1 Extremum Estimators
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1 FINC Financial Econometrics Handout Jialin Yu 1 Extremum Estimators Let θ 0 be a vector of k 1 unknown arameters. Extremum estimators: estimators obtained by maximizing or minimizing some objective functions. Why does this make sense? You want to find the best arameter which naturally involves maximization (of gain or minimization (of loss. Examles: 1. Maximum Likelihood: Maximizing the log-likelihood function; 2. Minimum Distance Estimators: LS, GMM, etc. Suose the objective function to be maximized is Q (θ, then the extremum estimator is defined by θ = arg max θ Θ Q (θ For well behaved objective functions, it is equivalent to say that the extremum estimator is defined by Q (θ = 0 θ= θ Examle: OLS. What is the objective function? What is the first order condition (FOC? We will rove consistency/asymtotic distribution at the general level for all extremum estimators. hen, we will secialize the argument to two (very imortant secial cases of MLE (maximum likelihood estimation and GMM (generalized method of moments. 1.1 Consistency of θ What is the issue here: you want to maximize Q 0 (θ but can maximize only its samle counterart Q (θ. Under what conditions is the maximizer 1
2 of the samle counterart close to the maximizer of the true objective function? For concreteness, think the OLS examle. Definition 1 (Uniform Convergence in Probability Q (θ converges uniformly in robability to Q 0 (θ if su Q (θ Q 0 (θ 0 θ Θ heorem 2 If there is a function Q 0 (θ such that (i Q 0 (θ is uniquely maximized at θ 0 (ii Θ is comact (iii Q 0 (θ is continuous (iv Q (θ converges uniformly in robability to Q 0 (θ, then θ θ0. Proof: For any ε > 0, we have with robability aroaching 1 that (a Q ( θ > Q (θ 0 ε/3 because θ is the maximizer of Q ; (b Q 0 ( θ > Q ( θ ε/3 by the uniform convergence of Q (θ to Q 0 (θ (c Q (θ 0 > Q 0 (θ 0 ε/3 again by the uniform convergence. herefore, with robability aroaching 1, Q 0 ( θ > Q ( θ ε/3 > Q (θ 0 2ε/3 > Q 0 (θ 0 ε Let N be an oen neighborhood of θ 0. θ 0 is the unique maximizer of Q 0 (θ imlies su θ Θ N C Q 0 (θ = Q 0 (θ < Q 0 (θ 0 for some θ Θ N C. Choose ε = Q 0 (θ 0 su θ Θ N C Q 0 (θ, it follows that with robability aroaching 1, Q 0 ( θ > su θ Θ N C Q 0 (θ and therefore θ N. Counterexamle: 1. if Θ is not comact, e.g. Θ = [0, 1 {2}, and assuming the objective function to be maximized is f (x = x if x [0, 1, f (x = 1 if x = 2 (by the way, is this objective function continuous?; 2. If Q 0 (θ is not continuous, f (x = Cos (x if x [0, 2π, f (x = 0 if x = 2π. Θ = [0, 2π] 2
3 3. What about non-uniqueness if the maximum? his relates also to the issue of identification. What is non-identification? E.g., if you are to estimate β = u v, can you searately identify u and v (even with infinite number of observations? he following lemma hels to easily verify the uniform convergence condition (iv in ractice. Lemma 3 (Uniform Law of Large Numbers If the data are i.i.d., Θ is comact, a (x i, θ is continuous at each θ Θ with robability one, and there is d (x such that a (x, θ d (x for all θ Θ and E [d (x] <. then E [a (x, θ] is continuous and su θ Θ n 1 i=1 n a (x i, θ E [a (x, θ] 0 Verify OLS consistency using both classical and this method. 1.2 Asymtotic Normality of θ aylor exand Q (θ θ= θ = 0 around θ 0 0 = Q ( θ = Q (θ 0 + Q (θ ( θ θ 0 for some θ in between θ 0 and θ (recall the imlicit assumtions involved here in this exansion. ( θ θ 0 [ 1 = Q (θ ] Q (θ 0 By CL Q (θ 0 d N (0, V (θ 0 θ θ0 and θ is in between θ 0 and θ imly θ θ 0. herefore, [ ] 1 Q (θ [ 1 Q (θ 0 ] S (θ0 1 (1 3
