# Predicting Goal-directed Attention Control Using Inverse-Reinforcement
Learning

Gregory J. Zelinsky  Department of Psychology, Stony Brook University, Stony
Brook, NY 11794 USA  Department of Computer Science, Stony Brook University,
Stony Brook, NY 11794 USA<EMAIL_ADDRESS>Yupei Chen
Department of Psychology, Stony Brook University, Stony Brook, NY 11794 USA
Seoyoung Ahn  Department of Psychology, Stony Brook University, Stony Brook,
NY 11794 USA  Hossein Adeli  Department of Psychology, Stony Brook University,
Stony Brook, NY 11794 USA  Zhibo Yang  Department of Computer Science, Stony
Brook University, Stony Brook, NY 11794 USA  Lihan Huang  Department of
Computer Science, Stony Brook University, Stony Brook, NY 11794 USA  Dimitrios
Samaras  Department of Computer Science, Stony Brook University, Stony Brook,
NY 11794 USA  Minh Hoai  Department of Computer Science, Stony Brook
University, Stony Brook, NY 11794 USA

###### Abstract

Understanding how goal states control behavior is a question ripe for
interrogation by new methods from machine learning. These methods require
large and labeled datasets to train models. To annotate a large-scale image
dataset with observed search fixations, we collected 16,184 fixations from
people searching for either microwaves or clocks in a dataset of 4,366 images
(MS-COCO). We then used this behaviorally-annotated dataset and the machine
learning method of Inverse-Reinforcement Learning (IRL) to learn target-
specific reward functions and policies for these two target goals. Finally, we
used these learned policies to predict the fixations of 60 new behavioral
searchers (clock = 30, microwave = 30) in a disjoint test dataset of kitchen
scenes depicting both a microwave and a clock (thus controlling for
differences in low-level image contrast). We found that the IRL model
predicted behavioral search efficiency and fixation-density maps using
multiple metrics. Moreover, reward maps from the IRL model revealed target-
specific patterns that suggest, not just attention guidance by target
features, but also guidance by scene context (e.g., fixations along walls in
the search of clocks). Using machine learning and the psychologically-
meaningful principle of reward, it is possible to learn the visual features
used in goal-directed attention control.

## Introduction

Ever since Yarbus’ seminal demonstration of how a goal can control attention
[1], understanding goal-directed attention control has been a core aim of
psychological science. This focus is justified. Goal-directed attention
underlies everything that we try to do, making it key to understanding
cognitively-meaningful behavior. Like Yarbus, we too demonstrate goal-directed
control of eye-movement behavior, but here these overt attention movements are
made by a deep-network model that has learned different goals.

Three factors distinguish our approach from previous work. First, it is image-
computable and uses learned, rather than handcrafted, features. Our model
therefore inputs an image but is not told anything about its features
("vertical", "clock", etc.), which all must be learned. This factor
distinguishes the current model from most others in the behavioral literature
on attention control [2, 3, 4], and makes our approach more aligned with
recent computational work. [5, 6] Second, the goal-directed behavior that we
study is categorical search, the visual search for any exemplar of a target-
object category. [7, 8, 9] We adopt this paradigm because categorical search
is the simplest (and therefore, best) goal-directed behavior to
computationally model—there is a target-object goal and the task is to find
it. A third and unique contribution of our approach is that we predict
categorical-search fixations using a policy that was learned, through many
observations of search-fixation behavior during training, to maximize the
goal-specific receipt of reward. Using inverse-reinforcement learning (IRL),
we obtain these reward functions and use them to prioritize spatial locations
to predict the fixations made by new people searching for the learned target
categories in new images. Doing this required the creation of a search-
fixation-annotated image dataset sufficiently large to train deep-network
models (see Methods). We show that this model successfully captured several
patterns observed in goal-directed search behavior, not the least being the
guidance of overt attention to the target-category goals.

### Inverse-Reinforcement Learning

IRL is an imitation-learning method from the machine-learning literature that
learns, through observations of an expert, a reward function and policy for
mimicking expert performance. We extend this framework to goal-directed
behavior by assuming that the image locations fixated by searchers constitute
the expert performance that the model learns to mimic. The specific IRL
algorithm that we use is Generative Adversarial Imitation Learning (GAIL
[10]), which makes reward proportional to the model’s ability to generate
State-Action pairings that imitate observed State-Action pairings. Here, the
Action is a shift of fixation location in a search image (the model’s
saccade), and the State is the search context (all the information available
for use in the search task). The State includes, but is not limited to, the
visual features extracted from an image and the learned visual representation
of the target category. Over training, and through the greedy maximization of
total-expected reward, the model learns a Policy for mapping States to Actions
that can be used to predict new Actions (saccades) given new States (search
images).

## Methods

### Model Methods

The IRL model framework is illustrated in Fig. 1. Model training can be
conceptualized as a Policy Generator (G) and a Discriminator (D) locked in an
adversarial process [10]. The Generator generates fake eye movements (Actions)
with the goal of fooling the Discriminator into believing that these actions
were made by a person, while the Discriminator’s goal is to discriminate the
real eye movements from the fake. More specifically, the Generator consists of
an Actor-Critic model [11] that learns a policy for maximizing total expected
reward over all possible sequences of fixations, with greater reward given to
the Generator when it produces person-like actions that the Discriminator
miss-classifies as real (the logarithm of the Discriminator output). This
reward-driven adversarial process plays out during training using Proximal
Policy Optimization (PPO) [12], with the result being a Generator that becomes
highly adept at imitating the behavioral fixations made during categorical
search. At testing, this learned Policy for mimicking people’s categorical
search fixations is used to predict the fixation behavior of new people
searching for the same target categories in new images. These fixation
predictions are quantified by what we call a saccade map, which is a priority
map reflecting the total reward expected if saccades were to land at all the
different locations in an image input.

Figure 1: The model’s adversarial imitation learning algorithm. During
training it learned from fixation-annotated images a reward function and
policy for predicting new search fixations in unseen test images.

### States and Actions: Cumulative Movements of a Foveated Retina

Broadly speaking, the State is the internal visual representation that is used
for search, and a big part of this are the features extracted from the image
input. To obtain a robust core State representation we pass each image through
a pre-trained ResNet-50 [13] to get a reasonably-sized feature map output
(1024x10x16). However, human search behavior is characterized by movements of
a foveated retina, and each of these search fixations dramatically changes the
State by re-positioning the high-resolution fovea in the visual input. We
captured this fixation-dependent change in State in two steps. First, we gave
the IRL model a simplified foveated retina. We did this using the method from
Geisler and Perry [14] to compute a retina-transformed version of the image
input (ReT-image), which in our implementation is an image having high
resolution within a central $3^{\circ}$ "fovea" (32x32 in the resized 512x320
pixel image) but is blurred outside of this fovea to approximate the loss of
resolution that occurs with increasingly eccentric viewing in the visual
periphery. Second, we accumulate these high-resolution foveal views, each a
different ReT-image, over 6 new fixations in a process that we refer to as
cumulative foveation. With each new "eye movement", the fovea is re-positioned
in the image, thereby progressively de-blurring what was an initially fairly
blurred visual input. Note that by adopting this cumulative-foveation State
encoder we are not suggesting that people have a similar capacity to maintain
high-resolution visual information once the fovea moves on, and indeed this is
known not to be the case [15]. Rather, we used this fixation-by-fixation State
encoder simply as a tool to integrate a dynamically changing State into the
IRL method. Figure 2 shows cumulative ReT-images obtained at three successive
fixation locations (0,1,2) for a sample scene, with the 7-fixation sequence of
these images comprising a dynamic State representation that is input to the
IRL model. The pre-trained ResNet-50 was dilated and fine-tuned on ReT-images
prior to this State encoding.

Figure 2: The formation of a cumulative retina-transformed image over the
first three fixations (0,1,2).

The IRL model learns to associate States with Actions, but these Actions must
also be defined in some space. We obtain an Action space by first resizing a
ReT-image input to 512x320 pixels, which we then discretize into a 10x16 grid
of 32x32 pixel cells. The center of each cell becomes a potential fixation
location, a computational necessity imposing a resolution limit on the model’s
oculomotor behavior. For each of the 6 new fixations generated by the model,
the cumulative ReT-image input is prioritized by the saccade map and one of
the 160 possible grid locations is selected for an eye movement.

### The Microwave-Clock Search Dataset

The currently most predictive models of complex fixation behavior are in the
context of a free-viewing task, where the best of these models (e.g., DeepGaze
II [16]) are trained on SALICON [17]. SALICON is a crowd-sourced dataset
consisting of images that were annotated with human mouse clicks indicating
salient image locations. Without SALICON, DeepGaze II and models like it would
not have been possible, and our understanding of free-viewing behavior, widely
believed to reflect bottom-up attention control (i.e., control solely by
features extracted from the visual input), would be diminished. To date,
however, there has been no comparable dataset for categorical search, and this
has hindered the computational modeling of goal-directed attention control.
Those suitably-sized and fixation-annotated image datasets that do exist
either did not use a standard search task [18, 19], used a search task but had
people search for multiple targets simultaneously [20], or used only one
target category (people) [21]. Here we introduce the Microwave-Clock Search
(MCS) dataset, which is now among the largest datasets of images that have
been annotated with goal-directed fixations. The MCS dataset makes it possible
to train deep-network models on human search fixations to predict how people
will move their attention in the pursuit of different target-object goals.

Figure 3: Representative training images (top) and testing images (bottom) in
the MCS dataset.

