# Contemporaneous regressors in the Error-correction Mechanism

I am estimating an Error Correction Model using the two step approach.

In the usual form the ECM second-step regression includes lagged first differences of independent variables $X_{1}, ..., X_{k}$ :

$\Delta Y_t = \alpha + \beta_1\Delta X_{1,t-1} + ... + \beta_k\Delta X_{k,t-1} + \lambda resid_{t-1} + \varepsilon_t$ $(1)$

I have seen examples where contemporaneous first differences of $X_{1}, ..., X_{k}$ are included in the second-step equation. Under which assumptions is it possible to include $\Delta X_{1t}, ..., \Delta X_{kt}$ in equation $(1)$? Any textbook reference would be greatly appreciated. Thanks.

The problem with including contemporaneous terms $\Delta X_{i,t}$ on the right hand side of equation $(1)$ is that they may be correlated with the error terms $\varepsilon_t$. In other words, (loosely speaking) $\Delta X_{i,t}$ might be endogenous (precise definition of endogeneity may be a bit more complicated, but that does not hurt the intuition of the argument). In presence of correlation, the OLS estimators for the coefficients on $\Delta X_{i,t}$ will be inconsistent.
The condition under which it is fine to include $\Delta X_{i,t}$ on the right hand side of equation $(1)$ is that $\Delta X_{i,t}$ is uncorrelated with $\varepsilon_t$.