# Bridging the Communication Gap: Artificial Agents Learning Sign Language
through Imitation

Federico Tavella1,2, Aphrodite Galata2 and Angelo Cangelosi1,2 *Corresponding
author<EMAIL_ADDRESS>Centre for Robotics and
AI, University of Manchester, Manchester, M13 9PL, UK2Department of Computer
Science, University of Manchester, Manchester, M13 9PL, UK

###### Abstract

Artificial agents, particularly humanoid robots, interact with their
environment, objects, and people using cameras, actuators, and physical
presence. Their communication methods are often pre-programmed, limiting their
actions and interactions. Our research explores acquiring non-verbal
communication skills through learning from demonstrations, with potential
applications in sign language comprehension and expression. In particular, we
focus on imitation learning for artificial agents, exemplified by teaching a
simulated humanoid American Sign Language. We use computer vision and deep
learning to extract information from videos, and reinforcement learning to
enable the agent to replicate observed actions. Compared to other methods, our
approach eliminates the need for additional hardware to acquire information.
We demonstrate how the combination of these different techniques offers a
viable way to learn sign language. Our methodology successfully teaches 5
different signs involving the upper body (i.e., arms and hands). This research
paves the way for advanced communication skills in artificial agents.

##  I Introduction

The development of artificial agents like humanoid robots has opened up a wide
range of possibilities in various scenarios, spanning from industrial to
domestic settings. Consequently, communication is crucial for human-robot
interaction, whether verbal [1] [2] or non-verbal [3] [1] [4] [2]. Verbal
communication is the default mode for most users, but it is not always
suitable or accessible for everyone. Deaf individuals, for instance, rely
heavily on non-verbal communication methods such as sign language. This
necessitates the development of robotic sign language [5] [6] [7] [8] [9],
where robots are designed to understand and speak sign languages. This field
aims to bridge the communication gap between the deaf community and the
broader population, allowing seamless interaction and engagement.

According to the World Federation of Deaf “there are more than 70 million deaf
people worldwide” [10], while according to WHO “by 2050 nearly 2.5 billion
people are projected to have some degree of hearing loss and at least 700
million will require hearing rehabilitation” [11], it is clear that also
machines need to learn how to convey messages using a medium different from
sound. The societal benefits of having robots that can speak sign language are
extensive and hold immense potential across various fields. For example, in
the field of education, these robots could assist in classrooms by providing
real-time sign language interpretation, enabling deaf students to fully
participate and engage in lessons.

Recently, researchers started approaching the problem of robotics sign
language, focusing on imitation [12] [7] or translation [8]. However, neither
of these approaches enables the robot to build an embodied representation of
different signs based on visual data, as they focus on retargeting rather than
learning. Moreover, they tend to use using additional hardware instead of
being vision based. [5] showed how a combination of computer vision and
reinforcement learning can be used to learn fingerspelling, but their work
does not attempt to address signs involving the full body.

In conclusion, our work is motivated by a gap in the current state-of-the-art
in the field of robotics sign language. In particular, there is no research
which approaches the problem of whole body sign language acquisition in
artificial agents based on visual data imitation. To summarise, the
contributions of our research (illustrated in Figure 1) are manifold:  1. Sign
Language Acquisition:we address the challenging problem of sign language
acquisition from RGB video for words, focusing on enabling a machine to
imitate sign language gestures which involve both hands and arms;  2. URDF
Model Development:we create a URDF model of a simulated character, which is
noteworthy as it is the first known model capable of imitating the whole body
and both hands in sign language gestures;  3. Experimental
Validation:Extensive experiments are conducted to optimise parameters for
rewards and hyperparameters for training models;  4. Successful Imitation of
Signs:In the end, we achieve a significant milestone by identifying a specific
reward structure and set of hyperparameters that enables our approach to
successfully learn how to imitate 5 different sign language words, indicating
the practical viability of our method.

Figure 1: Overview of our proposal. Given a RGB video of a person performing a
sign, we use a deep learning approach to extract body information and use such
information to teach a simulated humanoid how to perform a specific sign
(i.e., above).

##  II Background

###  II-A Sign language in robotics

Despite all the recent progress, the field of robotics sign language is still
very scarce. Most of the research so far focused on robot tutors to aid sign
language acquisition in humans [13, 14] [15] [16] [17] [18] [19] [20] [21]
[22]. In comparison, very few tried to make robots speak sign language [6] [7]
[8] [9]. Notably, [6] present an imitation system to teach a humanoid robot to
perform sign language by replicating observed demonstrations. By using a
Kinect camera and Leap motion sensor, the robot can mimic demonstrated signs
based on 3D keypoints and inverse kinematics. Alternatively, in [7], the user
wears a motion capture suit and performs a sign multiple times to train a set
of parallel Hidden Markov Models to encode each sign. Additionally, the
authors ensured signs comprehensibility and collision avoidance via a special
mapping from the user’s workspace to the robot’s joint space. The performance
of the system was assessed by teaching 10 signs in Persian Sign Language (PSL)
to the robot and involving PSL users to investigate how easy was for them to
recognise the signs. 8 participants out of ten managed to recognise the signs
during the first attempt, while 10 out of 10 did so during the second attempt.
[12] propose a method for robots to “learn” sign language by combining
learning and optimisation-based motion retargeting. They establish a mapping
relationship between the latent space and the robot motion space with a graph
decoder. Using the difference between the human and robot motion, they search
for the optimal latent code that minimises the gap by gradient descent.
Practically, they build an autoencoder and define multiple losses to minimise
the difference between human and robot skeletons in terms of joints
orientation and end effectors positions. Hence, despite the advancements in
robotic technologies, there remains a glaring lack of research focused on
robots that can acquire sign language through imitation, a crucial learning
mechanism for humans.

###  II-B Characters animation based on demonstrations

In the recent literature, we can find several examples of learning-based
approaches to control humanoid characters. For instance, [23] present a real-
time character control mechanism using a novel neural network architecture
called a Phase-Functioned Neural Network. In their new approach, they provide
as additional input the phase (i.e., a cyclic function like sine or cosine),
so that the character can cyclically reproduce the behaviour. For example,
walking forward is the repetition of two basic movements in repetition:
alternating moving one foot forward after another. In addition to the phase,
the authors provide as an input the user’s control (e.g., move left, right)
and the geometry of the scene to generate motions that reflect the desired
control. Most notably, [24] [25] developed a state-of-the-art approach to
skills acquisition. In [24], the authors trained a policy (i.e., a phase-
functioned neural network [23]) using reinforcement learning based on motion
capture data. The motion capture data are acquired from an online library and
include skills like running, jumping, doing a backflip, etc. Authors define
ad-hoc weighted rewards for different components (e.g., pose, movement
velocity, end effectors positions), which indicate to the agent the value of
the chosen action. Finally, in [25], the authors included a pose estimator and
a motion reconstruction component to extract data from RGB videos from YouTube
and use them to train the control policy. [26] develop a technique for
learning controllers for a large set of different behaviours. They divide the
library of motions into clusters of similar motions and train a different
policy for each cluster, which combined can reproduce the whole library. [27]
try to obviate the need to manually design rewards and tools for motion
selection by using an automated approach based on adversarial imitation
learning. The adversarial RL procedure automatically selects which motion to
perform, dynamically interpolating and generalising from the dataset. Finally,
[28] combine techniques from adversarial imitation learning and unsupervised
reinforcement learning to develop skill embeddings that produce life-like
behaviours. Most interestingly, they achieve significant speed-up by
parallelising the training by leveraging a novel GPU-based simulator,
decreasing the extensive time required to train a single policy. Similarly to
our proposal, [5] focuses on teaching robots sign language fingerspelling
through imitation learning from RGB videos. They develop a model for a robotic
hand and compare two learning algorithms (PPO [29] and SAC [30]). The study
successfully imitates six different fingerspelled letters with performance
similar to ideal retargeting.

##  III Methods

We begin by discussing how we construct our whole body model by combining
existing body and hand models. We then provide a brief overview of how we use
a deep learning model to analyse RGB videos and extract data necessary for
learning to imitate – namely, joint rotations. Next, we describe how we use
reinforcement learning to imitate sign language based on the information
extracted from the videos.

###  III-A Whole body model

We re-adapt a humanoid model available in the literature [24] to our scenario.
It is composed of 13 joints (2 shoulder, elbows, hips, knees, ankles and 1
neck, chest and waist/root). 9 out of 13 joints have 3 degrees of freedom,
while the remaining 4 (2 knees and 2 elbows) have only 1 Degree of Freedom
(DoF), for a total of 31 DoFs. It is 1.62cm tall and weighs 45 kg. As for the
hands, we reuse the model we describe in [5], but we duplicate the model and
mirror it to reflect the orientation of the right and left hands. Moreover, we
change the shape of the wrist from a small cube to a parallelepiped in order
to make it resemble the palm of a hand. Figure 2 shows our final model from
different points of view.

(a) Front view

(b) Side view

(c) Hand zoom in

Figure 2: Whole body model. We integrated an available body model [24] with a
hand model [5], replicated and mirrored to obtain both left and right hands.

We simulate the humanoid using PyBullet [31], an open-source physics
simulator. For each joint motor, the simulated controller calculates the error
as $\varepsilon=k_{p}\Delta P+k_{d}\Delta V$, where $k_{p}$ and $k_{d}$ are,
respectively, the position and velocity gains, and $\Delta P=P-\hat{P}$ and
$\Delta V=V-\hat{V}$ are the position and velocity errors (i.e., the
difference between the desired and the actual value respectively).

###  III-B Motion extraction and imitation

The work by Peng et al. [25] employs Human Motion Reconstruction [32] in
conjunction with a motion reconstruction method to derive rotational
information from videos, subsequently feeding it into DeepMimic [24]. It
should be noted that their study focused on specific, unconventional movements
such as backflips, which are typically absent from datasets used to train such
models. In contrast, our particular context involves more general motions,
prompting us to utilise FrankMocap [33] to extract 3D rotations and keypoints
for both the upper body and hands. Both models were used in the literature to
extract information from sign language videos [34] [35]. Figure 3 provides a
visual depiction of the extracted keypoints and rotations for the upper body
and hands. However, it is important to highlight that, given our emphasis on
sign language, we do not consider information related to the lower body,
specifically the legs.

(a) Body joints

(b) Hand joints

Figure 3: Body and hand keypoints extracted using FrankMocap. The body is
composed of 24 keypoints, while each hand has 21 keypoints. (Both reprinted
from [36])

Lately, imitation learning and behavioural cloning [37] took advantage of many
techniques from Reinforcement Learning (RL). RL is commonly represented as a
Markov Decision Process (MDP) [38], which comprises four key components: a set
of states $S$, a set of actions $A$, a reward function $R$, and the transition
probability $P_{a}(s,s^{\prime})$. An agent engaged in learning observes the
environment and its current state $s$. Based on these observations (i.e., the
state), it selects an action $a$ to transition to a new state $s^{\prime}$
according to the probability given by
$P_{a}(s,s^{\prime})=P(s_{t+1}=s^{\prime}|s_{t}=s,a_{t}=a).$ This transition
yields a reward $R_{a}(s,s^{\prime})$, or $r$, indicating the desirability of
the chosen action. The ultimate goal is to acquire a policy, a mapping from
states to actions, that maximises the expected cumulative reward.
Additionally, a policy can be parametric, denoted as $\pi_{\theta}$. In this
context, the policy needs to identify the optimal parameters $\theta^{*}$ that
maximise the expected cumulative reward.

Various algorithms are available for discovering an optimal policy, with the
selection typically influenced by the nature of the action and/or state spaces
(discrete or continuous). For more comprehensive information, readers are
directed to [37]. In our specific case, we employ the Proximal Policy
Optimization (PPO) algorithm [29] to estimate a policy for our problem.

PPO is an optimisation algorithm used when both action and state spaces are
continuous. It is a policy gradient method, where the gradient of the expected
cumulative reward is calculated using trajectories $\tau$ – i.e., sequences of
$(s,a,r)$ over a set of contiguous time steps – sampled by following the
policy. Thus, given a parametric policy $\pi_{\theta}$ and $T$ steps, the
expected reward is

$J(\theta)=\mathbb{E}_{\tau\sim
p_{\theta}(\tau)}\left[\sum_{t=0}^{T}\gamma^{t}r_{t}\right]$ (1)

where
$p_{\theta}(\tau)=p(s_{0})\prod_{t=0}^{T-1}p(s_{t+1}|s_{t},a_{t})\pi_{\theta}(a_{t}|s_{t})$
is the distribution over all possible trajectories
$\tau=(s_{0},a_{0},s_{1},a_{1},...,a_{T-1},s_{T})$ induced by the policy
$\pi_{\theta}$, $p(s_{0})$ being the initial state distribution and
$\gamma\in[0,1]$ is a discount factor used to ensure that the reward has an
upper bound. The policy gradient can be estimated as
$\nabla_{\theta}J(\theta)=\mathbb{E}\left[\nabla_{\theta}log(\pi_{\theta}(a_{t}|s_{t}))\mathcal{A}_{t}\right]$
where $\mathcal{A}_{t}=R_{t}-V(s_{t})$ represents the advantage of taking an
action $a_{t}$ from a given state $s_{t}$, $R_{t}$ is the cumulative reward by
a particular trajectory starting from state $s_{t}$ at time t, and
$V(s_{t})=\mathbb{E}_{t}\left[R_{t}|\pi_{\theta},s_{t}\right]$ is the value
function that estimates the average reward for starting at $s_{t}$ and
following the policy for all subsequent steps.

