ECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors

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1 ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur

2 Asymptotic theory The asymptotic properties of a estimator cocers the properties of the estimator whe sample size grows large. For the eed ad uderstadig of asymptotic theory, we cosider a example. Cosider the simple liear regressio model with oe explaatory variable ad observatios as yi = 0 + xi + εi, E ( εi) = 0, Var ( εi) =, i =,,...,. The OLSE of is ( xi x)( yi y) i= b = x x ad its variace is i= If the sample size grows large, the the variace of b gets smaller. The shrikage i variace implies that as sample size icreases, the probability desity of OLSE b collapses aroud its mea because Var(b) becomes zero. Let there are three OLSEs, ad which are based o sample sizes, ad respectively such that, 3 < < say. If c ad δ ( ) Var ( b ) =. are some arbitrarily chose positive costats, the the probability that the value of b lies withi the iterval ca be made to be greater tha δ for a large value of. This property is the cosistecy of b which esure that eve if the sample is very large, the we ca be cofidet with high probability that b will yield a estimate that is close to. i b b b 3 3 ± c ( )

3 3 Probability i limit Let be a estimator of based o a sample of size. Let be ay small positive costat. The for large, the requiremet that b takes values with probability almost oe i a arbitrary small eighborhood of the true parameter value ˆ is which is deoted as lim P ˆ < γ = γ plim ˆ = ˆ ˆ ad it is said that coverges to i probability. The estimator is said to be a cosistet estimator of. A sufficiet but ot ecessary coditio for to be a cosistet estimator of is that ˆ ad lim E ˆ = lim Var ˆ = 0.

4 4 Cosistecy of estimators Now we look at the cosistecy of the estimators of (i) Cosistecy of b Uder the assumptio that have lim Vb ( ) lim This implies that OLSE coverges to X ' X lim = X ' X = = lim = 0. Thus OLSE is a cosistet estimator of ad exists as a ostochastic ad osigular matrix (with fiite elemets), we i quadratic mea... This holds true for maximum likelihood estimators also. Same coclusio ca also be proved usig the cocept of covergece i probability. The cosistecy of OLSE ca be obtaied uder the followig weaker assumptios: ' (i) plim X X = exists ad is a osigular ad ostochastic matrix. (ii) X ' ε plim = 0.

5 5 Sice b = ( X ' X) X ' ε So X ' X = X ' ε. X ' X X ' ε plim( b ) = plim plim =.0 = 0. Thus b is a cosistet estimator of. The same is true for maximum likelihood estimator also.

6 6 (ii) Cosistecy of s Now we look at the cosistecy of s as a estimate of s = ee ' k = ε' Hε k k ' ' ( ' ) X X X X ' = εε ε ε k εε ' ε' X X ' X X ' ε =. εε '. We have Note that cosists of terms εi ad { εi, i =,,..., } is a sequece of idepedetly ad idetically i= distributed radom variables with mea. Usig the law of large umbers εε ' plim = ε' X X ' X X ' ε ε'x X ' X X ' ε plim = plim plim plim = 0..0 = 0 plim( s ) = ( 0) 0 =. Thus s is a cosistet estimator of. The same hold true for maximum likelihood estimators also.

7 7 Asymptotic distributios Suppose we have a sequece of radom variables { } { } α F α α with a correspodig sequece of cumulative desity fuctios for a radom variable with cumulative desity fuctio F. The coverges i distributio to if F coverges to F poit wise. I this case, F is called the asymptotic distributio of. α α Note that sice covergece i probability implies the covergece i distributio, so ( Note that plim D α = α α α α ted to α i distributio), i.e., the asymptotic distributio of α is F which is the distributio of α. E ( α ) : Var( α ): lim E( α ) : Mea of asymptotic distributio Variace of asymptotic distributio Asymptotic mea lim E α lim E( α ) : Asymptotic variace.

8 8 Asymptotic distributio of sample mea ad least squares estimatio Let α = Y = Yi i= of populatio mea plimy = Y Y be the sample mea based o a sample of size. Sice sample mea is a cosistet estimator, so which is costat. Thus the asymptotic distributio of is the distributio of a costat. This is ot a regular distributio as all the probability mass is cocetrated at oe poit. Thus as sample size icreases, the distributio of Y Y collapses. Suppose cosider oly the oe third observatios i the sample ad fid sample mea as The ad Y Var Y 3 3 = Yi. ( ) E Y i = = Y 9 3 ( ) = Var ( Yi) i= 9 = 3 3 = 0 as.

9 9 plim Y Thus ad has the same degeerate distributio as. Sice Var Y > Var Y, so is preferred over = Y Y. Now we observe the asymptotic behavior of ad Cosider a sequece of radom variables Thus for all, we have α α Y = ( Y Y) = ( Y Y ) ( α) ( ) ( α) ( ) E = EY Y = 0 E = E Y Y = 0 ( ) ( ) Var α = E Y Y = = 3 Var ( α) = E ( Y Y ) = = 3. Assumig the populatio to be ormal, the asymptotic distributio of ( ) ( 0,3 ) Y is N 0, Y is N Y. Y ( ) ( ) α Y { }. Y

10 0 Y So ow is preferable over. The cetral limit theorem ca be used to show that will have a asymptotically ormal distributio eve if the populatio is ot ormally distributed. Also, sice Z = Y ( ) ~ ( 0, ) ( Y Y) Y Y N ( ) ~ N 0, ad this statemet holds true i fiite sample as well as asymptotic distributios. Cosider the ordiary least squares estimate b= X ' X X ' y of i liear regressio model. If X is ostochastic the the fiite covariace matrix of b is ( ) α y = X + ε Vb ( ) = ( X' X). The asymptotic covariace matrix of b uder the assumptio that by X ' X lim ( X ' X) = lim lim lim X ' X = Σ xx exists ad is osigular. It is give =.0. Σ xx = 0 which is a ull matrix.

11 Cosider the asymptotic distributio of. The eve if is ot ecessarily ormally distributed, the asymptotically ( ) ~ N( 0, Σxx ) b ( ) ' Σ ( b ) b xx ~ k. ( ) b X ' X If is cosidered as a estimator of Σ xx, the χ ε X ' X ( ) ' ( b ) ( b ) ' X ' X ( b ) b is the usual test statistic as is i the case of fiite samples with = ( ( ) ) b~ N, X ' X.

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