A Comparison of Linear Forecasting Models and Neural Networks:
An Application to Euro inflation and Euro Divisia
Short title: Linear Models versus Neural Networks in Macroeconomic Forecasting
Jane M. Binner (corresponding author)
Department of Information Management and Systems
Nottingham Business School
Nottingham Trent University
Nottingham, NG1 4BU, UK
Email: jane.binner@ntu.ac.uk
Tel: +44 (0) 115 848 2429
Fax: +44 (0) 115 848 6175

Rakesh K. Bissoondeeal
Department of Information Management and Systems
Nottingham Business School
Nottingham Trent University
Nottingham, NG1 4BU, UK
Email: rakesh.bissoondeeal@ntu.ac.uk
Tel: +44 (0) 115 848 4089
Fax: +44 (0) 115 848 4707

Thomas Elger
Department of Economics
Lund University
P. O. Box 7082
Lund, Sweden

Email: thomas.elger@nek.lu.se
Tel: +46 46 222 7919
Fax: +46 46 222 4118

Alicia M. Gazely
Department of Information Management and Systems
Nottingham Business School
Nottingham Trent University
Nottingham, NG1 4BU, UK
Email: alicia.gazely@ntu.ac.uk
Tel: +44 (0) 115 848 2416
Fax: +44 (0) 115 848 6175

Andrew W. Mullineux
Department of Accounting and FinanceBirmingham Business School
The University of Birmingham
Birmingham, B15 2TT, UK
Email: A.W.MULLINEUX@bham.ac.uk
Tel: +44 (0) 0121 414 6642
Fax: +44 (0) 0121 414 6678

1

Abstract
Linear models reach their limitations with nonlinearities in the data. This paper
provides new empirical evidence on the relative macroeconomic forecasting
performance of linear and nonlinear models. The well established and widely usedunivariate autoregressive integrated moving average (ARIMA) and multivariate
vector autoregressive (VAR) models are used as linear forecasting models whereas
neural networks (NN) are used as nonlinear forecasting models. We endeavour tokeep the level of subjectivity in the NN building process to a minimum in an attempt
to exploit its full capability. This paper also investigates whether the historically poor
performance of the theoretically superior measure of the monetary services flow,
Divisia, relative to the traditional Simple Sum measure could be attributed to a certain
extent to the evaluation of these indices within a linear framework. The results
obtained suggest that nonlinear models provide better within-sample and out-ofsample forecasts and linear models are simply a subset of them. The Divisia index
also outperforms the Simple Sum index when evaluated in a nonlinear framework.
1. Introduction
This paper is concerned with forecasting the inflation rate in the Euro area. Nonlinear
NN are compared to the more traditional time series univariate ARIMA andmultivariate VAR forecasting models. The objectives are two-fold:(1) Primarily to display how various time series forecasting methods compare in theirforecasting accuracy of Euro inflation.
(2) Secondly, given the ubiquitous relationship between inflation and money, tocompare the inflation forecasting ability of the Euro M3 and a Euro Divisia M3 inboth linear and nonlinear frameworks.
The reason for carrying out such an empirical analysis is that it can provide useful
information to monetary policy decision makers of the ECB whose primary objective
is to maintain price stability in the Euro area. The ECB organises its assessment ofrisks of price stability under two pillars (see ECB, 1999a, b and 2000). Firstly, given
the widely held belief that development in the price level is a monetary phenomenon,
development in the amount of money held by the public may reveal useful
information about future price development and be a useful leading indicator of
inflation. Therefore in 1998, in its first pillar of monetary policy strategy, the ECB
decided to give broad monetary aggregate Euro M3 a prominent role. The second
pillar analyses a broad range of other economic and financial time series indicators
relevant to future price developments.
Our first objective is particularly relevant to the second pillar of the ECB’s monetary
policy strategy where inflation forecasts play a very important role. In order to enable
the monetary authorities to tackle appropriately inflationary pressures that may arisein the future it is necessary and crucial to produce accurate and reliable forecasts ofinflation. A large body of research is devoted to inflation forecasting (see, forexample, De Brouwer and Ericsson, (1998) for Australia, Stock and Watson (1999)
for the US, Drake and Mills (2002) for the Euro area). One question that lies in theheart of every forecasting exercise is which forecasting method to use? The
overwhelming majority of studies on inflation forecasting divide forecasts in 2 maincategories(1) forecasts from time series models such as ARIMA models and(2) forecasts from macroeconomic models such as the VAR models.

2However, such models are based on the assumption of linearity in the data and there is
now growing evidence that macroeconomic series contain nonlinearities (see forexample, Tiao and Tsay (1994) and Stanca (1999) and thus, though linear models
have been reasonably successful as a practical tool for analysis and forecasting, they
are inherently limited in the presence of nonlinearities in data and consequently
forecasts, as well as other conclusions drawn, from them could be misleading. In viewof the limitations of the linear models, nonlinear time series have gained muchattention in the recent decades. Several nonlinear models, such as the thresholdautoregressive (TAR) models (Tong, 1990) and the exponential autoregressive model(EXPAR) (Haggan and Ozaki, 1981), have been developed. However, an immediateproblem encountered while opting for such nonlinear models in preference to linear
models is that there exists no unified theory that can be applied to all such nonlinear
models as they require the imposition of assumptions concerning the precise form of
nonlinearity. But there are too many possible nonlinear patterns in a particular data set
and the prespecified nonlinear model may not be broad enough to capture all essential
characteristics. An alternative way to deal with nonlinearities in data is to use NN. In
contrast to the above model-based nonlinear methods, NN are data driven and are thus
capable of producing nonlinear models without prior beliefs about the functional
forms. NN are also highly flexible as they can approximate any continuous function to
any degree of accuracy (Hornik et al., 1989). Thus from a statistical viewpoint thenonlinear NN would be expected to perform better than the linear models in inflation
forecasting and since no such work has been carried out for the Euro area weinvestigate the performance of NN vis a vis linear models in forecasting Euro
inflation.
In the first pillar of the ECB’s monetary policy strategy, Euro M3, constructed bysimple summation, plays a very important role (see ECB 1999a,b, 2000 for moredetails). However, there is some debate (see, Barnett, 1980, 1982) that monetary
aggregates constructed by simple summation are flawed and the weighted Divisia
monetary aggregates are a theoretically superior measure of monetary services flow
and hence Drake et al. (1997) among others have suggested the use of a Euro Divisia
M3 instead of Euro M3 in the first pillar of the ECB’s monetary policy strategy.
However, despite its theoretical superiority the Divisia index does not always
outperform its Simple Sum counterpart empirically (Chrystal and MacDonald, 1994,
Herrmann et al., 2000), explaining the reluctance of the ECB and other central banks
in adopting such a monetary index for their monetary policy strategy. To provide an
explanation for the historically poor performance of the Divisia index, researchers
have focussed on measurement problems (see for example Drake et al., 1997).
However, none of them have put the validity of the linear statistical methods used toevaluate them into question despite the fact that, as stated earlier, there is increasing
evidence of nonlinearity in macroeconomic data and more importantly the fact that
Barnett and Chen (1986, 1988a, b), Barnett and Hinich (1992, 1993), Chen (1988),
and DeCoster and Mitchell (1991) have provided evidence of nonlinear structuresinherent in the Divisia index. Thus if the Divisia index performs poorly relative to the
Simple Sum index when compared in a linear framework one cannot say whether it is
the Divisia index to be blamed or the linear models which may not be able to capture
nonlinear behaviour. Given the ubiquitous relationship between inflation andmonetary aggregates this study can therefore be used to shed some light on the issueof whether the historical poor performance of the Divisia index relative its Simple

