Volatility prediction based on
scheduled macroeconomic
announcements
Athanassios Petralias, Petros Dellaportas
SYRTO WORKING PAPER SERIES
Working paper n. 13 | 2014
This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and
demonstration under grant agreement n° 320270.
This documents reflects only the author's view. The European Union is not liable for any use that may be made of the information
contained therein.
Volatility prediction based on scheduled
macroeconomic announcements
Athanassios Petraliasa , Petros Dellaportasa,∗
a
Department of Statistics, Athens University of Economics and Business
Patission 76, Athens 10434, Greece
Abstract
We investigate the impact of scheduled macroeconomic announcements on the volatility
of exchange rates by introducing a flexible model formulation. For each macroeconomic
index we estimate cutoff points in the surprise component of the announcement that
specify the degree the volatility process is affected. This degree is quantified by jumps of
unknown size that occur before and at the time of the announcement and then die out
exponentially with unknown rate. We make inferences using a population Markov chain
Monte Carlo reversible jump algorithm and illustrate our methodology by predicting exchange rates volatility using fifteen U.S. macroeconomic announcements.
Keywords: Bayesian methods, Exchange rates, GARCH models, Model selection,
Nonlinear time series, Nonparametric methods, Population Markov chain Monte Carlo,
Reversible jump, Thresholds, Volatility forecasting
MSC 2010: 62C10, 62J10, 62P20, 62F15, 62G05
Correspondence to: Prof. Petros Dellaportas, Department of Statistics, Athens University of Economics and Business, Patission 76, Athens 10434, Greece. Tel: +30 2108203567, Fax: +30 2108203565.
Email addresses: petralia@aueb.gr (Athanassios Petralias), petros@aueb.gr (Petros
Dellaportas)
∗
Preprint submitted to Elsevier
July 22, 2014
1. Introduction
Scheduled announcements of macroeconomic indices have been found to affect decisively the volatility of stocks and exchange rates in a wide range of studies. In this study
we propose a rich model formulation that is capable of exploring how announcement surprises, defined as the absolute percentage differences of the realised minus the Bloomberg
consensus reported values, affect the volatility process of exchange rates.
We propose a model formulation based on two ingredients. The first is a GARCHtype process that captures the usual stylised facts such as heavy tails and volatility
clustering. The second is a nonparametric, threshold-based process that captures the
impact of announcement surprises. Due to the predictive nature of the GARCH process
and the fact that the announcements are scheduled, our model specification may serve as
a volatility forecasting tool.
The non-linear structure possesses the following characteristics. First, it assumes that
for each macroeconomic index there is a different pre-announcement effect, quantified by
a jump of unknown size in the volatility process. Second, for each index the degree of
announcement surprise has a different impact on the volatility process. To accommodate
this, we define regions in the announcement surprise data that are specified by threshold
points. Depending on both the index and the region of the announcement surprise, a
volatility jump of unknown size that decays exponentially with unknown rate is added to
the volatility process. The number of regions and their corresponding threshold points
are unknown and different for each index.
To make the most of our flexible model framework, we need a powerful inference
tool to explore a large number of models. We adopt Bayesian inference and construct a
population reversible jump Markov chain Monte Carlo (MCMC) algorithm in order to
efficiently sample from both the model and the parameter space. The resulting samples
of posterior summaries of interest can serve as forecasting tools through model averaging.
The methodology proposed here is illustrated in an empirical study in which we use
data of three exchange rates of the U.S. dollar and fifteen U.S. macroeconomic announcements. The results indicate that our proposed model formulation provides more accurate
forecasts than typical conditional volatility models, and this is so even if we enrich the
existing models appropriately so that macroeconomic announcements are incorporated.
Furthermore, we identify the macroeconomic announcements that mostly affect exchange
2
rates volatility, and provide insights on the way each announcement affects volatility.
The rest of the paper proceeds as follows. In Section 2 there is a short review of studies
investigating the effect of news announcements on volatility. In Section 3 we present a
new class of models for volatility based on macroeconomic announcements. Section 4
describes the details of the MCMC implementation whereas Section 5 presents the data
and the results. Finally, Section 6 concludes with a short discussion.
2. Announcements and volatility
The impact of macroeconomic announcements on the asset and foreign exchange
(Forex) markets volatility has been the subject of extensive research in past years. A
large number of related studies is based on GARCH type models (Engle, 1982; Bollerslev, 1986), where additional explanatory variables are used to capture the effect of news
announcements. A common formulation for these models is given as
yt | Ft−1 = µt + ǫt ,
ǫt ∼ N (0, Gt Ht )
(1)
where yt , t = 1, . . . , T , denotes the exchange rate return, Ft denotes the information set
up to time t, µt is the expected value of the mean process at time t, Ht is a function of
some explanatory variables related to news announcements affecting the volatility process
and Gt is a GARCH-type process.
The main difference among these studies stands in the selection of explanatory variables affecting Ht . In fact, it is not unusual to differ not only on the type of announcements but also in the temporal effect on volatility, for example adopting formulations
specifying before and after announcement effects. In this context, a large number of
studies use indicator variables to account for macroeconomic announcements (Jones et al.
(1998); Bomfim (2000); Kim et al. (2004); Bauwens et al. (2005); Nikkinen et al. (2006)).
In DeGennaro & Shrieves (1997) news announcements are captured by the number of
headlines in each news category on the Reuters money news-alerts, while Hautsch &
Hess (2002) and Brenner et al. (2009) use explanatory variables that capture surprises
on headline figures.
Significant empirical results are also drawn from studies analysing the effect of news
announcements directly on the absolute returns (Ederington & Lee (1993); Mitchell
& Mulherin (1994)); on the implied volatility (Ederington & Lee (1996); Nikkinen &
3
Sahlström (2001)); or on the order flows stemming from daily and intraday trades (Evans
& Lyons (2005, 2008)). Flexible Fourier forms have been also adopted by incorporating ARCH, calendar and macroeconomic announcement effects (Andersen & Bollerslev
(1998b); Bollerslev et al. (2000); Laakkonen (2004)). Andersen et al. (2007a) by measuring separately the continuous sample path variation and the discontinuous jump part of
the quadratic variation process, observe that significant jumps in the realized volatility
process tend to coincide with the release of macroeconomic indices. In statistics literature, significant jump patterns in daily volatility similar to those advocated in this paper
have been uncovered in, for example, Barndorff-Nielsen & Shephard (2001) and Roberts
et al. (2004).
Although the empirical results from the above studies are in some cases contradictory, there are several points of agreement. First, all relevant studies agree that news
announcements are affecting decisively the volatility of exchange rates and stock markets,
when analysed on either a daily or an intraday basis. Furthermore, all studies underline
that the effect on the volatility is related to the type of announcement and the content
of news, rather than the very act of releasing information. From all the types of news
considered, most studies find that scheduled macroeconomic announcements are the most
important, and their effect turns out to be stronger when compared to ARCH or calendar
effects (DeGennaro & Shrieves (1997); Andersen & Bollerslev (1998b); Bollerslev et al.
(2000); Bauwens et al. (2005)). The documented volatility autocorrelation and the dayof-the-week volatility patterns seem to depend strongly on the news generating process
and the timing of major macroeconomic announcements (see Ederington & Lee (1993)
for intraday evidence and Ederington & Lee (1996); Jones et al. (1998) for daily data).
