GMM

Callback-driven generalized method of moments

from _api_doc_utils import *

1 Where it fits

Group: Estimation interfaces

GMM solves moment restrictions of the form

\[ \mathbb E[g_i(\theta)] = 0. \]

The user supplies a Python callback returning the per-observation moment matrix. In exactly identified cases the class can solve by Gauss-Newton; in overidentified cases it can use identity or two-step weighting and report sandwich covariance estimates.

2 Python API

Constructor: cm.GMM

Construct with GMM(moment_fn, jacobian_fn=None, max_iterations=100, tolerance=1e-6, ridge=1e-8, fd_eps=1e-6). fit(data, theta0, weighting='auto') stores the fitted parameters. fit_sketch(...) row-sketches the moment problem. summary(vcov='sandwich', omega='iid', lags=None, clusters=None) controls inference.

print(inspect.signature(cm.GMM))

(moment_fn, jacobian_fn=None, max_iterations=100, tolerance=1e-06, ridge=1e-08, fd_eps=1e-06)

cls = cm.GMM
display(HTML(html_table(["Public method"], public_methods(cls))))

Public method
fit(self, /, data, theta0, weighting='auto')
fit_sketch(self, /, data, theta0, sketch_size, weighting='auto', seed=None)
summary(self, /, vcov='sandwich', omega='iid', lags=None, clusters=None)

3 Minimal example

def moments(theta, data):
    resid = data["y"] - data["x"] * theta[0]
    return data["z"] * resid[:, None]

def jac(theta, data):
    return -(data["z"].T @ data["x"][:, None]) / data["x"].shape[0]

rng=np.random.default_rng(20); n=300; z=rng.normal(size=(n,3)); v=rng.normal(size=n); x=z@np.array([.9,.4,-.3])+v; y=1.2*x+.5*v+rng.normal(size=n)*.3
model=cm.GMM(moments, jacobian_fn=jac, max_iterations=200); model.fit({"x":x,"y":y,"z":z}, np.array([0.0]), weighting="identity")
print(model.summary()["coef"])
print(model.summary()["j_stat"])

[1.16176096]
0.7350528644433012

4 summary() contract

The table below is generated by fitting the live class in this repository and then inspecting summary(). Shapes are shown because most values are plain NumPy arrays or scalars.

def moments(theta, data):
    resid=data['y']-data['x']*theta[0]
    return data['z']*resid[:,None]
def jac(theta,data):
    return -(data['z'].T@data['x'][:,None])/data['x'].shape[0]
rng=np.random.default_rng(120); n=120; z=rng.normal(size=(n,3)); v=rng.normal(size=n); x=z@np.array([.9,.4,-.3])+v; y=1.2*x+.5*v+rng.normal(size=n)*.3
model=cm.GMM(moments,jacobian_fn=jac,max_iterations=200); model.fit({'x':x,'y':y,'z':z},np.array([0.0]),weighting='identity')
summary = model.summary()
display(HTML(html_table(["summary() key", "shape"], summary_shape_rows(summary))))

summary() key	shape
coef	(1,)
se	(1,)
vcov	(1, 1)
criterion	()
nit	()
weighting	()
vcov_type	()
omega_type	()
weight_matrix	(3, 3)
nobs	()
n_moments	()
original_n_moments	()
sketch_size	()
j_stat	()
j_df	()