**3. Methodologies**

For the forecasting analysis, we used a linear regression model. The model featured an intercept and an autoregressive term as its core components. The autocorrelation functions for the realized volatility (for the estimator that we used in our empirical research, see Equations (5)) plotted in Figure 4 showed that this simple autoregressive model should suffice to capture the main elements of the persistence of *RV* (and its "good" and "bad" counterparts).

Based on the suggestion of an anonymous referee, we also compared the in-sample performance of our benchmark autoregressive *RV* model with that of the best-fitting GARCH model, namely the Exponential GARCH (EGARCH), in predicting *RV* and found that the former produced a lower root mean square error (RMSEs) than the latter, which is not surprising given the insignificant coefficient in the volatility equation corresponding to the lagged GARCH term, highlighting the inability of the model to adequately capture volatility at a quarterly frequency. Complete details of these results are available upon request from the authors.

**Figure 4.** Autocorrelation functions of realized volatility. Dashed horizontal lines: bounds of the 95% confidence interval. *RVB*: Downside ("bad") *RV*. *RVG*: Upside ("good") *RV*.

In addition, we considered the various uncertainties, *U*, and international spillovers,*S*, as additional predictors. In our empirical application, this gave four forecasting models:

$$RV\_{t+h} = \beta\_0 + \beta RV\_t + \varepsilon\_{t+h\prime} \tag{1}$$

$$RV\_{t+h} = \beta\_0 + \beta RV\_t + \sum\_{j=1}^{n\_u} \beta\_{u,j} lI\_{t,j} + \varepsilon\_{t+h\prime} \tag{2}$$

$$RV\_{t+h} = \beta\_0 + \beta RV\_t + \sum\_{j=1}^{n\_s} \beta\_{s,j} S\_{t,j} + \epsilon\_{t+h\prime} \tag{3}$$

$$RV\_{t+h} = \beta\_0 + \beta RV\_t + \sum\_{j=1}^{n\_u} \beta\_{u,j} lI\_{t,j} + \sum\_{j=1}^{n\_s} \beta\_{s,j} S\_{t,j} + \epsilon\_{t+h\prime} \tag{4}$$

where the index *h* denotes the forecast horizon, and (for *h* > 1) *RVt*+*<sup>h</sup>* denotes the average realized variance over the forecast horizon being studied, with *h* = 1, 2, and 4 in our context. When computing out-of-sample forecasts, we constructed the data matrix in such a way that the number of forecasts was the same for all forecast horizons. In addition, *nu* and *ns* denote the number of uncertainties and international spillovers being studied, and  *t*+*h* denotes an error term.

Figures 5 and 6 plot the autocorrelation functions for the uncertainties and international spillovers. The figures show that all uncertainty measures exhibited a certain degree of persistence, while we observed persistence in the case of the international spillovers mainly for Canada, China, the United Kingdom, and the United States only.

As the dependent variable, we used the classical estimator of *RV* (Andersen and Bollerslev, 1998). In our case, we used the sum of the squared daily returns per quarter. We have

$$RV\_t = \sum\_{i=1}^{M} r\_{t,i\prime}^2 \tag{5}$$

where *rt*,*<sup>i</sup>* is the daily return, which is defined as the log-difference in prices as observed on two consecutive days, and *i* = 1, ... , *M* is the number of quarterly observations. As a robustness check, we shall also study whether uncertainties and international spillovers help to forecast √*RV*, which researchers also often call "volatility" in empirical finance applications.

