Estimation Methods in Econometric Analysis of Panel Data

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Explore the concepts of IV and GMM estimation in panel data analysis as presented in William Greene's work, covering topics like instrumental variable estimation, exogeneity, GMM, fixed effects, and more.

  • Panel Data
  • Econometrics
  • Instrumental Variable
  • GMM Estimation
  • Panel Analysis
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  1. Part 8: IV and GMM Estimation [ 1/51] Econometric Analysis of Panel Data William Greene Department of Economics University of South Florida

  2. Part 8: IV and GMM Estimation [ 2/51] Agenda Single equation instrumental variable estimation Exogeneity Instrumental Variable (IV) Estimation Two Stage Least Squares (2SLS) Generalized Method of Moments (GMM) Panel data Fixed effects Hausman and Taylor s formulation Application Arellano/Bond/Bover framework

  3. Part 8: IV and GMM Estimation [ 3/51] Structure and Regression Earnings (structural) equation x it + 'true'education measurable only with error measured 'schooling' = E the "causal" impact of education Reduced form x x Estimation problem = + y E S E , i,t" = sibling t in family i it it it = = = it + w , w=measurement error it it it it + for least squares (OLS or GLS) w )] 0 elies on this covariance equaling 0. How to estimate (consistently)? = + + y = (S S + ( w ) it it it it it w ) it it it 2 w = Cov[S ,( Consistency r it it it

  4. Part 8: IV and GMM Estimation [ 4/51] Least Squares y = X + y|X X X = ) X X + E[ b ] X X y E[ |X X X X X X ] ) X X X /N) ( plim( Q 1 1 = = + ( ( ( ) 1 + = plim = = + ( /N) /N) plim( 1 + b X /N) 1 Useful insight: LS converges to something, just not the parameter we are hoping to estimate.

  5. Part 8: IV and GMM Estimation [ 5/51] Exogeneity and Endogeneity Structure E[ Regression Projection "Regression of on " y X( + ) + w; The problem: X is not exogenous. y = X + |X y = X + g(X) + - g(X) = E[y|X] + u, = X + w y X ] = g(X) 0 [ ] ]= E[ u|X 0 + = | Exogeneity: E[ Strict Exogeneity: E[ (We assume no corr | x ] 0 (current period) , ,..., x x it it elation across individuals.) = x ] 0 (all periods) it i1 i2 iT

  6. Part 8: IV and GMM Estimation [ 6/51] The IV Estimator The variables : One endogenous variable. X x ,x ,...x , The Model Assumption z z = = [ x ], Z [ x ,x ,...x , z ] 1 2 K-1 K 1 2 K-1 it = = = E[ (Using "n" to denote E[(1/n i,t ) E[(1/n) Z'y The Estimator : | ] 0, E[ | z ] E[ z (y x )| z ] 0 it it it it it it it it E[(1/n) Z'X T) i i it = = z | z ] z (y x )| z ] 0 it it ] = E[(1/n) Replicate this condition (if possible) Find so (1/n) = (1/n) Z'y it i,t it it it ] Z'X

  7. Part 8: IV and GMM Estimation [ 7/51] A Moment Based Estimator Estimating Equation Find so (1/n) Instrumental Variable Estimator Z'y = (1/n) Z'X -1 = Z' ( X ) Z' y (Not equivalent to replacing x with .) z K

  8. Part 8: IV and GMM Estimation [ 8/51] Cornwell and Rupert Data Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years Variables in the file are EXP WKS OCC IND SOUTH SMSA MS FEM UNION ED LWAGE = work experience, EXPSQ = EXP2 = weeks worked = occupation, 1 if blue collar, = 1 if manufacturing industry = 1 if resides in south = 1 if resides in a city (SMSA) = 1 if married = 1 if female = 1 if wage set by union contract = years of education = log of wage = dependent variable in regressions These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp. 149-155. See Baltagi, page 122 for further analysis. The data were downloaded from the website for Baltagi's text.

