6. Symbolic noncausality test

The present section reviews the symbolic noncausality test (SNC) and discusses the differences with the classical Granger noncausality test. As in the case of independence test, the main idea here is to derive the asymptotic distribution for the statistic when there is no causality between the series. A full explanation of the test is shown in [60].

Let us consider that X and Y are two independent random time series sized T + 1 and the symbolized time series can be expressed as Sx = {sx1,sx2,..,sxT + 1} and Sy = {sy1,sy2,…, syT + 1}. To test causality, we have to define two new series, grouping Sx and Sy in the following way:

$$\begin{aligned} \text{(1) } \mathrm{Sxy} &= \{ \text{(sx}\_1, \text{sy}\_2), \text{(sx}\_2, \text{sy}\_3), \dots, \text{(sx}\_{t-1}, \text{sy}\_t), \dots, \text{(sx}\_T \text{sx}\_{T+1}) \} \\\\ \text{(2) } \mathrm{Syx} &= \{ \text{(sx}\_1, \text{sy}\_2), \text{(sx}\_2, \text{sy}\_3), \dots, \text{(sx}\_{t-1}, \text{sy}\_t), \dots, \text{(sx}\_T \text{sx}\_{T+1}) \} \end{aligned}$$

If the alphabet is composed by three symbols, the combination (sxt�<sup>1</sup>, syt) takes a value from the set of nine possible events {(1,1), (1,2), (1,3), (2,1), (2,2),(2,3),(3,1),(3,2),(3,3)}. Note that each event should be independent with probability 1/9 (Sx and Sy are random). Only if at least one event were deviated from 1/9, would there be evidence of noncausality.

An alphabet of a = 3 symbols determines n = 3<sup>2</sup> = 9 possible events in the set of pairs {(xt-1,yt)} or {(yt�<sup>1</sup>, xt)}. Considering "a" symbols and the events n = a 2 , the vector of the n frequencies Exyi/T and Eyxi/T could be approximated by a multivariate normal distribution N(1/n,σ<sup>2</sup> Ω) where σ<sup>2</sup> is (1/nT) and Ω is a idempotent matrix as in (9).

$$
\mathcal{O}\_{nm} \equiv \begin{bmatrix}
(n-1)/n & -1/n & \dots & -1/n \\
\vdots & \vdots & \ddots & \vdots \\
\end{bmatrix} \tag{9}
$$

Following a similar approach as in Section 5, the statistics for the both hypothesis can be defined as in (10) and (11).

$$\left\{\frac{\sum\_{i=1}^{i=n} \varepsilon xy\_i^2}{\sigma^2}\right\}\tag{10}$$

$$\left\{\frac{\sum\_{i=1}^{i=n} \varepsilon yx\_i^2}{\sigma^2}\right\}\tag{11}$$

The term in brackets in (10), (11) are quadratic forms in random normal variables. Applying the theorem presented in [59], in the present case where vector X = (ε1/σ,ε2/σ,…,εn/σ) is distributed multivariate normal N(ø,Ω). As mentioned in Section 5, tr(ΑΩ) = n-1, thus X'ΑX distributes Chi-square with (n�1) degrees of freedom. In this case, Α is the identity matrix I and Ω is symmetric, singular, and idempotent.

Note that we derive the test assuming that X and Y are random processes. However, we can apply the test for stationary time series and optionally apply an autoregressive process if we want to remove linear dependence and testing the noncausality between the residuals of the two series.

6. Symbolic noncausality test

116 Time Series Analysis and Applications

following way:

the series. A full explanation of the test is shown in [60].

(1) Sxy = {(sx1, sy2), (sx2, sy3),…,(sxt�<sup>1</sup>,syt),…,(sxT,sxT + 1)} (2) Syx = {(sx1, sy2), (sx2, sy3),…,(sxt�<sup>1</sup>,syt),…,(sxT,sxT + 1)}

{(yt�<sup>1</sup>, xt)}. Considering "a" symbols and the events n = a

is (1/nT) and Ω is a idempotent matrix as in (9).

