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I'm estimating the equation:

$$i_{t}=\beta_0+\beta_1\tilde\pi_t+\beta_2\tilde y_t+\varepsilon_t$$

Where $\hat\pi_t=\pi_t-\pi^{target}$ and $\hat y=\ln y_t-\ln y^{\ast}$, are the inflation deviations from target and output gap (data equivalent of cycle component of the series) respectively. And $i_t$ is the policy interest rate. When estimating this equation with time series for the three variables I get serial correlation of errors. Not a surprise really, nonetheless I've had trouble "fixing" it.

I tried estimating the equation (using lagged dependant and independent variables as regressors):

$$i_{t}=\beta_0+\beta_1\tilde\pi_t+\beta_2\tilde y_t+\beta_3i_{t-1}+\beta_4\tilde\pi_{t-1}+\beta_5\tilde y_{t-1}+\beta_6i_{t-2}+\beta_7\tilde\pi_{t-2}+\beta_8\tilde y_{t-2}+\omega_t$$

more compactly: $i_t=\beta_0+\left(\beta_1+\beta_4L+\beta_7L^2\right)\tilde\pi_t+\left(\beta_2+\beta_5L+\beta_8L^2\right)\tilde y_t+\left(\beta_3L+\beta_6L^2\right)i_t+\omega_t$

which yields that there's no evidence of $\omega_t$ being serially correlated. Nevertheless, I don't know if this approach is correct. And if not, what would be a better one. I'd really appreciate your help. Thank you.

PD: The series used are, for $\tilde y_t$ is the the hp-filtered cycle component of logged real output; for $\tilde\pi_t$ is observed inflation at $t$ minus policy target at the same $t$; and $i_t$ is the monetary policy interest rate target.

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