ECONOMETRICS I
ECON GR5411
Lecture 15 – Finishing Restricted Least
Squares and Generalized Regression
Model
by
Seyhan Erden
Columbia University
mammogram
mam ogram
Hypothesis Testing Example 2:
. reg testscr str el_pct comp_stu
Source | SS df MS Number of obs = 420
-------------+---------------------------------- F(3, 416) = 106.29
Model | 66004.0238 3 22001.3413 Prob > F = 0.0000
Residual | 86105.5698 416 206.984543 R-squared = 0.4339
-------------+---------------------------------- Adj R-squared = 0.4298
Total | 152109.594 419 363.030056 Root MSE = 14.387
------------------------------------------------------------------------------
testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
str | -.8489998 .3932246 -2.16 0.031 -1.621955 -.0760449
el_pct | -.6303601 .039997 -15.76 0.000 -.7089814 -.5517387
comp_stu | 27.26961 11.62113 2.35 0.019 4.426158 50.11307
_cons | 677.0642 8.303396 81.54 0.000 660.7424 693.3861
------------------------------------------------------------------------------
Lecture 13 GR5411 by Seyhan Erden 211/4/20 mammograms
Finite Sample Properties of Restricted LS:
Under the linear model ! = #$ + &
with usual assumptions,' &|# = 0*+, &|# = ' &&-|# = ./012 and !2 are iid and their 4th moments are finite.' #-# = 344 is positive definite.
11/4/20 3Lecture 15 GR5411 by Seyhan Erdenmammograms
First some useful properties:
Let 5 = #-# 678 8- #-# 678 678- #-# 67
Then,1. 8- ;$ − , = 8- #-# 67#-&2. ;$> − $ = #-# 67#- − 5#-3. ̂&> = 0 − A + #5#′ &4. 0 − A + #5# is symmetric and idempotent5. E, 0 − A + #5# = F − G + H
11/4/20 4Lecture 15 GR5411 by Seyhan Erdenmammogr ms
Proof of these useful properties:1. 8- ;$ − , = 8- #-# 67#-&8- ;$ − , = 8- #-# 67#-! − ,= 8- #-# 67#- #$ + & − ,= 8′$ + 8- #-# 67#-& − ,= 8- #-# 67#-&
11/4/20 5Lecture 15 GR5411 by Seyhan Erdenmammograms
Proof of these useful properties:2. ;$> − $ = #-# 67#- − 5#- &;$> = ;$ − #-# 678 8- #-# 678 67 8- ;$ − ,= $ + #-# 67#-& − #-# 678 8- #-# 678 678- $ + #-# 67#-& − ,= $ + #-# 67#-& − #-# 678 8- #-# 678 678-$ + 8- #-# 67#-& − ,= $ + ( #-# 67#- − #-# 678 8- #-# 678 678- #-# 67#-)&= $ + #-# 67#- − 5#- &
11/4/20 6Lecture 15 GR5411 by Seyhan Erdenmammograms
Proof of these useful properties:3. ̂&> = 0 − A + #5#′ &̂&> = ! − # ;$>= ! − # $ + #-# 67#- − 5#- &= #$ + & − #$ − # #-# 67#- − 5#- &= & − # #-# 67#-& + #5#-&= 0 − A + #5#′ &
11/4/20 7Lecture 15 GR5411 by Seyhan Erdenmammograms
Proof of these useful properties:4. 0 − A + #5#′ is symmetric and idempotent0 − A + #5#′ 0 − A + #5#′= 0 − A + #5#′ − A + A − A#5#′+#5#′ − A#5#′ + #5#′#5#= 0 − A + #5#′
Since, A#5#- = # #-# 67#-#5#- = #5#′ and the last
term:#5#-#5# = # #-# 678 8- #-# 678 678- #-# 67#-# #-# 678 8- #-# 678 678- #-# 67#-= # #-# 678 8- #-# 678 678-#-# 678 8- #-# 678 678- #-# 67#-= #5#- 8Lecture 15 GR5411 by Seyhan Erdenmammograms
Proof of these useful properties:5. E, 0 − A + #5#′ = F − G + HE, A = E, # #-# 67#′ = E, #-# 67#-# = GE, #5#-= E, # #-# 678 8- #-# 678 678- #-# 67 #-= E, 8- #-# 678 678- #-# 67 #-# #-# 678= E, 8- #-# 678 678- #-# 678 = H
Thus,E, 0 − A + #5#′ = F − G + H
11/4/20 9Lecture 15 GR5411 by Seyhan Erdenmammograms
Finite Sample Properties of Restricted LS:
From property 2;$> = $ + #-# 67#- − 5#- &
Hence,' ;$>|# = ' $ + #-# 67#- − 5#- & |#= $ + ' ' #-# 67#- − 5#- &|#= $ + #-# 67#- − 5#- ' &|#= $
