GMM

Callback-driven generalized method of moments

Code 
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 and raises when the iteration budget is exhausted without meeting the convergence tolerance. fit_sketch(...) row-sketches the moment problem. summary(vcov='sandwich', omega='iid', lags=None, clusters=None) controls inference.

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

3 Minimal example

Code 
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([0.9, 0.4, -0.3]) + v
y = 1.2 * x + 0.5 * v + rng.normal(size=n) * 0.3
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'])

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.

Code 
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([0.9, 0.4, -0.3]) + v
y = 1.2 * x + 0.5 * v + rng.normal(size=n) * 0.3
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))))