3

Sum counterpart could be attributed to a certain extent to incorrectly using linear
statistical models in evaluating them. More specifically, we construct multivariate
linear and nonlinear forecasting models and interchange Euro M3 and Euro Divisia
M3 to investigate their comparative forecasting performance. Hence this study allows
us to evaluate the forecasting potentials of linear univariate and multivariate models
and nonlinear NN in predicting Euro inflation and to compare the relative
performance of Euro M3 and Euro Divisia M3 in linear and nonlinear frameworks.
This paper is organised as follows: The next section provides a brief review of theliterature comparing linear and nonlinear forecasting models. Section 3 describes thedata and associated preliminary analysis. In section 4 the different models are
specified and estimated. Forecasts results are presented and discussed in section 5
while conclusions and suggestions for future development are offered in section 6.
2. Literature Comparing Forecasting Method EffectivenessThis section attempts to provide a brief review of recent research on comparing linear
models, like ARIMA and VAR models to nonlinear NN, but makes no attempt to be
exhaustive.
NN have gained enormous popularity in the recent years, especially in time series
forecasting. Most applications, however, are in areas where data are abundant as NN
are very data intensive. In macroeconomics, due to the scarcity of large data samples,
there exists only a few studies involving the use of NN that can be used to gauge its
usefulness in the field. Recent ones include that of Johnes (2000) and Moshiri and
Cameron (2000). Johnes (2000) contrasts models of the UK economy constructedusing NN and a variety of econometric models. Moshiri and Cameron (2000) use NN
to forecast Canadian inflation and compare the results to those from time series andeconometric models. The results in these studies, based on out-of-sample forecasts, do
not permit a demarcation between the linear models and NN as the latter is able tojustify its theoretical superiority in only some of the cases. In fact, these observations
reflect the results of quite a large number of such comparative studies across different
fields. This has led to questions being raised on whether studies implement NN insuch a way that it stands a reasonable chance of performing well (Adya and Callopy,1998). Indeed, the risks of making bad decisions are extremely high while building a
NN as there are no established procedures available to decide on the choice of theparameters of the NN, which basically is problem dependent. Although there havebeen attempts in several studies to develop guidelines in making these choices (see,for example, Balkin and Ord (2000), Gorr et al. (1994)), so far this matter, is still
subject to trial and error. Thus, despite the many satisfactory characteristics of theNN, building a NN for forecasting a particular problem is a nontrivial task.
Consequently, tedious experiments and time-consuming trial and error procedures areinevitable. However, this has not been the case in most of the comparative studies asin the absence of any a priori information about the parameters of the NN, their choice
has involved a lot of subjectivity (Nag and Mitra, 2002). Such an approachconsiderably reduces the possibilities of exploiting the true potentials of the NN andultimately leads to results from a large number of studies being dubious. For example,
Moshiri and Cameron (2000) perform some experimentation in finding the optimum
number of hidden units, however their choice for the amount of training required,another equally critical parameter, is rather subjective, thereby limiting the power of
the NN. In this study, we endeavour to keep the level of subjectivity to a minimum

