3 Basic Asymptotic Theory This chapter summarizes some definitions and limit theorems that are important for studying large-sample theory Most claims are stated without proof, as several require tedious epsilon-delta arguments We prove some results that build on fundamental definitions and theorems A good, general reference for background in asymptotic analysis is White (1984) In Chapter 12 we introduce further asymptotic methods that are required for studying nonlinear models 3.1 Convergence of Deterministic Sequences Asymptotic analysis is concerned with the various kinds of convergence of sequences of estimators as the sample size grows We begin with some definitions regarding nonstochastic sequences of numbers When we apply these results in econometrics, N is the sample size, and it runs through all positive integers You are assumed to have some familiarity with the notion of a limit of a sequence definition 3.1: (1) A sequence of nonrandom numbers faN : N ¼ 1; 2; g converges to a (has limit a) if for all e > 0, there exists Ne such that if N > Ne then jaN À aj < e We write aN ! a as N ! y (2) A sequence faN : N ¼ 1; 2; g is bounded if and only if there is some b < y such that jaN j a b for all N ¼ 1; 2; : Otherwise, we say that faN g is unbounded These definitions apply to vectors and matrices element by element Example 3.1: (1) If aN ẳ ỵ 1=N, then aN ! (2) If aN ẳ 1ị N , then aN does not have a limit, but it is bounded (3) If aN ¼ N 1=4 , aN is not bounded Because aN increases without bound, we write aN ! y definition 3.2: (1) A sequence faN g is OðN l Þ (at most of order N l ) if N Àl aN is bounded When l ¼ 0, faN g is bounded, and we also write aN ¼ Oð1Þ (big oh one) (2) faN g is oðN l Þ if N Àl aN ! When l ¼ 0, aN converges to zero, and we also write aN ¼ oð1Þ (little oh one) From the definitions, it is clear that if aN ẳ oN l ị, then aN ¼ OðN l Þ; in particular, if aN ¼ oð1Þ, then aN ẳ O1ị If each element of a sequence of vectors or matrices is OðN l Þ, we say the sequence of vectors or matrices is OðN l Þ, and similarly for oðN l Þ Exampleffi 3.2: (1) If aN ẳ logNị, then aN ẳ oN l ị for any l > (2) If aN ¼ pffiffiffiffi 10 ỵ N , then aN ẳ ON 1=2 ị and aN ẳ oN 1=2ỵgị ị for any g > 36 3.2 Chapter Convergence in Probability and Bounded in Probability definition 3.3: (1) A sequence of random variables fxN : N ¼ 1; 2; g converges in probability to the constant a if for all e > 0, P½jxN À aj > eŠ ! as N ! y p We write xN ! a and say that a is the probability limit (plim) of xN : plim xN ¼ a (2) In the special case where a ¼ 0, we also say that fxN g is op ð1Þ (little oh p one) p We also write xN ẳ op 1ị or xN ! (3) A sequence of random variables fxN g is bounded in probability if and only if for every e > 0, there exists a be < y and an integer Ne such that P½jxN j b be Š < e for all N b Ne We write xN ẳ Op 1ị (fxN g is big oh p one) If cN is a nonrandom sequence, then cN ẳ Op 1ị if and only if cN ẳ O1ị; cN ẳ op 1ị if and only if cN ẳ o1ị A simple, and very useful, fact is that if a sequence converges in probability to any real number, then it is bounded in probability lemma 3.1: matrices p If xN ! a, then xN ¼ Op ð1Þ This lemma also holds for vectors and The proof of Lemma 3.1 is not di‰cult; see Problem 3.1 definition 3.4: (1) A random sequence fxN : N ¼ 1; 2; g is op ðaN Þ, where faN g is a nonrandom, positive sequence, if xN =aN ¼ op ð1Þ We write xN ¼ op ðaN Þ (2) A random sequence fxN : N ¼ 1; 2; g is Op ðaN Þ, where faN g is a nonrandom, positive sequence, if xN =aN ¼ Op 1ị We write xN ẳ Op aN ị We could have started by defining a sequence fxN g to be op ðN d Þ for d A R if p N Àd xN ! 