# Granger causality and non-linear regression

I’m new to Granger Causality concept. I know that the “Granger causality” is a statistical concept of causality that is based on prediction. According to Granger causality, if a time series X "Granger-causes" (or "G-causes") a time series Y, then past values of X should contain information that helps predict Y above and beyond the information contained in past values of Y alone.

Suppose we have two time series;

X={1,2,3,4.5,5,6,7.2}; and Y={2,4,5,6.5,7.5,9,13}.

The following table shows samples of X,Y over time:

I would like to estimate the causality (or causality ratio) using non-linear regression model. Can anyone helps me to find if X "Granger-causes" Y using non-linear regression.

• Can you explain why you prefer a nonlinear model over a standard VAR? – Christoph Hanck Jan 19 '16 at 13:18
• In reality, of course, many causal relationships are more or less nonlinear, raising some doubts as to the applicability and usefulness of purely linear methods. So, I'm intended to estimate the causality ratio and feedback ratio using 10th order regression model, namely k = 10. – Omar14 Jan 19 '16 at 13:35
• With seven observations? – Christoph Hanck Jan 19 '16 at 13:38
• That's slightly off-topic, but by "10th order regression" do you mean including regressors $x,x^2,\dotsc,x^{10}$? That is asking for trouble unless there is a good subject-matter motivation for such high powers. @ChristophHanck, would there be any trouble carrying the notion of Granger causality over to models other than VAR? Say, we just stick to the basic principle that a model allows for a contribution of lagged $x$ towards predicting $y$ (besides lagged $y$'s own contribution), and that is testable. If $x$'s contribution is rejected, Granger causality is rejected. What about that? – Richard Hardy Jan 19 '16 at 20:33
• Why I said "asking for trouble": because the estimation variance and model sensitivity will likely be very high and you will be very likely to pick up noise in place of signal when estimating the model. – Richard Hardy Jan 19 '16 at 20:39

A non-linear Granger causality test was implemented by Diks and Panchenko (2006). The code can be found here and it is implemented in C. The test work as follows:

Suppose we want to infer about the causality between two variables $X$ and $Y$ using $q$ and $p$ lags of those variables, respectively. Consider the vectors $X_t^q = (X_{t-q+1}, \cdots, X_t)$ and $Y_t^p = (Y_{t-p+1}, \cdots, Y_t)$, with $q, p \geq 1$. The null hypothesis that $X_t^q$ does not contain any additional information about $Y_{t+1}$ is expressed by

$$H_0 = Y_{t+1}|(X_t^q;Y_t^p) \sim Y_{t+1}|Y_t^p$$

This null hypothesis is a statement about the invariant distribution of the vector of random variables $W_t = (X_t^q, Y_t^p, Z_t)$, where $Z_t=Y_{t+1}$. If we drop the time indexes, the joint probability density function $f_{X,Y,Z}(x,y,z)$ and its marginals must satisfy the following relationship:

$$\frac{f_{X,Y,Z}(x,y,z)}{f_Y(y)} = \frac{f_{X,Y}(x,y)}{f_Y(y)} \cdot \frac{f_{Y,Z}(y,z)}{f_Y(y)}$$

for each vector $(x,y,z)$ in the support of $(X,Y,Z)$. Diks and Panchenko (2006) show that, for a proper choice of weight function, $g(x,y,z)=f_Y^2(y)$, this is equivalent to

\begin{align} q = E[f_{X,Y,Z}(X,Y,Z)f_Y(Y) - f_{X,Y}(X,Y)f_{Y,Z}(Y,Z)]. \end{align}

They proposed the following estimator for $q$:

\begin{align} T_n(\varepsilon) = \frac{(n-1)}{n(n-2)} \sum_i (\hat{f}_{X,Y,Z}(X_i,Y_i,Z_i) \hat{f}_Y(Y_i) - \hat{f}_{X,Y}(X_i,Y_i) \hat{f}_{Y,Z}(Y_i,Z_i)) \end{align}

where $n$ is the sample size, and $\hat{f}_W$ is a local density estimator of a $d_W$-variate random vector $W$ at $W_i$ based on indicator functions $I_{ij}^W = I(\|W_i - W_j\| < \varepsilon)$, denoted by

\begin{align} \hat{f}_W(W_i) = \frac{(2 \varepsilon)^{-d_W}}{n-1} \sum_{j,j \neq i} I_{ij}^W. \end{align}

In the case of bivariate causality, the test is consistent if the bandwidth $\varepsilon$ is given by $\varepsilon_n = Cn^{-\beta}$, for any positive constant $C$ and $\beta \in (\frac{1}{4}, \frac{1}{3})$. The test statistic is asymptotically normally distributed in the absence of dependence between the vectors $W_i$. For the choice of the bandwidth, Diks and Panchenko (2006) suggest $\varepsilon_n = max(C_n^{-2/7},1.5)$, were $C$ can be calculated based on the ARCH coefficient of the series.

There are other tests of non-linear Granger causality such as in Hiemstra and Jones (1994), but this test in particular suffers from lack of power and over-rejection problems, as stated by Diks and Panchenko here.

As pointed out by @RichardHardy, you should be careful about using local density estimation in small samples. Since Diks and Panchenko showed that in samples smaller than 500 observations their test may under-reject, it would be wise to make further investigations in case the test does not reject the null hypothesis.

# References

Diks, C., & Panchenko, V. (2006). A new statistic and practical guidelines for nonparametric Granger causality testing. Journal of Economic Dynamics and Control, 30[9–10], 1647-1669.

Hiemstra, C., & Jones, J. D. (1994). Testing for Linear and Nonlinear Granger Causality in the Stock Price- Volume Relation. The Journal of Finance, 49(5), 1639–1664.

• Since the test involves density evaluation, I presume it is suited for relatively large samples. Would it make sense to use it for small samples as well? (The OP has samples of sizes between 15 and 100.) Also, what do you mean by the ARCH coefficient? Is it the $\alpha_1$ (a.k.a. "ARCH") coefficient from an ARCH(1) model? – Richard Hardy Jan 21 '16 at 17:21
• Exactly, it is the $\alpha_1$ coefficient. Since you have two series, you can estimate two ARCHs an take the mean of the alphas as a proxy for C. For series smaller than 500 obs the test may under-reject. There are some simulations in the Dicks and Panchenko paper. – Regis A. Ely Jan 21 '16 at 17:41

Transferring the notion of Granger causality from VAR to other models does not seem to have intrinsic contradictions. The basic principle when testing for Granger non-causality is to build a model that allows for a contribution of lagged $x$ towards predicting $y$ (besides lagged $y$'s own contribution), and testing the null hypothesis that the contribution is zero (by checking its statistical significance). If $x$'s contribution is too large to be written off to pure chance (is found to be statistically significantly different from zero), Granger non-causality is rejected; otherwise it is not.

The answer by @RegisA.Ely provides a very comprehensive way of testing, and I appreciate it. However, I have some reservations over how effective it might be when applied in small samples. My approach is more general and therefore should be applicable in small samples, too. That is, for a small sample, a "small" model would be used to avoid estimating complicated structures that need large sample size for decent estimation precision.

• I agree, but the problem of the test in small samples is under rejection, so I would say use Dicks and Panchenko test. If the test is significant, there is probably a non linear relationship. If the test is not significant, you need further investigation. – Regis A. Ely Jan 21 '16 at 18:40
• @RegisA.Ely, OK, I did not intend to pick on Dicks and Panchenko too much. I am just sceptical about local density estimation in small samples (and especially in more than one dimension) in general. – Richard Hardy Jan 21 '16 at 18:43