4and appropriately deal with other issues prone to affect the performance of NN in an
attempt to obtain the best possible NN models. Since it is beyond our reach toevaluate the performance of NN against the entire class of linear models, we chose thewell-established and extensively used ARIMA and VAR models as ourrepresentatives for linear models in recognition of their ability to produce reliable
forecasts.
3. Data and Preliminary Analysis
Many economic indicators help predict inflation. For example Stock and Watson(1999) have recently shown that 168 variables can be used to forecast US inflation. In
our study instead of using so many variables, we limit the list of variables to thosethat are more closely linked to inflation by economic theory or that have been
regularly used in previous empirical studies. Thus, in keeping with previous studies
such as Hendry and Doornik (1994) and Lutkepohl and Wolters (1998)), the variables
required for multivariate forecasting are: inflation, monetary aggregates- Euro M3 andEuro Divisia M3, GDP, GDP deflator and the opportunity cost variables of thecorresponding Divisia and the Simple Sum aggregates. These are quarterly seasonallyadjusted data, for the period 1980Q1 to 2000Q4, defined by the availability of theEuro-area data. Data on monetary assets, their respective rates of return, GDP and
GDP deflator have been obtained from Stracca (2001). After allowing for lags andtransformations estimation is conducted using data from 1981Q2 to 1998Q2, while
the remaining 10 observations (1998Q3 to 2000Q4) are kept for forecast evaluation(testing). The Simple Sum index is constructed by simply summing over the monetarycomponents while the Divisia index is constructed using equation (9) in Barnett et al.
(1992, pp. 2097). We use the Divisia price dual (Barnett, 1980) as the opportunity
cost variable for Divisia. Following Lutkephol and Wolters (1998) the opportunity
cost variable for the Simple Sum aggregate is calculated as ( Rt − rt ) where Rt is along term interest rate and rt is the own rate of M3. The log of all variables have beentaken and thus m3t is the log of real Simple Sum M3, d3t is log of real Euro Divisia
M3, y t is the log of real GDP, dualm3t is the log of the opportunity cost variable forEuro M3 and duald3t is the log of the opportunity cost variable for Euro Divisia M3.
pt is the logarithm of the GDP deflator and ∆pt = pt − pt −1 is the quarterly inflation
rate.
[Figure 1 about here]
m3t and d3t are contrasted in figure 1. The Simple Sum aggregate begins to increase
faster than its Divisia counterpart in 1980 and diverges significantly afterwards as theliterature suggests (see, for example, Drake et al. (2000, pp. 52). To check thestationarity properties of the series the Augmented Dickey and Fuller (1979) unit root
test is used. The results reported in table 1 show that for the majority of the variables
the null hypothesis of unit root and hence nonstationarity in the levels cannot berejected i.e., most variables are not I(0). The variables dualm3t and duald3t are moremarginal, however, as the hypothesis of stationarity in levels is not rejected at 5%
level but rejected at the 1% level. Unit root tests on the first differences of the variable
reveal that all of them are stationary. Hence all variables are I(1) at the 1% level.
[Table 1 about here]54. Model Specification and Estimation
In this section we present our main decisions regarding the specification andestimation of the three classes of models (univariate ARIMA, multivariate VAR and
NN). While the ARIMA and VAR methods are widely used, the NN method is arelatively new method. Thus, we provide only brief accounts for the ARIMA andVAR methods and we give a more detailed account for the NN method.
4.1 Univariate Time Series Model
The ARIMA is a general class of univariate time series models which represents
current values of a time series by past values of itself (autoregressive term (AR)) andpast values of stochastic errors (moving average terms (MA)). The acronym I refers to
the number of times (d) the time series has to be differenced to render it stationary. A
nonseasonal1 ARIMA(p,d,q) process can be represented asφ ( L)(1 − L) d y t = δ + θ ( L)ε t(1)
where ε t is independent and normally distributed with zero mean constant varianceand δ is a constant. φ (L) and θ (L) are the AR and MA polynomials, respectively
withorders
p
and
q
such
that
φ ( L) = 1 − φ1 L − L − φ p L p andθ ( L) = 1 − θ 1 L − L − θ q Lq , where L represents the backshift operator such thatLs y t = y t − s . A slightly modified Box and Jenkins approach (Box and Jenkins, 1970)
is used for identifying the best model for ARIMA forecasting. Thus, instead ofinspecting ACF and PACF in the identification stage we estimate a range of models,represented in table 2, with d = 2 ( for pt from section 3) and values of p and q
varying from 0 to 3 in a first step and retain the models which pass the diagnostic tests(no autocorrelation and conditional heteroscedasticity, significance of parameters). Ina second step the best ARIMA model is chosen to be the one which provides the best
out-of-sample forecast. We also estimate an ARIMA model with the orders of p and q
equal to 6. We then use Hendry’s (1993) general-to-specific methodology to obtain amore parsimonious model.
[Table 2 about here]
Thus after the first step only 4 ARIMA models were retained as the others exhibit
insignificant parameters and out of the 4 remaining models the ARIMA(0,2,1) is ourpreferred ARIMA specification because it outperforms the others in terms of out-ofsample forecasting accuracy. The estimated model2 is given below and the test
statistics given are computed from the residuals of the estimated models3. JB
represents the Jarque-Bera test for normality, LM(k), represents the test for
autocorrelation of order k, and ARCH(k) representing the test for conditional
heteroscedasticity of order k (for more details on this tests, see for example, Hendry(1995)). None of the diagnostic tests is significant at conventional levels and, hence,
the residuals appear to be normally distributed and free from autocorrelation andautoregressive conditional heteroscedasticity.
(1 − L) 2 pt = ε t − 0.510ε t −1(2)(0.102)R 2 = 0.23
J .B = 0.16 (0.92)
S.E. of regression = 0.002525
LM (1) = 0.00 (1.00)

LM (4) = 1.61 (0.81)

LM (8) = 4.91(0.77)

ARCH (1) = 0.40 (0.52)

ARCH (4) = 7.28 (0.12)

ARCH (8) = 10.81(0.21)

6

4.2 Multivariate Vector Autoregressive (VAR) Models
The advantage of VAR models over ARIMA models is that they can incorporate more
information in terms of other time series instead of just past observations and errors of
the series to be forecast. Having established in section 3 that the variables entering the
VAR are I(1), we first proceed to investigate whether they are cointegrated, that is,verify whether some linear combination of these nonstationary variables is stationary.
In the absence of cointegration between the variables a common forecastingprocedure would be to conduct a VAR on the first differences. However, if
cointegrating relationships can be established between the variables, the VAR should
also include the lagged cointegrating error term (vector error correction models(VECM)) (Granger, 1981). This prevents neglecting long run information containedin the levels of the variables and it has been shown that such an approach leads toimproved forecasting accuracy (Lesage(1990), Shoesmith (1992, 1995)).
4.2.1 Testing for Cointegration
To check for cointegration, the Johansen (1988) procedure is used. Thus let z t be aq ×1

vector,here

z t = ( M t , y t , Rtopp , ∆pt ) T

where

M t = m3t / d3t andR = dualm3t / duald 3t then z t can be formulated as the first difference of a VAR
model of lag length k∆z t = Γ1 ∆z t −1 + L + Γk −1 ∆z t −( k −1) + Ψz t −k + µ + δt t + φD + ε t
(3)
opp
t

where µ is a constant and the error term, ε t , is independently and normally
distributed, Γ1 , L, Γk −1 , Ψ and φ are coefficient matrices, D consists of dummy
variables4 and t is a trend variable. If rank (Ψ ) = r , where 0 < r < q, implies existenceof q × r matrices α and β such that Ψ = αβ ' and β ' z t is I(0) (Johansen and Juselius,
1990). r is the number of cointegrating relationships and each column of β is a
cointegrating vector. In the current study, trace tests are used to determine thecointegration rank (see Johansen, 1995)5.
The results, however, are sensitive to the choice of the lag length (k). Some model
selection criteria could be used in determining the lag length but different criteria
often suggest different orders. A more appropriate method is to combine this with
misspecification tests by choosing the lag length to ensure that the underlying
assumptions of the VAR model are satisfied (Johansen, 1995). More specifically, tocheck whether the residuals in the Johansen VAR are free from serial correlation,
conditional heteroscedasticity and the distribution of the residuals is normal. Our
experimentation uses a range of lags (VAR lag lengths of 1-8) and is based on the
misspecifications tests. For a VAR of order 6, the LM and JB tests, represented intable 3, do not show any sign of misspecification nor do the univariate ARCH tests6.
[Table 3 about here]
In testing for cointegration, the question of whether a constant and trend should enter
the long run relationship also arises. There are in general 5 possible ways ofincorporating these deterministic components into the analysis (see Johansen 1992,
Hansen and Juselius, 1995) but generally the most and the least restrictive ones are
excluded (see, for example, Drake (1996)). Therefore, the three models of interest to
us have the following specification. The model that we refer to as model 1 is the mostrestrictive model. It does not allow for linear trends in the data. The only
deterministic components in the model are the intercepts in the cointegration relations.

