We develop two joint tests for the parametric drift and volatility functions of a diffusion model based on empirical processes. One key feature of our joint tests is that they account for different convergence rates of parameter estimators. The tests are of classical Kolmogorov–Smirnov and Cramér–von Mises types, and are asymptotically distribution free. The proposed tests have nontrivial power against a class of local alternatives with different convergence rates for the drift and volatility terms. Monte Carlo simulations show that the tests perform quite well in finite samples and outperform the nonparametric test of Hong and Li. The new tests are applied to EUR/USD exchange rate data and generate some interesting empirical findings that are consistent with our theoretical results and simulation studies.

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