I have been working on the following question for a while but I'm stuck at a certain point.
Let the true model be $y_t=μ_t+\epsilon _t,t= 1,...,T,$ with $y_t$ a scalar, and $t$ time. You will show below how to derive the Hodrick-Prescott filter that extracts a time-varying mean $μ_t$ using ridge regression. The entire parameter vector is $μ= (μ_1,...,μ_T)′$, and the entire data is $y=(y_1,...,y_T)′$.
I have done part a and b and found $$ \hat{\mu} = (I_T + \lambda D'D)y $$ where $I_T$ is the identity matrix with dimension T, $\lambda$ is tuning parameter and $D$ is the second order difference matrix which is a tridiagonal and Toeplitz matrix of which the first row is [1, −2, 1, 0,... , 0].
However, I got stuck in part c and d.
c) Assume that $T^{−1/2}∑^T_{t=1}\epsilon _t → N(0,σ^2)$. Derive the probability limit of $\hat{μ}$ keeping $λ$ constant,and letting $μ_t=μ_0$.
d) Same as c), but instead of keeping $λ$ constant, let $T^αλ→0$. What should $α$ be equal to such that $\hat{μ_1}→μ_0$ and $T^{1/2}(\hat{μ_1}−μ_0)→N(0,σ^2)$? Here, $\hat{μ_1}$ is the first element of $μ$.
Any help would be highly appreciated.