I'm working on a time-series problem. Consider the model below:

$$Y = \beta_0 + \beta_1*t + \beta_2 * \sin(wt) + \beta_3 * \cos(wt) + \epsilon$$

For now, assume that $\epsilon$ is AR(2) with KNOWN parameters $\phi_1$ and $\phi_2$. I want to estimate $\beta$ coefficients above and I want to use pre-whitenning approach. So, what I do is:

$\epsilon_t = w_t + \phi_1 \epsilon_{t-1} + \phi_2 \epsilon_{t-2} \rightarrow w_t = \epsilon_t - \phi_1 \epsilon_{t-1} - \phi_2 \epsilon_{t-2}$

So, I define: $Bt = t-1$ and $B^2t = t-2$ and I write:

$\Phi(B) = 1 - \phi1B - \phi 2B^2$ and I multiply this operator function to the equation above:

$$\Phi(B)Y = \Phi(B)\beta_0 + \Phi(B)\beta_1*t + \Phi(B)\beta_2 * \sin(wt) + \Phi(B)\beta_3 * \cos(wt) + \Phi(B)\epsilon$$

and I get: $$\tilde{Y} = \tilde{\beta_0} + \tilde{\beta_1}*t + \tilde{\beta_2} * \tilde{\sin(wt)} + \tilde{\beta_3} * \cos(wt) + w_t$$ where $w_t$ is white noise.

I can then simply fit this recent model since noises are white noise. My question what is the relation between estimated $\tilde{\beta}$ and my original $\beta$'s?



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