Does an Autoregression model with two variables exist? When I look at the pattern of rainfall I realize that it has both deterministic and stochastic components. It's possible to fit a polynomial or an $AR(p)$ model to the data set.
My thinking is that there is a path say $f(t,y)$ which is a function of both raifall and time and also deterministic. The errors must be a function of time and Brownian motion.
My aim is to model $f(t, y)$ as an $AR(p)$ that is a function of both time and rainfall.
My question is, can I write something like
$$Y_t=AtY_{t-1}+Bt^2Y_{t-2}+Ct^3+Dt^4+e_t$$
Is it possible to come up with Autoregression model of this nature?

Can you use this model for time series analysis? Theoretically, yes. But you would have to test your assumptions here before using this model for practical application. Here, you are assuming a deterministic component $Ct^3+Dt^4$ and a stochastic component which is modeled using an AR(2) process $$X_t=a_1X_{t-1}+a_2X_{t-2} + \epsilon_t.$$ You are also assuming the parameters of AR(2) process varying with time such that $a_1=At$ and $a_2=Bt^2$.
But I doubt that rainfall time series actually follows this model. First of all, there are periods with zero rain which are not captured by this model. Second, the deterministic component does not really capture the seasonality of rainfall. Here I suggest, using something like LOESS smoother to remove the seasonal component. Third, your data may contain long-term persistence (LTP). Therefore, I suggest using a more involved model such as FARIMA which can capture the LTP behavior. FARIMA also contains AR(p) model as a special case - parameters are constant in time.
There is huge amount of work done for stochastic modeling of rainfall time series. You might find an adequate model in the literature. For example, Neyman-Scott model has been shown to preserve the first few moments of rainfall time series.
Also note that AR processes exhibit qualitatively very different behavior for different parameter values. You may look at unstable AR processes for an example. The point is that if you are varying the parameters of the AR model with time, you are likely to get very different behaviors in time.