  9. Part 8: IV and GMM Estimation [ 9/51] Wage Equation with Endogenous Weeks Worked lnWage= 1+ 2 Exp + 3 ExpSq + 4OCC + 5 South + 6 SMSA + 7 WKS + Weeks worked (WKS) is believed to be endogenous in this equation. We use the Marital Status dummy variable MS as an exogenous variable. Wooldridge Condition (Exogeneity) (5.3) Cov[MS, ] = 0 is assumed. Auxiliary regression: For MSto be a valid, relevant instrumental variable, In the regression of WKS on [1,EXP,EXPSQ,OCC,South,SMSA,MS] MSsignificantly explains WKS. A projection interpretation: In the projection xitK = 1 xit1 + 2xit2+ + K-1xit,K-1 + Kzit +u, K 0.

  10. Part 8: IV and GMM Estimation [ 10/51] Auxiliary Projection of WKS on (X,z) Ordinary least squares regression LHS=WKS Mean = 46.81152 -------------------------------------------------------------- Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] -------------------------------------------------------------- Constant 45.4842872 .36908158 123.236 .0000 EXP .05354484 .03139904 1.705 .0881 EXPSQ -.00169664 .00069138 -2.454 .0141 OCC .01294854 .16266435 .080 .9366 SOUTH .38537223 .17645815 2.184 .0290 SMSA .36777247 .17284574 2.128 .0334 MS .95530115 .20846241 4.583 .0000 Stock and Staiger (and others) test for weak instrument, z2 > 10. 4.5832 = 21.004. We do not expect MS to be a weak instrument.

  11. Part 8: IV and GMM Estimation [ 11/51] IV for WKS in Lwage Equation - OLS Ordinary least squares regression. LWAGE | Residuals Sum of squares = 678.5643 | Fit R-squared = .2349075 | Adjusted R-squared = .2338035 | +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Constant 6.07199231 .06252087 97.119 .0000 EXP .04177020 .00247262 16.893 .0000 EXPSQ -.00073626 .546183D-04 -13.480 .0000 OCC -.27443035 .01285266 -21.352 .0000 SOUTH -.14260124 .01394215 -10.228 .0000 SMSA .13383636 .01358872 9.849 .0000 WKS .00529710 .00122315 4.331 .0000

  12. Part 8: IV and GMM Estimation [ 12/51] IV (2SLS) for WKS +----------------------------------------------------+ | LHS=LWAGE Mean = 6.676346 | | Standard deviation = .4615122 | | Residuals Sum of squares = 13853.55 | | Standard error of e = 1.825317 | +----------------------------------------------------+ -------------------------------------------------------------- |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | -------------------------------------------------------------- Constant -9.97734299 3.59921463 -2.772 .0056 EXP .01833440 .01233989 1.486 .1373 EXPSQ -.799491D-04 .00028711 -.278 .7807 OCC -.28885529 .05816301 -4.966 .0000 SOUTH -.26279891 .06848831 -3.837 .0001 SMSA .03616514 .06516665 .555 .5789 WKS .35314170 .07796292 4.530 .0000 OLS------------------------------------------------------ WKS .00529710 .00122315 4.331 .0000

  13. Part 8: IV and GMM Estimation [ 13/51] Generalizing the IV Estimator-1 Define a partitioned regression for n observations = + + = ( K K variables plim(1/n) is exogenous plim(1/n) is not 2 X 0, X = plim(1/n) ( is exogenous) W' 0 W y X X X , X ) + = X + 1 1 2 2 1 2 1 2 2 = X 0, X 1 1 exogenous K +K exogenous variables (Variables in are relevant) W'X 0 There exists a set of M plim(1/n) W such that 1 2 W

  14. Part 8: IV and GMM Estimation [ 14/51] Generalizing the IV Estimator - 2 Define the set of exogenous variables , typically Z X Z Z = = 1 1 = K linear combinations of the M "Exclusion restructions" There are at least K variables in = P 2 an MxK matrix. = [ , ]= [ , Why must M be K +K ? So can have full column rank. For to have full column rank, there must be at least K excl usions. (There must be at least one exclusion for each endogenous variable in .) X W s. Z WP 2 2 2 W that are not in is used to predict Z X . 2 1 WP is NxK , X WP Z X 2 2 2 Z Z Z X Z ]= [ ] 1 2 1 2 1 1 2 Z 2 2

