An Introduction to Generalized Method of Moments. Chen,Rong aronge.net

Transcription

1 An Introduction to Generalized Method of Moments Chen,Rong aronge.net Asset Pricing, 2012

2 Section 1 WHY GMM? 2

3 Empirical Studies 3

4 Econometric Estimation Strategies 4

5 5

6 Maximum Likelihood Estimation 6

7 7

8 Large-Sample Properties of ML Estimator 8

9 Tests 0 : H c q 9

10 The Pros and Cons Consistency; efficiency; invariance Broad applications: heteroskedasticity, autocorrelation, system of simultaneous equations, time series, panel Full information about distribution is required (QMLE) Computational burdens conditional MLE the hypothesis of strictly stationary process 10

11 Least Square Estimation (Projection) Least square estimation is used when (1) we are interested in whether certain variables are correlated with each other or if one variable can predict another; (2) we reject a theory and want to find new variables to potentially explain the variation of (e.g.,) foreign exchange risk premium. Even if without enough knowledge of the joint distribution of the variables of interest, researchers could still conduct the study by (linearly) projecting one variable onto others to see if they are related. 11

12 Least Square Estimation (Projection) 12

13 13

14 An Example: Multiple Linear Regression 14

15 15

16 Test T-test F-test R bar square 16

17 The Pros and Cons Simple, straightforward, little information requirement Good small sample properties if classical hypotheses hold Many hypotheses could be relaxed: heteroskedasticity, autocorrelation, system of simultaneous equations, time series, panel Only linear relationship or first-order information Normal distribution Not make full use of information 17

18 GMM GMM is a nonlinear procedure Limited information Consistent under weak assumptions no distributional assumptions like MLE Asymptotically efficient in the class of models that uses the same amount of information Many estimators are special cases of GMM unifying framework for comparing estimators GMM is a large sample estimator Moment conditions 18

19 Section 2 IDEA OF GMM 19

20 The idea of GMM The restrictions on the joint distribution of variables of interest could be summarized as a set of restrictions on the moments of functions. These moments restrictions may be either unconditional or conditional. GMM estimator is just the one makes all the moments equal to zero (unique solution) or a linear combination of all the moments functions zero (no solution) 20

21 An Example: linear regression 21

22 What if we have more moment restrictions? 22

23 To make the R moments as close to zero as possible...how? 23

24 Section 3 THE GMM ESTIMATION 24

25 Moment Restrictions 25

26 26

27 The GMM Formula: Unconditional Case 27

28 Optimal GMM Estimation: Unconditional Case 28

29 29

30 Estimation in Practice 30

31 Distribution of GMM T 31

32 The GMM Formula: Conditional Case 32

33 The optimal A t 33

34 Section 4 SPECIFICATION TESTS 34

35 J-test 35

36 Test 0 : H c q T-test Newey and West (1987) LR test: use the variance of unrestricted case LM Test Wald Test LR J J 2 # of restrictions GMM R U LM T g S d d S g T R 1 U 1 var R T U c U c U W Tc q var c q U 2 var Ag # of restrictions T Ag T T T T 1 2 # of restrictions 36

37 Section 5 SPECIAL CASES 37

38 OLS Moment Restriction 0 K1 E X E X Y X 1 g T X Y X T 0 K1 K moment equations and K unknown parameters GMM T X X XY 1 38

39 The asymptotic variance (unconditional case) 1 1 ˆ GMM T N 0, T Ad ASA Ad T T T T 1 X Y X g T T 1 d plim plim plim XX KK T plim plim plim plim T T T T GMM Var T plim d S d plim X X X X XX 1 T S Var T g Var X E X X X X Homoskedasticity and no autocorrelation Var GMM T ˆ X X

40 Heteroskedasticity and Autocorrelation GMME is a consistent and unbiased estimator GMM 1 T X X XY GMM Var T plim XX XX XX White or Newey-West robust standard error: estimate a more accurate S (or Ω). 40

41 GLS: Heteroskedasticity and Autocorrelation 41

42 An Example: Autocorrelation 42

43 43

44 44

45 GLS is an Efficient Estimator GMM Var T XX X X XX GLS Var T X X 1 XX XXXX X X XX X X X X XX X X X X A A is a positive definite matrix KTTTTK 45

46 GLS-GMM Moment Restriction 1 0 K1 E X E X Y X 1 1 g T X Y X T 0 K1 GMM estimator GMM 1 1 T X X X Y 1 46

47 Standard error 1 1 ˆ GMM T N 0, T Ad ASA Ad T T T T 1 1 X Y X g T T 1 1 d plim plim plim X X KK T S plimvar T g plim Var X plim E X X plim X X T T T T 1 plim X X T GMM Var 1 1 T plim d S d 47

