Linear regression for time-series prediction Say we have $N$ time series $X_t^i$ for $i=1...N$and we want to predict a separate time series $Y_t$. Let's consider the following model: $Y_t = \sum_{i} \beta_i X_{t-1}^i $
I am just trying to figure when such a model makes sense


*

*Does this model have a name? Is there an interpretation in terms of random processes? Are there any known applications?

*Does it make sense to solve the $\beta_i$ by using regression approaches (ridge, lasso, etc) using the training points $(<x_{t-1}^1... x_{t-1}^N>, y_t)$?

 A: I'd look at this as a problem in least squared minimization.
so you're trying  to minimize:
$$ \langle \epsilon^2 \rangle_t= \langle ( Y^t - \sum_i \beta_i X^{t-1}_{i})^2 \rangle_t = \langle (Y_t - \vec{\beta} \cdot X^{t-1})^2 \rangle_t$$
I tend to interpret this type of problem as a Gaussian statistics problem since the solution only involves first and second moments.  The idea is that there is a Gaussian joint distribution for $p(y, x_1, x_2 \dots)$ with an arbitrary correlation matrix;  you are trying to estimate that correlation matrix, and then computing the conditional distribution $p(y \vert X)=p(y,X)/p(X)$.
In some contexts, this type of problem may be referred to as a Wiener Filtering problem.
A: I know you already accepted an answer, but just to add something for completeness in relation  to whether this model is used anywhere:
If you in addition would assume each $X_i$ to follow an AR(p) process, then your model is one of the most commonly used for analyzing predictability of stock returns in the financial econometrics literature. See for example Stambaugh (1999), which led to a lot of other research on this model by noticing the bias of the OLS estimator of $\beta$ in such settings. More recent papers include Chen and Deo (2009) and Amihud et al. (2010).