Since ' &|# = 0
11/4/20 10Lecture 15 GR5411 by Seyhan Erdenmammograms
Finite Sample Properties of Restricted LS:
From property 2 ;$> − $ = #-# 67#- − 5#- &
Hence,*+, ;$>|# = ' ;$> − $ ;$> − $ -|#= ' #-# 67#- − 5#- & #-# 67#- − 5#- & - |#= ' #-# 67#- − 5#- &&- #-# 67#- − 5#- -|#= #-# 67#- − 5#- ' &&-|# #-# 67#- − 5#- -= ./ #-# 67 − #-# 67#-#5 − 5#-# #-# 67 + 5#-#5= ./ #-# 67 − 5
Since ' &&-|# = ./0 and 5#-#5 = 5
11/4/20 11Lecture 15 GR5411 by Seyhan Erdenmammogr ms
Finite Sample Properties of Restricted LS:*+, ;$>|# = ./ #-# 67 − 5
can be estimated by K*+, ;$>|# = L>/ #-# 67 − 5
where L>/ = 1F − G + HM2N7O ̂&>/ = 1F − G + H ̂&>- ̂&>
11/4/20 12Lecture 15 GR5411 by Seyhan Erdenmammograms
Finite Sample Properties of Restricted LS:L>/ = 1F − G + HM2N7O ̂&>/ = 1F − G + H ̂&>- ̂&>
Note that, from property 3 above, we havê&> = 0 − A + #5#′ &
Then, ̂&>- ̂&> = &- 0 − A + #5#- - 0 − A + #5#′ &
From property 4, we know the term in parenthesis is
idempotent, then̂&>- ̂&> = &- 0 − A + #5#′ &
11/4/20 13Lecture 15 GR5411 by Seyhan Erdenmammograms
Is P>/ unbiased estimator of ./? ' P>/|# = ' 1F − G + H ̂&>- ̂&>|#= 1F − G + H ' E, &- 0 − A + #5#′ &= 1F − G + H ' E, 0 − A + #5#′ &&- |#= F − G + HF − G + H ' &&-|#= ./
Since, ̂&>- ̂&> = &- 0 − A + #5#′ &
And E, 0 − A + #5#′ = F − G + H from property 5.
11/4/20 14Lecture 15 GR5411 by Seyhan Erden
mammograms
Distributional Properties:
By linearity of property 2 above, conditional on #, ;$> − $ is normal. Given the mean and the variance
above, we deduce,;$> ~ R $, ./ #-# 67 − 5
We know that ̂&> = 0 − A + #5#′ & is linear in &,
so is also conditionally normal.
Since 0 − A + #5#′ #-# 67#- − 5#- - = 0, ̂&>
and ;$> are uncorrelated and thus independent. Thus, P>/ and ;$> are independent.
11/4/20 15Lecture 15 GR5411 by Seyhan Erdenmammograms
Distributional Properties:
Since, ̂&>- ̂&> = &- 0 − A + #5#′ & and the fact that 0 − A + #5#′ is idempotent with rank (F − G + H), it
follows (F − G + H)P>/./ ~ TO6UVW/E = ;$W − $WPX ;$W ~ R(0,1)TO6UVW/ /(F − G + H) ~ EO6UVW
Since there are H restrictions, there are G − H free parameters
instead of G Estimating a model with G coefficients and H
restrictions is equivalent to estimation with G − H
coefficients
16Lecture 15 GR5411 by Seyhan Erdenmammograms
Is ;$> more efficient?
An interesting relationship under homoscedastic
regression model:Z[\ ;$>, ;$ − ;$> = ' ;$ − ;$> ;$> − $ - = 0
You will show in the problem set, it is easy using
properties 2 and the fact that 5#-#5 = 5.
One corollary is ][\ ;$>, ;$ = \+, ;$>
Second corollary is\+, ;$ − ;$> = \+, ;$ + \+, ;$> − 2][\ ;$, ;$>= \+, ;$ − \+, ;$>
11/4/20 17Lecture 15 GR5411 by Seyhan Erdenmammograms
Hausman Equality:
Note that the Second corollary is known as Hausman
Equality\+, ;$ − ;$> = \+, ;$ − \+, ;$>
This will appear again in GLS and IV estimators this
semester.
The expression says that the variance of the
difference between the estimators is equal to the
difference between variances.
It occurs (generally) when we are comparing an
efficient and an inefficient estimator.
11/4/20 18Lecture 15 GR5411 by Seyhan Erden
]
mammogram
Is ;$> more efficient?\+, ;$ − ;$> = \+, ;$ − \+, ;$>= ./ #-# 67 − ./ #-# 67 − 5= ./5
Recall that A is a positive semi definite matrix.
