Model multivariate time series with copula - concepts I have a question regarding some time series concepts:
Suppose I have some "time series" data with cross correlation. Suppose I am able to fit a copula, say to capture dependencies between data of $\{t,t-1,...,t-n\}$ - i.e. for some time-lag. Now, if I want to draw some random numbers of this time series, I could do the following (ignoring the marginal distributions):
If I have the sample of the time-lag, e.g. $X_{t-1},...,X_{t-n}$, then I could draw $X_t|X_{t-1},...,X_{t-n}$, if I know the conditional distribution (the easiest case is a multivariate normal).
What is the closest time series concept to this? Is there any buzzword I have to look for? I used this concept to do a quick analysis of the data, but I'd like to know if this is a common and reasonable approach.
 A: In the simplest linear case, a state-space model is composed of two equations, the state equation (or transition equation),
$$ \boldsymbol{\theta_t} = \boldsymbol{T_t}\boldsymbol{\theta_{t-1}} + \boldsymbol{R_t}\boldsymbol{\nu_t}$$
and the observation equation,
$$\boldsymbol{Y_t} = \boldsymbol{F_t}\boldsymbol{\theta_t} + \boldsymbol{\epsilon_t}.$$
Vector $\boldsymbol{\theta_t}$ is termed the state-vector, and embodies all information that describes the state of the system at moment $t$. It evolves over time following the dynamics given by the state-equation.
The state vector is in principle made of latent, unobservable variables; what you do observe is $\boldsymbol{Y_t}$, which are linear combinations of elements of the state vector corrupted by noise $\boldsymbol{\epsilon_t}$. The specification is completed with the distribution of the state and observation noises ($\boldsymbol{\nu_t}$ and $\boldsymbol{\epsilon_t}$), usually gaussian distributions in the linear case.
Now, if you consider the linear time-invariant case ($\boldsymbol{T_t} =\boldsymbol{T}, \boldsymbol{F_t} = \boldsymbol{F}$ for all $t$), you have something very close to what you want; you would estimate the needed parameters, in case there are some. The best estimator of $\boldsymbol{Y_t}$
would be $\boldsymbol{F}\boldsymbol{\hat\theta_t}$ and $\boldsymbol{\hat\theta_t}$ can be estimated in terms of $\boldsymbol{Y_{t-1},\ldots}$ using the Kalman filter.
This is only a telegraphic account. For a good discussion see for instance Durbin and Koopman or Anderson and Moore among many books on state-space time series methods.
