Consistency and asymptotic normality

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1 Consistency an asymtotic normality Class notes for Econ 842 Robert e Jong March Stochastic convergence The asymtotic theory of minimization estimators relies on various theorems from mathematical statistics. The objective of this section is to exlain the main theorems that unerin the asymtotic theory for minimization estimators. 1.1 Infimum an suremum, an the limit oerations Suremum an infimum The suremum of a set A, if it exists, is the smallest number y for which x y for every x A. Analogously, the infimum of a set A, if it exists, is the largest number y for which x y for every x A. For an oen set such as A = (0, 1), the suremum su A an the infimum inf A are well-efine as 1 an 0, resectively. If the suremum is attaine for some element in A, the suremum is also calle the maximum; analogously, the minimum of a set is efine as the infimum if it is attaine for some element of A. Therefore, the maximum an minimum of A = (0, 1) are not efine, because they are attaine at 0 an 1, which are not insie (0, 1). If we efine the extene real line as R {, + }, the su an inf are always efine on the extene real line. Deartment of Economics, Ohio State University, 429 Ars Hall, Columbus, OH 43210, ejong@econ.ohio-state.eu 1

2 1.1.2 The limit oeration For a eterministic sequence x n, we say that it converges to x, notation lim x n = x, if for all ε > 0, there exists an N ε such that x n x < ε for all n N ε. The limsu (suerior limit, limes suerior in Latin) is efine as lim su x n = lim x n = inf su x m. n 1 m n The limsu is the eventual uer limit of a sequence. For examle, consier x n = n 1. Then lim su x n = inf su m 1 = inf n 1 m n n 1 n 1 = 0, just as one woul exect. From the earlier iscussion, it is now clear that the limsu is always efine on the extene real line. The limsu of a sequence is the eventual uer boun of a sequence. For examle, lim su sin(n) = 1, while the limit is obviously unefine. Analogously, we efine lim inf x n = lim x n = su n 1 For x n = n 1, we obtain lim inf x n = su n 1 inf x m. m n inf x m = su 0 = 0, m n n 1 as one woul exect from an eventual lower limit. If an only if the limsu an liminf of a sequence are equal, then the limit of the sequence exists. If, after noting that log(1 + x) x, we rea something like lim n2 log(φ(n)) lim n 2 (Φ(n) 1) = 0 where Φ( ) enotes the stanarnormal istribution function, we shoul realize the following. In rincile, when writing own the left-han sie exression, it is ossible for this exression to be unefine. The correct interretation of the above equation is that the inequality fins an uer boun if the exression exists; but formally the existence of the left-han sie remains unroven. The assertion that the lim su is less than or equal to zero is always true however if the limsu is efine on the extene real line. 2

3 1.2 Almost sure convergence, convergence in robability, an convergence in istribution Definitions of stochastic convergence concets While for a eterministic sequence X n it is relatively straightforwar to efine convergence of X n to a number X, the situation becomes more comlicate in the case where X n an/or X is allowe to be a ranom variable. There are four tyes of convergence for ranom variables that are use frequently in the econometric literature. They are convergence in istribution, convergence in robability, convergence in L, an almost sure convergence. We say that X n converges to X in istribution if lim P(X n y) = P(X y) for all y for which P(X y) is continuous in y. To see why we only want this requirement to hol only for continuity oints of P(X y), consier X n = n 1. This is a eterministic sequence an we shoul hoe that it converges in istribution to X = 0. However, setting y = 0, P(X n 0) = P(n 1 0) = 0 P(X 0) = 1. Because the convergence in istribution requirement exclues iscontinuity oints however, X n = n 1 oes converge in istribution to zero. X n converges in robability to X if for all δ > 0, lim P( X n X > δ) = 0. We tyically write X n X or lim X n = X to enote convergence in robability. X n is sai to converge almost surely to X if P( lim X n = X) = 1 which is equivalent to the conition that for all δ > 0, lim P(su X m X > δ) = 0. m n as We then write X n X or lim X n = X almost surely. Convergence in L of X n to X for 1 hols if lim E X n X = 0. 3

