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Forecasting in Big Data Environments: an Adaptable and Automated Shrinkage Estimation of Neural Networks (AAShNet)

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This paper considers improved forecasting in possibly nonlinear dynamic settings, with high-dimension predictors (big data environments). To overcome the curse of dimensionality and manage data and model complexity, we examine shrinkage estimation of a back-propagation algorithm of deep neural nets with skip-layer connections. We expressly include both linear and nonlinear components. This is a high-dimensional learning approach including both sparsity L 1 and smoothness L 2 penalties, allowing high-dimensionality and nonlinearity to be accommodated in one step. This approach selects significant predictors as well as the topology of the neural network. We estimate optimal values of shrinkage hyperparameters by incorporating a gradient-based optimization technique resulting in robust predictions with improved reproducibility. The latter has been an issue in some approaches. This is statistically interpretable and unravels some network structure, commonly left to a black box. An additional advantage is that the nonlinear part tends to get pruned if the underlying process is linear. In an application to forecasting equity returns, the proposed approach captures nonlinear dynamics between equities to enhance forecast performance. It offers an appreciable improvement over current uni-variate and multivariate models by RMSE and actual portfolio performance. https://arxiv.org/abs/1904.11145
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Forecasting in Big Data Environments: an Adaptable and
Automated Shrinkage Estimation of Neural Networks (AAShNet)
Ali HabibniaEsfandiar Maasoumi
ABSTRACT
This paper considers improved forecasting in possibly nonlinear dynamic set-
tings, with high-dimension predictors (big data environments). To overcome
the curse of dimensionality and manage data and model complexity, we exam-
ine shrinkage estimation of a back-propagation algorithm of a neural net with
skip-layer connections. We expressly include both linear and nonlinear compo-
nents. This is a high-dimensional learning approach including both sparsity L1
and smoothness L2penalties, allowing high-dimensionality and nonlinearity
to be accommodated in one step. This approach selects significant predictors
as well as the topology of the neural network. We estimate optimal values
of shrinkage hyperparameters by incorporating a gradient-based optimization
technique resulting in robust predictions with improved reproducibility. The
latter has been an issue in some approaches. This is statistically interpretable
and unravels some network structure, commonly left to a black box. An
additional advantage is that the nonlinear part tends to get pruned if the un-
derlying process is linear. In an application to forecasting equity returns, the
proposed approach captures nonlinear dynamics between equities to enhance
forecast performance. It offers an appreciable improvement over current uni-
variate and multivariate models by RMSE and actual portfolio performance.
Key Words: Nonlinear Shrinkage Estimation, Gradient-based Hyperparam-
eter Optimization, High-dimensional Nonlinear Time Series, Neural Networks
JEL classification: C45, C51, C52, C53, C61.
Department of Economics, Virginia Tech. Email: habibnia@vt.edu
Department of Economics, Emory University.
arXiv:1904.11145v1 [econ.EM] 25 Apr 2019
I. Introduction
An important step in designing modern predictive models is to cope with
high-dimensional data, presenting large numbers of (cor)related variables and
complex properties. “Big data” is both an increase in the number of samples
collected over time, and an increase in the number of potential explanatory
variables and predictors. When dimension grows, the specificities of high-
dimensional spaces and data must then be taken into account in the design of
predictive models. While this is valid in general, its importance is heightened
when using nonlinear tools such as artificial neural networks. Most nonlin-
ear models involve more parameters than the dimension of the data space
which may result in a lack of identifiability, lead to instability, and overfitting
(Huber (2011);Cherkassky et al. (1994); Moody (1991)). Selection of signifi-
cant predictors, and model complexity are the key tasks of designing accurate
predictive models in data-rich environments.
Feature extraction and feature selection are broadly the two main ap-
proaches to dimensionality reduction. Extraction transforms the original fea-
tures into a lower dimensional space preserving all its fundamentals. Fea-
ture selection methods select a small subset of the original features without
a transformation. Extraction methods include principal component analysis -
Pearson (1901); Eckart and Young (1936); factor analysis - Spearman (1904);
canonical correlations analysis - Hotelling (1936), and several others1.). Fea-
ture selection is accomplished by such methods as Ridge - Hoerl and Kennard
(1970); LASSO - Tibshirani (1996) and Elastic Net - Zou and Hastie (2005)).
