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 ^ - 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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