income = α +β( age )+γ( Di )+δ( Di ∗ age )+ ε
- E ( income 1 ) ( age ) ( age ) ( ) ( ) ( age ) Di
= =α^ +β +γ+δ =α+γ +δ+^ β
E incomeDi = 0 =α + β age
- Therefore, interaction terms help change slopes
Structural Change
o We generally assume that β is the same for all Y. However, it may look as follows:
o H 0 : Rβ=q (wil tell us if β applies to all y)
Unrestricted model : (^)
×
×
×
×
×
×
×
×
21
11
21
11
21
11
21
11
2
1
2
1
2
1
2
1 0
n
n
n
n
n
n
n
n X
X
y
y
- Yields (^1111) 1 b 1 (^) = ( X 1 ′ X 1 ) X ′ Y → e ′ e − and
2 2 2 2
1 b 2 (^) = ( X 2 ′ X 2 ) X ′ Y → e ′ e − and the total residual residual
sum of squares: e ′ e = e 1 ′ e 1 + e 2 ′ e 2
Restricted Model
- If the restriction is β 1 =β 2 we can impose k restrictions since there are 2k parameters (doesn’t violate assumption that
J<K):
1 2
2
1 1
1
1 2 k k
o [ ]
R k Ik Ik k
o
1 2
2
1 1
1
k k
R q
- In restricted scenario only have one set of β:
×
×
×
×
×
×
21
11
21
11
21
11 2
1
2
1
2
1
n
n
n
n
n
n X
X
y
y
( )
[ e e ( n n k )]
ee ee k F (^) k n n k 1 2 2
** ( , 1 22 ) ′ + −
x
y
o ~ ( 0 , 1 ) 2 ˆ
N
w w
T
rk
r t ∑ =+
o (^) ∑ =+
T
rk
wr w T k 1
2 ( ) ( 1 )
σˆ and (^) ∑ − =+
T
rk
wr T k
w ( ) 2
Specification Analysis
o Omitting relevant variables Biased parameters Variance is smaller than true model, therefore get higher t-ratios s 2 is a biased estimator of σ 2
o Including an irrelevant variable Unbiased parameters Variance is greater than true variance s 2 is an unbiased estimator of σ 2
Model Building
o R2 or adjusted R2 (slowly add variables to increase it)
o Akaike Information Criterion
kn AIC k sy R e
2 2 2 / ( )= ( 1 − ) (choose the lower
AIC)
o Bayesian Information Criterion:
kn BIC k sy R e
2 2 / ( )= ( 1 − )
Non-Nested or Competing Models
o Macroeconomics makes use of this
o H 0 : y = X β +ε x H 1 : y = Z γ+ ε z
o Encompassing Test
(Look for variables in common) W = X ∩ Z , X and Z are those remaining in each model
y = X β + Z γ+ w δ+ ε test if β = 0 or Z = 0
o J-Test y = ( 1 −α ) X β+α Z γ+ ε
Regress Y on Z, get γˆ
Construct Z γˆ
Run y = ( 1 −α ) X β+α( Z γˆ)+ε, ( )
N
SE α
o Cox Test This was not discussed in class. (There’s a complex discussion in the book.)
w
t
time
breaks Stable model
unstable model
2
Assumption 2: Matrix X, has rank K
Multicollinearity o Two cases: Perfect multcollinearity, solution: drop variable (if possible) Near or high multicollinearity o Detect with:
Variance Inflation Factor: ( ) 2 1
− R k
(15-20 is a large number)
Characteristic roots: SCR
LCR
(if > 20, then problems) where LCR
and SCR are the largest and smallest characteristic roots. o Fix: Remove observation (however, not always possible due to theory, etc.) Missing observations o Ignorable – data are unavailable for unknown reasons Case A: YA, XA nA observations on X and Y are available Case B: — , XB nB observations missing on Y Case C: YC, — nC observations missing on X
- Zero Order Regression: use X to replace XC o bLS will not change (recall that for simple regression:
∑
∑
=
=
A
A
n
i
i
n
i
i i
x x
y y x x
b
1
2
1
so when we add an observation ( X ) it makes no addition since: ( x − x ) = 0 ).
