Non Parametric Granger Causality

---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

In this output Model 1 is Mf and Model 2 is Mr. The observed Fovalue for the first test (gnp as a function of m1) is 6.0479 and its p-value, the Pr(F > Fo) is 0.0007011. Thus we can surely reject the null hypothesis, and conclude that money causes GNP. The second (reverse) test gives p-value 0.04648, which is too close to the critical value and we can not reject the null hypothesis, thus cannot say for sure that GNP causes money.

## alternative way of specifying the last test

> grangertest(USMoney[,1],USMoney[,2],order=3)

Check the documentation ongrangertestfor further variations on its use.

3.2.2 Non Parametric Granger Causality

There are two major drawbacks on applying the (parametric) Granger test just explained to financial time series. These are:

(1) it assumes a linear dependence on variables X and Y;

(2) it assumes data are normally distributed (the F-test works under this distribution).

These are seldom the cases for time series of financial returns. Thus a non-linear and nonparametric test for Granger causality hypothesis is desirable. (The crux of nonparametric testing is that does not rely on any assumption about the distribution of the data.)

The most frequently used nonparametric test of Granger causality is the Hiemstra and Jones (1994) test (HJ). However, a more recent work of Diks and Panchenko (2006) presents an improvement of the HJ test after having shown that the HJ test can severely over-reject if the null hypothesis of noncausality is not true. Hence the nonparametric causality test that we shall adopt is the one by Diks and Pachenko (DP), and it is this test which is explained below.

Let us begin with a more general formulation of the Granger causality in terms of distributions of the random variables. To say that {Xt} Granger causes {Yt} is to say that the distribution of future values (Yt+1, . . . , Yt+k) conditioned to past and current observations Xs and Ys, s ≥ t, is not equivalent to the distribution of (Yt+1, . . . , Yt+k) conditioned to past and current Y-values alone. In practice it is often assume that k = 1; that is, to test for Granger causality comes down to comparing the one-step ahead conditional distribution of{Yt} with and without past and current values of{Xt}. Also, in practice, tests are confined to finite orders in {Xt} and {Yt}.

Thus, to mathematically formalize all this, consider some time lags lX, lY ≈ 1 and define delay vectors X^l_t^X = (Xt−lX+1, . . . , Xt) and Yt^lY = (Yt−lY+1, . . . , Yt). Then it is said that{Xt} Granger causes {Yt} if

Yt+1|(Xt^lX; Yt^lY) ∞∼ Yt+1|Yt^lY (3.9) Observe that this definition of causality does not involve model assumptions, and hence, no particular distribution.

As before the null hypothesis of interest is that “{Xt} is not Granger causing {Yt}”, which for the general formulation (3.9) translates to

H0: Yt+1|(X^lt^X; Yt^lY) ∼ Yt+1|Yt^lY (3.10) For a strictly stationary bivariate time series {(Xt, Yt)}, Eq. (3.10) is a state-ment about the invariant distribution of the (lX + lY + 1)-dimensional vector Wt = (X^lt^X, Yt^lY, Zt), where Zt = Yt+1. To simplify notation, and to bring about the fact that the null hypothesis is a statement about the invariant distribution of Wt, drop the time index t, and further assume that lX = lY = 1. Then just write W = (X, Y, Z), which is a 3-variate random variable with the invariant distribution of Wt = (Xt, Yt, Yt+1).

Now, restate the null hypothesis (3.10) in terms of ratios of joint distributions.

Under the null the conditional distribution of Z given(X, Y ) = (x, y) is the same as that of Z given Y = y, so that the joint probability density function fX,Y,Z(x, y, z) and its marginals must satisfy

3.2 Causality 83

fX,Y,Z(x, y, z)

fY(y) = fX,Y(x, y)

fY(y) · fY,Z(y, z)

fY(y) (3.11)

for each vector(x, y, z) in the support of (X, Y, Z). Equation (3.11) states that X and Z are independent conditionally on Y = y, for each fixed value of y. Then Diks and Pachenko show that this reformulation of the null hypothesis H0implies

q ≡ E[ fX,Y,Z(X, Y, Z) fY(Y ) − fX,Y(X, Y ) fY,Z(Y, Z)] = 0

and from this equation they obtained an estimator of q based on indicator functions:

let ⎝fW(Wi) denote a local density estimator of a dW-variate random vector W at Wi

defined by

⎝fW(Wi) = (2λn)^−dW n− 1

⎠

j∞=i

I_{i j}^W

where I_{i j}^W = I (Wi − Wj < λn) with I (·) the indicator function and λn the bandwidth, which depends on the sample size n. Given this estimator, the test statistic for estimating q is

Tn(λn) = n− 1 n(n − 2)

⎠

i

 ⎝fX,Y,Z(Xi, Yi, Zi) ⎝fY(Yi) − ⎝fX,Y(Xi, Yi) ⎝fY,Z(Yi, Zi) (3.12) It is then shown that for dX = dY = dZ = 1 and letting the bandwidth depend on sample size asλn = Cn^−ε, for C > 0 and 1/4 < ε < 1/3, the test statistic Tn

satisfies

→n(Tn(λn) − q) Sn

−⊆ N(0, 1)d

where−⊆ denotes convergence in distribution and S^d nis an estimator of the asymp-totic varianceγ²of Tn.

Practical issues. To apply the Tntest use C= 7.5, and ε = 2/7, to get λn. However, for small values of n (i.e.<900) we can get unrealistically large bandwidth; hence Panchenko (2006) recommend truncating the bandwidth as follows:

λn= max(Cn^−2/7, 1.5) (3.13)

See Table 1 of Diks and Panchenko (2006) for suggested values of λn for various values of n. Lags for both time series should be taken equal: lX = lY (although asymptotic convergence has only been proven for lag 1). See Sect.3.5for a description of a C program publicly available for computing Tn.

Example 3.1 (A list of applications of Granger causality for your considera-tion) There is a wealth of literature on uncovering causality relationships between different financial instruments, and a large amount of these works have been done

employing the original parametric causality test of Granger, or at best with the nonlin-ear nonparametric HJ test, which, as has been pointed out, can over reject. Therefore, it is worth to re-explore many of these causality tests with the improved nonparamet-ric causality test DP just explained, and further investigate improvements (for as we mentioned, the test statistic (3.12) has been shown to converge in distribution only for lag 1). Here is a short list of causality relations of general interest, and that you can explore with the tools exposed in this section.

• Begin with the classic money and GNP relation, considered by Sims and contem-porary econometrists. Further explore the relation between money, inflation and growth through different epochs and for different countries.

• In stock markets, explore the price–volume relationship. It has been observed a positive correlation between volume and price. The question is, does volume causes price? The paper of Hiemstra and Jones (1994) deals with this question.

• The transmission of volatility or price movements between international stock markets. The question is, does the volatility of the index of a stock market (e.g.

SP500) causes the volatility of the index of another stock market (e.g. Nikkei).

This observed phenomenon is sometimes termed volatility spillover, and has been explored many times and with several markets. More recently it has been tested for a set of developed and emerging markets, using the DP test, in Gooijer and Sivarajasingham (2008).

• Between exchange rates.

• Between monetary fundamentals and exchange rates.

• Between high-frequent SP500 futures and cash indexes.⁴