4 1.2.2 Differences between stochastic convergence concets There are subtle ifferences among these convergence concets; however, convergence in istribution is in some ways in a ifferent category from the other stochastic convergence concets. The imortant thing to realize about convergence in istribution is that if X n converges to X, an X is a stanarnormally istribute ranom variable, then X n also converges to Y, which is a ranom variable that is also stanarnormally istribute. For examle, a sequence X n of i.i.. N(0, 1) ranom variables converges in istribution to a N(0, 1) ranom variable as n. This may be counterintuitive, because X n will never get closer to any articular ranom variable; but the imortant thing to realize is that only the istribution of X n nees to converge for convergence in istribution. While convergence in istribution only says something about istributions, convergence in robability, almost sure convergence an convergence in L are statements imlying that X n an X are asymtotically close. Almost sure convergence imlies convergence in robability, as can be easily seen from my efinitions above. Convergence in L imlies convergence in robability. This follows from the Markov inequality P(Y > δ) δ 1 E Y, because P( X n X > δ) = P( X n X > δ ) δ E X n X. Convergence in robability imlies convergence in istribution, an convergence in istribution to a constant imlies convergence in robability to that constant; I will leave this unroven here. No other relations between concets hol without further assumtions An o examle A classical examle of a sequence X n that converges in robability to zero but not in L is the following. Let X n be a ranom variable that takes the value ex(n) with robability 1/n an zero otherwise. Then for all δ > 0, lim P( X n > δ) lim P(X n = ex(n)) = lim n 1 = 0, so X n converges in robability to 0. However, for any 1, E X n = ex(n)p(x n = ex(n)) = n 1 ex(n), so X n oes not converge in L. Another examle where convergence in robability hols but convergence in L oes not is the case where X n = Y/n where Y is Cauchy. In that case, we have convergence in robability, but not in L, because moments of orer 1 an higher of the Cauchy istribution o not exist. 4

5 If in aition X n is ineenent across n, we also have an examle of a sequence that converges in robability, but not almost surely. This is because while X n 0, su m n X m oes not converge to zero in robability. After all, su m n X m ex(n) if one of the X m for which m n is nonzero; that is, su m n X m 1 only if X m = 0 m n. Therefore, for all 0 < δ < 1, P(su X m δ) P( m n : X m = 0) m n = P(X m = 0) = m=n = ex( (1 1/m) m=n log(1 1/m)) = 0. m=n To show that last result, note that because log(1 + x) x for x > 1, lim ex( K log(1 1/m)) lim ex( K (1/m)) = 0 K K m=n m=n because m 1 is not summable from 1 to infinity. This imlies that X n converges in robability to zero, but not almost surely Averages Less exotic statistics that we will encounter in econometrics are averages of i.i.. or weakly eenent ranom variables. Consier the following simle examle. Let X n = n 1/2 (z i Ez i ), where the z i are i.i.. an Ezi 2 <. Then X n converges in istribution to a normally istribute ranom variable. For each eterministic sequence a n such that lim a n = 0 we as have a n X n 0. However, it turns out that a n X n 0 is only true if a n (log log n) 1/2 0; this follows from the so-calle law of the iterate logarithm. 5