0, in which case we obtain the denition of op 1ị when d ẳ This is where the one in op ð1Þ comes from A similar remark holds for Op ð1Þ pffiffiffiffi ffi Example 3.3: If z is a random variable, then xN N z is Op N 1=2 ị and xN ẳ op ðN d Þ for any d > lemma 3.2: If wN ẳ op 1ị, xN ẳ op 1ị, yN ẳ Op 1ị, and zN ẳ Op 1ị, then (1) wN ỵ xN ẳ op 1ị; (2) yN ỵ zN ẳ Op 1ị; (3) yN zN ẳ Op 1ị; and (4) xN zN ẳ op 1ị In derivations, we will write relationships to as op 1ị ỵ op 1ị ẳ op 1ị, Op 1ị ỵ Op 1ị ¼ Op ð1Þ, Op ð1Þ Á Op ð1Þ ¼ Op 1ị, and op 1ị Op 1ị ẳ op 1ị, respectively Be- Basic Asymptotic Theory 37 cause a op ð1Þ sequence is Op ð1Þ, Lemma 3.2 also implies that op 1ị ỵ Op 1ị ẳ Op 1ị and op 1ị op 1ị ẳ op 1ị All of the previous definitions apply element by element to sequences of random vectors or matrices For example, if fxN g is a sequence of random K  random p p vectors, xN ! a, where a is a K  nonrandom vector, if and only if xNj ! aj , p j ¼ 1; ; K This is equivalent to kxN À ak ! 0, where kbk ðb bÞ 1=2 denotes the p Euclidean length of the K  vector b Also, ZN ! B, where ZN and B are M  K, p is equivalent to kZN À Bk ! 0, where kAk ½trðA Aފ 1=2 and trðCÞ denotes the trace of the square matrix C A result that we often use for studying the large-sample properties of estimators for linear models is the following It is easily proven by repeated application of Lemma 3.2 (see Problem 3.2) lemma 3.3: Let fZN : N ¼ 1; 2; g be a sequence of J  K matrices such that ZN ¼ op ð1Þ, and let fxN g be a sequence of J  random vectors such that xN ¼ Op 1ị Then ZN xN ẳ op 1ị The next lemma is known as Slutsky’s theorem lemma 3.4: Let g: R K ! R J be a function continuous at some point c A R K Let p fxN : N ¼ 1; 2; g be sequence of K  random vectors such that xN ! c Then p gðxN Þ ! gðcÞ as N ! y In other words, plim gxN ị ẳ gplim xN Þ ð3:1Þ if gðÁÞ is continuous at plim xN Slutsky’s theorem is perhaps the most useful feature of the plim operator: it shows that the plim passes through nonlinear functions, provided they are continuous The expectations operator does not have this feature, and this lack makes finite sample analysis di‰cult for many estimators Lemma 3.4 shows that plims behave just like regular limits when applying a continuous function to the sequence definition 3.5: Let ðW; F; PÞ be a probability space A sequence of events fWN : N ¼ 1; 2; g H F is said to occur with probability approaching one (w.p.a.1) if and only if PðWN Þ ! as N ! y c Definition 3.5 allows that WN , the complement of WN , can occur for each N, but its chance of occuring goes to zero as N ! y corollary 3.1: Let fZN : N ¼ 1; 2; g be a sequence of random K  K matrices, p and let A be a nonrandom, invertible K  K matrix If ZN ! A then 38 Chapter (1) ZÀ1 exists w.p.a.1; N p (2) ZÀ1 ! AÀ1 or plim ZÀ1 ¼ Ầ1 (in an appropriate sense) N N Proof: Because the determinant is a continuous function on the space of all square p matrices, detðZN Þ ! detðAÞ Because A is nonsingular, detðAÞ 0 Therefore, it follows that PẵdetZN ị 0 ! as N ! y This completes the proof of part Part requires a convention about how to define ZÀ1 when ZN is nonsingular Let N WN be the set of o (outcomes) such that ZN ðoÞ is nonsingular for o A WN ; we just showed that PðWN Þ ! as N ! y Define a new sequence of matrices by ~ ZN ðoÞ ZN ðoÞ when o A WN ; ~ ZN ðoÞ IK when o B WN p ~ ~ p Then PZN ẳ ZN ị ẳ PðWN Þ ! as N ! y Then, because ZN ! A, ZN ! A The p ~ inverse operator is continuous on the space of invertible matrices, so ZÀ1 ! AÀ1 N p This is what we mean by ZÀ1 ! AÀ1 ; the fact that ZN can be singular with vanishing N probability does not aÔect