In this work, our main focus is on feature selection techniques. We apply
shrinkage approaches (usually referred to as regularization in machine learning
literature). We embed feature selection in the backpropagation algorithm as
part of its overall operation. Accordingly, we extend our loss function to
include L1norm for the weights of the dense network, and L2norm for the
1Cunningham and Ghahramani (2015) surveyed the literature on linear dimensionality
reduction in their work.
2
weights in the skip-layer. The dense network corresponds to a multilayer
neural network, whereas the skip-layer denotes the direct connection from
each of the input variables to each of the output variables, which is similar to
a linear regression model.
Shrinkage is an implicitly embedded feature selection. It is an example of
model selection since only a subset of variables contributes to the final pre-
dictor. It has frequently been observed that L1shrinkage produces many zero
parameters, leading to some features being dropped and a sparse model. Only
those parameters whose impact on the empirical risk is considerable appear
in the fitted model Ng (2004). Shrinkage is a proper means of controlling
complexity in the nonlinear component. From an optimization point of view
we have a neural network learned/estimated by LASSO. This prevents hidden
units from getting stuck near zero and/or exploding weights.
Simultaneously, we employ the L2shrinkage on the skip-layer connections
(linear part of the model), in order to penalize groups of parameters, and
encourage the sum of the squares of the parameters to be small. Therefore we
will not drop specific features from linear component, making it possible to
interpret the marginal impact of predictors on the target variable. It is worth
mentioning that the linear part of the model can be interpreted as a Ridge
regression.
There are other benefits to shrinkage/regularization. Empirically, penal-
izing the magnitude of network parameters is also a way to reduce overfitting
and to increase prediction accuracy Ng (2004). This is especially true in the
state-of-art models, such as deep learning models with large number of pa-
rameters. Our proposed algorithm combines the neural networks advantage
of describing the nonlinear process with the superior accuracy of feature se-
lection that is provided by a penalized loss function that combines L1and L2
norms.
Many studies have suggested neural networks as a promising alternative
to linear regression models. Empirical evidence on out-of-sample forecasting
3
performance is, however, mixed. It is challenging to determine linear or non-
linear components. Linearity tests do often suggest that real world series are
rarely purely linear or nonlinear.
We consider the possibility that the series (yt) contain both a linear com-
ponent, (Lt), and a nonlinear component (Nt).
yt=Lt+Nt(I.1)
Neural network alone is not best suited to handle both linear and nonlinear
components, especially when the linear component is superior to the nonlinear
component.
Two different approaches to model and forecast series with both linear and
nonlinear patterns are available. The first approach is a two step methodology
to combine linear time series models and neural network models. In this
approach, the first step residuals are obtained from the fitted linear model
ˆet=ytˆ
Lt. In the second step a nonlinear model (e.g., GARCH, neural
nets) is trained on the residuals of the first step. In principle, this “hybrid”
two step approach can provide superior predictions when both the linear and
neural network model are well specified. In practice, however, two types of
model specification errors are introduced without an ability to assess their
mutual impact.
The alternative approach that we are proposing in this paper models both
linear and nonlinear components adoptively. It is based on a neural network
with skip-layer connections including both linear and nonlinear structures.
The rest of the paper is organized as follows. Section II provides the basic
framework of the proposed model. In Section III we investigate proper esti-
mation of shrinkage hyperparameters and introduce gradient-based techniques
based on reverse-mode automatic differentiation (RMAD) to accomplish this.
Section IV presents an application to US financial returns. Section V contains
some concluding remarks.
4
II. The Model
In this study, we examine a feedforward neural network with one hidden
layer, known as a dense network. Neural network models can be seen as
generalizations of linear models, when one allows direct connections from the
input variables to the output layer with a linear transfer function2, that we
refer to as the skip-layer. The model is expressed as
yt= Φ(x;w) = X
ik
xitwik +X
jk
φjX
ij
xitwij wj k +εt,(II.1)
where Φ describes the network by a vector function. We associate subscript
iwith the input layer, subscript jwith the hidden layer, and subscript kwith
the output layer. xit = (x1t, x2t, ..., xmn) is the value of the ith input node,
which can be a constant input representing biases, a matrix of lagged values
of ytand some exogenous variables. φj(.) and Jare activation functions and
number of neurons used at the hidden layer. A single-hidden-layer neural
network with skip-layer connections is shown in Figure II.1. A network with
only one hidden layer and skip-layer connections has three sets of weights:
those for direct connections between the inputs and the output (wik), those
connecting the inputs to the hidden layer (wij), and those connecting the
output of the hidden layer to the final output layer(wjk ).
