I am trying to forecast the product two time series. That is, given $\{x_t\}_{t=0}^{T-1}, \{y_t\}_{t=0}^{T-1}$, forecast $x_T\cdot y_T$. The two time series have minimal but nontrivial correlation ~0.1. I can apply standard time series forecasting techniques individually and then multiply two independent forecasts but that seems to introduce a lot of bias due to the correlation.
What is standard practice in this case? The data can be assumed to be stationary.
Estimate individual ARIMA models, obtain their residuals (innovations) $e_{xt}$ and $e_{yt}$.
From here you have a couple of options. The simplest one is to get the correlation of the residuals $\rho$, then use for forecasting. For instance, you can use the standard Cholesky decomposition method to produce correlated paths $e_{xt}$, $e_{yt}$ for Monte Carlo simulation, then plug them into ARIMA to forecast future values.
Of course, if $\rho$ is very small, you may consider ignoring it. Look at the statistical significance of your correlation coefficient. The chances are that it'll be not significant. Note, that correlation of series $x_t,y_t$ doesn't have to be the same as of $e_x,e_y$. This is especially true if you have exogenous variables, i.e. use ARIMAX model, because in this case the exogenous variables may capture the large part of correlation. Note also, that even if the series do not share exogenous variables, the variables could be correlated. For instance, GDP is correlated with Unemployment.
The second option is to use Copulas to forecast innovations. This may be a better solution in some cases.
If you have multiple correlated time series, you will usually model them using Vector autoregression, which generalizes AR models to include lags of different series.
In addition, as @Aksakal notes, think about why your time series are correlated. If there are common drivers, then account for these in ARIMAX models and look whether the residuals are still meaningfully correlated. If not, then your external variables capture all the correlation in your series, and you are done.