Half of the MCS dataset consists of COCO2014 images [22] depicting either a
microwave or a clock (based on COCO labels), from which we created disjoint
training and testing datasets. In selecting the training images we excluded
scenes depicting people and animals (to avoid attention biases to these
categories), and digital clocks in the case of the clock target category. This
latter constraint was introduced because the features of analog and digital
clocks are very different, and we were concerned that this would introduce
unwanted variability in the search behavior. No additional exclusion criteria
were used to select the training images, with our goal being to include as
many images for training as possible. These criteria left 1,494 analog clock
images and 689 microwave images, which we should note varied greatly in terms
of their search difficulty (see Fig. 3, top). Selection of the test images was
more tightly controlled, resulting in the test dataset being far smaller
(n=40). In addition to the exclusion criteria used for the training images,
test images were further constrained to have: (1) depictions of both a
microwave and a clock (enabling different targets to be designated in the
identical images, the perfect control for differences in bottom-up saliency),
(2) only a single instance of the target, (3) a target area less than 10% of
the image area, and (4) targets that do not appear at the image’s center (no
overlap between the target and the center cell of a 5x5 grid). The latter two
criteria were aimed at excluding really large targets or targets appearing too
close to the center starting-gaze position, with the goal of both being to
achieve a moderate level of search difficulty (see Fig. 3, bottom).

Figure 4: The categorical search paradigm used for behavioral data collection.

The above-described selection criteria were specific to target-present (TP)
images, but an equal number of target-absent (TA) images (n=2183) were
selected as well so as to create a standard TP versus TA search context. These
images were selected randomly from COCO, with the constraints that: (1) none
depicted the target, and (2) all depicted at least two instances of the target
category’s siblings. COCO defines the siblings of a microwave to be: ovens,
toasters, refrigerators, and sinks, all under the parent category of
“appliances”. Clock siblings are defined as: books, vases, scissors,
hairdryers, toothbrushes, and teddy bears, under the parent category of
“indoor”. Sibling membership was used as a selection criterion so as to
discourage TA responses from being based on scene type (e.g., a street scene
is unlikely to contain a microwave), and this criterion seemed to work well;
the overwhelming majority of selected TA scenes were kitchens that did not
depict a target.

The large size of the training dataset (4366 images) required data collection
to be distributed over groups of searchers. Each microwave training image was
searched by 2-3 people (n=27); each clock training image was searched by 1-2
people (n=26). After removing incorrect trials and TP trials in which the
target was not fixated (it is not desirable to train on these), 16,184 search
fixations remained for model training. Test images were each searched by a new
group of 60 participants, 30 searching for a microwave target and the other 30
searching the same images for a clock target in a between-subjects design. To
achieve a power and effect size of .8, based on a t-test comparing target
guidance to chance (see Fig. 5), we determined that a sample of 25
participants per target condition would be adequate. However, we chose to test
30 participants per condition in case of loss due to attrition or unusable
eye-tracking data.

### Behavioral Search Procedure

A standard categorical search paradigm was used for both training and testing
(see Fig. 4). TP and TA trials were randomly interleaved within target type,
and searchers made a speeded TP or TA manual response terminating each trial.
Search display visual angles were $54^{\circ}\times 35^{\circ}$ for testing;
for training angles ranged from $12^{\circ}\times 28.3^{\circ}$ in width and
$8^{\circ}\times 28.3^{\circ}$ in height. Eye position was sampled at 1000 Hz
using an EyeLink 1000 (SR Research) in tower-mount configuration (spatial
resolution $0.01^{\circ}$ rms). All participants provided informed consent in
accordance with policies set by the institutional review board at Stony Brook
University responsible for overseeing research conducted on human subjects.

## Results

### Search Behavior

Table 1 provides the mean button-press errors and the average number of
fixations made before the button-press response (which includes the starting
fixation) on correct search trials. Note that the roughly doubled error rates
in the training data should be interpreted with caution, as many of these
errors were due to incorrectly labelled target-object regions in COCO that
would cause errors given correct search judgments. Rather than correcting
these mislabelled objects (which would be changing COCO), we instead decided
to tolerate an inflated error rate and to exclude these error trials from all
analyses and interpretation.

| Target Category | Training Dataset | Testing Dataset
---|---|---|---
Error (%) | Mean (SD) Fixations | Error (%) | Mean (SD) Fixations
target-present | microwave | 18 | 5.46 ($\pm 2.6$) | 9 | 6.76 ($\pm 2.1$)
clock | 15 | 4.52 ($\pm 3.5$) | 6 | 5.33 ($\pm 1.8$)
target-absent | microwave | 8 | 7.95 ($\pm 4.1$) | 4 | 14.36 ($\pm 2.5$)
clock | 10 | 11.14 ($\pm 6.8$) | 5 | 15.85 ($\pm 2.3$)
Table 1: Summary statistics showing mean errors and number of search fixations
in the Microwave-Clock Search dataset.

Focusing first on the TP test data, Figure 5 plots the cumulative probability
of fixating the target with each saccade made during search. The central
behavioral data pattern (solid lines) is that attention, as measured by overt
gaze fixation, is strongly guided to both the microwave and clock targets.
This guidance is evidenced by the fact that 24% of the initial saccades landed
on targets (averaged over microwaves and clocks). This probability of target
fixation is well above chance, which we quantified using two object-based
chance baselines consisting of: (1) the probability of fixating the clock when
searching for a microwave (clock baseline), and (2) the probability of
fixating the microwave when searching for a clock (microwave baseline). We
confirmed above-chance target guidance by comparing the slopes of regression
lines fit to the target and baseline data (microwave: target slope = 0.15,
baseline slope = 0.03, t(58) = 26.31, p = 6.20e-34 < .001; clock: target slope
= 0.17, baseline slope = 0.004, t(58) = 52.65, p = 1.14e-50 < .001). Also
evident from this analysis is the importance of the first six saccades made
during the search tasks. If the target was going to be fixated, it is highly
likely that this would happen by the sixth eye movement. Collectively, these
results indicate that there are strong microwave and clock guidance signals in
the behavioral test data to predict.

Figure 5: Cumulative probability of fixating the microwave (red) or clock
(blue) target on target-present trials for behavioral participants (solid
lines) and the model (dashed lines). Bottom lines are object-based random
baselines (see text for details).

### IRL Model

To determine whether the model’s behavior is reasonable, we conducted two
initial qualitative analyses. The top row in Figure 6 shows cumulative ReT-
images for the starting fixation (0 in the yellow scanpath) and the fixations
following the first two saccades (1, 2). Note that the left ReT-image, because
it was computed based on a center initial fixation position, is blurred on
both the left and right sides. The middle and right ReT-images were computed
based on the landing positions of the first and second saccades, respectively.
The microwave target is indicated in each panel by the red box. The bottom row
shows the saccade maps corresponding to these ReT-images, where a bluer color
indicates greater total reward expected by moving fixation to different image
locations. The model initially expected the greatest total reward by fixating
the stove (left saccade map), but after that saccade, and the resulting change
in State (top middle), the model then selected the microwave target as the
location offering the greatest expected reward (bottom middle), which was
fixated next (right panels). Note that the model, because it was forced to
make six saccades (discussed below), continued to prioritize space even after
fixating the target. This qualitative analysis shows that the model learned an
association between a State (which includes the features of a microwave) and
an Action, and this enabled it to guide its fixations during the search of a
new image for this target-category goal.

Figure 6: Cumulative ReT-images (top row) and corresponding saccade maps
(bottom row) for the initial and first two new fixations (left to right) made
by the model in a microwave search task.

Figure 7 shows another qualitative evaluation, this time comparing Fixation-
Density Maps (FDMs) from people searching for a microwave (n=30) or a clock
(n=30) to FDMs generated by the model (sampling from probabilistic policy) as
it searched for the same targets in the same two test images. In both
examples, the model and behavioral searchers efficiently found the target
(bright red). More interesting, however, is that they both searched the scenes
differently depending on the target category. When searching for a microwave
(leftmost four panels) the model and behavioral searchers tended to look at
counter-tops, but when searching for a clock (rightmost four panels) they
tended to look higher up on the walls. Future work will more fully explore the
potential to learn and predict these effects of scene context on search.

Figure 7: Model and behavioral Fixation Density Maps computed for microwave
(left four) and clock (right four) searches in two trials (top, bottom).

We directly compared the model and participant search behavior in several
analyses of the test data. This comparison occurred on an image-by-image and
fixation-by-fixation basis, but was limited to the first six movements of
gaze. We introduced this limitation on the number of search saccades to reduce
model computation time, but believe that it is justified given the clear
adequacy of the first six saccades in revealing the goal-directed behavior of
interest, as shown in Figure 5.

Figure 8A (left plot) shows that the model was able to predict the behavioral
FDMs for microwave and clock targets, using an AUC metric where the scale is
between 0 and 1 and higher values indicate better predictive success. We also
include Subject models, computed using the leave-one-out method, to obtain a
practical noise limit on a model’s ability to predict group behavior [23].
This analysis shows that the IRL model was able to predict the spatial
distribution of behavioral fixations in the test images as well as could be
expected based on variability among the participants in their search behavior.
FDMs, however, are purely spatial, but search fixations are also made over
time, ultimately producing a scanpath. Because the IRL model also makes
sequences of fixations, we were able to compare its 6-saccade scanpaths to the
6-saccade scanpaths from the behavioral searchers (right plot). Based on
average MultiMatch similarity [24], excluding the fixation duration component,
the model did a very good job in predicting the spatio-temporal sequences of
fixations made by the behavioral searchers in their first six saccades, again
as well as could be expected from the behavioral data, and it did this for
both microwave and clock targets.

Figure 8: Model Performance Results. (A) Left: Success in predicting FDMs by
the IRL model and a Subject model using the Area-Under-the-Curve metric.
Right: Corresponding predictions of search scanpaths using average MultiMatch
similarity. (B) Left: Proportion of trials in which the target was fixated in
the first six saccades (fixated-in-6 accuracy). Right: Average number of
saccades to the target on the fixated-in-6 trials. Note that in the (A) plots
“Subj” refers to a Subject model whereas in the (B) plots "Subj" refers to
behavioral data.