In addition, there is evidence that while the surprising factor plays a key role, even news
that come out as expected seem to affect the volatility process (Laakkonen (2004)).
Studies seem to diverge with respect to the direction with which pre-announcement
effects affect volatility. Nikkinen & Sahlström (2001); Hautsch & Hess (2002); Bauwens
et al. (2005) found that volatility increases before news announcements. Contradictory
results providing evidence that the volatility decreases before news announcements appeared in DeGennaro & Shrieves (1997); Bomfim (2000); Brenner et al. (2009).
There are also different results concerning the duration of the effect of macroeconomic announcements. Most of the studies using intraday data find that the effect of
4
macroeconomic announcements on the intraday volatility is short lived, lasting from several minutes to one or two hours. On the other hand Evans & Lyons (2005) find that
currency markets are still absorbing news for several days after an announcement while
Brenner et al. (2009) estimate that volatility shifts do not offset each other over a threeday window around the announcement. In a related context, Evans & Lyons (2008)
find that arrival of macro news accounts for the 36% of the total daily variance, while
they point that macro news contribute far more to price variation in the long term than
previously thought.
3. The Proposed Models
3.1. A Flexible Threshold-GARCH model
We propose the following variation of the general model (1). Assume there are K
macroeconomic indices affecting the volatility process, and at time periods denoted by t∗i
there are announcements of index i, i = 1, . . . , K. Then, we propose the model
ǫt ∼ N (0, σ 2 Gt Ht )
ǫ2
= (1 − α1 − α2 ) + α1 2 t−1 + α2 Gt−1
σ Ht−1
J
K
i
Iij (t∗i )γij exp (−rij (t − t∗i )) + si 1 (t = t∗i − 1)
= 1+
yt | Ft−1 = µ + ǫt ,
Gt
Ht
i=1
(2)
(3)
(4)
j=1
Iij (t∗i ) = 1 cij ≤ Zit∗i < ci,j+1 .
(5)
The positive scalar σ 2 is considered as the global static error variance whereas the
GARCH process Gt has the form used in the Spline-GARCH model of Engle & Rangel
(2008) with E(Gt ) = 1 and positive parameters 0 < α1 + α2 < 1 . The threshold process
Ht > 0 describes the way that announcements affect the volatility. In particular, for
each index i = 1, . . . , K there are Ji regions. Iij (t∗i ) is a region-based indicator variable
indicating that region j, j = 1, . . . , Ji , affects volatility through a jump of size γij > −1
that occurred at time t∗i ≤ t, and through an exponential decay with rate rij > 0. The
announcement surprises of index i at time t∗i are expressed through a variable Zit∗i and
the degree of surprise is determined by threshold points cij , where ci,Ji := ∞. Finally, our
model specification is completed by allowing pre-announcement volatility jumps si > −1
occurring one time period before the announcements.
5
We call the above model formulation Flexible Threshold-GARCH model and we emphasize its rich flexibility: there are many models with varying number of parameters
specified by the number of regions Ji and the number of indices K that affect volatility.
Call m such a model, M the set of all models and use superscript m to denote the corresponding parameters within each model. Then, for each m ∈ M , the parameter vector
is
m m
m
m
.
θm = µm , σ m , α1m , α2m , K m , Jim , γijm , sm
i , rij , cij , i = 1, . . . , K , j = 1, . . . , Ji
Therefore, our target under a Bayesian inference setup is to estimate the posterior probability of each model m and the posterior densities of θm within each model m.
An interesting feature of the above formulation is the way Ht is modelled in (4).
Instead of using a functional form such as an exponential function that ensures positivity
of Ht , we prefer to directly use a formulation which results to an immediate sensible
interpretation of the parameters γij , rij and si with respect to the changes occurred to
σ 2 . A related additive formulation was suggested in Dellaportas et al. (2007). As a side
effect, the prior densities of γij and si are restricted to the regions that make Ht positive.
The effect of the specification (4) in the volatility process is depicted in Figure 1. There
are macroeconomic announcements at time points 3 and 10 of index 1, with different
degrees of surprise, and at time 7 of index 2. Assume that pre-announcement jumps
do not exist. The left panel depicts the three components that are added to produce
the process Ht at the right panel. Note that jump I12 corresponds to a decrease in the
volatility following the release of a macroeconomic index, since we allow positive and
negative jumps (γij ) occurring in the volatility.
Figure 1: The effect of explanatory variables in the threshold process.
6
3.2. A Spline-GARCH model
Another way to capture, in a non-linear fashion, the volatility shifts caused by news
announcements is to use a modification of the Spline-GARCH model of Engle & Rangel
(2008). We propose an amendment of this model by applying the exponential spline
function to the time instances in which announcements occur and not to equally spaced
time intervals as in Engle & Rangel (2008). The number of spline knots, denoted by K,
is assumed to be unknown. Hence, our proposed Spline-GARCH model is based on (2)
and (3), but (4) and (5) are replaced by
K
wi ((t − t∗i )+ )2 + γi Zit∗i + si 1 (t = t∗i − 1) ,
Ht = exp w0 t +
(6)
i=1
where (t − t∗i )+ = (t − t∗i ) if t > t∗i and zero otherwise. The parameter vector in this
model, for each model m ∈ M , is
m
θm = (µm , σ m , α1m , α2m , w0m , K m , wim , γim , sm
i , i = 1, . . . , K ) .
In this way when a variable Zit∗i is included in the model a new knot is generated,
causing shifts in the variance process. By performing model selection on the number of
variables included in the model we automatically determine the number of knots K m in
this quadratic spline. Thus, the more variables included in the model the more frequent
shifts take place, while their sharpness depends on wi . As in Engle & Rangel (2008), the
values of explanatory variables Zit∗i and the pre-announcement effect affect exponentially
the volatility process.
4. Inference
4.1. Priors
Our prior specification is non-informative, in the sense that vague proper priors are
proposed, but special care is taken to avoid issues of Lindley’s paradox (Bartlett, 1957)
and model identifiability. For example, as rij increases so that the exponent in (4)
approaches zero, Ht as well as the likelihood function remain constant. Thus, the integrability of the posterior density conditional on any given model requires the use of
proper priors. We performed the following (necessary) Bayesian data analysis exercise.
For the models used in the empirical analysis, we increased the standard deviations of
all prior densities by a factor of ten. We then examined the robustness of our results to
7
the prior dispersion by inspecting posterior model probabilities and posterior summaries
of interest such as volatility predictions. Although the posterior model probabilities do
change slightly, the predictive variance densities based on model averaging remain identical. This certifies that our prior densities are vague enough not to affect the posterior
summaries of interest. The details of these experiments can be found in Section 5.3.