o R
2 will be lower:
(recall that: YMY
ee R 0
2 1 ′
= − , additional yi adds other
value)
- Modified Zero Order Regression: fill missing spaces with zeros and add a dummy variable that takes on value of 1 for missing values (algebraically identical to filling the gaps with X. o bLS will change o R
2 will change o Systematic – when there is a reason (sample selection bias issue) Outliers or Influential Observations
Let Xn and Yn be matrices whose elements are r.v. and plim Xn = A, plim Yn = B:
plim X
n = A -
- plim (Xn∙Yn) = A∙B (if conformable) o Convergence in Distribution xn converges in distribution to x with a CDF F(x) if:
lim (^) n → ∞ F ( xn )− F ( x )= 0 at all continuity points of F(), notation:
x x
d n →
Rules (let x x
d n →^ and^ p^ lim^ yn = c )
d n n →
d n + n → +
d n /^ n →^ / if c ≠ 0
- Let g(xn) be a continuous function: g ( x ) g ( ) x
d n →
- If p lim( xn − yn ) = 0 then xn has the same limiting
distribution as yn
Least Squares:
o bLS is a consistent estimator of β, plim b = β
Use assumption that 1
1 lim −
−
=
′^
Q
n
XX
p
−
n
X
n
XX
b
1 therefore
−
−
n
X
Q p n
X
n
XX
p b p
lim limβ lim
1
1
lim (^) = 0
n
X
p
somehow… using convergence in mean square:
n
X
E
, lim (^) n →∞ 0 = 0
XX
n
E X X
n n
X
Var
′^
2
2
, and
lim 0 ( ) 0
2 = =
→ ∞ Q
n
XX
n
n
σ
o Therefore p lim b = β
o B has a limiting distribution that is normal o Stabilizing transformation:
This, n ( b −β), converges in distribution to a normal distrbituion
Lindberg-Levy Univariate Central Limit Theorem
- Let x 1 , …, xn are a random sample from a probability distribution with finite mean μ and variance σ 2 and
n
i
i x n
x
=
∑ , then ( ) ( 0 , ) 2
n b β N σ
d − → (this is the
distribution of the statistic)
−
−
n
X
p nb Q p
n
X
n
XX
nb
β , lim lim
1
1
here we need to show that the latter term is distributed normally since Q is a constant:
o (^) = 0
n
X
E
, by assumptions 3 and 5
o ( ) n
XX
E X X
n n
X
Var
′^12
therefore
Q
n
XX
p 2 2
limσ = σ
o Therefore, ( 0 , ) 2 N Q n
X d
o And,
( )
( )
( ( ) ) 2 1
2 1
2 1
1 1 2 1
−
−
−
− − −
b N X X
n
XX
n
b N
nb N Q
nb N Q Q QQ
d
d
d
d
o Implications for t and f-tests
T-test:
p lim s = σ
[ ]
[ ]
2 2 2
1 2
1
2 1
lim lim
∑
−
−
−
i
i n
p s p
n
X
n
XX
n
X
n k n
n s
X XX X
n k
I X XX X
n k
M
n k n k
ee s
- In large sample context t-statistic becomes z-statistic (NOTES LESS FROM NOW ON) F-Test:
( )
2 1 1 ˆ (^) ( ) ( ) − − ′ −
= ′ = XX
n k
ee
Var β OLS s XX
( ) 2 1 1 ˆ (^) ˆ ( ) ( ) − − ′
= ′ = XX
n
ee
Var β MLE σ XX
- So, in small sample context they are different, but in large sample context they are the same.