We also studied the predictive value of uncertainties and international spillovers for upward ("good", *RVG*) and downward ("bad", *RVB*) realized variance. Thus, we also forecast *RVG* and *RVB* with our forecasting equations. In line with Barndorff-Nielsen et al. (2010), we computed the bad and good realized volatility as described by the following two equations (**1** = indicator function):

$$RVG\_t = \sum\_{i=1}^{M} r\_{t,i}^2 \mathbf{1}\_{[(r\_{t,i}) > 0]^\prime} \tag{6}$$

$$RVB\_t = \sum\_{i=1}^{M} r\_{t,i}^2 \mathbf{1}\_{[(r\_{t,i}) \prec 0]}.\tag{7}$$

For the estimation of our forecasting model, we used the least absolute shrinkage and selection operator (Lasso) estimator. Our choice of the Lasso as our preferred estimation technique reflects the fact that the dimension of the forecasting model became quite large (relative to the size of our sample period) when we added the various uncertainties and international spillovers to the core components of the model. The Lasso technique chose the coefficients, *β*, *β<sup>u</sup>*,1, *β<sup>u</sup>*,2, ... , *β<sup>s</sup>*,1, *β<sup>u</sup>*,2, ..., so as to minimize the following expression (for a detailed discussion of the Lasso, see, e.g., Hastie et al. [30]):

$$\sum\_{t=1}^{N} \left( RV\_{t+h} - \beta \mathfrak{e}\_{0} - \beta RV\_{t} - \sum\_{j=1}^{n\_{u}} \mathfrak{f}\_{u,j} lI\_{t,j} - \sum\_{j=1}^{n\_{d}} \mathfrak{f}\_{s,j} \mathfrak{S}\_{t,j} \right)^{2} + \lambda \left( |\beta| + \sum\_{j=1}^{n\_{u}} |\mathfrak{f}\_{u,j}| + \sum\_{j=1}^{n\_{d}} |\mathfrak{f}\_{s,j}| \right), \tag{8}$$

where *N* denotes the number of observations used for estimation of the model. Hence, the Lasso shrinking used the *L*1 norm of the coefficient vectors to shrink the dimension of the estimated model. Depending on the magnitude of the shrinkage parameter, *λ*, the Lasso estimator shrinked and even set to zero some of the coefficients and, thus, can be viewed as a predictor-selection technique.

**Figure 5.** Autocorrelation functions of uncertainties. Dashed horizontal lines: bounds of the 95% confidence interval.

We selected the value of the shrinkage parameter, *λ*, to minimize the minimum mean cross-validated error when we used 10-fold cross validation. For estimation of the Lasso

models, we used the R package "glmnet" [31]. For the R environment for statistical computing, see the R Core Team [32].

**Figure 6.** Autocorrelation functions of international spillovers. Dashed horizontal lines: bounds of the 95% confidence interval.

In order to compute out-of-sample forecasts, we primarily used a recursively expanding estimation window (with a training period of 10 years to initialize the estimations) and, as a robustness check, a fixed-length rolling estimation window. We, then, evaluated the forecasts by means of the Clark and West [33] test. The null hypothesis was that the models being compared had an equal out-of-sample mean-squared prediction error (MSPE). ˆ

The Clark–West test requires regressing the quantity *ft*+*<sup>h</sup>* = (*RVt*+*<sup>h</sup>* − *RV*<sup>ˆ</sup> *<sup>A</sup>*,*t*+*<sup>h</sup>*)<sup>2</sup> − [(*RVt*+*<sup>h</sup>* − *RV*<sup>ˆ</sup> *<sup>B</sup>*,*t*+*<sup>h</sup>*)<sup>2</sup> − (*RV*<sup>ˆ</sup> *A*,*t*+*h* − *RV*<sup>ˆ</sup> *<sup>B</sup>*,*t*+*<sup>h</sup>*)<sup>2</sup>] on a constant, where a hat denotes the forecast of *RV*, and the subindices *A* and *B* denote the two models under scrutiny (*B* denotes the larger model). The Clark–West test is based on an adjusted difference of the MSPEs implied by Models A and B. The test rejects the null hypothesis if the t-statistic of the constant in this regression model is significantly positive (one-sided test; we used Newey–West robust standard errors to study the significance of the t-statistic).

#### **4. Empirical Results**

In Table 1, we report the baseline forecasting results for WTI and Brent. The table gives the *p*-values of the Clark–West test. The key message to take home from the results given in the table is that the Lasso model that included uncertainty and/or international spillovers outperformed in our out-of-sample forecasting exercise for the core model at the intermediate and long forecasting horizon. We obtained this key result for both WTI and Brent crude oil-price realized variances.