6 1.3 The law of large numbers Statement of the law of large numbers Consistency of estimators is erive using a law of large numbers (LLN). The LLN asserts that (z i Ez i ) n 1 converges to zero. It can be shown that if E z i < an z i is i.i.. (ineenent an ientically istribute), then the above exression converges to zero almost surely (an therefore also in robability; the almost sure version is Kolmogorov s strong law of large numbers). If E z i is not finite, the LLN oes not nee to hol; a counterexamle woul be the Cauchy istribution, for which z n is istribute ientically to z i. If convergence to zero hols almost surely, we talk about the strong law of large numbers, while if the converge to zero is in robability, we refer to the result as a weak law of large numbers Weak LLN uner ineenence If z i has mean zero, is i.i.. an Ezi 2 <, then V (n 1 z i ) = n 2 V (z i ) = n 1 V (z i ) 0, so a weak law of large numbers will hol for z i in this case. The conition Ezi 2 < can be weakene to E z i < by a truncation argument. This can be one as follows. Define zi K = z i I( z i > K) an z ik = z i I( z i K). Then E n 1 z i E n 1 (z ik Ez ik ) + E n 1 (zi K Ezi K ) (E n 1 (z ik Ez ik ) 2 ) 1/2 + 2n 1 E zi K (n 1 V (z ik )) 1/2 + 2E z i I( z i > K) (n 1 K 2 ) 1/2 + 2 z F(z), z >K an by taking first the limit as n an then the limit as K, the convergence in L 1 follows. To rove Kolmogorov s strong law of large numbers is much more involve; this tyically requires a result known as the three series theorem. I will not ursue that here. 6

7 1.3.3 Discussion There exist many formulations of the LLN that are ifferent from the above. For examle, if z i is uncorrelate, Ez 2 i < an Var(n 1 (z i Ez i ) = n 2 Var(z i ) 0, then the weak LLN will also hol, in site of the fact that the i.i.. assumtion has not been mae. Also, the ineenence assumtion can be relaxe further to allow for ranom variables z i that have faing memory roerties. To illustrate this, note that in general Var(n 1 (z i Ez i )) = n 2 cov(z i, z s ) s=1 2n 2 2n 2 cov(z i, z s ) s=i cov(z i, z i+m ), m=0 an if the last exression converges to zero somehow, then a weak LLN can be obtaine as well. Often, alternatives to the classical LLN for i.i.. variables involve assumtions on the existence of moments E z i 1+δ for some δ > The central limit theorem Statement of the central limit theorem The central limit theorem (CLT) states that n 1/2 (z i Ez i ) N(0, Var(z i )). If z i is i.i.. an Ez 2 i <, then the above result hols. 7

8 1.4.2 CLT an characteristic function The characteristic function of a ranom variable X is efine as ψ(r) = E ex(irx), where i satisfies i 2 = 1. There is a one-to-one corresonence between characteristic functions an istribution functions, an because ψ( ) 1, showing that the characteristic function converges to the characteristic function of a normal istribution suffices to show convergence in istribution to a normal. The reasoning for roving a CLT for a mean zero, i.i.. sequence z i with a variance σ 2 then starts by observing that E ex(irn 1/2 z i ) = n E ex(irn 1/2 z i ) by ineenence. By the Taylor series exansion, ex(x) 1+x+0.5x 2 an log(1+x) x, an therefore n E ex(irn 1/2 z i ) E(1 + (irn 1/2 z i ) + 0.5(irn 1/2 z i ) 2 ) = n (1 0.5r 2 n 1 σ 2 ) = ex( log(1 0.5r 2 n 1 σ 2 )) ex( 0.5r 2 σ 2 ), which is the characteristic function of the N(0, σ 2 ) istribution. Of course, the analytical ifficulty is to justify the signs in the above reasoning Discussion For the CLT, lots of alternative secifications are also ossible. It is ossible to allow for some heterogeneity of the istributions of z i, an also the requirement of ineenence can be relaxe to some tye of faing memory assumtion. Heterogeneity of the istributions of z i refers to the fact that the istributions F i (x) of z i are not ientical for all i. 8

