When a continuous time model is estimated from its non-recursive discrete approximation, the presence of identities and exogenous variables in the system does not preclude the use of standard procedures. However, if we wish to use the exact discrete model for estimation purposes, the treatment of identities and exogenous variables is not so straightforward. It is found that the procedure based on the exact discrete model is unlikely to be affected by the presence of identities, but when exogenous variables occur in the system some sort of approximation is usually necessary before the model can be estimated with discrete data. An approximate model is constructed to deal with the latter case and the asymptotic properties of estimators derived from this model are investigated. UNDER CERTAIN CONDITIONS, a stochastic model represented by a system of continuously distributed lags can be regarded as the solution of a system,of linear stochastic differential equations. Two general approaches are available if we wish to estimate the parameters of such a system by conventional methods and with discrete data.2 The first approach (see [1 and 2]) is to take a discrete approximation to the model and estimate the approximate model by standard methods. The second approach makes use of the discrete model which is known to correspond to the continuous time model in the sense that observations at equidistant points in time that are generated by the latter system also satisfy the former. The main advantage of the second approach is that no specification error is involved, so that it is possible in some cases to obtain consistent and asymptotically efficient estimators of the parameters in the model. In addition to the arguments of asymptotic theory, the results of a previous study [8] have given some recommendation to the second approach on the basis of small sampling performance. However, the model used in the sampling experiment of this study was relatively simple and it is the aim of the present paper to discuss the use of the second approach in more complicated models. The complications with which we will be concerned are the presence of identities and exogenous variables; both these complications may be expected to occur in more realistic economic, models. Before the procedure is viable when there are identities in the model, we must ascertain whether the disturbance in the exact discrete model has a non-singular

[It is now popular to construct economic models in differential equation form. Perhaps the most serious econometric problem faced when dealing with a differential equation system is the practical difficulty of finding consistent estimates of the important structural parameters. In this paper a simple three-equation Phillips model is considered and consistent estimates of the structural parameters are provided by the minimum-distance procedure. The small-sample distributions of these estimates are investigated by the Monte Carlo method; and the results are then compared with those of the three-stage least-squares estimates found by making a discrete approximation to the system of differential equations.]

Government debt can be rolled over forever without primary surpluses in some stochastic economies, including some economies that are dynamically efficient. In an overlapping‐generations model with constant growth rate, g , of labor‐augmenting productivity, and with shocks to the durability of capital, we show that along a balanced growth path, the maximum sustainable ratio of bonds to capital is attained when the risk‐free interest rate, r f , equals g . Furthermore, this maximal ratio maximizes utility per capita along a balanced growth path and ensures that the economy is dynamically efficient.

We consider the problem of constructing confidence intervals (CIs) for a linear functional of a regression function, such as its value at a point, the regression discontinuity parameter, or a regression coefficient in a linear or partly linear regression. Our main assumption is that the regression function is known to lie in a convex function class, which covers most smoothness and/or shape assumptions used in econometrics. We derive finite‐sample optimal CIs and sharp efficiency bounds under normal errors with known variance. We show that these results translate to uniform (over the function class) asymptotic results when the error distribution is not known. When the function class is centrosymmetric, these efficiency bounds imply that minimax CIs are close to efficient at smooth regression functions. This implies, in particular, that it is impossible to form CIs that are substantively tighter using data‐dependent tuning parameters, and maintain coverage over the whole function class. We specialize our results to inference on the regression discontinuity parameter, and illustrate them in simulations and an empirical application.