We used robust standard errors to compute the *p*-values because (as one would have expected) the forecast errors were autocorrelated at the longer forecast horizons due to the overlapping forecast horizons. As suggested by an anonymous reviewer, we also tested whether the forecast errors had a unit root. A standard unit root test (Kwiatowski et al. [34] showed that the forecast errors could be regarded as stationary time series. Detailed results are not reported to save journal space but are available from the authors upon request.

There was also evidence of predictive value when we further extended the forecast horizon to six and eight quarters. Further, when we studied the natural logarithm of *RV*, we observed improvements in the forecasting performance at the longer and, depending on the model specification, at the intermediate forecasting horizon. Detailed results are available from the authors upon request.


**Table 1.** Baseline forecasting results.

CW test: *p*-value (based on Newey–West robust standard errors) of the Clark–West test. Training period used to initialize the recursive-estimation scheme: 40 quarters.

In order to shed further light on the relative forecasting performance of the model, we document in Table 2, for both WTI and Brent, the forecasting gains expressed as the percentage increase (or decrease) in the ratio of the root-mean-squared-forecasting error (RMSFE) of the benchmark (that is, autoregressive) model and the alternative models. A positive forecasting gain, thus, shows that the RMSFE of the benchmark model exceeded the RFMSFE of the alternative model, implying that the alternative model yielded better forecasts under a standard quadratic loss function.

We observed positive forecasting gains mainly at the intermediate and especially at the long forecasting horizon. The forecasting gains were the largest when we combined uncertainty and international spillovers (*h* = 2, 4). The autoregressive benchmark model, in turn, tended to fare better than the alternative models at the short forecasting horizon. Taken together, the results corroborated the results of the Clark–West test. The correlation between the forecasting gains reported in Table 2 and the *p*-values of the Clark–West test given in Table 1 was significantly negative (coefficient of correlation = −0.48, t-statistic of −2.19, *p*-value = 0.04), showing that higher forecasting gains tended to be associated with lower *p*-values and, thus, significant test results.


**Table 2.** Forecasting gains.

Note: The forecasting gains are defined as 100 × (RMSFE0/RMSFE1 − <sup>1</sup>), where the index 0 denotes the benchmark (autoregressive) model, and the index 1 denotes the alternative models (including uncertainty and/or spillovers). RMSFE: root-mean-squared-forecasting error. Training period used to initialize the recursiveestimation scheme: 40 quarters.

A major exception arose in the case of Brent and the spillovers model and *h* = 2, where the forecasting gain was negative (and large in absolute terms), while the Clark–West test (which is, as described in the methodology in Section 3, based on the adjusted difference of the out-of-sample MSPEs generated by the two models being compared) yielded a significant result.

Next, we summarize in Table 3 the results (Clark–West test) for the good and bad realized variances, again for both WTI and Brent. The results corroborated that uncertainty and/or international spillovers added to the forecasting performance of the model estimated on data for bad realized variance at the intermediate and long forecast horizon. For good realized variance, we observed insignificant test results in the case of uncertainty for WTI and significant test results for international spillovers at the intermediate and long forecast horizon. In addition, the test results for both uncertainty and international spillovers were insignificant at the short and the intermediate, but not at the long, forecast horizons for Brent.


**Table 3.** Forecasting results for upside and downside volatility.

Note: CW test: *p*-value (based on Newey–West robust standard errors) of the Clark–West test. Training period used to initialize the recursive-estimation scheme: 40 quarters.

Table 4 gives the forecasting results for √*RV*. We use the terms "realized volatility" and "realized variance" interchangeably in this paper, while researchers in the empiricalfinance literature often use the term "volatility" to refer to √*RV*. The results for WTI showed that uncertainty had predictive value at the long forecast horizon, but international spillovers did not add to the forecasting performance of the model. The test results for Brent, in turn, were significant for uncertainty at the intermediate and the long forecast horizon, and for international spillovers at the long forecast horizon.

