On the Calculation of Critical Values of the Durbin–Watson Test

In this work, we present an accurate and efficient algorithm (based on neural networks) for computing the lower and upper critical values of the Durbin–Watson test for first-order autocorrelation, tailored for large sample sizes T and a large number of regressors k - a setting increasingly common in big-data applications. The critical values are obtained by numerically inverting the cumulative distribution function (CDF) using an adaptive Newton scheme. Evaluation of the CDF relies on numerical integration of the characteristic function of the underlying random variables. Because computing this characteristic function becomes expensive for large T, we develop a computationally cheap yet accurate approximation of the integrand. Since the convergence of the root-finding procedure depends strongly on the choice of initial values, we train neural networks on a precomputed dataset of critical values. For any given pair (T, k), the trained model provides accurate initial guesses, which significantly accelerate the adaptive Newton method. We validate our approach by recomputing previously published critical values and show that several of them are imprecise or internally inconsistent. Finally, we demonstrate that the proposed algorithm computes critical values within a fraction of a second on a standard laptop, even for sample sizes as large as \(2 \cdot 10^6\) $2·{10}^6$ .