LM(4) = 0.479 [0.751] JB = 0.196 [0.906] ARCH(4) = 0.970 [0.434]

Here JB refers to the Jarque-Bera Test for nonnormality, RESET is the usual

test for general nonlinearity and misspeci¬cation, LM(4) denotes a Lagrange-

6.2 Money Demand 111

Multiplier test for autocorrelation up to order 4, ARCH(4) is a Lagrange-

Multiplier test for autoregressive conditional heteroskedasticity up to order

4. Numbers in brackets are corresponding p-values. Given these diagnostic

statistics, the regression seems to be well speci¬ed. There is a mean shift in

98:3 and the impulse dummies capture the fact, that the structural change in

GNP occurs two months before the change in real money. The in¬‚ation rate is

not signi¬cant and is therefore not included in the equation.

The implied income elasticity of money demand is 0.47/(1“0.53) = 1 and the

interest rate elasticity is “0.13/(1“0.53) = “0.28. These are quite reasonable

magnitudes. Equation (6.10) can be transformed into the following error cor-

rection representation:

= ’0.47 · (mrt’1 ’ yt’1 + 0.28rt’1 )

∆mrt

+ 0.47∆yt ’ 0.13∆rt + deterministic terms + ut (6.11)

Stability tests for the real money demand equation (6.10) are depicted in ¬gure

6.3. The Cusum of squares test indicates some instability at the time of the

Asian crises, and the coe¬cients of lagged real money and GNP seem to change

slightly after the crisis. A possibility to allow for a change in these from 1998

on is two introduce to additional right-hand-side variables: lagged real money

multiplied by the step dummy ds9803 and GNP multiplied by ds9803. The

respective coe¬cients for the period 1998:3-2002:3 can be obtained by summing

the coe¬cients of lagged real money and lagged real money times step dummy

and of GNP and GNP times step dummy, respectively. This reveals that the

income elasticity stays approximately constant (0.28/(1“0.70)=0.93 until 98:02

and (0.28+0.39)/(1-0.70+0.32)=0.92) from 98:3 to 2002:3 and that the interest

rate elasticity declines in the second half of the sample from “0.13/(1“0.70)=“

0.43 to “0.13/(1-0.79+0.32)=“0.21:

0.697 mrt’1 + 0.281 yt ’ 0.133 rt

mrt =

(7.09) (2.39) (’6.81)

’ 0.322 mrt’1 · ds9803t + 0.288 yt · ds9803t

(’2.54) (2.63)

+ 0.133 ’ 0.032 s1t ’ 0.041 s2t ’ 0.034 s3t

(0.25) (’2.49) (’3.18) (’2.76)

+ 0.110 ds9802t + 0.194 di9801t + ut (6.12)

(2.04) (5.50)

112 6 Money Demand Modelling

Figure 6.3: Stability Test for the Real Money Demand Equation (6.10)

R2 = 0.989

T = 51 (1990:1 - 2002:3) RESET(1) = 4.108 [0.050]

LM(4) = 0.619 [0.652] JB = 0.428 [0.807] ARCH(4) = 0.408 [0.802]

Accordingly, the absolute adjustment coe¬cient µ in the error correction rep-

resentation increases from 0.30 to 0.62.

It can be concluded that Indonesian money demand has been surprisingly sta-

ble throughout and after the Asian crisis. A shift in the constant term and

two impulse dummies that correct for the di¬erent break points in real money

and real output are su¬cient to yield a relatively stable money demand func-

tion with an income elasticity of one and an interest rate elasticity of “0.28.

However, a more ¬‚exible speci¬cation shows that the adjustment coe¬cient µ

increases and that the interest rate elasticity decreases after the Asian crisis.

6.3 Fuzzy Model Identi¬cation 113

6.3 Fuzzy Model Identi¬cation

6.3.1 Fuzzy Clustering

Ruspini (1969) introduced a notion of 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 pro-

cedure to a Fuzzy ISODATA clustering technique. Bezdek (1981) generalized

Dunn™s approach to obtain an in¬nite family of algorithms known as the Fuzzy

C-Means (FCM) algorithm. He generalized the fuzzy objective function by

introducing the weighting exponent m, 1 ¤ m < ∞;

n c

(uik )m d2 (xk , vi ),

Jm (U, V ) = (6.13)

k=1 i=1

where X = {x1 , x2 , . . . , xn } ‚ Rp is a subset of the real p-dimensional vector

space Rp consisted of n observations, U is a randomly fuzzy partition matrix

of X into c parts, vi is the cluster centers in Rp , d(xk , vi ) = xk ’ vi =

(xk ’ vi )T (xk ’ vi ) is an inner product induced norm on Rp , uik is referred

to as the grade of membership or belonging of xk to the cluster i. This grade

of membership satis¬es the following constraints:

0 ¤ uik ¤ 1, for 1 ¤ i ¤ c, 1 ¤ k ¤ n, (6.14)

n

for 1 ¤ i ¤ c,

0< uik < n, (6.15)

k=1

c

for 1 ¤ k ¤ n.

uik = 1, (6.16)

i=1

The FCM uses an iterative optimization of the objective function, based on the

weighted similarity measure between xk and the cluster center vi .

More on the steps of the FCM algorithm are discussed 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 classes c, and the clustering

114 6 Money Demand Modelling

method. The method of initialization requires a good estimate of the clusters

and its application is dependent, so the cluster validity problem is reduced to

the choice of an optimal number of classes c. Several cluster validity measures

have been developed in the past by Bezdek and Pal (1992).

6.3.2 Takagi-Sugeno Approach

Takagi and Sugeno (1985) formulated the membership function of a fuzzy set A

as A(x), x ∈ X. All the fuzzy sets are associated with linear membership func-

tions. Thus, a membership function is characterized by two parameters giving

the greatest grade 1 and the least grade 0. The truth value of a proposition ”x

is A and y is B” is expressed by

x is A and y is B = A(x) § B(y)

Based on above de¬nition, the a¬ne Takagi-Sugeno (TS) fuzzy model consists

of rules Ri with the following structure:

If

x is Ai (6.17)

Then

yi = ai x + bi , i = 1, 2, ..., r. (6.18)

This structure consists of two parts, namely x is Ai as antecedent part and

yi = ai x + bi as consequent part.

Note : x ∈ X ‚ Rp is a crisp input vector. Ai is a (multidimensional) fuzzy set:

µAi (x) : X ’ [0, 1], yi ∈ R is an output of the i-th rule, ai ∈ Rp is a parameter