Regression model of large, correlated, heavy-tailed data [closed]

I have a large panel-like data set: about 15,000 individuals, on average 350 time points, two dozen variables (plus some more variables we left out for context-specific reasons). What I want to achieve is a model explaining one dependent variable $y$ by (some of) the other (i.e. explanatory) variables $x$. As a first step, this should be something like a linear regression model.

Properties of the data:

• $y$ follows a non stationary process (not rejected by Dickey–Fuller test) and increases linearly over time on average.
• One of the explanatory variables $x_1$, with $x_1(t) \geq x_1(t-1)$, is way stronger correlated with $y$ than any other variable.
• Most of the variables do not follow a Gaussian distribution, some (including $y$ and $y^\prime$) have a heavy tailed distribution.
• Some of the variables are strongly correlated.

What I've done/calculated so far:

1. Pearson correlation of $y$ with each $x_k$ separately.
2. Pearson correlation of $y$ with each $x_k$, controlling for $x_1$.
3. Sampling only every $n^{th}$ time point ($n$ chosen so that the autocorrelation of $y$ has decayed below 1/e), and doing simple linear regression on the sample, using time as just one more variable. Regression of $y$ on $x_1$ yields $R^2=0.36$; including all variables yields $R^2=0.40$.
4. Taking $y^\prime$, i.e. $y(t+1)-y(t)$ as dependent variable. The autocorrelation of $y^\prime$ decays really fast. Regression of $y^\prime$ on $x_1^\prime$ yields $R^2=1.5\cdot 10^{-4}$, regression of $y^\prime$ on all $x_k$ yields $R^2=2\cdot 10^{-4}$.
5. Taking $\log(y(t+1)/y(t))$ ('log return') as dependent variable. Regression on all $x_k$ yields $R^2=0.003$; regression on $\log(x_1(t+1)/x_1(t))$ yields $R^2=0.0012$.

Despite the low $R^2$, the F-test yields a very low p-value and most p-values for the coefficients are very low. All regressions include a constant term.

What I'm looking for:

• In the best case a (pointer to a) 'standard' way to derive a model from data of this size (and kind), and a quantification of how much sense the resulting model makes.
• Criteria which of the explanatory variables to use in a model (the size of the data shows even really small effects significantly).
• What happens if you regress the log-return $y$ on log-return $x_1$? Or does that not make sense for this data? Commented Jul 31, 2013 at 18:49
• Doesn't make too much sense, I guess; did it anyway for curiosity. Had an error in 4., see update. Commented Aug 1, 2013 at 18:44