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Here is my problem:

I basically have 20 or so variables (I have 1000 of these values over an increasing time axis). I want to calculate the weights of these input variables. I am going to try Linear regression to estimate the weights. Is this the correct way to start thinking about it?

If I have an output variable which depends on these input variables, I could run a linear regression. But I just have 20 variables with different values at different points in time, and I want to estimate weights to estimate what value a variable will have at a later date (no output variable)

Any help will be appreciated.

Note: My dataset is a 1000*20 set

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I know my answer is late, but might help others.

To answer your first question, yes each time series could be studied independently as univariate time series, by obtaining the mean and the autocovariance function for each series. However this approach doesn't take into account the possible dependence between the series.

By doing a linear regression to each series you are estimating the trend of the series, which does come into play when trying to fit a model for forecasting.

I will outline a very general approach for univariate time series that can be extended to multivariate time series:

  • Plot the series: check for trend and seasonality, changes in behaviour, outliers, etc.

  • Estimate the trend: a) with a smoothing procedure such as moving averages (no estimates) or b) model the trend with a regression equation.

  • "De-trend" the series. For additive models subtract the trend. For multiplicative models divide the series by the trend values.

  • Determine seasonal factors. The usual method is to average the "de-trended" values for a specific season.

  • Determine the random (residuals) component:

    For an additive model: random = series - trend For a multiplicative model: random = series/(trend*seasonal).

  • Choose a model to fit the residuals, using sample autocorrelation function.

  • Use residuals to forecast and then invert the transformations described above to arrive at forecasts of the original series.

You can check out Rob J Hyndman Forecasting Principles and Practice and the The Little Book of Time Series for a better exposition.

With respect to your second question, you might want to use multivariate time series, in which a vector / matrix approach is used to mostly fit vector autoregresive models (VAR). You can find a much better explanation in Vector Autoregresive Models for Multivariate Time series, and how to use the R package vars in VAR, SVAR, SVEC Models: Implementation within R Package vars

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You can try an autoregressive model, where each variable at time $t$ is regressed on the variables at time $t-1$. It is easy to fit using linear regression, by constructing a dataset where there is one row for each timestep $t$, 20 columns holding the variables at time $t-1$, and one extra column for the variable to predict at time $t$. There are 20 such datasets, one per variable to predict.

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