First term in Eq.(II.1) represents a linear regression term. The second
term, denoting the dense network of the two layers, hidden and output, is
usually referred to as a multi-layer perceptron in the literature. It has been
shown to be able to perform well with nonlinear complex data. A greater ca-
pacity of the dense network, compared to the skip-layer, is realized by stacking
two layers, enabling it to model more complex data. A differentiable nonlinear
2Using linear function for the output unit activation function (in conjunction with
nonlinear activations amongst the hidden units) allows the network to perform a powerful
form of nonlinear regression. So, the network can predict continuous target values using a
linear combination of signals that arise from one layer of nonlinear transformations of the
input.
5
activation function φis used in the hidden units. εtis a random disturbance
term which captures all other factors influencing ythan the x. A linear com-
ponent term moves the model in the linear direction. This aids statistical
interpretation and unravels the structure behind the network, otherwise left
to a black box. This simultaneous approach has the advantage, when we apply
shrinkage techniques to estimate network parameters for an essentially linear
process, of pruning the hidden neurons.
Figure II.1. A single-hidden-layer neural network with skip-layer connec-
tions
Estimation of network elementary parameters based on prediction error
minimisation is known as training/learning. The most common cost/risk func-
tion is the mean squared prediction error (MSE), E=1
nPn
t=1(ytˆyt)2. Given
target values ytand network estimated outputs ˆyterror functions are obtained
for each parameter set, followed by tuning of the parameters.
The error surface becomes increasingly complicated with the number of
input variables and network parameters. It is common to employ the con-
ventional feed-forward neural network, trained with the popular and revo-
lutionary gradient-descent-type algorithm known as backpropagation. The
backpropagation algorithm was first introduced by Bryson et al. (1979) and
6
popularized in the field of artificial neural network research by Werbos (1988)
and Rumelhart et al. (1986). Error function’s sensitivity to network param-
eters is assessed via Gradient Descent optimization. Gradient is normally
defined as the first order derivative of the error function with respect to each
of the model parameters. Working out the gradients can be performed in a
completely mechanical way known as Automatic Differentiation Baydin et al.
(2017). AD employs the Jacobian matrix of gradients for each parameter wito
identify directions that decrease the height of the error surface (see Appendix).
In fact backpropagation is only a specific case of reverse-mode AD that is ap-
plied to an objective function errors as functions of model parameters. The
weight adjustment is given by
wnew =wold η∂E(w)
∂w (II.2)
Where the constant ηis the learning rate (step size) for updating elemen-
tary parameters, its value falls between zero and one. By iteratively repeating
this mechanism, the network can be trained in a way that converges to the
optima. The set of new elementary parameters are repeatedly presented to
the network until the error value is minimized. Around the optimum point,
all the elements of the gradient would be very small, leading to tiny changes
in new parameters.
We add the L1and L2penalties in training our modelto the loss func-
tion e
E(.), the original MSE. The following optimization problem is used for
training:
w= argmin
we
E(w|λ, X) = argmin
w
E(w|λ, X) + Ω(w, λ)(II.3)
where the regularization term Ω(w, λ) is a combination of the L1 norm and
the L2 norm of the parameter vector. λsets the impact of shrinkage on the
loss, with larger values resulting in more penalization. Using the regularized
7
objective causes the training procedure to be inclined to smaller parameter
values; unless larger parameters considerably improve the original error value
(MSE). Assuming a fixed λ, to learn w, we only need to include the derivative
of Ω(w, λ) in our derivatives:
∆ = E(w)
∂w +Ω(w)
∂w
wnew =wold η
(II.4)
Where ∆ is the gradient of the regularized loss function. λ > 0 is propor-
tional to complexity of the model but is not a parameter that appears in the
model. It is a hyperparameter. In the next section, we explain the impact of
hyperparameters and elaborate on our procedure for tuning them.