9 1.5 Moment conitions Many articles an books in econometrics contain so-calle moment conitions of the tye E z i <, or lim su n 1 or assume that for some δ > 0, E z i 2+δ <. E z i 2 <, By now, you robably realize that these conitions tyically originate from the verification of conitions necessary for obtaining a law of large numbers or central limit theorem. For examle, n 1 n (z as i Ez i ) 0 if the z i are i.i.. an E z i <. So, if somewhere along the line we nee the result n 1 n (z as i Ez i ) 0, we simly imose E z i < as a regularity conition. The same hols for the central limit theorem. Finally, note that E z i < is imlie by E z i q < for some q > ; this is because for 1 q, (E X ) 1/ (E X q ) 1/q. An examle of a istribution for which moment conitions can fail to hol is the Stuent istribution. If z i has a Stuent istribution with k egrees of freeom, then E z i k oes not exist; remember that a t istribution with one egree of freeom is the Cauchy istribution. 1.6 The ominate convergence theorem A tool that is frequently use in asymtotic theory in econometrics is the ominate convergence theorem. Theorem 1 If X n then lim EX n = EX. X an X n Y a.s. for some ranom variable Y an EY <, This result is wiely use in econometrics; notably, it is use for asymtotic normality of minimization estimators when the mean value in the Taylor series exansion argument is relace by its limit. 9

10 1.7 A joint convergence result The following result lays a key role in asymtotic theory for justifying variance estimation: Theorem 2 If X n X an Y n both arguments, then f(x n, Y n ) c for some constant c an f(x, y) is continuous in f(x, c). Using the above theorem, we can reason as follows. If n 1 n z2 i N(0, σ 2 ), then n 1/2 n z i n 1 n z2 i N(0, 1) A beginner s mistake However, it is not allowe to reason n 1/2 n z i n 1 n z2 i = n 1/2 n 1 n z i n 1 n z2 i n 1/2 0 σ 2 = 0 σ 2 an n 1/2 n z i as many beginners ten to o. You cannot fin the continuous function f(.,.) that will allow you to o this trick; you woul nee f(x, y) = n 1/2 x/y, but eenence of f on n is not allowe. The correct reasoning islaye reviously assume f(x, y) = x/y an clearly, this is allowe. 1.8 The multivariate case Often when convergence in istribution of a ranom vector is shown, a result know as the Cramèr-Wol evice is use. The Cramèr-Wol evice is a tool that can be use to simlify the roblem of showing convergence in istribution of a vector to convergence in istribution X if an only if for all of scalars. It asserts that for a vector X n of imension k, X n k-vectors ξ, ξ X n ξ X. This imlies that using the Cramèr-Wol evice, the roblem of showing asymtotic normality of a vector can be translate into a roblem involving an alication of the CLT for scalars. 1.9 Why asymtotic normality? The reason econometricians are intereste in establishing asymtotic normality of estimators is that this roerty allows us to justify inference roceures such as t-tests an F-tests. 10

11 Of course, t-tests that are justifie base on asymtotic theory are really asymtotically stanarnormal, an F-tests that are justifie base on asymtotic theory are really asymtotically chi-square. Asymtotic normality of a k-vector X n, i.e. X n N(0, V ), imlies that if ˆV V, X nj ˆVjj N(0, 1), which will imly valiity of t-values. A similar argument can be use to show that F-tests are asymtotically vali if we have asymtotic normality of our estimator Exercises 1. Assume that x n is a nonecreasing, boune eterministic sequence of scalars. Show that lim x n exists. 2. Assume that X n is an increasing sequence of ranom variables such that X n X. Show that in this case, X n X also. 3. Assume that X n an Y n are sequences of ranom variables such that X n X an Y n Y. Show that Y n + X n X + Y. 4. Assume that X n an Y n are sequences of ranom variables such that X n X an Y n Y. Fin an examle that shows that Y n + X n X + Y oes not nee to hol. 5. Assume that X n is a sequence of ranom variables such that X n c for a constant c. Show that X n c also. 6. In section 1.2.3, what haens to the o examle if the n 1 in its efinition is relace by n 2? 7. Let x i be i.i.. an N(0, 1) istribute. Using characteristic functions, show that n 1/2 n x i is also N(0, 1) istribute. 8. Formally rove the ominate convergence theorem. 11