7A less restrictive model is referred to as model 2. The model allows for linear trends
in the data, but it is assumed that there are no trends in the cointegration relations. It
also has a non-zero intercept. The model referred to as model 3 is the least restrictive.
It also allows for a linear trend in the cointegration space.
In order to determine which of the three possible deterministic specifications is themost appropriate in the cointegration, Johansen (1992) suggests applying the Pantula(1989) principle. In so doing, the rank order and the presence of the deterministic
components are jointly determined. In practice this involves estimating all the 3models outlined above and conducting the trace test to determine the cointegration
rank sequentially from the most restrictive to the least restrictive specification. The
first time the null hypothesis of r cointegrating vectors is not rejected indicates boththe cointegration rank and the appropriate specification for the deterministic
components. Results from the application of the Pantula (1989) principle, reported intable 4, suggest that model 2 should be used for the Simple Sum system and the rank
is 2 whereas model 2 should be used and the rank is 3 for Divisia system. The
cointegrating vectors for both systems are presented in table 5.
[Table 4 about here][Table 5 about here]
4.2.2 Short-run Equations for Inflation
In this section we present estimation results for single error correction equations ofinflation. For both Simple Sum and Divisia, the corresponding second cointegrating
vector is used for specifying their short-run equations of the form (3) for inflation forthe period 1981Q2 to 1998Q2. These cointegrating vectors have been chosen since
the signs of the coefficients of their components are consistent with economic theory(see Doornik et al., 1998). Money affects prices with long lags, approximately two
years (Drake et al., 2000) and hence 7 lags of each of the independent variables havebeen used.7 Following the general to specific methodology (Hendry, 1993),
parameters insignificant at the 5% significance level were deleted and the equations
rerun, using the ordinary least squares method, until just significant parameters
remained. The error correction terms were kept in the equations at all the times andeliminated in the final stage if they were not significant. This strategy eventually
resulted in the equations given by equations (4) and (5) for the Euro M3 and EuroDivisia M3 respectively. Here also the diagnostic tests do not show any signs of
misspecification.
Simple Sum∆2 pt = 0.121∆m3t −2 − 0.533∆2 pt −1 + 0.000062 Re st −1 + ε t(0.054)(0.102)(0.000026)

Res = is 2nd cointegrating vector for a VAR lag = 7
R 2 = 0.30

J .B = 0.41 (0.81)

S.E. of regression = 0.002454

LM (1) = 0.38 (0.54)

LM (4) = 1.61 (0.81)

LM (8) = 8.61(0.38)

ARCH (1) = 1.70 (0.19)

ARCH (4) = 1.61 (0.81)

ARCH (8) = 2.22 (0.97)

8(4)

Divisia∆2 pt = 0.141∆d 3t −3 − 0.537∆2 pt −1 + 0.000070 Re st −1 + ε t(0.054)(0.102)(0.000027)(5)Res = is the 2nd cointegrating vector for a VAR lag = 7
R 2 = 0.32

J .B = 0.24 (0.88)

S.E. of regression = 0.002908

LM (1) = 1.34 (0.25)

LM (4) = 3.18 (0.53)

LM (8) = 5.94(0.65)

ARCH (1) = 0.05 (0.83)

ARCH (4) = 5.64 (0.23)

ARCH (8) = 8.81 (0.36)

4.3 Nonlinear Models: Neural Networks
Neural networks are composed of highly interconnected processing elements (nodes)that work simultaneously to solve specific problems. In time series analysis they areused as nonlinear function approximators. They take in a set of inputs and produce a
set of outputs according to some mapping rules predetermined in their structure.
[Figure 2 about here]
This paper considers the most popular form of NN called the feedforward network.
Figure 2 depicts such a network that consists of layers of nodes. The input layer andoutput layer represent the input and output variables of the model. Between them lie
one or more hidden layers that progressively transform the original input stimuli to
final output and hold the networks ability to learn nonlinear relationships. For afeedforward NN with one hidden layer, the general prediction equation, given by
Faraway and Chatfield (1998), for computing a forecast of y t using an input vector(x1 , x2 ,L, x m ) may be written in the formyˆ t = f ( wco + ∑ who g ( wch + ∑ wih xi ))
h

(6)

i

where wch denote the weights for the connections between a constant input, usuallytaken as 1, and the hidden nodes and wco denotes the weight of the direct connection
between the constant input and the output. The weights wih and who denote theweights for the other connections between the input and hidden nodes and between
hidden and the output nodes respectively. The two functions f and g denote theactivation functions used in the hidden layer and the output layer respectively.
NN have to be trained in order to be able to use them to perform certain tasks likepredicting a response corresponding to a new input pattern. The training procedure
involves iteratively modifying the randomly initialised weights of the NN to minimise
1some kind of error function usually the mean square error (MSE), ∑ ( y t − yˆ t ) 2 .
nVarious standard optimisation techniques such as the conjugate gradient and quasiNewton methods exist for minimising the error function, however, in application
studies, the backpropagation algorithm (Rumelhart et al., 1986) developed by the9neural network community is the most popular training algorithm used. Standard
optimisation techniques tend to converge faster than the backpropagation algorithm
but this advantage is overshadowed by the fact that the latter is computationally more
efficient (Monterola et al., 2002). Moreover, the backpropagation algorithm generally
has better generalisation (performs well on unseen data) than standard optimisation
techniques (Cubiles-de-la-Vega et al., 2002), hence is our preferred algorithm despite
the greater time required for convergence.
However, it is well known that the backpropagation algorithm used for training
suffers from the local minimum problem. Randomly selecting initial weights fortraining is a common approach, however, if these initial weights are located close to
local minima, the algorithm is likely to converge to a local minimum. Some
researchers have tried to overcome this problem by, for example, using genetic
algorithms (Shazly and Shazly, 1999) and simulated annealing (Masters, 1993). Even
then there is no assurance that such measures will help the optimisation algorithm toconverge to a global minimum. We follow the most commonly used method to findthe best local minimum or even the global minimum, more specifically, we restart thetraining with different weights. The actual number of restarts employed in practice isgenerally limited by the computing time required to train a NN (Plasmans et al.,1998). In this work we therefore use 10 restarts.
4.3.1 Designing the Neural Networks
Apart from the weights of the NN, there are many other parameters, like the numberof input variables, the combination of input variables, the number of hidden layers
and hidden nodes, the types of activation functions in the hidden and output layers,
the value of the learning rate and the momentum rate and the amount of training
which are also unknown. As we mentioned earlier there are no established rules tohelp us in choosing the appropriate values of these parameters and we have to resort
to trial and error to obtain their appropriate values. Clearly, experimenting over thewhole parameter space of the parameters is beyond the scope of the paper. In thisstudy, therefore, we focus on experimenting with the different values of key
parameters like initial weights, the number of hidden nodes, amount of training
required, different sets of input variables and we draw attention to the other issues thatneed to be considered while making the choices for the remaining parameters of theNN.