We employ L1and L2shrinkage on the parameters of the dense network
and skip-layer, respectively; as is depicted by following optimization problem:
w= argmin
w
E(w|λ, X) + λ2
2X
ik
w2
ik +λ1(X
ij|wij |+X
jk|wjk |) (II.5)
which can be realized by iteratively adjusting the parameters using the
updating rules below
wnew
ik =wold
ik η(∂E (w|λ,X)
∂wik +λ2wold
ik )
wnew
ij =wold
ij η(∂E (w|λ,X)
∂wij +λ1sgn(wold
ij ))
wnew
jk =wold
jk η( E(w|λ,X )
∂wj k +λ1sgn(wold
jk ))
(II.6)
Where λ1and λ2are non-negative values known as shrinkage hyperpa-
rameters. L1sparsity norm and L2smoothing norm are two closely related
regularizers that can be used to impose a penalty on the complexity of the
model that is to be learned. Shrinkage estimation of the model can be seen
as an implementation of Occam’s razor, introducing a controllable trade-off
between fitting data and model complexity, enabling us to have models of less
complexity with adequate generalization capability. Regularization in neural
networks limits the magnitude of network parameters by adding a penalty
8
for weights to the model error function. In this study, L2shrinkage penalizes
parameters in skip-layer connections by adding sum of their squared values
to the error term. L1shrinkage penalizes parameters in the dense network
to encourage the topology of the learned network to be sparse. The relative
importance of the compromise between finding small weights and minimizing
the original risk function depends on the size of λ.
To use L2shrinkage, we add a λ2wterm to the gradient as the derivative of
w2is 2w.L2shrinkage works with all forms of learning algorithms, but does
not provide implicit feature selection. The derivative of the absolute value of
wis w/|w|, however L1norm is not differentiable at zero and hence poses a
problem for gradient-based methods.
The problem can be solved using the exact gradient, which is discontinuous
at zero. We can also solve the problem by the smooth approximation approach
which will allow us to use gradient descent. To smooth out the L1norm using
an approximation, we use w2+in place of |w|, where is a smoothing
parameter which can also be interpreted as a sort of sparsity parameter. When
is large compared to w, the expression w+is dominated by and taking
the squared root yields approximately . Lee et al. (2006)
III. Gradient-based Hyperparameter
Optimization
The major drawback of shrinkage is that it introduces additional hyper-
parameters. In practice we have two set of parameters: model elementary
parameters (network weights and biases), and learning algorithm hyperpa-
rameters (magnitude of L1and L2penalties, and learning rate). We would
ideally like to determine these hyperparameters to get optimal generaliza-
tion3. As opposed to elementary parameters, these hyperparamters cannot
3Generalization means building a model on one set of training data and hope that it
makes effective predictions on a different set of test data.
9
be directly trained by the data. Whereas the elementary parameters specify
how to transform the input data into the desired output, the hyperparameters
define how our model and algorithm are actually structured.
The performance and robustness of neural networks relies to a large ex-
tent on hyperparameters. Tuning these hyperparameters not only makes the
investigation of methods difficult, but also hinders reproducibility (Bergstra
et al. (2011b)). Transparent tuning of hyperparameters can be part of an Hy-
perparameter Optimization (HPO), as an outer loop in training procedures.
The de-facto na¨ıve approach of searching through combinations of poten-
tial values of hypergradients and choosing the one that performed the best
(a.k.a. grid search) is very time-consuming and becomes quickly infeasible as
the dimension of hyperparameter space grows. In many practical applications
manually searching the space of hyperparameter settings is tedious and tends
to lead to unsatisfactory outcomes. Bergstra and Bengio (2012) show empir-
ically and theoretically that random search more efficient than grid search.
Statistical techniques such as cross-validation Wahba (1990), bootstrapping
Efron and Tibshirani (1994), and Bayesian methods MacKay (1992) can also
assist in determining hyperparameters.
