# How do mobility restrictions and social distancing during COVID-19 affect
the crude oil price?

Asim K. Dey  Kumer P. Das  Princeton University,Princeton, NJ, USA  The
University of Texas at Dallas, Dallas, TX, USA  University of Louisiana at
Lafayette,Lafayette, LA,USA

###### Abstract

We develop an air mobility index and use the newly developed Apple’s driving
trend index to evaluate the impact of COVID-19 on the crude oil price. We use
quantile regression and stationary and non-stationary extreme value models to
study the impact. We find that both the air mobility index and driving trend
index significantly influence lower and upper quantiles as well as the median
of the WTI crude oil price. The extreme value model suggests that an event
like COVID-19 may push oil prices to a negative territory again as the air
mobility decreases drastically during such pandemics.

###### keywords:

Apple Mobility, Air Mobility, Crude Oil, Quantile Regression, Extreme Value
Theory

##  1 Introduction

Understanding the real-time fuel demand is very important for many reasons.
Energy merchants want to get a trading edge from all possible resources such
as thermal images from cameras on pipelines, satellite data tracking worldwide
oil tankers, and so on. Another way of quantifying the real-time demand is by
tracking the transportation sector which accounts for the largest share of
U.S. petroleum consumption. In 2019, the transportation sector accounts for
68% of all petroleum consumption in the U.S. Moreover, finished motor gasoline
accounts for about 45% of total U.S. petroleum consumption (U.S. Energy
Information Administration). Human mobility trends introduced by Apple Inc. in
mid-April can also be used to understand on-the-spot gasoline consumption data
as it captures user activity in searching for directions on smartphones. In
particular, it is of great interest to use such mobility data during and after
the COVID-19 lockdowns where the fuel demand fluctuates vigorously [4, 23,
30].

Apple has introduced the tool to support the impactful work happening around
the globe to mitigate the spread of COVID-19 [2]. Using aggregated data
collected from Apple maps, this trend is generated by counting the number of
requests made for directions.

To quantify mobility restrictions and social distancing we use Apple mobility
trends, in particular driving trend in US [2]. The other important component
of fuel consumption is jet fuel consumption which has been decreased
drastically due to COVID-19. Although an average of 24,900 commercial
passenger flights departed U.S. airports each day in January 2020, by July
2020, flight volume had declined to 13,700 per day–51% of the July 2019 level
[16]. We introduce a new air mobility index based on temporal airline network
to evaluate the dynamics of daily U.S. flight volumes. A low score of the two
indices indicates that people are following the government’s guidelines on
mobility restrictions more and practicing more social distancing. A high score
of the two indices implies lower mobility restrictions and less social
distancing. The objective of this study is to understand the impact of the
Apple driving trend index and the newly developed air mobility index on the
crude oil price.

In this paper, first, we investigate the effect of driving trend and air
mobility on the different quantiles of the crude oil price based on the
quantile regression model. Second, we evaluate the tail of the crude oil price
using an extreme value model. Generally, extreme value models focus on the
upper tail of the distribution. However, we study the lower tail of the crude
oil price and analyze different determinants of extreme lower crude oil price
based on non-stationary extreme value model as discussed in Dey et al. (2020)
[14].

The rest of the paper is organized as follows. In Section 2 we introduce
different measures of mobility restrictions and social distancing. In Section
3 we describe the data. We discuss the two methods, namely, quantile
regression and extreme value theory, used in the study in Section 4. We report
the findings and describe the results in Section 5. Finally, we conclude in
Section 6.

##  2 Quantification of mobility restrictions and social distancing

To control the spread of COVID-19 different governments have been implementing
a variety of mobility restrictions and social distancing measures [8, 19, 31,
35]. To quantify the impact of such control measures on human mobility
behavior we utilize the two following mobility indices.

###  2.1 Apple’s driving trend index

The Apple driving trend represents a relative driving volume of Apple users
compared to a baseline volume on January 13, 2020. In our study we consider
weekly average of the Apple driving trend between January 13, 2020, and August
25, 2020 as a measure of mobility. We name this measure as Apple driving trend
index ($ADI$). We normalize $ADI$ in week $t$ by adjusting the average $ADI$
and corresponding standard deviation as

$\displaystyle H_{t}={{ADI_{t}-\mu_{ADI}}\over{\sigma_{ADI}}},$ (1)

where, $\mu_{ADI}$ and $\sigma_{ADI}$ are the mean and standard deviation of
$ADI$ between January 13, 2020, and August 25, 2020.

###  2.2 Air mobility index

In our study, we consider daily flight volumes in the U.S. as a measure of air
mobility. To evaluate daily flight volumes, we introduce a complex network
analysis. We define a graph $G=(V,E)$ as a model for a airline network, with
node set $V$ and set of edges $E\subset V\times V$ such that $(u,v)\in E$
represents an edge from node $u$ to $v$, $u,v=1,2,\cdots,n$. Here, the nodes
represent the U.S. airports and if there is a direct flight between two nodes
(i.e., airports) they are connected by an edge. Here $|V|$ is the number of
nodes and $|E|$ is the number of edges in the network. The elements of the
$n\times n$-symmetric adjacency matrix, $A$, of $G$ can be written as

$A_{ij}=\begin{cases}1,&\mbox{if}\;(i,j)\in E\\\
0,&\mbox{otherwise}.\end{cases}$ (2)

We assume that $G$ is undirected i.e., for all $e_{uv}\in E$, $e_{uv}\equiv
e_{vu}$. The degree, $d_{u}$, of a node $u$ is the number of edges incident to
$u$ i.e., for u,v $\in$ V and e $\in$ E, $d_{u}=\sum_{u\neq v}e_{u,v}$. We
define $f_{d}$ to be the fraction of nodes $u\in V$ with degree $d_{u}=d$. The
collection $\\{f_{d}\\}$, $d\leq 0$ is called the degree distribution of $G$
[27].

Temporal Network is an extension of network analysis which appears in many
domains of knowledge such as finance [3, 38], epidemiology [12, 15, 18, 36]. A
temporal network is a network structure that changes in time, which can be
represented with a time indexed graph $G^{t}=(V(t),E(t))$, where, $V(t)$ is
the set of nodes in the network at time $t$, $E(t)\subset V(t)\times V(t)$ is
a set of edges in the network at time $t$. Here, $t$ is either discrete or
continuous. To quantify the impact of COVID-19 on flight volumes which
necessarily determines the jet fuel demand, we construct two temporal airline
network ($G^{t}$) for two different time scale.

First, we develop a weekly airline network ($G^{t}_{w}$) in each week Sunday
day ($t$) between January 13, 2020 and August 25, 2020:
$\mathbb{G}_{w}=\\{{G}^{1}_{w},\ldots,{G}^{\mathcal{T}}_{w}\\}$, where
$\mathcal{T}=32$. We consider the airline network on each Sunday is an average
representation of that week. The number of nodes ($|V^{w}_{t}|$), number of
edges ($|E^{w}_{t}|$), and degree distribution are evaluated in each
$G^{t}_{w}$.

Figure 1 illustrates the U.S. flight volumes on Feb 2, 2020, and on May 3,
2020. Table 1 shows how does the degree distribution of the US Airline network
change (decrease) due to the COVID-19 pandemic. For example, the mean number
of departed and landed flights in US airport on Feb 2, 2020, is 111, which
reduces to 31 on May 3, 2020. The maximum number of departed and landed
flights in a US airport on Feb 2, 2020, was 1,748, whereas, the maximum number
of departed and landed flights in a US airport on May 3, 2020, was only 683.

(a) US airline network on Feb 2, 2020.

(b) US airline network on May 3, 2020.

Figure 1: Impact of COVID-19 on US airlines. Red points represent nodes (airports) and green lines represent edges. (a) Shows normal US airline network with 340 nodes and 18,805 edges. (b) ) Represents reduced US airline network with 319 nodes and 4,980 edges.  Table 1: Degree distribution of U.S. Airline networks on Feb 2, 2020 and on May 3, 2020. Network | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max.
---|---|---|---|---|---|---
$G^{02-02-2020}$ | 2 | 6 | 19 | 111 | 73 | 1748
$G^{05-03-2020}$ | 1 | 3 | 7 | 31 | 20 | 683

Second, we construct a monthly airline network ($G^{t}_{m}$) on $15^{th}$ of
each month ($t$) between January 2000 and August 2020:
$\mathbb{G}_{m}=\\{{G}^{1}_{m},\ldots,{G}^{\mathcal{T}}_{m}\\}$, where
$\mathcal{T}=248$. We consider the airline network on $15^{th}$ of each month
is an average representation of that month. Similarly as $\mathbb{G}_{w}$, we
evaluate the number of nodes ($|V^{m}_{t}|$), number of edges ($|E^{m}_{t}|$),
and degree distribution in each $G^{t}_{m}$.

In our study, we consider the number of edges in an airline network at time
$t$ as a measure of air mobility. We name the metrics as air mobility index
(AMI). Therefore we define the weekly and monthly air mobility index as
$AMI^{w}_{t}=|E^{w}_{t}|$ and $AMI^{m}_{t}=|E^{m}_{t}|$, respectively. Similar
to Eq. 1, we normalize $AMI^{w}$ and $AMI^{m}$ as

$\displaystyle
K^{w}_{t}={{AMI^{w}_{t}-\mu_{AMI^{w}}}\over{\sigma_{AMI^{w}}}}\quad\mathrm{and}\quad
K^{m}_{t}={{AMI^{m}_{t}-\mu_{AMI^{m}}}\over{\sigma_{AMI^{m}}}},$ (3)

where $\mu_{AMI^{w}}$, $\mu_{AMI^{m}}$ and
$\sigma_{AMI^{w}}$,$\sigma_{AMI^{m}}$ are the mean and standard deviation of
$AMI^{w}$ and $AMI^{m}$ respectively.

###  2.3 COVID-19 and Mobility trend

We study the impact of COVID-19 variables (C), e.g., weekly U.S. new cases,
weekly U.S. new deaths on U.S. mobility dynamics and social distancing. We
standardized each COVID-19 variable as

$\displaystyle O_{t}={{C_{t}-\mu_{C}}\over{\sigma_{C}}},$ (4)

where, $C_{t}$ is a COVID-19 variable (weekly US new cases, weekly US new
deaths) in week $t$, $\mu_{C}$ and $\sigma_{C}$ are the mean and standard
deviation of the corresponding variable between January 13, 2020, and August
25, 2020.

Figure 2 shows the time plots of normalize weekly average of COVID-19 new
cases and deaths in 2020, along with the two mobility indices, i.e., weekly
driving trend and weekly air mobility in 2020. Notice that in the first gray
region, between late March and middle of May, when the COVID-19 new cases and
deaths jump, the driving trend and air mobility collapse. However, in the
second gray region, between the middle of July and the end of August, even
though there are surges in new cases and deaths, driving trend and air
mobility increase gradually. That is, initially a sudden increase in COVID-19
new cases and deaths negatively influence the driving trend and air mobility,
but after a certain period, high COVID-19 new cases and deaths do not cause
lower driving and decrease air mobility.

Figure 2: Time plots of normalize weekly average of COVID-19 variables (new
cases and new deaths in the US) and normalize weekly average mobility metrics
from January 2020 to September 2020.

##  3 Data

The WTI crude oil price ($ per Barrel) is obtained from the U.S. Energy
Information Administration [17]. We compute weekly average of WTI crude oil
price, $P^{\prime}$, between January 13, 2020, and August 25, 2020. We
normalize weekly WTI crude oil price $P^{\prime}_{t}$ in week $t$ as

$\displaystyle
P_{t}={{P^{\prime}_{t}-\mu_{P^{\prime}}}\over{\sigma_{P^{\prime}}}},$ (5)

where, $\mu_{P^{\prime}}$ and $\sigma_{{P^{\prime}}}$ are the mean and
standard deviation of $P^{\prime}$ between January 13, 2020, and August 25,
2020.

Apple mobility data are obtained from Apple mobility trends reports [2]. Air
mobility index is based on two types of data: airline On-Time Performance Data
and airport coordinate data. The flight data are obtained from the Bureau of
Transportation Statistics (BTS) [5]. And, U.S. airport coordinate data are
obtained from a geospatial data management platform named Koordinates [28]. We
obtain U.S. COVID-19 data from Our World in Data [37].

##  4 Methodology

We evaluate the impact of driving trends and air mobility on WTI crude oil
prices. First, we apply the quantile regression methodology to evaluate the
effects of driving trend and air mobility volume on WTI crude oil price.
Second, we quantify the left tail or extreme lower quantiles of the WTI crude
oil price associated with the COVID-19 pandemic with nonstationary extreme
value models.

###  4.1 Quantile Regression

Quantile regression is a generalization of linear regression. Whereas the
classical least squares regression estimates the conditional mean of the
response variable, quantile regression estimates the conditional median and
other quantiles of the response variable. That is, quantile regression
constructs a set of regression lines for different quantiles of the
conditional distribution of the dependent variable. Unlike least squares
regression, quantile regression does not depend on the particular parametric
assumption for the dependent variable and it also avoids constant variance
assumption [20, 25, 26, 29].

We can model a $\tau^{th}$ quantile of a response variable $y$ given a
particular value of the predictor variable, $\mathbf{X}=\mathbf{x}$, as

$\displaystyle
Q_{Y}(\tau|\mathbf{x})=\beta_{0}(\tau)+\beta_{1}(\tau)x_{1}+\beta_{2}(\tau)x_{2},$
(6)

where $\tau\in$ (0, 1), $Y$ stands for weekly WTI crude oil price ($P$),
$X_{1}$ is driving trend ($H$), and $X_{2}$ is air mobility ($K^{w}$) [24].

The parameters can be estimated by solving the equation

$\displaystyle\hat{V}(\tau)=\operatorname*{\arg\\!\min}_{b}\sum\limits_{i=1}^{n}\rho_{\tau}(y_{i}-x_{i}^{T}b)$

where,

$\rho_{\tau}(z)=\begin{cases}\tau z,&\mbox{if}\;z\geq 0\\\
(\tau-1)z,&\mbox{if}\;z<0.\end{cases}$

Koenker and Machado (1999) define a goodness of fit of the quantile regression
model by comparing its error sum of squares with error sum of squares of a
restricted form of the model. In our study, we define a restricted model which
only include an “intercept” as $Q_{Y}(\tau|\mathbf{x})=\beta_{0}(\tau)$ with
$\tilde{V}(\tau)=\operatorname*{\arg\\!\min}_{b_{0}}\sum\limits_{i=1}^{n}\rho_{\tau}(y_{i}-x_{i}^{T}b_{0})$.
We compute the Pseudo $R^{2}$ of the model defined in Equation 6 as

$\displaystyle R^{2}(\tau)=1-{\hat{V}(\tau)\over\tilde{V}(\tau)}.$ (7)

###  4.2 Extreme Value Model

Extreme value models evaluate the tail of a distribution and they have a wide
range of applications in climate and atmospheric science to industrial risks,
geosciences, finance, economics, and insurance [7, 6, 9, 11, 13, 33, 32].

For a sequence of independent random variables, $Y_{1},Y_{2},...,Y_{n}$, with
a common distribution function $G$, let
$M_{n}=\max\left\\{Y_{1},Y_{2},...,Y_{n}\right\\}$ be the maximum of the
process over $n$ time units (i.e., block) of the observations. For
$\\{a_{n}>0\\}$ and $\\{b_{n}\\}$ then
$Pr\left\\{\left(M_{n}-b_{n}\right)/a_{n}\leq z\right\\}\rightarrow G(z)$ as
$n\rightarrow\infty$, where $G$ is a non-degenerate function. According to the
Fisher and Tippett extremal theorem [21], $G$ belongs to the family of
Generalized Extreme Value (GEV) distribution. The distribution function of a
non-stationary GEV model can be written as

$\displaystyle F\left(z;\mu(t),\sigma(t),\xi(t)\right)$ $\displaystyle=$
$\displaystyle\exp\left\\{-\Big{[}1+\xi(t)\Big{(}\frac{z-\mu(t)}{\sigma(t)}\Big{)}^{-\frac{1}{\xi(t)}}\Big{]}\right\\},$
(8)

The GEV parameters $\mu(t)$, $\sigma(t)$, and $\xi(t)$ are functions of time
or/and other covariates. The $t$ denotes the time (e.g., months, year, etc.)
over which the maximum is chosen. A more detail about the nonstationary GEV
model and its’ parameters can be found in [10, 14]. The $r$-year return level,
i.e., the level expected to be exceeded once every $r$ year, can be defined as

$\displaystyle
z_{r}=\mu(t)+{\sigma(t)\over\xi(t)}\left[{\left(-\log\left(1-r^{-1}\right)\right)^{-\xi(t)}}-1\right].$
(9)

The probability of an extreme event ($z$) can be obtained as

$\displaystyle
P\left(Z>z\right)=1-\exp\left\\{-\left(1-\xi(t)\left(\frac{\mu(t)-z}{\sigma(t)}\right)^{-\frac{1}{\xi(t)}}\right)\right\\}.$
(10)

If $\mu(t)=\mu$, $\sigma(t)=\sigma$ and $\xi(t)=\xi$, the non-stationary GEV
model becomes stationary GEV model. We can estimate the GEV model parameters
using maximum likelihood estimation (MLE). The concept of return period and
return level is used to quantify the likelihood of extreme events.

In our study, we are interested in modeling the lower tail of the WTI crude
oil price, $P$, to evaluate the impact of social distancing and mobility
restrictions (i.e, $ADI$ and $AMI$) on it. That is, we focus on the minima of
WTI crude oil price over blocks rather than its maxima. The same GEV model
settings can be applied here based on the relation,

$\displaystyle
M^{\prime}_{n}=\min\left\\{Y_{1},Y_{2},...,Y_{n}\right\\}=-\max\left\\{-Y_{1},-Y_{2},...,-Y_{n}\right\\}.$
(11)

By rearranging, we find

$\displaystyle-M^{\prime}_{n}=-\min\left\\{Y_{1},Y_{2},...,Y_{n}\right\\}=\max\left\\{-Y_{1},-Y_{2},...,-Y_{n}\right\\}.$
(12)

Therefore, to evaluate the lower tail of WTI crude oil price we use the GEV
model on negative block minima of WTI crude oil price [22].

##  5 Results

We examine the effect of mobility restrictions and social distancing due to
COVID-19 on WTI crude oil price. We quantify mobility restrictions and social
distancing based on Apple’s driving trend Index and a newly proposed Air
mobility index.

Figure 3: Time plots of normalize weekly average crude oil price ($ per
Barrel) and normalize weekly average mobility metrices from January 2020 to
August 2020.

Figure 3 depicts the dynamics of WTI crude oil price, Apple’s driving index,
and Air mobility index from January 2020 to September 2020. It is clear from
the figure that the divining trend and air mobility significantly decreased in
March and April of 2020, primarily due to the surge of COVID-19 and related
stay-at-home orders at different states in the U.S. during that period. These
confinement measures slumped the demand for oil and oil products, which
results in a collapse in WTI crude oil price. Although the driving trend
increased significantly after restrictions eased, air mobility did not
increase much primarily because of people’s tendency to avoid public
transportation. WTI crude oil price gradually increased after lockdown and
other restrictions were relaxed.

We now evaluate the effect of driving and air mobility tend on WTI crude oil
price. We develop a quantile regression model for each of the quantiles
$\tau=(0.01,0.05,0.1,0.5,0.9,0.95,0.99)$ based on Eq. 8. We are particularly
interested in the lower and upper quantile of the WTI crude oil price. Table 2
summarizes the outputs of the quantile regression models.

We find that both driving trend and air mobility significantly affects lower
and upper quantiles as well as the median of the WTI crude oil price. The
coefficients of driving trend and air mobility are all positive. Therefore, a
decrease (increase) in driving trend or air mobility leads to a decrease
(increase) in $\tau^{th}$ quantiles of the WTI crude oil price, where
$\tau\in(0.01,0.05,0.1,0.5,0.9,0.95,0.99)$. Notice that the coefficient of
driving trend tends to gradually decrease from the lower quantiles to upper
quantiles. On the other hand, the coefficient of air mobility tends to
gradually increase from the lower quantiles to the upper quantiles. That is,
the driving trend has a greater influence on the lower quantiles of the WTI
crude oil price. On the other hand, air mobility has a higher influence on the
upper quantiles of the WTI crude oil price.

The values of $R^{2}$ are higher in lower quantiles. That is, driving trend
and air mobility able to explain the variability of lower quantiles of WTI
crude oil price better compared to its upper quantiles. In our study, we focus
on how does driving trend and air mobility shape the collapse of WTI crude oil
price during the COVID-19 pandemic. The $R^{2}$ values reveal that the driving
trend and air mobility greatly influence the extreme decrease of WTI crude oil
price (lower quantiles) during the pandemic. Figure 4 depicts the regression
lines of WTI crude oil price for different quantiles.

Table 2: Estimates of quantile regression models for crude oil price with standard errors. ${}^{***}~{}p<0.01$, ${}^{**}~{}p<0.05$, ${}^{*}~{}p<0.1$. |  |  |  | Quantile ($\tau$) |  |  |
---|---|---|---|---|---|---|---
| 0.01 | 0.05 | 0.10 | 0.50 | 0.90 | 0.95 | 0.99
Constant ($\beta_{0}$) | -0.649∗∗∗ | -0.649∗∗∗ | -0.568∗∗∗ | -0.071 | 0.405∗∗∗ | 0.585∗∗∗ | 0.585∗∗∗
| (0.041) | (0.097) | (0.114) | (0.185) | (0.102) | (0.144) | (0.074)
Driving trend ($\beta_{1}$) | 0.846∗∗∗ | 0.846∗∗∗ | 0.786∗∗∗ | 0.527∗∗∗ | 0.559∗∗∗ | 0.273∗ | 0.273∗
| (0.034) | (0.081) | (0.090) | (0.178) | (0.151) | (0.229) | (0.169)
Air mobility ($\beta_{2}$) | 0.298∗∗∗ | 0.297∗∗ | 0.229∗ | 0.358∗∗ | 0.514∗∗∗ | 0.617∗∗∗ | 0.617∗∗∗
| (0.077) | (0.145) | (0.170) | (0.199) | (0.085) | (0.088) | (0.039)
Pseudo $R^{2}$ | 0.738 | 0.722 | 0.687 | 0.536 | 0.540 | 0.523 | 0.526
Figure 4: Normalize WTI crude oil prices regression models for different
quantile levels.

Now we turn our analysis to evaluate the crash of WTI crude oil price
resulting from the COVID-19 pandemic based on the extreme value model. We
study monthly minimum crude oil price (Y) for the last twenty years from 2000
to 2020 and model left tail, i.e., extreme lower quantile of the WTI crude oil
price. However, due to the unavailability of Apple’s driving trend index, we
only use monthly air mobility $K^{m}$ as a covariate in the extreme value
modeling. Figure 5 shows the time plot of monthly minimum crude oil price and
monthly air mobility (on the $15^{th}$ of every month) in the U.S. from
January 2000 to August 2020.

Figure 5: Monthly minimum crude oil price and monthly air mobility from
January 2000 to August 2020.

We experiment a set of of non-stationary extreme value models,
$Y^{\prime}_{t}\sim GEV(\mu(t),\sigma(t),\xi(t))$, where
$Y^{\prime}_{t}=-Y_{t}$, $Y_{t}$ is the monthly minimum crude oil price in
month $t$. We assess the different combinations of time trends and covariate
(air mobility) effect on the monthly minimum crude oil price. We select two
best models based on (smaller) Akiake Information Criterion (AIC) [1] and
Bayesian information criterion (BIC) [34] values. Table 3 describes the
selected models along with a baseline stationary model. Model 1 allows a
linear time trend in the location parameter. In Model 2 location parameter
depends linearly on time trend and air mobility. There is a time trend in the
log-scale parameter in both models.

Table 3: Different stationarity/non-stationary models with corresponding AIC and BIC. Model | Model descriptions | AIC/BIC
---|---|---
Model 0 | $Y^{\prime}_{t}\sim GEV(\mu,\sigma,\xi)$ | $AIC=2285.212$
(Stationary) |  | $BIC=2295.751$
| $Y^{\prime}_{t}\sim GEV(\mu(t),\sigma(t),\xi)$ | $AIC=2164.935$
Model 1 | $\mu(t)=\beta_{0}+\beta_{1}t$ | $BIC=2182.503$
| $log~{}\sigma(t)=\gamma_{0}+\gamma_{1}t$ |
| $Y^{\prime}_{t}\sim GEV(\mu(t),\sigma(t),\xi)$ | $AIC=2129.58$
Model 2 | $\mu(t)=\beta_{0}+\beta_{1}t+\delta K^{m}$ | $BIC=2150.66$
| $log~{}\sigma(t)=\gamma_{0}+\gamma_{1}t$ |

Table 4 shows the estimated parameters of Model 1 and Model 2. In Model 2 the
covariate air mobility has a negative effect on monthly minimum WTI crude oil
price, where the estimated value of $\delta$ is -5.190, with a standard error
of 0.974. Therefore, one unit increase in air mobility moves monthly minimum
WTI crude oil price in opposite direction (decrease) by 5.190 units when
covariate ($t$) held constant. That is, an increase in air mobility results in
an increase in overall WTI crude oil price.

Figure 8 in Appendix represents the goodness of fit plots of the two models.
We find that Quantile-Quantile (Q-Q) plots for both Model 1 and Model 2
approximately $45$-degree line. The observed versus fitted density plot for
both models also yield that both Model 1 and Model 2 capture the variability
of the monthly minimum WTI crude oil price.

Table 4: Estimated parameters of nonstationary GEV models described in Table 3, standard errors are in parenthesis. | $\hat{\beta_{0}}$ | $\hat{\beta_{1}}$ | $\hat{\delta}$ | $\hat{\gamma}_{0}$ | $\hat{\gamma}_{1}$ | $\hat{\xi}$
---|---|---|---|---|---|---
Model 1 | -22.11 | -0.387 |  | 1.974 | 0.010 | -0.606
| (0.651) | (0.014) |  | (0.056) | ($<$ 0.001) | (0.0443)
Model 2 | -27.965 | -0.298 | -5.190 | 2.035 | 0.008 | -0.631
| (0.909) | (0.015) | (0.974) | (0.053) | ($<$ 0.001) | (0.049)

We compute the return level, i.e., the extreme lower quantile of WTI crude oil
price which is expected to fall below on average once every $r$ year using Eq.
9. The return levels $z_{r}$ based on Model 1 depend on time $t$. That is, for
certain values of $t$ we can compute return levels of crude oil price for
different return periods. The return levels based on Model 2 depend on time
$t$ as well as on monthly air mobility. Table 5 represents two scenarios of
the covariates and their corresponding 10, 20, 50, and 100 years return
levels.

Table 5: Estimated return levels for the GEV distribution fitted to the monthly minimum crude oil price. $t=193$ (Jan-16), $t=244$ ( Apr-20). | Covariates | 10-yr return | 20-yr return | 50-yr return | 100-yr return
---|---|---|---|---|---
| values | level | level | level | level
Model 1 |  $t=193$ (Jan 2016) | 42.042 | 35.387 | 30.139 | 27.754
|  $t=244$ (Apr 2020) | 27.826 | 17.049 | 8.5499 | 4.687
Model 2 |  $t=193$, $K^{m}=15308$ (Jan 2016) | 35.253 | 29.911 | 25.779 | 23.939
|  $t=244$, $K^{m}=8643$ (Apr 2020) | 12.687 | 4.517 | -1.801 | -4.615

Based on Model 1, for $t$ corresponds to April 2020, the estimate of the crude
oil price that we expect to fall below once every 10, 20, 50, and 100 years
are 27.826, 17.049, 8.5499, and 4.687, respectively. Notice that the minimum
crude oil price in April 2020 was 6.65 and the $100$-year return level is
4.687. Therefore, based on Model 1, we can say that the minimum oil price in
April 2020 is a $100$-year event, which is also clear in Figure 6. However,
when we include monthly air mobility as a covariate in the model (Model 2), we
find that return levels fall dramatically. For the values of $t$ and $N$ in
April 2020, the 10 and 20 years return levels are 12.687 and 4.517,
respectively. Therefore, lower air mobility ( Model 2) makes the minimum oil
price in April 2020 a $20$-year event. That is, lower air mobility increases
the risk of occurring a very low oil price. Figure 7 depicts how a very low
oil demand (i.e., oil demand in April 2020) causes frequent lower oil prices.

We have also found that Model 2 predicts 50-yr return and 100-yr return levels
as -1.801 and -4.615, respectively. Therefore, there is a risk of occurring a
negative oil price -1.801 in every 50 years, and a negative oil price -4.615
in every 100 years, if there is very low air mobility, i.e., $K^{m}=8,643$
(with a time trend $t=244$).

Figure 6: Return levels of minimum crude oil prices for different return
periods, based on Model 1. Figure 7: Return levels of minimum crude oil prices
for different return periods, based on Model 2.

##  6 Conclusion

To the best of the author’s knowledge, the proposed air mobility index and
Apple’s driving trend index have not been used to study the impact of COVID-19
on crude oil prices. We use the concept of temporal network to develop the air
mobility index. We use quantile regression to evaluate the effects of driving
trend and air mobility volume on WTI crude oil price. Based on the results
obtained from quantile regression, we claim that driving trend has a higher
influence on lower quantiles of crude oil prices whereas air mobility has a
higher influence on upper quantiles of crude oil prices. We also quantify the
tail or extreme quantiles of the WTI crude oil price associated with the
COVID-19 pandemic by stationary and non-stationary GEV models. We conclude
that with COVID-19, an already volatile oil market has reached a flashpoint.
Although oil prices have recovered significantly in the past couple of months,
it is unlikely that there will be the same buoyancy in prices as witnessed
following the previous global economic recessions such as the Great Recession
in 2008.

##  7 Appendix

(a) Q – Q plot for Model 1.

(b) Density plot, Model 1.

(c) Q – Q plot for Model 2.

(d) Density plot, Model 2.

Figure 8: Diagnostic plots for Model 1 and Model 2.

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