Content uploaded by Ali Habibnia

Author content

All content in this area was uploaded by Ali Habibnia on Aug 09, 2017

Content may be subject to copyright.

NONLINEAR FORECASTING WITH MANY PREDICTORS

BY NEURAL NETWORK FACTOR MODELS

ALI HABIBNIA

DEPARTMENT OF STATISTICS,LONDON SCHOOL OF ECONOMICS

ABSTRACT

•This study proposes a nonlinear forecasting tech-

nique based on an improved factor model with

two neural network extensions. This model

would be able to capture both non-linearity and

non-normality of a high-dimensional dataset.

•Speciﬁcation (architecture) of the neural network

Factor model is determined on the basis of sta-

tistical inference and special emphasis is given to

data-driven speciﬁcation.

•Linear factor models can be represented as a spe-

cial case of this neural network factor model.It

means that, if there is no non-linearity between

variables, it will work like a linear model.

INTRODUCTION

Forecasting with factor models are a two-step process:

•Factor Estimation, which summarizes the infor-

mation contained in a large data set in a small

number of factors.

Xit

observations

= Λift

factors

+ξit

idiosyncratic

(1)

•Forecasting Equation, which is the prediction of

the variable of interest by using common factors.

yt+1|t

one of the X

=λ ft+1|t+εt+1 (2)

Common factors and the idiosyncratic component can

be forecast simultaneously or separately.

CONTRIBUTION & FORMULATION

Figure 1: The standard auto-associative neural network archi-

tecture for nonlinear PCA (combination of two feed-forward NNs)

The ﬁrst extension proposes a NLPCA (neural network

principal component analysis) as an alternative for fac-

tor estimation, which allows the factors to have a non-

linear relationship to the input variables. NLPCA non-

linearly generalizes the classical PCA method by a non-

linear mapping from data to factors. Both neural net-

work parameters and unobservable factors (f) can be op-

timised simultaneously to minimise the reconstruction

error e:

e=ˆ

X−X, MSE =E(|| ˆ

X(f)−X||2)(3)

•Second extension is a nonlinear factor augmented

forecasting equation based on a single hidden

layer feed-forward neural network model which

can be built in a similar fashion as a statistical

model.

•A neural network model can be deﬁned as:

yt=G(xt;ψ) + εt=α0˜xt+

h

X

i=1

λiF(˜ω0

ixt−βi) + εt

(4)

•The function F(˜ω0

ixt−βi), often called the activa-

tion function, is a logistic function.

Figure 2: Artiﬁcial Neuron conﬁguration

REFERENCES

[1] J. H. Stock and M. W. Watson. Forecasting using principal components from a large number of predictors. American Statistical Association,

97:1167–1179, 2002.

[2] M. Forni et al. The generalized dynamic factor model: Estimation and forecasting. American Statistical Association, 100:830–840, 2005.

[3] M. C. Medeiros and T. Terasvirta. Building neural network models for time series: A statistical approach. J of Forecasting, 25:49–75, 2006.

[4] C. M. Kuan and H. White. Artiﬁcial neural networks: An econometric perspectiv. Econometric Reviews, 13:1–91, 1994.

[5] M. Deistler and E. Hamann. Identiﬁcation of factor models for forecasting returns. Financial Econometrics, 3(2):256–281, 2005.

[6] A. N. Gorban and B. M. Kegl(Eds.). Principal Manifolds for Data Visualization and Dimension Reduction. Springer, 2008.

[7] M. A. Kramer. Nonlinear principal component analysis using autoassociative neural networks. AIChE, 37:233–243, 1991.

RESULTS

Out-of-sample forecast evaluation results based on dif-

ferent criteria (RMSE, Hit-Rate and Theil) showed that

the proposed neural network factor model (NNFM)

signiﬁcantly outperformed linear factor model and

Random-Wald approach.

CONTACT INFORMATION

Web http://personal.lse.ac.uk/habibnia/

Email a.habibnia@lse.ac.uk

Phone +44 (0)7737842985

NLPCA ON FINANCIAL RETURNS

Figure 3: Nonlinear PCA can describe the inherent structure of the data by a curved subspace.

CONSTRUCTION OF FNN

Three stages of model building:

•Variable selection

by linearizing the model (approximate NN model

by a polynomial of sufﬁciently high order) and

applying well-known techniques of linear vari-

able selection to this approximation.

•Parameter estimation

Estimate the parameters by maximum likelihood,

making use of the normality assumptions made

on residual.

•Determining the number of hidden units (neu-

rones)

Applying Lagrange multiplier type tests. One

possibility is to begin with a small model and se-

quentially add hidden units to the model.

FINANCIAL FORECASTING

Financial returns present special features and share the

following stylised facts: comovements, non-linearity,

non-gausianity (skewness and heavy tails) and lever-

age effect, which makes the modelling of this variable

hard.

Figure 4: monthly return observations of the 52 companies in

S&P100 index