HPO must be guided by some performance metric, typically measured by
cross-validation (CV) on the training set, or evaluation on a held-out vali-
dation set. The rationale behind CV is to split the data into the training
samples used for learning the algorithm, and the validation samples (one or
several folds) for estimating the risk of each algorithm and for evaluation of
its performance. CV consists of averaging several hold-out estimators (folds)
of the risk corresponding to different splits of the data, and selecting the al-
gorithm with the smallest estimated risk. Within each fold, hyperparameters
are fixed and we only estimate model elementary parameters. The validation
samples play the role of new unseen data as long as the data are i.i.d.4For
a general description of the CV see Geisser (1975), and Arlot and Celisse
4This assumption can be relaxed. see: Chu and Marron (1991).
10
(2010) for a comprehensive review on cross-validation procedures and their
applications in different algorithms and frameworks. Several studies such as
Rivals and Personnaz (1999) show cases in which CV performance is less than
satisfactory.
Recently, automated approaches for estimation of hyperparameters have
been proposed which can provide substantial improvements and transparency.
Although one may also “hyperparameterize” certain discrete choices in design
of the model (e.g. number of hidden units), we focus only on the continuous
hyperparameters in this work. There are a number of gradient-free automated
optimization methods (Hutter et al. (2011); Bergstra et al. (2011a); Bergstra
et al. (2013); Snoek et al. (2012)), all of which rely on multiple complete
training runs with varied fixed hyperparameters. Hyperparameters are chosen
to optimize the validation loss after complete training of the model parameters.
Gradient-based HPO approaches, proposed by Larsen et al. (1996) and
Andersen et al. (1997), emerged in the 1990s. We can distinguish two main
approaches of gradient-based optimization: Implicit differentiation and itera-
tive differentiation.
Implicit differentiation, first proposed by Larsen et al. (1996), computes
the derivative of the cost Lvalid with respect to λbased on the observation
that, under some regularity conditions, the implicit function theorem can be
applied in order to calculate the gradients of the loss function. In particular,
the cost function is assumed to smooth and converge to local minima. The
inner optimization w(λ)argminwLtrain can be characterized by the implicit
equation wLtrain = 0. Bengio (2000) derived the gradients for unconstrained
cost function and applied the algorithm to L2 shrinkage for linear regression.
The method has also been used to find kernel parameters of Support Vector
Machines Keerthi et al. (2007). Pedregosa (2016) proposes HOAG which uses
inexact gradients, allowing the gradient with respect to hyperparameters to
be computed approximately.
In iterative differentiation, first proposed by Domke (2012), the gradi-
11
ent for hyperparameters are calculated by differentiating each iteration of
the inner optimization loop and using the chain rule to aggregate the results.
However, the problem with this reverse-mode approach is that one must retain
the entire history of elementary parameter updates, making a na¨ıve implemen-
tation impractical due to memory constraints. Reverse-mode differentiation
requires intermediate variables to be maintained in the memory for the reverse
pass and evaluation of validation loss needs hundreds or thousands of inner op-
timization iterations. Maclaurin et al. (2015) later extended this for setting of
stochastic gradient descent via reverse mode automatic differentiation of vali-
dation loss.The burden of storing the entire training trajectory w1,··· , wTis
avoided by an algorithm that exactly reverses SGD with momentum to com-
pute gradients with respect to all training parameters, only using a relatively
small memory footprint, making a solution feasible for large-scale big data
machine learning problems.
We defined the updating rule for elementary parameters as wt+1 =wt
ηLtrain where Ltrain =e
E(wt|λ, Xtrain) is the regularized loss value on train
data. To calculate hypergradients we rely on the unregularized loss function,
that is Lvalid =E(wt|λ, Xvalid), as the actual generalization performance of
the model, on unseen data points, does not directly depend on regularizers;
otherwise the model with no regularization would be always selected:
λ= argminλLvalid
s.t. w(λ)argminwLtrain
(III.1)
There are cases where SGD can become very slow. The method of momen-
tum is designed to accelerate learning, especially in the face of high curvature,
small but consistent gradients, or noisy gradients Goodfellow et al. (2016). We
modify our training (Algorithm 1) to include a velocity variable vstoring the
momentum by calculating exponentially decaying moving average of past gra-
dients.
where γtis the momentum decay rate. The training procedure starts
12
Algorithm 1 Stochastic gradient descent with momentum
1: input: initial w1, decays γ, learning rates η, loss Ltrain
2: initialize v1=0
3: for t= 1 to Tdo
4: gt=wLtrain
5: vt+1 =γtvt(1 γt)gt
6: wt+1 =wt+ηtvt
7: end for
8: output trained parameters wT
with elementary parameters velocity v1= 0 and w1and ends with vTand
wT=wT1+ηT1vT1. Algorithm 2 is then used to calculate the gradients
of validation loss with regard to the hyperparameters.