The common practice has been to construct NN using the same input variables as in
VAR models to allow direct comparison between them. However, such a procedure isbiased towards the linear model as the regressors from the linear equation tell us about
linear correlation and this is not appropriate for nonlinear relationships modelled by
the NN (Zhang et al., 1998). For these reasons, we feel justified to use the ‘best’ set of
input variables for the NN. A modified version of the preferred model of therelationship between inflation and money of Binner et al., (2002) adapted originallyfrom Dorsey (2000, pp.34) given by equation (7) below is used. However, forcomparative purposes we also construct NN using the set of input variables of theVAR models and using the set of input variables of the VAR models from which theerror correction term has been excluded.
∆2 pt = f (∆M t −1 , ∆M t − 2 , ∆M t −3 , ∆M t − 4 , ∆2 pt −1 ) + ε t(7)10

Hidden layers play a very important role for the successful applications of the NN asthey allow NN to perform nonlinear mapping between the input and the output.
Without hidden nodes, NN are equivalent to linear statistical model (see, for example,
Warner and Misra (1996)). It has been shown that a 3 layer NN, i.e., a NN with only
one hidden layer can approximate any function to any degree of accuracy (Hornik etal., 1989). Two hidden layer NN could be more beneficial to certain problems(Barron, 1994), however, given our relatively small sample and the fact that the
number of parameters increases rapidly with each layer (Tkacz, 2001), we focus on 3
layer NN in the present study.
The choice is more complicated for the number of hidden nodes. Usually few hidden
nodes are preferred as there is less likelihood of overfitting, i.e. encountering
problems of drawing too many characteristics from the data used for training, and atendency to yield better generalisation. But NN with too few hidden nodes may nothave enough power to model and learn the richness of the data (Church and Curram,
1996). Similar problems are encountered if the NN are not trained to the right degree.
Inadequately training a NN will lead to missing patterns in the data while excessive
training will result in overfitting. We use a grid search to jointly determine theappropriate number of hidden nodes and the amount of training required (Gorr et al.,1994). We consider 5 networks with hidden units between 1 and the number of input
variables (Balkin and Ord, 2000) that is 5 in our case. Preliminary investigation overthe number of training ranging from 10,000 to 50,000, suggested that better results are
obtained in the range 15000 to 20000 for the Simple Sum NN models and in the range
10000 to 15000 for the Divisia model. Therefore, extensive experimentation isconstrained to these ranges with increments of 1000. Since we perform 10 restarts foreach point in our grid, this means that 300 NN for each set of input variables andmonetary aggregate are investigated, i.e. a total of 1800 NN are run in thisinvestigation.
The logistic function f ( x) = 1 /(1 + e − x ) is the most popular activation function among
researchers for the hidden layer. However, we use the hyperbolic tangent (tanh)function, f ( x) = (e x − e − x ) /(e x + e − x ) as it has been used very successfully ininflation forecasting experiments (see, for example, Binner et al., (2002)). It is also
generally held that tanh gives rise to faster convergence of training algorithms thanlogistic functions (Bishop, 1995). For the output layer, we follow the recommendationof Rumelhart et al. (1995) who suggest the use of the linear function f ( x) = x fortime series prediction with continuous output. The remaining parameters for the NN,
the learning and momentum rates for the backpropagation algorithm are set as thedefault values of Matlab 6.0, i.e. 0.01 and 0.9 respectively (see, for example, Bishop(1995), for more details on these parameters).
In addition to the parameters of the NN, there are some other factors such as the datanormalisation and performance measures that affect the performance of NN (Zhang etal., 1988). In practice NN training can be made more efficient by preprocessing thedata as this enables the network to extract valuable information (Gately, 1996) and to
significantly reduce the time necessary to complete training (Krunic et al., 2000). In
this paper we use one of the most common forms of preprocessing which consists ofrescaling the data in the range [-1, 1] so that they have similar values. This choice ismotivated by the fact that the input variables used for NN modelling differ by several

11

orders of magnitude and the sizes of variables do not necessarily reflect their relative
importance in finding out the required outputs (Bishop, 1995). Another issue ofconcern is related to performance measures. There are several measures of accuracy
but each of them has advantages and limitations (Makridakis et al., 1983). For thisreason none of them is universally accepted as the best measure of accuracy andhence in this study we shall be making use of a number of performance measures.
5. Predictive Performance Assessment
The NN should be tested on a validation set after they have been trained and the one
leading to the minimum forecast error in the validation set should provide the best
generalisation and is normally retained to evaluate its forecasting performance on atest sample. However, one of the main disadvantages of NN, as mentioned above, isthat it requires an enormous amount of data, if the series are short or not
representative of the process being modelled the NN might not perform well (Balkin
and Ord, 2000). Thus in studies with small data sets it is common to use the test set
for both validation and testing purposes (Zhang et al., 1998). That is the routefollowed in this paper given that the data set available to us is quite modest by thestandards of NN analysis.