Ωnxn �

and Ω is symmetric, singular, and idempotent.

defined as in (10) and (11).

event were deviated from 1/9, would there be evidence of noncausality.

and Eyxi/T could be approximated by a multivariate normal distribution N(1/n,σ<sup>2</sup>

The present section reviews the symbolic noncausality test (SNC) and discusses the differences with the classical Granger noncausality test. As in the case of independence test, the main idea here is to derive the asymptotic distribution for the statistic when there is no causality between

Let us consider that X and Y are two independent random time series sized T + 1 and the symbolized time series can be expressed as Sx = {sx1,sx2,..,sxT + 1} and Sy = {sy1,sy2,…, syT + 1}. To test causality, we have to define two new series, grouping Sx and Sy in the

If the alphabet is composed by three symbols, the combination (sxt�<sup>1</sup>, syt) takes a value from the set of nine possible events {(1,1), (1,2), (1,3), (2,1), (2,2),(2,3),(3,1),(3,2),(3,3)}. Note that each event should be independent with probability 1/9 (Sx and Sy are random). Only if at least one

An alphabet of a = 3 symbols determines n = 3<sup>2</sup> = 9 possible events in the set of pairs {(xt-1,yt)} or

ð Þ n � 1 =n �1=n … �1=n �1=n nð Þ � 1 =n … �1=n ⋮ ⋮⋱⋮

�1=n �1=n … ð Þ n � 1 =n

Following a similar approach as in Section 5, the statistics for the both hypothesis can be

σ2 ( )

σ2 ( )

The term in brackets in (10), (11) are quadratic forms in random normal variables. Applying the theorem presented in [59], in the present case where vector X = (ε1/σ,ε2/σ,…,εn/σ) is distributed multivariate normal N(ø,Ω). As mentioned in Section 5, tr(ΑΩ) = n-1, thus X'ΑX distributes Chi-square with (n�1) degrees of freedom. In this case, Α is the identity matrix I

P<sup>i</sup>¼<sup>n</sup> <sup>i</sup>¼<sup>1</sup> <sup>ε</sup>xy<sup>2</sup> i

P<sup>i</sup>¼<sup>n</sup> <sup>i</sup>¼<sup>1</sup> <sup>ε</sup>yx<sup>2</sup> i

2

, the vector of the n frequencies Exyi/T

Ω) where σ<sup>2</sup>

(9)

(10)

(11)

$$\mathbf{x}\_{t} = \alpha\_{0} + \alpha\_{1}\mathbf{x}\_{t-1} + \mu\mathbf{x}\_{t} \tag{12}$$

$$y\_t = \beta\_0 + \beta\_1 y\_{t-1} + uy\_t \tag{13}$$

Finally, the statistics of noncausality SNC(X ! Y) and SNC(Y ! X) are defined as in (14) and (15).

$$\text{SNC}(X \to Y) \equiv nT \left\{ \sum\_{i=1}^{i=n} \left( \frac{\text{Ex}y\_i}{T} - \frac{1}{n} \right)^2 \right\} \\ \text{assumptionally distributed } \chi^2\_{n-1} \tag{14}$$

$$\text{SNC}(Y \to X) \equiv nT \left\{ \sum\_{i=1}^{i=n} \left( \frac{\text{Eyx}\_i}{T} - \frac{1}{n} \right)^2 \right\} \\ \text{assumptionally distributed} \; \chi^2\_{n-1} \tag{15}$$

Note that in practice, computing the statistic is very simple. In summary, the test works as follows:

Step 1: Consider time series {xt}t = <sup>1</sup>,2,…,T + <sup>2</sup> and {yt}t = <sup>1</sup>,2,…,T + <sup>2</sup> we can optionally apply an AR (1) to both series as in (12) and (13) in order to eliminate autocorrelation and define the new residuals time series {uxt}t = <sup>1</sup>,2,…,T + <sup>1</sup> and {uyt}t = <sup>1</sup>,2,…,T + 1. Note that 1 observation is lost after applying AR(1).

Step 2: In {uxt}t = <sup>1</sup>,2,…,T + <sup>1</sup> and {uyt}t = <sup>1</sup>,2,…,T + <sup>1</sup> apply a partition in "a" equiprobable regions and translate the series into {sxt}t = <sup>1</sup>,2,…,T + <sup>1</sup> and {syt}t = <sup>1</sup>,2,…,T + 1.

Step 3: According to the two hypothesis, X ! Yand Y ! X define the two sets Sxy = {(sx1, sy2), (sx2,sy3),…,(sxt-1,syt),…,(sxT,sxT + 1)} and Syx = {(sx1,sy2), (sx2,sy3),…,(sxt-1,syt),…,(sxT, sxT + 1)}.

Step 4: For Sxy and Syx, compute the frequency of the n=a<sup>2</sup> different events Exyi/T and Eyxi/T considering i = 1,2,…, a<sup>2</sup> .

Step 5: Taking into account Eqs. (14) and (15) compute the SNC(<sup>X</sup> ! <sup>Y</sup>) = nT{Σ[(Exyi/T)–(1/n)]<sup>2</sup> } and SNC(<sup>Y</sup> ! <sup>X</sup>) = nT{Σ[(Eyxi/T) � (1/n)]<sup>2</sup> }.