11/4/20 19Lecture 15 GR5411 by Seyhan Erdenmammograms
Is ;$> more efficient?\+, ;$ − \+, ;$>= ./ #-# 678 8- #-# 678 678- #-# 67 ≥ 0
Hence, \+, ;$ ≥ \+, ;$>
in the positive definite sense. Thus Restricted LS is
more efficient than OLS estimator (in the linear
homoscedastic model)
11/4/20 20Lecture 15 GR5411 by Seyhan Erdenmammograms
Why do we need Generalized Linear
Regression Model?
The assumption of i.i.d. sampling fits many applications.
For example, ! and # may contain information about
individuals, such as wages, education, personal
characteristics. If individuals are selected by simple random
sampling ! and # will be i.i.d.
Because #2, !2 and #W, !W are independently distributed
for _ ≠ H and &2 and &W are independently distributed for _ ≠H.
In the context of Gauss – Markov assumptions, the
assumption ' &&-|# is diagonal therefore is appropriate if
the data is collected in a way that makes the observations
independently distributed.
11/4/20 21Lecture 15 GR5411 by Seyhan Erdenmammograms
Some sampling schemes encountered in econometrics do
not, however, result in independent observations and
instead can lead to error terms that are correlated. The
leading example is time series data.
The presence of correlated errors creates two problems
for inference based on OLS.
1. Neither the heteroskedasticity-robust nor
homoskedasticity-only standard errors produced by
OLS provide valid basis for inference.
2. If the error term is correlated across observations,
then ' &&-|# is not diagonal, hence ' &&-|# ≠./0, thus OLS is not BLUE.
11/4/20 22Lecture 15 GR5411 by Seyhan Erdenmammograms
In this lecture we will study an estimator, generalized
least squares (GLS), that is BLUE (at least
asymptotically) when conditional covariance matrix of
errors is no longer proportional to the identity matrix
(errors are non-spherical, ' &&-|# ≠ ./0).
A special case of GLS is weighted least squares (WLS)
in which the conditional covariance matrix of errors is
diagonal and the _ab diagonal element is a function of #2.
Like WLS, GLS transforms the regression model so that
the errors of the transformed model satisfy Gauss-
Markov conditions. The GLS estimator is the OLS
estimator of the transformed model.
11/4/20 23Lecture 15 GR5411 by Seyhan Erden
^
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mammogram
Recall Gauss-Markov Conditions for
Multiple Regression:1. ' &|# = 0(F×F d+E,_1 [e fX,[P)2. ' &&′|# = ./0(F×F d+E,_1g_Eℎ ./[F EℎX i_+j[F+kP)3. # has a full column rank
Recall that under these conditions OLS is BLUE
11/4/20 24Lecture 15 GR5411 by Seyhan Erdenmammogr ms
Generalized Linear Regression Model:
Recall that when &|# is not spherical, the model
is ! = #$ + &' &|# = 0*+, &|# = ' &&′|# = Ω
where Ω is an F×F positive definite matrix that
can depend on X. When errors are spherical we
have the special case that Ω = ./0
Two leading cases for Ω ≠ ./0 are
heteroskedasticity and autocorrelation.
11/4/20 25Lecture 15 GR5411 by Seyhan Erdenmammograms
Generalized Linear Regression Model:
Heteroskedasticity arises when errors do not have the
same variances. This can happen with cross section as
well as with time series data. For example volatile high-
frequency data such as daily observations of financial
market and in cross section data where the scale of
observations depend on the level of the regressor.
Disturbances are still assumed to be uncorrelated across
observations under heteroskedasticity so Ω would be
Ω = .7/ 0 00 .// …⋮0 ⋮0 ⋱0
00⋮.O/
11/4/20 26Lecture 15 GR5411 by Seyhan Erdenmammograms
Generalized Linear Regression Model:
Autocorrelation is more of a time-series data
issue, let’s how Ω would look like if we have the
following auto correlation: Let’s say &7 = \7
But thereafter, the errors follow an AR(1) model:&a = p&a67 + \a
where \a is i.i.d. with mean zero and variance 1.
Hence, &a = p p&a6/ + \a67 + \a= p/&a6/ + p\a67 + \a
11/4/20 27Lecture 15 GR5411 by Seyhan Erdenmammograms
Generalized Linear Regression Model:= p/ p&a6q + \a6/ + p\a67 + \a= pq&a6q + p/\a6/ + p\a67 + \a= pq p&a6r + \a6q + p/\a6/ + p\a67 + \a= pr&a6r + pq\a6q + p/\a6/ + p\a67 + \a= ⋯ .= pa67\7 + pa6/\/ + pa6q\q + ⋯+ p/\a6/+p\a67 + \a=M2N7a pa62 \2
11/4/20 28Lecture 15 GR5411 by Seyhan Erdenmammogr ms

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