This paper studies a class of models where full identification is not necessarily assumed. We term such models partially identified. It is argued that partially identified systems are of practical importance since empirical investigators frequently proceed under conditions that are best described as apparent identification. One objective of the paper is to explore the properties of conventional statistical procedures in the context of identification failure. Our analysis concentrates on two major types of partially identified model: the classic simultaneous equations model under rank condition failures; and time series spurious regressions. Both types serve to illustrate the extensions that are needed to conventional asymptotic theory if the theory is to accommodate partially identified systems. In many of the cases studied, the limit distributions fall within the class of compound normal distributions. They are simply represented as covariance matrix or scalar mixtures of normals. This includes time series spurious regressions, where representations in terms of functionals of vector Brownian motion are more conventional in recent research following earlier work by the author.

We tabulate the limiting cumulative distribution and probability density functions of the least-squares estimator in a first-order autoregressive regression when the true model is near-integrated in the sense of Phillips. The results are obtained using an exact numerical method which integrates the appropriate limiting moment generating function. The adequacy of the approximation is examined for various first-order autoregressive processes with a root close to unity.

Let Yt satisfy the stochastic difference equation for t = 1,2,…, where et are independent and identically distributed random variables with mean zero and variance σ2 and the initial conditions (Y−p+1,…, Y0) are fixed constants. It is assumed that the process is invertible and that the true, but unknown, roots m1,m2,…,mp of satisfy the hypothesis Hd: m1 = … = md = 1 and |mj| < 1 for j = d + 1,…,p. We present a reparameterization of the model for Yt that is convenient for testing the hypothesis Hd. We consider the asymptotic properties of (i) a likelihood ratio type “F-statistic” for testing the hypothesis Hd, (ii) a likelihood ratio type t-statistic for testing the hypothesis Hd against the alternative Hd−1. Using these asymptotic results, we obtain two sequential testing procedures that are asymptotically consistent.

Wu introduced a new technique for proving consistency of least-squares estimators in nonlinear regression. This paper extends his results in three directions. First, we consider the minimization of arbitrary functions (M-estimators instead of least squares). Second, we use an improved type 2 inequality. Third, an extension of Kronecker's lemma yields a more powerful result.