We also analyzed search accuracy and the number of saccades that were made
during search. Note that “Accuracy” refers here to the proportion of trials in
which the target was fixated in the first six eye movements (fixated-in-6
accuracy), and “Avg Saccades” refers to the mean number of saccades needed to
find the target on the accurate fixated-in-6 trials. Figure 8B shows that the
model was slightly less successful than behavioral searchers in locating
targets within six saccades (left plot), but when it did find the target it
tended to do so about as efficiently as our participants, needing only about
half a fixation more in the case of clocks (right plot). Chance fixated-in-6
accuracy is less than .25, based on a shuffling of eye data and images within
each participant, and this is far lower than fixated-in-6 accuracy for the IRL
model (microwave: t(58) = -31.74, p = 2.34e-38 <.001, Cohen’s d = 8.20; clock:
t(58) = -75.87, p = 9.73e-60 <.001, Cohen’s d = 19.59). But perhaps the
clearest measure of search efficiency is the cumulative probability of target
fixation over saccades. As indicated by the dashed lines in Figure 5, the
model’s search efficiency, although generally lower (reflecting the difference
in fixated-in-6 accuracy), was strongly guided to targets much like the
participants’ search behavior. We interpret this as meaning that the IRL model
learned goal-specific attention control, as measured by a gold-standard
metric.

## Discussion

Models of search behavior have traditionally aimed at describing relatively
coarse patterns (e.g., set-size effects) in highly simplified contexts [2, 3],
limitations that were imposed by a reliance on handcrafted features to create
a guidance signal. In this study we adopted the radically different approach
of training a model simply on many observations of search behavior, and showed
that the Policy learned by this model predicted multiple overt measures of
goal-directed attention control. The success of these predictions is
significant in that it requires a re-setting of the goal posts with respect to
model evaluation. While once computational methods limited attention models to
fitting patterns of search data in simple contexts, with deep networks it is
possible to predict individual fixations made in the search for categories of
objects in realistic scenes.

Training this model required creating the Microwave-Clock Search dataset,
which is among the only datasets of goal-directed attention (search fixations)
large enough to train deep-network models. We encourage people to download
this dataset from https://you.stonybrook.edu/zelinsky/datasetscode/ and use it
in their own predictive-modeling work, citing this publication. Our hope is
that the availability of this dataset will promote greater model development
and comparison, which given the pace of recent advances might meaningfully
advance the understanding of goal-directed attention control.

The visual search for an object category is a goal-directed behavior of unique
importance, shared by pigeons and people and most species in between. Because
of its fundamental role in survival, search is likely to use the most basic of
control processes—reward. [25] Using the MCS dataset and Inverse-Reinforcement
Learning, we showed that the target-specific reward functions learned by our
model predicted the goal-directed fixations made by new people searching new
images for the learned target categories. Machine learning has made it
possible to learn the reward functions underlying goal-directed attention
control. In ongoing work we are expanding our search dataset to 18 target
categories so as to begin characterizing how reward functions vary over common
real-world objects and to more fully explore scene context effects. In future
work we also plan to manipulate different types of reward used in training,
and apply IRL to questions in individual-difference learning.

## Data availability

The dataset described in this paper is available in the Microwave-Clock Search
(MCS) dataset repository: https://you.stonybrook.edu/zelinsky/datasetscode/.

## References

  * [1] Yarbus, A.  Eye movements and vision plenum.  _New York_ (1967).
  * [2] Wolfe, J. M.  Guided search 2.0 a revised model of visual search.  _Psychonomic bulletin & review_ 1, 202–238 (1994).
  * [3] Zelinsky, G. J.  A theory of eye movements during target acquisition.  _Psychological review_ 115, 787 (2008).
  * [4] Bundesen, C.  A theory of visual attention.  _Psychological review_ 97, 523 (1990).
  * [5] Zhang, M. _et al._ Finding any waldo with zero-shot invariant and efficient visual search.  _Nature communications_ 9, 3730 (2018).
  * [6] Zelinsky, G. J. _et al._ Benchmarking gaze prediction for categorical visual search.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops_ (2019).
  * [7] Schmidt, J. & Zelinsky, G.  Search guidance is proportional to the categorical specificity of a target cue.  _The Quarterly Journal of Experimental Psychology_ 62, 1904–1914 (2009).
  * [8] Eimer, M.  The neural basis of attentional control in visual search.  _Trends in cognitive sciences_ 18, 526–535 (2014).
  * [9] Zelinsky, G. J., Peng, Y., Berg, A. C. & Samaras, D.  Modeling guidance and recognition in categorical search: Bridging human and computer object detection.  _Journal of Vision_ 13, 30–30 (2013).
  * [10] Ho, J. & Ermon, S.  Generative adversarial imitation learning.  In _Advances in Neural Information Processing Systems_ , 4565–4573 (2016).
  * [11] Konda, V. R. & Tsitsiklis, J. N.  Actor-critic algorithms.  In _Advances in neural information processing systems_ , 1008–1014 (2000).
  * [12] Schulman, J., Wolski, F., Dhariwal, P., Radford, A. & Klimov, O.  Proximal policy optimization algorithms.  _arXiv preprint arXiv:1707.06347_ (2017).
  * [13] He, K., Zhang, X., Ren, S. & Sun, J.  Delving deep into rectifiers: Surpassing human-level performance on imagenet classification (2015).
  * [14] Perry, J. S. & Geisler, W. S.  Gaze-contingent real-time simulation of arbitrary visual fields.  In _Human vision and electronic imaging VII_ , vol. 4662, 57–70 (International Society for Optics and Photonics, 2002).
  * [15] Irwin, D. E.  Integrating information across saccadic eye movements.  _Current Directions in Psychological Science_ 5, 94–100 (1996).
  * [16] Kummerer, M., Wallis, T. S., Gatys, L. A. & Bethge, M.  Understanding low-and high-level contributions to fixation prediction.  In _Proceedings of the IEEE International Conference on Computer Vision_ , 4789–4798 (2017).
  * [17] Jiang, M., Huang, S., Duan, J. & Zhao, Q.  Salicon: Saliency in context.  In _The IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ (2015).
  * [18] Mathe, S. & Sminchisescu, C.  Actions in the eye: Dynamic gaze datasets and learnt saliency models for visual recognition.  _IEEE transactions on pattern analysis and machine intelligence_ 37, 1408–1424 (2014).
  * [19] Papadopoulos, D. P., Clarke, A. D., Keller, F. & Ferrari, V.  Training object class detectors from eye tracking data.  In _European conference on computer vision_ , 361–376 (Springer, 2014).
  * [20] Gilani, S. O. _et al._ Pet: An eye-tracking dataset for animal-centric pascal object classes.  In _2015 IEEE International Conference on Multimedia and Expo (ICME)_ , 1–6 (IEEE, 2015).
  * [21] Ehinger, K. A., Hidalgo-Sotelo, B., Torralba, A. & Oliva, A.  Modelling search for people in 900 scenes: A combined source model of eye guidance.  _Visual cognition_ 17, 945–978 (2009).
  * [22] Lin, T.-Y. _et al._ Microsoft coco: Common objects in context.  In _European conference on computer vision_ , 740–755 (Springer, 2014).
  * [23] Bylinskii, Z., Judd, T., Oliva, A., Torralba, A. & Durand, F.  What do different evaluation metrics tell us about saliency models?  _IEEE transactions on pattern analysis and machine intelligence_ 41, 740–757 (2018).
  * [24] Dewhurst, R. _et al._ It depends on how you look at it: Scanpath comparison in multiple dimensions with multimatch, a vector-based approach.  _Behavior research methods_ 44, 1079–1100 (2012).
  * [25] Anderson, B. A.  A value-driven mechanism of attentional selection.  _Journal of vision_ 13, 7–7 (2013).

## Acknowledgements

We would like to thank the National Science Foundation for their generous
support through award IIS-1763981, and members of the EyeCog Lab for their
help with data collection and invaluable feedback.

## Author contributions

M.H., D.S., and G.J.Z. conceptualized the research; H.A., Y.C., and G.J.Z.
collected the dataset, L.H., Z.Y., D.S., and M.H. implemented the model. All
authors analyzed and interpreted data, but especially Z.Y., Y.C., L.H., and
S.A. G.J.Z., Y.C., S.A., and H.A. wrote the paper.

## Additional information

Competing interests.

The authors declare no competing interests.


# Unconventional Policies Effects on Stock Market Volatility: A MAP Approach

Demetrio Lacava111University of Messina, email<EMAIL_ADDRESS>Giampiero M.
Gallo222Italian Court of Audits (Corte dei conti, Milan – disclaimer) and New
York University in Florence; email<EMAIL_ADDRESS>Edoardo
Otranto333Corresponding author. University of Messina and CRENoS, Via dei
Verdi 75, 98122 Messina, email<EMAIL_ADDRESS>
###### Abstract

Taking the European Central Bank unconventional policies as a reference, we
suggest a class of Multiplicative Error Models (MEM) taylored to analyze the
impact such policies have on stock market volatility. The new set of models,
called MEM with Asymmetry and Policy effects (MAP), keeps the base volatility
dynamics separate from a component reproducing policy effects, with an
increase in volatility on announcement days and a decrease unfolding
implementation effects. When applied to four Eurozone markets, a Model
Confidence Set approach finds a significant improvement of the forecasting
power of the proxy after the Expanded Asset Purchase Programme implementation;
a multi–step ahead forecasting exercise estimates the duration of the effect,
and, by shocking the policy variable, we are able to quantify the reduction in
volatility which is more marked for debt–troubled countries.

Keywords: Financial markets, Impulse response function, Model Confidence Set,
Multiplicative Error Model, Realized volatility.

JEL Codes: C32, C58, E44, E52, E58, G17

##  1 Introduction

During the Great Recession, with policy interest rates reaching the zero lower
bound, many central banks, the Federal Reserve (FED), the European Central
Bank (ECB) and the Bank of England (BoE) in particular, adopted unconventional
actions in order to stimulate the real economy. Among unconventional monetary
policies, one finds the central bank balance sheet expansion – generally
through Asset Purchase Programs (APP) – which affects the real economy by
reducing long–term yields and the credit spread, and by modifying inflation
rate expectations during periods in which the liquidity trap makes the
conventional policy no longer effective. Since 2009, the ECB resorted to
so–called Quantitative Easing (QE) in order to inject liquidity into the
financial system. Geared toward preventing a systemic collapse due to the
deterioration of credit ratings, but also to an adverse market attitude toward
debt–ridden economies, these policies have been credited with successfully
reducing market uncertainty and increasing Eurozone stability.

