*! version 1.0.0 20may2026 Dr Merwan Roudane (merwanroudane920@gmail.com)

Title

mixi01_fmiv -- Fully Modified IV, GIVE, and GMM estimation

Syntax

mixi01_fmiv depvar indepvars [if] [in]
[,
options]

Model specification
iv(varlist) instrumental variables

i1(varlist) I(1) regressors and instruments

i0(varlist) I(0) regressors and instruments

Estimation method
method(string) iv (FM-IV), give (FM-GIVE), or
gmm (FM-GMM); default: iv

Long-run covariance estimation
kernel(string) kernel function; default: bartlett

bw(#|auto) bandwidth; default: auto

prewhiten apply VAR(1) prewhitening

GMM options
wmatrix(string) weight matrix for GMM: unadjusted,
robust, hac; default: hac

iterate(#) maximum number of GMM iterations; default: 2

Sargan test
sargan perform FM-Sargan test for instrument validity

Deterministic components
noconstant suppress constant

trend include time trend

Reporting
level(#) confidence level; default: level(95)

first display first-stage regression results

detail display long-run covariance estimates

Description

mixi01_fmiv estimates the regression

y_t = A x_t + u_0t

using FM-IV, FM-GIVE, or FM-GMM estimation as developed in Kitamura and
Phillips (1997). The regressors x_t may be endogenous and may contain
both I(1) and I(0) components with unknown cointegrating linkages. The
instruments z_t may also be a mixture of I(1) and I(0) processes, possibly
cointegrated with the regressors.

The FM-IV estimator (equation (10) in Kitamura and Phillips) takes the form:

A_tilde = (Y^{+′} P_Z X − T Delta^+_0z (Z'Z)^{−1} Z'X) (X' P_Z X)^{−1}

where P_Z = Z(Z'Z)^{−1}Z' and the FM corrections adjust for endogeneity and
serial correlation using the "a" subscript convention (Kitamura and Phillips,
Table 1).

FM-GIVE (Section 5.2 of KP97) applies a GLS-type transformation using the
spectral density of u_{0t} ⊗ z_{1t} at frequency zero to achieve efficiency
gains for the stationary components while maintaining optimal estimation of
the nonstationary components.

FM-GMM (Section 5.1 of KP97) uses an optimal distance matrix S_ in a
linear GMM framework extended to allow for nonstationary regressors and
instruments.

Key result (Theorem 4.3, KP97): Under Assumptions EC, IV and LR,

(a) sqrt(T)(A_tilde − A)H_1 →_d N(0, J_ S_ J'_) — normal for
the stationary components.

(b) T(A_tilde − A)H_2 →_d MN(0, Omega_{00.b} ⊗ (integral B_2 B_2')^{−1})
— mixed normal for the nonstationary components.

Options

Model specification

iv(varlist) specifies the instrumental variables z_t. The instrument
set should include q >= m variables, where m is the number of regressors.
Both I(1) and I(0) instruments are allowed.

i1(varlist) specifies which regressors and instruments are I(1).

i0(varlist) specifies which regressors and instruments are I(0).

Estimation method

method(iv) uses FM-IV (default). This applies FM corrections to
standard IV, providing consistency and median-unbiased estimation for both
stationary and nonstationary components.

method(give) uses FM-GIVE. This applies a GLS-type transformation to
achieve asymptotically efficient estimation of the stationary coefficients
(when instruments are strictly exogenous). The limit theory for the
nonstationary components is unchanged.

method(gmm) uses FM-GMM with an optimal distance matrix. Under the
Assumption NF (no feedback from errors to stationary instruments), FM-GMM
achieves the same efficiency as GIVE for the stationary components.

Sargan test

sargan performs the FM-Sargan test for over-identifying restrictions
(Section 6 of KP97). The test statistic is asymptotically chi-squared(q−m)
under the null of valid instruments.

GMM options

wmatrix(string) specifies the initial weight matrix for iterative GMM.

iterate(#) sets the maximum number of GMM iterations.

Remarks

Cointegrated instruments. Kitamura and Phillips (Section 4.2) show that
when the I(1) instruments and regressors are cointegrated — as arises naturally
when lagged regressors serve as instruments — the FM-IV estimator remains
valid with only minor changes to the limit theory (subscripts "b" replaced
by "c2").

Assumption NF (No Feedback). For FM-GMM and FM-GIVE efficiency, one
needs E[u_{0,s+j} ⊗ z_{1t}] = 0 for j >= 1. This is automatic when
instruments are strictly exogenous or when the orthogonality arises from
rational expectations.

FM-Sargan test. The FM version of the Sargan test adjusts for the
presence of nonstationary instruments by using FM residuals in place of
standard IV residuals. The resulting test is asymptotically chi-squared
under the null.

Examples

Example 1: Basic FM-IV

. mixi01_fmiv y x1 x2, iv(z1 z2 z3) i1(x1 z1 z2) i0(x2 z3) method(iv)

Example 2: FM-GMM with Sargan test

. mixi01_fmiv y x1 x2 x3, iv(z1 z2 z3 z4 z5) method(gmm) sargan kernel(qs) bw(auto)

Example 3: FM-GIVE with first-stage output

. mixi01_fmiv y x1 x2, iv(z1 z2 z3) i1(x1 z1 z2) i0(x2 z3) method(give) first

Stored results

mixi01_fmiv stores the following in e():

Scalars

e(N) number of observations

e(k) number of regressors

e(k_iv) number of instruments

e(k_i1) number of I(1) regressors/instruments

e(k_i0) number of I(0) regressors/instruments

e(bw) bandwidth used

e(sargan) FM-Sargan test statistic (if sargan)

e(sargan_p) p-value for FM-Sargan test

e(sargan_df) degrees of freedom for FM-Sargan test

Macros

e(cmd) mixi01_fmiv

e(depvar) dependent variable name

e(indepvars) regressor names

e(instruments) instrument names

e(method) iv, give, or gmm

e(kernel) kernel function

Matrices

e(b) coefficient vector

e(V) variance–covariance matrix

e(Omega) long-run covariance matrix

e(Delta) one-sided long-run covariance

e(Sigma_uu) error covariance Sigma_00

e(Omega_002) conditional long-run variance Omega_{00.2}

e(S_z) distance matrix for GMM (if method(gmm))

e(first_F) first-stage F statistics (if first)

Functions

e(sample) marks estimation sample

References

Hansen, L. P. (1982). Large sample properties of generalized method of
moments estimators. Econometrica, 50(4), 1029–1054.

Kitamura, Y. and P. C. B. Phillips (1997). Fully modified IV, GIVE and GMM
estimation with possibly non-stationary regressors and instruments.
Journal of Econometrics, 80(1), 85–123.

Phillips, P. C. B. and B. E. Hansen (1990). Statistical inference in
instrumental variables regression with I(1) processes.
Review of Economic Studies, 57(1), 99–125.

Sargan, J. D. (1958). The estimation of economic relationships using
instrumental variables. Econometrica, 26(3), 393–415.

Author

Dr Merwan Roudane

Department of Economics (Independent Researcher)

merwanroudane920@gmail.com

Also see

Master help — mixi01.

Sibling commands — mixi01_fmols, mixi01_fmvar,
mixi01_fmiv, mixi01_acl, mixi01_svar,
mixi01_vecm, mixi01_irf, mixi01_test.