Estimating production functions with time series data A while back I asked How do we estimate production functions?

The answers given address cases when dealing with cross-sectional data, However most of the data I've been seeing is given by a time series.

For example if we have a (logged) VAR(1) with two variables $x_{t,1}$ and $x_{t,2}$, the appropriate model is:

$$x_{t,1}=\alpha_1+\beta_{11}x_{t-1,1}+\beta_{12}x_{t-1,2}+\mu_{t,1}$$
$$x_{t,2}=\alpha_2+\beta_{21}x_{t-1,2}+\beta_{22}x_{t-1,1}+\mu_{t,2}$$

How do we develop concise production functions from a set of equations like these?
How do we develop concise production functions from a set of equations
  like these?


It goes the other way around.

VAR's are  a special group of models for time series data, developed mainly for atheoretical forecasting purposes. In a VAR approach we do not assume some structural economic model, we just want to use the past of a variable and its interelations with other variables, to form a good quality predictor (and it is for this reason that when they were first introduced and propagated by Sims,  they created a lot of controversy).

In general, estimating a production function of a single firm using time-series data does not alter the econometric approach described in the answer to your other question, only here the assumption we make is that the unknown parameters refer to a single firm (and not to many firms), and so stay constant across time (and not across firms on the same time period.

A new econometric issue is that with time series data we are forced to deal with the aspect of serial correlation (in a cross-sectional setting, "serial" correlation makes no sense because we can permutate the index without consequences). Serial correlation may emerge in two ways: a) In the regressors  b) in the error term.

A new economic issue in a production function setting is the intertemporal change in Total Factor Productivity.