Quantile Double AR Time Series Models for
Financial Returns
Yuzhi Cai
∗
, Swansea University
Gabriel Montes-Rojas, City University London
Jose Olmo, Centro Universitario de la Defensa de Zaragoza
Abstract
In this paper we develop a novel quantile double AR model for modelling finan-
cial time series. This is done by specifying a generalized lambda distribution to the
quantile function of the location-scale double autoregressive model developed in Ling
(2004, 2007). Model parameter estimation uses MCMC Bayesian methods. A novel
simulation technique is introduced for forecasting the conditional distribution of fi-
nancial returns m-periods ahead, and hence any predictive quantities of interest. The
application to forecasting Value-at-Risk at different time horizons and coverage prob-
abilities for Dow Jones Industrial Average shows that our method works very well in
practice.
Key words: Bayesian methods, density forecasts, generalized lambda distribution, quantile
function, quantile forecasts.
∗
Address for correspondence: Dr Yuzhi Cai, School of Business and Economics, Swansea University,
Swansea, SA2 8PP, United Kingdom. Email: y.cai@swansea.ac.uk
1
1 Introduction
Consider Ling (2004, 2007) double AR(p) model defined by
y
t
= a
0
+ a
1
y
1
+ ···a
p
y
t−p
+ η
t
b
0
+ b
1
y
2
t−1
+ ··· + b
p
y
2
t−p
, (1)
where b
i
> 0 (i = 0, . . . , p), η
t
is an independent random sequence and η
t
∼ N(0, 1), and
y
s
is independent of η
t
for s < t. This model is a special case of ARMA-ARCH models
proposed by Weiss (1984), but it is different from the ARCH models proposed by Engle
(1982) if a
i
= 0. This model encompasses a large proportion of applications in empirical
economics and finance where volatility plays an important role in modeling autoregressive
series (further discussion on the motivation for the double AR(p) models can be found in
Weiss (1984) and Ling (2004, 2007)).
It is worth mentioning that model (1) has only been investigated for the conditional
mean. Moreover, the normality requirement on the error term η
t
of the model is quite
restrictive as many economic and financial time series are non-Gaussian. This motivated
us to develop a novel quantile double AR model corresponding to model (1) that also allows
us to deal with general non-Gaussian time series easily. In order to illustrate our approach
we apply the developed model to the Dow Jones Industrial Average (DJIA).
In many areas of research studying extreme quantiles is of fundamental importance.
An example is Value-at-Risk (VaR) in economics and finance. Statistical inference on ex-
treme quantiles can be made once the probability distribution or density function of the
innovations η
t
is known. However, a direct quantile approach to statistical modelling has
recently become more popular. One of the methods for estimating conditional quantiles
of y
t
is to use the quantile regression techniques (see Koenker and Bassett (1978) and
Koenker (2005)), which allow us to obtain a sequence of conditional quantiles by using a
semi-parametric model, that is, without imposing strong distributional assumptions on η
t
.
The development in this area is rapid. For example, Koenker and Zhao (1996) extended
2
quantile regression to linear ARCH models and Engle and Manganelli (2004) developed
a different conditional autoregressive VaR model. Xiao and Koenker (2009) developed a
two-step approach of quantile regression estimation for linear GARCH time series. Tay-
lor (2008) proposed the exponentially weighted quantile regression for estimating time-
varying quantiles, and Giot and Laurent (2003) model VaR using ARCH models based on
skewed t-distribution. Galvao (2009, 2011) considered unit root quantile autoregression
testing and quantile regression dynamic panel model with fixed effects. Cai and Stander
(2008) proposed a quantile self-exciting autoregressive time series model and developed a
Bayesian approach for parameter estimation. Cai (2007, 2010c) also proposed forecasting
methods for such models.
However, one of the problems associated with the above models is that the extreme
quantiles (corresponding to extreme risks) may not be properly estimated, for example, the
estimated quantile curves may cross over (non-monotonicity). This is because when the
probability τ approaches the extremes (i.e. 0 or 1), the estimated τth conditional quantile
becomes less and less precise.