The prior probabilities of the discrete densities on the model space and on threshold
points are set to be discrete uniforms over all possible models and distinct observed
values of announcement surprises respectively. Within each model, prior densities for the
parameters of the Flexible Threshold-GARCH model are chosen to be as follows. We
first apply transformations to the real line, gij = log(γij + 1), ςi = log(si + 1) and ρij =
log(rij ), i = 1, . . . , K, j = 1, . . . , Ji . We then place non-informative priors µ ∼ N (0, 1),
σ 2 ∼ IG(10−5 , 10−5 ), α1 , α2 ∼ U (0, 1) with α1 + α2 < 1, gij ∼ N (0, 0.42 ), ςi ∼ N (0, 0.42 ),
ρij ∼ N (0, 22 ). Recall from Section 3.1 that these prior densities are constrained so
that Ht > 0. For the Spline-GARCH model we use wi ∼ N (0, 0.0032 ), γi ∼ N (0, 32 ),
si ∼ N (0, 0.52 ).
4.2. The population reversible jump MCMC algorithm
In problems with complex multi-modal distributions standard vanilla MCMC samplers
may fail to efficiently move around the support of the target distribution. A way to deal
with these problems is to adopt Population-based MCMC methods (Geyer (1991); Gilks
et al. (1994); Liu et al. (2000); Liu (2001); Liang & Wong (2001); Jasra et al. (2007a)).
In this setting, MCMC operates by embedding the target density into a sequence of
ℓ = 1, . . . , L independent distributions obtained by simulating L parallel chains, whilst
allowing the chains to interact via various moves. Jasra et al. (2007b) proposed an
extension of these methods to transdimensional parameter spaces and we adopt ideas
from this paper to develop the proposed inference algorithm.
Our population reversible jump algorithm includes the following basic moves. An
exchange move is used to swap information between two adjacent (in terms of temperature) chains by exchanging all variables and associated parameters between them. In
a crossover move only a fraction of the variables with their associated parameters is exchanged between two randomly chosen chains. A mutation move is used to update a
chain according to a reversible jump step as suggested by Green (1995), which includes
addition, deletion, replacement, split and merge moves. The basic steps of the Flexible
8
Threshold-GARCH reversible jump algorithm that obtains samples (θm , m), m ∈ M , on
spaces of varying dimension follows. The specific details can be found in the Appendix.
The reversible jump Algorithm
• Initialize the chain and sweep over the following:
1. Randomly add, delete or replace an index variable.
2. For all index variables, randomly propose to split or merge
the current threshold points.
3. Update all remaining parameters in the current model through
adaptive random walk Metropolis Hastings kernels.
The above algorithm is enriched by applying a population algorithm. Denote by π be the
reversible jump invariant distribution with states (m, θm ). We construct two auxiliary
distributions πℓ ∝ π ζℓ , 1 = ζ1 > ζ2 > 0, with ζℓ denoting the inverse temperature
parameter in chain ℓ.
The population Algorithm
• Run 3 parallel Markov chains each one with target densities π ζℓ .
• Every 10 iterations choose randomly between
– an exchange move which changes the states (m, θm ) between two
randomly chosen chains which are adjacent in terms of
temperature.
– a crossover move which changes a randomly chosen subset of
variables included in m and its associated parameters θm
between two randomly chosen chains.
• In the rest of the iterations perform a mutation move which updates
the chains according to the reversible jump Algorithm.
The key intuition behind the population algorithm is that we want enough chains to
explore efficiently the model space but not too many so that the algorithm is expensive
in terms of CPU time. In our data the first (untempered) chain had addition/deletion
acceptance rates for different exchange rates between 4.7% and 5.4%, whereas in the third
9
tempered chain the corresponding proportions were 7% to 9% respectively. This led to
satisfactory mixing of the chain; see the MCMC diagnostics in Section 5.3. For a sufficient intercommunication between chains we followed Jasra etal. (2007b) and exchanged
information using either an exchange or a crossover move every ten iterations.
The temperature ladder is specified as ζℓ = z ℓ−1 , where the scalar z, 0 < z < 1, is
calibrated during the burn-in period of the algorithm as follows. We started from z = 0.8
and every 50 exchange move proposals we set z ′ = z + δ(0.5 − α), where δ > 0 is a
pre-specified sensitivity parameter and α is the acceptance rate of the exchange move
during the latest 50 exchange move sweeps. The constant δ = 0.1 is chosen so that the
exchange move is accepted about half of the time (Liu, 2001).
4.3. Prediction
We base all our predictions to Bayesian model averaging (see Raftery et al. (1997);
Liu & Maheu (2009)). The posterior distribution of a quantity ∆, which in our case
corresponds to the model averaging variance forecasts, given the data D and the models
m ∈ M , is given as
f (∆ | D) =
f (∆ | m, D)f (m | D),
m∈M
which is an average of the posterior predictive distribution under each model m,
f (∆ | m, D) = f (∆ | θm , m, D)f (θm | m, D)dθm
weighted by the posterior model probabilities f (m | D).
In our context each state of the Markov chain corresponds to a model m and associated
parameters θm , so each calculated volatility is being averaged across both model and
parameter posterior densities. To avoid confusion, we call the Flexible Threshold-GARCH
and Spline-GARCH as ’model specifications’. Model averaging takes place with respect
to models m within a model specification.
In the empirical analysis we produce one day ahead out-of-sample forecasts based
on monthly updated in-the-sample parameter estimates. This means that we reran the
MCMC algorithm twelve times to produce daily out-of-sample forecasts for a year. Since
we deal with scheduled announcements, when forming the out-of-sample forecasts the
day of occurrence of each announcement is known a priori, thus we can evaluate the preannouncement effect. For the days following the release, when the announcement outcome
10
is known, we can directly perform predictions based on the estimated predictive density
conditional on the announced surprise. In the exact date of the release the outcome is
not known a priori, and therefore we cannot predict Iij (t∗i ), thus we propose two ways to
perform forecasting. One, adopted in our main analysis, is to use an ‘empirical Bayes’
type estimator that estimates the probability of the not yet observed Iij (t∗i ) with its
past (in-the-sample) sample average. The second estimator relies on the hypothesis that
the news announcement will lie close to the ‘Bloomberg consensus’, so that Ii1 (t∗i ) =
1 ci1 ≤ Zit∗i < ci,2 . In our empirical study it turned out that the former estimator is
slightly better than the latter; but we suggest that choice between them should be based
on the degree of belief one has to the first or the second assumption underlying their
choices.
To evaluate the predictive performance of each model specification we rely first on
robust loss functions evaluated based on a true volatility proxy. We adopt the MSE and
QLIKE loss functions defined as
T
2
1 2
σ̂t − ht
MSE =
T t=1
QLIKE =
T
σ̂ 2
1
log ht + t
T t=1
ht
where ht denotes variance forecast for day t and the proxy σ̂ 2 is the realized variance obtained using 1-minute high frequency data; see Andersen & Bollerslev (1998a); BarndorffNielsen & Shephard (2002, 2004). Patton (2011) notes that MSE and QLIKE loss functions are robust in the sense that the ranking of any two volatility forecasts is the same
whether it is done using the true conditional variance or some conditionally unbiased
volatility proxy.
A second criterion used to evaluate the predictive ability of the models is based on
the optimal pools methodology of Geweke & Amisano (2011), also employed by Jensen &
Maheu (2010). Model comparison under the optimal prediction pool methodology mixes
the predictive densities of each model specification and sets the mixing weights equal to
the value that maximizes the log pooled predictive score. The larger the mixing weight
the more important the model specification is in predicting future outcomes. Denote
with f (yt |Ft−1 , A) the probability density for the exchange rate series yt under a model
specification A and the available information set Ft−1 . The respective log-predictive score
11
is given as
LS =
T
log(f (yt |Ft−1 , A)).
t=1
Then, for a pool model specifications A1 , ..., An , we seek to find the weights wi∗ that
maximize the log predictive score function
wi∗ = arg max
wi
T
t=1
log
n
i=1
wi f (yt |Ft−1 , Ai ) .