  15. Part 8: IV and GMM Estimation [ 15/51] The Best Set of Instruments = [ What is the best to use (the best way to combine the exogenous instruments)? (a) If M = K +K , it makes no difference. (b) If M < K +K , there are too few instruments to (c) If M > K +K , there are many possible linear combinations and one best combination, 2SLS. Z Z Z , ]= [ X Z P , ]= [ X WP , ] 1 2 1 2 1 1 2 continue 1 2 1 2

  16. Part 8: IV and GMM Estimation [ 16/51]

  17. Part 8: IV and GMM Estimation [ 17/51]

  18. Part 8: IV and GMM Estimation [ 18/51] Two Stage Least Squares 1 A Class of IV estimators: = [ 2SLS is defined by (1) Regress ( 1 X Z Z , Z ] = [ X WP , ], =[ Z X ] Z y 1 2 1 and X ) on all of W , column by column, and ) = 2 X X 2 X and compute predicted values, ( X . 1 X Z = X X is rep roduced perfectly by regressing . is a linear combination of ) ( , ) 2 2 1 2 X X Z Z on itself so = X 1 1 1 X 1 1 = = Z = WP X and the extra variables) 2 2 2 2 2 1 = = X Z X = ( , 1 (2) Regress y on to estimate .

  19. Part 8: IV and GMM Estimation [ 19/51] 2SLS Estimator By the definitions, is a set of instrumental variables. [ ] is consistent and asymptotically normally distributed. ( )'( ) n-K Assuming ho moscedasticity and no autocorelation, Est.Asy.Var[ ] [ X'X Z -1 = X'X X'y n-K X X y X y X ( y )'( y ) 2 = , not X'X X'X = 2 -1 -1 ] [ ]

  20. Part 8: IV and GMM Estimation [ 20/51] 2SLS Algebra ) ) ( ( X'X = Z Z'Z -1 ( ) Z'X 'X ( -1 = X' Z Z'Z ( ) Z' X ) ( ) -1 -1 = X' Z Z'Z ( ) Z' Z Z'Z ( ) Z' X = X'X Therefore [ ] X'X X'X X'X X'X -1 -1 1 = [ ] [ ] n-K ( y X )'( y X ) X'X -1 = Est.Asy.Var[ ] [ ]

  21. Part 8: IV and GMM Estimation [ 21/51] 2SLS for Panel Data Fixed Effects (FE2SLS) 1 ( ( )( )( ) ( ) ( ) 1 i i i = N i 1 = N i 1 = N i 1 = b X M Z Z M Z Z M X D D D 2sls,lsdv i i i ) 1 i i i N i 1 = N i 1 = N i 1 = X M Z Z M Z Z M y D D D i i i 1 1 1 XZ Z Z Z X XZ Z Z) Z y =[( Convert data to group mean deviations, then use 2SLS Random Effects (RE2SLS) )( ) ( )] [( )( ( )] 1 ( ( )( )( ) ( ) ) ) 1 i i i = N i 1 = 1 N i 1 = 1 N i 1 = 1 b X Z Z Z Z X 2sls,RE i i i i i i ( 1 i i i N i 1 = 1 N i 1 = 1 N i 1 = 1 X Z Z Z Z y i i i i i i Step 1:Use conventional pooled 2SLS and FE2SLS. Estimate variance components. Step 2: Convert data to partial deviations from means, then use 2 SLS.

  22. Part 8: IV and GMM Estimation [ 22/51] CREATE ; id = trn(7,0)$ SETPANEL ; Group = id $ NAMELIST NAMELIST FE2SLS ; Lhs = lwage ; Rhs = X ; Inst = z ; Panel$ RE2SLS ; Lhs = lwage ; Rhs = X ; Inst = z ; Panel$ ; x = one,exp,expsq,occ,south,smsa,wks$ ; z = one,exp,expsq,occ,south,smsa,ms,union,ed$

  23. Part 8: IV and GMM Estimation [ 23/51] CREATE ; id = trn(7,0)$ SETPANEL ; Group = id $ NAMELIST NAMELIST FE2SLS ; Lhs = lwage ; Rhs = X ; Inst = z ; Panel$ RE2SLS ; Lhs = lwage ; Rhs = X ; Inst = z ; Panel$ ; x = one,exp,expsq,occ,south,smsa,wks$ ; z = one,exp,expsq,occ,south,smsa,ms,union,ed$