Generalized Regression Model: Y = X β + ε, E ( ε | X )= 0 , = Ω
2
Var (ε | X ) σ
o Heteroskedasticity:
= Ω× =
2
2 11 11 2 2
nn nn
Var X nn
Often found in cross-section data, also in high frequency o Autocorrelation (memory, persistence)
−
−
×
1
2 1
1 1
1 2 1
2 2
n
n
Var X nn
o Least Squares in General Context:
( ) ( )
2 1 1 ( | )
− −
Var bLS X = σ X ′ X X ′Ω X X ′ X
Not efficient estimator
Assume
lim Q n
X X
p (^) =
′Ω^
, then p lim bLS = β, therefore
consistent Also asymptotically normal:
− 1 − 1
2 , lim Q n
X X
Q p n
b N
d LS
β CAN BE SHOWN?
MAYBE DO?
However, asymptotically inefficient, does not achieve CRLB o Omega knowledge: Ω is known
- Use spectral decomposition ( Ω = C Λ C ′), transform parameters, and run least-squares on them (Weighted Least Squares).
- Use GLS for smaller samples
- Use MLE for larger samples Ω is unknown but its structure is known
- E.g. if variance is a function of firm size then can run
regression, forecast and yield (^) Ωˆ^ which can be used:
σ i = f ( firmsize )=α+β si + ν i
2
. If this regression has
spherical disturbances (no autocorr. or heterosked.) then ˆ^2
σ i is a consistent and asymptotically efficient estimator.
Ω is completely unknown
- Since Ω is n x n and it is symmetric it has: 2
n ( n + 1 )
parameters which is greater than n (i.e. it cannot be estimated)
- However, X ′Ω X has (^) n
k k <
parameters, therefore
X ′Ω^ X can be approximated
o Define (^) ∑∑ = =
n
i
n
j
ij xixj n
Q
1 1
σ which approximates
X ′Ω X
o White’s (Heteroskedasticity) Consistent Covariance matrix
defined: (^) ∑
×
n
i
k k eixixi
n
s
1
2 0
p lim s 0 = Q
( ) ( )
1
1
2
2 1 1 1 1 ( )
−
=
−
= ′ ∑ XX n
e xx n
XX
n n
Varb
n
i
LS i i i
using White’s heteroskedasticity consistent covariance matrix… o Newey-West Autocorrelation Consistent Covariance Matrix
( )( )
= + ∑ ∑ ′ + ′ = =+
− − −
L
l
T
tl
wl etetl xtxtl xtlxt n
Q s 1 1
0
where 1
L
l wl , where L is the lag length
o Generalized Least Squares:
Transform data in such a way that assumption 4, Var(ε|X) = σ
2 I, is satisfied Model: Y=Xβ + ε, Var(ε|X) = σ
2 Ω, transform data into:
Y = X β + ε and Var X I
Spectral Decomposition:
- Consistent
- Asymptotic Normality
- Asymptotically Efficient
2 * *^1211
lim
lim
− − −
−
N X X N X X
X X Q
n
X X p n
p
d
β GLS βσ β σ
Since βˆ MLE^ achieves Cramer-Rao Lower Bound, so does βˆ GLS
Good to use in smaller-sample case, MLE for larger samples
o Weighted Least Squares
Size of industry case, divide all observations by size:
nn s^ n
s
2
1
2 11
n
nk n
n n
n
k
s
x s
s s
x
s
x s
s s
x
X PX
1
1
1 1
1 1
11
o What is the structure of Ω? (Case when Ω is unknown but its structure is
known) White’s Test
- When running LS model, e should be a consistent estimator of ε, therefore
- Run regression of
ei = f ( 1 , x , x , x , x , x , x ,( x 1 x 2 ),( x 1 x 3 ),( x 2 x 3 )) + υ i
2 3
2 2
2 1 2 3 1
2
2 = σ
2 , H 1 : not H 0.
2 ~ χ
2 (P), where P is the number of regressors in f( ) above.
