A natural approach to survival analysis in many settings is to model the subjectâ€™s â€œhealthâ€ status as a latent stochastic process, where the terminal event is represented by the first time that the process crosses a threshold. â€œThreshold regressionâ€ models the covariate effects on the latent process. Much of the literature on threshold regression assumes that the process is one-dimensional Wiener, where crossing times have a tractable inverse Gaussian distribution but where the process characteristics are fixed at baseline. This framework is not easily extended to incorporate time-varying covariates or dependent competing risks. We introduce a novel approach for performing threshold regression with time-dependent covariates in a discrete time setting, where the process is a Gaussian random walk, with time-varying drift as a parameterized function of time-varying covariates. We estimate model parameters using the E-M algorithm, and present numerical algorithms for efficiently evaluating the observed and complete data likelihoods and estimating information matrices. We discuss results of applying this method to both simulated data and to the Freddie Mac residential mortgage data set. In the latter case we quantify associations between macroeconomic conditions and mortgage default and prepayment behavior.