Consider the ARDL(1,1) model: $y_t=\beta_1y_{t-1}+\beta_2x_{t}+\beta_3x_{t-1}+\eta_t$.

Assume that I know $x_t$ is stationary around a linear trend: $x_t=\delta t+z_t$

Then: $y_t=\beta_1y_{t-1}+\beta_2z_{t}+\beta_3z_{t-1}+\beta_2\delta t+\beta_3\delta (t-1)+\eta_t$

I then regress: $y_t=\alpha_1y_{t-1}+\alpha_2z_{t}+\alpha_3z_{t-1}+\nu_t$ So I will get consistent estimates of the alphas.

My question is: will the estimate of $\alpha_2$ converge to $\beta_2$ or am I getting a different coefficient as the term $\beta_2\delta t+\beta_3\delta (t-1)$ is missing from the regression?



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