Algorithm 2 Reverse-mode differentiation of SGD
1: input: wT,vT,γ,η, train loss Ltrain, validation loss Lvalid
2: initialize dv=0,dλ=0,t=0,=0
3: initialize dw=wLvalid
4: for t=Tcounting down to 1do
5: t=dwTvt
6: wt1=wtηtvt
7: gt=wL(wt,λ, t)
8: vt1= [vt+ (1 γt)gt]t
9: dv=dv+ηtdw
10: t=dvT(vt+gt)
11: dw=dw(1 γt)dvwwLtrain
12: dλ=dλ(1 γt)dvλwLtrain
13: dv=γtdv
14: end for
15: output gradient of Lvalid w.r.t λ
The velocity vtis needed to reverse the path, otherwise without momen-
tum, gtand ηtalone would not be able to recover wt1. Notice that the loss of
information caused by finite precision arithmetic in computers leads to failure
of this algorithm. For this reason, we need to store the bits lost in vtwhen
multiplied by γt.
Given this powerful gradient-based mechanism for finding hyperparam-
eters, a natural extension to our model is to introduce a hyperparameter α
denoting the contribution of skip-layer and dense-network in producing predic-
tions with higher generalization. That is to say, our model can be reformulated
13
as:
yt= Φ(x;w) = αX
ik
xitwik + (1 α)X
jk
φjX
ij
xitwij wj k +εt,(III.2)
where αassumes a value between zero and one. Appreciating that the
skip-layer and the dense network have unbalanced effect on the outcome, one
can see how this may result in faster convergence of training procedure. More
importantly, αcan be interpreted as the activation of skip-layer and dense
network and can point to linearity or nonlinearity components.
IV. Case Study: Return Prediction
Research into modelling and forecasting financial returns has a long his-
tory. Several models are described in Tsay (2005) and Campbell et al. (1996)
that attempt to explain return time series using linear combinations of one or
more financial market factors. The most widely studied single factor model is
the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965)
that relates the expected return of equities to the expected rate of return on
a market index (such as the Standard and Poors 500 Index). The empirical
performance of CAPM is poor as it cannot explain the behaviour of asset re-
turns, see Fama and French (2004). This failure is perhaps due to the absence
of multiple factors. Arbitrage pricing theory (APT) is a general model pro-
posed by Ross (1976) to account for these deficiencies. APT presents a linear
approximate model of expected asset returns based on an unknown number
of macroeconomic “factors” or market indices. The relationship between the
factors and historical returns is routinely determined linearly.
Return time series present characteristics such as comovement, nonlin-
earity, non-Gausianity (skewness and heavy tails), volatility clustering and
leverage effect. This makes the modelling task very challenging, see Hsieh
(1991); Bollerslev et al. (1994); Brooks (1996); Cont (2001).
The data are daily returns of m= 418 equities on the S&P 500 index from
14
03.01.2006 through 28.09.2018, for a total of 3208 observations. The initial
sample 03.01.2006 - 28.09.2017 is used for estimation (training), with T=
2957 in-sample size. The holdout sample period 01.10.2017 - 28.09.2018 (251
observations) is employed to examine the models’ out-of-sample forecasting
performance. 1-step (here one day) ahead forecasts of targets (ˆyit+1|t) are
based on a rolling estimation window. Parameter estimates are updated every
five steps.
We believe accounting for comovements between financial returns is im-
portant in forecasting returns. Consequently, the lags of other equities are
included as predictors for any return series. We examine the nonlinear high-
dimensional forecasting model described in the prior sections (AAShNet model)
as well as several competing models and benchmarks.
