Momentvoorwaarden

Momentvoorwaarden, or moment conditions, are equations that express constraints that must hold for the true model parameters, based on theoretical assumptions about the data-generating process. They take the form E[g(X, theta0)] = 0, where g is a function of the observed data X and the parameter vector theta0. In words, the expected value of certain functions of the data and the parameters should be zero if the model is correct.

These conditions underpin estimation methods such as the method of moments and the generalized method of moments (GMM). In practice, one uses the sample analogs by forming m_n(theta) = (1/n) sum_i g(X_i, theta). Estimation proceeds by either solving m_n(theta) = 0 (in exactly identified cases) or by minimizing a quadratic form m_n(theta)' W m_n(theta) with an appropriate weighting matrix W to obtain efficient estimates.

Moment conditions can be identified or overidentified. Identification requires that the moment functions provide enough information to pin down theta. When there are more moment conditions than parameters, the model is overidentified; this allows overidentification tests such as the Hansen J-test to assess the validity of the instruments or moment restrictions.

A common example occurs in instrumental variables estimation, where the instruments Z satisfy E[Z_i (y_i - X_i' beta)] = 0. Here, the moments link the observed data to the parameters through exogeneity assumptions. Moment conditions are central to many econometric models and are sensitive to specification; misspecification can lead to biased or inefficient estimates.