A model specification test for nonlinear stochastic diffusions with delay

This paper investigates model specification problems for nonlinear stochastic differential equations with delay (SDDE). Compared to the model specification for conventional stochastic diffusions without delay, the observed sequence does not admit a Markovian structure so that the classical testing procedures may not be applicable. To overcome this difficulty, a moment estimator is newly proposed based on the ergodicity of SDDEs and its asymptotic properties are established. Based on the proposed moment estimator, a testing procedure is proposed for our model specification testing problems. Particularly, the limiting distributions of the proposed test statistic are derived under null hypotheses and the test power is examined under some specific alternative hypotheses. Finally, a Monte Carlo simulation is conducted to illustrate the finite sample performance of the proposed test.