For the bond markets, the effects of monetary policy actions are studied in
two directions: an increase in prices due to increased direct demand (and
hence a reduction in yields) and a lower volatility (because of the increased
stability brought to the system). Among the empirical analyses we cite Eser
and Schwaab (2016) and De Santis (2020); Ghysels et al. (2017) use intra–daily
high frequency data to document an asymmetric effect due to the presence of
Security Market Programme (SMP) purchases in a GARCH(1,1). They find that
unconventional monetary policies by ECB had a highly statistically significant
dampening impact on yield volatility for most countries and
maturities.444Rather than lengthening the citations of this rich literature in
the bond markets, as they address issues and transmission channels which are
not the main focus of this paper, we prefer to summarize some of them in a
table in the supplemental material.

As per the stock market, the directional effect is less pronounced: event
studies pinpoint to single positive effects on stock prices (Haitsma et al.,
2016); the transmission of monetary policy to the stock market can be traced
to the portfolio-rebalancing channel (Georgiadis and Gräb, 2016) in
redetermining asset allocations as an indirect result of bond price movements;
inflation expectations have been less of a concern for at least the past
fifteen years, while the confidence channel mitigates general uncertainty with
what we consider a price direction–neutral effect.

The focus of this paper is the impact of the QE on the volatility of stock
markets: while the methodology of choice in other papers is a GARCH model
(Engle, 1982; Bollerslev, 1986), in which some policy related variable
(usually one or more dummies) is inserted in the conditional variance
equation, in what follows, we exploit the established merits of working with a
realized measure of volatility (Andersen et al., 2006). Being a
positive–valued process, this measure is suitable to be modeled for
forecasting purposes as a Multiplicative Error Model, MEM (Engle, 2002; Engle
and Gallo, 2006), in which volatility is the product of a time-varying factor
(following a GARCH–type process) and a positive random variable ensuring
positiveness without resorting to logs. Within the MEM class, Brownlees et al.
(2012) propose a model - the Composite AMEM (ACM) - in which the conditional
expected volatility is the sum of a short– and a long–run component. This
model suggests a way to separate two volatility components, one representing
the usual autoregressive dynamics of volatility and the other where we gather
both the announcement and the implementation effects of unconventional
policies. As a by–product of estimating the latent factors, we can calculate
the share of the policy–related component on total expected volatility. To
extend this model, labeled MEM with Asymmetry and Policy effects (MAP), we
consider the case in which the two latent factors, rather than combining
additively, are modified to be a product of each other (with two
specifications adopted).

The empirical application analyzes the impact of unconventional monetary
policies by the ECB on stock market volatility, taking four Eurozone countries
(France, Germany, Italy and Spain) as our leading case, considering the latter
two as representative of debt–burdened markets. We proxy for unconventional
policies by using the ratio between the securities purchased by the ECB for
unconventional policy purposes and the ECB total asset (similarly to D’Amico
et al., 2012; Voutsinas and Werner, 2011). Our main results show that
announcements cause an immediate peak in volatility, whereas the
implementation of unconventional policies has a mitigating effect along time;
moreover, including these variables in the model, improves the out–of–sample
forecasting, in particular after the EAPP announcements.

The paper is organized as follows. Section 2 describes data as well as the
stylized facts on volatility with respect to the unconventional policies
implementation. Section 3 introduces the novel MEM–based models employed in
our empirical analysis, in Section 4. We discuss model estimation and
inference in Sub–section 4.1, model comparisons including an analysis of which
models enter Model Confidence Set in various subsamples (in 4.2) and a
multi–step forecasting exercise to determine the estimated duration of the
effects and the volatility response to a shock to the policy variable (in
4.3). Finally, Section 5 concludes with some remarks.

##  2 Proxies for unconventional monetary policies

In investigating the impact of unconventional monetary policies by ECB we
consider two different variables, which refer to announcement and
implementation effects on volatility, respectively. The former is measured by
means of a dummy variable taking value of 1 on days in which ECB releases
communications regarding a monetary policy
decision555https://www.ecb.europa.eu/press/pr/activities/mopo/html/index.en.html.
To proxy for the implementation effect we use the amount of securities held by
ECB as a fraction of total asset666Data are taken from the ECB website and
Datastream. (cf. D’Amico et al., 2012; Voutsinas and Werner, 2011).

Our analysis is based on a dataset consisting of 2686 daily observations of
annualized realized kernel volatility777Data from https://realized.oxford-
man.ox.ac.uk/data/download (RV hereafter) which is a robust estimator of the
volatility, in particular with respect to microstructure noise (Barndorff-
Nielsen et al., 2008). The analysis relates to four Eurozone market indices
(CAC40 for France, DAX30 for Germany, FTSE MIB for Italy and IBEX35 for Spain
– referred to by country in what follows) for the periods between June 1, 2009
and December 31, 2019.

We analyze this dataset by a simple visual inspection of the graphs,
highlighting the main announcements in the time span considered, and by
analyzing some stylized facts that justify the model proposed in Section 3.

###  2.1 Graphical Analysis of Announcements

Figure 1: France, Germany, Italy and Spain RV. Sample period: June 1, 2009 to
December 31, 2019. Number of observations: 2686\. The vertical lines represent
relevant events for policy actions (see text).

Figure 1 shows the evolution of our series. All series seem to follow a
similar pattern, with a period of low volatility in the first year of the
sample. However, while the RV series seem not to respond to the unconventional
monetary policies established by ECB at the beginning of the crisis,888These
are the 12-month Longer Term Refinancing Operations program - the LTRO - and
the Covered Bond Purchase Programme - the CBPP - which aimed at containing the
liquidity crisis and the consequent credit crunch for the Eurozone. starting
from May 2010, in all the series, one can notice how the RV reacted to some
important events (reported as vertical lines in Figure 1), such as:

  * •

the SMP (Security Market Programme) announcement on May 10th, 2010. By means
of purchases of government bond in the secondary market, the ECB aimed to
control the increase in the credit spread and to restore the proper
functioning of monetary policy transmission channels. While the RV jumped on
the day when the SMP was announced, the subsequent implementation of this
program had a dampening effect on volatility.

  * •

the “Whatever it takes” declaration by Mario Draghi on July 26th, 2012, which
served to reassure investors regarding the emerging denomination risk. Through
this declaration, the ECB announced the Outright Monetary Transaction (OMT),
which replaced the SMP successfully, depressing volatility until the end of
2014.

  * •

the EAPP (Expanded Asset Purchase Programme) announcement on January 22nd,
2015. It was established mainly to improve monetary policy transmission
mechanisms as well as to adjust the inflation rate toward the target level of
2%. It refers to a series of unconventional measures such as the Assed Backed
Securities Purchase Programme (ABSPP), the CBPPs and the Corporate and Public
Sector Purchase Programme (CSPP and PSPP, respectively), through which ECB
conducted monthly securities purchases.

  * •

March 10th, 2016. The amount of securities purchased within the EAPP passed
from the initial level of € 60 billion to € 80 billion per month, causing a
downward trend in the RV series.

  * •

October 26th, 2017. Volatility increases after the announcement through which
ECB communicated the cut in the monthly purchases, which were reduced to € 15
billion. In contrast to the previous announcement, it caused an increasing
trend in all the considered markets.

A simple visual inspection shows the decrease in volatility in correspondence
of these events; moreover, we notice also an effect caused by the amount
purchased by ECB. We aim to quantify this effect and its weight on the level
of volatility.

###  2.2 Stylized facts

In order to analyze the effects of unconventional monetary policy on
volatility, one would have to build a structural model detailing the channels
of transmission onto financial markets, isolating the nature of the
innovations. What is preferred in empirical investigations is a reduced–form
approach in which the net effect of certain variables is examined without
resorting to a structural explanation, also not to enter in the subjective
choice about which theory to adopt (Ghysels et al., 2017). We adhere to this
strategy, starting from some stylized facts which may serve as the basis for
making time series modeling choices later in the paper. We will not delve on
the well–known persistence features of realized volatility and its asymmetric
behavior relative to upward and downward movements in a market. We rather
suggest a few results about the relationship between volatility and policy
variables.

The first stylized fact is about higher volatility on the days of
announcements. Not being able to repeat a visual inspection of our set of 144
ECB announcements in the time span considered (as we did in Figure 1), we
suggest to see whether there is a pattern in terms of volatility behavior
before and after an announcement, by calculating the average realized
volatility 5 days before and 5 days after the announcements;999The only
exception relates to first announcement occurred at $t=4$, so that the average
in that case involves only 3 terms before and after. Table 1 shows how these
averages compare to the volatility on the day of the announcement in relative
percentage terms. As all the variations are negative, the announcement days
mark, on average, a peak of volatility, with a subsequent reduction after the
announcement.

Table 1: Percentage variation of the average of volatility (5 days before and after the announcement date) with respect to the volatility of the announcement day for four volatility time series. | France | Germany | Italy | Spain
---|---|---|---|---
before | -4,40 | -5,92 | -9,79 | -8,56
after | -5,52 | -5,39 | -7,39 | -5,02

A second empirical regularity is about the direction of the causal
relationship (in a Granger’s sense) between the realized volatility and the
unconventional policy proxy (allowing also for a dummy on announcement days).
The non–causality tests101010In view of the stationarity of the realized
volatility and of the nonstationarity of the policy variable, we adopted the
procedure suggested by Toda and Yamamoto (1995) which envisages a VAR(6) in
levels with five lags on either side to be cross–tested and an extra one (not
to be tested). seem to support the idea that the unconventional policy affects
the realized volatility, but not for the reverse.

A third issue is whether there is a systematic difference between average
returns on announcement and non announcement days, simply assessed by
regressing returns standardized by the realized volatility on the dummy for
announcement days. Since the corresponding coefficient is never significant,
we proceed with the empirical regularity that there is no systematic impact of
the announcements on returns.