We formulate the sign language acquisition problem as a control problem. We
model it as an MDP and use PPO [29] to learn a policy for each different
motion. However, as we added a full body model, it was not clear whether it
could be possible to reuse the scaling factors from [24] and [5] for the
subrewards. One option would be to recalibrate the weights of each different
sub-reward, but that would imply additional experiments to find the optimal
balance between different rewards. In a resource eager scenario like when
using reinforcement learning, additional experiments are a major drawback.
Hence, we take inspiration from [26] and define our reward as a multiplicative
reward (rather than an additive) as $r_{t}=r_{t}^{p}\cdot r_{t}^{v}\cdot
r_{t}^{e}\cdot r_{t}^{r}$ where $r_{t}^{x}$ is the reward at time $t$ for
quantity $x$ (which can be body, velocity, end effectors and root), calculated
as $r_{t}^{x}=e^{-k^{x}\varepsilon_{t}^{x}}$ with $\varepsilon_{t}^{x}$ being
the error for quantity $x$ at time $t$, and $k^{x}$ being a factor used to
balance the reward based on the scale of the error, as described in [24]. In
addition, we redefine $r_{t}^{p}$ and $r_{t}^{v}$ as
$r_{t}^{p}=r_{t}^{p,b}\cdot r_{t}^{p,h}$ and $r_{t}^{v}=r_{t}^{v,b}\cdot
r_{t}^{v,h},$ with $b$ indicating the body and $h$ the hands. We choose to
divide the rewards (and the errors) for body and hands as we believe this give
us a more clear indication of what the algorithm can (not) learn. As for the
end-effectors reward $r_{t}^{e}$, we consider just the position of the wrists,
as opposed to [5] in which they considered the position of the fingertips.
Table I summarises the differences between the three approaches.
Complementary, scaling factors are the same from [24], for both body and
hands.

| DeepMimic | SiLa | Ours
---|---|---|---
Number of joints | 13 | 16 | 45
Number of DoFs | 34 | 15 | 50
State space size | 197 | 210 | 509
Action space size | 36 | 15 | 57
TABLE I: Comparison between DeepMimic [24], SiLa [5] for fingerspelling and
our whole body approach.

##  IV Experimental setup

###  IV-A Controller tuning

As for the controllers tuning, we do not need to go through the whole
procedure described in [5] for the simulated hand, as we can reuse the values
of the controllers pose gain $k_{p}$ and velocity gain $k_{d}$ from [24] for
the body and from [5] for the hands. However, we need to refine the values for
the arms, as the fact that we attached end effectors to them means that we
also changed the dynamics of the whole arms. As opposed to [5] approach to
tuning the controller parameters, we do not deploy a completely automated
approach, but rather an empirical one. Considering we aim to adapt only the
values for 2 joints (i.e., shoulder and elbow), we find it easier to manually
adapt the values balancing the body and hand error rather than develop a
sophisticated function to find an acceptable equilibrium between the two
errors.

###  IV-B Motion imitation

We exploit a motion file from [33] for the hyperparameter tuning of the
multilayer perceptron representing our policy. However, preliminary results
immediately showed that the scaling factors for the hands (that were the same
as for [24]) did not yield satisfactory results. In fact, as the reward is a
product of sub-rewards, if one of the sub-rewards is 0, then the total reward
will be 0 as well. Thus, we run additional searches for the parameter for hand
pose and velocity. We use a mixture of automated and manual approaches,
alternating hyperparameters and scaling factors explorations. We test the
permutation of the following values for pose and velocity: [2, 1, 0.5, 0.2]
and [$10^{-1}$, $10^{-2}$, $10^{-3}$, $10^{-4}$] respectively. Intuitively,
given that the reward $r_{t}^{x}=e^{-k^{x}\varepsilon_{t}^{x}}$, tuning the
scaling factor $k^{x}$ makes the reward more (or less) lenient towards the
error $x$.

We perform a hyperparameter search based on Bayesian optimisation to find the
ideal hyperparameters. When performing such a search, we train our algorithm
for 25 million steps, which is 50% of how long we train to achieve imitation
for the final policies. The set over which we perform the search is summarised
in Table II. Finally, we test the generalisability of such hyperparameters
over five different signs, repeating each attempt 10 times using different
seeds and training for 50 million steps. We choose the different signs via a
qualitative evaluation of the output of FrankMocap. We choose a small amount
of different motions to imitate due to extensive time required to train and
statistically validate multiple policies. In addition, when choosing the signs
we also take into account the limitations of the model from [5] (e.g., only 1
DoF for the elbow, wrist or phalanges of each finger). The lemmas
corresponding to the five signs are above, snow, father, mother and yes. Their
respective identifiers from WLASL [39] are 00433, 52861, 69318, 69402 and
69546, which we use as labels in our results.

Parameter | Values
---|---
hidden layers size | (512-1024-512), (256-512-256)
learning rate | (1, 3, 10, 30, 100) x $10^{-6}$
number of steps | 512, 1024, 4096
batch size | 128, 256, 512
log std dev | -5, -3, -2, -1
discount factor | 0.9, 0.95
number of epochs | 3, 5, 10
TABLE II: Values of different hyperparameters for PPO.

##  V Results

Our fist steps aim at finding the ideal scaling factors $k^{p,h}$ and
$k^{v,h}$ for the rewards $r_{t}^{p,h}$ and $r_{t}^{v,h}$. We explore the
permutations of $k^{p,h}$ = [2, 1, 0.5, 0.2] and $k^{v,h}$ = [$10^{-1}$,
$10^{-2}$, $10^{-3}$, $10^{-4}$] using a Bayesian approach. We find two
combinations of parameters which provide the best results: one with
$k^{p,h}=0.5$ and $k^{v,h}=5\cdot 10^{-4}$, and one with $k^{p,h}=0.2$ and
$k^{v,h}=10^{-4}$. This means that, in order to enable our algorithm to be
able to learn, our reward needs to be more lenient with the errors regarding
the pose and velocity of the hands, as the previous values used in [5] were
$k^{p,h}=0.5$ and $k^{v,h}=0.05$. Consequently, we perform two hyperparameters
searches to test different combinations of values $k^{p,h}$ and $k^{v,h}$. In
particular, we test $k^{p,h}=0.5$ and $k^{v,h}=5\cdot 10^{-4}$ versus
$k^{p,h}=0.2$ and $k^{v,h}=10^{-4}$. The former does not provide any
satisfactory result (i.e., max reward after 25 million steps around 0.02),
while the latter provides some interesting candidates, as a couple of runs
have a reward higher than 0.4 at 50% of the training. Thus, we confirm that
the ideal parameters are $k^{p,h}=0.2$ and $k^{v,h}=10^{-4}$.

We test these parameters on the sign above. We analyse the results by checking
the error of the imitation results – i.e., the difference between the
reference motion and the simulated one. By doing so, we observe a delay
between the replicated and reference position of elbows. Thus, we modify the
velocity gain $k_{d}$ for the shoulder and elbow, in order to improve the
responsiveness of our model. By changing the value from 40 to 8 for the
shoulder and 30 to 6 for the elbow, we notice a significant improvement
(Figure 4) without significantly affecting the rest of the connected joints,
like the wrist.

Figure 4: Elbow dynamics with $k_{d}=40$ (top) vs $8$ (bottom).

In addition, as we changed one of the parameters of the controller and
slightly increased the velocity error, we decreased the scaling factor via
some manual tuning. We change $k^{v,b}$ from $10^{-1}$ to $5\cdot 10^{-3}$ so
that the reward related to the body velocity error is higher. In the end, we
select the configuration summarised in Table III.

hidden layers size | learning rate | N steps
---|---|---
256, 512, 256 | $3\cdot 10^{-6}$ | 1024
batch size | log std dev | discount | N epochs
128 | -3 | 0.95 | 10
TABLE III: Final set of hyperparameters selected using the tuning motion.

Table IV summarises the different scaling factors we explore. In addition, we
provide an approximation of the component $r^{p}\cdot r^{v}$ of the reward,
based on the error measured while replicating the original movement. We can
see how changing $k^{b,v}$ can bring a (hypothetical) increment of
approximately 10%. Given that we identified reward and hyperparameters, we
proceed to test the generalisability of such values over different motions.

| Default | Run 1 | Run 2 | Run 3
---|---|---|---|---
$k^{p,b}$ | 2 | 2 | 2 | 2
$k^{p,h}$ | 2 | $5\cdot 10^{-1}$ | $2\cdot 10^{-1}$ | $2\cdot 10^{-1}$
$k^{v,b}$ | $10^{-1}$ | $10^{-1}$ | $10^{-1}$ | $5\cdot 10^{-3}$
$k^{v,h}$ | $10^{-1}$ | $5\cdot 10^{-4}$ | $10^{-4}$ | $10^{-4}$
$r^{p}\cdot r^{v}$ | 0 | 0.297 | 0.694 | 0.790
TABLE IV: Estimated sub-rewards for different scaling values, calculate using
the tuning motion.

Figure 5 shows the results for the five different signs. Each sign is
calculated as mean (and standard deviation) over 10 different seeds. We can
see that, out of five different signs, only three achieve a cumulative reward
of around 1800. The two lemmas for which the algorithm does not converge to
acceptable results are father and mother. The signs are similar, as both
involve placing the hand fully open, with the former attaching the thumb
forehead and the latter to the chin.

Figure 5: First run of 5 signs, the average cumulative reward is calculated
over 10 different seeds

By looking at the single runs, we notice that the error never converges over 7
different runs with different seeds. Considering we change the reference
motion, we roll back to test two previous configurations for the scaling
factors. In this way, we can double-check whether $k_{p,h}$ and $k_{v,b}$
actually contribute to the learning process. We carry out a hyperparameters
search over the following configurations, summarised by the first two columns
of Table V.

| Run 1 | Run 2 | Run 3 | Run 4
---|---|---|---|---
$k^{p,b}$ | 2 | 2 | 1 | $5\cdot 10^{-1}$
$k^{p,h}$ | 2 | $2\cdot 10^{-1}$ | $2\cdot 10^{-1}$ | $2\cdot 10^{-1}$
$k^{v,b}$ | $5\cdot 10^{-3}$ | $10^{-1}$ | $10^{-3}$ | $5\cdot 10^{-3}$
$k^{v,h}$ | $10^{-4}$ | $5\cdot 10^{-4}$ | $10^{-4}$ | $10^{-4}$
$r^{p}\cdot r^{v}$ | 0.656 | 0.060 | 0.785 | 0.715
TABLE V: Estimated sub-rewards for different scaling values, calculate using
the father motion.

The first one aims to reduce the relevance of the reward regarding of the pose
of the hands, while the second one targets the body and hand velocity.
However, neither of the sweeps produces a single run in which the different
error components converge, and thus the reward is approximately 0. Hence, we
try to modify the scaling factors according to the last 2 columns of Table V.
In the first run, we change all the factors except for the one associated with
the pose of the hands. We can see a significant improvement over the previous
sweep. However, we notice that the reward for the body pose oscillates
considerably, while the reward for the body velocity has a high value but with
very few variations. Thus, we try to address these issues with our second
sweep. We reduce the scaling factor associated with the body pose and increase
the one associated with the body velocity. From this final sweep, we select a
set of hyperparameters to test over the other signs. Table VI summarises the
best set of hyperparameters we discover in our extensive exploration. Overall,
the only difference is the parameters regarding the number of epochs and the
number of steps, which changed from 10 to 5 and 1024 to 512 respectively.

hidden layers size | learning rate | N steps
---|---|---
256, 512, 256 | $3\cdot 10^{-6}$ | 512
batch size | log std dev | discount | N epochs
128 | -3 | 0.95 | 5
TABLE VI: Selected hyperparameters using the motion representing the sign
father. These hyperparameters are selected following quantitative (i.e.,
reward) and qualitative (i.e., visually analysing the training curve)
evaluations.

Finally, we train our model with the selected hyperparameters over the
different signs. Figure 6 shows the learning curves as average (and standard
deviation) over 10 different seeds. We can see how, when compared to Figure 5,
the learning curve is less steep. However, all five signs achieve a cumulative
reward above 1500, as opposed to our previous attempt in which two of them did
not surpass 750. Table VII provides a convenient comparison of the results of
the two different runs and the performance ceiling of ideal retargeting.

Figure 6: Final run, the average cumulative reward is calculated over 10 different seeds Motion | Run 1 | Run 2 | Retargeting
---|---|---|---
00433 | $1843\pm 16$ | $1838\pm 13$ | 1976
52861 | $1777\pm 62$ | $1840\pm 9$ | 1961
69318 | $745\pm 408$ | $1568\pm 41$ | 1954
69402 | $600\pm 685$ | $1685\pm 16$ | 1961
69546 | $1732\pm 27$ | $1780\pm 16$ | 1967
TABLE VII: Cumulative rewards over 2000 steps, comparing the two different
sets of hyperparameters and ideal retargeting

##  VI Discussion

Our paper introduces a novel approach to address the problem of sign language
acquisition from demonstration. We have successfully developed a URDF model of
a humanoid with two hands, each equipped with dexterous fingers. Our model
stands as a novel solution enabling the simultaneous imitation of body and
hand movements. Leveraging off-the-shelf pre-trained pose estimation models,
such as FrankMocap [33], we eliminate the need for additional hardware to
capture spatio-temporal information. Furthermore, we elucidate the methodology
for modelling rewards, facilitating imitation via a well-established
reinforcement learning algorithm, such as PPO. Our extensive experimentation
on reward parameters and model hyperparameters illustrates the iterative
process required when using reinforcement learning. We identified an optimal
set of parameters that enables our approach to effectively imitate five
different sign language signs. Consequently, we can assert that our proposal
offers a viable avenue for acquiring sign language.

Looking ahead, our work opens up to several compelling future directions.
Firstly, enhancing our approach involves increasing the DoFs for each hand
joint from 1 to 2, addressing a limitation identified in the model provided by
[5]. This enhancement would broaden the range of sign language signs that our
system can effectively imitate. Another avenue for improvement lies in
devising mechanisms for leveraging previous experiences to expedite the
learning process for new signs. Finally, the ultimate frontier in deploying
our research lies in the transition from a simulated environment to a real-
world setting. This would entail the integration of our policies into a
physical robot capable of performing signs, with the added challenge of human
recognition of the signs executed by the robot. This would be a significant
milestone in making sign language more accessible to individuals who rely on
it as their primary mode of communication.

## References

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$\displaystyle\leq
T^{-1}\left\|(\hat{\bm{\phi}}_{i\cdot}^{L}-\bm{\phi}_{i\cdot})\mathbf{X}\right\|_{2}^{2}+2T^{-1}|(\hat{\bm{\phi}}_{i\cdot}^{L}-\bm{\phi}_{i\cdot})\mathbf{X}\mathbf{u}_{i\cdot}^{\prime}|$
$\displaystyle\leq
T^{-1}\left\|(\hat{\bm{\phi}}_{i\cdot}^{L}-\bm{\phi}_{i\cdot})\mathbf{X}\right\|_{2}^{2}+2T^{-1}\|\hat{\bm{\phi}}_{i\cdot}^{L}-\bm{\phi}_{i\cdot}\|_{1}\|\mathbf{X}\mathbf{U}^{\prime}\|_{\max}$
$\displaystyle\lesssim
s_{i}\lambda^{2}+s_{i}\lambda\lambda=2s_{i}\lambda^{2},$

where the last inequality holds by Proposition 1 $(b)$$(c)$ and Lemma 3. To
evaluate the second term, putting $x=\sqrt{2\nu c_{uu}^{2}T^{-1}\log N}$ in
Lemma 1 $(a)$ yields

$\displaystyle\operatorname{\mathbb{P}}\left(\max_{i}\left|\frac{1}{T}\sum_{t=1}^{T}\left(u_{it}^{2}-\operatorname{\mathbb{E}}u_{it}^{2}\right)\right|\gtrsim\sqrt{\frac{\log
N}{T}}=o(\lambda)\right)=O(N^{-\nu}).$

Combining these inequalities gives $(ii)\lesssim
s_{i}\lambda^{2}+o(\lambda)=o(\lambda)$ since $s_{i}\lambda=o(1)$ by the
assumed condition $\bar{r}_{i}=o(1)$.

Consequently we obtain

$\displaystyle(a)~{}~{}\left\\{(ii)+\sigma_{i}^{2}\right\\}(i)+(ii)\omega_{j}^{2}$
$\displaystyle\lesssim\left(M_{\omega}^{2}\lambda+M_{\omega}^{3-2r}\lambda^{2-r}s_{\omega}\right)+o(\lambda)$
$\displaystyle\lesssim M_{\omega}^{2}\lambda+\min_{i\in[N]}\bar{r}_{i}^{2}$

and

$\displaystyle(b)~{}~{}\left\\{(ii)+\sigma_{i}^{2}\right\\}(i)^{\prime}+(ii)\omega_{j}^{2}$
$\displaystyle\lesssim M_{\omega}^{2}\lambda+o(\lambda)\asymp
M_{\omega}^{2}\lambda.$

Denote
$m_{ij}=\sigma_{i}\sqrt{{\bm{\omega}}_{j}^{\prime}\bm{\Sigma}_{x}{\bm{\omega}}_{j}}=\sigma_{i}\omega_{j}$
and $\hat{m}_{ij}$ is either
$\hat{\sigma}_{i}\sqrt{\hat{\bm{\omega}}_{j}^{\prime}\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}}$
or $\hat{\sigma}_{i}\hat{\omega}_{j}$. For any $r$ such that
$|\hat{m}_{ij}^{2}-m_{ij}^{2}|\leq r$, we have $\hat{m}_{ij}^{2}\geq
m_{ij}^{2}-r$, and hence, $\hat{m}_{ij}\geq m_{ij}-\sqrt{r}$. Therefore,

$\displaystyle r\geq\left|m_{ij}^{2}-\hat{m}_{ij}^{2}\right|$
$\displaystyle=\left|m_{ij}\hat{m}_{ij}+\hat{m}_{ij}^{2}\right|\left|\frac{m_{ij}}{\hat{m}_{ij}}-1\right|$
$\displaystyle\geq\left|m_{ij}\left(m_{ij}-\sqrt{r}\right)+m_{ij}^{2}-r\right|\left|\frac{m_{ij}}{\hat{m}_{ij}}-1\right|$
$\displaystyle=\left|2m_{ij}^{2}-O\left(\sqrt{r}\right)\right|\left|\frac{m_{ij}}{\hat{m}_{ij}}-1\right|,$

which gives the inequality,

$\displaystyle\left|\frac{m_{ij}}{\hat{m}_{ij}}-1\right|\lesssim\frac{r}{\left|m_{ij}^{2}-O(\sqrt{r})\right|}.$

Since $\min_{i,j}m_{ij}^{2}\geq\gamma^{2}$ by Condition 5, this completes the
proof. ∎

######  Lemma 8.

Suppose $\nu>2$ and $\max_{i\in[N]}\bar{v}_{i}=O(T^{-\kappa_{1}})$ for some
constants $\kappa_{1}\in(0,1/2)$. If Conditions 1–5 are assumed, then for each
$(i,j),(k,\ell)\in\mathcal{H}$ and any $x>0$, the following inequalities hold:

$\displaystyle(a)~{}$
$\displaystyle~{}\operatorname{\mathbb{P}}(\mathcal{Z}_{ij}\geq
x+\delta_{1})-O((N\vee T)^{-\nu+2})$
$\displaystyle\qquad\qquad\leq\operatorname{\mathbb{P}}(\texttt{T}_{ij}-\sqrt{T}\phi_{ij}/\hat{m}_{ij}\geq
x)\leq\operatorname{\mathbb{P}}(\mathcal{Z}_{ij}\geq x-\delta_{1})+O((N\vee
T)^{-\nu+2}),$ $\displaystyle(b)~{}$
$\displaystyle~{}\operatorname{\mathbb{P}}(\texttt{T}_{ij}-\sqrt{T}\phi_{ij}/\hat{m}_{ij}\geq
x,\texttt{T}_{k\ell}-\sqrt{T}\phi_{k\ell}/\hat{m}_{k\ell}\geq x)$
$\displaystyle\qquad\qquad\leq\operatorname{\mathbb{P}}(\mathcal{Z}_{ij}\geq
x-\delta_{1},\mathcal{Z}_{k\ell}\geq x-\delta_{1})+O((N\vee T)^{-\nu+2}),$

where $(\mathcal{Z}_{ij},\mathcal{Z}_{k\ell})$ is a bivariate standard normal
random vector with the covariance (correlation)
$\rho_{(i,j),(k,\ell)}=\sigma_{ik}\omega_{j\ell}/(\sigma_{i}\sigma_{k}\omega_{j}\omega_{\ell})$
and $\delta_{1}=O(T^{-\kappa})$ for some $\kappa\in(0,\kappa_{1}]$ is some
positive sequence.

###### Proof of Lemma 8.

Prove $(b)$. Fix $(i,j)$ and $(k,\ell)$ arbitrary and let
$\tilde{z}_{ij}=T^{-1/2}\sum_{t=1}^{T}\xi_{t}^{(i,j)}$ with
$\xi_{t}^{(i,j)}=u_{it}\mathbf{x}_{t}^{\prime}{\bm{\omega}}_{j}/m_{ij}$ and
$m_{ij}=\sigma_{i}\omega_{j}$. From Theorem 1 we have

$\displaystyle\operatorname{\mathbb{P}}(\texttt{T}_{ij}-\sqrt{T}\phi_{ij}/\hat{m}_{ij}\geq
x,\texttt{T}_{k\ell}-\sqrt{T}\phi_{k\ell}/\hat{m}_{k\ell}\geq x)$
$\displaystyle\qquad\leq\operatorname{\mathbb{P}}(\tilde{z}_{ij}\geq
x-\bar{v}_{i},\tilde{z}_{k\ell}\geq x-\bar{v}_{k})+O((N\vee T)^{-\nu+2}).$

Note that
$(\tilde{z}_{ij},\tilde{z}_{k\ell})=T^{-1/2}\sum_{t=1}^{T}(\xi_{t}^{(i,j)},\xi_{t}^{(k,\ell)})$
is the sum of the square integrable martingale difference sequence with
respect to the filtration,
$\mathcal{F}_{t}=\sigma\\{\mathbf{u}_{t-s},\mathbf{x}_{t-s+1}:s=0,1,\dots\\}$.
Kifer (2013, Theorem 3.2) establishes the strong approximation that without
changing its distribution the vector sequence
$(\xi_{t}^{(i,j)},\xi_{t}^{(k,\ell)})$ can be redefined on a richer
probability space where there exists a bivariate standard normal random vector
$(\mathcal{Z}_{ij},\mathcal{Z}_{k\ell})$ with the covariance (correlation)
$\rho_{ijk\ell}=\sigma_{ik}\omega_{j\ell}/(m_{ij}m_{k\ell})$ such that
$(\tilde{z}_{ij},\tilde{z}_{k\ell})=(\mathcal{Z}_{ij},\mathcal{Z}_{k\ell})+O(T^{-\kappa_{2}})$
holds a.s. for some $\kappa_{2}>0$, provided that the sufficient conditions
(3.4) and (3.5) in his paper are true. Therefore, the proof of $(b)$ completes
if these two conditions are verified.

Check Kifer’s condition (3.4). For any $(i,j)$ and $(k,\ell)$, we obtain

$\displaystyle\left|T^{-1}\sum_{t=1}^{T}\left(\operatorname{\mathbb{E}}[\xi_{t}^{(i,j)}\xi_{t}^{(k,\ell)}\mid\mathcal{F}_{t-1}]-\frac{\sigma_{ik}{\bm{\omega}}_{j}^{\prime}\bm{\Sigma}_{x}\bm{\omega}_{\ell}}{m_{ij}m_{k\ell}}\right)\right|=\frac{|\sigma_{ik}|}{m_{ij}m_{k\ell}}\left|{\bm{\omega}}_{j}^{\prime}T^{-1}\sum_{t=1}^{T}(\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}-\bm{\Sigma}_{x})\bm{\omega}_{\ell}\right|$
$\displaystyle\leq\frac{\|{\bm{\omega}}_{j}\|_{1}\|\bm{\omega}_{\ell}\|_{1}}{\gamma^{2}}\left\|T^{-1}\mathbf{X}\mathbf{X}^{\prime}-T^{-1}\operatorname{\mathbb{E}}\mathbf{X}\mathbf{X}^{\prime}\right\|_{\max}.$

Thus Lemma 4 with Conditions 4 and 5 entails that the upper bound is
$O(M_{\omega}^{2}T^{-1/2}\log^{3/2}(N\vee T))$ with probability at least
$1-O((N\vee T)^{-\nu})$. By the assumed condition, we have

$\displaystyle M_{\omega}^{2}T^{-1/2}\log^{3/2}(N\vee
T)\lesssim\max_{i\in[N]}\bar{v}_{i}=O(T^{-\kappa_{1}})$

for some constant $\kappa_{1}\in(0,1/2)$. This verifies condition (3.4).

Check Kifer’s condition (3.5). By Condition 5, we observe that

$\displaystyle\operatorname{\mathbb{E}}\left[\|(\xi_{t}^{(i,j)},\xi_{t}^{(k,\ell)})\|_{2}^{4}\right]\lesssim\max_{i,j}\operatorname{\mathbb{E}}\left[|\xi_{t}^{(i,j)}|^{4}\right]\leq\max_{i,j}\frac{\operatorname{\mathbb{E}}[u_{it}^{4}]}{\sigma_{i}^{4}\omega_{j}^{4}}\max_{j}\operatorname{\mathbb{E}}\left[(\mathbf{x}_{t}^{\prime}{\bm{\omega}}_{j})^{4}\right],$

where $\max_{i}\max_{t}\operatorname{\mathbb{E}}[u_{it}^{4}]<\infty$ by
Condition 1 and $\min_{i,j}\sigma_{i}^{4}\omega_{j}^{4}\geq\gamma^{4}$.
Without loss of generality, suppose $K=1$. By the VMA($\infty$) representation
in (14), we obtain

$\displaystyle\operatorname{\mathbb{E}}\left[(\mathbf{x}_{t}^{\prime}{\bm{\omega}}_{j})^{4}\right]\lesssim\sum_{\ell=0}^{\infty}\operatorname{\mathbb{E}}\left[(\bm{\omega}_{j}^{\prime}\mathbf{B}_{\ell}\mathbf{u}_{t-\ell})^{4}\right]+\left(\sum_{\ell=0}^{\infty}\operatorname{\mathbb{E}}\left[(\bm{\omega}_{j}^{\prime}\mathbf{B}_{\ell}\mathbf{u}_{t-\ell})^{2}\right]\right)^{2}=:(I)+(II).$

Recall that
$\bm{\Sigma}_{x}=\operatorname{\mathbb{E}}[\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}]=\sum_{\ell=0}^{\infty}\mathbf{B}_{\ell}\bm{\Sigma}_{u}\mathbf{B}_{\ell}^{\prime}$.
Thus we have

$\displaystyle(II)=\left(\sum_{\ell=0}^{\infty}\bm{\omega}_{j}^{\prime}\mathbf{B}_{\ell}\bm{\Sigma}_{u}\mathbf{B}_{\ell}^{\prime}\bm{\omega}_{j}\right)^{2}=(\bm{\omega}_{j}^{\prime}\bm{\Sigma}_{x}\bm{\omega}_{j})^{2}=\omega_{j}^{4}\leq
1/\gamma^{2}.$

Khintchine’s inequality for a weighted sum of i.i.d. subG random variables
(Vershynin, 2018, Exercise 2.6.5) yields

$\displaystyle(I)$
$\displaystyle=\sum_{\ell=0}^{\infty}\operatorname{\mathbb{E}}\left[(\bm{\omega}_{j}^{\prime}\mathbf{B}_{\ell}\mathbf{u}_{t-\ell})^{4}\right]\lesssim\sum_{\ell=0}^{\infty}\left(\bm{\omega}_{j}^{\prime}\mathbf{B}_{\ell}\bm{\Sigma}_{u}\mathbf{B}_{\ell}^{\prime}\bm{\omega}_{j}\right)^{2}$
$\displaystyle\leq\left(\sum_{\ell=0}^{\infty}\bm{\omega}_{j}^{\prime}\mathbf{B}_{\ell}\bm{\Sigma}_{u}\mathbf{B}_{\ell}^{\prime}\bm{\omega}_{j}\right)^{2}=\omega_{j}^{4}\leq
1/\gamma^{2}.$

Therefore for given constant $\kappa_{1}\in(0,1/2)$, we have

$\displaystyle\sum_{t=1}^{\infty}t^{\kappa_{1}-1}\operatorname{\mathbb{E}}\left[\|(\xi_{t}^{(i,j)},\xi_{t}^{(k,\ell)})\|^{2}1\left\\{\|(\xi_{t}^{(i,j)},\xi_{t}^{(k,\ell)})\|^{2}\geq
1/t^{\kappa_{1}-1}\right\\}\right]$
$\displaystyle\leq\sum_{t=1}^{\infty}t^{2\kappa_{1}-2}\operatorname{\mathbb{E}}\left[\|(\xi_{t}^{(i,j)},\xi_{t}^{(k,\ell)})\|^{4}\right]\lesssim\frac{\operatorname{\mathbb{E}}[u_{it}^{4}]}{\gamma^{8}}\sum_{t=1}^{\infty}t^{2\kappa_{1}-2}<\infty,$

which verifies condition (3.5). This completes the proof of $(b)$.

Prove $(a)$. Note that

$\displaystyle\operatorname{\mathbb{P}}\left(\texttt{T}_{ij}\geq x\right)$
$\displaystyle\geq\operatorname{\mathbb{P}}\left(\tilde{z}_{ij}\geq
x+\bar{v}_{i}\right)-\operatorname{\mathbb{P}}\left(\max_{i,j}|v_{ij}|>\max_{i\in[N]}\bar{v}_{i}\right)$
$\displaystyle=\operatorname{\mathbb{P}}\left(\tilde{z}_{ij}\geq
x+\max_{i\in[N]}\bar{v}_{i}\right)-O((N\vee T)^{-\nu+2}),$

where the last equality follows from Theorem 1. The remainder of the proof of
$(a)$ is the same as that of $(b)$. This completes all the proofs. ∎

######  Lemma 9.

For any bivariate standard normal random vector
$(\mathcal{Z}_{1},\mathcal{Z}_{2})$ with correlation $\rho\in(-1,1)$, it holds
that

$\displaystyle\sup_{z\in[0,\infty)}\frac{\operatorname{\mathbb{P}}(\mathcal{Z}_{1}\geq
z,\mathcal{Z}_{2}\geq z)}{Q(z)^{2}}\leq\frac{1}{\sqrt{1-\rho^{2}}}.$

###### Proof of Lemma 9.

Let $(x,y)\mapsto\phi(x,y;\rho)$ denote the density function of the bivariate
standard normal random vector with correlation $\rho\in(-1,1)$. A simple
calculation yields for any $z\geq 0$,

$\displaystyle\operatorname{\mathbb{P}}\left(\mathcal{Z}_{ij}\geq
z,\mathcal{Z}_{k\ell}\geq
z\right)=\int_{z}^{\infty}\\!\\!\\!\int_{z}^{\infty}\phi(x,y;\rho)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
$\displaystyle=\frac{1}{\sqrt{1-\rho^{2}}}\int_{z}^{\infty}\\!\\!\\!\int_{z}^{\infty}\phi(x,y;0)\exp\left\\{-\frac{2-\rho^{2}}{2(1-\rho^{2})}\left(x^{2}+y^{2}\right)+\frac{2\rho}{2(1-\rho^{2})}xy\right\\}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
$\displaystyle\leq\frac{1}{\sqrt{1-\rho^{2}}}\int_{z}^{\infty}\\!\\!\\!\int_{z}^{\infty}\phi(x,y;0)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\max_{0\leq
x,y<\infty}\exp\left\\{-\frac{2-\rho^{2}}{2(1-\rho^{2})}\left(x^{2}+y^{2}\right)+\frac{2\rho}{2(1-\rho^{2})}xy\right\\}.$

Note that
$\int_{z}^{\infty}\int_{z}^{\infty}\phi(x,y;0)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y=Q(z)^{2}$
and

$\displaystyle\max_{0\leq
x,y<\infty}\left\\{-\frac{2-\rho^{2}}{2(1-\rho^{2})}\left(x^{2}+y^{2}\right)+\frac{2\rho}{2(1-\rho^{2})}xy\right\\}\leq\max_{0\leq
x,y<\infty}\frac{-2+\rho+\rho^{2}}{2(1-\rho^{2})}(x^{2}+y^{2})=0$

uniformly in $\rho\in(-1,1)$. Combining the results gives the desired uniform
bound. ∎

######  Lemma 10.

If Conditions 1 and 2 are true, then we have

$\displaystyle\operatorname{\mathbb{P}}\left(\max_{i\in[N]}\max_{t\in[T]}\max_{k\in[K]}|y_{i,t-k}|^{m}\gtrsim\sqrt{\log^{m}NT}\right)=O((N\vee
T)^{-\nu}).$

###### Proof of Lemma 10.

In this proof, we ignore $\max_{k\in[K]}$ and set $k=1$ without loss of
generality because $K$ is finite. It does not affect the result up to some
positive constant factor. For
$\mathbf{y}_{t}=\sum_{\ell=0}^{\infty}\mathbf{B}_{\ell}\mathbf{u}_{t-\ell}$,
let $\mathbf{b}_{\ell,i}$ denote the $i$th column vector of
$\mathbf{B}_{\ell}^{\prime}$ and
$b=\sum_{\ell=T}^{\infty}\|\mathbf{B}_{\ell}\|_{\infty}$ as before. By the
truncation argument as in the proof of Lemma 3, we have for any $x>0$,

$\displaystyle\operatorname{\mathbb{P}}\left(\max_{i\in[N]}\max_{t\in[T]}|y_{i,t-1}|>x\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}\left(\max_{t\in[T]}\sum_{\ell=0}^{\infty}\|\mathbf{B}_{\ell}\|_{\infty}\|\mathbf{u}_{t-1-\ell}\|_{\infty}>x\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}\left(\max_{t\in[T]}\sum_{\ell=0}^{T-1}\|\mathbf{B}_{\ell}\|_{\infty}\|\mathbf{u}_{t-1-\ell}\|_{\infty}>x/2\right)+\operatorname{\mathbb{P}}\left(\max_{t\in[T]}\sum_{\ell=T}^{\infty}\|\mathbf{B}_{\ell}\|_{\infty}\|\mathbf{u}_{t-1-\ell}\|_{\infty}>x/2\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}\left(\max_{t\in[T]}\max_{\ell\in[T]}\|\mathbf{u}_{t-\ell}\|_{\infty}>\frac{x}{2b}\right)+\frac{2}{x}\sum_{\ell=T}^{\infty}\|\mathbf{B}_{\ell}\|_{\infty}\operatorname{\mathbb{E}}\left[\max_{t\in[T]}\|\mathbf{u}_{t-\ell}\|_{\infty}\right]$
$\displaystyle\lesssim
NT\exp\left(-\frac{x^{2}}{4c_{u}b^{2}}\right)+\frac{\delta^{T}}{x(1-\delta)}\log^{1/2}NT,$

where $\delta\in(0,1)$ and $c_{u}>0$ have been given in Lemma 2 and Condition
1. Thus for any finite $m\in\mathbb{N}$, we have

$\displaystyle\operatorname{\mathbb{P}}\left(\max_{i\in[N]}\max_{t\in[T]}|y_{i,t-1}|^{m}>x\right)=\operatorname{\mathbb{P}}\left(\max_{i\in[N]}\max_{t\in[T]}|y_{i,t-1}|>x^{1/m}\right)$
$\displaystyle\lesssim
NT\exp\left(-\frac{x^{2/m}}{4c_{u}b^{2}}\right)+\frac{\delta^{T}}{x^{1/m}(1-\delta)}\log^{1/2}NT.$

Setting $x=\\{4c_{u}b^{2}(\nu+2)\log(N\vee T)\\}^{m/2}$ for any $\nu>0$ leads
to the result. ∎

######  Lemma 11.

Define $\lambda^{*}\asymp\sqrt{T^{-1}\log^{3}(N\vee T)}$. If Conditions 1–3
and 7 are true and $\bar{s}\lambda=o(1)$ holds, then the event,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\|T^{-1}\mathbf{U}^{*}\mathbf{X}^{\prime}\|_{\max}\gtrsim\lambda^{*}\right)=O((N\vee
T)^{-\nu}),$

occurs with probability at least $1-O((N\vee T)^{-\nu})$.

###### Proof of Lemma 11.

Recall the notation that
$\mathbf{U}^{*}=(\hat{\mathbf{u}}_{1}\zeta_{1},\dots,\hat{\mathbf{u}}_{T}\zeta_{T})$
and
$\hat{\mathbf{u}}_{t}=\mathbf{y}_{t}-\hat{\bm{\Phi}}_{{\textsf{L}}}\mathbf{x}_{t}=\mathbf{u}_{t}+(\bm{\Phi}-\hat{\bm{\Phi}}_{{\textsf{L}}})\mathbf{x}_{t}$.
We have

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\left\|T^{-1}\sum_{t=1}^{T}\mathbf{x}_{t}\mathbf{u}_{t}^{*\prime}\right\|_{\max}>x\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\left\|T^{-1}\sum_{t=1}^{T}\mathbf{x}_{t}\mathbf{u}_{t}^{\prime}\zeta_{t}\right\|_{\max}>x/2\right)+\operatorname{\mathbb{P}}^{*}\left(\left\|T^{-1}\sum_{t=1}^{T}\mathbf{x}_{t}(\hat{\mathbf{u}}_{t}-\mathbf{u}_{t})^{\prime}\zeta_{t}\right\|_{\max}>x/2\right).$
(A.20)

Bound the first term of (A.20). Since $\zeta_{t}$ is a sequence of i.i.d. subG
random variables by Condition 7, applying the union bound and Hoeffding’s
inequality under $\operatorname{\mathbb{P}}^{*}$ yields for any $x>0$,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\left\|T^{-1}\sum_{t=1}^{T}\mathbf{x}_{t}\mathbf{u}_{t}^{\prime}\zeta_{t}\right\|_{\max}>x/2\right)$
$\displaystyle\leq
2KN^{2}\max_{i,j\in[N]}\max_{k\in[K]}\exp\left(-\frac{x^{2}T}{4c_{\zeta}T^{-1}\sum_{t=1}^{T}y_{i,t-k}^{2}u_{jt}^{2}}\right)$
$\displaystyle\leq
2KN^{2}\exp\left(-\frac{x^{2}T}{4c_{\zeta}\max_{i,j\in[N]}\max_{t\in[T]}\max_{k\in[K]}y_{i,t-k}^{2}u_{jt}^{2}}\right)$
$\displaystyle\lesssim
2KN^{2}\exp\left(-\frac{x^{2}T}{4c_{\zeta}\log^{2}NT}\right),$

where the last inequality holds with probability at least $1-O((N\vee
T)^{-\nu})$ by Lemma 10.

By Proposition 1(b) and Lemma 10, the second term of (A.20) is bounded as

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\left\|T^{-1}\sum_{t=1}^{T}\mathbf{x}_{t}(\hat{\mathbf{u}}_{t}-\mathbf{u}_{t})^{\prime}\zeta_{t}\right\|_{\max}>x/2\right)$
$\displaystyle=\operatorname{\mathbb{P}}^{*}\left(\left\|T^{-1}\sum_{t=1}^{T}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\zeta_{t}(\bm{\Phi}-\hat{\bm{\Phi}}_{{\textsf{L}}})^{\prime}\right\|_{\max}>x/2\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\left\|T^{-1}\sum_{t=1}^{T}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\zeta_{t}\right\|_{\max}\max_{i\in[N]}\left\|\hat{\bm{\phi}}_{i\cdot}^{{\textsf{L}}}-\bm{\phi}_{i\cdot}\right\|_{1}>x/2\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\left\|T^{-1}\sum_{t=1}^{T}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\zeta_{t}\right\|_{\max}\bar{s}\lambda\gtrsim
x/2\right)$ $\displaystyle\leq
2K^{2}N^{2}\max_{k,\ell\in[K]}\max_{i,j\in[N]}\exp\left(-\frac{x^{2}T}{4c_{\zeta}\bar{s}^{2}\lambda^{2}T^{-1}\sum_{t=1}^{T}y_{i,t-k}^{2}y_{j,t-\ell}^{2}}\right)$
$\displaystyle\lesssim
2K^{2}N^{2}\exp\left(-\frac{x^{2}T}{4c_{\zeta}\bar{s}^{2}\lambda^{2}\log^{2}NT}\right),$

with probability at least $1-O((N\vee T)^{-\nu})$. Combining the obtained
results with taking

$\displaystyle
x=2\sqrt{c_{\zeta}(\nu+2)T^{-1}\log^{3}NT}~{}(\,\asymp\lambda^{*})$

gives the desired statement since $\bar{s}\lambda=o(1)$. This completes the
proof. ∎

######  Lemma 12.

Recall $\lambda^{*}\asymp\sqrt{T^{-1}\log^{3}(N\vee T)}$, defined in Lemma 11.
If Conditions 1–3 and 7 are true, then the event,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i\in[N]}\left\|\hat{\bm{\phi}}_{i\cdot}^{{\textsf{L}}*}-\hat{\bm{\phi}}_{i\cdot}^{{\textsf{L}}}\right\|_{1}\gtrsim\bar{s}\lambda^{*}\right)=O((N\vee
T)^{-\nu+1})$

occurs with high probability.

###### Proof of Lemma 12.

Define
$\bm{\delta}_{i\cdot}^{*}=\hat{\bm{\phi}}_{i\cdot}^{{\textsf{L}}*}-\hat{\bm{\phi}}_{i\cdot}^{{\textsf{L}}}$.
Derive the non-asymptotic error bound for the Lasso estimator. For
$\lambda^{*}$ such that
$b\lambda/2\leq\lambda^{*}\asymp\sqrt{T^{-1}\log^{3}(N\vee T)}$, define two
events,

$\displaystyle\mathcal{E}_{1}^{*}=\left\\{\left\|T^{-1}\mathbf{X}{\mathbf{U}^{*}}^{\prime}\right\|_{\max}\leq\lambda^{*}/2\right\\},~{}~{}~{}\mathcal{E}_{2}^{*}=\left\\{\left\|T^{-1}\mathbf{X}\mathbf{X}^{\prime}-\operatorname{\mathbb{E}}[T^{-1}\mathbf{X}\mathbf{X}^{\prime}]\right\|_{\max}\leq\lambda^{*}\right\\}.$

Lemmas 11 and 4 guarantee that the event,

$\displaystyle\left\\{\operatorname{\mathbb{P}}^{*}\left(\mathcal{E}_{1}^{*c}\right)=O((N\vee
T)^{-\nu})\right\\}\cap\mathcal{E}_{2}^{*},$

occurs with probability at least $1-O((N\vee T)^{-\nu})$. Then, following the
proof of Proposition 1, we obtain the desired error bound,
$\|\bm{\delta}_{i\cdot}^{*}\|_{1}\lesssim\hat{s}_{i}\lambda^{*}$, on event
$\mathcal{E}_{1}^{*}\cap\mathcal{E}_{2}^{*}$. Finally an application of
Theorem 1(b) of Zhu and Liu (2020) gives $\hat{s}_{i}\lesssim
s_{i}\leq\bar{s}$ with high probability. The assertion is obtained by the
union bound. This completes the proof. ∎

######  Lemma 13.

Recall $\lambda^{*}\asymp\sqrt{T^{-1}\log^{3}(N\vee T)}$, defined in Lemma 11.
If Conditions 1–3 and 7 are true and $\bar{s}\lambda=o(1)$ holds, then for
$\bar{\tau}_{1}=\lambda^{*}+\bar{s}^{2}\lambda^{*2}$ the event,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i\in[N]}\left|\hat{\sigma}_{i}^{*2}-\hat{\sigma}_{i}^{2}\right|\gtrsim\bar{\tau}_{1}\right)=O((N\vee
T)^{-\nu+1}),$

occurs with probability at least $1-O((N\vee T)^{-\nu+1})$.

###### Proof of Lemma 13.

Recall that $u_{it}^{*}=\hat{u}_{it}\zeta_{t}$,
$\sigma_{i}^{*2}=T^{-1}\sum_{t=1}^{T}{u}_{it}^{*2}$, and
$\hat{\sigma}_{i}^{2}=T^{-1}\sum_{t=1}^{T}\hat{u}_{it}^{2}$. By the
construction, we have
$\sigma_{i}^{*2}-\hat{\sigma}_{i}=T^{-1}\sum_{t=1}^{T}\hat{u}_{it}^{2}(\zeta_{t}^{2}-1)$,
where $\\{\hat{u}_{it}^{2}(\zeta_{t}^{2}-1)\\}_{t}$ (for each $i\in[N]$) is a
sequence of independent centered sub-exponential random variables under
$\operatorname{\mathbb{P}}^{*}$. Thus for some constant $c_{\zeta}>0$,
Bernstein’s inequality with the union bound gives for any $x\in(0,1]$,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i\in[N]}\left|\sigma_{i}^{*2}-\hat{\sigma}_{i}^{2}\right|>x\right)=\operatorname{\mathbb{P}}^{*}\left(\max_{i\in[N]}\left|T^{-1}\sum_{t=1}^{T}\hat{u}_{it}^{2}(\zeta_{t}^{2}-1)\right|>x\right)$
$\displaystyle\leq
2N\max_{i}\exp\left(-\frac{x^{2}T}{c_{\zeta}T^{-1}\sum_{t=1}^{T}\hat{u}_{it}^{4}}\right)\leq
2N\exp\left(-\frac{x^{2}T}{c_{\zeta}\max_{i}\max_{t}\hat{u}_{it}^{4}}\right).$

Denote by ${\bm{\delta}}_{i}\in\mathbb{R}^{KN}$ the $i$th column vector of
$\bm{\Delta}^{\prime}=(\hat{\bm{\Phi}}^{L}-\bm{\Phi}^{0})^{\prime}$. Then we
have

$\displaystyle\hat{u}_{it}^{4}=(u_{it}-\mathbf{x}_{t}^{\prime}\bm{\delta}_{i})^{4}\lesssim
u_{it}^{4}+(\mathbf{x}_{t}^{\prime}\bm{\delta}_{i})^{4}\lesssim
u_{it}^{4}+\|\mathbf{x}_{t}\|_{\infty}^{4}\max_{i\in[N]}\|\bm{\delta}_{i}\|_{1}^{4}.$

Therefore, by Proposition 1(b) and Lemma 10, it holds that
$\max_{i}\max_{t}{u}_{it}^{4}\lesssim\log^{2}(NT)$ and
$\max_{t}\|\mathbf{x}_{t}\|_{\infty}^{4}\max_{i\in[N]}\|\bm{\delta}_{i}\|_{1}^{4}\lesssim\bar{s}\lambda\log^{2}(NT)$
occur with probability at least $1-O((N\vee T)^{-\nu+1})$. Therefore, taking

$\displaystyle x\asymp\sqrt{(\nu+1)T^{-1}\log^{3}(NT)}\asymp\lambda^{*}$

leads to the upper bound,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i}\left|\sigma_{i}^{*2}-\hat{\sigma}_{i}^{2}\right|\gtrsim\lambda^{*}\right)\lesssim(N\vee
T)^{-\nu},$

which holds with probability at least $1-O((N\vee T)^{-\nu+1})$.

Recall
$\bm{\delta}_{i\cdot}^{*}=\hat{\bm{\phi}}_{i\cdot}^{{\textsf{L}}*}-\hat{\bm{\phi}}_{i\cdot}^{{\textsf{L}}}$.
Then we have

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i}\left|\hat{\sigma}_{i}^{*2}-\hat{\sigma}_{i}^{2}\right|>x\right)$
$\displaystyle=\operatorname{\mathbb{P}}^{*}\left(\max_{i}\left|\sigma_{i}^{*2}-\hat{\sigma}_{i}^{2}-2T^{-1}\sum_{t=1}^{T}u_{it}^{*}\mathbf{x}_{t}^{\prime}\bm{\delta}_{i}^{*}+{\bm{\delta}_{i}^{*}}^{\prime}\hat{\bm{\Sigma}}_{x}\bm{\delta}_{i}^{*}\right|>x\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\max_{i}\left|\sigma_{i}^{*2}-\hat{\sigma}_{i}^{2}\right|+2\left\|T^{-1}{\mathbf{U}^{*}}^{\prime}\mathbf{X}\right\|_{\max}\max_{i}\left\|\bm{\delta}_{i}^{*}\right\|_{1}+\max_{i}\left\|\bm{\delta}_{i}^{*}\right\|_{1}^{2}\left\|\hat{\bm{\Sigma}}_{x}\right\|_{\max}>x\right),$

where $\|\hat{\bm{\Sigma}}_{x}\|_{\max}=O(1)$ with probability at least
$1-O((N\vee T)^{-\nu})$ by Lemma 4 and Condition 5. By Lemmas 11 and 12,
setting

$\displaystyle
x\asymp\lambda^{*}+\bar{s}\lambda^{*2}+\bar{s}^{2}\lambda^{*2}\asymp\lambda^{*}+\bar{s}^{2}\lambda^{*2}=\bar{\tau}_{1}$

with the union bound yields the desired result. This completes the proof. ∎

######  Lemma 14.

If Conditions 1–3 and 7 are true, then for
$\bar{\tau}_{2}=M_{\omega}^{2}\lambda$ the event,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\left|T^{-1}\sum_{t=1}^{T}({u}_{it}^{*2}-\hat{u}_{it}^{2})\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}\right|\gtrsim\bar{\tau}_{2}\right)=O((NT)^{-\nu}),$

occurs with probability at least $1-O((N\vee T)^{-\nu})$.

###### Proof of Lemma 14.

Because $\operatorname{\mathbb{E}}^{*}{u}_{it}^{*2}=\hat{u}_{it}^{2}$,
$({u}_{it}^{*2}-\hat{u}_{it}^{2})\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}$
is a sequence of i.i.d. sub-exponential random variables under
$\operatorname{\mathbb{P}}^{*}$, Bernstein’s inequality is applied. For any
$x\in(0,1]$, we obtain

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\left|T^{-1}\sum_{t=1}^{T}({u}_{it}^{*2}-\hat{u}_{it}^{2})\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}\right|>x\right)\lesssim\exp\left(-\frac{x^{2}T}{T^{-1}\sum_{t=1}^{T}(\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j})^{2}}\right)$
$\displaystyle\lesssim\exp\left(-\frac{x^{2}T}{\max_{j}\max_{t}\|\hat{\bm{\omega}}_{j}\|_{1}^{4}\|\mathbf{x}_{t}\|_{\infty}^{4}}\right)$
$\displaystyle\lesssim\exp\left(-\frac{x^{2}T}{M_{\omega}^{4}\log^{2}(NT)}\right),$

where the last inequality holds with probability at least $1-O((N\vee
T)^{-\nu})$. Thus taking

$\displaystyle x=\sqrt{\nu M_{\omega}^{4}T^{-1}\log^{3}(NT)}\asymp
M_{\omega}^{2}\lambda=\bar{\tau}_{2}$

leads to the upper bound to be $O((NT)^{-\nu})$. This completes the proof. ∎

######  Lemma 15.

If Conditions 1–3 are true and $\bar{s}\lambda+M_{\omega}^{2}\lambda=o(1)$
holds, then for

$\displaystyle\bar{\tau}_{3}:=\max\left\\{M_{\omega}^{3-2r}\lambda^{1-r}s_{\omega}\log^{2}(N\vee
T),~{}\bar{s}\lambda\log(N\vee T),~{}M_{\omega}^{2}\lambda\log(N\vee
T)\right\\},$

we have

$\displaystyle\operatorname{\mathbb{P}}\left(\left|T^{-1}\sum_{t=1}^{T}\left(\hat{u}_{it}^{2}-\hat{\sigma}_{i}^{2}\right)\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}\right|\gtrsim\bar{\tau}_{3}\right)=O((N\vee
T)^{-\nu}).$

###### Proof of Lemma 15.

We have

$\displaystyle\operatorname{\mathbb{P}}\left(\left|T^{-1}\sum_{t=1}^{T}\left(\hat{u}_{it}^{2}-\hat{\sigma}_{i}^{2}\right)\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}\right|>x\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}\left(\left|T^{-1}\sum_{t=1}^{T}\left(u_{it}^{2}-\sigma_{i}^{2}\right){\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}{\bm{\omega}}_{j}\right|>x/4\right)$
$\displaystyle\quad+\operatorname{\mathbb{P}}\left(\max_{i}\max_{t}\left|u_{it}^{2}-\sigma_{i}^{2}\right|\left|\mathbf{x}_{t}^{\prime}(\hat{\bm{\omega}}_{j}\hat{\bm{\omega}}_{j}^{\prime}-{\bm{\omega}}_{j}{\bm{\omega}}_{j}^{\prime})\mathbf{x}_{t}\right|>x/4\right)$
$\displaystyle\quad+\operatorname{\mathbb{P}}\left(\max_{i}\max_{t}\left|\hat{u}_{it}^{2}-u_{it}^{2}\right|\hat{\bm{\omega}}_{j}^{\prime}\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}>x/4\right)+\operatorname{\mathbb{P}}\left(\max_{i}\left|\sigma_{i}^{2}-\hat{\sigma}_{i}^{2}\right|\hat{\bm{\omega}}_{j}^{\prime}\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}>x/4\right)$

We first evaluate the second to fourth probabilities. By (the proof of) Lemma
7 with Condition 5, the inequalities

$\displaystyle\max_{i}\max_{j}\max_{t}\left|u_{it}^{2}-\sigma_{i}^{2}\right|\mathbf{x}_{t}^{\prime}(\hat{\bm{\omega}}_{j}\hat{\bm{\omega}}_{j}^{\prime}-{\bm{\omega}}_{j}{\bm{\omega}}_{j}^{\prime})\mathbf{x}_{t}$
$\displaystyle\quad\leq\max_{i}\max_{j}\max_{t}|u_{it}^{2}+\sigma_{i}^{2}|\|\mathbf{x}_{t}\|_{\infty}^{2}\|\hat{\bm{\omega}}_{j}(\hat{\bm{\omega}}_{j}-{\bm{\omega}}_{j})^{\prime}+(\hat{\bm{\omega}}_{j}-{\bm{\omega}}_{j}){\bm{\omega}}_{j}^{\prime}\|_{1}$
$\displaystyle\quad\lesssim\|\hat{\bm{\omega}}_{j}-{\bm{\omega}}_{j}\|_{1}\|{\bm{\omega}}_{j}\|_{1}\log^{2}(N\vee
T)\lesssim M_{\omega}^{3-2r}\lambda^{1-r}s_{\omega}\log^{2}(N\vee T),$
$\displaystyle\max_{i}\max_{j}\max_{t}\left|\hat{u}_{it}^{2}-u_{it}^{2}\right|\hat{\bm{\omega}}_{j}^{\prime}\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}$
$\displaystyle\quad\lesssim
2\max_{i}\max_{t}|u_{it}|\|\mathbf{x}_{t}\|_{\infty}\|\bm{\delta}_{i}\|_{1}+\max_{t}\|\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\|_{\infty}\max_{i}\|\bm{\delta}_{i}\|_{1}^{2}$
$\displaystyle\quad\lesssim\bar{s}\lambda\log(N\vee T),$
$\displaystyle\max_{i}\max_{j}\left|\hat{\sigma}_{i}^{2}-\sigma_{i}^{2}\right|\hat{\bm{\omega}}_{j}^{\prime}\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}\lesssim\bar{s}\lambda^{2}$

simultaneously hold with probability at least $1-O((N\vee T)^{-\nu})$.

Next we evaluate the first probability. For
$\bar{c}_{3}=M_{\omega}^{2}\log^{2}(N\vee T)$, the Azuma-Hoeffding inequality
with the union bound and Lemma 10 yield

$\displaystyle\operatorname{\mathbb{P}}\left(\max_{i,j}\left|T^{-1}\sum_{t=1}^{T}({u}_{it}^{2}-\sigma_{i}^{2}){\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}{\bm{\omega}}_{j}\right|>x\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}\left(\max_{i,j}\left|T^{-1}\sum_{t=1}^{T}({u}_{it}^{2}-\sigma_{i}^{2}){\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}{\bm{\omega}}_{j}\right|>x\mid\max_{i,j,t}\left|{u}_{it}^{2}-\sigma_{i}^{2}\right|{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}{\bm{\omega}}_{j}\lesssim\bar{c}_{3}\right)$
$\displaystyle\qquad+\operatorname{\mathbb{P}}\left(\max_{i,j,t}\left|{u}_{it}^{2}-\sigma_{i}^{2}\right|{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}{\bm{\omega}}_{j}\gtrsim\bar{c}_{3}\right)$
$\displaystyle\leq
2KN^{2}\exp\left(-\frac{x^{2}T}{2\bar{c}_{3}^{2}}\right)+O((N\vee T)^{-\nu}),$

where the upper bound becomes $O((N\vee T)^{-\nu})$ by setting

$\displaystyle x=2(\nu+2)\bar{c}_{3}T^{-1/2}\log^{1/2}(N\vee T)\asymp
M_{\omega}^{2}\lambda\log(N\vee T).$

If we set

$\displaystyle x$
$\displaystyle\asymp\max\left\\{M_{\omega}^{3-2r}\lambda^{1-r}s_{\omega}\log^{2}(N\vee
T),~{}\bar{s}\lambda\log(N\vee
T),~{}\bar{s}\lambda^{2},~{}M_{\omega}^{2}\lambda\log(N\vee T)\right\\}$
$\displaystyle=\max\left\\{M_{\omega}^{3-2r}\lambda^{1-r}s_{\omega}\log^{2}(N\vee
T),~{}\bar{s}\lambda\log(N\vee T),~{}M_{\omega}^{2}\lambda\log(N\vee
T)\right\\}=\bar{\tau}_{3},$

the result follows. This completes the proof. ∎

######  Lemma 16.

If Conditions 1–5 and 7 are true and
$\bar{s}\lambda+M_{\omega}^{2}\lambda=o(1)$ holds, then for

$\displaystyle\bar{\mu}_{1}:=\max\left\\{\bar{\tau}_{1},\bar{\tau}_{2},\bar{\tau}_{3}\right\\},$

the event,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i\in[N]}\max_{j\in[KN]}\left|\frac{\tilde{m}_{ij}^{*}}{\hat{m}_{ij}^{*}}-1\right|\gtrsim\bar{\mu}_{1}\right)=O((N\vee
T)^{-\nu}),$

occurs with probability at least $1-O((N\vee T)^{-\nu})$.

###### Proof of Lemma 16.

For any $x>0$, a simple calculus yields

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i\in[N]}\max_{j\in[KN]}\left|\frac{\tilde{m}_{ij}^{*}}{\hat{m}_{ij}^{*}}-1\right|>x\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\max_{i}\max_{j}\left|\tilde{m}_{ij}^{*2}-\hat{m}_{ij}^{*2}\right|>x\hat{m}_{ij}^{*}\left(\tilde{m}_{ij}^{*}+\hat{m}_{ij}^{*}\right)\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\max_{i}\max_{j}\left|\tilde{m}_{ij}^{*2}-\hat{m}_{ij}^{*2}\right|>x\min_{i}\min_{j}\hat{m}_{ij}^{*2}\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\max_{i}\max_{j}\left|\tilde{m}_{ij}^{*2}-\hat{m}_{ij}^{*2}\right|\gtrsim\gamma^{3}x\right)+\operatorname{\mathbb{P}}^{*}\left(\min_{i}\min_{j}\hat{m}_{ij}^{*2}\lesssim\gamma^{3}\right)$

We see that $\tilde{m}_{ij}^{*2}\geq 0$ a.s. and, by Lemmas 7 and 13 with
Condition 5, we obtain

$\displaystyle\hat{m}_{ij}^{*2}=\hat{\sigma}_{i}^{*2}\hat{\bm{\omega}}_{j}^{\prime}\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}$
$\displaystyle\geq\hat{\sigma}_{i}^{2}\hat{\bm{\omega}}_{j}^{\prime}\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}-|\hat{\sigma}_{i}^{*2}-\hat{\sigma}_{i}^{2}|\hat{\bm{\omega}}_{j}^{\prime}\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}$
$\displaystyle\gtrsim\sigma_{i}^{2}\omega_{j}^{2}-M_{\omega}^{2}\lambda-\bar{\tau}_{1}(\omega_{j}^{2}-M_{\omega}^{2}\lambda)\geq\gamma^{2}-O(M_{\omega}^{2}\lambda).$

uniformly in $(i,j)\in[N]\times[KN]$ with probability at least $1-O((N\vee
T)^{-\nu+1})$. Thus it suffices to evaluate the upper bound of

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i}\max_{j}\left|\tilde{m}_{ij}^{*2}-\hat{m}_{ij}^{*2}\right|>x\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\max_{i}\max_{j}\left|\tilde{m}_{ij}^{*2}-\hat{m}_{ij}^{2}\right|>x/2\right)+\operatorname{\mathbb{P}}^{*}\left(\max_{i}\max_{j}\left|\hat{m}_{ij}^{*2}-\hat{m}_{ij}^{2}\right|>x/2\right).$
(A.21)

The second probability of (A.21) is bounded by the same argument just above;
we obtain

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i}\max_{j}\left|\hat{m}_{ij}^{*2}-\hat{m}_{ij}^{2}\right|\gtrsim\bar{\tau}_{1}\right)$
$\displaystyle=O((N\vee T)^{-\nu+1}),$ (A.22)

which occurs with probability at lease $1-O((N\vee T)^{-\nu+1})$. Next bound
the first probability of (A.21). Set

$\displaystyle
x_{2}:=c\bar{\tau}_{2}+{\tau}_{ij,3}~{}~{}\text{with}~{}~{}\bar{\tau}_{2}=M_{\omega}^{2}\lambda,~{}~{}~{}{\tau}_{ij,3}=\left|T^{-1}\sum_{t=1}^{T}\left(\hat{u}_{it}^{2}-\hat{\sigma}_{i}^{2}\right)\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}\right|$

for some constant $c>0$. Then Lemma 14 establishes that the event,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\left|\tilde{m}_{ij}^{*2}-\hat{m}_{ij}^{2}\right|>x_{2}\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\left|T^{-1}\sum_{t=1}^{T}\left({u}_{it}^{*2}-\hat{u}_{it}^{2}\right)\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}\right|+\left|T^{-1}\sum_{t=1}^{T}\left(\hat{u}_{it}^{2}-\hat{\sigma}_{i}^{2}\right)\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}\right|>x_{2}\right)$
$\displaystyle\leq\operatorname{\mathbb{P}}^{*}\left(\left|T^{-1}\sum_{t=1}^{T}\left({u}_{it}^{*2}-\hat{u}_{it}^{2}\right)\hat{\bm{\omega}}_{j}^{\prime}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}\right|\gtrsim\bar{\tau}_{2}\right)=O((N\vee
T)^{-\nu}),$ (A.23)

holds with probability at least $1-O((N\vee T)^{-\nu})$. By Lemma 15,
${\tau}_{ij,3}$ is bounded by

$\displaystyle{\tau}_{ij,3}\lesssim\bar{\tau}_{3}=\max\left\\{M_{\omega}^{3-2r}\lambda^{1-r}s_{\omega}\log^{2}(N\vee
T),~{}\bar{s}\lambda\log(N\vee T),~{}M_{\omega}^{2}\lambda\log(N\vee
T)\right\\}$

with probability at least $1-O((N\vee T)^{-\nu})$.

Combine the two probabilities. Since $\lambda\asymp\lambda^{*}$, taking

$\displaystyle x$
$\displaystyle=\max\left\\{\bar{\tau}_{1},x_{2}\right\\}\asymp\max\left\\{\bar{\tau}_{1},\bar{\tau}_{2},\bar{\tau}_{3}\right\\}=\bar{\mu}_{1}$

in (A.21) establishes the desired inequality up to some positive constant
factor. This completes the proof. ∎

######  Lemma 17.

If Conditions 1–5 and 7 are true, then for

$\displaystyle\bar{\mu}_{2}:=\max\left\\{\lambda^{2-r}M_{\omega}^{2-2r}s_{\omega},\
\bar{s}\lambda M_{\omega}\log^{3/2}(N\vee T)\right\\},$

the event,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i\in[N]}\max_{j\in[KN]}\left|\frac{R_{ij}^{*}}{\hat{m}_{ij}^{*}}\right|\gtrsim\bar{\mu}_{2}\right)=O((N\vee
T)^{-\nu}),$

occurs with probability at least $1-O((N\vee T)^{-\nu})$.

###### Proof of Lemma 17.

Since we have established that $\hat{m}_{ij}^{*}$ is uniformly lower bounded
by a positive constant by Lemmas 7 and 13 with Conditions 5 and
$M_{\omega}^{2}\lambda=o(1)$ in the proof of Lemma 16. Thus it is sufficient
to derive the upper bound of $\max_{i}\max_{j}|R_{ij}^{*}|$. Observe that

$\displaystyle|R_{ij}^{*}|\leq\|T^{-1/2}\mathbf{u}_{i}^{*}\mathbf{X}^{\prime}\|_{\infty}\|\hat{\bm{\omega}}_{j}-{\bm{\omega}}_{j}\|_{1}+\|\sqrt{T}(\hat{\bm{\phi}}_{i}^{L}-\hat{\bm{\phi}}_{i}^{L*})\|_{1}\|\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}-\mathbf{e}_{j}\|_{\infty}.$

By the proof of Proposition 2, the event,

$\displaystyle\left\\{\max_{j\in[N]}\|\hat{\bm{\omega}}_{j}-{\bm{\omega}}_{j}\|_{1}\lesssim(M_{\omega}^{2}\lambda)^{1-r}s_{\omega}\right\\}\cap\left\\{\max_{j\in[N]}\|\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}-\mathbf{e}_{j}\|_{\infty}\lesssim
M_{\omega}\lambda\right\\},$

occurs with probability at lease $1-O((N\vee T)^{-\nu})$. Conditional on this
event with setting

$\displaystyle x$
$\displaystyle=(\lambda^{*}\max_{j}\|\hat{\bm{\omega}}_{j}-{\bm{\omega}}_{j}\|_{1})\vee(\sqrt{T}\bar{s}\lambda^{*}\max_{j}\|\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}-\mathbf{e}_{j}\|_{\infty}),$

we have by Lemmas 11 and 12,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\max_{i,j}|R_{ij}^{*}|>x\right)$
$\displaystyle\qquad\leq\operatorname{\mathbb{P}}^{*}\left(\max_{i}\|T^{-1/2}\mathbf{u}_{i}^{*}\mathbf{X}^{\prime}\|_{\infty}\max_{j}\|\hat{\bm{\omega}}_{j}-{\bm{\omega}}_{j}\|_{1}>x/2\right)$
$\displaystyle\qquad\qquad+\operatorname{\mathbb{P}}^{*}\left(\max_{i}\|\sqrt{T}(\hat{\bm{\phi}}_{i}^{L}-\hat{\bm{\phi}}_{i}^{L*})\|_{1}\max_{j}\|\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}-\mathbf{e}_{j}\|_{\infty}>x/2\right)$
$\displaystyle\qquad\leq\operatorname{\mathbb{P}}^{*}\left(\max_{i}\|T^{-1/2}\mathbf{u}_{i}^{*}\mathbf{X}^{\prime}\|_{\infty}\gtrsim\lambda^{*}\right)+\operatorname{\mathbb{P}}^{*}\left(\max_{i}\|\hat{\bm{\phi}}_{i}^{L}-\hat{\bm{\phi}}_{i}^{L*}\|_{1}\gtrsim\bar{s}\lambda^{*}\right)$
$\displaystyle\qquad=O((N\vee T)^{-\nu+1})$

with high probability. Finally we see that by the proof of Proposition 2

$\displaystyle x$
$\displaystyle=(\lambda^{*}\max_{j}\|\hat{\bm{\omega}}_{j}-{\bm{\omega}}_{j}\|_{1})\vee(\sqrt{T}\bar{s}\lambda^{*}\max_{j}\|\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}-\mathbf{e}_{j}\|_{\infty})$
$\displaystyle\lesssim\left(\lambda^{2-r}M_{\omega}^{2-2r}s_{\omega}\right)\vee\left(\bar{s}\lambda
M_{\omega}\log^{3/2}(N\vee T)\right)=\bar{\mu}_{2},$

with high probability. This completes the proof. ∎

######  Lemma 18.

If Conditions 1–5 and 7, and $M_{\omega}\leq T^{1/2}/\log^{3}(N\vee T)$ are
true, then the events,

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\frac{S_{ij}^{*}}{\tilde{m}_{ij}^{*}}>\texttt{t}\right)=Q(\texttt{t})\left\\{1+O\left(\frac{M_{\omega}\log
NT}{T^{1/2}}\right)\left(1+\texttt{t}\right)^{3}\right\\}$

and

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\frac{S_{ij}^{*}}{\tilde{m}_{ij}^{*}}<-\texttt{t}\right)=Q(\texttt{t})\left\\{1+O\left(\frac{M_{\omega}\log
NT}{T^{1/2}}\right)\left(1+\texttt{t}\right)^{3}\right\\}$

for $\texttt{t}\in[0,\bar{\texttt{t}}]$ uniformly in $(i,j)\in\mathcal{H}$,
occurs with probability at least $1-O((N\vee T)^{-\nu})$.

###### Proof of Lemma 18.

Apply Jing et al. (2003, Theorem 2.3) to the self-normalized sum,
$S_{ij}^{*}/\tilde{m}_{ij}^{*}$ under $\operatorname{\mathbb{P}}^{*}$. By
Lemmas 15 and 7 with Conditions 5 and 7, we have

$\displaystyle B_{ij}^{2}$
$\displaystyle:=T^{-1}\sum_{t=1}^{T}\operatorname{\mathbb{E}}^{*}{u}_{it}^{*2}(\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j})^{2}=T^{-1}\sum_{t=1}^{T}\hat{u}_{it}^{2}(\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j})^{2}$
$\displaystyle\geq\hat{\sigma}_{i}^{2}\hat{\bm{\omega}}_{j}^{\prime}\hat{\bm{\Sigma}}_{x}\hat{\bm{\omega}}_{j}-O(\bar{\tau}_{3})\geq\gamma^{3}(1-o(1))$

with probability at least $1-O((N\vee T)^{-\nu})$. Furthermore, since
$\operatorname{\mathbb{E}}^{*}|\zeta|^{3}$ is bounded by a finite constant
under Condition 7, we have

$\displaystyle L_{ij}$
$\displaystyle:=T^{-3/2}\sum_{t=1}^{T}\operatorname{\mathbb{E}}^{*}|{u}_{it}^{*}|^{3}|\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}|^{3}\asymp
T^{-3/2}\sum_{t=1}^{T}|\hat{u}_{it}|^{3}|\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}|^{3}$
$\displaystyle\leq
T^{-3/2}\sum_{t=1}^{T}|\hat{u}_{it}|^{2}|\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}|^{2}\max_{t}|\hat{u}_{it}||\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}|=T^{-1/2}B_{ij}^{2}\max_{t}|\hat{u}_{it}||\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}|,$

where Proposition 1 and (the proofs of) Lemmas 7 and 10 give

$\displaystyle\max_{t}|\hat{u}_{it}||\mathbf{x}_{t}^{\prime}\hat{\bm{\omega}}_{j}|\lesssim\max_{t}\left(|u_{it}|+\|\mathbf{x}_{t}\|_{\infty}\|\bm{\delta}_{i}\|_{1}\right)\|\mathbf{x}_{t}\|_{\infty}\|\hat{\bm{\omega}}_{j}\|_{1}$
$\displaystyle\leq\left(\max_{i}\max_{t}|u_{it}|+\max_{t}\|\mathbf{x}_{t}\|_{\infty}\|\bm{\delta}_{i}\|_{1}\right)\max_{t}\|\mathbf{x}_{t}\|_{\infty}\max_{j}\|\hat{\bm{\omega}}_{j}\|_{1}\lesssim
M_{\omega}\log(NT)$

with probability at least $1-O((N\vee T)^{-\nu})$. Thus

$\displaystyle
d_{ij}:=\frac{B_{ij}}{L_{ij}^{1/3}}\gtrsim\frac{T^{1/6}B_{ij}^{1/3}}{M_{\omega}^{1/3}\log^{1/3}(NT)}\geq\frac{T^{1/6}\gamma^{1/2}(1+o(1))}{M_{\omega}^{1/3}\log^{1/3}(NT)}\geq\log^{2/3}(N\vee
T),$

where the lower bound holds uniformly in $(i,j)$ since $M_{\omega}\leq
T^{1/2}/\log^{3}(N\vee T)$ is implied by the condition $\bar{\mu}=o(1)$.
Therefore, by Jing et al. (2003, Theorem 2.3), we conclude that

$\displaystyle\operatorname{\mathbb{P}}^{*}\left(\frac{S_{ij}^{*}}{\tilde{m}_{ij}^{*}}>\texttt{t}\right)=Q(\texttt{t})\left\\{1+O\left(\frac{M_{\omega}\log
NT}{T^{1/2}}\right)\left(1+\texttt{t}\right)^{3}\right\\}$

holds with high probability for $\texttt{t}\in[0,\bar{\texttt{t}}]$, where
$\bar{\texttt{t}}\asymp\log^{1/2}(N\vee T)=o(d_{ij})$, uniformly in
$(i,j)\in\mathcal{H}$. This completes the proof. ∎

##  Appendix D Precision Matrix Estimation

The construction of the debiased lasso estimator requires a precision matrix
estimator. In a low dimensional setting, it is obtained as the inverse of the
sample covariance matrix of $\mathbf{X}$, but it becomes less accurate or even
infeasible as the dimensionality tends to large. To this end, we need some
regularized estimator. In this paper we employ the CLIME method of Cai et al.
(2011) for the estimation of $\bm{\Omega}$.

Define $\hat{\bm{\Theta}}=(\hat{\theta}_{ij})\in\mathbb{R}^{KN\times KN}$ as
the solution to the following minimization problem:

$\displaystyle\min\|\bm{\Theta}\|_{1}~{}~{}~{}\text{subject
to}~{}~{}~{}\|\hat{\bm{\Sigma}}_{x,\varepsilon}\bm{\Theta}-\mathbf{I}_{KN}\|_{\max}\leq\lambda_{1},$

where $\lambda_{1}$ is a positive regularization parameter and
$\hat{\bm{\Sigma}}_{x,\varepsilon}=\hat{\bm{\Sigma}}_{x}+\varepsilon\mathbf{I}_{KN}$
for $\varepsilon\geq 0$. Note that
$\hat{\bm{\Sigma}}_{x,0}=\hat{\bm{\Sigma}}_{x}$. The CLIME estimator of
$\bm{\Omega}$ is then obtained by symmetrization with

$\displaystyle\hat{\omega}_{ij}=\hat{\omega}_{ji}=\hat{\theta}_{ij}1\\{|\hat{\theta}_{ij}|\leq|\hat{\theta}_{ji}|\\}+\hat{\theta}_{ji}1\\{|\hat{\theta}_{ij}|>|\hat{\theta}_{ji}|\\}.$

The non-asymptotic error bound is derived in Lemma 7, which is of independent
interest. For related theoretical results of stationary time series, see Shu
and Nan (2019).

When the lasso is debiased (desparsified) in a linear regression context, the
nodewise regression (van de Geer et al., 2014) is frequently used as an
alternative. However, it might not suitable for the VAR models since it
repeats the regression of each variable in $\mathbf{X}$ on the others, which
includes regressions of a “past” variable on “future” ones.

##  Appendix E Additional Experimental Results

###  E.1 Results with $\hat{m}_{ij}=\hat{\sigma}_{i}\hat{\omega}_{j}$

The DGP is identical to the one used in Table 1.

Table E1: Directional FDR and power using asymptotic and bootstrap thresholds
for $q=0.1$, with cross-sectionally uncorrelated errors $m=2$ ($\max s_{i}=5$)
---
|  |  | $T=200$ |  | $T=300$
|  |  | asymptotic |  | bootstrap |  | asymptotic |  | bootstrap
|  |  | dFDR | dPWR |  | dFDR | dPWR |  | dFDR | dPWR |  | dFDR | dPWR
Standard Normal Error
| $N=50$ |  | 6.0 | 96.6 |  | 6.7 | 96.9 |  | 6.0 | 99.7 |  | 6.9 | 99.7
| $N=100$ |  | 6.3 | 92.8 |  | 6.9 | 93.2 |  | 6.1 | 99.1 |  | 7.3 | 99.2
| $N=200$ |  | 6.0 | 89.6 |  | 7.5 | 90.7 |  | 6.0 | 98.5 |  | 8.2 | 98.8
| $N=300$ |  | 5.7 | 86.7 |  | 6.7 | 87.4 |  | 5.5 | 98.0 |  | 8.5 | 98.4
Mixture Normal Error
| $N=50$ |  | 6.8 | 92.7 |  | 5.8 | 92.0 |  | 6.6 | 98.7 |  | 6.2 | 98.7
| $N=100$ |  | 7.7 | 86.8 |  | 5.6 | 84.7 |  | 7.3 | 97.2 |  | 6.5 | 96.9
| $N=200$ |  | 8.5 | 84.2 |  | 6.7 | 82.5 |  | 7.7 | 96.3 |  | 7.7 | 96.3
| $N=300$ |  | 8.4 | 80.8 |  | 5.5 | 77.3 |  | 7.7 | 95.1 |  | 7.7 | 95.1
$m=4$ ($\max s_{i}=9$)
|  |  | $T=200$ |  | $T=300$
|  |  | asymptotic |  | bootstrap |  | asymptotic |  | bootstrap
|  |  | dFDR | dPWR |  | dFDR | dPWR |  | dFDR | dPWR |  | dFDR | dPWR
Standard Normal Error
| $N=50$ |  | 5.8 | 87.1 |  | 6.7 | 87.9 |  | 5.8 | 96.0 |  | 6.8 | 96.4
| $N=100$ |  | 5.8 | 81.2 |  | 8.3 | 83.6 |  | 5.9 | 93.7 |  | 8.4 | 94.8
| $N=200$ |  | 5.9 | 75.3 |  | 9.3 | 78.3 |  | 5.6 | 90.5 |  | 9.4 | 92.2
| $N=300$ |  | 6.1 | 69.3 |  | 8.5 | 71.7 |  | 5.6 | 87.3 |  | 9.7 | 89.5
Mixture Normal Error
| $N=50$ |  | 6.0 | 82.9 |  | 5.8 | 82.6 |  | 5.9 | 93.8 |  | 6.1 | 94.0
| $N=100$ |  | 6.4 | 77.3 |  | 7.1 | 78.1 |  | 6.3 | 91.1 |  | 7.6 | 91.8
| $N=200$ |  | 7.2 | 71.6 |  | 8.2 | 72.6 |  | 6.8 | 87.2 |  | 9.0 | 88.5
| $N=300$ |  | 7.4 | 65.8 |  | 6.8 | 65.1 |  | 6.8 | 83.7 |  | 8.9 | 85.0
$m=7$ ($\max s_{i}=15$)
|  |  | $T=200$ |  | $T=300$
|  |  | asymptotic |  | bootstrap |  | asymptotic |  | bootstrap
|  |  | dFDR | dPWR |  | dFDR | dPWR |  | dFDR | dPWR |  | dFDR | dPWR
Standard Normal Error
| $N=50$ |  | 5.1 | 66.4 |  | 6.0 | 67.8 |  | 5.2 | 78.8 |  | 6.1 | 79.9
| $N=100$ |  | 5.6 | 58.1 |  | 8.0 | 61.0 |  | 5.6 | 72.3 |  | 8.0 | 74.7
| $N=200$ |  | 5.7 | 50.2 |  | 8.8 | 53.3 |  | 5.4 | 65.4 |  | 9.1 | 68.7
| $N=300$ |  | 5.8 | 47.1 |  | 8.8 | 49.8 |  | 5.2 | 62.6 |  | 9.8 | 66.2
Mixture Normal Error
| $N=50$ |  | 5.0 | 63.1 |  | 4.9 | 62.9 |  | 4.9 | 76.2 |  | 5.1 | 76.5
| $N=100$ |  | 5.9 | 55.5 |  | 6.7 | 56.6 |  | 5.9 | 69.6 |  | 7.2 | 71.0
| $N=200$ |  | 7.5 | 48.2 |  | 7.5 | 48.0 |  | 6.6 | 62.8 |  | 8.1 | 64.3
| $N=300$ |  | 6.8 | 45.7 |  | 7.3 | 46.2 |  | 6.2 | 60.4 |  | 9.0 | 62.8

###  E.2 Results for cross-sectionally correlated errors

The DGP is identical to the one used in Table 1, except the errors are cross-
sectionally correlated: first generate
${\bm{\Sigma}}_{u}^{\ast}=(\sigma_{ij}^{\ast})$:
$\sigma_{i}^{*2}\sim\text{i.i.d.}U(0.1,1.9)$ and
$\sigma_{ij}^{\ast}=\sigma_{i}^{\ast 2}-\left|i-j\right|/10$ for $i\neq j$
such that $\sigma_{i}^{\ast 2}-\left|i-j\right|/10>0$ and
$\left|i-j\right|\leq 10$; otherwise $\sigma_{ij}^{\ast}=0$. If
$\lambda_{\min}\left({\bm{\Sigma}}_{u}^{\ast}\right)>0$, set
${\bm{\Sigma}}_{u}={\bm{\Sigma}}_{u}^{\ast}$; otherwise
${\bm{\Sigma}}_{u}=({\bm{\Sigma}}_{u}^{\ast}+c\mathbf{I}_{n})/\bar{d}$, where
$c=-\lambda_{\min}\left({\bm{\Sigma}}_{u}^{\ast}\right)+0.1$ and $\bar{d}$ is
the average of the diagonal elements of
${\bm{\Sigma}}_{u}^{\ast}+c\mathbf{I}_{n}$.

Table E2: Directional FDR and Power using asymptotic and bootstrap thresholds
for $q=0.1$, with cross-sectionally correlated errors, with
$\hat{m}_{ij}=\hat{\sigma}_{i}\sqrt{\hat{\bm{\omega}}_{j}^{\prime}\bm{\Sigma}_{x}\hat{\bm{\omega}}_{j}}$
$m=2$ ($\max s_{i}=5$)
---
|  |  | $T=200$
|  |  | asymptotic |  | bootstrap
|  |  | dFDR | dPWR |  | dFDR | dPWR
Standard Normal Error
| $N=50$ |  | 9.3 | 97.5 |  | 6.8 | 96.8
| $N=100$ |  | 10.8 | 94.7 |  | 7.0 | 93.1
| $N=200$ |  | 10.6 | 91.9 |  | 7.7 | 90.5
| $N=300$ |  | 10.4 | 89.6 |  | 6.8 | 87.3
Mixture Normal Error
| $N=50$ |  | 10.5 | 94.3 |  | 5.8 | 91.8
| $N=100$ |  | 12.6 | 89.5 |  | 5.5 | 84.6
| $N=200$ |  | 13.1 | 87.0 |  | 6.6 | 82.4
| $N=300$ |  | 14.1 | 84.4 |  | 5.5 | 77.6
Table E3: Directional FDR and Power using asymptotic and bootstrap thresholds
for $q=0.1$, with cross-sectionally correlated errors, with
$\hat{m}_{ij}=\hat{\sigma}_{i}\hat{\omega}_{j}$ $m=2$ ($\max s_{i}=5$)
---
|  |  | $T=200$
|  |  | asymptotic |  | bootstrap
|  |  | dFDR | dPWR |  | dFDR | dPWR
Standard Normal Error
| $N=50$ |  | 6.0 | 96.6 |  | 6.7 | 96.9
| $N=100$ |  | 6.3 | 92.9 |  | 6.9 | 93.2
| $N=200$ |  | 6.2 | 89.6 |  | 7.6 | 90.5
| $N=300$ |  | 5.7 | 86.8 |  | 6.7 | 87.4
Mixture Normal Error
| $N=50$ |  | 6.9 | 92.8 |  | 5.7 | 92.0
| $N=100$ |  | 7.7 | 86.9 |  | 5.6 | 84.8
| $N=200$ |  | 8.3 | 84.2 |  | 6.7 | 82.7
| $N=300$ |  | 8.5 | 80.9 |  | 5.5 | 77.4

##  Appendix F List of Variable Names in Empirical Applications

Table F1: Full list of FRED-MD macroeconomic variables. FREDMD ID and Variable
Descriptions
---
Group 1: Output and Income | Group 5: Interest and Exchange Rate
1 | Real Personal Income | 68 | Federal Funds Rate
2 | Real personal income ex transfer receipts | 69 | 3-Month AA Financial Commercial Paper Rate
3 | IP Index | 70 | 3-Month Treasury Bill
4 | IP: Final Products and Nonindustrial Supplies | 71 | 6-Month Treasury Bill
5 | IP: Final Products (Market Group) | 72 | 1-Year Treasury Rate
6 | IP: Consumer Goods | 73 | 5-Year Treasury Rate
7 | IP: Durable Consumer Goods | 74 | 10-Year Treasury Rate
8 | IP: Nondurable Consumer Goods | 75 | Moody’s Seasoned Aaa Corporate Bond Yield Aaa bond
9 | IP: Business Equipment | 76 | Moody’s Seasoned Baa Corporate Bond Yield Baa bond
10 | IP: Materials | 77 | 3-Month Commercial Paper Minus FEDFUNDS CP-FF spread
11 | IP: Durable Materials | 78 | 3-Month Treasury C Minus FEDFUNDS 3 mo-FF spread
12 | IP: Nondurable Materials | 79 | 6-Month Treasury C Minus FEDFUNDS 6 mo-FF spread
13 | IP: Manufacturing (SIC) | 80 | 1-Year Treasury C Minus FEDFUNDS 1 yr-FF spread
14 | IP: Residential Utilities | 81 | 5-Year Treasury C Minus FEDFUNDS 5 yr-FF spread
15 | IP: Fuels | 82 | 10-Year Treasury C Minus FEDFUNDS 10 yr-FF spread
16 | Capacity Utilization: Manufacturing | 83 | Moody’s Aaa Corporate Bond Minus FEDFUNDS
Group 2: Labor Market | 84 | Moody’s Baa Corporate Bond Minus FEDFUNDS
17 | Help-Wanted Index for United States | 85 | Trade Weighted U.S. Dollar Index: Major Currencies Ex rate
18 | Ratio of Help Wanted/No. Unemployed | 86 | Switzerland / U.S. Foreign Exchange Rate
19 | Civilian Labor Force | 87 | Japan / U.S. Foreign Exchange Rate
20 | Civilian Employment | 88 | U.S. / U.K. Foreign Exchange Rate
21 | Civilian Unemployment Rate | 89 | Canada / U.S. Foreign Exchange Rate
22 | Average Duration of Unemployment (Weeks) | Group 6: Prices
23 | Civilians Unemployed -Less Than 5 Weeks | 90 | PPI: Finished Goods
24 | Civilians Unemployed for 5-14 Weeks | 91 | PPI: Finished Consumer Goods
25 | Civilians Unemployed -15 Weeks & Over | 92 | PPI: Intermediate Materials
26 | Civilians Unemployed for 15-26 Weeks | 93 | PPI: Crude Materials
27 | Civilians Unemployed for 27 Weeks and Over | 94 | Crude Oil
28 | Initial Claims | 95 | PPI: Metals and metal products:
29 | All Employees: Total nonfarm | 96 | CPI : All Items
30 | All Employees: Goods-Producing Industries | 97 | CPI : Apparel
31 | All Employees: Mining and Logging: Mining | 98 | CPI : Transportation
32 | All Employees: Construction | 99 | CPI : Medical Care
33 | All Employees: Manufacturing | 100 | CPI : Commodities
34 | All Employees: Durable goods | 101 | CPI : Durables
35 | All Employees: Nondurable goods | 102 | CPI : Services
36 | All Employees: Service-Providing Industries | 103 | CPI : All Items Less Food
37 | All Employees: Trade | 104 | CPI : All items less shelter
38 | All Employees: Wholesale Trade | 105 | CPI : All items less medical care
39 | All Employees: Retail Trade | 106 | Personal Cons. Expend.: Chain Index
40 | All Employees: Financial Activities | 107 | Personal Cons. Exp: Durable goods
41 | All Employees: Government | 108 | Personal Cons. Exp: Nondurable goods
42 | Avg Weekly Hours : Goods-Producing | 109 | Personal Cons. Exp: Services
43 | Avg Weekly Overtime Hours : Manufacturing | Group 7: Money and Credit
44 | Avg Weekly Hours : Manufacturing | 110 | M1 Money Stock
45 | Avg Hourly Earnings : Goods-Producing | 111 | M2 Money Stock
46 | Avg Hourly Earnings : Construction | 112 | Real M2 Money Stock
47 | Avg Hourly Earnings : Manufacturing | 113 | St. Louis Adjusted Monetary Base
Group 3: Consumption, Orders and Inventories | 114 | Total Reserves of Depository Institutions
48 | Real personal consumption expenditures | 115 | Reserves Of Depository Institutions
49 | Real Manu. and Trade Industries Sales | 116 | Commercial and Industrial Loans
50 | Retail and Food Services Sales | 117 | Real Estate Loans at All Commercial Banks
51 | New Orders for Consumer Goods | 118 | Total Nonrevolving Credit
52 | New Orders for Durable Goods | 119 | Nonrevolving consumer credit to Personal Income
53 | New Orders for Nondefense Capital Goods | 120 | MZM Money Stock
54 | Un.lled Orders for Durable Goods | 121 | Consumer Motor Vehicle Loans Outstanding
55 | Total Business Inventories | 122 | Total Consumer Loans and Leases Outstanding
56 | Total Business: Inventories to Sales Ratio | 123 | Securities in Bank Credit at All Commercial Banks
57 | Consumer Sentiment Index | Group 8: Stock Market
Group 4: Housing | 124 | S&P’s Common Stock Price Index: Composite
58 | Housing Starts: Total New Privately Owned | 125 | S&P’s Common Stock Price Index: Industrials
59 | Housing Starts Northeast | 126 | S&P’s Composite Common Stock: Dividend Yield
60 | Housing Starts Midwest | 127 | S&P’s Composite Common Stock: Price-Earnings Ratio
61 | Housing Starts South | 128 | VXO
62 | Housing Starts West |  |
63 | New Private Housing Permits (SAAR) |  |
64 | New Private Housing Permits Northeast |  |
65 | New Private Housing Permits Midwest |  |
66 | New Private Housing Permits South |  |
67 | New Private Housing Permits West |  |
Table F2: Full list of the Local Area Districts (LADs) for the UK regional house price Scotland |  | London Area |  | Wales
---|---|---|---|---
ID | Region Names |  | ID | Region Names |  | ID | Region Names
1 | Clackmannanshire |  | 32 | City.of.London |  | 65 | Isle.of.Anglesey
2 | Dumfries.and.Galloway |  | 33 | Barking.and.Dagenham |  | 66 | Gwynedd
3 | East.Ayrshire |  | 34 | Barnet |  | 67 | Conwy
4 | East.Lothian |  | 35 | Bexley |  | 68 | Denbighshire
5 | Na.h-Eileanan.Siar |  | 36 | Brent |  | 69 | Flintshire
6 | Falkirk |  | 37 | Bromley |  | 70 | Wrexham
7 | Highland |  | 38 | Camden |  | 71 | Ceredigion
8 | Inverclyde |  | 39 | Croydon |  | 72 | Pembrokeshire
9 | Midlothian |  | 40 | Ealing |  | 73 | Carmarthenshire
10 | Moray |  | 41 | Enfield |  | 74 | Swansea
11 | North.Ayrshire |  | 42 | Greenwich |  | 75 | Neath.Port.Talbot
12 | Orkney.Islands |  | 43 | Hackney |  | 76 | Bridgend
13 | Scottish.Borders |  | 44 | Hammersmith.and.Fulham |  | 77 | Vale.of.Glamorgan
14 | Shetland.Islands |  | 45 | Haringey |  | 78 | Cardiff
15 | South.Ayrshire |  | 46 | Harrow |  | 79 | Rhondda.Cynon.Taf
16 | South.Lanarkshire |  | 47 | Havering |  | 80 | Caerphilly
17 | Stirling |  | 48 | Hillingdon |  | 81 | Blaenau.Gwent
18 | Aberdeen.City |  | 49 | Hounslow |  | 82 | Torfaen
19 | Aberdeenshire |  | 50 | Islington |  | 83 | Monmouthshire
20 | Argyll.and.Bute |  | 51 | Kensington.and.Chelsea |  | 84 | Newport
21 | City.of.Edinburgh |  | 52 | Kingston.upon.Thames |  | 85 | Powys
22 | Renfrewshire |  | 53 | Lambeth |  | 86 | Merthyr.Tydfil
23 | West.Dunbartonshire |  | 54 | Lewisham |  |  |
24 | West.Lothian |  | 55 | Merton |  |  |
25 | Angus |  | 56 | Newham |  |  |
26 | Dundee.City |  | 57 | Redbridge |  |  |
27 | East.Dunbartonshire |  | 58 | Richmond.upon.Thames |  |  |
28 | Fife |  | 59 | Southwark |  |  |
29 | Perth.and.Kinross |  | 60 | Sutton |  |  |
30 | Glasgow.City |  | 61 | Tower.Hamlets |  |  |
31 | North.Lanarkshire |  | 62 | Waltham.Forest |  |  |
|  |  | 63 | Wandsworth |  |  |
|  |  | 64 | Westminster |  |  |

##  Appendix G Additional Results of Empirical Applications

Figure G1: Dynamic network visualization: macroeconomic variables, asymptotic
$t_{0}$, $q=0.05$ Figure G2: Dynamic network visualization: UK regional house
prices, asymptotic $t_{0}$, $q=0.05$