# Taylor rule estimation with OLS serial autocorrelation

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.