Topic 3: The Generalized Method of Moments

Dr Pham Thi Bich Ngoc Hoa Sen University ngoc.phamthibich@hoasen.edu.vn  Simple moment conditions Population E[ ]  Sample  cov[ X ,  ]   T T ˆ t  X ˆ 0 ' t t 0  OLS as a MM estimator y  X   , so that ˆ  y  Xˆ  Moment conditions: T E[ ]   T1  ( yt  X t ˆ )  t 1 E[ X '  ]   X ' ( y  Xˆ )   MM estimator 1 ˆ ˆ X ' y  X ' X      X ' X  X ' y   IV is a MM estimator cov[X ,  ]  0, but cov[Z ,  ]  Moment condition: E[ Z '  ]   Z ' ( y  Xˆ IV )   MM estimator: Z ' y  Z ' Xˆ IV   ˆ  Z ' X  Z ' y 1 IV  In the previous IV estimator we have considered the case where the number of instruments is equal to the number of coefficients we want to estimate    Size of Z is the same as the size of IV What happens if the number of instruments is greater than the number of coefficients? Essentially, the number of equations is greater than the number of coefficients you want to estimate: model is over-identified   Maintain the moment condition as before Variance of moment condition is: var[Z '  ]  E[Z '  ' Z ]   2 Z ' Z  W  Minimise ‘weighted’ distance: V   ' ZW Z '  1 ˆ  ( y '  ' X ' ) Z ( Z ' Z ) Z ' ( y  Xˆ ) 1   First order conditions: V  X ' Z ( Z ' Z ) 1 Z ' ( y  Xˆ )  ˆ  MM estimator (looks like an IV estimator with more instruments than parameters to estimate): 1 1 1 ˆ   ( X ' Z Z ' Z  Z ' X ) X ' Z Z ' Z  Z ' y Index: i = 1, ,N for individuals g = 1, ,G for equations (this would be t=1, T for a panel) Data matrices: G rows,  y i1   x i1     y i2    yi  ,X    i      y iG   0 K1 0 x i2 0 K2 0   β1   i1       0  β i2     ,β = , εi =            x iG   βG   iG  K G columns y i  X iβ + ε i  zi1    Zi     0 0 zi2 L1 L2 0 0   0  , G rows (1 for each equation)   xiG  L G columns Such that  zi1,1i1        z  0 i1,2 i1    E  zi1i1   E   for L1 instrumental variables         z i1,L1 i1    Same for zi2i2 ,  zi1i1    L1 rows  z     L rows 0 i2 i2    E[Ziε]  E  , for observation i         ziGiG    L G rows Summing over i gives the orthogonality condition,   zi1i1     L1      N  zi2 i2     L 1 N   E  i=1Ziε   E i=1  N      N          ziGiG     L G rows rows rows 10 (1) In case of TRIANGLE RELATIONSHIP: IV is an external variable  STATA: ◦ ivregress gmm depvar1 [varlist1] (depvar2 = varlistiv) Estat endog / estat overid ◦ ivreg2 depvar1 [varlist1] (depvar2 = varlistiv), (gmm2s) ◦ EG: ivregress gmm Y [X2] (X1 = Z1 Z2 Z3) Or: gmm (Y - {b1}*X2 - {b2}*X1 - {b0}), instruments(X1 Z1 Z2 Z3) wmatrix(robust) 11 (1) In case of TRIANGLE RELATIONSHIP: IV is an external variable  STATA: ◦ gmm ([eqname1:]) ([eqname2:]) [in] [weight] [, options] [if] where is the substitutable expression for the jth moment equation ◦ EG: ivregress gmm Y [X2] (X1 = Z1 Z2 Z3) Or: gmm (Y - {b1}*X2 - {b2}*X1 - {b0}), instruments(X1 Z1 Z2 Z3) wmatrix(robust) 12 (2) In case IV is the lagged endogenous variable:  STATA: ◦ ivreg2 depvar1 [varlist1] (depvar2 = varlistiv), (gmm2s)  varlistiv will included lagged variables ◦ gmm (Y - {b1}*X2 - {b2}*X1 - {b0}), instruments(X1 Z1 Z2 Z3) wmatrix(robust) ◦ (Z1 Z2 Z3) can be replaced for (L.Z1 L.Z2 L2.Z3) 13 Linear generalized method of moments (GMM) Arellano-Bond (1991); Arellano-Bover (1995)/ Blundell-Bond(1998) yit = αyi,t-1 + x’it β + αi + εit Reasons: True dependence, observed or unobserved heterogeneity, error correlation yi,t-1 is correlated with αi OLS – inconsistent (Nickell, 1981) Remove αi (yi,t-1 - yi,t-2 ) is correlated with (εit - εi,t-1 ) FE – inconsistent (Bond, 2002) But yi,t-2 is uncorrelated with (εit - εi,t-1 ) GMM – consistent (Hansen, 1982) Principle: E[Z’Ê] =  2SLS Σ yi,t-2 SYS – GMM êit* = for each t>=3 DIF – GMM collapsed makes an assumption that first difference of instruments variables are uncorrelated with the fixed effects: yi1  Δwi2 … Main tests are for serial correlation, over-identifying restrictions (Sargan, Hansen) 14  STATA/panel data ◦ xtabond/xtabond2 depvar varlist [if exp] [in range] ◦ Options:  gmmstyle(varlist [, laglimits(# #) collapse orthogonal equation({diff | level | both}) passthru {cmdab:sp: d:)}  ivstyle(varlist [, equation({diff | level | both}) passthru mz]) 15  Common Obtions: ◦ noleveleq specifies that level equation should be excluded from the estimation, yielding difference rather than system GMM ◦ small requests t statistics instead of z statistics and an F test instead of a Wald chi-squared test of overall model fit thng ke t dung f test thay vi m test 16  Common Obtions: ◦ robust For one-step