Estimation of the local trend models (State Space) through ML

Tsay, R. S. (2010), Analysis of Financial Time Series, 2nd Edition, discusses on page 504 the estimation of local trend models (state space). The measurement and the transition equations are as follows

(1) $Y_t = \mu_t +e_t$
(2) $\mu_t = \mu_{t-1} + u_t$

with $e_t$ ~ $N(0,\sigma^2_e)$ and $u_t$ ~ $N(0,\sigma^2_u)$!

Based on properties of forecast errors, the Kalman filter provides an efficient way to evaluate the likelihood function of the data for estimation. Specifically, the likelihood function under normality is

(3) $p(y_1.....y_T|\sigma_e^2,\sigma_u^2) = P(y_1\sigma_e^2,\sigma_u^2){\displaystyle \prod_{t=2}^{T}(y_t|F_{t-1},\sigma_e^2,\sigma_u^2)}$ $= P(y_1\sigma_e^2,\sigma_u^2){\displaystyle \prod_{t=2}^{T}(v_t|F_{t-1},\sigma_e^2,\sigma_u^2)}$

where $y_1$ ∼ $N(\mu_{1|0},V_1)$ and $v_t = (y_t −\mu_{t|t−1})$∼$N(0,V_t)$.

Consequently,assuming $\mu_{1|0}$ and $\Sigma_{1|0}$ are known, and taking the logarithms, we have

(4) $\ln[L(\sigma_e^2, \sigma_u^2)] = -\frac{T}{2}\ln(2\pi)-\frac{1}{2} \sum_{t=1}^{T}(\ln(V_t)+\frac{v_t^2}{V_t})$

where ${v_t^2}$ is $y_1 −\mu_{1|0}$.

My question is, how did Tsay arrive from (3) to (4)? To be more precise, where is $(\ln(V_t)+\frac{v_t^2}{V_t})$ coming from?

• Could you give a full reference for Tsay (2010)? Commented May 20, 2016 at 9:29
• You have strange notation here Commented May 20, 2016 at 20:50

The density of a normal distribution is $\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}$. With $\mu=0, \sigma^2=V_t, x=v_t$ and the sum of it (log of product gets to sum of logs) you get
$$\log\left\{\prod_{t=1}^T\frac{1}{\sqrt{2\pi V_t}}\exp\left\{-\frac{v_t^2}{2V_t}\right\}\right\} = \sum_{t=1}^T \left(-\frac{1}{2}\log\{\sqrt{2\pi}\})-\frac{1}{2}\log\{V_t\}-\frac{1}{2}\frac{v_t^2}{V_t}\right)$$
And that equals $-\frac{T}{2}\log\{\sqrt{2\pi}\}-\frac{1}{2}\sum_{t=1}^T\left(\log\{V_t\}+\frac{v_t^2}{V_t}\right)$. Where $\log$ equals $\ln$.