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Nonlinear forecasting with many predictors by neural network factor models (Deep Learning in Finance)

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Abstract

This study proposes a nonlinear generalization of factor models based on artificial neural networks for forecasting financial time series with many predictors. http://eprints.lse.ac.uk/62916/
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
XX, 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
ixtβi) + εt
(4)
The function F(˜ω0
ixtβi), 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
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Artificial neural networks: An econometric perspectiv
  • C M Kuan
  • H White
C. M. Kuan and H. White. Artificial neural networks: An econometric perspectiv. Econometric Reviews, 13:1-91, 1994.