5. Empirical study
5.1. The data
Our data set consists of 3131 daily observations from 1/1/2002 (i.e. since the introduction of the euro currency) up to 31/12/2013 of the Euro-dollar (EURUSD), British
pound-dollar (GBPUSD) and Dollar-Swiss franc (USDCHF) exchange rates. The data
were obtained from Bloomberg. The diagnostics with respect to the macroeconomic announcements effects on the volatility of exchange rates are obtained on basis of the full
sample. The predictive ability of the model specifications is evaluated for 261 observations
in 2013 by forming one step ahead out-of-sample forecasts and estimating the parameters
and model probabilities on a rolling window basis, as described in Section 4.3. For this
exercise 1-minute high frequency data for the year 2013 (approximately 370,000 observations) over the three exchange rates were employed to calculate the required realized
volatilities.
We used surprises of 15 monthly U.S. scheduled macroeconomic announcements, that
we believed that they may affect the volatility process and/or have been found to explain
volatility fluctuation by previous empirical studies. In the full sample (2002-2013) there
are 144 observations for each macroeconomic announcement. The surprises are defined as
the absolute percentage difference of the realised minus the Bloomberg consensus reported
values. The macroeconomic announcements considered are presented in Table 1.
Under the Flexible Threshold-GARCH specification, model selection takes place with
respect to the fifteen macroeconomic announcements and to the number and location
of threshold points. We report in the last column of Table 1 the maximum number
of regions, which coincides with the number of different (distinct) observations of each
macroeconomic announcement.
12
Table 1: List of the macroeconomic announcements (K) and surprises (Zit∗i ) descriptive statistics; minimum values are equal to zero.
Macroeconomic announcement
Mean
St. dev.
Median
Maximum
Maximum number of regions (Ji )
(×100)
(×100)
(×100)
(×100)
(different observations out of 144)
1
GDP QoQ (Annualized)
0.33
0.34
0.20
1.70
15
2
Industrial Production
0.29
0.26
0.25
2.00
14
3
Durable Goods Orders
1.77
1.57
1.35
8.20
44
4
Wholesale Inventories
0.41
0.32
0.30
1.80
15
5
Advance Retail Sales
0.39
0.35
0.30
1.80
14
6
Housing Starts
5.47
4.44
4.58
29.56
140
7
ISM Manufacturing
2.99
2.56
2.31
14.23
130
8
ISM Non-Manufacturing
3.72
2.95
3.29
15.10
138
9
Leading Indicators
0.14
0.13
0.10
0.50
6
10
Consumer Confidence
6.01
6.11
4.00
31.99
135
11
Consumer Price Index (MoM)
0.10
0.09
0.10
0.40
5
12
Producer Price Index (MoM)
0.37
0.33
0.30
1.70
15
13
Trade Balance
5.84
5.07
4.74
27.78
139
14
Unemployment Rate
0.12
0.11
0.10
0.50
6
15
Personal Income
0.19
0.27
0.10
1.80
13
5.2. Results
We first report the results with respect to the effect of the macroeconomic announcements on the volatility of the three exchange rates under the Flexible Threshold-GARCH
and Spline-GARCH specifications. The four models with the highest estimated posterior probability within each specification and exchange rate are presented in Table 2.
Although most of the macroeconomic announcements are included in the four highest
posterior density models of all specifications, we observe that the Consumer Confidence
(no. 10) is present in most of the models across all specifications and exchange rates and
that Unemployment Rate (no. 14) is present in the four best models under the Flexible
Threshold-GARCH specification in both EURUSD and USDCHF currencies.
A quantity usually adopted to help identifying important regressors in high dimensional regression type problems is the marginal probability of inclusion (see Barbieri &
Berger, 2004), calculated as the percentage of times a variable is observed in the transdimensional MCMC sample. Denoting with fˆ(m | D) the estimated posterior probability
of a model m, the probability of inclusion of macroeconomic announcement i is given as
pi =
fˆ(m | D),
m∈M i
13
Table 2: The four models with the highest posterior probability for each exchange rate and their estimated
posterior probabilities under the Flexible Threshold-GARCH and Spline-GARCH specifications.
Flexible Threshold-GARCH
Spline-GARCH
EURUSD
Model
Variables
Probability
No. of parameters*
Variables
Probability
No. of parameters
First
6,10,13,14
0.0253
402
7
0.1860
8
Second
4,10,11,13,14
0.0238
301
7,10
0.1483
11
Third
10,11,13,14
0.0195
290
6,7
0.1258
11
Fourth
10,11,14
0.0167
153
6,7,10
0.0536
14
GBPUSD
First
7,10
0.1035
270
3,10
0.2601
11
Second
3,7,10
0.0639
323
10
0.2502
8
Third
7,8,10
0.0281
423
3,4,10
0.1671
14
Fourth
8,10
0.0264
244
4,10
0.0921
11
USDCHF
First
6,8,10,14
0.0945
392
3,4,10
0.2161
14
Second
6,7,8,10,14
0.0701
543
10
0.0911
8
Third
6,8,10,14,15
0.0362
425
7,10
0.0646
11
Fourth
6,8,10,11,14
0.0241
419
3,10
0.0597
11
*The number of parameters under the threshold point combination with the highest posterior probability is reported.
where M i denotes all the models visited by the algorithm in which macroeconomic announcement i is present.
Table 3 lists the macroeconomic announcements according to their marginal probability of inclusion in the two model specifications. The results are in accordance to those
analysed on basis of the four most probable models of Table 2. It is interesting to notice that different announcements affect each exchange rate. For example Unemployment
rate, found as an important announcement in previous studies (Ederington & Lee, 1993;
Andersen & Bollerslev, 1998b; Nikkinen & Sahlström, 2001), has marginal probability
of inclusion 100% in EURUSD and USDCHF (Flexible Threshold-GARCH specification), but below 9% for GBPUSD. However, we note that Consumer Confidence has high
probability of inclusion in all exchange rates. Another interesting observation is that
announcements related to actual growth such as GDP, Industrial Production and Personal Income display low probability of inclusion (below 30%) under both specifications
for all currencies. On the other hand sample surveys for the condition of the economy,
such as ISM Manufacturing, ISM Non-Manufacturing and Consumer Confidence seem to
affect more the volatility of exchange rates. Presumably, this can be attributed to the
14
Table 3: Marginal probability of inclusion of macroeconomic announcements
Macroeconomic announcement
Flexible Threshold-GARCH
Spline-GARCH
EURUSD
GBPUSD
USDCHF
EURUSD
GBPUSD
USDCHF
1
GDP QoQ (Annualized)
0.1398
0.1264
0.1034
0.0688
0.0331
0.0272
2
Industrial Production
0.2525
0.0803
0.0596
0.0474
0.0210
0.0311
3
Durable Goods Orders
0.0349
0.3581
0.1961
0.0245
0.5851
0.5134
4
Wholesale Inventories
0.5648
0.2021
0.0753
0.0204
0.3262
0.4945
5
Advance Retail Sales
0.1628
0.1761
0.0558
0.0345
0.0308
0.0461
6
Housing Starts
0.3817
0.1312
0.7811
0.3130
0.0066
0.1439
7
ISM Manufacturing
0.3917
0.7699
0.5246
0.9393
0.0326
0.4247
8
ISM Non-Manufacturing
0.1400
0.2710
1.0000
0.0395
0.0199
0.0157
9
Leading Indicators
0.2412
0.1151
0.2330
0.0813
0.0195
0.0757
10
Consumer Confidence
0.8311
1.0000
0.7555
0.4167
0.9999
0.9293
11
Consumer Price Index (MoM)
0.5086
0.1085
0.2919
0.0430
0.0321
0.0374
12
Producer Price Index (MoM)
0.0703
0.0658
0.1406
0.0168
0.0160
0.0156
13
Trade Balance
0.4673
0.1072
0.0699
0.0048
0.0043
0.0209
14
Unemployment Rate
1.0000
0.0885
1.0000
0.0677
0.0471
0.0314
15
Personal Income
0.1088
0.0765
0.2829
0.1403
0.0154
0.0503
The two announcements with the highest marginal probability of inclusion for each case are denoted with bold.
