## Estimating Geweke’s (1982) Measure of Instantaneous Feedback

Category : Research

### Estimating Geweke’s (1982) Measure of Instantaneous Feedback

#### Abstract

This note describes the gwke82 program, which implements instantaneous feedback measure for two time series following Geweke (1982).
Keywords: Geweke, Granger, causality, VAR, time series, instantaneous feedback

#### Introduction

Tests of statistical causality are really tests of whether lags of one variable can be used to predict current values of another variable. If data are measured frequently, and the causality isn’t instantaneous, then this is adequate. However, if data are measured infrequently (ex: most macroeconomic data are measured yearly), then standard Granger (1969) type causality tests may miss much of the contemporaneous correlation between variables. Geweke (1982) proposed a measure of instantaneous correlation, calculated from the residuals of standard Granger (1969) type causality tests, that captures “instantaneous feedback.” This program, gwke82, quickly estimates such Geweke-type instantaneous feedback between pairs of variables.

Granger type causality is quite common in financial economics research as well as other disciplines. Geweke type causality however has not received the type of research attention as Granger type causality (Interestingly, however, Geweke type causality has received attention from neurosciences: ex. Zhang (2010)). We think that this is because of the lack of econometric software coverage of the Geweke type causality. Our suggested command –gwke82– intends to fill this void.

While the Geweke (1982) method applies to any vector-valued linear function, the intuition behind the technique can be more easily seen for a system of two random variables which can be estimated using the standard VAR methodology (We follow the notations in Geweke (1982)):

$x_{t} =\sum_{s=1}^p E_{1s}x_{t-s}+u_{1t}, ~~~ var(u_{1t})= \Sigma_{1}$

$y_{t} =\sum_{s=1}^p G_{1s}y_{t-s}+v_{1t}, ~~~ var(v_{1t})= T_{1}$

$x_{t} =\sum_{s=1}^p E_{2s}x_{t-s}+\sum_{s=1}^p F_{2s}y_{t-s}+u_{2t}, ~~~ var(u_{2t})= \Sigma_{2}$

$y_{t} =\sum_{s=1}^p G_{2s}y_{t-s}+\sum_{s=1}^p H_{2s}x_{t-s}+v_{2t}, ~~~ var(v_{2t})= T_{2}$

$x_{t} =\sum_{s=1}^p E_{3s}x_{t-s}+\sum_{s=0}^p F_{3s}y_{t-s}+u_{3t}, ~~~ var(u_{3t})= \Sigma_{3}$

$y_{t} =\sum_{s=1}^p G_{3s}y_{t-s}+\sum_{s=0}^p H_{3s}x_{t-s}+v_{3t}, ~~~ var(v_{3t})= T_{3}$

If for instance, all the coefficients for the lags of $x$ ($H_2$) are statistically significant, then it is said that “$x$ Granger-causes $y$.” Such estimation, however, potentially leaves a lot of correlation between $x$ and $y$ unexploited. Specifically, if $y_{t}$ is correlated with $x_{t}$ after controlling for their lags, then there is instantaneous correlation left between them. This is the basis of the Geweke (1982) measure of instantaneous feedback. Geweke proposed that the variance/covariance matrix of residuals from the VAR estimation be used to estimate the linear feedback between $y$ to $x$, and $x$ to $y$, and the instantaneous linear feedback between $x$ and $y$.

If $x$ does not Granger-cause $y$, then (3) and (4) can be rewritten as (1) and (2), respectively. Comparing equations (1) and (3), then, gives us an estimate of the impact of $y$ on $x$. Specifically, Geweke (1982) proposed the following as measures of linear feedback:

$n \cdot F_{X \to Y} = n \cdot ln(T_{1}/T_{2}) ~~~ \sim \chi_{p}^{2}$

$n \cdot F_{Y \to X} = n \cdot ln(\Sigma_{1}/\Sigma_{2}) ~~~ \sim \chi_{p}^{2}$

$n \cdot F_{X \cdot Y} = n \cdot ln(T_{2} \cdot \Sigma_{2} / |\Upsilon|)$

$= n \cdot ln(\Sigma_{2}/\Sigma_{3})$

$= n \cdot ln(T_{2}/T_{3}) ~~~ \sim \chi_{1}^{2}$

$n \cdot F_{X,Y} = n \cdot ln(\Sigma_{1} \cdot T_{1} / |\Upsilon|) ~~~ \sim \chi_{(2p+1)}^{2}$

where $|\Upsilon|=\left[ \begin{array}{cc}\Sigma_{2} & C \\ C & T_{2} \\ \end{array} \right]$ and $C = covar(u_{2t},v_{2t})$.

$F_{X \to Y}$ and $F_{Y \to X}$ are the Granger type causations.

$n$ is the number of observations for the unrestricted estimations.

$F_{X \cdot Y}$ is the measure of instantaneous causation (instantaneous feedback).

$F_{X,Y}$ is the measure of total feedback between x and y.

$F_{X,Y}$ also equals to $F_{X \to Y} + F_{Y \to X} + F_{X \cdot Y}$.

Geweke (1982) showed that the measures above are asymptotically distributed as F-distributions. He also proved that equations (9), (10) and (11) are equal implying that instantaneous causality can be verified directly by comparing equations (5) to (3) and (6) to (4) or indirectly by using the variance-covariance matrix of equations (3) and (4). Computationally, the gwke82 command uses equation (9) for instantaneous causality.

Geweke (1982) generalized the above results to include vector valued functions (so that the measures are asymptotically Chi-squared), allowed for more than two endogenous variables, and allowed for the inclusion of exogenous variables as well. The gwke82 command does not allow for vector valued functions, but it does allow for exogenous variables and more than two endogenous variables.

#### References

Geweke, J. 1982. Measurement of linear dependence and feedback between multiple time series. Journal of the American Statistical Association 77, 378: pp. 304-313.

Granger, C. W. J. 1969. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 3: pp. 424-438.

#### The gwke82 command

##### Options
• m is the number of lags.
• exog denotes exogenous variable(s).
• detail stores all estimations’ results.

##### Example #1: Usage

The estimation reveals that dln_inc Granger-causes dln_cons-p. There is evidence of instantaneous feedback between dlin_inv and dln_cons-p, and between dln_inc and dln_cons-p. The total correlation between dln_inc and dln_cons-p is statistically significant.

The causality statistics and corresponding p-values that are reported after gwke82 command are also saved as return matrices.

The detail option stores all estimation’s results. These stored estimations can be accessed by estimates dir command.