12 2 Asymtotics of the linear moel: the scalar case 2.1 The consistency argument In this section we will investigate the asymtotic behavior of least squares estimators. The urose is only to serve as an introuction to the more general case, where we will only assume that our estimator is some minimization estimator, for which no close form solution is available. To unerstan the rinciles involve better, we will focus on the case of a scalar regressor x i in this section. In the case of the simle linear moel y i = θ 0 x i + ε i, where x i R, the close form solution for the least squares estimator is ˆθ n = argmin θ Θ n 1 (y i θx i ) 2 = n y ix i n. x2 i The argmin is efine here as the value at which the function over which the argmin is takes is minimize. For examle, argmin x R (1 + x 2 ) = 0. Note that this solution for ˆθ n was obtaine by ifferentiation with resect to θ. Next, we note that - if the true moel is correct - it is ossible to write n ˆθ n = θ 0 + ε ix i n. x2 i After this ste, we have to make some assumtion on the behavior of x i : is it a eterministic or a stochastic sequence? Consistency argument: the eterministic regressor case First, we assume that x i is eterministic. Then note that n ε ix i n = n 1 n ε ix i x2 i n 1 n, an therefore ˆθ n as θ 0 if 1. n 1 n ε ix i as 0; x2 i 2. n 1 n x2 i Q, where Q > 0. 12

13 2.1.2 Consistency argument: the ranom regressor case Uner the assumtion that x i is ranom, sufficient conitions for ˆθ n 1. n 1 n ε ix i as 0; 2. n 1 n x2 i as Q, where Q > 0. as θ 0 are Both conitions follow, by Kolmogorov s law of large numbers, from the conitions 1. (x i, ε i ) is i.i..; 2. Ex 2 i < ; 3. Eε i x i = 0; 4. E ε i x i <. In the case that ε i an x i are ineenent of each other, conitions 1-4 are imlie by 1. (x i, ε i ) is i.i..; 2. Ex 2 i < ; 3. Eε i = 0; 4. E ε i <. 2.2 The asymtotic normality argument Fining the limit istribution of ˆθ n requires some more work than the simle consistency result. If we assume that the ε i are i.i.. normal with variance σ 2 an x i is eterministic, then n ε ix i n x2 i is istribute N(0, σ 2 ( n x2 i ) 1 ) for any samle size. Therefore, if n 1 n x2 i Q, we get n 1/2 (ˆθ n θ 0 ) N(0, σ 2 Q 1 ). The asymtotic version of this argument simly relaces the exact normality by asymtotic normality, an uses a CLT as a relacement for the exact normality assumtion. 13

14 2.2.1 Asymtotic normality argument: the eterministic regressor case Sufficient conitions for the asymtotic normality result n 1/2 (ˆθ n θ 0 ) 1. n 1/2 n ε ix i N(0, σ 2 Q); 2. n 1 n x2 i Q where Q > 0. N(0, σ 2 Q 1 ) are It is ossible to use a central limit theorem for ineenent, but not ientically istribute ranom variables to establish Asymtotic normality argument: the ranom regressor case At this oint, our interest will be in relaxing the assumtion that the x i are eterministic. If we assume 1. (x i, ε i ) is i.i..; 2. Eε i x i = 0; 3. Ex 2 i < an Eε 2 ix 2 i < ; then by the central limit theorem, n 1/2 x i ε i N(0, Eε 2 ix 2 i), an in aition, n 1 x 2 i Q, an therefore by Theorem 2, Therefore, n 1/2 n x iε i n 1 n x2 i n 1/2 (ˆθ n θ 0 ) where P = Eε 2 i x2 i. N(0, Q 2 Eε 2 ix 2 i). N(0, Q 2 P), 14

15 If we assume in aition that ε i an x i are ineenent, then P = Eε 2 i x2 i = Eε2 i Ex2 i = σ2 Q, an then our result becomes n 1/2 (ˆθ n θ 0 ) N(0, σ 2 Q 1 ). The above result also follows from assuming that E(ε 2 i x i ) = σ 2 (a homoskeasticity assumtion). However, lease note that asymtotic normality is also obtaine without homoskeasticity assumtion. In the sections that follow, we will always assume that the x i sequence is stochastic. 15

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