How to forecast time series analysis for more then one dependent variables?
I think you do not need to "predict" counter_id and country_code as variables, what you need is to produce results to every counter and country (and sometimes these results will be zero).

The technique you are looking for is called VAR (Vector Autoregression)

VAR

The vector autoregression (VAR) model is one of the most successful, flexible, and easy to use models for the analysis of multivariate time series. It is a natural extension of the univariate autoregressive model to dynamic multivariate time series.

The equation you propose is solvable by VAR:

$Y_t = C + A_1 Y_{t-1} + A_2 Y_{t-2} + ... + A_n Y_{t-n}$

Where $Y_t = \begin{bmatrix} Y_{i=1,j=1,t} \\ Y_{i=1,j=2,t} \\
\vdots \\
Y_{i=1,j=m,t} \\
\vdots \\
Y_{i=n,j=m,t} \end{bmatrix}$

Where, $i$ is every country, $j$ is every counter.

The usage is very similar to univariate time series analysis, which you are probably familiar with.