11.3 The Fuzzy Approach to Money Demand


11.3.1 Fuzzy Clustering

Ruspini (1969) introduced fuzzy partition to describe the cluster structure of a data set and suggested an algorithm to compute the optimum fuzzy partition. Dunn (1973) generalized the minimum-variance clustering procedure to a Fuzzy ISODATA clustering technique. Bezdek (1981) used Dunn's (1973) approach to obtain an infinite family of algorithms known as the Fuzzy C-Means (FCM) algorithm. He generalized the fuzzy objective function by introducing the weighting exponent $ m$, $ 1 \leq m < \infty$:

$\displaystyle J_m(U,V) = \sum_{k=1}^n \sum_{i=1}^c (u_{ik})^m d^2 (x_k,v_i),$ (11.11)

where $ {\cal X} = \{x_1, x_2, \ldots, x_n\} \subset \mathbb{R}^p$ is a subset of the real $ p$-dimensional vector space $ \mathbb{R}^p$ consisting of $ n$ observations, $ U$ is a random fuzzy partition matrix of $ {\cal X}$ into $ c$ parts, $ v_i$'s are the cluster centers in $ \mathbb{R}^p$, and $ d(x_k,v_i) = \Vert x_k - v_i \Vert = \sqrt{(x_k -
v_i)^\top (x_k - v_i)}$ is an inner product induced norm on $ \mathbb{R}^p$. Finally, $ u_{ik}$ refers to the degree of membership of point $ x_k$ to the $ i$th cluster. This degree of membership, which can be seen as a probability of $ x_k$ belonging to cluster $ i$, satisfies the following constraints:
$\displaystyle 0 \le u_{ik} \le 1,$   $\displaystyle \textrm{for} \; 1 \le i \le c, 1 \le k \le n,$ (11.12)
$\displaystyle 0 < \sum_{k=1}^n u_{ik} < n,$   $\displaystyle \textrm{for} \; 1 \le i \le c,$ (11.13)
$\displaystyle \sum_{i=1}^c u_{ik} = 1,$   $\displaystyle \textrm{for} \; 1 \le k \le n.$ (11.14)

The FCM uses an iterative optimization of the objective function, based on the weighted similarity measure between $ x_k$ and the cluster center $ v_i$. More details on the FCM algorithm can be found in Mucha and Sofyan (2000).

In practical applications, a validation method to measure the quality of a clustering result is needed. Its quality depends on many factors, such as the method of initialization, the choice of the number of clusters $ c$, and the clustering method. The initialization requires a good estimate of the clusters and the cluster validity problem can be reduced to the choice of an optimal number of clusters $ c$. Several cluster validity measures have been developed in the past by Bezdek and Pal (1992).


11.3.2 The Takagi-Sugeno Approach

Takagi and Sugeno (1985) proposed a fuzzy clustering approach using the membership function $ \mu_A(x): \mathcal{X} \to [0,1]$, which defines a degree of membership of $ x \in \mathcal{X}$ in a fuzzy set $ A$. In this context, all the fuzzy sets are associated with piecewise linear membership functions.

Based on the fuzzy-set concept, the affine Takagi-Sugeno (TS) fuzzy model consists of a set of rules $ R_i, i=1,\ldots,r$, which have the following structure:

IF $ x$ is $ A_i$, THEN $ y_i = a_i^{\top} x + b_i$.

This structure consists of two parts, namely the antecedent part ``$ x$ is $ A_i$'' and the consequent part `` $ y_i = a_i^{\top} x + b_i$,'' where $ x \in \mathcal{X} \subset \mathbb{R}^p$ is a crisp input vector, $ A_i$ is a (multidimensional) fuzzy set defined by the membership function $ \mu_{A_i}(x): \mathcal{X} \rightarrow [0,1]$, and $ y_i \in \mathbb{R}$ is an output of the $ i$-th rule depending on a parameter vector $ a_i \in \mathbb{R}^p$ and a scalar $ b_i$.

Given a set of $ r$ rules and their outputs (consequents) $ y_i$, the global output $ y$ of the Takagi-Sugeno model is defined by the fuzzy mean formula:

$\displaystyle y = \frac {\sum_{i=1}^r \mu_{A_i}(x) y_i}{\sum_{i=1}^r \mu_{A_i}(x)}.$ (11.15)

It is usually difficult to implement multidimensional fuzzy sets. Therefore, the antecedent part is commonly represented as a combination of equations for the elements of $ x=(x_1,\ldots,x_p)^\top$, each having a corresponding one-dimensional fuzzy set $ A_{ij}, j=1,\ldots,p$. Using the conjunctive form, the rules can be formulated as:

IF $ x_1$ is $ A_{i,1}$ AND $ \cdots$ AND $ x_p$ is $ A_{i,p}$, THEN $ y_i = a_i^{\top} x + b_i$,

with the degree of membership $ \mu_{A_i}(x) = \mu_{A_i,1}(x_1) \cdot \mu_{A_i,2} (x_2) \cdots \mu_{A_i,p}(x_p)$. This elementwise clustering approach is also referred to as product space clustering. Note that after normalizing this degree of membership (of the antecedent part) is:

$\displaystyle \phi_i (x) = \frac{\mu_{A_i}(x)} {\sum_{j=1}^{r} \mu_{A_j}(x)}.$ (11.16)

We can also interpret the affine Takagi-Sugeno model as a quasilinear model with a dependent input parameter (Wolkenhauer; 2001):

$\displaystyle y = \left( \sum_{i=1}^r \phi_i (x) a_i^\top \right) x + \sum_{i=1}^r \phi_i (x) b_i = a^\top (x) + b(x).$ (11.17)


11.3.3 Model Identification

The basic principle of model identification by product space clustering is to approximate a nonlinear regression problem by decomposing it to several local linear sub-problems described by IF-THEN rules. A comprehensive discussion can be found in Giles and Draeseke (2001).