4 (this ste is almost correct. Asymtotic normality follows from Slutsky ( d ( θ θ 0 N 0, S (θ 0 1 V (θ 0 [S ] (θ 0 1 heorem 4 If the estimator satisfies θ θ and (i θ 0 Interior (Θ (ii Q (θ is twice continuously differentiable in a neighborhood N of θ 0 (iii Q (θ 0 d N (0, Σ (or equivalently relace this assumtion with your favorite central limit theorem (iv there is H (θ continuous at θ 0 such that su θ N Q (θ H (θ 0 (this makes the almost correct ste correct; (v H = H (θ 0 is nonsingular. hen ( θ θ0 d N ( 0, H 1 ΣH 1 Condition (iv makes (1 valid. Condition (iv can be verified simiar to verifying uniform law of large numbers (Lemma 3. Examle: consistency and normality of OLS 4
5 2 Maximum Likelihood Estimator Let f (x i, θ be the robability density of observation x i. likelihood estimator (MLE is defined as he maximum θ = max θ Θ log f (x i, θ i=1 I.e., you choose the arameter which is most likely to generate the observations. It is not obvious now why this is otimal. But we will show that MLE has a number of otimality roerties. Let s i (θ = log f (x i, θ. S (θ = s i (θ i=1 is called the Score. he MLE estimator sets the score to 0. 1 = f (x, θ dx Differentiate once, f (x, θ 0 = dx log f (x, θ = f (x, θ dx = S (θ f (x, θ dx Differentiate again, 0 = = = S (θ f (x, θ dx + S (θ f (x, θ dx + S (θ f (x, θ dx + f (x, θ S (θ dx log f (x, θ S (θ f (x, θ dx S (θ 2 f (x, θ dx 5
6 herefore, I (θ = E [ S (θ ] = E [S (θ 2] where I (θ is the information matrix. I i (θ = E [ s i (θ ] = E [s i (θ 2] denotes the information in the i-th observation. I (θ = E [S (θ 2] denotes the information in the samle, and denotes the average information. I = 1 I i (θ With indeendent samling, the s i are indeendent of each other and I (θ = I i (θ. With i.i.d. samling I i (θ = I j (θ = I (θ = 1 I (θ I (θ. 2.1 Consistency of the MLE estimator A feature of MLE estimator is that identification is sufficient to guarantee the log-likelihood function has a unique maximum at the true arameter θ 0. Lemma 5 If θ 0 is identified (θ θ 0 imlies f (x, θ f (x, θ 0 with ositive robability and E [ log f (x, θ ] < for all θ then Q 0 (θ = E [log f (x, θ] has a unique maximum at θ 0. 6
7 Proof: By the strict Jensen s inequality [ ] f (x, θ Q 0 (θ 0 Q 0 (θ = E log f (x, θ 0 [ ] f (x, θ > log E f (x, θ 0 f (x, θ = log f (x, θ 0 f (x, θ 0 dx = 0 heorem 6 Suose the observations are i.i.d. with.d.f. f (x i, θ 0 and (i f (x, θ f (x, θ 0 with ositive robability if θ θ 0 (ii θ 0 Θ comact (iii logf (x, θ is continuous at each θ with robability one (iv E [su θ Θ log f (x, θ ] <. hen θ MLE θ0. Proof: he theorem is roved by verifying the conditions of theorem 2 using the uniform law of large numbers. 2.2 Asymtotic Normality of the MLE estimator heorem 7 Assume the conditions for theorem 6 are satisfied and (i θ 0 Interior (Θ (ii f (x, θ is twice continuously differentiable with resect to θ and f (x, θ > 0 in a neighborhood N of θ 0 (iii su θ N f (x, θ dx < and su 2 θ N f (x, θ dx < (iv I 2 i (θ (the information matrix of [ ] an individual observation is not singular (v E su 2 θ N log f (x, θ < 2. hen d ( θmle θ 0 N (0, I i (θ 1 7
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