There are several statistical models that can be used to analyze the relationship between time series variables. Some common approaches include:

It's worth noting that there is no one "best" model for analyzing the relationship between time series variables, and the appropriate model will depend on the specifics of your data and the research question you are trying to answer.

As I understand your question I would agree with the above linear regression approach and after ordinary regression model building I would optimize it with adaptive lasso methods because of these methods so called Oracle property. I have attached a recent paper that you may find helpful.. Best wishes David Booth

It depends on you core interest several statistical tools can be employed depending on the distribution of the data...however I will suggest you look into regression techniques, ARIMA

LINEAR REGRESSION MODEL, MULTIVARIATE MODEL, ECONOMETRIC MODEL

Of course everything depends on your variables. If they are nonstationary, it would change the recommendations. Think first to possible theoretical relations between the variables. Like several of my predecessors, I would suggest starting with linear regression. Look at the autocorrelations among the residuals, and if they are small enough, at the cross-correlations between them and the explanatory variables (in difference, if necessary). Add lags and introduce an ARMA model for the errors if needed. Proceed step by step and avoid a model that is too complex. Good luck!

There are several statistical methods that you can use to identify the independent variable that explains the most variation in the dependent variable. One method is to use Pearson's correlation coefficient to measure the strength and direction of the linear relationship between each independent variable and the dependent variable. You can then rank the independent variables by the magnitude of their correlation coefficients to see which one is most strongly correlated with the dependent variable.

Another method you can use is multiple regression analysis, which allows you to assess the combined effect of all the independent variables on the dependent variable. You can use this method to identify the independent variable that has the largest individual effect on the dependent variable, even if it is not the most strongly correlated with the dependent variable.

Keep in mind that these methods only assess the strength of the linear relationship between the variables, and they may not be appropriate if the relationship is non-linear. In such cases, you may need to use other statistical techniques, such as non-linear regression or machine learning algorithms, to identify the independent variable that explains the most variation in the dependent variable.

Depends on what you want to study. It sounds like ARIMA for economic series or a survival analysis could fit

It depends on your specific study. There exists a large number of statistical models to analyse the relationships between time series variables. In your case of multivariate time series analysis (few time series variables) that might be options from linear models (VAR, VECM, AECM, GARCH...) to nonlinear ones (TVAR, SETAR, TVECM, NARDL, GAM....). Moreover, the choice depends on time series features.

All the answers above are excellent, I would suggest the following article for your approach using python

Feature Selection for times series forecasting with python ( machine learning mastery.com )

From the way you propose your experimental design, I read that the ones they suggest are very good options. You are finally considering the relationship between two variables, one of which would be an external regressor. So it will depend on the distribution of the variables and how you work it to be able to analyze them. With Granger causal regression, VAR, simple lagged models...it all depends on the length of the data. Fundamentally, however, I think that less complicated models also solve our problems. Regards.

One-way error component model which is similar to linear regression ( you have one dependent and many independent variabLes) and suites panel data ( the case of having one dependent and many independent measured in many sittings/ time points)

Maybe you want to consider the recursive least squares algorithm (RLS). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), acquired data is weighted according to its age, with increased weight given to the most recent data.

Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...

I would try cross-correlation first.

Probably, Principal component analysis will be helpfull.

You might try a partially non-model approach via "prewhitening":

see for example https://online.stat.psu.edu/stat510/lesson/9/9.1

... or search for the term.

Basically, you apply apply the same filter to both your input and output series, get the residuals from these filters and look at the cross-correlation between them. The filter is selected to get approximate white-noise from your input series. The size of the cross-correlation might thought of as measuring the contribution of perturbation in the input series to changes in the output series. The spread and lag of the cross-correlation function here may also have some useful information for comparing different input series.

The above idea is closely related to the "transfer function" approach to time-series modelling, but you would be getting an empirical estimate of the transfer function without actually fitting a full transfer-function model.

I agree with David A. Jones but there is no reason to use the same filter to the series since it can be impossible. What is important is that one of the output series are approximately transformed into white noise series and the input series are made stationary.