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.

  • $\begingroup$ You probably want to look at state-space models, which provide a way of doing what you want. $\endgroup$ – F. Tusell May 5 '14 at 10:41
  • $\begingroup$ Could you elaborate on this a little bit more? Thank you. $\endgroup$ – user13655 May 5 '14 at 11:28
  • $\begingroup$ I cannot elaborate much more in the space of a comment; see answer below. Note I describe the simplest case; neither linearity nor gaussianity are required (although life is simpler with those two assumptions). $\endgroup$ – F. Tusell May 5 '14 at 14:36

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.

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