What is GMM and why is it so common in asset pricing?

Q: What is GMM and why is it so common in asset pricing?

GMM estimates parameters by making a set of 'moment conditions' hold on average. Moment conditions are functions g(x, theta) with theoretical expectation zero under the true theta. GMM picks theta to make the sample average of g as close to zero as possible, weighted optimally.\n\nZuzanna Kiprotich at Nairobi Asset Pricing Research estimates the consumption-CAPM. The Euler equation says E[beta * (C_{t+1}/C_t)^(-gamma) * (1+R_{t+1}) - 1] = 0 for each asset. With 12 test assets, there are 12 moment conditions; with only 2 parameters (beta, gamma), the system is overidentified.\n\nGMM steps:\n1. Compute sample moments g_T(theta) = (1/T) * sum of g(x_t, theta)\n2. Minimize Q(theta) = g_T(theta)' * W * g_T(theta) for weighting matrix W\n3. Optimal W is the inverse of covariance of moments — gives efficient GMM\n4. Update W using first-stage estimates (two-step GMM) or iterate (iterated GMM)\n\nWhy finance loves GMM:\n- Doesn't require full distributional assumptions (unlike MLE)\n- Handles heteroskedasticity and autocorrelation via robust covariance (Newey-West HAC)\n- Overidentification test (J-statistic) checks model fit: J = T * Q(theta_hat) ~ chi^2 with (#moments - #params) df\n- Natural for Euler equations, term structure models, stochastic discount factor tests\n\nZuzanna's fit gives beta=0.985, gamma=18.2, with J-stat p-value=0.032 — borderline rejecting the C-CAPM. This replicates the classic equity premium puzzle: reasonable fits require implausibly high risk aversion. GMM made that puzzle rigorous and testable.

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GMM estimates parameters by making a set of moment conditions hold on average. Moment conditions are functions with theoretical expectation zero under the true theta...

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Zuzanna Kiprotich at Nairobi Asset Pricing Research estimates the consumption-CAPM. The Euler equation says E[beta * (C_{t+1}/C_t)^(-gamma) * (1+R_{t+1}) - 1] = 0 for each asset. With 12 test assets, there are 12 moment conditions; with only 2 parameters (beta, gamma), the system is overidentified.

GMM steps:
1. Compute sample moments g_T(theta) = (1/T) * sum of g(x_t, theta)
2. Minimize Q(theta) = g_T(theta)' * W * g_T(theta) for weighting matrix W
3. Optimal W is the inverse of covariance of moments — gives efficient GMM
4. Update W using first-stage estimates (two-step GMM) or iterate (iterated GMM)

Why finance loves GMM:
- Doesn't require full distributional assumptions (unlike MLE)
- Handles heteroskedasticity and autocorrelation via robust covariance (Newey-West HAC)
- Overidentification test (J-statistic) checks model fit: J = T * Q(theta_hat) ~ chi^2 with (#moments - #params) df
- Natural for Euler equations, term structure models, stochastic discount factor tests

Zuzanna's fit gives beta=0.985, gamma=18.2, with J-stat p-value=0.032 — borderline rejecting the C-CAPM. This replicates the classic equity premium puzzle: reasonable fits require implausibly high risk aversion. GMM made that puzzle rigorous and testable.

What is GMM and why is it so common in asset pricing?

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