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I would like to know what are the standard approach to construct a model to predict the value of a time series $y_t$ that depends from other time series $\bar{X}_t$. I use to see around that for this kind of task there is a large amount of models but all autoregressive in a way. I'm thinking about, for example VAR, SARIMAX, RNN, LSTM. I'm looking for a model, or at least an approach, where there is no my lagged target variable as predictors. Does anyone has some references ?

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A time series forecast model based upon lagged variables of that time series is commonly called auto-correlation. A time series forecast model based upon other series is called cross-correlation in the time-series literature. A general forecast that uses any number of lagged time series (including the lagged series itself) is called vector auto regression.

Any of the other ML and AI approaches can use non-linear methods to achieve the modelling and draw from an infinite amount of factors other than lagged time series. In finance, it's common to use factor based investing, for example (like fundamental value data). In Machine Learning, we are interested in finding those features (feature selection) that best model the time series outcome.

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