asymptotic analysis 3.3 Convergence in Distribution definition 3.6: A sequence of random variables fxN : N ¼ 1; 2; g converges in distribution to the continuous random variable x if and only if FN ðxÞ ! F ðxÞ as N ! y for all x A R where FN is the cumulative distribution function (c.d.f.) of xN and F is the (continud ous) c.d.f of x We write xN ! x d a When x @ Normalðm; s Þ we write xN ! Normalðm; s Þ or xN @ Normalðm; s Þ (xN is asymptotically normal ) In Definition 3.6, xN is not required to be continuous for any N A good example of where xN is discrete for all N but has an asymptotically normal distribution is the Demoivre-Laplace theorem (a special case of the central limit theorem given in Section 3.4), which says that xN sN Npị=ẵNp1 pị 1=2 has a limiting standard normal distribution, where sN has the binomial ðN; pÞ distribution definition 3.7: A sequence of K  random vectors fxN : N ¼ 1; 2; g converges in distribution to the continuous random vector x if and only if for any K  nond d random vector c such that c c ¼ 1, c xN ! c x, and we write xN ! x d When x @ Normalðm; VÞ the requirement in Definition 3.7 is that c xN ! d Normalðc m; c VcÞ for every c A R K such that c c ¼ 1; in this case we write xN ! a Normalðm; VÞ or xN @ Normalðm; VÞ For the derivations in this book, m ¼ Basic Asymptotic Theory lemma 3.5: 39 d If xN ! x, where x is any K  random vector, then xN ¼ Op ð1Þ As we will see throughout this book, Lemma 3.5 turns out to be very useful for establishing that a sequence is bounded in probability Often it is easiest to first verify that a sequence converges in distribution d lemma 3.6: Let fxN g be a sequence of K  random vectors such that xN ! x If d g: R K ! R J is a continuous function, then gðxN Þ ! gðxÞ The usefulness of Lemma 3.6, which is called the continuous mapping theorem, cannot be overstated It tells us that once we know the limiting distribution of xN , we can find the limiting distribution of many interesting functions of xN This is especially useful for determining the asymptotic distribution of test statistics once the limiting distribution of an estimator is known; see Section 3.5 The continuity of g is not necessary in Lemma 3.6, but some restrictions are needed We will only need the form stated in Lemma 3.6 d corollary 3.2: If fzN g is a sequence of K  random vectors such that zN ! Normalð0; VÞ then d (1) For any K  M nonrandom matrix A, A zN ! Normalð0; A VAÞ d a 2 (2) zN VÀ1 zN ! wK (or zN VÀ1 zN @ wK ) lemma 3.7: d Let fxN g and fzN g be sequences of K  random vectors If zN ! z p d and xN À zN ! 0, then xN ! z Lemma 3.7 is called the asymptotic equivalence lemma In Section 3.5.1 we discuss generally how Lemma 3.7 is used in econometrics We use the asymptotic equivalence lemma so frequently in asymptotic analysis that after a while we will not even mention that we are using it 3.4 Limit Theorems for Random Samples In this section we state two classic limit theorems for independent, identically distributed (i.i.d.) sequences of random vectors These apply when sampling is done randomly from a population theorem 3.1: Let fwi : i ¼ 1; 2; g be a sequence of independent, identically distributed G  random vectors such that Ejwig jị < y, g ẳ 1; ; G Then the PN p sequence satisfies the weak law of large numbers (WLLN): N À1 iẳ1 wi ! mw , where mw Ewi ị 40 Chapter theorem 3.2 (Lindeberg-Levy): Let fwi : i ¼ 1; 2; g be a sequence of independent, identically distributed G  random vectors such that Ewig ị < y, g ẳ 1; ; G, and Ewi ị ẳ Then fwi : i ¼ 1; 2; g satisfies the central limit theorem (CLT); that is, N À1=2 N X d wi ! Normal0; Bị iẳ1 where B ẳ Varwi Þ ¼ Eðwi wi0 Þ is necessarily positive semidefinite For our purposes, B is almost always positive definite 3.5 Limiting Behavior of Estimators and Test Statistics In this section, we apply the previous concepts to sequences of estimators Because estimators depend on the random outcomes of data, they are properly viewed as random vectors 3.5.1 Asymptotic Properties of Estimators ^ definition 3.8: Let fyN : N ¼ 1; 2; g be a sequence of estimators of the P  vector y A Y, where N indexes the sample size If ^ p yN ! y ð3:2Þ ^ for any value of y, then we say yN is a consistent estimator of y Because there are other notions of convergence, in the theoretical literature condition (3.2) is often referred to as weak consistency This is the only kind of consistency we will be concerned with, so we simply call condition (3.2) consistency (See White, 1984, Chapter 2, for other kinds of convergence.) Since we not know y, the consistency definition requires condition (3.2) for any possible value of y ^ definition 3.9: Let fyN : N ¼ 1; 2; g be a sequence of estimators of the P  vector y A Y Suppose that pffiffiffiffi ffi d ^ N ðyN À yÞ ! Normalð0; VÞ ð3:3Þ pffiffiffiffi ^ where V is a P  P positive semidefinite matrix Then we say that yN is Npffiffiffiffi ffi ^ asymptotically pffiffiffiffiffi normally distributed and V is the asymptotic variance of N ðyN À yÞ, ^N yị ẳ V denoted Avar N y ^ ^ Even though V=N ẳ VaryN ị holds only in special cases, and yN rarely has an exact ^N as if normal distribution, we treat y Basic Asymptotic Theory 41 ^ yN @ Normalðy; V=NÞ ð3:4Þ whenever statement (3.3) holds For this reason, V=N is called the asymptotic vari^ ance of yN , and we write ^ AvaryN ị ẳ V=N ð3:5Þ ^ However, the only sense in which yN is approximately normally distributed with mean y and variance V=N is contained in statement (3.3), and this is what is needed to perform inference about y Statement (3.4) is a heuristic statement that leads to the appropriate inference When we discuss consistent estimation of asymptotic variances—a topic that will pffiffiffiffi ffi ^ arise often—we should technically focus on estimation of V Avar N ðyN À yÞ In most cases, we will be able to find at least one, and usually more than one, consistent ^ ^ ^ estimator VN of V Then the corresponding estimator of AvarðyN Þ is VN =N, and we write ^N Av^ryN ị ẳ V =N a ^ ð3:6Þ The division by N in equation (3.6) is practically very important What we call the ^ asymptotic variance of yN is estimated as in equation (3.6) Unfortunately, there has not been a consistent usage of the term ‘‘asymptotic variance’’ in econometrics ^ ^ Taken literally, a statement such as ‘‘VN =N is consistent for AvarðyN Þ’’ is not very p ^ meaningful because V=N converges to as N ! y; typically, VN =N ! whether ^N or not V is not consistent for V Nevertheless, it is useful to have an admittedly ^ imprecise shorthand In what follows, if we say that ‘‘VN =N consistently estimates pffiffiffiffiffi ^N Þ,’’ we mean that VN consistently estimates Avar N ðyN À yÞ ^ ^ Avarðy pffiffiffiffiffi a ^ definition 3.10: If N ðyN À yÞ @ Normalð0; VÞ where V is positive definite with p ^ ^N jth diagonal vjj , and V ! V, then the asymptotic standard error of yNj , denoted ^Nj Þ, is ð^Njj =NÞ 1=2 seðy v In other words, the asymptotic standard error of an estimator, which is almost always reported in applied work, is the square root of the appropriate diagonal ele^N ment of V =N The asymptotic standard errors can be loosely thought of as estimating ^ the standard deviations of the elements of yN , and they are the appropriate quantities to use when forming (asymptotic) t