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We develop asymptotic approximations to the distribution of forecast errors from an estimated AR(1) model with no drift when the true process is nearly I(1) and both the forecast horizon and the sample size are allowed to increase at the same rate. We find that the forecast errors are the sums of two components that are asymptotically independent. The first is asymptotically normal whereas the second is asymptotically nonnormal. This throws doubt on the suitability of a normal approximation to the forecast error distribution. We then perform a Monte Carlo study to quantify further the effects on the forecast errors of sampling variability in the parameter estimates as we allow both forecast horizon and sample size to increase.

We study the semi-parametric estimation of the conditional mode of a random vector that has a continuous conditional joint density with a well-de…ned global mode. A novel full-system estimator is proposed and its asymptotic properties are studied. We speci…cally consider the estimation of vector autoregressive conditional mode models and of systems of linear simultaneous equations de…ned by mode restrictions. The proposed estimator is easy to implement and simulations suggest that it is reasonably behaved in …nite samples. An empirical example illustrates the application of the proposed methods, including its use to obtain multi-step forecasts and to construct impulse response functions.

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