Three traditional performance measures are first used to compare the fit andforecasting accuracy of alternative models: root mean square error (RMSE), mean
absolute error (MAE), and mean absolute percentage error (MAPE). Before
calculating these measures the NN forecasts are backtransformed to the same units astheir actual values to make them comparable.
Figures 3, 4 and 5 show the within-sample RMSE, MAE and MAPE performances
respectively and figures 6, 7 and 8 show the corresponding out-of-sample
performances of the Simple Sum NN constructed with the set of input variables as inequation 7. The patterns shown by the RMSE, MAE and MAPE, within-sample andout-of sample, for different number of hidden nodes and amount of training acrossdifferent sets of input variables are in general similar. A comparison of the withinsample RMSE to that of the out-of-sample RMSE reveals that as the number ofhidden nodes and amount of training are increased the within-sample forecast error
decreases but, as expected, the reversed pattern is observed with the out-of-sample
forecasts. This clearly demonstrates that with too many hidden nodes and excessive
training, poor generalisation will occur and hence the need to appropriately choosethese parameters. The MAE shows a similar pattern to that of the RMSE, however,
the movement across both surfaces is not always in congruence. The discrepancies in
the performance measures become more apparent as the RMSE and MAE arecompared to the MAPE. The differences are apparently due to the inherent limitations
in each of the performance measures. Therefore, these observations show thatchoosing the best model on the basis of just one performance measure would be
misleading and thus in the current study the best model is chosen to be the one which
consistently shows small forecast errors across each of the three performancemeasures and which also provides the best trade-offs between within-sample and outof-sample forecast errors. On this basis, the number of iterations and hidden nodes
chosen are reported in table 6 for each set of input variables (sets A, B and C also
defined table in 6) for each monetary aggregate. One noticeable pattern in thesevalues is that number of iterations or number of hidden nodes or both increase as thenumber of input variables increases. This could be due to the fact that the higher the12number of input variables, the higher the level of complexity of the NN and hence
more hidden nodes or/and training are required to learn the relationship between input
and output variables. The static forecasting performance of the ARIMA and VAR
models are reported in tables 7 and 8 respectively, while those of the NN based on thethree different sets of input variables are reported in tables 9, 10 and 11.
[Figures 3, 4, 5, 6, 7 and 8 about here][Tables 6, 7, 8, 9, 10 and 11 about here]
A comparison of the results from the ARIMA and VAR forecasts suggest themultivariate models provide more accurate forecasts of Euro inflation. Looking at
RMSE for example, the out-of-sample forecasting accuracy increases by about 9%
with VAR models when compared to ARIMA models and hence the VAR models are
retained as representatives for linear models for comparison with nonlinear NN. On
comparing the results from VAR modelling and NN constructed with the same input
variables (set B) as in the VAR models, it is not possible to discriminate between
them as both of perform equally well in 12 comparisons of the within-sample and outof-sample forecasts of the two monetary indices. NN constructed with the input
variables from set A show a better performance but are still outperformed by thelinear models in a few cases. However, a comparison of the results from the VAR
modelling to those from NN constructed input variables are from set C, reveals thatsuperior inflation forecasts are achieved using NN, both within-sample and out-ofsample, in every case examined. Looking again at the RMSE, for example, out-ofsample forecasting accuracy increases by approximately 10% with NN over VAR
models. These results demonstrate the sensitivity of the NN to the choice of input
variables and reveal that input variables used for building the linear models are notnecessarily the most appropriate ones for the nonlinear models.
We also evaluate the relative forecasting potential of the VAR and NN models byusing a simple encompassing test (Fair and Shiller, 1990). Such a test has someadvantages over the other performance measures (RMSE, MAE, MAPE) to compare
the forecasts. Firstly, it can differentiate between competing forecasting models even
if there are no big differences in the performance measures. Secondly, it helps todiscriminate between models in cases where the performance measures are in favourof a particular model while despite having larger performance measures othercompeting models might contain vital information unique to them. Thirdly, such a test
gives some statistical meaning to the forecasts of the NN relative to those of the linear
models. The test is carried out by regressing the actual values of the changes in
inflation on a constant, linear model forecasts ( f L ) and NN forecasts ( f N ) . If the ttests show that the coefficients of the forecasts of both models are significantly
different from zero, then both models contain independent information that havepower in forecasting the changes in inflation. If one of the coefficients of theforecasting models is significantly different from zero and the other one is not thenthe latter is just a subset of the former. In addition, the model with the significant
coefficient contains further relevant information. Finally, if none of the coefficientsare significantly different from zero then neither model is useful in forecasting thechanges in inflation. The best NN forecasts obtained by using the input variables in C,
are evaluated against the VAR forecasts. The results from the encompassing tests
carried out for within-sample and out-of-sample forecasts are given below. The JB,
LM and ARCH tests do not show any signs of misspecification. The results revealthat in every case only the coefficient of the NN forecast is significant at the13

conventional 5% significance level which implies that that NN forecasts are
statistically superior to the linear models forecasts and hence VAR forecasts are
simply a subset of the NN. These results further confirm that better macroeconomic
forecasts can be achieved with the use of nonlinear NN.
Simple Sum
Within-sample∆2 pt = 0.0000007 − 0.165 f L + 1.128 f N + ε t(0.000253)

R 2 = 0.49

(0.287)

J .B = 2.40 (0.3)(8)(0.232)

S.E. of regression = 0.002090

LM (1) = 2.47 (0.12)

LM (4) = 5.35 (0.25)

LM (8) = 12.30(0.14)

ARCH (1) = 0.26 (0.62)

ARCH (4) = 0.63 (0.96)

ARCH (8) = 3.80 (0.875)

Out-of-sample∆2 pt = −0.000615 − 0.0933 f L + 1.349 f N + ε t(0.0006)

R 2 = 0.41

(1.005)

J .B = 0.67 (0.72)(0.689)(9)S.E. of regression = 0.001375

LM (1) = 0.25 (0.62)

LM (4) = 3.60 (0.46)8

ARCH (1) = 1.46 (0.23)

ARCH (4) = 5.82 (0.21)

Divisia
Within-sample∆2 pt = 0.00000268 + 0.0647 f L + 0.960 f N + ε t(0.000272) (0.361)

R 2 = 0.42

J .B = 0.40 (0.82)(0.323)

S.E. of regression = 0.002241

LM (1) = 3.80 (0.15)

LM (4) = 3.83 (0.43)

LM (8) = 10.39(0.24)

ARCH (1) = 0.14 (0.71)

ARCH (4) = 3.69 (0.45)

ARCH (8) = 7.87 (0.45)

14

(10)

Out-of-sample∆2 pt = −0.000411 − 0.969 f L + 1.633 f N + ε t(0.000551) (0.925)

R 2 = 0.42

J .B = 0.16 (0.92)(0.830)(11)

S.E. of regression = 0.001365

LM (1) = 0.74 (0.39)

LM (4) = 6.26 (0.18)

ARCH (1) = 0.47 (0.49)

ARCH (4) = 5.93 (0.20)

Finally, on comparing the inflation forecasting performance of the two monetary
indices firstly within a linear framework, it is found that Euro Divisia M3 has better
within-sample convergence than its Simple Sum counterpart. However, the main
property sought here is better generalisation, i.e. better out-of-sample performancethat apparently Divisia fails to provide. When the impact of the two monetary indiceson the prediction accuracy is evaluated in a nonlinear framework, overall the Simple
Sum index has better within-sample convergence, however, the Divisia index clearly
outperforms it in terms of out-of-sample convergence. These results do seem tosuggest that one of the reasons for the poor historical performance of the Divisia
index against the Simple Sum index could be attributed to incorrectly choosing linear
models to evaluate the two monetary indices. These results corroborate the findings ofBinner and Gazely (1999), Binner et al., (2002, 2003) and Gazely and Binner (1998,
2000) who have consistently found that the Divisia index outperforms its Simple Sum
counterpart when evaluated using NN.
6. Summary and Conclusions
There is growing evidence that macroeconomic series contain nonlinearities but linear
models such as the ARIMA and VAR models are widely used for forecasting suchseries, despite the inability of linear models to cope with nonlinearities. In this paper
we provide new empirical evidence on the relative macroeconomic forecasting
performance of linear ARIMA and VAR models and the nonlinear NN. We also
investigate whether the poor performance of the theoretically superior measure ofmonetary services, Divisia, relative to its Simple Sum counterpart could be attributed
partly to the incorrect choice of linear models used to evaluate them.