statistics and confidence intervals Obtaining valid asymptotic standard errors (after verifying that the estimator is asymptotically normally distributed) is often the biggest challenge when using ffia new estimator pffiffiffiffi ^ If statement (3.3) holds, it follows by Lemma ffiffiffiffi that N yN yị ẳ Op 1ị, p3.5 p or ^ ^ yN À y ¼ Op ðN À1=2 Þ, and we say that yN is a N-consistent estimator of y N - 42 Chapter ^ consistency certainly implies that plim yN ¼ y, but it is much stronger because it tells us that the rate of convergence is almost the square root of the sample size N: ^ yN y ẳ op N c ị for any a c < In this book, almost everyffi consistent estimator pffiffiffiffi we will study—andpffiffiffiffiffi one we consider in any detail—is N -asymptotically norevery mal, and therefore N -consistent, under reasonable assumptions pffiffiffiffi ffi If one N -asymptotically normal estimator has an asymptotic variance that is smaller than another’s asymptotic variance (in the matrix sense), it makes it easy to choose between the estimators based on asymptotic considerations ^ ~ definition 3.11: Let yN and yN be pffiffiffiffiffi estimators of y each satisfying statement (3.3), pffiffiffiffi ffi ^ ~ with asymptotic variances V ¼ Avar N ðyN yị and D ẳ Avar N yN yị (these ^ generally depend on the value of y, but we suppress that consideration here) (1) yN is ~N if D À V is positive semidefinite for all y; (2) yN ^ asymptotically e‰cient relative ffiffiffiffiffi y pffiffiffiffi p to ~N are N-equivalent if N yN yN ị ẳ op ð1Þ ^ ~ and y pffiffiffiffiffi When two estimators are N -equivalent, they have the same limiting distribution (multivariate normal in this case, with the same asymptotic variance) This conclusion follows immediately from the asymptotic equivalence lemma (Lemma 3.7) pffiffiffiffi ffi ^ Sometimes, to find the limiting distribution of, say, N ðyN À yÞ, it is easiest to first pffiffiffiffiffi ~N À yÞ, and then to show that yN and yN are ^ ~ findffi pffiffiffiffi the limiting distribution of N ðy N -equivalent A good example of this approach is in Chapter 7, where we find the limiting distribution of the feasible generalized least squares estimator, after we have found the limiting distribution of the GLS estimator ^ ^ ^ definition 3.12: Partition yN satisfying statement (3.3) into vectors yN1 and yN2 ^N1 and yN2 are asymptotically independent if ^ Then y   V1 V¼ V2 pffiffiffiffiffi ^ where V1 is the asymptotic variance ffiffiffiffiffi N ðyN1 À y Þ and similarly for V2 In other p of ^ words, the asymptotic variance of N ðyN À yÞ is block diagonal Throughout this section we have been careful to index estimators by the sample size, N This is useful to fix ideas on the nature of asymptotic analysis, but it is cumbersome when applying asymptotics to particular estimation methods After this ^ chapter, an estimator of y will be denoted y, which is understood to depend on the ^ p sample size N When we write, for example, y ! y, we mean convergence in probability as the sample size N goes to infinity Basic Asymptotic Theory 3.5.2 43 Asymptotic Properties of Test Statistics We begin with some important definitions in the large-sample analysis of test statistics definition 3.13: (1) The asymptotic size of a testing procedure is defined as the limiting probability of rejecting H0 when it is true Mathematically, we can write this as limN!y PN (reject H0 j H0 ), where the N subscript indexes the sample size (2) A test is said to be consistent against the alternative H1 if the null hypothesis is rejected with probability approaching one when H1 is true: limN!y PN (reject H0 j H1 ị