This paper presents an alternative method to derive the limiting distribution of residual-based statistics. Our method does not impose an explicit assumption of (asymptotic) smoothness of the statistic of interest with respect to the model's parameters and thus is especially useful in cases where such smoothness is difficult to establish. Instead, we use a locally uniform convergence in distribution condition, which is automatically satisfied by residual-based specification test statistics. To illustrate, we derive the limiting distribution of a new functional form specification test for discrete choice models, as well as a runs-based tests for conditional symmetry in dynamic volatility models.