What are some tests for the predictability of time-series? I have 2500 time series which I want to test the predictability and based on that, choose the best one to forecast. 
Ideally I want to use a simple model like ARMA-GARCH for forecasting. Are there any tests for forecasting abilities which I can assess the time-series?
 A: Diebold & Kilian (2001) (a free version here) suggest a measure of predictability for a time series. Here is an excerpt from the abstract:

We propose a measure of predictability based on the ratio of the expected loss of a short-run forecast to the expected loss of a long-run forecast. This predictability measure can be tailored to the forecast horizons of interest, and it allows for general loss functions, univariate or multivariate information sets, and covariance stationary or difference stationary processes. We propose a simple estimator, and we suggest resampling methods for inference. 

The measure $P$ is defined on p. 659 as
$$ P(L,\Omega,j,k) = 1 - \frac{\mathbb{E}(L(e_{t+j,t}))}{\mathbb{E}(L(e_{t+k,t}))} $$
where $e_{t+j,t}$ is the forecast error for (short) horizon $j$; $e_{t+k,t}$ is the forecast error for (long) horizon $k$; $L$ is a loss function; $\Omega$ is an information set; and $\mathbb{E}$ is mathematical expectation conditional on the information set $\Omega$. 


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*If $P$ is high (close to 1), the series is highly predictable at horizon $j$ relative to $k$. 

*If $P$ is low (close to zero), the relative predictability is low.


See also Theil's $U$ statistic mention on p. 659. According to Diebold & Kilian, 

Theil's $U$ assesses 1-step forecast accuracy relative to that of a "naive" no-change forecast, whereas $P$ assesses 1-step accuracy relative to that of a long-horizon ($k$-step) forecast. 

See also section 5. Concluding remarks and directions for future research (p. 666-668) for some discussion and possible extensions.
References:


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*Diebold, Francis X., and Lutz Kilian. "Measuring predictability: theory and macroeconomic applications." Journal of Applied Econometrics 16.6 (2001): 657-669.