A considerable amount of research has been carried out in the recent years on NN.
However, despite their ability to capture nonlinear relationships, findings generally do
not allow any discrimination between conventional linear statistical techniques and
NN. One of the main reasons for this is that there are no well defined guidelines tobuild NN for solving a particular task and their construction has involved a lot of
subjectivity on the part of researchers, thereby considerably restricting the power ofNN and ultimately leading to the results of many studies being dubious. In order toobtain the best possible NN forecasting model in this study, we have tried to keep thelevel of subjectivity to a minimum, particularly we perform rigorous experiments todetermine the appropriate number of iterations, number of hidden nodes and theappropriate set of input variables. At the same time we have considered other issues
like data processing, local minima problems and limitations of performance measures.
Our best models for the NN outperform the traditionally used linear ARIMA andVAR models in macroeconomic forecasting and are statistically superior to them. The
gain in forecasting accuracy in the NN is very likely to have emerged from the15

capability of the NN to capture nonlinear relationships between macroeconomic
variables. The first conclusion to be drawn from this result is that despite being
constrained by the lack of large data samples in macroeconomics, NN can find be
successfully applied in the field, provided extreme care is taken in designing the
network. However, at this stage we would not recommend the policy makers, such as
the ECB who require inflation forecasts, to abandon the use of conventional statisticaltechniques in favour of NN. The latter still has some very serious limitations, e.g.,
particularly time consuming trial and error procedures and the lack of available
statistical techniques for analyzing the relationship between input and output
variables. However, till such problems are overcome, we would strongly suggest that
the ECB and macroeconomic forecasters use NN as a complementary tool forforecasting.
Another recommendation to the ECB involves the use of monetary aggregates fortheir monetary policy strategy. It is widely accepted that the Simple Sum procedure isinappropriate and the weighted Divisia index is a superior measure of monetaryservices flow. However, the Divisia index does not always outperform its Simple Sum
counterpart in empirical studies, explaining the reluctance of the ECB to use theweighted monetary aggregate instead of M3. The results of this study suggest that thepoor performance of the Divisia index can be attributed to a certain extent to theincorrect choice of linear statistical methods used to evaluate its performance relative
to the Simple Sum index, as the Divisia clearly outperforms the Simple Sum index
when evaluated in a nonlinear framework but not in a linear framework. However,these results should only be considered as indicative as we have not performed an
exhaustive experimentation to find the best Simple Sum and Divisia NN forecast
models, as the overriding aim was to evaluate linear ARIMA and VAR models andnonlinear NN models. Nonetheless, on the basis of the results at hand, and based on
similar results obtained by Binner and Gazely (1999), Binner et al., (2002, 2003) andGazely and Binner (1998, 2000), it appears that Divisia consistently outperforms the
Simple Sum index when evaluated in a nonlinear framework. Thus ourrecommendation to the ECB would be to at least pay more serious attention to the
behaviour of the Divisia monetary aggregate.
Finally, we end by providing a recommendation on how this work can be taken a stepfurther. Although NN considered in this paper outperform traditional econometricmodels in forecasting Euro inflation, we believe that the forecasting accuracy of NN
can still be improved. We performed a grid search to determine the optimum numberof hidden nodes and training required and performed some experimentation to find the
optimum set of input variables. However, ideally, a NN for a particular task has to be
optimised over the entire parameter space of the learning rate, momentum rate,
number of hidden layers and nodes, combination of input variables and activation
functions. In that respect, NN combined with genetic algorithm optimization
techniques (Nag and Mitra, 2002), can potentially be used for building more accurate
models for forecasting Euro inflation and is recommended for future research.
Acknowledgments
The authors gratefully acknowledge the help of Mr Livio Stracca (see Stracca (2001))
for making the Euro area data available and Professor David Edgerton from Lund
University, Sweden for providing very helpful comments.

16

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Figure Captions

1.
2.
3.
4.
5.
6.
7.
8.

Simple Sum M3 index versus Divisia M3 index
NN modelWithin-sample RMSE performance of NN for the Euro-Simple Sum M3
Within-sample MAE performance of NN for the Euro-Simple Sum M3
Within-sample MAPE performance of NN for the Euro-Simple Sum M3
Out-of-sample RMSE performance of NN for the Euro-Simple Sum M3
Out-of-sample MAE performance of NN for the Euro-Simple Sum M3
Out-of-sample MAE performance of NN for the Euro-Simple Sum M3

Table Captions
1. ADF unit root tests (1980:1-1998:2)
2. ARIMA models considered. Y represents Yes and N represents N
3. Multivariate Autocorrelation and Normality Tests (1980:1-1998:2)
4. Simultaneous choice of rank and deterministic components (1980:1-1998:2)
5. Cointegration vectors (1980:1-1998:2)
6. Number of iterations and hidden nodes used for the different networks
7. Within-sample and out-of-sample fit measures using the best ARIMA model
8. Within-sample and out-of-sample fit measures using VAR models
9. Within-sample and out-of-sample fit measures using NN constructed using thevariables in set A
10. Within-sample and out-of-sample fit measures using NN constructed using thevariables in set B
11. Within-sample and out-of-sample fit measures using NN constructed using thevariables in set C