##  3 Policy–related volatility components

Tha availability of high–frequency data has given rise to a stream of
literature on realized variance measures (Andersen et al., 2006) with models
based on the features of volatility traditionally investigated in the GARCH
framework (Engle, 1982; Bollerslev, 1986). Among these, a prominent place is
taken by Multiplicative Error Models (MEM) as defined by Engle (2002) and
successively revised by Engle and Gallo (2006) to allow for asymmetric effects
(AMEM).

The model we propose is an extension of the Composite AMEM (ACM) (Brownlees et
al., 2012, to which we refer for details) in which the conditional expected
volatility is the sum of a short– and a long–run component.111111Another model
treating volatility components additively is the Spillover AMEM (SAMEM) by
Otranto (2015), which captures volatility spillovers among markets in a
univariate framework. In this ACM we insert the effect of unconventional
policies as a latent factor, explicitly affecting the dynamics of the
volatility. More precisely, our model is based on the decomposition of the
volatility level in the sum of two components, representing the base
volatility component ($\varsigma$) and what can be related to the
unconventional policies ($\xi$), respectively, allowing us to quantify its
relative effect on the overall level of volatility.

Thus, our composite MEM with Asymmetry and Policy effects (MAP) consists of
four equations:

$\begin{array}[]{l}RV_{t}=\mu_{t}\epsilon_{t},\quad\epsilon_{t}|\Psi_{t-1}\sim
Gamma(\vartheta,\frac{1}{\vartheta})\\\ \mu_{t}=\varsigma_{t}+\xi_{t}\\\
\varsigma_{t}=\omega+\alpha RV_{t-1}+\beta\varsigma_{t-1}+\gamma
D_{t-1}RV_{t-1}\\\
\xi_{t}=\delta(E\left(x_{t}|\Psi_{t-1}\right)-x^{\star})+\varphi(\Delta_{t}-\Delta^{\star})+\psi\xi_{t-1}\end{array}$
(1)

where $\Psi_{t}$ is the information set available at time $t$. Following Engle
and Gallo (2006), we consider the error term $\epsilon_{t}$ as Gamma
distributed, in view of its flexible dependence on just one parameter,
$\vartheta$ and of its good empirical performance. From these assumptions, we
get not only a time–varying conditional mean
$E\left(RV_{t}|\Psi_{t-1}\right)=\mu_{t}$, but also a time–varying conditional
variance (volatility of volatility)
$Var\left(RV_{t}|\Psi_{t-1}\right)=\mu_{t}^{2}/\vartheta$; as $\varsigma_{t}$
follows a GARCH–type dynamics, the model is suitable to capture possible
clustering in volatility. In our model (1), $D_{t-1}$ represents a dummy
variable taking value of 1 if the return of the asset (index) at time $t-1$ is
negative, 0 otherwise.

The additional component $\xi_{t}$ is an AR(1) involving $x_{t}$ and
$\Delta_{t}$ as predetermined variables (in view of the stylized facts),
representing, respectively, the implementation and announcement effect;
finally, $x^{\star}$ is a constant, e.g. the unconditional mean or an initial
value (to accomodate a random walk process). Similar remarks apply to
$\Delta^{\star}$ and $\Delta_{t}$; this structure, under the hypothesis of
stationarity, ensures a zero unconditional mean for $\xi_{t}$. As for the
timing of these predetermined variables, since the days of monetary policy
announcements are put on the calendar well in advance, we can consider the
announcement variable, $\Delta_{t}$, at its contemporaneous time. By contrast,
the proxy representing the implementation effect has to enter in expected
terms: for simplicity, given its behavior assessed as a random walk, we adopt
the lagged value as an expectation. Thus, by also replacing $x^{\star}$ and
$\Delta^{\star}$ with the respective sample means $\bar{x}$ and
$\bar{\Delta}$, the last equation in model (1) for operational purposes will
become:
$\xi_{t}=\delta(x_{t-1}-\bar{x})+\varphi(\Delta_{t}-\bar{\Delta})+\psi\xi_{t-1}$.

Note that in Equation (1), when we set $\psi=\beta$, we get the X–MAP
specification for $\mu_{t}$ (which corresponds to an AMEM with an exogenous
regressor):

$\mu_{t}=\omega+\alpha RV_{t-1}+\beta\mu_{t-1}+\gamma
D_{t-1}RV_{t-1}+\delta(E\left(x_{t}|\Psi_{t-1}\right)-x^{\star})+\varphi(\Delta_{t}-\Delta^{\star}),$
(2)

which, in turn, nests the classical AMEM upon imposing $\delta=\varphi=0$.

For stationarity in the MAP, it is required that both components are
stationary in covariance, that is $(\alpha+\beta+\frac{\gamma}{2})<1$ and
$\psi<1$; positiveness, instead, requires that ($\varsigma_{t}+\xi_{t}>0)$ for
each $t$, which could be ensured even though $\delta$ is negative, as we
expect.

Finally, as argued by Engle and Lee (1999), the coefficient $\psi$ in
$\xi_{t}$ process is required to be less than $\beta$ to ensure the
identification of the model. Since one of the main ECB aims by means of
unconventional policies is to stabilize financial markets in the short run, we
expect an immediate effect of this kind of policy in reducing stock market
volatility. In other words, in our model, the part of volatility depending
directly on unconventional policies represents the short run component of
realized volatility as well as the proper volatility dynamics represents the
long–run component. From this assumption, we expect the long–run component to
have a higher persistence than the short one, that is $0<\psi<\beta<1$, as an
identification condition.

It is important to underline that $\xi_{t}$ is an unobservable signal, with
its own dynamics, which can be jointly estimated so as to quantify and plot
the separate effect of the unconventional ECB actions on the volatility
$RV_{t}$. As a by–product, we can also derive the share of this component
relative to the overall level of volatility, that is
$1-\frac{\varsigma_{t}}{\mu_{t}}=\frac{\xi_{t}}{\mu_{t}}$.

As shown by Engle (2002) for the MEM case, quasi maximum likelihood
maximization implies that the estimators of the unknown coefficients in model
(1) are consistent and asymptotically normal, under an assumption of correct
specification of the conditional mean equation, irrespective of the
distributional assumption, as discussed by Engle and Gallo (2006). When the
scale parameter $\theta$ is unknown, robust standard errors will shield
against the actual shape of the Gamma distribution.

Whether the policy impact on the general level of volatility should enter in
an additive way, as in model (1), or multiplicatively is always open to
question.121212Within the MEM class, a multiplicative component MEM is
suggested by Brownlees et al. (2011), adapting a GARCH approach by Engle and
Sokalska (2012).  In what follows we discuss two different specifications of
the multiplicative version of MAP which ensure the compliance with the
constraint that the unconditional mean of $\xi_{t}$ is equal to one so that
the model is identified.

In the first specification we allow $\xi_{t}$ to impact on $RV_{t}$ through a
logistic function - and by means of the delta method. The model, called
Logistic-MAP (L-MAP), is specified as in model (3)

$\begin{array}[]{l}RV_{t}=\mu_{t}\epsilon_{t},\quad\epsilon_{t}|\Psi_{t-1}\sim
Gamma(\vartheta,\frac{1}{\vartheta})\\\
\mu_{t}=2\varsigma_{t}(\frac{exp(\xi_{t})}{1+exp(\xi_{t})})\\\
\varsigma_{t}=\omega+\alpha RV_{t-1}+\beta\varsigma_{t-1}+\gamma
D_{t-1}RV_{t-1}\\\
\xi_{t}=\delta(E\left(x_{t}|\Psi_{t-1}\right)-x^{\star})+\varphi(\Delta_{t}-\Delta^{\star})+\psi\xi_{t-1}\end{array}$
(3)

Alternatively, we can add a constant term in the $\xi_{t}$ equation, providing
a Plain-MAP (P-MAP) specification, namely:

$\begin{array}[]{l}RV_{t}=\mu_{t}\epsilon_{t},\quad\epsilon_{t}|\Psi_{t-1}\sim
Gamma(\vartheta,\frac{1}{\vartheta})\\\ \mu_{t}=\varsigma_{t}\xi_{t}\\\
\varsigma_{t}=\omega+\alpha RV_{t-1}+\beta\varsigma_{t-1}+\gamma
D_{t-1}RV_{t-1}\\\
\xi_{t}=(1-\psi)+\delta(E\left(x_{t}|\Psi_{t-1}\right)-x^{\star})+\varphi(\Delta_{t}-\Delta^{\star})+\psi\xi_{t-1}\end{array}$
(4)

A way to characterize how the suggested models differ in terms of how the
policy proxy impacts expected volatility is by calculating the marginal
effects of a change in $x_{t-1}$ (respectively, $\Delta_{t}$) on
$\mu_{t+\tau}$ ($\tau$-steps ahead), as done in Table 2.131313The derivation
of the formulas is available as supplemental material.

Table 2: Marginal effects of policy variables $x_{t-1}$ and $\Delta_{t}$ on
$\mu_{t+\tau}$, $\tau\geq 0$.

Model | Marginal effect on $\mu_{t+\tau}$
---|---
MAP | $\kappa\psi^{\tau}$
X–MAP | $\kappa\beta^{\tau}$
L-MAP | $2\varsigma_{t+\tau}\kappa\psi^{\tau}\frac{exp(\xi_{t+\tau})}{(1+exp(\xi_{t+\tau}))^{2}}$
P-MAP | $\kappa\psi^{\tau}\varsigma_{t+\tau}$

  * •

Note: $\kappa=\delta$ for the marginal effects of the implementation variable
$x_{t-1}$;
$~{}\ ~{}\ \qquad\kappa=\varphi$ for the marginal effects of the announcement
variable $\Delta_{t}$.

##  4 Empirical application

###  4.1 Estimation results

The goal of this section is to show how different models for the realized
volatility series of the four indices take the policy variables into
consideration and how the effects are exerted on the level of volatility in
different specifications.

Parameter estimation of the different models are shown in Table 3.
Coefficients are highly significant in all cases with a volatility persistence
($\alpha+\beta+\frac{\gamma}{2}$) which decreases by about 2% passing from the
AMEM to our more complex MAP. Moreover, the impact of the more recent observed
values, measured by the $\alpha$ and $\gamma$ coefficients, is generally
higher in the AMEM. This indicates a lower influence of current shocks on
volatility projections, in line with the expected calming effect that the
unconventional monetary policies have on market volatility.