ẳ In practice, the asymptotic size of a test is obtained by finding the limiting distribution of a test statistic—in our case, normal or chi-square, or simple modifications of these that can be used as t distributed or F distributed—and then choosing a critical value based on this distribution Thus, testing using asymptotic methods is practically the same as testing using the classical linear model A test is consistent against alternative H1 if the probability of rejecting H1 tends to unity as the sample size grows without bound Just as consistency of an estimator is a minimal requirement, so is consistency of a test statistic Consistency rarely allows us to choose among tests: most tests are consistent against alternatives that they are supposed to have power against For consistent tests with the same asymptotic size, we can use the notion of local power analysis to choose among tests We will cover this briefly in Chapter 12 on nonlinear estimation, where we introduce the notion ffiffiffiffiffi p of local alternatives—that is, alternatives to H0 that converge to H0 at rate 1= N Generally, test statistics will have desirable asymptotic properties when they are based on estimators with good asymptotic properties (such as e‰ciency) We now derive the limiting distribution of a test statistic that is used very often in econometrics lemma 3.8: Suppose that statement (3.3) holds, where V is positive definite Then for any nonstochastic matrix Q  P matrix R, Q a P, with rankðRÞ ¼ Q, pffiffiffiffiffi a ^ N RðyN À yÞ @ Normalð0; RVR Þ and pffiffiffiffiffi pffiffiffiffiffi a ^ ^ ẵ N RyN yị ẵRVR ½ N RðyN À yފ @ wQ ^N In addition, if plim V ¼ V then pffiffiffiffiffi pffiffiffiffiffi ^ ^ ^ ẵ N RyN yị ẵRVN R ẵ N RyN yị a ^ ^ ^N ẳ yN yị R ẵRV =NịR ŠÀ1 RðyN À yÞ @ wQ 44 Chapter For testing the null hypothesis H0 : Ry ¼ r, where r is a Q  nonrandom vector, define the Wald statistic for testing H0 against H1 : Ry r as ^ ^ ^ WN ðRyN À rị ẵRVN =NịR RyN rị 3:7ị a ^ Under H0 , WN @ wQ If we abuse the asymptotics and treat yN as being distributed ^N =NÞ, we get equation (3.7) exactly as Normalðy; V lemma 3.9: Suppose that statement (3.3) holds, where V is positive definite Let c: Y ! RQ be a continuously diÔerentiable function on the parameter space Y H RP , where Q a P, and assume that y is in the interior of the parameter space Define CðyÞ ‘y cðyÞ as the Q  P Jacobian of c Then p a ^ N ẵcyN ị cyị @ Normalẵ0; CyịVCyị 3:8ị and p p a ^ ^ f N ẵcyN ị cyịg ẵCyịVCyị f N ẵcyN ị cyịg @ wQ ^ ^ ^ ^N Define CN CðyN Þ Then plim CN ¼ CðyÞ If plim V ¼ V, then pffiffiffiffi ffi pffiffiffiffiffi a ^ ^ ^ ^0 ^ f N ẵcyN ị cyịg ẵCN VN CN f N ẵcyN ị cyịg @ wQ ð3:9Þ Equation (3.8) is very useful for obtaining asymptotic standard errors for nonlin^ ^ ^ ^ ^0 ear functions of yN The appropriate estimator of AvarẵcyN ị is CN VN =NịCN ẳ ^ ^0 ^ ^ CN ẵAvaryN ފCN Thus, once AvarðyN Þ and the estimated Jacobian of c are obtained, we can easily obtain ^ ^ ^ ^0 AvarẵcyN ị ẳ CN ẵAvaryN ịCN 3:10ị The asymptotic standard errors are obtained as the square roots of the diagonal ^ ^ elements of equation (3.10) In the scalar case gN ẳ cyN ị, the asymptotic standard ^N Þ½AvarðyN ފ‘y cðyN Þ Š 1=2 ^ ^ ^ error of gN is ½‘y cðy Equation (3.9) is useful for testing nonlinear hypotheses of the form H0 : cyị ẳ against H1 : cyị 0 The Wald statistic is pffiffiffiffiffi pffiffiffiffi ffi ^ ^ ^ ^ ^0 ^ ^ ^ ^0 ^ 3:11ị WN ẳ N cyN ị ẵCN VN CN N cyN ị ẳ cyN ị ẵCN VN =NịCN cyN ị a Under H0 , WN @ wQ The method of establishing equation (3.8), given that statement (3.3) holds, is often called the delta method, and it is used very often in econometrics It gets