fact that changes in actual growth have been mainly absorbed by results of indices that
provide advance signs for the condition of the economy, such as confidence and sentiment
reports, or other announcements with this characteristic such as Durable Goods Orders
or Housing Starts, the latter being also related to the financial crisis of 2008.
It is interesting to investigate whether days with certain macroecoeconomic announcements provide better predictive ability when compared with non-announcements days.
We examined the one-day-ahead predictive likelihood of the model specifications as in
Jensen & Maheu (2013), but we did not observe any significant difference in days with
largest one-day-ahead predicitive likelihoods in any announcement. As an example, see
Figure 2 in which the Consumer Confidence and Unemployment rate announcements are
depicted together with predictive likelihoods.
Table 4 presents summary statistics for the posterior densities of pre-announcement
effects under the highest posterior probability model of each specification. There is a
sign agreement in relevant specifications that represents a drop of volatility before the
announcement of Consumer Confidence. It is evident that the pre-announcement effect
depends on the announcement and the related exchange rate and can be positive or negative. To this respect we observe an increase in the volatility of EURUSD and USDCHF
15
5
4.5
4
3.5
3
2.5
2
1.5
01/2013
04/2013
07/2013
10/2013
01/2014
Figure 2: One-day-ahead predictive likelihood under the Flexible Threshold-GARCH model; EURUSD
currency. Circles represent the announcement days of the Consumer Confidence index and squares of
the Unemployment rate.
the day before the announcement of Unemployment rate.
Figures 3 and 4 depict estimates of the posterior means of jump sizes γij and decay
rates rij across all observed threshold points for the model with the highest posterior
probability of Table 2, under the Flexible Threshold-GARCH specification for EURUSD
exchange rate. It seems that the hypothesis that the larger the degree of surprise (defined
on basis of the threshold points cij ), the larger the size of jump (γij ) in the volatility is
not confirmed. As noted in several event studies (Jones et al. (1998); Hautsch & Hess
(2002); Laakkonen (2004); Andersen et al. (2007b)) since the effect of an announcement
is dependent to past or anticipated announcements, it is hard to isolate it in a changing
macroeconomic environment.
However certain interesting features are observed. The 197 estimated size of jump
parameters range from -0.66 (66% decrease in the volatility at the day of announcement),
up to 2.02 (202% increase). The 90% of these parameters range from -0.50 to 1.17.
Respectively the 90% of rates of decay rij estimates range from 0.04 up to 29.4. The
lower the value of the rate of decay the longer the impact of a given jump in volatility.
Half of them lie below the unity, which corresponds to an impact of about 36.8% of
the original jump at the day following the announcement, 13.5% two days after the
16
Table 4: Pre-announcement effect (ςi ) on the volatility based on models with the highest posterior
probability; posterior standard deviations are in brackets.
Macroeconomic announcement
Flexible Threshold-GARCH
EURUSD
3
Durable Goods Orders
4
Wholesale Inventories
6
Housing Starts
7
ISM Manufacturing
8
ISM Non-Manufacturing
GBPUSD
USDCHF
Spline-GARCH
EURUSD
GBPUSD
USDCHF
-0.161 [0.130]
-0.395 [0.115]
-0.222 [0.116]
0.373 [0.023]
0.0001 [0.0004]
-0.088 [0.080]
0.325 [0.127]
0.149 [0.0003]
10
Consumer Confidence
-0.340 [0.007]
13
Trade Balance
-0.008 [0.010]
14
Unemployment Rate
0.447 [0.012]
-0.664 [0.086]
-0.337 [0.014]
-0.661 [0.125]
-0.377 [0.128]
0.111 [0.012]
announcement and 5% three days after an announcement. The 25% of estimates lie
below the value 0.23, which corresponds to an effect with rather long memory, having
10% of its initial magnitude even 10 days after an announcement. These results are in
line with the findings of Evans & Lyons (2005, 2008) and Brenner et al. (2009), that found
macroeconomic releases to affect volatility several days following the announcement.
We now turn to forecasting aspects of our specifications. We stress that forecasts are
not based on one model but on model averaging. This fact allows borrowing strength between parameter estimates across models and therefore obtain robust volatility forecasts.
For comparative reasons besides the Flexible-Threshold GARCH and Spline-GARCH, we
report the results under a typical GARCH(1,1) specification.
Figure 5 depicts in-the-sample (2002-2012) variance estimates and out-of-sample (2013)
variance predictions under each model specification for the EURUSD currency. These
can be visually compared with squared residuals and realized volatility respectively. It is
clear that specifications which use information from the macroeconomic announcements
are more spiky than the usual smooth GARCH estimates. This is in line with the findings
of various empirical studies (DeGennaro & Shrieves, 1997; Andersen & Bollerslev, 1998b;
Bollerslev et al., 2000; Bauwens et al., 2005), in which the effects of news on the volatility
are found to be stronger than that of GARCH effects.
In Figure 6 the out-of-sample volatility predictions are displayed for the three exchange
rates under the Flexible Threshold-GARCH specification. The ‘empirical Bayes’ and
‘Bloomberg consensus’ estimators are nearly indistinguisable. In Table 5 we present the
17
Housing Starts
Consumer Confidence
2
1
1.5
0.5
γij
1
γij
0.5
0
0
−0.5
−0.5
0
0.05
0.1
0.15
0
0.05
0.1
0.15
cij
cij
Trade Balance
0.2
0.25
0.3
Unemployment Rate
0.8
1.5
0.6
1
γ
ij
0.4
γ
0.5
ij
0.2
0
0
−0.2
−0.4
−0.5
0
0.05
0.1
cij
0.15
0.2
0
1
2
3
cij
4
5
−3
x 10
Figure 3: Posterior means of jump size parameters γij against announcement surprises threshold points,
under the first best model of the Flexible Threshold-GARCH specification for the EURUSD exchange
rate.
forecast errors, log-predictive scores and optimal pool weights for all exchange rates under all specifications. We observe that under all criteria the Flexible Threshold-GARCH
specification dominates over the Spline-GARCH and GARCH specifications for the EURUSD and USDCHF exchange rates. For GBPUSD the Flexible Threshold-GARCH and
Spline-GARCH specifications perform approximately equally well when evaluated on basis of the QLIKE loss function and the log-predictive score. Flexible Threshold-GARCH
displays lower Mean Square Errors (MSEs) than Spline-GARCH, but the latter has higher
optimal pool weight. In view of that, one could use both specifications to predict GBPUSD volatility on a model averaging basis. Under all criteria the two specifications
that use information from news announcements provide better forecasts compared to a
typical GARCH(1,1) specification for all three exchange rates. These differences in the
forecasting performance based on MSE and QLIKE among specifications do not change
if we consider only announcement days or if we exclude the announcement days.