Diagnostics on the standardized residuals is given by the Ljung-Box statistics
for lags 1, 5 and 10 (at the bottom of the table): we see how the AMEM is able
to capture the persistence in the volatilities (we fail to reject the null of
no serial correlation at 1% significance level, with the exception of Germany
at the highest lag, as well as Spain at lag 5). However, our additive model
(the MAP) seems to have a better statistical performance in this respect,
especially in the case of Spain, where we never reject the null of no
autocorrelation at 1% level.

Model performance in terms of estimation results can be compared in Table 4,
where we report the information criteria (AIC and BIC) and two loss functions
(Mean Square Error –MSE – and Quasi-Likelihood – QLike, consistent in the
sense of Patton, 2011) to evaluate the fitting capabilities of the models. For
all countries, the best performing model (the one in bold) is the L-MAP. The
only exception is Spain, where the L-MAP has better performance in terms of
the information criteria, even if the P-MAP is marginally better in terms of
MSE.

Table 3: Models estimation results. Sample: June 1, 2009 – December 31, 2019. | France | Germany
---|---|---
| AMEM | X–MAP | MAP | L–MAP | P–MAP | AMEM | X–MAP | MAP | L–MAP | P–MAP
Coefficient estimates (robust s.e. in parentheses)
$\omega$ | 0.857 | 1.136 | 1.056 | 1.011 | 1.025 | 0.957 | 1.081 | 1.034 | 1.026 | 1.034
| (0.046) | (0.121) | (0.065) | (0.007) | (0.053) | (0.017) | (0.052) | (0.016) | (0.059) | (0.519)
$\alpha$ | 0.171 | 0.165 | 0.154 | 0.151 | 0.153 | 0.193 | 0.191 | 0.183 | 0.182 | 0.183
| (0.017) | (0.020) | (0.015) | (0.015) | (0.017) | (0.010) | (0.011) | (0.014) | (0.012) | (0.055)
$\beta$ | 0.708 | 0.689 | 0.707 | 0.712 | 0.709 | 0.692 | 0.684 | 0.696 | 0.696 | 0.694
| (0.019) | (0.029) | (0.015) | (0.014) | (0.018) | (0.011) | (0.016) | (0.013) | (0.014) | (0.094)
$\gamma$ | 0.113 | 0.120 | 0.117 | 0.119 | 0.119 | 0.087 | 0.090 | 0.089 | 0.090 | 0.090
| (0.008) | (0.009) | (0.011) | (0.010) | (0.012) | (0.006) | (0.011) | (0.008) | (0.008) | (0.008)
$\delta$ |  | -0.636 | -1.836 | -0.297 | -0.161 |  | -0.454 | -1.328 | -0.219 | -0.115
|  | (0.075) | (0.326) | (0.057) | (0.029) |  | (0.035) | (0.283) | (0.052) | (0.057)
$\varphi$ |  | 1.297 | 2.817 | 0.464 | 0.231 |  | 1.067 | 2.317 | 0.378 | 0.188
|  | (0.381) | (0.539) | (0.093) | (0.045) |  | (0.311) | (0.470) | (0.073) | (0.073)
$\psi$ |  |  | 0.111 | 0.194 | 0.134 |  |  | 0.098 | 0.175 | 0.138
|  |  | (0.060) | (0.081) | (0.083) |  |  | (0.077) | (0.097) | (0.204)
$\theta$ | 7.559 | 7.728 | 7.817 | 7.827 | 7.820 | 9.460 | 9.610 | 9.700 | 9.719 | 9.714
| (0.222) | (0.228) | (0.231) | (0.230) | (0.231) | (0.290) | (0.298) | (0.301) | (0.301) | (0.301)
Loglik | -7825.1 | -7793.9 | -7778.1 | -7776.2 | -7777.4 | -7592.5 | -7570.6 | -7557.8 | -7555.0 | -7555.7
p–values for Ljung-Box statistics
lag 1 | 0.155 | 0.258 | 0.132 | 0.108 | 0.125 | 0.171 | 0.262 | 0.184 | 0.173 | 0.194
lag 5 | 0.111 | 0.167 | 0.215 | 0.192 | 0.207 | 0.028 | 0.063 | 0.056 | 0.051 | 0.054
lag 10 | 0.115 | 0.137 | 0.210 | 0.201 | 0.201 | 0.002 | 0.004 | 0.003 | 0.003 | 0.003
| Italy | Spain
| AMEM | X–MAP | MAP | L–MAP | P–MAP | AMEM | X–MAP | MAP | L–MAP | P–MAP
Coefficient estimates (robust s.e. in parentheses)
$\omega$ | 1.198 | 1.708 | 1.533 | 1.480 | 1.490 | 1.055 | 1.533 | 1.438 | 1.410 | 1.426
| (0.073) | (0.221) | (0.260) | (0.239) | (0.146) | (0.075) | (0.110) | (0.135) | (0.621) | (0.132)
$\alpha$ | 0.268 | 0.274 | 0.249 | 0.248 | 0.250 | 0.224 | 0.205 | 0.197 | 0.193 | 0.195
| (0.018) | (0.032) | (0.033) | (0.042) | (0.022) | (0.013) | (0.014) | (0.024) | (0.060) | (0.023)
$\beta$ | 0.608 | 0.568 | 0.604 | 0.605 | 0.603 | 0.671 | 0.656 | 0.671 | 0.674 | 0.671
| (0.023) | (0.043) | (0.043) | (0.049) | (0.030) | (0.007) | (0.020) | (0.027) | (0.091) | (0.028)
$\gamma$ | 0.084 | 0.086 | 0.086 | 0.089 | 0.088 | 0.078 | 0.087 | 0.085 | 0.086 | 0.086
| (0.005) | (0.011) | (0.019) | (0.022) | (0.008) | (0.022) | (0.007) | (0.015) | (0.017) | (0.009)
$\delta$ |  | -1.074 | -2.354 | -0.343 | -0.178 |  | -1.025 | -2.823 | -0.389 | -0.207
|  | (0.161) | (0.519) | (0.048) | (0.031) |  | (0.086) | (0.528) | (0.133) | (0.037)
$\varphi$ |  | 2.059 | 3.448 | 0.518 | 0.254 |  | 1.964 | 3.428 | 0.501 | 0.247
|  | (0.471) | (0.569) | (0.084) | (0.038) |  | (0.361) | (0.671) | (0.087) | (0.042)
$\psi$ |  |  | 0.051 | 0.089 | 0.058 |  |  | 0.058 | 0.123 | 0.072
|  |  | (0.083) | (0.055) | (0.073) |  |  | (0.090) | (0.298) | (0.112)
$\theta$ | 10.593 | 11.030 | 11.183 | 11.185 | 11.181 | 9.149 | 9.487 | 9.580 | 9.626 | 9.623
| (0.423) | (0.427) | (0.436) | (0.437) | (0.436) | (0.317) | (0.338) | (0.349) | (0.355) | (0.352)
Loglik | -7721.3 | -7665.3 | -7646.3 | -7646.1 | -7646.5 | -8117.0 | -8066.5 | -8052.9 | -8046.3 | -8046.7
p–values for Ljung-Box statistics
lag 1 | 0.299 | 0.691 | 0.300 | 0.287 | 0.324 | 0.010 | 0.025 | 0.011 | 0.006 | 0.008
lag 5 | 0.497 | 0.721 | 0.800 | 0.805 | 0.824 | 0.009 | 0.015 | 0.019 | 0.012 | 0.015
lag 10 | 0.336 | 0.472 | 0.647 | 0.625 | 0.624 | 0.047 | 0.086 | 0.119 | 0.089 | 0.100
Table 4: Model comparisons via Information Criteria (AIC and BIC) and forecasting capability (MSE and MAE) – Sample period: June 1, 2009 - December 31, 2019. Best model in bold.  | AMEM | X-MAP | MAP | L-MAP | P-MAP
---|---|---|---|---|---
France
AIC | 5.830 | 5.809 | 5.797 | 5.796 | 5.797
BIC | 5.841 | 5.824 | 5.815 | 5.814 | 5.815
MSE | 29.531 | 29.117 | 28.844 | 28.627 | 28.639
QLike | 0.068 | 0.066 | 0.065 | 0.065 | 0.065
Germany
AIC | 5.657 | 5.642 | 5.633 | 5.631 | 5.632
BIC | 5.668 | 5.658 | 5.651 | 5.649 | 5.649
MSE | 23.551 | 23.269 | 23.096 | 22.93 | 22.939
QLike | 0.054 | 0.053 | 0.052 | 0.052 | 0.052
Italy
AIC | 5.753 | 5.713 | 5.699 | 5.699 | 5.699
BIC | 5.764 | 5.728 | 5.717 | 5.717 | 5.717
MSE | 28.171 | 27.634 | 27.209 | 27.191 | 27.203
QLike | 0.048 | 0.046 | 0.045 | 0.045 | 0.045
Spain
AIC | 6.048 | 6.011 | 6.002 | 5.997 | 5.997
BIC | 6.059 | 6.027 | 6.02 | 6.015 | 6.015
MSE | 42.048 | 40.918 | 40.462 | 39.461 | 39.439
QLike | 0.056 | 0.054 | 0.053 | 0.053 | 0.053

For what concerns the unconventional policy proxies, coefficients are
significant at 1% level and they enter the model with the expected sign.
According to other results in this stream of research (see, for example,
Bomfim, 2003; Chan and Gray, 2018; Shogbuyi and Steeley, 2017), the
coefficient $\varphi$ of the dummy variable is positive, meaning that there is
an immediate reaction in the market on announcement days, in line with our
stylized facts. Conversely, $\delta$ is negative, therefore the unconventional
policies implementation successfully reduces stock market volatility, as
expected.