  24. Part 8: IV and GMM Estimation [ 24/51] GMM Estimation Orthogonality Conditions General Model Formulation: ; plim[(1/n) M K Instrumental variables ; plim[(1/n) IV formulation implies M orthogonality conditions E[z (y )] x' 0 for each instrumental variable z . 2SLS uses only K of these in the form E[x (y )] 0 where x = x' + y = X X' ] 0 (possibly) K regressors in Z X Z' ] = 0. = m m M j=1 = z k k jk j = X'X X'y -1 Solution is Consider an estimator that uses all M equations wh The orthogonality condition to mimic is E[(1/n) z (y )]=0, m=1,...,M This is M equations in K unknowns each of the form E[g ( )]=0. ( ) en M > K i n i=1 im x i m

  25. Part 8: IV and GMM Estimation [ 25/51] GMM Estimation - 1 i n i=1 im z (y E[(1/n) E[g ( )]=0, m = 1,...,M. Sample counterparts - finding the estimator: (1/n) x )=0 (a) If M = K, the one exact solution is 2SLS (b) If M < K, there are t (c) If M > K, there are excess equations. How to reconcile them? x )]=0, m=1,...,M i m i N i=1 im z (y i oo few equations. No solution. First Pass: "Least Squares" ( ) 2 : M i n i=1 im z (y = Try Minimizing wrt (1/n) x ) ( ) g 'g ( ) i m=1

  26. Part 8: IV and GMM Estimation [ 26/51] GMM Criterion Function: Minimize with respect to 7 elements of this sum of 9 terms. 2 2 2 it it it + + + ) x ) 2 lwag e exp lwage ex sq p lwag e 1 x x x ( ( ( ) 1 1 1 it it it it it it it it 4195 4195 4195 2 2 it it + + + occ lwage south lwage smsa lwage x x ( ) ( ) ( ) 1 1 1 i t it it it it it it it it it 4195 4195 4195 2 2 2 it it it + + ms lwage union lwage ed lwage x x x ( ) ( ) ( ) 1 1 1 it it it it it it it it it 4195 w here x 4195 4195 = + 1 + + + + + exp expsq occ south s msa w ks 1 2 3 4 5 6 7 it it it it it it it 9 7 The overidentification (redundancy) arises because we could use any of the = 36 possible sets of 7 terms out of the 9 to obtain the estimates. GMM provides a way to combine these efficiently. NAMELIST NAMELIST 2SLS NLSQ ; x ; z ; lhs ; fcn ; labels = b1,b2,b3,b4,b5,b6,b7 ; start = b ? (Starting values are 2SLS) ; inst = Z ; pds = 0 $ ? (Use White Estimator) = one,exp,expsq,occ,south,smsa,wks$ = one,exp,expsq,occ,south,smsa,ms,union,ed$ = lwage ; RHS = X ; INST = Z $ = lwage-b1'x ? (Linear function begins with b1)

  27. Part 8: IV and GMM Estimation [ 27/51] GMM Estimation - 2 ( ( ) ( ) ( ) = ( ) g 'g g ' I Minimum Distance Esti ) 2 = M i n i=1 im z (y = the minimizer of (1/n) x ) ( ) ( ), M g 'g K i m=1 -1 g ( ) mator This is a " More generally: Let Results: For any positive definite matrix , is consistent and asymptotically normally distributed. " with weight matrix = . I the minimizer of ( ) = g ' A g ( ) A 1 ( ) g ' ( ) g ' ( ) Asy.Var[ ]= A Asy.Va r[ ( )] g A (See JW, Ch. 14 for analysis of asymptotic properties.)

