Figure 4.1 is the flow chart of a Bayesian early warning system, named BEWS-A. The system takes advantage of the filter model to make feature selection. The research effort is devoted to the investigation of indirect per-formance measures, such as noise removal, data reduction, and information measure, in selecting features. The system, BEWS-A is characterized by the nature of the filter model. It does not rely on a particular classifier’s bias, but on the intrinsic properties of the data, so the selected features can be used to learn different classifiers.
In step A, we first collect monthly data over period 1994M01 to 2003M12 for a sample of 28 countries and each country has 48 different financial vari-ables. A key assumption of our thesis is that countries in the sample don’t share common characteristics as far as openness to capital flows. Therefore, for each country, we will find out the leading indicators of its own and build a Bayesian early warning system, BEWS-A according to its particular char-acteristic. Thus the following steps from B to H will be described for a single
Figure 4.1: Flow chart of BEWS-A
country. In step B, we develop a Bayesian analysis model which is applied to detect regime shifts in a time series. A regime shift occurs when a significant difference exists between the mean value of the variable before and after a certain point. It implies that there are disturbance and change throughout the normal operation. So the detected regime shifts show signs of financial crises. As Figure 4.2 shows, the inputs of a Bayesian analysis model are the original values of all variables. They are transformed into posterior proba-bilities as the outputs which are representative of the probability that shifts occur.
Figure 4.2: Inputs and Outputs for Bayesian analysis model
After building the Bayesian analysis model, the obtained posterior prob-ability series, Pstep(t) will be used to evaluate both of the F statistic and the Spearman correlation coefficient for each of the variables. In step C and step D, the purpose is to extract key variables from all of the variables. The difference is that step C makes the first stage of extraction, and step D makes the second stage of extraction. Recall that, in chapter 1, we had introduced a forward crisis variable Yt−j which is given with the aim of predicting four months ahead of time. In step C, it is treated as a response variable and the
F values could tell us the degree of how a variable affect the response vari-able Yt−j. The higher the F values for a variable, the more we believe that it is a key variable. In step D, the Spearman correlation coefficients make it possible to find out the relationship between variables and Yt−j. Modulus of a correlation coefficient reflects the strength of the relationship. This helps us to determine whether a variable is important or not as well. Figure 4.3 illustrates the procedure of variable extraction in step C and step D, where the columns of the matrix represent variables and the rows represent sam-ples. Among these variables, the bold ones are key variables extracted by means of the F statistics and the shaded ones are key variables extracted by means of the Spearman correlation.
Figure 4.3: variable extraction in step C and D
In BEWS-A, We use both of the F statistics and the Spearman correlation to compare the results so as to find out key variables. However, those key vari-ables are not leading indicators yet because they are intercorrelated. Thus there must be some redundancy existing among those key variables. In step E, we continue to employ factor analysis to classify key variables. Factor analysis is capable of finding relationships between key variables and
sepa-rating those key variables into a smaller number of factors. In each factor, the factor loading of each key variable is a measure of significance. That is, the higher the factor loading is, the more significant the associated variable is. In this way, a variable with the highest Spearman correlation coefficient in a specific factor will be regarded as a leading indicator, and consequently the number of leading indicators we pick is as the same as the number of extracted factors. Therefore, the step F is achieved through step C, step D, amd step E. Table 4.1 is an example of leading indicator identification.
Assuming that there are 8 key variables extracted from step C and step D.
In the procedure of factor analysis, the 8 key variables are condensed into 3 factors and the associated factor loadings for each key variable are also calculated. As the table shows, the 8th key variable has the highest factor loading in factor 1. The 7th and the 3th key variables have the highest factor loadings in factor 2 and factor 3 respectively. Therefore, the 3th, 7th, and 8th variables will be identified as leading indicators.
Table 4.1: An example of leading indicator identification Key Factor Factor Factor
Variables 1 2 3
Key Var.1 0.14 0.59 0.35 Key Var.2 0.85 0.06 0.21 Key Var.3 0.18 0.01 0.91∗ Key Var.4 0.14 0.80 0.17 Key Var.5 0.89 0.16 0.04 Key Var.6 0.23 0.56 0.11 Key Var.7 0.16 0.91∗ 0.05 Key Var.8 0.93∗ 0.14 0.05
Owing to the property of factor analysis, the identified leading indica-tors are minimally correlated with each other. They are different in sig-nificance according to the performance of predicting crises. So a weight-ing procedure is necessary to buildweight-ing a robust system. As we have men-tioned before, the Spearman correlation coefficient quantifies the direction and magnitude of correlation, and the modulus of a correlation coefficient can reflect the strength of the relationship. The higher the modulus of a correlation coefficient is, the more important the variable is. Therefore, in step G, the leading indicators will be weighted with their Spearman correla-tion coefficients. In step H, we use the posterior probabilities of the leading indicators obtained from the Bayesian analysis model to create an index probability variable with their Spearmon correlation coefficients. Suppose Z = {Zi(t), i = 1, 2, . . . ,d} are the posterior probabilities of the leading indicators. R = {Ri, i = 1, 2, . . . ,d} are their correlation coefficients and P(t) is the index probability variable. Then the equation is defined as follows:
P(t) = Z1(t)R1+ Z2(t)R2+ · · · + Zd(t)Rd
R1+ R2+ · · · + Rd (4.1)
In addition, a threshold value T is determined based on the training samples.
For BEWS-A, the calculated index variable P(t) will be ultimately put to use in anticipating crises in this form:
Y(t) =
where Y(t) = 1 represents that crises may occur within future four months and then an alarm will be sent. This system has accorded with our expec-tation to give an early warning. As long as new information is available, the new posterior probabilities of leading indicators will be calculated, and then P(t) will be updated according to the present state of information.