Let us now discuss identification and estimation of the fuzzy model in case of multivariate data. Suppose

$\displaystyle y = f(x_{1}, x_{2}, ..., x_{p}) + \varepsilon$ (11.18)

where the error term $ \varepsilon$ is assumed to be independent, identically and normally distributed around zero. The fuzzy function $ f$ represents the conditional mean of the output variable $ y$. In the rest of the chapter, we use a linear form of $ f$ and the least squares criterion for its estimation. The algorithm is as follows.

Step 1
For each pair $ x_r$ and $ y$, separately partition $ n$ observations of the sample into $ c_r$ fuzzy clusters by using fuzzy clustering (where $ r = 1, ..., p)$.

Step 2
Consider all possible combinations of $ c$ fuzzy clusters given the number of input variables $ p$, where $ c = \prod^p_{r=1} c_r$.

Step 3
Form a model by using data taken from each fuzzy cluster:

$\displaystyle y_{ij} = \beta_{i0} + \beta_{i1} x_{1ij} + \beta_{i2} x_{2ij} + ... + \beta_{ip} x_{pij} + \varepsilon_{ij}$ (11.19)

where observation index $ j = 1,\ldots,n$ and cluster index $ i = 1, \ldots, c$.

Step 4
Predict the conditional mean of $ x$ by using:

$\displaystyle \hat{y}_k = \frac {\sum_{i=1}^c (b_{i0} + b_{i1} x_{1k} + ... + b_{ip} x_{pk}) w_{ik}} {\sum_{i=1}^c w_{ik}}, \qquad k = 1, \ldots, n,$ (11.20)

where $ w_{ik} = \prod^p_{r=1} \delta_{ij} \mu_{rj}(x_k), i = 1,\ldots, c, $ and $ \delta_{ij}$ is an indicator equal to one if the $ j$th cluster is associated with the $ i$th observation.

The fuzzy predictor of the conditional mean $ y$ is a weighted average of linear predictors based on the fuzzy partitions of explanatory variables, with a membership value varying continuously through the sample observations. The effect of this condition is that the non-linear system can be effectively modelled.

The modelling technique based on fuzzy sets can be understood as a local method: it uses partitions of a domain process into a number of fuzzy regions. In each region of the input space, a rule is defined which transforms input variables into output. The rules can be interpreted as local sub-models of the system. This approach is very similar to the inclusion of dummy variables in an econometric variable. By allowing interaction of dummy-variables and independent variables, we also specify local sub-models. While the number and location of the sub-periods is determined endogenously by the data in the fuzzy approach, they have been imposed exogenously after visual data inspection in our econometric model. However, this is not a fundamental difference because the number and location of the sub-periods could also be determined automatically by using econometric techniques.


11.3.4 Modelling Indonesian Money Demand with Fuzzy Techniques

In this section, we model the M2 money demand in Indonesia using the approach of fuzzy model identification and the same data as in Section 11.2. Like in the econometric approach logarithmic real money demand ($ mr_t$) depends on logarithmic GNP ($ y_t$) and the logarithmic long-term interest-rate ($ r_t$):

$\displaystyle mr_t = \beta_0 + \beta_1 \ln Y_t + \beta_2 r_t.$ (11.21)

The results of the fuzzy clustering algorithm are far from being unambiguous. Fuzzy clustering with real money and output yields three clusters. However, real money and output clusters overlap, such that it is difficult to identify three common clusters. Hence, we arrange them as four clusters. On the other hand, clustering with real money and the interest rate leads to two clusters. The intersection of both clustering results gives 4 different clusters.

Table 11.2: Four clusters of Indonesian money demand data
Cluster Observations $ \beta_0$ $ \beta_1$ ($ y_t$) $ \beta_2$ ($ r_t$)
    (t-value) (t-value) (t-value)
$ 1$ 1-15 3.9452 0.5479 -0.2047
    (3.402) (5.441) (-4.195)
$ 2$ 16-31 1.2913 0.7123 0.1493
    (0.328) (1.846) (0.638)
$ 3$ 34-39 28.7063 -1.5480 -0.3177
    (1.757) (-1.085) (-2.377)
$ 4$ 40-51 -0.2389 0.8678 0.1357
    (-0.053) (2.183) (0.901)

The four local models are presented in Table 11.2. In the first cluster that covers the period 1990:1-1993:3 GNP has a positive effect on money demand, and the interest rate effect is negative. The output elasticity is substantially below one, but increases in the second cluster (1993:4-1997:3). The interest rate has no significant impact on real money in the second period. The third cluster from 1997:4 to 1998:4 covers the Asean crisis. In this period, the relationship between real money and output breaks down while the interest rate effect is stronger than before. The last cluster covers the period 1999:4-2002:3, in which the situation in Indonesia was slowly brought under control as a result of having a new government elected in October 1999. The elasticity of GNP turned back approximately to the level before the crisis. However, the effect of the interest rate is not significant.

Figure 11.4: Fitted money demand (dotted line): econometric model (dashed line) and fuzzy model (solid line).

\includegraphics[width=1.05\defpicwidth]{fuzzyplot.ps}

The fit of the local sub-models is not as good as the fit of the econometric model (Figure 11.4). The main reasons for this result are that autocorrelation and seasonality of the data have not been considered in the fuzzy approach, mainly for computational reasons. Additionally, the determination of the number of different clusters turned out to be rather difficult. Therefore, the fuzzy model for Indonesian money demand described here should be interpreted as an illustrative example for the robustness analysis of econometric models. More research is necessary to find a fuzzy specification that describes the data as well as the econometric model.