  28. Part 8: IV and GMM Estimation [ 28/51] Example: Suppose we have 9 positive weights V...V GMM Criterion Function 1 9 : Minimize with respect to 7 elements of this sum of 9 terms. 2 2 2 it i it + + + ) x ) lwage exp lwage e xpsq lwage 1 x x x V ( V ( V ( ) 1 1 1 1 2 3 it it it it i t t it it it 4195 4195 4195 2 2 2 it + + + occ lwage south lwage sms a lwa ge x x V ( ) V ( ) V ( ) 1 1 1 4 5 6 it it it it it it it it it it it 4195 4195 4195 2 2 2 it it it + + ms l wag e union lwage e d lwage x x x V ( ) V ( ) V ( ) 1 1 1 7 8 9 it it it it it it it it it 4195 419 5 4195 where x = + 1 + + + + + exp expsq occ so uth sm a s wks 1 2 3 4 5 6 7 it it it it it it it This is ( ) g ' A g ( ) 0 0 V 1 0 0 V 2 A where = 0 0 0 0 V 9

  29. Part 8: IV and GMM Estimation [ 29/51] An Optimal Weighting Matrix = the minimizer of ( ) For any positive definite matrix , is consistent and asymptotically normally distributed. g ' A g ( ) A 1 ( ) g ' ( ) g ' ( ) Asy.Var[ ]= A Asy.Var[ ( )] g A Is The most efficient estimator in the GMM class has there a 'best' matrix? A ( ) 1 A = Asy.Var[ ( )] g . ( ) 1 the minimizer of ( ) Asy.Var[ ( )] = g ' g ( ) g GMM

  30. Part 8: IV and GMM Estimation [ 30/51] The GMM Estimator ( ) 1 the minimizer of q = ( ) Asy.Var[ ( )] = g ' g ( ) g GMM 1 ( ) g ' ( ) g ' ( ) 1 Asy.Var[ ]= Asy.Var[ ( )] g GMM ( ) g ' = For IV estimation, ( ) = (1/n) ( g Z' y - X ), (1 /N) Z'X i 2 N i 1 = 2 2 2 = Asy.Var[ ( )] g (1 / n) z z =( / n ) Z'Z i 1 1 2 2 -1 2 -1 ]= ( (1 / n) ] ( (1 / n) Asy.Var[ X'Z )[( / n ) Z'Z Z'X ) = X'Z Z [ 'Z ] Z'X ) !!!!! GMM IMPLICATION: 2SLS is not just efficient for IV estimators that use a linear c ombination of the columns of Z. It is efficient among all estimators that use the columns of Z.

  31. Part 8: IV and GMM Estimation [ 31/51] Step 1. Minimize with respect to 7 elements of this sum of 9 terms. 2 2 2 it it it + + + ) x t ) x lwage exp lw age expsq lwage 1 x x x ( ( ( ) 1 1 1 it it it it it it it it 4195 4195 4195 2 2 2 it + + + occ lwage south lwage smsa lwage x ( ) ( ) ( ) 1 1 1 it it it i it it it it it it it 4195 4195 4195 2 2 2 it it it + + ms lwag e union lwa ge ed lwage x x x ( ) ( ) ( ) 1 1 1 it it it it it it it it it 4195 where x 4195 4195 = + 1 + + + + + ex p expsq occ s outh sms a wk s . 1 2 3 4 5 6 7 it it it it it it it Compute GMM Weighting Matrix After Step 1 Optimization. 1 expsq 1 1 exp expsq occ south smsa exp expsq oc south smsa ms union ed it it it it c it it 1 4195 it = = x [ ( )] g 2 lwage A . . ( ) Est AsyVar GMM it it it 2 4195 = 1 it it it ms it it uni on it it ed it it it = + 1 + + + + + exp occ south smsa wks x where 1 2 3 4 5 6 7 it it it it it it

  32. Part 8: IV and GMM Estimation [ 32/51] GMM with Optimal Weighting Matrix Minimize with respect to the 7 elements of this sum of 81 terms. (Note it is symmetric in j and m so there are 45 unique terms in the sum.) 9 9 ( ) it it 1 x A x q = ( ) ( ) z lwage it m z lwage 1 1 , , it i t j it GMM it it 4195 4195 jm = = 1 1 j m ( ) = ms union ed z x 1 = + exp expsq expsq + occ occ + south south + smsa smsa + it it it it it it it it it + wks 1 exp 1 2 3 4 5 6 7 it it it it it it it This can be treated as a conventional optimization problem.