We compare our proposed model with a benchmark, the sample mean of
ytover the in-sample window, as the 1-step ahead forecast. This corresponds
to assuming the log daily price of follows a random walk (RW) with drift. It
is almost equivalent to the “zero forecast” when the in-sample window is large
enough. Furthermore, a buy-and-hold (B&H) strategy in the market portfolio
(S&P 500 Index) has been considered as another benchmark. To understand
whether allowing nonlinearity improves portfolio performance we examine the
AAShNet algorithm (with Ridge and Lasso) optimized by cross-validation.
Since predictability of financial returns has major consequences for finan-
cial decision making, the model with minimal forecast error is deemed op-
timal. However, the model with minimum forecast error does not necessar-
ily guarantee profit maximization, the primary objective of financial decision
makers. Armstrong and Collopy (1992), Pesaran and Timmermann (1995,
2000), Granger and Pesaran (2000) and Engle and Colacito (2006) argue that
a forecast evaluation criterion should be related to decision making and judge
predictability of financial returns in terms of portfolio simulation. More specif-
ically, a trading (portfolio) simulation approach assumes that all competing
models are applied with stock market virtual investment decisions, and out-
15
of-sample portfolio performances are used to evaluate the predictability of
alternative models.
Consequently, this paper examines both statistical and portfolio perfor-
mance measures (the out-of-sample RMSE and the portfolio performance dur-
ing the out-of-sample period). Figure IV.1 illustrates portfolio excess returns
for the out-of-sample period for the proposed model (ASShNet) against com-
peting approaches. We randomly selected 50 stocks out of 418 stocks to
construct the portfolio. However, the forecast of each selected stock is based
on the lags of all 418 equities.
Figure IV.1. Comparison of AAShNet, and the competing models based on
the portfolio excess returns in the out-of-sample period.
Consider a passive, equally weighted (1/M) portfolios with short selling.
This portfolio is known to be a very stringent benchmark that many opti-
mization models fail to outperform (see DeMiguel et al. (2009). We compute
the portfolios out-of-sample excess returns and volatility as well as the Sharpe
ratio. Sharpe ratio measures risk-adjusted returns, a portfolio with a greater
Sharpe ratio offers greater returns for the same risk. If a portfolio with lower
Sharpe ratio has returned better over a time period than another portfolio
with a higher ratio, the risk of losing by investing in the former fund will be
higher.
The proposed penalized neural net behaves noticeably better in this empir-
16
Table I
Portfolio Return Sharp Ratio Ave(RMSE)
AAShNet 10.2% 1.143 0.01558
B&H 9.55% 0.767 -
RW 8.17% 0.770 0.01564
Ridge 5.45% 0.561 0.01613
Lasso 6.87% 0.741 0.01590
ical analysis. Table I provides evidence for out-of-sample forecasting ability
of this model vis-`a-vis competing approaches in terms of the Sharp ratio.
AAShNet also offers an appreciable improvement over linear shrinkage mod-
els and benchmarks based on RMSE and actual portfolio performance. In the
Ridge and Lasso regressions, the best model is selected by cross-validation.
We perform generalized cross-validation, which is an efficient leave-one-out
cross-validation.
AAShNet produces higher returns (10.24%) at the end of the out-of-sample
period, with a Sharpe ratio of (1.143) that is superior to alternative models.
This indicates that significantly improved forecast is obtained by modelling
nonlinear dynamics among variables. One should note that Random Walk
with drift and AR(1) are special cases of shrinkage models and AAShNet
when there is no dependence on other equities.
V. Concluding Remarks
Forecasting with many predictors has received a good deal of attention
in recent years. Shrinkage methods are one of the most common approaches
for forecasting with many predictors. Such methods have generally ignored
nonlinear dynamic relations among predictors and the target variable.
In this study, we suggested an Adaptable and Automated Shrinkage Esti-
mation of Neural Networks (AAShNet). We explained how skip-layer connec-
tions move the model in the right direction when the data contains both linear
and nonlinear components. To overcome the curse of dimensionality and to
manage model complexity, we penalized the model loss function with L1and
17
L2norms. Setting the size of shrinkage is still an open question. Recent
studies have proposed automated approaches for estimation of algorithm hy-
perparameters. We employed the gradient-based automated approaches which
treat shrinkage hyperparameters in the same manner as the network weights
during training, and simultaneously optimize both sets of parameters.