This implementation effect can also be seen in Figure 2, which plots the
evolution of the two volatility components (the blue line for the base
component, left axis, and the red dotted-line, for the policy component, right
axis141414The line representing the $\xi$ process is defined in three
different levels: the lowest depends on the characteristic path of the $\xi$
component; the highest represents, instead, volatility spikes due to the
announcement effect; finally, the intermediate line, represents the level of
volatility on days after a monetary policy announcement, when volatility comes
back to its previous level.). This effect is more evident starting from
October 2014 - when the ECB implicitly communicated to the market that it
would purchase corporate (next to government) bonds - coming in the form of a
change in the slope of $\xi$ equation, lasting for the entire period of the
program. This is one of the most interesting results, as it implies that the
effectiveness of these policies rests on the total amount of securities held
for monetary policy purposes, relative to ECB total asset. Moreover, an
increase in this volatility component is observed in April 2017, which
coincides with the reduction in the amount of securities purchased by ECB, set
to € 60 billion per month from the previous level of € 80 billion.

Figure 2: France, Germany, Italy and Spain Base (left axis) and Policy (right
axis) component of volatility from the MAP. Sample period: June 1, 2009 to
December 31, 2019

Results remain by and large the same, when we consider the Multiplicative MAP
specifications. Once again, the proxies enter the models with the expected
sign and with the highest level of significance. In both cases, coefficients
are fairly lower, in view of the multiplicative way components combine: in
other words, while in the MAP we generally have a negative $\xi_{t}$, within
the multiplicative versions the effect by unconventional policies of dampening
volatility is had when the $\xi_{t}$ process is less than 1 (by construction
it is always positive): this requires lower proxies’ coefficients, and this is
achieved without imposing any kind of constraints.

In order to compare economic effects across models, it is instructive to
compute the corresponding marginal effects of the policy variables ($x_{t-1}$
and $\Delta_{t}$) on $\mu_{t}$: while in the additive specification they are
constant and equal to the estimated coefficients, in the multiplicative
specifications marginal effects are time varying. Looking at the average
values151515For what concerns $\Delta_{t}$, the marginal effects are
considered only with respect to announcement days, and, as such, the average
relates just to such days. (Table 5), the higher marginal effect of the
implementation proxy is associated to the P-MAP. More specifically, a marginal
increase in the proxy leads to a range of reductions in realized volatility
comprised between -1.577 (Germany) and -3.382 (Spain); conversely, on
announcement days volatility marginally increases, on average, by 2.603 and
4.146 points, in Germany and Spain, respectively. Overall, while
unconventional policies had a higher impact on debt–troubled countries, the
effect is present also for the others.

Table 5: Average marginal effects of policy variables on $\mu_{t}$ – Sample
period: June 1, 2009 - December 31, 2019 (parameter estimates by country as in
Table 3).

| Marginal effect of $x_{t-1}$ on $\mu_{t}$ | Marginal effect of $\Delta_{t}$ on $\mu_{t}$
---|---|---
| MAP | L-MAP | P-MAP | MAP | L-MAP | P-MAP
France | -1.836 | -1.571 | -2.172 | 2.817 | 2.236 | 3.177
Germany | -1.328 | -1.158 | -1.577 | 2.317 | 1.821 | 2.603
Italy | -2.354 | -2.033 | -2.693 | 3.448 | 2.785 | 3.953
Spain | -2.823 | -2.489 | -3.382 | 3.428 | 2.912 | 4.146

  * •

Note: the average marginal effect of $\Delta_{t}$ refers to the announcement
days (see text).

The evolution of the marginal effects associated to the two multiplicative
component specifications is shown in Figures 3 and 4: for both models and for
all the markets, marginal effects have a specular behaviour, to a certain
extent mirroring the behavior of the realized volatility measure. In
particular, the marginal effects of $\Delta_{t}$ have peaks in correspondence
of volatility spikes, whereas the period of lower marginal effects of the
proxy corresponds to periods of low volatility in the market. This result
validates the dynamics in our models with the policy variables we adopted.

In the case of the MAP, the impact of unconventional policies on stock market
volatility can also be measured as the ratio of the policy–related component
$\xi_{t}$ to the general level of volatility $\mu_{t}$. The average of such a
ratio signals a reduction in volatility associated with the ECB unconventional
policies between -0.6% and -1.1%, when measured for Germany and France, which
becomes more marked for Italy (-1.24%) and Spain (-1.38%).

As far as the dynamics of the $\xi_{t}$ component is concerned, the
persistence effects, driven by the coefficient $\psi$, are fairly weak: such a
coefficient is significant at 5$\%$ only for France across models and only
marginally so for Germany for the L-MAP model. As a result, the data show that
the policy–related volatility component follows the evolution of the policy
proxy (with parameter $\delta$) and the changes occurring on announcement days
(with parameter $\varphi$).

Figure 3: France, Germany, Italy and Spain: marginal effects from the L-MAP.
Sample period: June 1, 2009 to December 31, 2019  Figure 4: France, Germany,
Italy and Spain: marginal effects from the P-MAP. Sample period: June 1, 2009
to December 31, 2019

###  4.2 Out–of–sample Performance

In order to compare the one-step-ahead out-of-sample forecasting performance
across the estimated models, we compute the Model Confidence Set (MCS, Hansen
et al., 2011) using the MSE and the QLike loss functions at the 10%
significance level.

For this purpose, we split the sample at the end of a year and we compute the
one-step-ahead out-of-sample forecasts for the following year. The choice of
the splitting dates is driven by the need to exclude the most important QE
program, the EAPP, so that the first subsample considered stops at the end of
2014. We then consider each additional year in turn until 2018, in order to
consider the full period of the EAPP.

As shown in Table 6, both loss functions provide similar results, confirming
that the proxy–augmented models capture the features of the EAPP in delivering
an improved forecasting performance. This outcome is more apparent when
considering the subsamples individually. Before the EAPP announcement, the MCS
results in the 2015 column of Table 6 show, for example, that all the models
enter the best set of models (according to QLike, represented by the symbol
$\Circle$) in 3 out of 4 cases, and even that the AMEM is the best model
($\CIRCLE$) in France, Germany and Spain and it belongs to the best set in
Italy, for which the best model is the MAP (results are more mixed for the
MSE).

Similar results derive from the forecasting period ending in 2016. For both
loss functions, this time, the AMEM is the best model in 3 out of 4 cases,
whereas in Spain such a role goes to the X-MAP: no component model is present
in the best set in Germany, where the AMEM is the only model belonging to the
best set (together with the P-MAP if we consider the MSE).

This good performance of the base AMEM is seriously challenged by our models
when we move to the forecasting period ending in 2017, characterized by a
remarkable change in the ECB balance sheet composition: during this period the
ECB purchased assets at a pace of € 80 billion per month until March 2017 and
€ 60 billion per month until the end of the year. This time the
policy–augmented models have a higher forecasting power and the AMEM is always
excluded from the best set of models. Importantly, focusing on the QLike, the
MAP enters the best set in 3 out of 4 cases, whereas the P-MAP is the best
model for Germany and Spain.

This latter model becomes the best across all cases considering the
forecasting periods ending in 2018 and 2019, respectively. The difference in
the results of these two sub-samples is in the best set of models, which is
larger in the last year, possibly due to the fact that no APP was implemented
until November 2019, when the ECB purchased new assets at a level of € 20
billion per month. Overall, however, unconventional policies seem to have
played a crucial role in reducing stock market volatility.

Table 6: The Model Confidence Set results. P-value 10%

France
---
| 2015 | 2016 | 2017 | 2018 | 2019
AMEM |  $\square$ $\CIRCLE$ |  $\blacksquare$ $\CIRCLE$ |  |  |
X-MAP |  $\square$ $\square$ |  $\square$ $\Circle$ |  $\blacksquare$ $\CIRCLE$ |  |  $\square$ $\Circle$
MAP |  |  $\square$ $\Circle$ | $\Circle$ |  |  $\square$ $\square$
L-MAP |  $\blacksquare$ $\Circle$ |  $\square$ $\Circle$ |  |  |  $\square$ $\square$
P-MAP |  $\square$ $\Circle$ |  $\square$ $\Circle$ | $\Circle$ |  $\blacksquare$ $\CIRCLE$ |  $\blacksquare$ $\CIRCLE$
Germany
| 2015 | 2016 | 2017 | 2018 | 2019
AMEM |  $\blacksquare$ $\CIRCLE$ |  $\blacksquare$ $\CIRCLE$ |  | $\Circle$ |
X-MAP |  $\square$ $\Circle$ |  |  |  |
MAP |  $\square$ $\Circle$ |  |  |  |
L-MAP |  $\square$ $\Circle$ |  |  |  |  $\square$ $\Circle$
P-MAP |  $\square$ $\Circle$ |  $\square$ $\square$ |  $\blacksquare$ $\CIRCLE$ |  $\blacksquare$ $\CIRCLE$ |  $\blacksquare$ $\CIRCLE$
Italy
| 2015 | 2016 | 2017 | 2018 | 2019
AMEM |  $\square$ $\Circle$ |  $\blacksquare$ $\CIRCLE$ |  |  |
X-MAP |  $\square$ $\Circle$ |  $\square$ $\Circle$ |  $\blacksquare$ $\CIRCLE$ |  $\blacksquare$ $\square$ |  $\square$ $\CIRCLE$
MAP |  $\blacksquare$ $\CIRCLE$ |  $\square$ $\Circle$ | $\Circle$ |  |  $\square$ $\Circle$
L-MAP |  $\square$ $\Circle$ |  $\square$ $\Circle$ |  |  |  $\square$ $\Circle$
P-MAP |  $\square$ $\Circle$ |  $\square$ $\Circle$ |  |  $\square$ $\CIRCLE$ |  $\blacksquare$ $\Circle$
Spain
| 2015 | 2016 | 2017 | 2018 | 2019
AMEM | $\CIRCLE$ |  $\square$ $\Circle$ |  |  |
X-MAP | $\Circle$ |  $\blacksquare$ $\CIRCLE$ |  $\blacksquare$ $\Circle$ | $\Circle$ |
MAP |  | $\Circle$ | $\Circle$ | $\Circle$ |
L-MAP | $\Circle$ | $\Circle$ | $\Circle$ |  |  $\square$ $\square$
P-MAP |  $\blacksquare$ $\Circle$ |  $\square$ $\Circle$ |  $\square$ $\CIRCLE$ |  $\blacksquare$ $\CIRCLE$ |  $\blacksquare$ $\CIRCLE$

  * •

Note: $\square$($\Circle$) indicates models belonging to the best set
according to the MSE(QLike) loss function; $\blacksquare$($\CIRCLE$)
represents the best model.