18
Table 5: Forecast errors, log-predictive scores and optimal pool weights. Bold denote the specification
that provides better forecasts under the respective criterion; results based on ‘empirical Bayes’ estimators;
‘Bloomberg consensus’ estimators in brackets (see Section 4.3).
Model
Flexible Threshold-GARCH
Spline-GARCH
GARCH
EURUSD
In-the-sample errors based on squared residuals
MSE
(×10−10 )
QLIKE
49.5646
51.2658
51.7309
-9.2718
-9.2330
-9.2214
Out-of-sample errors based on squared residuals
MSE (×10−10 )
QLIKE
10.7935 [10.8402]
10.9648 [10.9693]
11.0077
-9.7958 [-9.7908]
-9.7759 [-9.7755]
-9.7724
Out-of-sample errors based on realized volatility
MSE (×10−10 )
QLIKE
1.9276 [1.9330]
2.1548 [2.1613]
2.1743
-9.5981 [-9.5958]
-9.5843 [-9.5835]
-9.5839
Out-of-sample log-predictive score
LS (×103 )
1.0385 [1.0379]
1.0359 [1.0358]
1.0355
Out-of-sample optimal pool weights
wi∗
0.0000
1.0000
0.0000
GBPUSD
In-the-sample errors based on squared residuals
MSE (×10−10 )
QLIKE
44.8740
45.8647
46.0003
-9.4958
-9.4660
-9.4519
Out-of-sample errors based on squared residuals
MSE (×10−10 )
QLIKE
12.2226 [12.2605]
12.2693 [12.2563]
12.3659
-9.7226 [-9.7199]
-9.7230 [-9.7241]
-9.7142
Out-of-sample errors based on realized volatility
MSE (×10−10 )
QLIKE
1.5697 [1.5774]
1.5739 [1.5709]
1.6054
-9.6897 [-9.6885]
-9.6897 [-9.6900]
-9.6874
Out-of-sample log-predictive score
LS (×103 )
1.0290 [1.0286]
1.0290 [1.0291]
1.0278
Out-of-sample optimal pool weights
wi∗
0.3479
0.5461
0.1060
USDCHF
In-the-sample errors based on squared residuals
MSE (×10−10 )
QLIKE
314.3459
341.6850
346.6429
-9.1037
-9.0463
-9.0308
Out-of-sample errors based on squared residuals
MSE (×10−10 )
18.9276 [19.0211]
19.0142 [19.0284]
19.0126
QLIKE
-9.4724 [-9.4682]
-9.4655 [-9.4651]
-9.4640
Out-of-sample errors based on realized volatility
MSE (×10−10 )
QLIKE
4.8349 [4.8923]
4.9494 [4.9245]
5.0590
-9.2579 [-9.2521]
-9.2536 [-9.2533]
-9.2502
Out-of-sample log-predictive score
LS (×103 )
0.9963 [0.9957]
0.9954 [0.9954]
0.9952
Out-of-sample optimal pool weights
wi∗
0.0000
1.0000
19
0.0000
Housing Starts
rij
Consumer Confidence
40
40
30
30
rij
20
10
0
20
10
0
0.05
0.1
c
0
0.15
0
0.05
0.1
0.15
c
ij
0.2
0.25
0.3
ij
Trade Balance
Unemployment Rate
40
4
30
3
r
ij
r
ij
20
2
10
0
1
0
0.05
0.1
cij
0.15
0
0.2
0
1
2
3
cij
4
5
−3
x 10
Figure 4: Posterior means of decay rates rij against announcement surprises threshold points, under the
first best model of the Flexible Threshold-GARCH specification for the EURUSD exchange rate.
5.3. MCMC diagnostics and sensitivity analysis
Table 6 presents the associated acceptance rates and CPU time required under each
specification for all exchange rates. As expected, the two auxiliary tempered chains
display higher addition, deletion and replacement acceptance rates. The calibration of
the temperature ladder seems to work well since the exchange move is accepted about half
of the times. Finally, the adaptive MCMC works rather well with acceptance rates close
to the targeted 0.234 in all cases. Figure 7 presents posterior model probabilities and
associated traces for the Flexible Threshold-GARCH specification under all exchange
rates, indicating that the mixing of the MCMC chain was good, with models varying
between one and thirteen variables.
A proper Bayesian data analysis requires a sensitivity analysis on the effect of prior
densities to the posteriors. As reported in Section 4.1, this is achieved by increasing
the standard deviation of all parameter prior densities by factors of two, five and ten,
and inspecting the changes in the resulting posterior densities. Table 7 demonstrates
20
Table 6: MCMC diagnostics
EURUSD
GBPUSD
USDCHF
Flexible Threshold-GARCH
CPU time (hours)
138.6
59.3
99.2
1st chain
2nd chain
3rd chain
1st chain
2nd chain
3rd chain
1st chain
2nd chain
3rd chain
Addition
0.0539
0.0676
0.0879
0.0513
0.0665
0.0794
0.0478
0.0576
0.0720
Deletion
0.0522
0.0700
0.0877
0.0505
0.0656
0.0809
0.0466
0.0589
0.0710
Replacement
0.0059
0.0096
0.0160
0.0048
0.0083
0.0115
0.0042
0.0061
0.0099
Split
0.4114
0.3990
0.3943
0.4851
0.4933
0.5097
0.4398
0.4411
0.4367
Merge
0.4186
0.4070
0.4047
0.4819
0.4908
0.5023
0.4521
0.4546
0.4562
0.2347
0.2350
0.2339
0.2326
0.2348
0.2341
0.2332
0.2326
0.2362
0.2323
0.2338
0.2376
0.2345
0.2325
0.2346
0.2361
0.2340
0.2327
σm
0.2351
0.2357
0.2338
0.2338
0.2343
0.2301
0.2348
0.2340
0.2329
Update sm
i
0.2351
0.2309
0.2343
0.2350
0.2327
0.2278
0.2313
0.2336
0.2332
m
γij
m
rij
0.2339
0.2347
0.2324
0.2371
0.2283
0.2320
0.2341
0.2333
0.2342
0.2367
0.2354
0.2344
0.2311
0.2349
0.2282
0.2302
0.2354
0.2232
Acceptance rate
Update
µm
m
Update αm
1 , α2
Update
Update
Update
Exchange
0.5054
Crossover
0.1798
0.5116
0.5031
0.2068
0.1232
Spline-GARCH
CPU time (hours)
132.6
133.9
128.6
1st chain
2nd chain
3rd chain
1st chain
2nd chain
3rd chain
1st chain
2nd chain
3rd chain
Addition
0.0360
0.0533
0.0682
0.0326
0.0471
0.0653
0.0342
0.0467
0.0595
Deletion
0.0371
0.0554
0.0900
0.0327
0.0497
0.0724
0.0344
0.0526
0.0815
Replacement
0.0036
0.0076
0.0146
0.0025
0.0043
0.0094
0.0034
0.0063
0.0118
0.2349
0.2353
0.2330
0.2336
0.2341
0.2346
0.2355
0.2355
0.2336
0.2314
0.2332
0.2332
0.2312
0.2319
0.2334
0.2333
0.2316
0.2363
σm
0.2330
0.2333
0.2331
0.2344
0.2346
0.2332
0.2345
0.2339
0.2358
Update w0m
0.2339
0.2359
0.2347
0.2320
0.2341
0.2349
0.2378
0.2314
0.2352
wim
γim
sm
i
0.2356
0.2336
0.2307
0.2337
0.2334
0.2402
0.2364
0.2318
0.2382
0.2331
0.2333
0.2370
0.2353
0.2353
0.2302
0.2320
0.2430
0.2311
0.2362
0.2321
0.2364
0.2321
0.2354
0.2341
0.2348
0.2330
0.2338
Acceptance rate
Update
µm
m
Update αm
1 , α2
Update
Update
Update
Update
Exchange
0.4964
0.4936
0.5099
Crossover
0.2751
0.2991
0.2300
Note: Results based on a single run on the full sample. All algorithms ran for 100.000 iterations with 10.000 iterations burn-in on
an Intel Corei7 CPU. For the out-of-sample forecasts we ran the algorithms 12 times on a rolling window basis (see Section 4.3).