  33. Part 8: IV and GMM Estimation [ 33/51] Extended GMM Estimation 1 n 1 n i n n = ( )= g z (y x ) z i i = i 1 = i 1 i i Assuming homoscedasticity and no autocorrelation. 2SLS is the efficient GMM estimator. What if there is heteroscedasticity? 1 Asy.Var[ ( )] , est n based on 2SLS residuals e. (This is precisely the matrix we computed earlier.) The GMM estimator minimizes 1 1 n n i i n 2 i n 2 i = g z z imated with e z z = i 1 = i 1 i i 2 i 1 1 n 1 1 n n 1 n i i i n n 2 i n = q z (y x ) ' e z z z (y x ) . = i 1 = i 1 = i 1 i i i i i -1 This is not 2SLS because the weighting matrix is not ( Z'Z ) .

  34. Part 8: IV and GMM Estimation [ 34/51] Application - GMM NAMELIST NAMELIST 2SLS NLSQ ; x = one,exp,expsq,occ,south,smsa,wks$ ; z = one,exp,expsq,occ,south,smsa,ms,union,ed$ ; lhs = lwage ; RHS = X ; INST = Z $ ; fcn = lwage-b1'x ; labels = b1,b2,b3,b4,b5,b6,b7 ; start = b ? 2sls starting values ; inst = Z ; pds = 0 $ White. If > 0, uses Newey-West)

  35. Part 8: IV and GMM Estimation [ 35/51] 2SLS GMM with Heteroscedasticity

  36. Part 8: IV and GMM Estimation [ 36/51] A General Minimum Distance Es timator Suppose you had M independent estimators of the same parameter vector, each with asymptotic covariance matrix An inefficient way to average these would be 1 would have Asy.Var Avg m m M m V m 1 m m = V = Avg m 2 = = M 1 1 m A more efficient way to average them would be a matrix weighted average For example, consider 2SLS with each of several instruments. 1 1 m m m = 1 1 1 V V V would have Asy.Var = MD m m m MD m = = = 1 1 1 m m m

  37. Part 8: IV and GMM Estimation [ 37/51]

  38. Part 8: IV and GMM Estimation [ 38/51]

  39. Part 8: IV and GMM Estimation [ 39/51] 1 s M j = 1 j M j = = w 1 s j j = 1 j M j = 1 j Not optimal, but better than a simple average.

  40. Part 8: IV and GMM Estimation [ 40/51] Analysis of Fannie Mae Fannie Mae The Funding Advantage The Pass Through

  41. Part 8: IV and GMM Estimation [ 41/51] First Stage Rate Difference = MortgageRate + "new home" (dummy variable) + "small loan" (dummy variable) + "up front fees paid" (dummy variable) + "mortgage bank" vs. depository inst. (dummy variable) + 1,i i,t 1,i i,t i,t + "i"= state,year grouping "t"= individual loan in specified state,year Nearly all "conforming" loans (under $317,000) are held by Fannie Mae. Expect to be > 0 as Fannie Mae is able to finance at lower cost than other institutions, and Fannie Mae does not finance Jumbo loans. Interest is in "pass through" of the cost advantage. +"loan to value ratio terms" i,t 0 JumboLoan ( dummy variable for loan > $3 JumboLoan ( dummy variable for loan > $317,000) 17,000) 1,i

  42. Part 8: IV and GMM Estimation [ 42/51] Second Stage Pass Through a = + "market characteristics" + "state" and "quarter" dummy variables + w + "Estimated Capital Cost Advantage" 1,i 0 1 i Primary interest is in capital cost advantage that is passed through to mortgagees. which is the amount of the 1 Result: Less than half of cost advantage was passed through to borrowers.

  43. Part 8: IV and GMM Estimation [ 43/51] A Minimum Distance Estimator Estimates of 1 Second step based on 370 observations. Corrected for "heteroscedasticity, autocorrelation, and monthly clustering." Four estimates based on different estimates of corporate credit spread: 0.07 (0.11) 0 .31 (0.11) 0.17 (0.10) 0.10 (0.11) Reconcile the 4 estimates with a minimum distance estimator (0.07 ) Minimize wrt : .11 2 2 2 2 2 2 2 2 (0.31 ) (0.17 ) (0.10 ) + + + 1 1 1 1 1 .11 .10 .11 Estimated mortgage rate reduction: About 17 basis points. .17%.

