1. Part 12: Asymptotics for the Regression Model 12-1/39 Econometrics I Professor William Greene Stern School of Business Department of Economics

2. Part 12: Asymptotics for the Regression Model 12-2/39 Econometrics I Part 12 – Asymptotics for the Regression Model

3. Part 12: Asymptotics for the Regression Model 12-3/39 Setting The least squares estimator is (XX) -1 Xy = (XX) -1  i x i y i =  + (XX) -1  i x i ε i So, it is a constant vector plus a sum of random variables. Our ‘finite sample’ results established the behavior of the sum according to the rules of statistics. The question for the present is how does this sum of random variables behave in large samples?

4. Part 12: Asymptotics for the Regression Model 12-4/39 Well Behaved Regressors A crucial assumption: Convergence of the moment matrix XX/n to a positive definite matrix of finite elements, Q What kind of data will satisfy this assumption? What won’t? Does stochastic vs. nonstochastic matter? Various conditions for “well behaved X”

5. Part 12: Asymptotics for the Regression Model 12-5/39 Probability Limit

6. Part 12: Asymptotics for the Regression Model 12-6/39 Mean Square Convergence E[b|X]=β for any X. Var[b|X]  0 for any specific well behaved X b converges in mean square to β

7. Part 12: Asymptotics for the Regression Model 12-7/39 Crucial Assumption of the Model

8. Part 12: Asymptotics for the Regression Model 12-8/39 Consistency of s 2

9. Part 12: Asymptotics for the Regression Model 12-9/39 Asymptotic Distribution

10. Part 12: Asymptotics for the Regression Model 12-10/39 Asymptotics

11. Part 12: Asymptotics for the Regression Model 12-11/39 Asymptotic Distributions  Finding the asymptotic distribution  b  β in probability. How to describe the distribution? Has no ‘limiting’ distribution  Variance  0; it is O(1/n)  Stabilize the variance? Var[  n b] ~ σ 2 Q -1 is O(1)  But, E[  n b]=  n β which diverges  n (b - β)  a random variable with finite mean and variance. (stabilizing transformation) b apx. β +1/  n times that random variable

12. Part 12: Asymptotics for the Regression Model 12-12/39 Limiting Distribution  n (b - β)=  n (X’X) -1 X’ ε =  n (X’X/n) -1 (X’ ε/ n) Limiting behavior is the same as that of  n Q -1 (X’ ε/ n) Q is a fixed matrix. Behavior depends on the random vector  n (X’ ε/ n)

13. Part 12: Asymptotics for the Regression Model 12-13/39 Limiting Normality

14. Part 12: Asymptotics for the Regression Model 12-14/39 Asymptotic Distribution

15. Part 12: Asymptotics for the Regression Model 12-15/39 Asymptotic Properties  Probability Limit and Consistency  Asymptotic Variance  Asymptotic Distribution

16. Part 12: Asymptotics for the Regression Model 12-16/39 Root n Consistency  How ‘fast’ does b  β?  Asy.Var[b] =σ 2 /n Q -1 is O(1/n) Convergence is at the rate of 1/  n  n b has variance of O(1)  Is there any other kind of convergence? x 1,…,x n = a sample from exponential population; min has variance O(1/n 2 ). This is ‘n – convergent’ Certain nonparametric estimators have variances that are O(1/n 2/3 ). Less than root n convergent. Kernel density estimators converge slower than  n

17. Part 12: Asymptotics for the Regression Model 12-17/39 Asymptotic Results  Distribution of b does not depend on normality of ε  Estimator of the asymptotic variance (σ 2 /n)Q -1 is (s 2 /n) (X’X/n) -1. (Degrees of freedom corrections are irrelevant but conventional.)  Slutsky theorem and the delta method apply to functions of b.

18. Part 12: Asymptotics for the Regression Model 12-18/39 Test Statistics We have established the asymptotic distribution of b. We now turn to the construction of test statistics. In particular, we based tests on the Wald statistic F[J,n-K] = (1/J)(Rb - q)’[R s 2 (XX) -1 R] -1 (Rb - q) This is the usual test statistic for testing linear hypotheses in the linear regression model, distributed exactly as F if the disturbances are normally distributed. We now obtain some general results that will let us construct test statistics in more general situations.

19. Part 12: Asymptotics for the Regression Model 12-19/39 Full Rank Quadratic Form A crucial distributional result (exact): If the random vector x has a K-variate normal distribution with mean vector  and covariance matrix , then the random variable W = (x -  )  -1 (x -  ) has a chi-squared distribution with K degrees of freedom. (See Section 5.4.2 in the text.)