21
−4
−5
Flexible Threshold−GARCH (in−the−sample)
x 10
x 10
Flexible Threshold−GARCH (out−of−sample)
8
4
6
4
2
2
0
2002
2003
2004
−4
2005
2006
2008
2009
2010
2011
2012
01/13 02/13 03/13 04/13 05/13 06/13 07/13 08/13 09/13 10/13 11/13 12/13
−5
Spline−GARCH (in−the−sample)
x 10
x 10
Spline−GARCH (out−of−sample)
8
4
6
4
2
2
0
2002
2003
2004
2005
−4
2006
2008
2009
2010
2011
2012
01/13 02/13 03/13 04/13 05/13 06/13 07/13 08/13 09/13 10/13 11/13 12/13
−5
GARCH (in−the−sample)
x 10
x 10
GARCH (out−of−sample)
8
4
6
4
2
2
0
2002
2003
2004
2005
2006
2008
2009
2010
2011
2012
01/13 02/13 03/13 04/13 05/13 06/13 07/13 08/13 09/13 10/13 11/13 12/13
Figure 5: Estimated EURUSD volatilities in-the-sample and out-of-sample. Solid: volatility estimates
(empirical Bayes estimator), dotted: squared residuals (left panel) and realized volatility (right panel)
that in the Flexible Threshold-GARCH specification, only marginal changes occur in the
associated forecast errors. Moreover, the out-of-sample volatility forecasts remain almost
identical under the two extreme prior specifications, see Figure 9. Similar results hold
for the Spline-GARCH specification. In Table 7 we also report results based on varying
MCMC samples that provide evidence that convergence has been achieved with sample
size 100, 000. Plots of volatility forecasts (not reported here) illustrate that the forecasts
are visually identical between 5, 000 and 100, 000 MCMC sample sizes.
6. Conclusions
We presented a new class of flexible threshold models for predicting volatilities of
exchange rates that incorporate information from scheduled news announcements. For
the dataset analysed, we have found strong evidence that suggests the use of such extra
information can enrich the GARCH structures commonly used.
22
−5
EURUSD
x 10
8
6
4
2
01/13
02/13
03/13
04/13
05/13
−5
06/13
07/13
08/13
09/13
10/13
11/13
12/13
08/13
09/13
10/13
11/13
12/13
08/13
09/13
10/13
11/13
12/13
GBPUSD
x 10
6
4
2
01/13
02/13
03/13
04/13
05/13
−5
06/13
07/13
USDCHF
x 10
10
8
6
4
2
01/13
02/13
03/13
04/13
05/13
06/13
07/13
Figure 6: Out-of-sample volatility estimates under the Flexible Threshold GARCH specification for the
three exchange rates. Solid: volatility estimates based on ‘empirical Bayes’ estimator, dashed: volatility
estimates based on ‘Bloomberg consensus’ estimator, dotted: realized volatility
The use of our proposed threshold model is intuitively appealing, although it may
suffer from the usual problem of over-fitting. For this purpose, we have advocated the
use of model averaging prediction that exploits modern MCMC strategies that sample
in a transdimensional space. Our detailed empirical analysis provided evidence that our
model predicts rather well when compared with conditional volatility models and other
nonparametric formulations such as splines.
Our statistical framework can be easily extended to the multivariate case, where
given the variance estimates, once can employ a Dynamic Conditional Correlation (Engle,
2002) or a Cholesky-type parametrization (Dellaportas & Pourahmadi, 2012) to obtain
covariance estimates; see Petralias (2010) for illustrations of such multivariate extensions.
23
No. of variables histogram
No. of variables sample
EURUSD
15
No. of variables
Posterior probability
EURUSD
0.3
0.2
0.1
0
10
5
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
No. of variables
2
10
4
x 10
GBPUSD
0.3
0.2
0.1
0
10
5
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
No. of variables
2
USDCHF
4
6
Iterations
8
10
4
x 10
USDCHF
15
No. of variables
Posterior probability
8
15
No. of variables
Posterior probability
GBPUSD
4
6
Iterations
0.3
0.2
0.1
0
10
5
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
No. of variables
2
4
6
Iterations
8
10
4
x 10
Figure 7: Estimated posterior probability for the number of variables in the model under the Flexible
Threshold GARCH specification (left panel) and the number of variables across iterations (right panel).
Acknowledgments
We are grateful to Omiros Papaspiliopoulos for many useful comments that greatly
improved a previous version of the paper. The second author was supported by the
European Union, Seventh Framework Programme FP7/2007-2013 under grant agreement
SYRTO-SSH-2012-320270.
References
Andersen, T. G., & Bollerslev, T. (1998a). Answering the skeptics: Yes, standard volatility
models do provide accurate forecasts. International Economic Review , 39 , 885–905.
Andersen, T. G., & Bollerslev, T. (1998b). Deutsche mark-dollar volatility: Intraday activity
24
−5
x 10
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
01/13
02/13
03/13
04/13
05/13
06/13
07/13
08/13
09/13
10/13
11/13
12/13
Figure 8: Out-of-sample EURUSD volatility forecasts of the Flexible Threshold-GARCH specification
under different MCMC samples; solid: 100,000 sample, dashed: 10,000 sample; dotted: 5,000 sample. A
burn-in period of 10,000 is used in all cases.