###  4.3 Volatility responses to policy actions

Figure 5: Multi-step forecast. Sample period: June 1, 2009 to December 31,
2017. Forecast Period: January 1, 2018 - December 31, 2018

In order to further analyze the dynamic effects of the policy actions on
volatility, we graphically appraise the multi–step forecasts by model, as well
as the profile of the impulse responses to a one standard deviation shock to
the policy variable. For either case, we can evaluate the persistence of the
effects and the time taken by different models in converging to the
unconditional level of volatility. This puts us in the position to address two
substantive questions: i) for how long do QE policies affect stock market
volatility? ii) what would be the volatility response to a higher QE shock?

We focus on the sub-sample for the estimation period ending in 2017, with the
following year used as the out–of–sample period and apply multi–step ahead
forecasting,161616The derivation of the formulas is available in the
supplemental material. obtaining the results shown in Figure 5. As expected,
the duration of the effect - measured as the number of days the volatility
needs to reach the unconditional level, i.e. the unconditional mean171717For
practical purposes, convergence is considered achieved when subsequent
forecasts do not differ by more than 1 basis point of volatility. \- depends
on the country: we get an average duration between 41 business days (estimated
in Italy via P-MAP) and 96 business days (from the MAP in Germany).
Furthermore, regardless of the duration, once the unconventional policy effect
is completely absorbed by the market, volatility converges to its
unconditional level, which does not depend on the different model
specifications; however, due to the different way they account for the policy
effect, a slight difference between models presents itself, more marked for
the AMEM, which neglects those effects altogether. As expected, in all the
cases the convergence path is upward sloping, with volatility reaching a
higher level when the downward impact of unconventional policies ceases. Such
an upward profile is shared by the AMEM as well, given that volatility is
lower than its unconditional level at the beginning of the forecasting period.

Finally, we analyze how volatility would have responded to a higher QE shock.
For this purpose we have increased the QE proxies by one standard deviation
(0.26). Taking the previous multi-step forecasting values as a baseline
scenario, in Figure 6 we represent the results as the difference between the
two scenarios. The role played by unconventional policies in reducing stock
market volatility is clear: a higher unconventional policy shock would have
had a stronger downward effect, with volatility that would have been lower
down to 2.5 points in debt–troubled countries (2 points in France and Germany)
relative to the baseline scenario.

Figure 6: Impulse Response Function: the effect on volatility of a 1 standard
deviation unconventional policy shock. Sample period: June 1, 2009 to December
31, 2017. Forecast Period: January 1, 2018 - December 31, 2018

##  5 Concluding Remarks

In this paper, we suggested a new class of models to analyze how
unconventional monetary policies affect realized volatility. In detail, we
suggested variants of multiplicative error models with two components, one
representing what depends directly on QE policies, the other reproducing base
volatility dynamics: such components may combine additively or
multiplicatively. As shown by our results on ECB policies for four important
Eurozone markets, what matters for the effectiveness of these policies is the
balance sheet composition (as argued by Curdia and Woodford, 2011). Indeed, an
increase in securities held by ECB for monetary policy purposes relative to
total asset reduces volatility in both debt–troubled countries (Italy and
Spain, in our analysis) and in benchmark countries (France and Germany), with
more benefits for the former, in general. Admittedly, our policy proxies do
not allow us to go deeper into the specific impact of each implemented policy,
so that we cannot identify which of these extraordinary measures is more
effective. This, of course, is an issue worth pursuing in future analysis, as
well as the possibility to control also for spillovers among countries in a
multivariate context, which could also highlight the presence of a common
response across markets.

The presence of our proxy in the models entering the Model Confidence Set when
evaluating the out–of–sample forecasting performance underlines the importance
of the EAPP. This is also confirmed by the multi-step forecast procedure,
given that, on the one hand, unconventional policies have lasting effects in
lowering volatility for at most 90 business days, and, on the other, when
shocking unconventional policies by one standard deviation stock market
volatility decreases by up to 2.72 points.

Exploiting the component structure of our models, we can extract a separate
and distinctive signal related to the policy effects on volatility: between
the additive and multiplicative way of combining the components, the results
show a preference for the latter in estimation and out–of–sample, especially
in 2018 and 2019. In economic terms, such results document that the
unconventional monetary policy has a mitigating effect on market volatility at
times of distress, when the interest rates are close to the zero lower bound,
as a further tool available to central banks in order to restore the proper
functioning of the economy.

SUPPLEMENTARY MATERIAL

Title:

Unconventional Policies Effects on Stock Market Volatility: A MAP Approach.
Supplemental Material

Literature Review:

A summary of few empirical analysis concerning the effects of unconventional
policies mainly on the bond market.

Marginal effects:

Derivation of marginal effects of the policy variables on volatility (Table 2,
Section 3 in the main text).

Multi-step forecasting:

Formulas used in the Multi–step forecasting procedure (Section 4.3 in the main
text)

## References

  * Andersen et al. (2006) Andersen, T. G., Bollerslev, T., Christoffersen, P. F. and Diebold, F. X. (2006) Volatility and correlation forecasting, in Handbook of Economic Forecasting (Eds.) G. Elliott, C. W. J. Granger and A. Timmermann, North Holland.
  * Barndorff-Nielsen et al. (2008) Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. and Shephard, N. (2008) Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise, Econometrica, 76, 1481–1536.
  * Bollerslev (1986) Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, 307–327.
  * Bomfim (2003) Bomfim, A. N. (2003) Pre-announcement effects, news effects, and volatility: monetary policy and the stock market, Journal of Banking & Finance, 27, 133–151.
  * Brownlees et al. (2011) Brownlees, C. T., Cipollini, F. and Gallo, G. M. (2011) Intra-daily volume modeling and prediction for algorithmic trading, Journal of Financial Econometrics, 9, 489–518.
  * Brownlees et al. (2012) Brownlees, C. T., Cipollini, F. and Gallo, G. M. (2012) Multiplicative error models, in Volatility Models and Their Applications (Eds.) L. Bauwens, C. Hafner and S. Laurent, Wiley, pp. 223–247.
  * Chan and Gray (2018) Chan, K. F. and Gray, P. (2018) Volatility jumps and macroeconomic news announcements, Journal of Futures Markets, 38, 881–897.
  * Curdia and Woodford (2011) Curdia, V. and Woodford, M. (2011) The central-bank balance sheet as an instrument of monetary policy, Journal of Monetary Economics, 58, 54–79.
  * D’Amico et al. (2012) D’Amico, S., English, W., López-Salido, D. and Nelson, E. (2012) The Federal Reserve’s large-scale asset purchase programmes: rationale and effects, The Economic Journal, 122, F415–F446.
  * De Santis (2020) De Santis, R. A. (2020) Impact of the asset purchase programme on euro area government bond yields using market news, Economic Modelling, 86, 192–209.
  * Engle (1982) Engle, R. F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50, 987–1007.
  * Engle (2002) Engle, R. F. (2002) New frontiers for ARCH models, Journal of Applied Econometrics, 17, 425–446.
  * Engle and Gallo (2006) Engle, R. F. and Gallo, G. M. (2006) A multiple indicators model for volatility using intra-daily data, Journal of Econometrics, 131, 3–27.
  * Engle and Lee (1999) Engle, R. F. and Lee, G. J. (1999) A permanent and transitory component model of stock return volatility, in Cointegration, Causality, and Forecasting: A Festschrift in Honor of Clive W. J. Granger (Eds.) R. F. Engle and H. White, Oxford University Press, Oxford, pp. 475–497.
  * Engle and Sokalska (2012) Engle, R. F. and Sokalska, M. E. (2012) Forecasting intraday volatility in the US equity market. Multiplicative component GARCH, Journal of Financial Econometrics, 10, 54–83.
  * Eser and Schwaab (2016) Eser, F. and Schwaab, B. (2016) Evaluating the impact of unconventional monetary policy measures: Empirical evidence from the ECB’s Securities Markets Programme, Journal of Financial Economics, 119, 147–167.
  * Georgiadis and Gräb (2016) Georgiadis, G. and Gräb, J. (2016) Global financial market impact of the announcement of the ECB’s asset purchase programme, Journal of Financial Stability, 26, 257–265.
  * Ghysels et al. (2017) Ghysels, E., Idier, J., Manganelli, S. and Vergote, O. (2017) A high-frequency assessment of the ECB Securities Markets Programme, Journal of the European Economic Association, 15, 218–243.
  * Haitsma et al. (2016) Haitsma, R., Unalmis, D. and de Haan, J. (2016) The impact of the ECB’s conventional and unconventional monetary policies on stock markets, Journal of Macroeconomics, 48, 101–116.
  * Hansen et al. (2011) Hansen, P. R., Lunde, A. and Nason, J. M. (2011) The model confidence set, Econometrica, 79, 453–497.
  * Otranto (2015) Otranto, E. (2015) Capturing the spillover effect with multiplicative error models, Communications in Statistics-Theory and Methods, 44, 3173–3191.
  * Patton (2011) Patton, A. J. (2011) Volatility forecast comparison using imperfect volatility proxies, Journal of Econometrics, 160, 246 – 256.
  * Shogbuyi and Steeley (2017) Shogbuyi, A. and Steeley, J. M. (2017) The effect of quantitative easing on the variance and covariance of the UK and US equity markets, International Review of Financial Analysis, 52, 281–291.
  * Toda and Yamamoto (1995) Toda, H. Y. and Yamamoto, T. (1995) Statistical inference in vector autoregressions with possibly integrated processes, Journal of Econometrics, 66, 225–250.
  * Voutsinas and Werner (2011) Voutsinas, K. and Werner, R. A. (2011) New evidence on the effectiveness of “Quantitative Easing” in Japan, Tech. rep., CFS working paper.