20. Part 12: Asymptotics for the Regression Model 12-20/39 Building the Wald Statistic-1 Suppose that the same normal distribution assumptions hold, but instead of the parameter matrix  we do the computation using a matrix S n which has the property plim S n = . The exact chi-squared result no longer holds, but the limiting distribution is the same as if the true  were used.

21. Part 12: Asymptotics for the Regression Model 12-21/39 Building the Wald Statistic-2 Suppose the statistic is computed not with an x that has an exact normal distribution, but with an x n which has a limiting normal distribution, but whose finite sample distribution might be something else. Our earlier results for functions of random variables give us the result (x n -  ) S n -1 (x n -  )   2 [K] (!!!)VVIR! Note that in fact, nothing in this relies on the normal distribution. What we used is consistency of a certain estimator (S n ) and the central limit theorem for x n.

22. Part 12: Asymptotics for the Regression Model 12-22/39 General Result for Wald Distance The Wald distance measure: If plim x n = , x n is asymptotically normally distributed with a mean of  and variance , and if S n is a consistent estimator of , then the Wald statistic, which is a generalized distance measure between x n converges to a chi- squared variate. (x n -  ) S n -1 (x n -  )   2 [K]

23. Part 12: Asymptotics for the Regression Model 12-23/39 The F Statistic An application: (Familiar) Suppose b n is the least squares estimator of  based on a sample of n observations. No assumption of normality of the disturbances or about nonstochastic regressors is made. The standard F statistic for testing the hypothesis H0: R  - q = 0 is F[J, n-K] = [(e*’e* - e’e)/J] / [e’e / (n-K)] where this is built of two sums of squared residuals. The statistic does not have an F distribution. How can we test the hypothesis?

24. Part 12: Asymptotics for the Regression Model 12-24/39 JF is a Wald Statistic F[J,n-K] = (1/J)  (Rb n - q)[R s 2 (XX) -1 R’] -1 (Rb n - q). Write m = (Rb n - q). Under the hypothesis, plim m=0.  n m  N[0, R(  2 /n)Q -1 R’] Estimate the variance with R(s 2 /n)(X’X/n) -1 R’] Then, (  n m )’ [Est.Var(  n m)] -1 (  n m ) fits exactly into the apparatus developed earlier. If plim b n = , plim s 2 =  2, and the other asymptotic results we developed for least squares hold, then JF[J,n-K]   2 [J].

25. Part 12: Asymptotics for the Regression Model 12-25/39 Application: Wald Tests read;nobs=27;nvar=10;names= Year, G, Pg, Y, Pnc, Puc, Ppt, Pd, Pn, Ps $ 1960 129.7.925 6036 1.045.836.810.444.331.302 1961 131.3.914 6113 1.045.869.846.448.335.307 1962 137.1.919 6271 1.041.948.874.457.338.314 1963 141.6.918 6378 1.035.960.885.463.343.320 1964 148.8.914 6727 1.032 1.001.901.470.347.325 1965 155.9.949 7027 1.009.994.919.471.353.332 1966 164.9.970 7280.991.970.952.475.366.342 1967 171.0 1.000 7513 1.000 1.000 1.000.483.375.353 1968 183.4 1.014 7728 1.028 1.028 1.046.501.390.368 1969 195.8 1.047 7891 1.044 1.031 1.127.514.409.386 1970 207.4 1.056 8134 1.076 1.043 1.285.527.427.407 1971 218.3 1.063 8322 1.120 1.102 1.377.547.442.431 1972 226.8 1.076 8562 1.110 1.105 1.434.555.458.451 1973 237.9 1.181 9042 1.111 1.176 1.448.566.497.474 1974 225.8 1.599 8867 1.175 1.226 1.480.604.572.513 1975 232.4 1.708 8944 1.276 1.464 1.586.659.615.556 1976 241.7 1.779 9175 1.357 1.679 1.742.695.638.598 1977 249.2 1.882 9381 1.429 1.828 1.824.727.671.648 1978 261.3 1.963 9735 1.538 1.865 1.878.769.719.698 1979 248.9 2.656 9829 1.660 2.010 2.003.821.800.756 1980 226.8 3.691 9722 1.793 2.081 2.516.892.894.839 1981 225.6 4.109 9769 1.902 2.569 3.120.957.969.926 1982 228.8 3.894 9725 1.976 2.964 3.460 1.000 1.000 1.000 1983 239.6 3.764 9930 2.026 3.297 3.626 1.041 1.021 1.062 1984 244.7 3.707 10421 2.085 3.757 3.852 1.038 1.050 1.117 1985 245.8 3.738 10563 2.152 3.797 4.028 1.045 1.075 1.173 1986 269.4 2.921 10780 2.240 3.632 4.264 1.053 1.069 1.224