Step 6: Finally, two null hypotheses must be contrasted: X does not cause Y, and Y does not cause X. In the first case SNC(X ! Y) should be compared with a Chi-2 with n-1 degree of freedom at 0.05 of significance, if SNC(X ! Y) is larger than the critical value the null hypothesis is rejected. The same should be done with SNC(Y ! X).

### 7. Symbolic noncausality and Granger noncausality

The concept of causality into the experimental practice is due to Clive Granger. The classical approach of Granger causality is based on temporal properties. Although the principle was formulated for wide classes of systems, the autoregressive modeling framework proposed by Granger was basically a linear model, and as mentioned in [61] the choice was made due to practical reasons. Granger noncausality test is among the most applied tool testing causality. Three limitations should be noted: (1) the classical test has a good performance when the process is linear. This is because it is based on the vector autoregressive model (VAR); (2) there are extension of the classical test to consider nonlinear causality but they are related with a particular nonlinear model; (3) some authors assert that empirical time series are generally contaminated with noise producing what is known as spurious causality or not allowing to detect the causality.

SCN test presented in [60] is a nonparametric noncausality test based on the symbolic time series analysis. The idea is to develop a complementary test to the Granger noncausality, showing strengths in the points where the Granger test is weak. In this sense, the proposed SNC test performs well detecting nonlinear processes, in particular the chaotic processes. In addition, the mentioned problem related with spurious causality should be alleviated. In fact, according to some experiments nonlinear models such as NLAR model, Lorenz map, and models with exponential terms are not detected by Granger test but the SNC identifies these processes. The test is based on information theory considering an approximation of the entropy as the measure of uncertainty of a random variable. Information theory is considered to be a subset of communication theory. However, in [62] is consider that it is much more. It has fundamental contributions to make in statistical physics, computer science, and statistical inference, and in probability and statistics. It is important to highlight and is an important idea relating symbolic analysis, information theory, and the concept of noise. Information theory considers that communication between A and B is a physical process in an imperfect ambient contaminated by noise. Another important concept is the discrete channel, defined as a system consisting of an input alphabet X and output alphabet Yand a probability transition matrix p(y|x) that expresses the probability of observing the output symbol y given that we send the symbol x.

To compare the performance between the classical Granger noncausality and the proposed SNC test, the following stochastic and deterministic models were simulated:


Table 2 shows the results of the power experiments applying the SNC and the Granger noncausality test to 10,000 Monte Carlo simulations for the four models and for different sample sizes (T = 50, 100, 500, 1000, and 5000).


Table 2. Simulated power of the SNC and the Granger non causality statistic.

formulated for wide classes of systems, the autoregressive modeling framework proposed by Granger was basically a linear model, and as mentioned in [61] the choice was made due to practical reasons. Granger noncausality test is among the most applied tool testing causality. Three limitations should be noted: (1) the classical test has a good performance when the process is linear. This is because it is based on the vector autoregressive model (VAR); (2) there are extension of the classical test to consider nonlinear causality but they are related with a particular nonlinear model; (3) some authors assert that empirical time series are generally contaminated with noise producing what is known as spurious causality or not allowing to

SCN test presented in [60] is a nonparametric noncausality test based on the symbolic time series analysis. The idea is to develop a complementary test to the Granger noncausality, showing strengths in the points where the Granger test is weak. In this sense, the proposed SNC test performs well detecting nonlinear processes, in particular the chaotic processes. In addition, the mentioned problem related with spurious causality should be alleviated. In fact, according to some experiments nonlinear models such as NLAR model, Lorenz map, and models with exponential terms are not detected by Granger test but the SNC identifies these processes. The test is based on information theory considering an approximation of the entropy as the measure of uncertainty of a random variable. Information theory is considered to be a subset of communication theory. However, in [62] is consider that it is much more. It has fundamental contributions to make in statistical physics, computer science, and statistical inference, and in probability and statistics. It is important to highlight and is an important idea relating symbolic analysis, information theory, and the concept of noise. Information theory considers that communication between A and B is a physical process in an imperfect ambient contaminated by noise. Another important concept is the discrete channel, defined as a system consisting of an input alphabet X and output alphabet Yand a probability transition matrix p(y|x) that expresses the probability of

To compare the performance between the classical Granger noncausality and the proposed

1. AR(1). We consider two independent series generated by autoregressive (AR) processes: Xt = 0.2 + 0.45Xt�<sup>1</sup> + ε1<sup>t</sup> and Yt = 0.8 + 0.5Yt-<sup>1</sup> + ε2t. Where ε1<sup>t</sup> and ε2<sup>t</sup> are i.i.d. and normally

3. NLAR (Autoregressive Nonlinear). Xt = 0.2jXt-1j/(2 + jXt-1j) + ε1<sup>t</sup> and Yt = 0.7jYt-1j/(1 + jXt-1j)

Table 2 shows the results of the power experiments applying the SNC and the Granger noncausality test to 10,000 Monte Carlo simulations for the four models and for different

generated randomly. This is a discrete version of the Lorenz process as in [63].

Yt�<sup>1</sup> + ε1<sup>t</sup> and

<sup>t</sup>�1; with initial conditions X1, Y<sup>1</sup>

observing the output symbol y given that we send the symbol x.

SNC test, the following stochastic and deterministic models were simulated:

2. Nonlinear with exponential component. Xt = 1.4–0.5Xt�1e

Yt = 0.4 + 0.23Yt�<sup>1</sup> + ε2t; where ε1<sup>t</sup> and ε2<sup>t</sup> are i.i.d. normal(0,1).

+ ε2t; where ε1<sup>t</sup> and ε2<sup>t</sup> are i.i.d. normal(0,1).

sample sizes (T = 50, 100, 500, 1000, and 5000).

4. Lorenz: Xt <sup>=</sup> <sup>1</sup>.96Xt�1�0.8Xt�<sup>1</sup>Yt�1; Yt <sup>=</sup> <sup>0</sup>.2Yt�<sup>1</sup> <sup>+</sup> <sup>0</sup>.8X<sup>2</sup>

detect the causality.

118 Time Series Analysis and Applications

distributed (0,1).

Following [60], a 60% acceptance or rejection of the null hypothesis is considered as a threshold. SNC and Granger noncausality correctly identifies noncausality in AR(1) process. Table 2 suggests that SNC is more conservative in the rejection of causality with percentages less than 5%. The nonlinear model with an exponential component implies causality from Y to X. Note that SNC detects the causality when the sample size is 500 or larger. However, Granger test does not detect causality in any case. As asserted by [58] the NLAR process is very difficult to detect. Note that SCN is the only one detecting the causality when T = 5000. The Lorenz discrete map is also chaotic, and it is detected by SNC starting from T = 100. However, note that Granger test never detects the causality. In particular, is highlighted that Granger test is not able to detect the model with an exponential component, the NLAR model and the chaotic Lorenz map.

Finally, we compare both tests with real data from US. In particular, we consider two wellknown relationships in economics: the Phillips curve [64] about the relation between unemployment and inflation rates, the Okun's law [65] establishing a relation between unemployment and economic rate. We take annual data for the US unemployment rate, inflation rate, and economic growth for the period 1948–2016 representing a total of 69 observations. Table 3 shows the results of the Granger noncausality test and the symbolic test considering a partition of two symbols.

The results are similar for both tests. On one hand, Granger and symbolic tests detect causality from inflation to unemployment in the Phillips curve. On the other hand, the two tests detect causality running from economic growth to unemployment in the Okun's law. The economic theory suggests that inflation increases unemployment while economic growth reduces it. Note that STSA allows thinking about causality in a more general way, whereas Granger noncausality needs to think of continuous measured variables, this should not be a problem for STSA. Let us consider the following example; we now can test the hypothesis of causality from economic growth (G) and inflation (P) to unemployment (U). The main problem is that we have to test causality from a two-dimensional variable to a one dimensional. Symbolization permits to transform the two-dimensional problem in one dimensional and then to apply the symbolic test as explained. We can follow a similar approach as in [66] where STSA is applied


\* Indicates rejection of the null hypothesis at the 5% level significance.

Table 3. SNC and the Granger non causality for the Phillips Curve and Okun's Law in US.

Figure 2. Two-dimensional variable (economic growth and inflation) is transformed into a four symbol variable.

to dynamic regimes. Figure 2 shows the transformation of the variable (G, P) in a symbolic variable with an alphabet of four symbols (I: low economic growth and low inflation, II: low economic growth and high inflation, III: high economic growth and high inflation, IV: high economic growth and low inflation) considering as partition the mean of each variable. Note that now the application of symbolic causality is easy, the hypothesis that the economic regime (G, P) does not cause unemployment is rejected since the SNC is 31.76 and Chi-2 with 15 degree of freedom (42 –1) at 95% is 25.00. The opposite hypothesis is not rejected because the SNC is 24.71. It is not possible to test this type of causality with the traditional Granger noncausality test.
