proof for spectral representation sample version:

Assume $T$ is odd and define $M = (T-1)/2$. Then we define $\omega_1 = 2\pi/T, \omega_2 = 4\pi/T,\dots, \omega_M = 2M\pi/T$. (Thus, at least use 2 points to define a cycle, $T$ contains at most $T/2$ cycles). Note $\cos(\omega_j t)$ and $\sin(\omega_j t)$ and $1_T$ constitute $2M +1 = T$ linearly independent vectors and $x_t$ lives in $\mathbb{R}^T$. then we can find coeffcients for each vector. Specifically, the desgin matrix is

\[M_t = \left(1, \cos(\omega_1(t-1)), \sin(\omega_1(t-1)), \dots ,\cos(\omega_M(t-1)), \sin(\omega_M(t-1)) \right)\]

with $t$ acts as the row index. Thus, the coeffcients will be

\[\hat\theta = \left(\sum_{i=1}^T M_t^TM_t \right)^{-1} \left(\sum_{t=1}^T M_t x_t \right)\]

The exact form of the $M^TM$ will be

\(\sum_{t=1}^T M_t^T M_t = \begin{bmatrix} T & 0^T \\ 0 & \frac{T}{2} I_{T-1} \end{bmatrix}\) thus, the parameters become \(\hat{\theta} = \begin{bmatrix} T & 0^T \\ 0 & \frac{T}{2} I_{T-1} \end{bmatrix} \begin{bmatrix} \sum_{t=1}^T x_t \\ \sum_{t=1}^T x_t\cos(\omega_1(t-1)) \\ \sum_{t=1}^T x_t\sin(\omega_1(t-1)) \\ \vdots \\ \sum_{t=1}^T x_t\cos(\omega_M(t-1))\\ \sum_{t=1}^T x_t\sin(\omega_M(t-1)) \end{bmatrix}\)

we then see that

\[\begin{aligned} &\hat{\mu} = \frac{1}{T} \sum_{t=1}^T x_t \\ &\hat\alpha_j = \frac{2}{T} \sum_{t=1}^T x_t \cos(\omega_j(t-1)) \\ &\hat\beta_j = \frac{2}{T} \sum_{t=1}^T x_t \sin(\omega_j(t-1)) \end{aligned}\]

which is just an ordinary sample Fourier decomposition.

because the decomposition is exact we have

\[\sum_{t=1}^T (\hat x_t - x_t)^2 = 0\]

for $\hat{x}_t = M_T \hat\theta $. By some calculation, we could find that

\[\frac{1}{T} \sum_{t=1}^T (x_t - \hat\mu)^2 = \frac{1}{2}\sum_{j=1}^M(\hat\alpha_j^2 + \hat\beta_j^2)\]

we can further write it in the periodigram as

\[\frac{1}{T} \sum_{t=1}^T (x_t - \hat\mu)^2 = \frac{1}{2}\sum_{j=1}^M(\hat\alpha_j^2 + \hat\beta_j^2) = \frac{4\pi}{T} \sum_{j=1}^M\hat S(\omega_j)\]

Estimating spectral density

Goal: given the observations $(x_1, \dots, x_T)$ estimate $S(\omega)$.

Naive idea: use sample periodogram $\hat{S}(\omega)$ to estimate $S(\omega)$.

Consider $X_t = \sum_{j=0}^\infty \psi_j \epsilon_{t-j}$ where $\sum_{j=0}^\infty |\psi_j| < \infty$ Asssume $s(\omega) >0$ , for $\forall \omega$ , then $$ \begin{aligned} &\frac{2\hat S(\omega) }{ S(\omega)} \rightarrow \chi^2(2)
&\hat{S}(\omega) \text{ and } \hat{S}(\lambda) \text{ asymptotically independent} \end{aligned}

$$ This shows

\[\mathbb{E} \left[\frac{2\hat{S}(\omega)}{S(\omega)} \right] \sim 2\]

this means $\hat{S}(\omega)$ is unbiased. But there is no variance reduction asymptotically.

To fix, there are two ways: parametric models and nonparametric models.

parametric estimation

Assum $ARMA(p,q)$ model, we have

\[x_t = \phi_0 + \phi_1 x_{t-1} + \dots + \phi_p x_{t-p} +\epsilon_t - \theta_1\epsilon_{t-1} - \theta_2\epsilon_{t-2} -\dots\]

Step1: estimate the parameters $(\phi_1,\dots,\phi_p)$

Step2: esimate the spectral density by closed form formula:

\[\hat S(\omega) = \frac{\hat{\sigma}^2}{2\pi }\]


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