26. Part 12: Asymptotics for the Regression Model 12-26/39 Data Setup Create; G=log(G); Pg=log(PG); y=log(y); pnc=log(pnc); puc=log(puc); ppt=log(ppt); pd=log(pd); pn=log(pn); ps=log(ps); t=year-1960$ Namelist;X=one,y,pg,pnc,puc,ppt,pd,pn,ps,t$ Regress;lhs=g;rhs=X$

27. Part 12: Asymptotics for the Regression Model 12-27/39 Regression Model Based on the gasoline data: The regression equation is g =  1 +  2 y +  3 pg +  4 pnc +  5 puc +  6 ppt +  7 pd +  8 pn +  9 ps +  10 t +  All variables are logs of the raw variables, so that coefficients are elasticities. The new variable, t, is a time trend, 0,1,…,26, so that  10 is the autonomous yearly proportional growth in G.

28. Part 12: Asymptotics for the Regression Model 12-28/39 Least Squares Results +----------------------------------------------------+ | Ordinary least squares regression | | LHS=G Mean = 5.308616 | | Standard deviation =.2313508 | | Model size Parameters = 10 | | Degrees of freedom = 17 | | Residuals Sum of squares =.003776938 | | Standard error of e =.01490546 | | Fit R-squared =.9972859 | | Adjusted R-squared =.9958490 | | Model test F[ 9, 17] (prob) = 694.07 (.0000) | | Chi-sq [ 9] (prob) = 159.55 (.0000) | +----------------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant -5.97984140 2.50176400 -2.390.0287 Y 1.39438363.27824509 5.011.0001 9.03448264 PG -.58143705.06111346 -9.514.0000.47679491 PNC -.29476979.25797920 -1.143.2690.28100132 PUC -.20153591.07415599 -2.718.0146.40523616 PPT.08050720.08706712.925.3681.47071442 PD 1.50606609.29745626 5.063.0001 -.44279509 PN.99947385.27032812 3.697.0018 -.58532943 PS -.81789420.46197918 -1.770.0946 -.62272267 T -.01251291.01263559 -.990.3359 13.0000000

29. Part 12: Asymptotics for the Regression Model 12-29/39 Covariance Matrix

30. Part 12: Asymptotics for the Regression Model 12-30/39 Linear Hypothesis H 0 : Aggregate price variables are not significant determinants of gasoline consumption H 0 : β 7 = β 8 = β 9 = 0 H 1 : At least one is nonzero

31. Part 12: Asymptotics for the Regression Model 12-31/39 Wald Test Matrix ; R = [0,0,0,0,0,0,1,0,0,0/ 0,0,0,0,0,0,0,1,0,0/ 0,0,0,0,0,0,0,0,1,0] ; q = [0 / 0 / 0 ] $ Matrix ; m = R*b - q ; Vm = R*Varb*R' ; List ; Wald = m' m $ Matrix WALD has 1 rows and 1 columns. 1 +-------------- 1| 66.91506

32. Part 12: Asymptotics for the Regression Model 12-32/39 Restricted Regression Compare Sums of Squares Regress; lhs=g;rhs=X; cls:pd=0,pn=0,ps=0$ +----------------------------------------------------+ | Linearly restricted regression | | Ordinary least squares regression | | LHS=G Mean = 5.308616 | | Standard deviation =.2313508 | | Residuals Sum of squares =.01864365 |.00377694 | Standard error of e =.3053166E-01 | | Fit R-squared =.9866028 |.9972859 without restrictions | Adjusted R-squared =.9825836 | | Model test F[ 6, 20] (prob) = 245.47 (.0000) | | Restrictns. F[ 3, 17] (prob) = 22.31 (.0000) | Note: J(=3)*F = Chi-Squared = 66.915 from before | Not using OLS or no constant. Rsqd & F may be < 0. | | Note, with restrictions imposed, Rsqd may be < 0. | +----------------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant -4.46504223 4.77789711 -.935.3631 Y 1.05851456.55196204 1.918.0721 9.03448264 PG -.15852276.05008100 -3.165.0057.47679491 PNC.21765564.18336687 1.187.2516.28100132 PUC -.24298315.10328032 -2.353.0309.40523616 PPT -.12617610.10436708 -1.209.2432.47071442 PD.000000......(Fixed Parameter)....... -.44279509 PN.222045D-15......(Fixed Parameter)....... -.58532943 PS -.444089D-15......(Fixed Parameter)....... -.62272267 T.02944666.02126600 1.385.1841 13.0000000

33. Part 12: Asymptotics for the Regression Model 12-33/39 Nonlinear Restrictions I am interested in testing the hypothesis that certain ratios of elasticities are equal. In particular,  1 =  4 /  5 -  7 /  8 = 0  2 =  4 /  5 -  9 /  8 = 0