  44. Part 8: IV and GMM Estimation [ 44/51] The Minimum Distance Estimator 0.07 (0.11) 0.31 (0.11) .017 (0.10) 0.10 (0.11) Reconcile the 4 estimates with a minimum distance estimator 1 1 2 1 3 1 4 1 ( ( ( ( - ) - ) - ) - ) 1 1 1 2 3 4 -1 1 - ),( 1 1 1 Minimize [( - ),( - ),( - )]' 1 1 1 1 1 1 2 .07 /.11 = 1 + 2 2 2 2 + + + (1 /.11 ) (1 /.11 ) (1 /.10 ) (1 /.11 ) 2 .31 /.11 + 2 2 2 2 + + + (1 /.11 ) (1 /.11 ) (1 /.10 ) (1 /.11 ) + ... Approximately .17%.

  45. Part 8: IV and GMM Estimation [ 45/51] Testing the Overidentifying Restrictions ( ) 1 q = ( ) Asy.Var[ ( )] Under the hypothesis that E[ ( )] = , q [M K] g ' g ( ) g g 0 d 2 M = number of moment equations K = number of parameters estimated (In our example, M = 9, K = 7.) M - K = number of 'extr needed to identify the parameters. For the example, | Value of the GMM criterion: | | e(b)tZ inv(ZtWZ) Zte(b) = 537.3916 | a' moment equations. More than are

  46. Part 8: IV and GMM Estimation [ 46/51] Inference About the Parameters 1 i i N i=1 i 2 -1 N i=1 i 2 -1 = ( X'Z )[ z z ] ( Z'X ) ( X'Z )[ z z ] ( Z'y ) GMM i i 1 i N i=1 i 2 -1 Est.Asy.Var[ ]= ( X'Z )[ z z ] ( Z'X ) GMM i Restrictions can be tested using Wald statistics; H : ( )= H :Not H r h 0 1 0 ) ( ( ) ( ) 1 = ( Wald r ( )- ) h ' R E st.Asy.Var[ ] R r ( )- h GMM GMM GMM r ( )- h GMM = R GMM E.g., for a simple test, H : =0, this is the square of the t-ratio. 0 k

  47. Part 8: IV and GMM Estimation [ 47/51] Extending the Form of the GMM Estimator to Nonlinear Models Very little changes if the regression function is nonlinear. 1 Asy.Var[ ( )] n based on nonlinear 2SLS residuals e. The GMM estimator m 1 1 n n i i n 2 i n 2 i = g z z , estimated with e z z = i 1 = i 1 i i 2 inimizes i 1 1 n 1 1 n n 1 n i i i n n 2 i n = q z (y f( x )) ' e z z z (y f( x )) . = i 1 = i 1 = i 1 i i i i i The problem is essentially the same.

  48. Part 8: IV and GMM Estimation [ 48/51] A Nonlinear Conditional Mean i i = f( E[ (y Nonlinear instrumental variables (2SLS) minimizes (y exp( )) [ x z ' Z'Z x z ) exp( exp( x x ) ))] = 0 i i ( Nonlinear GMM then minimizes 1 (y exp( n = n ) ( ) i i n i=1 1 n i=1 ] z (y exp( x )) i i i i 1 n i i i n i=1 2 n 2 1 n i=1 x z )) '[(1 /n) z z ] z (y exp( x )) = i 1 i i i i i i 1 1 n W i i n i=1 1 n i=1 (y exp( x z )) ' z (y exp( x ) ) i i i i

  49. Part 8: IV and GMM Estimation [ 49/51] Nonlinear Regression/GMM NAMELIST NAMELIST ? Get initial values to use for optimal weighting matrix NLSQ ; lhs = lwage ; fcn=exp(b1'x) ; inst = z ; labels=b1,b2,b3,b4,b5,b6,b7 ; start=7_0$ ? GMM using previous estimates to compute weighting matrix NLSQ (GMM) ; fcn = lwage-exp(b1'x) ; inst = Z ; labels = b1,b2,b3,b4,b5,b6,b7 ; start = b ; pds = 0 $ (Means use White style estimator) ; x = one,exp,expsq,occ,south,smsa,wks$ ; z = one,exp,expsq,occ,south,smsa,ms,union,ed$

  50. Part 8: IV and GMM Estimation [ 50/51] Nonlinear Wage Equation Estimates NLSQ Initial Values

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