its name from its use of calculus The argument is as follows Because y is in the interior of Y, ^ ^ and because plim yN ¼ y, yN is in an open, convex subset of Y containing y with Basic Asymptotic Theory 45 probability approaching one, therefore w.p.a.1 we can use a mean value expansion ^ ^ cyN ị ẳ cyị þ CN Á ðyN À yÞ, where CN denotes the matrix CðyÞ with rows eval^N and y Because these mean values are trapped beuated at mean values between y ^ tween yN and y, they converge in probability to y Therefore, by Slutsky’s theorem, p € N ! CðyÞ, and we can write C pffiffiffiffiffi pffiffiffiffiffi ^ € ^ N ẵcyN ị cyị ẳ CN N yN yị p p ^ ^ ẳ Cyị N yN yị ỵ ẵCN Cyị N yN yị p p ^ ^ ẳ Cyị N yN yị ỵ op 1ị Op 1ị ẳ Cyị N yN yị ỵ op 1ị We can now apply the asymptotic equivalence lemma and Lemma 3.8 [with R Cðyފ to get equation (3.8) Problems 3.1 Prove Lemma 3.1 3.2 Using Lemma 3.2, prove Lemma 3.3 3.3 Explain why, under the assumptions of Lemma 3.4, gxN ị ẳ Op ð1Þ 3.4 Prove Corollary 3.2 3.5 Let fyi : i ¼ 1; 2; g be an independent, identically distributed sequence with Eyi2 ị < y Let m ẳ Eyi ị and s ẳ Varyi ị a LetyN denote the sample average based on a sample size of N Find p Varẵ N yN mị p b What is the asymptotic variance of N ðyN À mÞ? c What is the asymptotic variance of yN ? Compare this with VarðyN Þ d What is the asymptotic standard deviation of yN ? e How would you obtain the asymptotic standard error of yN ? pffiffiffiffiffi ^ 3.6 Give a careful (albeit short) proof of the following statement: If N yN yị ẳ ^N y ẳ op ðN Àc Þ for any a c < Op ð1Þ, then y pffiffiffiffi ffi ^ be a N -asymptotically normal estimator for the scalar y > Let 3.7 Let y ^ ^ g ẳ logyị be an estimator of g ẳ logyị ^ a Why is g a consistent estimator of g? 46 b Find the asymptotic variance of pffiffiffiffi ffi ^ N ðy À yÞ Chapter pffiffiffiffiffi N ð^ À gÞ in terms of the asymptotic variance of g ^ ^ ^ c Suppose that, for a sample of data, y ¼ and seyị ẳ What is g and its (asymptotic) standard error? d Consider the null hypothesis H0 : y ¼ What is the asymptotic t statistic for testing H0 , given the numbers from part c? ^ e Now state H0 from part d equivalently in terms of g, and use g and seð^Þ to test g H0 What you conclude? pffiffiffiffiffi ^ ^ ^ 3.8 Let y ẳ y1 ; y2 ị be a N -asymptotically normal estimator for y ¼ ðy1 ; y2 Þ , ^ ^ ^ with y2 0 Let g ¼ y1 =y2 be an estimator of g ¼ y1 =y2 ^ a Show that plim g ¼ g ^ b Find Avarð^Þ in terms of y and AvarðyÞ using the delta method  g  À:4 ^ ^ c If, for a sample of data, y ẳ 1:5; :5ị and Avaryị is estimated as , À:4 ^ find the asymptotic standard error of g pffiffiffiffiffi ^ ~ 3.9 Let y and y be two consistent, pNffi-asymptotically normal estimators of the pffiffiffiffiffi ffiffiffiffi ~ ^ P  parameter vector y, with Avar N y yị ẳ V1 and Avar N y yị ẳ V2 Dene a Q parameter vector by g ẳ gyị, where gị is a continuously diÔer^ ~ entiable function Show that, if y is asymptotically more e‰cient than y , then ^ g ^ ~ gðy Þ is asymptotically e‰cient relative to ~ gðy Þ g ... lemma and Lemma 3. 8 [with R Cðyފ to get equation (3. 8) Problems 3. 1 Prove Lemma 3. 1 3. 2 Using Lemma 3. 2, prove Lemma 3. 3 3. 3 Explain why, under the assumptions of Lemma 3. 4, gxN ị ẳ Op ð1Þ 3. 4... Theory 3. 5.2 43 Asymptotic Properties of Test Statistics We begin with some important definitions in the large-sample analysis of test statistics definition 3. 13: (1) The asymptotic size of a testing... The method of establishing equation (3. 8), given that statement (3. 3) holds, is often called the delta method, and it is used very often in econometrics It gets its name from its use of calculus