One way to deal with the crossing-over problem is to specify a parametric conditional
quantile function model (see Gilchrist (2000) for an excellent introduction to this paramet-
ric approach). This procedure allows to estimate the whole conditional quantile function of
y
t
directly using a wider class of distributions for η
t
, including those which are defined only
via their quantile functions and that may not have closed mathematical expressions for their
density or distribution functions. Our quantile double AR model follows this procedure and
it enables us to obtain valid estimation of extreme quantiles. Furthermore, quantile func-
tion properties allow us to construct the distribution of η
t
by combining several quantile
functions in a proper way (see Gilchrist, 2000), leading to an appropriate model for cap-
turing important features of economic and financial time series, including the occurrence
of extremes and volatility clustering. The flexibility of the generalized lambda distribution
(see Freimer et al. (1988)) motivated us to use this distribution in the construction of the
3
proposed quantile double AR model and this distribution because also enables us to study
the conditional quantile function of y
t
directly.
We also propose a Bayesian method to estimate the model parameters, which is a mod-
erate extension of the Bayesian method proposed by Cai (2009, 2010a, b). It will become
clear later in the paper that, unlike some other estimation methods, our Bayesian approach
also plays an important role in our proposed forecasting method which allows us to take
the uncertainty of the estimated model parameters into account when forecasting. Our
forecasting method can be used to obtain m-step ahead (m > 0) out-of-sample forecasts,
not just point forecasts but also the whole predictive distributions via the quantile function
models. Little work can be found on this in the literature.
Our results show that volatility clustering phenomenon observed in many financial re-
turns is reflected more parsimoniously in this model by the generalized lambda distribution
parameters. This indicates that despite the prominence in the literature of models to fore-
cast conditional volatility, it can be the case that current volatility is not so instrumental for
forecasting the conditional distribution of returns and researchers/practitioners need to look
at other parameters driving the behavior of the distribution tails. Furthermore, the flexibil-
ity of the quantile double AR model permits asymmetries in the dynamics of the tails of the
predictive conditional distribution of returns. This implies that this model provides a better
understanding of the impact of large negative/positive returns in the likelihood of future
losses/gains.
Therefore, the main contributions of this paper are: (a) It proposes a quantile double AR
model for economic and financial time series. Different from model (1), our model study
the conditional quantile function of y
t
, which allows us to model the extreme quantiles
directly and to study non-Gaussian time series easily. (b) Combined with our Bayesian
method we also propose a forecasting method for quantile function models, which enables
us to obtain m-step ahead out-of-sample predictive distributions, and hence any predictive
quantities of interest, including extreme quantiles.
4
The article is structured as follows. In Section 2, we propose the model and briefly
discuss the Bayesian estimation method for model fitting. Section 3 discusses an out-of-
sample forecasting method that also exploits features of MCMC Bayesian methods. Sec-
tion 4 applies these techniques to modeling and forecasting m-periods ahead the conditional
distribution of the Dow Jones Industrial Average (DJIA). Further discussion and comments
are found in Section 5.
2 The model and parameter estimation
Let y
1
, . . . , y
n
be an observed time series. The proposed quantile double AR(k
1
, k
2
) time
series model takes the form
Q
y
t
(τ | β, y
t−1
) = a
0
+ a
1
y
t−1
+ ···+ a
k
1
y
t−k
1
+
b
0
+ b
1
y
2
t−1
+ ··· + b
k
2
y
2
t−k
2
Q(τ, γ),
(2)
where (k
1
, k
2
) is the order of the model, β = (η
1
, η
2
, γ) is a vector of model parameters,
where η
1
= (a
0
, . . . , a
k
1
) and η
2
= (b
0
, . . . , b
k
2
), in which b
0
> 0, b
j
≥ 0, j = 1, . . . , k
2
and y
t−1
= (y
1
, . . . , y
t−1
)
⊤
. Finally, Q(τ, γ) is a well defined quantile function used to
describe the distribution of the error term of the model.
Note that model (2) is equivalent to the quantile process of model (1) if Q(τ, γ) is the
quantile function of N(0, 1). If we let
Q(τ, γ) =
τ
γ
1
− 1
γ
1
−
(1 − τ)
γ
2
− 1
γ
2
, γ
1
< 0, γ
2
< 0, (3)
then our proposed double AR quantile function model becomes
Q
y
t
(τ | β, y
t−1
) = a
0
+ a
1
y
t−1
+ ··· + a
k
1
y
t−k
1
+
b
0
+ b
1
y
2
t−1
+ ··· + b
k
2
y
2
t−k
2
τ
γ
1
−1
γ
1
−
(1−τ)
γ
2
−1
γ
2
,
(4)
5