The empirical application to forecasting daily returns of equities in the
S&P 500 index from 2006 to 2018 provides support for the out-of-sample fore-
casting ability of AAShNet algorithm vis-`a-vis some competing approaches,
both in terms of statistical criteria and trading simulation performance. Our
empirical results encourage further research toward other possible applications
of the proposed model.
VI. Appendix: Automatic Differentiation
There are three main approaches that computer can work out the deriva-
tives: Numerical, Symbolic and Automatic differentiation. Automatic dif-
ferentiation refers to a family of procedures to automatically calculate exact
derivatives of any function, including program subroutines, with time com-
plexity at most a small constant factor of the time complexity of the original
function. It is not inherently ill-conditioned and unstable similar to the nu-
merical method and has much less computational complexity. It also does not
suffer from expression swell problem of symbolic differentiation.
AD augments the standard computation with calculation of derivatives
whose combination through chain rule gives the derivative for overall compo-
sition. AD can be applied on evaluation trace of arbitrary program subrou-
tines which can be more than closed-form functions and are in fact capable
of incorporating complex control flows which do not directly alter values. An
automatic differentiator takes a code subroutine that computes a function of
several independent variables as input and gives as output a code that com-
putes the original function along the gradient of the function with respect
18
to the independent variables. As most of the functions are piece-wise dif-
ferentiable and control flows not directly interfering with calculations, chain
rule can be used repeatedly in such a way that gradients are calculated along
intermediate values being computed.
Based on modus operandi of automatic differentiation there can be two
implementations of this technique; the forward mode and the reverse mode.
We investigate each method, by applying them on the same trivial function
y=f(x1, x2) = x1x2cos(x1) at (x1, x2) = (6,3).
v1=x1= 6
v2=x2= 3
v3=v1v2= 6 ×3
v4=cos(v1) = cos(6)
v5=v3v4= 18 0.96
y=v5= 17.04
(VI.1)
In forward mode, we build a Forward Primal Trace of the values propa-
gating through the function and a corresponding Forward Tangent Trace. Eq.
VI.1 shows the forward evaluation of primals. Forward primal trace depicts
the natural flow of composition. Eq. VI.2 is the corresponding tangent trace
for ˙y=∂f
∂x1, that is the rate of change of the function fwith respect to the
input x1. Notice that both traces are evaluated as written, top to bottom. To
calculate the derivative with respect to ndifferent parameters, nforward mode
differentiations would be needed. This makes the forward-mode very ineffi-
cient for deep learning models where the number of parameters may amount
to millions.
19
˙v1= ˙x1= 1
˙v2= ˙x2= 0
˙v3= ˙v1v2+ ˙v2v1= 1 ×3+0×6
˙v4= ˙v1× −sin(v1) = 1 × −sin(6)
˙v5= ˙v3˙v4= 3 0.279
˙y= ˙v5= 2.72
(VI.2)
The reverse mode works by complementing each intermediate variable vi
with an adjoint ¯virepresenting the sensitivity of output yto changes in vi. In
reverse mode the code is executed and the trace is stored in memory at first
stage. At second stage, the adjoints are calculated in opposite direction of the
execution of the original function. The reverse adjoint trace corresponding to
Eq. VI.1 is depicted in VI.3.
¯v5= ¯y= 1
¯v4= ¯v5
∂v5
∂v4
= ¯v5× −1 = 1
¯v3= ¯v5
∂v5
∂v3
= ¯v5×1 = 1
¯v1= ¯v4
∂v4
∂v1
= ¯v4× −sin(v2) = 0.27
¯v2= ¯v3
∂v3
∂v2
= ¯v3×v1= 6
¯v1= ¯v1+ ¯v3
∂v3
∂v1
= ¯v1+ ¯v3×v2= 2.72
¯x1= ¯v1= 2.72
¯x2= ¯v2= 6
(VI.3)
For a function f:RnRmwhose number of operations to be eval-
uated is denoted by ops(f), the complexity of calculating the Jacobian by
forward and reverse modes are n×c×ops(f) and m×c×ops(f), respec-
tively, where it is guaranteed that c < 6Griewank and Walther (2008). That
is if nm, backward-mode is preferable, although it would have increased
memory requirements. And forward mode should be used when the number
20
of dependent variables is greater than the number of independent variables.
21
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