48 Instrument Variables 当 X 与 ε 相关时,OLSE 将是有偏和不一致的 48

49 Idea of IV 49

50 IV Estimator 50

51 51

52 52

53 IVE is a GMME Moment Restriction 0 K1 E Z E Z Y X 1 g Z Y X T T 0 K1 K moment equations and K unknown parameters 1 Z X ZY ˆGMM 53

54 The asymptotic variance (unconditional case) 1 1 ˆ GMM T N 0, T Ad ASA Ad T T T T 1 Z Y X g T T 1 dplim plim plim ZX KK T S plimvar T g plim Var Z plim E Z Z plim Z Z T T T T GMM Var T plim d S d plim Z X ZZXZ 1 T 54

55 MLE 以 GARCH(1,1) 模型为例 y x, v h h t t t t t t 2 h t 0 1 t1 2 t1 最大化似然函数就是一个矩条件 t 1 T T 2 t ht i1 i1 ht 2 t T 1 1 ln L ln2 ln ln h T t ln L ht 0 55

56 GMM 估计量方差 1 1 ˆ GMM T N 0, T Ad ASA T T T Ad T 1 ln L gt T g 2 T 1 ln L d plim T KK 2 1 ln L 1 ln L ln L 1 ln L S plimvar T g plim Var =plim E plim E T T T T GMM ln L ln L ln L Var T plim d S d plim E T 56

57 Section 6 GMM APPS 57

58 Example I 58

59 59

60 60

61 61

62 Example 2 CKLS (1992) dr ( r ) dt r dz r r r t t t t t1 t t t1 E t r t1 t t1 r t1 t t t r r 0 62

63 63

64 Example 3 Linear Factor Models e R a f T t t 1 e g R a f f 0 T t t t T T 1 e R e e R a bf R a bf T t t T t t 1 e e S E R a bf f R a bf f t t t t t t T t1 e t1 e R R There is no need to assume that the errors are iid and they are conditionally homoskedastic. 64

65 Example 4 Linear factor models in discount factor form m b f E p E mx E xf b T 1 g b xf b-p ( N 1) T T t t t t 1 g b T T 1 A dw W fx W T b T t 1 GMM 1 T 1 T 1 GMM b1t dd d ; b2t ds d ds p t T t1 T t1 p t 65

66 What s the difference between b and λ? e e e 0 E mr E R b cov f, R e e cov, var e cov R, f E R b R f b f var An example f f b Eff b Emf var 66

67 Example 5 Test whether a return on the frontier E a br R i p 0 Test whether a return span the frontier without R f for any two fixed values of a 1 and a 2 d d a br R 1 1 d i a br 1 1 R E 0 d d a 2 br 2 R d i a br 2 2 R 1 1 m a a br br

68 Example 6 Testing for characteristics N assets, N moment conditions g E m b x p y i i i i T T t1 t1 t t 68

69 Example 7 Conditional CAPM Linear conditional expectation model Thus t1 it mt E R E R t1 it t1 mt cov R, R var R t1 mt R Z u it t1 i it Z t1 m Z E uu t1 i 2 t 1 it mt E u t1 mt 2 E u Z E u u Z t1 mt t1 i t1 it mt t1 m 69

70 General GMM conditions R Z t t1 g E R Z 0 T T Z mt t1 m t1 2 u Z u uz mt t1 mt t t1 m Nm 1 m Nm conditions for m N 1 param. 70

71 Constant conditional beta( Ferson and Harvey (1991), Evans (1994), Ferson and Korafczyk (1995)) R Z t t1 R Z m t1 m g E 2 Z 0 T T t 1 u Z u uz mt t1 mt t t1 m R R t mt Nm 1 m Nm Nm conditions for m N 1 + N param. 71

72 Constant conditional reward to risk ( Campbell, 1987; Harvey, 1989) : Relative Risk Aversion var 0 1 conditions for 1 +1 param. t mt t mt t t T T m t m t t mt t t t m t m T T t t mt t mt t m E R R R Z g E R Z Z R u u Nm m Nm m N R Z R Z g E Z R u u u Z conditions for 1 +1 param. Nm m Nm m m N

73 Linear conditional beta (Ferson and Harvey, 1994, 1995)? i i R Z t t 1 Z t1 m m R Z m t 1 m Z t1 2 g E u R u 1 0 T T mt t mt t i Rt Z 1 t1 m 1 Rt Z t t1 m Nm 1 m N N N conditions and param.,,, Rt, i 2 i m it i 73

74 Example 8 Latent variable model with constant beta GMM conditions E r, E r 2 E r E r t1 2t t1 1t t1 1t t 1 t1 2t t 2 1 r Z 1t t1 1 g E r Z Z 0 T T 2t t1 2 t1 Z Z t1 2 t1 2 Km N K m N K m conditions for m Nm K N K param. 74

75 Example 9 Hansen-Jagannathan lower bound 1 E a Rb 0 t R f E Ra RRb 1 0 t t t 1 Em t 0 R f 2 2 E a Rb m t 0 t 1 N 1 1 conditions for N 2 param. One side J-test 75

76 The End.