22

Figures
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
Simple Sum index

0.7
80

82

84

86

88

90

Divisia index

92

94

96

98

00

Figure 1: Simple Sum M3 index versus Divisia M3 index

1

Input Layer

Hidden Layer

Output Layer

Figure 2: NN model

23

RMSE

RMSE
-3

-3

x 10

x 10

2.5

1.8

2.4

1.6

2.3

1.4
2.2

1.2

2.1
2
5

1
5
4

No. of hidden
units

3
2
1

1.5

1.6

1.7

1.8

1.9

2

4
No. of hidden
units

4

No. of iterations

x 10

3
2
1

Figure 3: Within-sample RMSE
performance of NN for the EuroSimple Sum M3

1.5

1.6

1.7

1.8

1.9

2
4

No. of iterations

x 10

Figure 6: Out-of-sample RMSE
performance of NN for the EuroSimple Sum M3

MAE

MAE

-3

-3

x 10

x 10

1.85

1.3

1.8

1.2

1.75

1.1

1.7
1

1.65

0.9

1.6
1.55
5

0.8
5
4

No. of hidden
units

3
2
1

1.5

1.6

1.7

1.8

1.9

2

4
No. of hidden
units

4

No. of iterations

x 10

3
2
1

Figure 4: Within-sample MAE
performance of NN for the EuroSimple Sum M3

1.5

1.6

1.7

1.8

1.9

2
4

No. of iterations

x 10

Figure 7: Out-of-sample MAE
performance of NN for the EuroSimple Sum M3

MAE

MAPE

200

200

190

180

180
160
170
140

160
150
5

120
5
4

No. of hidden
units

3
2
1

1.5

1.6

1.7

1.8

1.9

No. of iterations

2

4
No. of hidden
units

4

x 10

3
2
1

Figure 5: Within-sample MAPE
performance of NN for the EuroSimple Sum M3

1.5

1.6

1.7

1.8

1.9

No. of iterations

Figure 8: Out-of-sample MAE
performance of NN for the EuroSimple Sum M

24

2
4

x 10

Tables

Variable

ADF Test Statistics

Specification

m3
∆m3t

d 3t
∆d t
yt
∆y t
pt
∆pt

-1.503-4.834**-2.288-5.151**-2.280-6.932**-2.451-2.785[T, 1][C, 0][T, 1][C, 0][T, 4][C, 0][T, 2][T, 1]∆2 pt
dualsm3t∆dualsm3t
dualdm3t∆dualdm3t-14.316-3.825*-5.680**-3.781*-5.760**[C, 0][T, 1][C, 1][T, 1][C, 1]

Table 1: ADF unit root tests (1980:1-1998:2)
Notes:
T: constant and trend, C: represents constant[, n], n: the number of lags used**: significant at 1%, *: significant at 5%
Critical values are from MacKinnon (1991)
Models
ARIMA(0,2,1)
ARIMA(0,2,2)
ARIMA(0,2,3)
ARIMA(1,2,0)
ARIMA(1,2,1)
ARIMA(1,2,2)
ARIMA(1,2,3)
ARIMA(2,2,0)
ARIMA(2,2,1)
ARIMA(2,2,2)
ARIMA(2,2,3)
ARIMA(3,2,0)
ARIMA(3,2,1)
ARIMA(3,2,2)
ARIMA(3,2,3)
ARIMA(6,2,6)

Retained
Y
N
N
Y
N
Y
N
N
N
N
N
N
N
N
N
Y

Table 2: ARIMA models considered. Y represents Yes and N represents N.
25

Simple Sum

Divisia

Autocorrelation test:LM (1)LM (4)
Jarque Bera Normality test

15.04

21.92

(0.52)(0.15)

22.09

17.80

(0.14)(0.34)

7.47

11.22

(0.49)(0.19)

Table 3: Multivariate Autocorrelation and Normality Tests (1980:1-1998:2)
Note: Values in parenthesis are p-values. The LM-tests are asymptotically distributed
χ2(16), whilst the normality test is asymptotically distributed χ2(8).

Simple Sum
p-r

4
3
2
1

r

0
1
2
3

Divisia

Model 1Model 2Model 3Model 1Model 2Model 3

80.30

58.65

65.81

81.13

56.86

66.78

49.92

43.84

58.96

49.92

43.84

58.96

49.19

33.46

39.26

48.05

33.56

43.38

31.88

26.70

39.08

31.88

26.70

39.08

24.00

9.96

14.08

28.49

16.42

24.34

17.79

13.31

22.95

17.79

13.31

22.95

7.41

2.34

3.57

11.88

0.36

7.82

7.50

2.71

10.56

7.50

2.71

10.56

Table 4: Simultaneous choice of rank and deterministic components (1980:1-1998:2)
Note: Numbers in italics are 90 percent quantiles of the trace test tabulated in
Johansen (1995)

26

MoneyGDP

Inflation

Opp. cost

Trend

Simple Sum

-1
-1

1.181
1.451

-40.345-1.8090.633-0.055-

Divisia

-1
-1
-1

2.596
1.220
1.304

45.946-2.054-3.186-0.232-0.032
0.111-

Table 5: Cointegration vectors (1980:1-1998:2)

Simple Sum
Input Variables
Set A:∆m3t − 2 / ∆d 3t −3 ,∆2 pt −1
Set B:∆m3t − 2 / ∆d 3t −3 ,∆2 pt −1 , Re s t −1
Set C:∆M t −1 , ∆M t − 2 ,∆M t −3 , ∆M t − 4 ,Divisia

No. ofiterations
15000

No. of hidden
nodes
2

No. ofiterations
10000

No. of hidden
nodes
2

16000

4

10000

4

18000

5

12000

2

∆2 pt −1

Table 6: Number of iterations and hidden nodes used for the different networks

RMSE
MAE
MAPE

Within-Sample
0.002502
0.002070
139.7463

Out-of-Sample
0.001601
0.001145
103.6850

Table 7: Within-sample and out-of-sample fit measures using the best ARIMA model

27

RMSE
MAE
MAPE

Simple Sum
WithinOut-of-sample
sample0.002383 0.001456
RMSE
0.001906 0.001113
MAE
168.24% 157.34%
MAPE

Divisia
Withinsample
0.002334
0.001814
166.48%Out-of-sample
0.001495
0.001164
166.62%

Table 8: Within-sample and out-of-sample fit measures using VAR models

RMSE
MAE
MAPE

Simple Sum
WithinOut-of-sample
sample0.002381 0.001455
RMSE
0.001935 0.001112
MAE
166.04%
153.69%MAPE

Divisia
Withinsample
0.002383
0.001854
164.56%Out-of-sample
0.001466
0.001147
144.46%

Table 9: Within-sample and out-of-sample fit measures using NN constructed using
the variables in set A

RMSE
MAE
MAPE

Simple Sum
WithinOut-of-sample
sample0.002412 0.001179
RMSE
0.001924 0.000983
MAE
157.06%153.84%
MAPE

Divisia
Withinsample
0.002416
0.001865
154.53%Out-of-sample
0.001807
0.001324
161.21%

Table 10: Within-sample and out-of-sample fit measures using NN constructed using
the variables in set B

RMSE
MAE
MAPE

Simple Sum
WithinOut-of-sample
sample0.002050 0.001345
RMSE
0.001585 0.001071
MAE
152.41%150.62%
MAPE

Divisia
Withinsample
0.0021920.001689
142.19%Out-of-sample
0.001316
0.000999111.61%

Table 11: Within-sample and out-of-sample fit measures using NN constructed using
the variables in set C

28

Footnotes:
1

The reason for using nonseasonal ARIMA models is that the data provided to us
had already been seasonally adjusted.
2
Values in parentheses under the estimated coefficients are standard errors
3
The numbers in parentheses behind the values of the test statistics are thecorresponding p-values
4
The dummy variables are constructed to take into account the high peaks in the firstdifferences in the measures of money in 1990Q3 and in the opportunity cost variables
in 1994Q2.
5
The values of the trace statistics are found using CATS and RATS software.
6
The univariate ARCH test statistics are available from the authors upon request
7
The computations reported in this section were carried out on Eviews 4.0
8
Due to our short out-of-sample data set, misspecification tests could not be carriedout for order 829