Dynamic Vector Mode Regression

Abstract

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.

Testing Forecast Rationality for Measures of Central Tendency

“…As a building block for the above tests, we also present new tests for the rationality of mode forecasts. Mode regression has received some attention in the recent literature (see, e.g., Silva, 2012 andKemp et al, 2019), however tests for mode forecast rationality similar to those available for the mean and median (e.g., Mincer andZarnowitz, 1969 andGaglianone et al, 2011) are lacking. Direct analogs of existing tests are infeasible because the mode is not elicitable (Heinrich, 2014).…”

“…Following Kemp and Silva (2012) and Kemp et al (2019), we estimate the covariance matrix by its sample counterpart,…”

“…The constant k 1 equals the median absolute deviation which is a robust measure for the scale (standard deviation) of the data and follows the suggestion of Kemp and Silva (2012) and Kemp et al (2019). The choice of the bandwidth parameter should be proportional to the scale of the underlying data such that test results are robust to linear re-scaling.…”

“…In a similar manner, Eddy (1980), Kemp and Silva (2012) and Kemp et al (2019) propose estimation of the (conditional) mode through estimating the modal interval with an asymptotically shrinking length. We formalize these ideas in the decision-theoretical framework through the following definition.…”

“…We consider the (elicitable) "generalized modal interval," defined in detail in Section 2.2, and show that it converges to the mode for the class of strongly unimodal probability distributions. We combine these results with recent work on mode regression (Kemp and Silva, 2012;Kemp et al, 2019) to obtain mode forecast evaluation tests analogous to well-known tests for mean and median forecasts.…”