Table 7: Forecast errors of the Flexible Threshold-GARCH specification under different priors and
MCMC samples; displayed are the out-of-sample errors based on EURUSD realized volatility
MSE (×10−10 )
QLIKE
prior s.d. ×1, sample 100,000
1.9276
-9.5981
prior s.d. ×2, sample 100,000
1.9536
-9.5966
prior s.d. ×5, sample 100,000
1.9456
-9.5958
prior s.d. ×10, sample 100,000
1.9303
-9.5966
prior s.d. ×1, sample 5,000
1.9238
-9.5989
prior s.d. ×1, sample 10,000
1.9391
-9.5973
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Appendix: The Population Reversible Jump algorithm
Reversible jump Algorithm
For parameters θm associated with model m ∈ M , the goal is to sample from the
posterior distribution
f (θm , m | D) ∝ f (D | θm , m)f (θm | m)f (m),
where D represents the data based on T time points, f (D | θm ) is the likelihood of model
m, and f (θm | m) is the prior density of the parameters θm conditional on model m. The
likelihood based on model (2)-(5) is
f (D | θm , m) = (2πσ 2 )−T /2 |V |−1/2 exp −
ǫ′ V −1 ǫ
2σ 2
,
where ǫ = (ǫ1 , . . . ǫT )′ and V is a diagonal matrix with elements Gt Ht . The specification
of the priors is given in Section 4.1, while the general structure of the algorithm can be
found in Section 4.2.
Assume that at the current state (θm , m) of the Markov chain there exist K m index
+
−
, qm
variables. We select an addition, deletion or replacement move with probabilities qm
0
and qm
respectively. We used equal probabilities taking care to change them accordingly
when there are zero (K m = 0) or the maximum number of variables (K m = K max ) in the
current model m, with K max = 15 in the empirical application.
The addition move proceeds as follows. First we propose to move to model m′ by
adding an index variable not present in the current model m chosen with probability
j(m, m′ ) =
(K max
1
q+ .
− K m) m
′
The proposed parameters θm
′ = (cij , gij , ρij , ςi ) are sampled from the proposal density
′
′
′
′
q(θm
′ | θm , m, m ) which is taken to be equal to the prior distribution f (θm′ | m ) defined
in Section 4.1. The deletion move proceeds in a similar way, where in the replacement
move we select one variable present at the current model to be replaced with one not
present with probability
j(m, m′ ) =
1
K m (K max
− K m)
0
.
qm
′
′
In all cases the proposed move to (θm
′ , m ) is accepted with probability
′
′
′
′
′
′
f (D | θm
′ , m )f (θm′ | m )f (m )j(m , m)
α = min 1,
′
′
′
f (D | θm , m)f (θm | m)f (m)q(θm
′ | θm , m, m )j(m, m )
30
.
In the split move, for every i we propose to increase the number of threshold points
cij by one with uniform probability
q(c′ij | cij ) =
Jimax
1
,
− Jim
where Jimax the total number of possible threshold points, taken equal to the number of
distinct observations of each variable Zit∗i . To achieve better mixing of the Markov chain,
the parameters gij and ρij are drawn from a density that dependents on the values of the
current state of the chain. We draw u1 and u2 from the prior distributions of gij and ρij
respectively and set
g´ij
= gi,j−1 + u1
gi,j−1
´
= gi,j−1 − u1
ρ´ij
= ρi,j−1 + u2
ρi,j−1
´
= ρi,j−1 − u2 .
The move is then accepted with probability
′
′
′
′
′
′
M
f (D | θm
′ , m )f (θm′ | m )f (m )q(cij | cij )qc′
ij
2
|J| ,
α = min 1,
f (D | θm , m)f (θm | m)f (m)q(c′ij | cij )q(u1 | θm , m)q(u2 | θm , m)qcSij
where qcSij , qcMij are the probabilities to perform a split or merge move, which are set equal
(1/2) with appropriate changes when Jim = 1 or Jim = Jimax and the Jacobian term for
the gij parameters is
|J| =
∂(gi,j−1
´ , g´ij )
=
∂(gi,j−1 , u1 )
1 −1
1
= 2.
1
Since the same Jacobian is obtained for the ρij parameters, the term |J|2 is included
in the acceptance ratio. In the merge move we randomly select a threshold point to be
deleted taking care so that the first threshold point (j = 1) is always included in cij .
Then we set the new parameters as
gi,j−1 =
(g´ij + gi,j−1
´ )/2
ρi,j−1 = (ρ´ij + ρi,j−1
´ )/2.
When performing random walk metropolis to update {µ, σ, α1 , α2 , gij , ςi , ρij }, we found
useful to apply adaptive Metropolis methods, enforcing the diminishing adaptive condition which ensures asymptotic convergence and ergodicity and the bounded convergence
condition so that the convergence time of the kernel is bounded in probability, as described in Roberts & Rosenthal (2009). To this respect the scale of the proposal for each
31
parameter (denoted as τi for a parameter i) is adaptively updated as follows: Given a
starting value for the proposal scale (acquired through a pilot run), set a global maximum
(and minimum) bound, M = max(τi ) (in our case we set M = 10e+5 for all parameters),
so that − log(M ) ≤ log(τi ) ≤ log(M ). Then every b batches of 50 swaps add (subtract) to
the logarithm of the proposal scale a quantity, log(τi ) + δ(b), where δ(b) = min(b0 , b−1/2 )
and b0 = 0.1, if the acceptance rate is higher (lower) than 0.234. The necessary condition
Ht > 0 resulted to less than 5% rejection of the proposed values of gij , ςi and ρij .
Population Algorithm
In the exchange move we randomly select two adjacent, in terms of temperature,
ℓ
chains, and propose to swap their values. Denote with xℓ the state (mℓ , θm
) of chain ℓ,
ℓ
with target distribution πℓ (xℓ ) and πℓ ∝ π ζℓ (see Section 4.2). The Metropolis-Hastings
acceptance ratio used to swap the states of chains 1 and 2 is of the form
α = min 1,
π1 (x2 )π2 (x1 )
π1 (x1 )π2 (x2 )
.
We choose not to temper the prior distributions, thus they cancel out in the acceptance
ratio.
The crossover move takes a fraction of the variables, along with their associated parameters present in the current chain and places them in another randomly selected chain.
This move proceeds as follows. Select two random chains (not necessarily adjacent), say
1 and 2, with probability τ1 and τ2|1 respectively. Then
if both chains have the same variables reject the move, else:
- draw a discrete uniform random variable u ∼ DU (1, . . . , v), where v is the number
of variables in chain 1 not included in chain 2.
- delete randomly u variables with their associated parameters from chain 1.
- add the u variables with their associated parameters to chain 2.
- accept the new state with probability
α = min 1,
π1 (x′1 )π2 (x′2 )q(x1 , x2 | x′1 , x′2 )
π1 (x1 )π2 (x2 )q(x′1 , x′2 | x1 , x2 )
,
where q(x′1 , x′2 | x1 , x2 ) = q(u | v)τ1 τ2|1 , is the proposal density of the new states, q(u | v)
the uniform probability to select u from v available variables i, τ1 is the probability to
select chain 1 and τ2|1 is the probability to select chain 2 given we have selected chain 1.
32
The term τ1 τ2|1 cancels out in the acceptance ratio since these probabilities are set to be
uniform.
The algorithm is coded in Matlab and is available from the first author upon request.
33