34. Part 12: Asymptotics for the Regression Model 12-34/39 Setting Up the Wald Statistic To do the Wald test, I first need to estimate the asymptotic covariance matrix for the sample estimates of  1 and  2. After estimating the regression by least squares, the estimates are f 1 = b 4 /b 5 - b 7 /b 8 f 2 = b 4 /b 5 - b 9 /b 8. Then, using the delta method, I will estimate the asymptotic variances of f 1 and f 2 and the asymptotic covariance of f 1 and f 2. For this, write f 1 = f 1 (b), that is a function of the entire 10  1 coefficient vector. Then, I compute the 1  10 derivative vectors, d 1 =  f 1 (b)/  b and d 2 =  f 2 (b)/  b These vectors are 1 2 3 4 5 6 7 8 9 10 d 1 = 0, 0, 0, 1/b 5, -b 4 /b 5 2, 0, -1/b 8, b 7 /b 8 2, 0, 0 d 2 = 0, 0, 0, 1/b 5, -b 4 /b 5 2, 0, 0, b 9 /b 8 2, -1/b 8, 0

35. Part 12: Asymptotics for the Regression Model 12-35/39 Wald Statistics Then, D = the 2  10 matrix with first row d 1 and second row d 2. The estimator of the asymptotic covariance matrix of [f 1,f 2 ] (a 2  1 column vector) is V = D  s 2 (XX) -1  D. Finally, the Wald test of the hypothesis that  = 0 is carried out by using the chi- squared statistic W = (f-0)V -1 (f-0). This is a chi-squared statistic with 2 degrees of freedom. The critical value from the chi-squared table is 5.99, so if my sample chi- squared statistic is greater than 5.99, I reject the hypothesis.

36. Part 12: Asymptotics for the Regression Model 12-36/39 Wald Test In the example below, to make this a little simpler, I computed the 10 variable regression, then extracted the 5  1 subvector of the coefficient vector c = (b 4,b 5,b 7,b 8,b 9 ) and its associated part of the 10  10 covariance matrix. Then, I manipulated this smaller set of values.

37. Part 12: Asymptotics for the Regression Model 12-37/39 Application of the Wald Statistic ? Extract subvector and submatrix for the test matrix;list ; c =b(4:9)]$ matrix;list ; vc=varb(4:9,4:9) ? Compute derivatives calc ;list ; g11=1/c(2); g12=-c(1)*g11*g11; g13=-1/c(4) ; g14=c(3)*g13*g13 ; g15=0 ; g21= g11 ; g22=g12 ; g23=0 ; g24=c(5)/c(4)^2 ; g25=-1/c(4)$ ? Move derivatives to matrix matrix;list; dfdc=[g11,g12,g13,g14,g15 / g21,g22,g23,g24,g25]$ ? Compute functions, then move to matrix and compute Wald statistic calc;list ; f1=c(1)/c(2) - c(3)/c(4) ; f2=c(1)/c(2) - c(5)/c(4) $ matrix ; list; f = [f1/f2]$ matrix ; list; vf=dfdc * vc * dfdc' $ matrix ; list ; wald = f' * * f$

38. Part 12: Asymptotics for the Regression Model 12-38/39 Computations Matrix C is 5 rows by 1 columns. 1 1 -0.2948 -0.2015 1.506 0.9995 -0.8179 Matrix VC is 5 rows by 5 columns. 1 2 3 4 5 1 0.6655E-01 0.9479E-02 -0.4070E-01 0.4182E-01 -0.9888E-01 2 0.9479E-02 0.5499E-02 -0.9155E-02 0.1355E-01 -0.2270E-01 3 -0.4070E-01 -0.9155E-02 0.8848E-01 -0.2673E-01 0.3145E-01 4 0.4182E-01 0.1355E-01 -0.2673E-01 0.7308E-01 -0.1038 5 -0.9888E-01 -0.2270E-01 0.3145E-01 -0.1038 0.2134 G11 = -4.96184 G12 = 7.25755 G13= -1.00054 G14 = 1.50770 G15 = 0.000000 G21 = -4.96184 G22 = 7.25755 G23 = 0 G24 = -0.818753 G25 = -1.00054 DFDC=[G11,G12,G13,G14,G15/G21,G22,G23,G24,G25] Matrix DFDC is 2 rows by 5 columns. 1 2 3 4 5 1 -4.962 7.258 -1.001 1.508 0.0000 2 -4.962 7.258 0.0000 -0.8188 -1.001 F1= -0.442126E-01 F2= 2.28098 F=[F1/F2] VF=DFDC*VC*DFDC' Matrix VF is 2 rows by 2 columns. 1 2 1 0.9804 0.7846 2 0.7846 0.8648 WALD Matrix Result is 1 rows by 1 columns. 1 1 22.65

39. Part 12: Asymptotics for the Regression Model 12-39/39 Noninvariance of the Wald Test