How to interpret my time series regression model when doing a regression for each month individually (12 regressions)? I am investigating the warming trend for a given grid point in Europe. Local temperature for a point on Earth is mainly determined by atmospheric circulation, or bluntly by where the wind is coming from. For this I have monthly data. The regression model looks like this:

1 dependent variable: T_loc, local temperature (measured in Celsius)
T_Circ: 3 variables expressing the atmospheric circulation (measured in m/s) 
U_t => West-East component of wind (+ if West wind, - if East wind)
V_t => South-North component of wind (ditto)
 W_t => upward direction of wind (used as proxy for cloud cover)
T_noncirc:
The part of local temperature that is not determined by the atmospheric circulation. This great number of processes are proxied by the T_glob temperature.
Finally, there is a Memory term, which is the T_loc of the previous month.
What is crucial is that in the first instance, this regression is done individually for each month in itself to account for seasonality. For example: If we look at the month of May, we have data between 1950 and 2010, and T_loc at t = 0 local temperature in May 1950, t = 1 is May 1951, t = 2 is May 1952, etc. Therefore, the memory term is not the lag term T_{loc, t - 1} in an econometric sense, but rather a separate time series of variables (in the case of this example local temperature in April between 1950 and 2010).
U_t and its estimated coefficient on T_loc is what interests us in this regression. Climate models (computer models that make forecasts for the climate up to 2100) have structurally underestimated the warming trend for Europe, and 1 reason for this is that they have difficulty simulating both the distribution and the effect on the local temperature correctly.
The aim is therefore to do this regression on my 2 observational data sets, and on my 10 climate model data sets (simulated data sets). Then we can assess which of those 10 climate models come close to reality, so that we have more confidence in the future predictions of those relevant climate models.
*So my problem/question is the following:

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*What type of model is this? Since in my regression no variable is explained by lagged values ​​of itself or of the other variables, it is not a VAR(p) model in my opinion.

*Is this still time series regression at all if we filter the data per month and only regress for 1 month at a time?

*I find non stationarity for the global temperature trend, but not for the local one (which we do know there is a trend in it too - prob ADF and KPSS tests dont find non-stationarity due to the highly varying nature of this variable). How do I handle these variable(s) in my regression if I still want to have BLUE, consistent estimators, asymptotic normal behaving estimators etc?*


It's an AR(1) model.
Beyond that, it's usually a better idea to create one model instead of 12.  You might consider one SARIMA model with AR(1) and Seasonal(1) term for annual seasonality.
