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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.

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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 $\varepsilon_t$.

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  • $\begingroup$ So, in line of principle, if Xt is the lag of a macroeconomic variable (i.e. the past quarter's GDP growth) this will likely be uncorrelated with future error terms (εt). Is it right? $\endgroup$
    – Math
    Jan 31 '17 at 9:06
  • $\begingroup$ @Math, That is an assumption we make in these models. $\endgroup$ Jan 31 '17 at 9:24

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