Generalized Method of Moments (GMM) in StatsModels

Generalized Method of Moments (GMM) is a flexible estimation technique that uses moment conditions relationships expected to hold in the data to estimate model parameters. In StatsModels, GMM is implemented as a class that you subclass to define your own moment conditions. The process is especially useful for both linear and non-linear models, and works with or without instrumental variables.

How GMM Works in StatsModels

• Moment Conditions: The moment conditions are defined that relate the data and parameters. These are mathematical expressions expected to be zero when evaluated at the true parameter values.
• Subclassing: In StatsModels, you implement GMM by subclassing the GMM class and defining the momcond method, which returns your moment conditions.
• Estimation: StatsModels estimates parameters by minimizing the (weighted) sum of squared sample moments, iteratively updating the weighting matrix for efficiency.

Step-by-Step Implementation

Step 1: Import Libraries and Prepare Data

Let's create a simple linear model:

y = \beta_0 + \beta_1 x + \epsilon

We will estimate β_0 and β_1 using GMM.

• Import numpy as np: Load NumPy for array and math operations.
• from statsmodels.sandbox.regression.gmm import GMM: Import GMM class for estimation.
• np.random.seed(42): Set random seed for reproducibility.
• n = 100: Set sample size.
• x = np.random.normal(size=n): Generate 100 random x values.
• \beta_0 = 1.0; \beta_1 = 2.0: Set true intercept and slope.
• epsilon = np.random.normal(scale=1.0, size=n): Generate random noise.
• y = \beta_0 + \beta_1 x + \epsilon : Create y using the linear model.
• instruments = np.column_stack((np.ones(n), x)): Stack constant and x as instrument matrix.

import numpy as np
from statsmodels.sandbox.regression.gmm import GMM

# Simulate data
np.random.seed(42)
n = 100
x = np.random.normal(size=n)
beta_0 = 1.0
beta_1 = 2.0
epsilon = np.random.normal(scale=1.0, size=n)
y = beta_0 + beta_1 * x + epsilon

# Instruments (here, just use x and a constant as instruments)
instruments = np.column_stack((np.ones(n), x))

Step 2: Define the GMM Model by Subclassing

The LinearGMM class defines how the moment conditions are constructed: residuals multiplied by instruments.

class LinearGMM(GMM):
    def momcond(self, params):
        # params: [beta_0, beta_1]
        y_hat = params[0] + params[1] * self.exog[:, 1]
        error = self.endog - y_hat
        # Moment conditions: error * instruments
        return error[:, None] * self.instrument

Explanation:

• params are the parameters to estimate.
• error is the residual.
• Moment conditions are the product of residuals and instruments.

Step 3: Initialize and Fit the GMM Model

The model is initialized with the data and fit using the fit() method. start_params gives initial guesses for the parameters. maxiter controls the number of iterations for updating the weighting matrix.

# Prepare exog with constant and x
exog = np.column_stack((np.ones(n), x))

# Initialize model
model = LinearGMM(y, exog, instruments)

# Fit model (one-step)
results = model.fit(start_params=[0, 0], maxiter=2)

Output:

Step 4: View Output

• results.params gives the estimated coefficients (β_0 and β_1). results.bse provides their standard errors.
• Output Interpretation: The estimated parameters should be close to the true values used to generate the data (here, 1.0 and 2.0).
  

print("Estimated parameters:", results.params)
print("Standard errors:", results.bse)

Output:

Download the complete source code from here : Generalised Method of Moments