momentconditions

Moment conditions are restrictions that connect observable data to unknown model parameters through expected values. Formally, suppose a random vector X and a parameter vector theta satisfy E[m(X, theta0)] = 0 for the true parameter theta0, where m is a vector-valued function of X and theta. These conditions reflect the probabilistic implications of a model, such as zero mean errors or instruments being uncorrelated with the error term.

In estimation, one uses the population moment conditions as a guide to construct estimators from sample data. The sample analog is m_n(theta) = (1/n) sum m(X_i, theta). The Method of Moments solves m_n(theta) = 0 when possible. The Generalized Method of Moments (GMM) minimizes Q_n(theta) = m_n(theta)' W m_n(theta) with a positive definite weighting matrix W. The choice of W influences efficiency; the optimal W is the inverse of the covariance matrix of m_n.

Examples: In a simple linear regression with E[e|X] = 0, a basic moment condition is E[e] = 0, i.e., E[Y - beta X] = 0. In instrumental variable settings, one uses E[Z (Y - X'beta)] = 0, where Z is a valid instrument; these generate multiple moment conditions when Z has dimension greater than the parameter vector.

Identification and testing: The number of moment conditions K relative to the number of parameters p determines identification. If K > p, the model is overidentified and there are testable restrictions, such as Hansen’s J test. If K = p, the model is exactly identified. Misspecified moments can lead to inconsistent estimates; standard errors must account for heteroskedasticity and potential misspecification.

Applications: Moment conditions are central in econometrics, finance, and causal inference, especially for models where the likelihood is difficult to specify or exogeneity/instrumental relationships provide natural restrictions.