Likelihood function of DSGE model using Kalman filter

Question

In Frank Schorfheide's class notes on likelihood functions of DSGE models, he expresses the value of the likelihood function for a given vector of parameters $\theta$, and time series $Y^T$ as:

$$p(Y^{T}|\theta)=(2\pi)^{-nT/2}\left(\prod_{t=1}^{T}\left|F_{t|t-1}\right|\right)^{-1/2}exp\{-\frac{1}{2}\sum_{t=1}^{T}v_{t}F_{t|t-1}v_t\prime\}$$

where $v_t$ is the innovation in $y$

$$v_t=y_t-\hat{y}_{t|t-1}$$

and the marginal distribution of $y_t$ is

$$y_t|Y^{t-1}\sim\mathcal{N}\left(\hat{y}_{t|t-1}, F_{t|t-1}\right)$$

I've just got a few questions about what these terms look like. First, does anyone have an idea what $n$ is in the exponent of the first term in the first equation? I think it might be a misprint, but I'm not sure. Second, what does $F_{t|t-1}$ look like? For an $n\times 1$ vector $y$ I'm picturing an $n\times n$ matrix, but what would the values of $F_{j,k}$ be equal to? I'm picturing the covariance between $y_{t,j}$ and $\hat{y}_{t|t-1,k}$ - is that correct? Lastly, I'm assuming from the results of my code that the value the likelihood function returns is a scalar, but it doesn't look like the formula produces one -- for an $n\times 1$ vector $y$, wouldn't the second term in the first equation be $n\times n$? Or do you think it's meant to be the determinant of $F$?

Answer 1

$n$ is the dimension of the observation vector, as you mention in your question. $F$ is the covariance matrix of innovations; I think you are missing an exponent of -1 in the last term of the likelihood. It should read $v_t' F_{t|t-1}^{-1} v_t$ (using the convention that $v_t$ is a column vector; your notation seems to assume the opposite).

What you have in the second term of the likelihood is indeed the determinant of $F$.

You might want to peruse any among many excellent books on Kalman filter and state-space models, like Durbin-Koopman, Anderson-Moore or Harvey. Or may be just look at the Wikipedia the topic on Kalman filtering.