- Non-constructive test, because doesn’t tell you how to proceed (i.e. correct problem) Goldfield-Quandt Test
- Sort all data by size in ascending order
- Slice data into two samples, run regression on each slice
- H 0 : σ 1
2 = σ 2
2 , H 1 : not H 0.
1 1
n
sort
n x
x
x
x
, cut in half, sample 1: e 1 ′ e 1 , sample 2: e 2 ′ e 2
11 1
22 2 ( 2 , 1 ) ee n k
ee n k F (^) n kn k ′ −
− − = where^ e^2 ′^ e^2 is on
numerator because expect σ 1
2 < σ 2
2
- Could also split up data into 3 parts and test 1:33 to 66: (ignore middle piece).
- If find that a variable is causing a problem, then know what Ω is and can use weighted least squares. Breusch-Godfrey / Lagrange Multiplier Test
- ( 0 )
2 2
σ i = σ f α − α′ z i , where H 0 : σi
2 = σ
2 , H 1 : not H 0.
- LM = ½ explained sum of squares in the regression of ei ( e e / n ) 2 ′ on zi.
- LM ~ χ 2 (P), o Feasible GLS (a two step estimator)
(^) FGLS ( X X ) X Y
ˆ ˆ−^1 −^1 ˆ−^1
( ) ( )
2 1 1 ˆ (^) ˆ − −
Var β FGLS = σ X ′Ω X
Run regression:
( 50 ) ln ln ln( 50 )
2 2 2 2
e i σ f mupop ei σ λ mupop
λ = ⇔ = + , yields
( )
( )
2
2 1
lnˆ
lnˆ
e n
e
.
Conduct forecast, and if in log, exponentiate and put on diagonals of
Ωˆ o Maximum Likelihood Estimator Likelihood function of all parameters (including Ω) Likelihood ratio test:
2 − 2 ln LR −ln L 0 ~ χ( P ) where P = the number of α’s
Time-Series Model
o yt = d 0 + d 1 xt + ε t , note that (^) { } t Y
t
t
= ∞
= −∞
and that t = 1,…,T which is a “time-
window” o With time-series data we run into two problems Independence Randomness o Properties: Stationarity – expect mean, var, cov to be finite – we know they are well-behaved Ergodicity – (has to do with independence) Martingale Sequences – (fixes randomness problem) o With these three ideas we define Central Limit Theorem based on Martingale Sequences, we continue to maintain E ( ε (^) t | X ) = 0 and
Cov ( ε (^) t , ε t − s ) ≠ 0 where t ≠ s
o Ways to determine autocorrelation / capture covariance: Autoregressive Process:
(^) ∑
P
j
Q T rj 1
2 and
( )
∑
∑
=
=+
− = (^) T
t
t
T
jt
t t j
j e
ee
1
2
1
γ which is
essentially a correlation coefficient between residuals and lag residuals.
2
Q = Tr 1 ~ χ( 1 )
o Durbin-Watson Test
( )
∑
∑
=
=
= T
t
t
T
t
t t
e
e e D
1
2
2
2 1 (notice only one lag here)
T is the number of observations and K is the number of parameters Lower limit: d (^) L ( T , k )
Upper limit: dU ( T , k )
Hypothesis testing:
- If d < dL then reject H
- If d > dU then fail to reject H
- If dL < d < dU then it is inconclusive o Everything done so far relies on assumption that you don’t have a lagged dependent variable (i.e. X’s do not include Yt-1)
If it were included: yt = α +β 1 x 1 t +β 2 yt − 1 + ε t
(where ε t = P ε t − 1 + ut ) and
( )
t t t t t
t t t t t P y x y u
y x y P u
⇒ = − − − −
− −
− −
1 1 1 2 1
11 2 1 1
now ε
is dependent on x, therefore ε is not consistent. Can use the Durbin H test:
• Z
TS
T
h r C
- Where S (^) C 2 → Var ( β^ ˆ 2 )and
( )
∑
∑
=
=
− = (^) T
t
t
T
t
t t
e
ee
r
1
2
2
2 1
Moving Average:
