Granger causality
[1] Ordinarily, regressions reflect "mere" correlations, but Clive Granger argued that causality in economics could be tested for by measuring the ability to predict the future values of a time series using prior values of another time series.
Since the question of "true causality" is deeply philosophical, and because of the post hoc ergo propter hoc fallacy of assuming that one thing preceding another can be used as a proof of causation, econometricians assert that the Granger test finds only "predictive causality".
[2] Using the term "causality" alone is a misnomer, as Granger-causality is better described as "precedence",[3] or, as Granger himself later claimed in 1977, "temporally related".
[7] However, it remains a popular method for causality analysis in time series due to its computational simplicity.
[8][10] If a time series is a stationary process, the test is performed using the level values of two (or more) variables.
Then the null hypothesis of no Granger causality is not rejected if and only if no lagged values of an explanatory variable have been retained in the regression.
To test the null hypothesis that x does not Granger-cause y, one first finds the proper lagged values of y to include in a univariate autoregression of y: Next, the autoregression is augmented by including lagged values of x: One retains in this regression all lagged values of x that are individually significant according to their t-statistics, provided that collectively they add explanatory power to the regression according to an F-test (whose null hypothesis is no explanatory power jointly added by the x's).
The null hypothesis that x does not Granger-cause y is not rejected if and only if no lagged values of x are retained in the regression.
Multivariate Granger causality analysis is usually performed by fitting a vector autoregressive model (VAR) to the time series.
The non-parametric tests for Granger causality can be used as diagnostic tools to build better parametric models including higher order moments and/or non-linearity.
In fact, the Granger-causality tests fulfill only the Humean definition of causality that identifies the cause-effect relations with constant conjunctions.
[14] If both X and Y are driven by a common third process with different lags, one might still fail to reject the alternative hypothesis of Granger causality.
[15] Other possible sources of misguiding test results are: (1) not frequent enough or too frequent sampling, (2) nonlinear causal relationship, (3) time series nonstationarity and nonlinearity and (4) existence of rational expectations.
A method for Granger causality has been developed that is not sensitive to deviations from the assumption that the error term is normally distributed.
[20] A modified Granger causality test based on the GARCH (generalized auto-regressive conditional heteroscedasticity) type of integer-valued time series models is available in many areas.
[23] The methodology uses recursive techniques such as the Forward Expanding (FE), Rolling (RO), and Recursive Evolving (RE) windows to overcome the limitations of traditional Granger causality tests and understand changes in causal relationships across different periods.
[25] A long-held belief about neural function maintained that different areas of the brain were task specific; that the structural connectivity local to a certain area somehow dictated the function of that piece.
Collecting work that has been performed over many years, there has been a move to a different, network-centric approach to describing information flow in the brain.
Different methods of obtaining some measure of information flow from the firing activities of a neuron and its surrounding ensemble have been explored in the past, but they are limited in the kinds of conclusions that can be drawn and provide little insight into the directional flow of information, its effect size, and how it can change with time.
Previous Granger-causality methods could only operate on continuous-valued data so the analysis of neural spike train recordings involved transformations that ultimately altered the stochastic properties of the data, indirectly altering the validity of the conclusions that could be drawn from it.
In 2011, however, a new general-purpose Granger-causality framework was proposed that could directly operate on any modality, including neural-spike trains.
A temporal point process is a stochastic time-series of binary events that occurs in continuous time.
This type of binary-valued representation of information suits the activity of neural populations because a single neuron's action potential has a typical waveform.
Using this approach one could abstract the flow of information in a neural-network to be simply the spiking times for each neuron through an observation period.
[citation needed] One of the simplest types of neural-spiking models is the Poisson process.
Neurons, however, exhibit a fundamental (biophysical) history dependence by way of its relative and absolute refractory periods.
[citation needed] Software packages have been developed for measuring "Granger causality" in Python and R:
