8. Conclusion

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

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

Null hypothesis Granger SNC(2 symbols)

Unemployment does not cause inflation 0.04 1.53 Inflation does not cause unemployment 16.90\* 9.41\*

Unemployment does not cause economic growth 3.37 2.94 Economic growth does not cause unemployment 61.01\* 9.65\*

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

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

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

of two symbols.

120 Time Series Analysis and Applications

Phillips curve

Okun's law

STSA is a powerful tool being applied to many scientific fields. There are recent applications in robotic, biology, medicine, communication, and engineering. However, applications in social sciences are very recent. The main difficult is the few historical data produced by the social processes. Social sciences are used to applied statistical tests for proving their hypothesis. However, there is much work to do in developing statistical tests based on STSA to be applied in social sciences. There are some very recent efforts applied to economics and finance using STSA. In particular, we present a symbolic independence test, which seems to be powerful in detecting nonlinearities compared with well-known BDS and runs test. The symbolic test is better detecting models such as the chaotic Anosov and Logistic or some stochastic models such as NLMA or NLSIGN. A second symbolic test about causality detects complex processes such as NLAR, nonlinear exponential, or the Lorenz chaotic process when the traditional Granger noncausality cannot. The symbolic causality also enables causality to be tested in a more general perspective. The application of test from a two-dimensional economic variable to a one-dimensional economic variable is a clear example of the potential of STSA in economics and social sciences in general.

One future research line could be to develop a powerful nonlinear test for multidimensional variables. As it was explained, STSA permits to transform a multidimensional time series in a one-dimensional time series simplifying the analysis. This could have important applications in relationships involving vector functions. A more general line of research is to find methodologies to define the optimal partition. As mentioned before, equiprobable partition is generally applied but to find the right partition is still a theoretical and practical weakness in STSA.