estimation, robust specifies that the robust estimator of the covariance matrix of the parameter estimates be calculated The resulting standard error estimates are consistent in the presence of any pattern of heteroskedasticity and autocorrelation within panels tranh phuong sai thay doi va da cong tuyen In two-step estimation, the standard covariance matrix is already robust in theory but typically yields standard errors that are downward biased twostep robust requests Windmeijer’s finitesample correction for the two-step covariance matrix ◦ 17  Common Obtions: twostep twostep specifies that the two-step estimator is to be calculated instead of the one-step ngoac la nhung bien ko b noi sinh ivstyle() specifies a set of variables to serve as standard instruments, with one column in the instrument matrix per variable Normally, strictly exogenous regressors are included in ivstyle options, in order to enter the instrument matrix, as well as being listed before the main comma of the command line 18  Common Options: gmmstyle : specifies a set of variables to be used as bases for "GMM-style" instrument sets laglimits(a b): for the transformed equation, lagged levels dated t-a to t-b are used as instruments, while for the levels equation, the first-difference dated t-a+1 is normally used + a and b can each be missing ("."); a defaults to and b to infinity + E.g., gmm(w, lag(2 )), the standard treatment for an endogenous variable, is equivalent to gmm(L.w, lag(1 )), thus gmm(L.w) + the lag limits are a and b, then lags of the specified variables in differences dated t-b to t-a are used tat ca bien qua khu deu dung lam bien cong cu 19  Arellano-Bond test for AR(1) in first differences The null hypothesis: no autocorrelation If p-value 5%  instruments are valid 20 The augmented Cobb- Doughlas model: LnY = F (LnK, LnL, LnM, LnFDI) INPUT ENDOGENEITY PROBLEM 21 22 23 24 25 [...]... excluded from the estimation, yielding difference rather than system GMM ◦ small requests t statistics instead of z statistics and an F test instead of a Wald chi-squared test of overall model fit thng ke t dung f test thay vi m test 16  Common Obtions: ◦ robust For one-step estimation, robust specifies that the robust estimator of the covariance matrix of the parameter estimates be calculated The resulting... that the two-step estimator is to be calculated instead of the one-step trong ngoac la nhung bien ko b noi sinh ivstyle() specifies a set of variables to serve as standard instruments, with one column in the instrument matrix per variable Normally, strictly exogenous regressors are included in ivstyle options, in order to enter the instrument matrix, as well as being listed before the main comma of the. .. thus gmm(L.w) + the lag limits are a and b, then lags of the specified variables in differences dated t-b to t-a are used tat ca bien qua khu deu dung lam bien cong cu 19  Arellano-Bond test for AR(1) in first differences The null hypothesis: no autocorrelation If p-value =3 DIF – GMM collapsed makes an assumption that first difference of instruments variables are uncorrelated with the fixed effects: yi1  Δwi2 … Main tests are for serial correlation, over-identifying restrictions (Sargan, Hansen) 14  STATA/panel data ◦ xtabond/xtabond2 depvar varlist [if...(1) In case of TRIANGLE RELATIONSHIP: IV is an external variable  STATA: ◦ ivregress gmm depvar1 [varlist1] (depvar2 = varlistiv) Estat endog / estat overid ◦ ivreg2 depvar1 [varlist1] (depvar2 = varlistiv), (gmm2s) ◦ EG: ivregress gmm Y [X2] (X1 = Z1 Z2 Z3) Or: gmm (Y - {b1}*X2 - {b2}*X1 - {b0}), instruments(X1 Z1 Z2 Z3) wmatrix(robust) 11 (1) In case of TRIANGLE RELATIONSHIP: IV... hypothesis: no autocorrelation If p-value 5%  instruments are valid 20 The augmented Cobb- Doughlas model: LnY = F (LnK, LnL, LnM, LnFDI) INPUT ENDOGENEITY PROBLEM 21 22 23 24 25 ... In the previous IV estimator we have considered the case where the number of instruments is equal to the number of coefficients we want to estimate    Size of Z is the same as the size of. .. specifies that the robust estimator of the covariance matrix of the parameter estimates be calculated The resulting standard error estimates are consistent in the presence of any pattern of heteroskedasticity... size of IV What happens if the number of instruments is greater than the number of coefficients? Essentially, the number of equations is greater than the number of coefficients you want to estimate: