1 Introduction

International energy markets are highly volatile as the prices of energy are constantly fluctuating around shocks, crisis, and news of different kind and nature. This has been even more evident during the last twenty years, since the flow of funds corresponding to energy has experimented an exponential growth due to an increasing global energy demand that has fostered large energy trade volumes between countries He et al. (2018); Chen et al. (2021). Volatility in international oil markets is very relevant for the economy since oil is a core driver of modern economic activity Miller and Ratti (2009); Albulescu and Ajmi (2021); Basher and Sadorsky (2006). In finance, this topic is equally relevant as the stocks market react to oil shocks and volatility spillovers Nandha and Faff (2008); Malik and Ewing (2009); El-Sharif et al. (2005); Balcilar and Ozdemir (2013); Henriques and Sadorsky (2008); Nandha and Brooks (2009); Falzon and Castillo (2013).

With the arrival of globalization, financial markets have become increasingly interconnected. This situation has led to substantial benefits due to higher capital circulation from countries with excess savings to countries with a deficit of funds, generating economic growth Bernanke et al. (2011); Matsumoto (2022); Nguyen (2022). However, it also has presented some new challenges for the global economy. In particular, the increased connectedness has resulted in an important financial market integration, which means that markets comove and so the volatility generated by a shock in one part of the world can easily propagate to other areas. The present situation therefore has situated important challenges at the forefront of the analysis and modelling of financial and economic stability Bordo et al. (2016); Sánchez García and Cruz Rambaud (2023).

There exists a vast body of literature which has studied volatility shocks and spillovers from oil to stocks markets in different periods, places, and with different methodologies. For example, Chen et al. (2018) found unidirectional spillovers from crude oil future prices to energy stock market prices in China. Mensi et al. (2013) studied the correlations and volatility spillovers between energy, food, gold and beverages during 2000-2011 finding important correlations and spillovers for the SP500 and the rest of the involved variables, with important implications for optimal portfolio and portfolio management. Ashfaq et al. (2019) studied the volatility and spillover effects of oil prices on the stock returns of the biggest Asian exporting and importing economies. Jiang and Marsh (2019) analyzed the mean and volatility spillover linkages between the US oil market, the US stock market and the US Energy Pipeline Sector Index, including the impact of the financial crisis of 2007-2009. Sarwar et al. (2020) analyzed volatility spillovers between oil and returns for Karachi, Shangai and Bombay stock markets, by dividing the data in pre- and post-financial crisis samples to capture different volatility regimes. Finally, Wang (2020) analyzed volatility spillovers between international crude and stock markets, finding that the volatility spillovers are mainly driven by short-term spillovers and that interest rates are their primary drivers.

However, two important gaps in the relevant literature on this topic can be highlighted. First, there is an absence of literature which examines this relationship from a dynamic perspective, i.e., how volatility spillovers evolve over time, in different periods and under different volatility regimes. This is not true for the relationship between other macroeconomic variables, such as economic uncertainty and oil prices Antonakakis et al. (2014), currency markets and commodities Antonakakis and Kizys (2015), oil returns and policy uncertainty Wang and Lee (2020), and so on. Thus, this channel may be increased or reduced during periods of high volatility such as economic or financial crises, and so, this paper proposes a closer examination of this phenomenon. Second, there is very small evidence on the behavior of volatility spillovers from the oil to the financial markets and vice-versa, depending on whether the shock is real economy or financial sector oriented. In fact, the behavior of volatility spillovers may vary depending on the nature of the shock, and if so, volatility shocks and spillovers from stocks to oil markets may be higher than the reverse during financial crises, making oil markets a receiver of shocks and spillovers instead of a transmitter in net terms, and vice versa. Our study helps to build understanding of these two gaps, and differentiates from previous works with similar purposes, such as Elgammal et al. (2021), Mensi et al. (2021), and Jebabli et al. (2022) in that dynamic comparisons are applied, and the behavior of volatility spillovers only during crises is studied.

In order to tackle these research questions, this paper employs a combined GARCH Teräsvirta (2009), VAR Lütkepohl (2013), and spillover Diebold and Yilmaz (2009) approach. In particular, it employs the GARCH methodology to measure the volatility and its peaks, the impulse response functions of the VAR model to trace volatility shocks, and the forecasted error variance and the spillover index approach to measure static and dynamic spillovers with a recursive sampling strategy.

The rest of this paper is structured as follows. Section 2 presents the data and the methodology used to measure volatility and volatility spillovers. Section 3 shows the main volatility results and some simulations, the impulse response functions, the static spillovers between all variables, and the comparison between the dynamic behavior of volatility spillovers for the two highest periods of volatility in the millennium, the COVID-19 pandemic and the Great Recession. Finally, Section 4 summarizes and concludes.

2 Theoretical framework, data and methodology

2.1 Theoretical framework

Following Degiannakis et al. (2018), the main theoretical transmission mechanisms from oil prices to stocks markets are:

  1. 1.

    The stock valuation channel. This is the direct channel by which oil prices influence stock markets. Defining \(R_{i,t}\) as the stock returns:

    $$\begin{aligned} R_{i,t}=\log \frac{P_{i,t}}{P_{i,t-1}}, \end{aligned}$$

    where \(P_{i,t}\) denotes the price of the stock of a firm i at time t. By economic theory, which states that stock prices discount its future cash flows Huang et al. (1996), then

    $$\begin{aligned} P_{i,t}=\sum _{n=t+1}^{N}\frac{E(CF_n)}{[1+E(r)]^n}, \end{aligned}$$

    where \(CF_n\) is the cash flow at time n, E the mathematical expectation, and r is the discount rate. Stock returns are then affected by the expected cash flow and the discount rate, as well as the factors which alter them, such as oil prices. Depending on whether a firm is an oil-producer or demander, oil price can increase or decrease its cash flow.

  2. 2.

    The monetary channel. Inflation and interest rates affect the discount rate Mohanty et al. (2011), and therefore oil prices affect interest rates through this mechanism. Thus, rising oil prices turn into increased costs for firms, which are passed to the consumer. On the other hand, the increase in inflation is expected to cause a contractive monetary policy response Basher and Sadorsky (2006). Finally, the increase in the interest rates due to monetary policy will make stocks decline.

  3. 3.

    The output channel. Oil price fluctuations are assumed to affect aggregate output Hamilton (1983, 2003); Kilian (2008). The literature claims that positive oil price shocks have an income and production effect which leads to changes in the aggregate output. Specifically, increased oil prices decrease the marginal available income of households, as retail prices, as well as gasoline and heating, increase. The production channel is similar to channel 2.

  4. 4.

    The fiscal channel. Oil exporting countries finance their investments by using oil export revenues. Rising oil prices leads to a wealth transfer from the oil importing to the oil exporting countries Dohner (1981). Thus, the governments of the oil exporting countries will have higher purchasing power, which would lead to a higher demand. The companies will then increase their production to satisfy the demand, so private firms are expected to increase their cash flows and, as a result, stock prices will reach higher levels.

  5. 5.

    The uncertainty channel. It was introduced by Brown and Yücel (2002). Increasing oil prices cause higher uncertainty in the economy due to its effects on inflation, output, etc. As a result, increased oil prices decrease the demand of firms for sunken investments, which reduces expected cash flows and, therefore, stock prices. This uncertainty is also propagated to households which reduce their consumption for durable goods Bernanke (1983). As shown by Edelstein and Kilian (2009), the incentives of households to save rather than to consume also increase.

Volatility in oil prices is therefore expected to cause volatility in the stock markets through the aforementioned channels. Cash flows, gasoline prices, inflation, output, etc. will vary as the result of continuous variations in oil prices, which will be turned into unstable stock prices. In addition, we postulate that these transmission channels may be reversed if the origin of the shock is financial. If, for example, a round of defaults due to overindebtness diminishes the stock value of a group of companies located in oil exporting countries, this will reduce their expected cash flow. This will reduce the investment and operating capabilities of those companies, which will have an effect on oil prices via changes in oil supply. The empirical existence of this theoretical reverse volatility spillovers is tested in the next sections of this paper.

2.2 Data

The oil data consists of 256 monthly observations from the West Texas Intermediate Index (WTI) and the Brent Index (BI), with a sample period from the 1st of January 2000 to the 1st of April 2021, downloaded from the Federal Reserve Bank of St. Louis. The data are available to the public at the website https://fred.stlouisfed.org/.

The log-return corresponding to period t can be calculated as follows:

$$\begin{aligned} r_t = \log \frac{P_t}{P_{t-1}} = \log P_t - \log P_{t-1}, \end{aligned}$$
(1)

where \(P_t\) is the price at the end of period t, and \(P_{t-1}\) is the price at he end of period \(t-1\).

Figures 1 and 2 show the temporal evolution of the raw data and log-returns. At first sight, the series have a high chance of being cointegrated. The returns series show clear signs of volatility spikes being during the COVID-19 pandemic higher than in any other period of the millennium. The log-returns seem to be stationary.

Fig. 1
figure 1

Monthly prices of the WTI and BI international crude indices. Source: Own elaboration

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Fig. 2
figure 2

Logarithmic returns of WTI and BI prices according to Equation (1). Source: Own elaboration

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Table 1 exhibits the statistics of the oil returns for the WTI and BI. Observe that their means oscillate around 0.14%, and 0.16%, and their standard deviations around 0.046 and 0.043, respectively. The min-max path is slightly higher for the WTI, whilst the returns are negatively skewed and the excess of kurtosis indicates a leptokurtic distribution, pointing to a flat-type distribution with fat tails in both cases. The Jarque-Bera statistical test rejects a normal distribution, the ARCH(12) test indicates the presence of Autorregresive Conditional Heteroskedasticity, and the Augmented Dickey Fuller (ADF) test indicates stationary data, for every level of statistical significance in the three tests. Moreover, the Engle-Granger test indicates cointegration at every confidence level.

Table 1 Summary of statistics and statistical tests of the oil data. Source: Own elaboration
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The data corresponding to financial markets are standard values of the monthly closing price from the SP500, NASDAQ100, DAX30 and IBEX35 stock markets. The observation range is similar to the oil data, i.e., [2000-01: 2021-04]. The data are expressed in log-returns calculated by following Equation (1). The summary statistics are offered in Table 2.

As shown in Table 3, all variables are stationary at any level of significance as shown by an ADF-GLS test.

Table 2 Summary of statistics and statistical tests of the financial data. Source: Own elaboration
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2.3 Volatility methodology

The methodology for the volatility analysis employed in this paper consists of GARCH (‘sGARCH’) models. They have been computed with the ‘rugarch’ package Ghalanos (2020) of R software, and therefore follow a similar notation. The reader is referred to Teräsvirta (2009); Ghalanos (2020) for further technical inquiries. For a detailed application of GARCH models in macroeconomic and macrofinancial monthly time series, see Sánchez García et al. (2022).

A noteworthy problem in Econometrics is to approximate the distribution of a variable \(y_t\), conditional on another variable \(x_t\). In this context, most approaches involve estimates of the conditional mean \(\mu _t = E(y_t | x_t)\). In general, let \((y_t,x_t)\) (\(t=1,2,\dots ,n\)) be the observed sample, that is to say, the realization of some jointly stationary process (Y, X). We are interested in those distributions with density functions in the form:

$$\begin{aligned} f(y | \alpha (x_t,\theta )) := \frac{\textrm{d}}{\textrm{d}t} P(y_t \le y | x_t), \end{aligned}$$
(2)

where \(\theta \) is a finite-dimensional parameter vector, and

$$\begin{aligned} \alpha _t := \alpha (x_t,\theta ) \end{aligned}$$
(3)

is a low-dimensional parameter, depending on time, which describes the influence of \(x_t\) on the conditional distribution. This is why these models are called “autoregressive conditional density models” (in short, ARCD).

Table 3 ADF-GLS tests of the data. Source: Own elaboration
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However, for applied research, it is convenient to rewrite the density function in terms of location and scale parameters, specifically those cases where the location parameter is the conditional mean and the scale parameter is the conditional variance. Therefore, we start from the triple:

$$\begin{aligned} \alpha _t := (\mu _t,\sigma _t^2,\eta _t), \end{aligned}$$
(4)

where:

  • \(\mu _t = \mu (\theta ,x_t) = E(y_t | x_t)\) is the conditional mean,

  • \(\sigma _t^2 = \sigma ^2(\theta ,x_t) = E((y_t - \mu _t)^2 | x_t)\) is the conditional variance, and

  • \(\eta _t = \eta (\theta ,x_t)\) includes the rest of (shape) parameters of the conditional distribution.

Thus, the conditional mean and variance allow to define the normalized variable

$$\begin{aligned} z_t(\theta ) := \frac{y_t - \mu (\theta ,x_t)}{\sigma (\theta ,x_t)}, \end{aligned}$$
(5)

where, as usual, \(\sigma (\theta ,x_t)\) denotes the conditional standard deviation. For the treatment of spillovers in the next subsection, the parameter of interest is the conditional variance (\(\sigma _t^2\)), which is modelled as:

$$\begin{aligned} \sigma ^2_t = \left( \gamma + \sum _{j=1}^{m} \delta _{j}v_{jt}\right) + \sum _{j=1}^{q} \alpha _j \epsilon ^2_{t-j} + \sum _{j=1}^p \beta _j \sigma ^2_{t-j}, \end{aligned}$$
(6)

where \(\gamma \) is the intercept, \(v_j\) the m external regressors which are passed pre-lagged, and \(\epsilon _{t}\) the residuals of the conditional mean models computed as an AR(2) process selected to eliminate any form of autocorrelation in the residuals:

$$\begin{aligned} \mu = E(y_t|x_t) = \phi _0 + \phi _1 y_{t-1} + \phi _2 y_{t-2} + \epsilon _t. \end{aligned}$$
(7)

2.4 Spillover methodology

The spillover methodology used in this paper derives from VAR models Sims (1980); Pfaff (2008); Lütkepohl (2013) and the Spillover Index approach from Diebold and Yilmaz (2009), Diebold and Yilmaz (2012) and Antonakakis et al. (2014). It allows for the measurement of the magnitude of interdependence between the volatility of variables allowing to trace and decompose the origin and the receiver of the spillover effect.Footnote 1

The VAR processes adopt the form:

$$\begin{aligned} y_t = A_1 y_{t-1} + A_2 y_{t-2} + \cdots + A_p y_{t-p} + \epsilon _t, \end{aligned}$$
(8)

where \(y_t = (y_{1t}, y_{2t}, \dots , y_{kt})\) is the vector of the k endogenous variables, \(A_1, A_2, \dots , A_p\) are the \(k \times k\) coefficient matrices, and \(\epsilon _t\) is the vector of error terms, which follows a white noise.

The core of the volatility dynamics analysis comes from the moving average representation of the model:

$$\begin{aligned} y_t = \sum _{i=0}^\infty B_i \epsilon _{t-i}, \end{aligned}$$

where the \(k \times k\) parameter matrices are obtained by recursion. Then, the h step forecast is:

$$\begin{aligned} y_{T+h} - y_{T+h | T} = \Phi _0\epsilon _{T+h} + \Phi _1\epsilon _{T+h-1} + \cdots + \Phi _{h-1}\epsilon _{T+1} \end{aligned}$$
(9)

and letting \(\Sigma = I_{K}\), the Forecasted Error Variance Decomposition (FEVD) of the ith element of \(y_{T+h}\) is defined as Lütkepohl (2013):

$$\begin{aligned} \sigma _{i}^2 = \sum _{j=0}^{K} (\beta _{ij,0}^2 + \cdots + \beta _{ij,h-1}^2), \end{aligned}$$
(10)

where \(\beta _{ij}\) is the ij component of \(\Phi _{i}\).

The models employed in this paper use the Cholesky decomposition, making the order of the variables relevant. The Cholesky decomposition gives a k \(\times \) k matrix \(\beta = (\beta _{ij})\) (\(i,j = 1,\dots ,k\)), where each element represents the impact of the variable j on the variable i.

Consider the variance-covariance matrix \(\Sigma _\epsilon \), which is positive-definite and symmetric. By the Cholesky decomposition, given a symmetric positive-definite matrix, there exists a unique lower-triangular matrix C where the main diagonal equals I, and a diagonal matrix D whose main diagonal is composed of positive elements such that

$$\begin{aligned} \Sigma _\epsilon =CDC'. \end{aligned}$$

Then, the vector \(\epsilon _t\) admits the following linear transformation:

$$\begin{aligned} v_t=C^{-1}\epsilon _t \end{aligned}$$

with

$$\begin{aligned} \hbox {var}(v_t)=D, \end{aligned}$$

where the elements of \(v_t\) are uncorrelated. To get intuitive results, the elements of matrix \(\beta \) can be normalized by the rest of the elements of their row, i.e.,

$$\begin{aligned} \bar{\beta }_{ij} = \frac{\beta _{ij}}{\sum _{j=1}^{k} \beta _{ij}}, \end{aligned}$$
(11)

where \(\bar{\beta } := (\bar{\beta }_{ij})\) is the matrix composed by the normalized elements. The volatility spillover indices are then calculated as follows.

  • The Total Spillover Index:

    $$\begin{aligned} TSI = \frac{\sum _{{\mathop {i \ne j}\limits ^{i,j = 1}}}^{k} \bar{\beta }_{ij}}{k} \times 100. \end{aligned}$$
    (12)

  • The Directional Spillover Index:

    $$\begin{aligned} DSI_{i\leftarrow j} = \frac{\sum _{{\mathop {i\ne n}\limits ^{j=1}}}^{k} \bar{\beta }_{ij}}{n} \times 100, \end{aligned}$$
    (13)

    which can be calculated by pairs, if \(j=1\), or for all the remaining \(j=1,\dots ,k, i\ne j\) variables, and where n equals the FEVD horizon. In the case of pairs (\(j=1\)), the net directional spillovers can be calculated by the next index.

  • Net Pairwise Spillover Index:

    $$\begin{aligned} NPS_{ij} = \frac{\bar{\beta _{ji}} - \bar{\beta _{ij}}}{n} \times 100 \end{aligned}$$
    (14)

    or

    $$\begin{aligned} NPS_{ij} = DSI_{i\rightarrow j} - DSI_{i \leftarrow j}, \end{aligned}$$
    (15)

    i.e., the net spillovers between two variables are the difference between their gross spillovers.

Fig. 3
figure 3

\(\hat{\sigma }^{2}\)s calculated by the GARCH models (log scale). The blue line represents the Brent Index, whilst the red line represents the WTI Index. Source: Own elaboration

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2.5 Identification of the shocks

In this paper, the volatility shocks originated by crises are divided into two categories: financial and economic. It is implicitly considered the reasonable assumption that a financial crisis is reflected first by financial markets, and that subsequently other variables of the economic and financial system respond to the shock. Moreover, it is assumed that economic shocks driven by events which cause drastic demand or supply restrictions are more likely to be represented first by variables which are at the center of the productive structure of the real economy, such as energy prices.

To identify the shocks, a narrative identification similar to the way of Antolín-Díaz and Rubio-Ramírez (2018) is employed. In particular, two narrative identification restrictions are written for identifying the shocks:

Narrative Identification Restriction 1 Economic (financial) crises generated by drastic and sudden economic (financial) events are first reflected by variables associated to supply and demand disruptions in the economic (financial) system.

During the COVID-19 pandemic, drastic decreases in demand paralyzed the supply chain, pushing oil prices heavily downwards, as shown in Figure 2. This event had its roots in a sudden stop of the economic system due to lock-downs, mobility restrictions and so on. Thus, Restriction 1 implies that this behavior was first reflected by oil prices.

On the other hand, the financial crisis of 2008 was caused by an exponential credit boom Chari et al. (2008) and a housing frenzy Brunnermeier (2009), which leaded to a liquidity crunch due to defaulting and the lack of an adequate rating of financial products. In this way, an event with financial origins is expected to be anticipated by financial markets.

Narrative Identification Restriction 2 During economic (financial) crises, the economic (financial) variables of Restriction 1 suffer most volatility than in any other period of the sample.

Restriction 2 comes from empirical evidence. In Figures 2 and 3, it is possible to see that the sudden stop of the economy due to COVID-19 pandemic caused the highest peaks of volatility in the sample period. Furthermore, at its side, the spike of the financial crisis looks very similar to other random volatility spikes of the sample period. Similarly, the VIX index reached its maximum value in 2007, indicating that financial markets suffered from most volatility during this period in the recent economic history.

Even though crises with economic (financial) variables are expected to be anticipated by economic (financial) variables, the correlations between both variables may cause spillovers to the others, which may further cause feedback loops. However, the net spillover approach of Equation (15) allows to trace which variable leads the way as it nets the interactions, reflecting which variable dominates the relationship.

3 Results

3.1 Volatility measurement and simulations

Table 4 shows the two estimated GARCH models, which present statistically significant \(\mu \), \(\alpha \) and AR(1) parameters. All parameters have the expected sign, i.e., the conditional mean of the average returns (\(\hat{\mu }\)) is positive, the effect of past values of the conditional variance (\(\hat{\beta }\)) on the conditional variance of the current period is positive, and the effect of the squared errors on the conditional variance (\(\hat{\alpha }\)) is positive, which confirms the existence of ARCH errors. However, only the conditional mean and the ARCH errors are statistically significant in both models, presenting the latter the most powerful effects on the conditional variance by far with coefficients more than 10 times higher than any other parameter. The AR(1) and AR(2) parameters have been added because they capture all forms of statistically significant autocorrelation, but only the AR(1) parameter is statistically significant in the regression. It confirms that the expected returns of the last period have a positive effect on the returns of the present period. Similarly, there is no expected sign for the parameter \(\hat{\omega }\), since it is a control for the rest of the aspects of the conditional distribution.

The three information criteria show very similar values for both models. The weighted Ljung-Box statistic of 9 lags for the residuals and squared residuals shows no sign of autocorrelation or remaining Autorregresive Conditional Heteroskedastic processes, indicating that the two models have captured any form of autocorrelation or ARCH processes. This result is supported by the ARCH Lagrange Multiplier test for 7 lags. The Nyblom Stability tests for individual parameters indicate that all coefficients are stable in both models; however, the joint statistic indicates stability at every confidence level only for the WTI model. The Engle-Granger sign bias tests point to non-mispecification of the model in both cases. Additionally, the Pearson Goodness of Fit test accepts the normality of the standardized residuals in both models.

Table 4 Estimated GARCH models. Source: Own elaboration
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Figure 3 shows the volatility modelled by the GARCH specifications during the period from February 2000 to April 2021. At a first glance, it can be noticed that volatility in international oil markets has been higher during the COVID-19 pandemic than during the 2007-2009 financial crisis, as shown by both price indices. In fact, observe that the volatility corresponding to the first period has been nearly the double.  A second result is that, in contrast with the 2007-2009 financial crisis, where both indices practically co-moved, during the COVID-19 pandemic the volatility spikes were considerably higher in the case of the WTI. This gap appears mainly in the months of May and June of 2020, when the Brent Index recovers but the WTI does not. This may be related with the OPEC+ record cuts of oil supply happened in the month of May, with the COVID-19 and the economic shut-ins affecting directly to the United States and Canada. These two periods present the highest volatility of the millennium.

Figure 4 confronts the kernel density of the conditional \(\hat{\sigma }^{2}\) of the GARCH simulations with the actual densities. The kernel densities fit very well the actual data, whose probability density functions (PDFs) look very similar to an inverse chi-squared distribution.

Fig. 4
figure 4

Conditional Kernel Densities of the \(\hat{\sigma }\) of the GARCH simulations vs actual PDFs. Source: Own elaboration

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Fig. 5
figure 5

Impulse responses. Source: Own elaboration

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3.2 Impulse responses

Figure 5 shows the impulse response functions of shocks to one standard deviation from and to all variables in the system of equations. In particular, two VAR models were estimated by following Equation (8). The first one includes the WTI, the SP500 and the NASDAQ, whilst the second one, the Brent, the DAX, and the IBEX. Each system groups an oil price index with the stock market index affecting a country directly related to the former, i.e., which operates with that oil price index.

A common pattern can be noticed in oil price shocks. These have a little, statistically significant, positive effect on financial markets during the very short run, i.e., during two months. All stock markets share this same result, and this is consistent since the oil price indices are cointegrated. Therefore, a shock on one market may lead to a shock on the other of approximately the same magnitude. In fact, the impulse responses of the shocks from the WTI to the WTI and from the Brent to the Brent start and evolve in a very similar way.

Another pattern which can be easily recognized is how shocks from strong indices (highly correlated with the evolution of the other stock markets and the oil prices), i.e., the SP500 for the US and the DAX for Europe, have a statistically significant impact on the oil prices. In fact, this impact is more than ten times higher than the one from the oil prices to the stock markets, and more persistent, as it lasts for approximately four months. These findings may require further attention as some studies examine the impact of shocks and the evolution of oil prices on stock markets Nandha and Faff (2008); Bhar and Nikolova (2009, 2010), and others examine volatility movements and co-movements Malik and Hammoudeh (2007); Arouri et al. (2012); Phan et al. (2016), but a closer and global examination of the impact of shocks and the evolution of stock markets reacting to oil prices may lead to further findings in this bidirectional relationship, as monthly data indicates. There could also exist a distinction between indices which move oil prices and indices which are moved by oil prices, as the preliminary evidence in this study signs.

Shocks in weak indices (low correlated with other indices and oil prices) do have very small effects in the case of both strong indices and oil prices. It is noticeable that a shock in the NASDAQ index causes a small, significant, negative response of the WTI price, being this the only negative response of the system. As expected, shocks in the strong indices have substantial effects in the weak indices, very close to their own response, or even higher in the case of the SP500 to the NASDAQ.

Table 5 Directional total and net spillovers and total spillover indices. Source: Own elaboration
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3.3 Static spillovers

Table 5 presents the static spillovers calculated by equations from (12) to (15) for the sample period from 2000-2001 to 2021-2004. Each column represents the directional spillover of each variable j to the volatility of the other i variables, and the final row in each box represents the Total Spillover Index and the Total Net Spillover of each variable j.

The Total Spillover Index represents the overall impact of the source on the receiver variable. With respect to them, it is possible to see that the SP500 and the DAX have the highest values (72.4% and 65.5%, respectively). They are mainly distributed, firstly, by their influence on the NASDAQ and the IBEX (66% and 58%, respectively) and, secondly, by their influence on the WTI and the Brent (6.4% and 7.5%, respectively). This result is intuitive as they are leading indices which carry over other indices of their environment, and therefore have substantial spillovers on them. The NASDAQ and the IBEX have very little spillover effects on both, the oil prices and the rest of stocks. On the other hand, the WTI has a considerably higher (more than 3.5 times) Total Spillover Index (21.8%) than the Brent (5.7%), i.e., the North American reference crude oil has more spillovers on the stock markets than the European reference oil.

From the Net Spillovers it is possible to see that the strong indices are the main sources of volatility spillovers. In particular, the DAX is the main net transmitter of volatility (81.7%) followed by the SP500 (59.9%). This is caused mainly by the strong influence of DAX on the Brent (24% in net terms) and the reduced directional spillover from the Brent to the DAX (1.5%). It is important to mention here that the WTI is the third net transmitter of shocks (14.9%), unlike the Brent, which is a net receiver (−8.1%). This difference may require a closer examination, as it may be generated by the strong spillover impact of the DAX on the Brent, and may or not fade out when incorporating further stock market indices. Eventually, the NASDAQ and the IBEX share very close negative net spillover positions (−74.8% and −73.6%, respectively). Additionally, it may reveal the power of the United States crude oil in the world.

3.4 Dynamic spillovers

Figure 6 shows the comparison of the dynamic net spillovers between the different oil prices and stock market indices during the two periods of highest volatility in the oil markets of the millennium, respectively, the COVID-19 pandemic and the Great Recession (see Figure 3). The dynamic net spillovers have been calculated as in equations from (12) to (15) starting from the FEVD of the VAR model, by using the recursive sampling method.

First, volatility spillovers have been considerably higher in the case of the COVID-19 pandemic than in the case of the Great Recession in the US. That is, in the case of the US, when oil price volatility has been higher (in the COVID-19 pandemic, it has been nearly the double than in the Great Recession (see Figure 3)), the volatility spillovers from oil to the stock markets have been also higher, even three times higher in some periods. This suggests that the transmission of volatility increases when the volatility also increases. This evidence points to the fact that, in the US economy, higher volatility in oil markets produces higher volatility spillovers on the stock markets, potentially in a linear way.

Second, there is a structural difference between the COVID-19 pandemic and the Great Recession which is reflected in the behavior of the SP500 dynamic spillover. As the Great Recession was a financial crisis, the SP500 was a net transmitter of volatility to oil markets. On the other hand, being the COVID-19 pandemic a real-economy crisis with supply side restrictions and effects on oil production and distribution, the SP500 is acting as a receiver of volatility, and oil markets as a transmitter. This finding corroborates the theoretical reversion mechanism of volatility spillovers postulated in Section 2.1.

Third, in the European economy it is important to notice first the behavior of the dynamic spillover from the Brent to the DAX. In concordance with the static results, the DAX is the only net transmitter of volatility to oil markets in any period of high volatility. This strong influence of the DAX on the Brent seem to be even more robust taking into account its presence during the Great Recession. Furthermore, the dynamic net spillovers in European countries moves approximately in the same range during both crises. This is also true for the IBEX, and may reflect a generalized more stable behavior of the dynamic spillovers from oil to the European stock markets, which can be investigated in future research.

Fig. 6
figure 6figure 6

Dynamic volatility spillovers from oil to stocks markets during high volatility periods. Source: Own elaboration

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4 Conclusion and discussion

Price volatility in international oil markets is a recurrent phenomenon due to supply and demand shocks and frictions. This volatility increases considerably during periods of international crises, whereby this paper presents the following three questions. Do volatility spillovers from international oil prices to stock markets act different when the shock is financial or economic? Does the behavior of these spillovers differ during the most volatile periods? Are the relationships bidirectional, and if so, of which magnitude? In order to answer these questions, this study employs monthly data from the WTI and Brent oil price indices and the SP500, NASDAQ, DAX and IBEX stock market indices, for the period from 2000 to 2021, and combines the GARCH methodology for volatility measurement with the impulse response methodology for shocks, and the spillover index approach of Diebold and Yilmaz (2009, 2012) for the analysis of volatility spillovers.

During the COVID-19 pandemic, price volatility in international oil markets has been significantly higher than during the 2007-2009 financial crisis, reaching unprecedented levels in the millennium. The volatility spikes have been higher in the West Texas Intermediate index, specially during the months of May and June of 2020. This may be related to the economic shut-ins in the US economy and a reactivity to record high OPEC+ cuts. The main theoretical channel driving the volatility spillovers is potentially the uncertainty channel, due to the rising variability of oil prices. Additionally, oil markets are characterized for having a relatively rigid supply, so the drastic changes in demand suffered during the COVID-19 pandemic mainly caused changes in prices, which affected stock prices through the channels mentioned in Section 2.1. Additionally, this paper documents that this effect may be reversed if the origin of the crisis is due to financial factors.

The analysis of the impulse response functions shows very small increases of the log-returns in the stock markets in the very short term as a reaction to oil price shocks. Some authors such as Nandha and Faff (2008); Falzon and Castillo (2013) document a negative globally relationship between oil price movements and equity returns both for the US and the UK, but our study aligns with the work by El-Sharif et al. (2005) who find a positive relationship for the UK, and Mollick and Assefa (2013) who find a post-Great Recession positive relationship between oil prices and stocks returns. Cueppers and Smeets (2015) find that normal movements in the oil price do not affect the German DAX stock index, but oil price shocks do negatively. Our IRFs, although positive and small, also reflect an impact of this kind. However, what our study further documents is the impact of a shock from the DAX to the Brent, which is statistically significant, and more than ten times higher. Thus, it can be concluded that the relationship between both variables is bidirectional and higher in orders of magnitude in the case of the stocks to the oil market. This relationship may be decomposed into sectorial or company-level data for further insights.

With respect to the main objective of this paper, the evidence shows that volatility spillovers do exist, both in the static and the dynamic case. In fact, in the case of the US economy, dynamic spillovers are bigger the higher the volatility of oil prices, potentially in a linear way, as whilst volatility of the WTI price has doubled from the Great Recession to the COVID-19 pandemic, the net spillovers to financial markets have tripled. Additionally, the US seems to be very reactive to the source of the shock; i.e., when the shock is financial, financial markets act as a net volatility transmitter, and when the shock is economic, the reverse holds. Nevertheless, this is not the case for European indices, where the net volatility spillovers of the Brent to financial markets have been very similar during the COVID-19 pandemic and the Great Recession. Even the volatility of oil prices during the pandemic doubled the one corresponding to the Great Recession. In particular, one interesting result is how the DAX stock index constitutes an exception to the rest of indices, as it is a net transmitter of volatility to the Brent in the case of both a static and a dynamic analysis. Furthermore, this stock index is a net transmitter of volatility to crude oil markets even in a real-economy crisis such as the COVID-19 pandemic, revealing its strong spillover power. On the other hand, the IBEX-35 Spanish stock index is always a net receiver of volatility from the Brent oil index.

The results of this paper have important practical implications. Financial investors with open positions in the US market can expect higher volatility in their financial assets if there is a financial crisis, and similarly higher volatility in the US reference oil price in case of a real-economy crisis. They can, thus, select their positions at the beginning of a crisis depending on its origin. On the other hand, financial investors with positions open in European financial markets, in particular the German DAX, do not have to worry about the origin of a crisis to adopt positions.

As future research guidelines, a global study of which indices constitute net transmitters or receivers of shocks and volatility spillovers incorporating the main variables at a global level can be a fruitful task because, as shown in this paper, the results may vary substantially. In particular, Bayesian VAR models which allow for a big number of variables and lags maintaining robust the structural analysis Bańbura et al. (2010) can be an outstanding methodology to be employed in this case. Finally, a disaggregation of the results of this paper by sectors of the economy or companies is also presented as a venue of future research.