Table 5 summarizes the results for a rolling-estimation window. The test results were significant for all three forecasting horizons (at the 10% level) for uncertainty in the case of WTI. In addition, the test results for international spillovers were significant for WTI when we studied the long forecast horizon. As for Brent, the test results for uncertainty and international spillovers were significant for the long forecast horizon and, in addition, for the intermediate forecast horizon in the case of international spillovers.


**Table 4.** Forecasting results for the realized volatility (√*RV*).

Note: CW test: *p*-value (based on Newey–West robust standard errors) of the Clark–West test. Training period used to initialize the recursive-estimation scheme: 40 quarters.



Note: CW test: *p*-value (based on Newey–West robust standard errors) of the Clark–West test. Length of the rolling-estimation window: 40 quarters.

In order to illustrate how the Lasso estimator works, we plot in Figure 7 the importance of the uncertainty and international spillovers over time. The results are for WTI and a recursive-estimation window. We used a simple metric of importance. Specifically, we define importance as the number of nonzero coefficients estimated for uncertainty (international spillovers) divided by *nu* (*ns*). Hence, zero means that the Lasso sets all coefficients, for example, of uncertainty to zero in a given forecasting period, and one means that all coefficients of uncertainty are included in the model.

The results show that uncertainty tended to be of more importance on average than international spillovers at the short and the intermediate forecast horizon, while the importance of both categories of predictors was more or less balanced at the long forecast horizon. The results also illustrate that the importance of both uncertainty and international spillovers was not stable over the out-of-sample period, lending support to our decision to use a recursive- and a rolling-estimation window to analyze the forecasting properties of uncertainty and international spillovers for the realized volatility over time.

This result is not surprising but is indicative of the fact that uncertainty and its spillovers themselves are not constant and vary across time (as shown in Figures 2 and 3) depending on events that affect the macroeconomic uncertainty in these major economies and the associated spillovers, thereafter, to the rest of the world.

**Figure 7.** Importance of uncertainty and international spillovers. The results are for WTI and a recursive-estimation window. Importance is defined as the number of nonzero coefficients estimated for uncertainty divided *nu* and similarly for spillovers. Hence, zero means that all coefficients of, for example, uncertainty are zero in a given forecasting period, and one means that all coefficients of uncertainty/spillovers are nonzero. The time axis refers to the period in which a forecast is being made.

Table 6 summarizes, as a further robustness check, the results for a ridge-regression approach. A ridge regression also solves the minimization problem given in Equation (8) for the Lasso with the difference being that the penalty term multiplied by the *λ* coefficient used the L2 norm to shrink the estimated coefficients of the forecasting model. The results show that, at the intermediate forecast horizon, only the uncertainty improved the forecasting performance, whereas, for the long forecasting horizon, both uncertainty and international spillovers (Brent) helped to improve the forecast accuracy.

Our forecasting analysis confirmed the initial premise of our paper that the uncertainties of other important economies within the G7 in addition to the US and China also tend to drive oil market volatility due to the importance of their position as exporters and importers in the oil market. Especially in the longer-run, the spillovers of uncertainty from these major economies to other countries in the world are important in capturing the accurate size of the global demand in the oil market in the wake of increased uncertainty.

The relatively stronger long-run influence of spillovers on oil market volatility is understandable, since it takes time for uncertainty originating in the G7 and China to spread to the rest of the world, via various channels namely, trade, financial markets, and exchange rates [35,36], to the extent that it leads to uncertainty convergence over time [37]. In sum, our findings are indicative of the fact that accounting for the total amount

of uncertainty spillovers of the major economies, over and above their own uncertainties, allowed us to better model worldwide uncertainty and its influence on global oil demand, which, in turn, translated into more accurate forecasting of the realized variance capturing oil-market volatility.


**Table 6.** Forecasting results for a ridge regression.

Note: CW test: *p*-value (based on Newey–West robust standard errors) of the Clark–West test. Length of the recursive-estimation window: 40 quarters.

## **5. Conclusions**

Based on a dataset for the G7 countries and China, our results showed that uncertainty and international spillovers had predictive value in an the out-of-sample forecasting exercise for the realized variance of crude oil (West Texas Intermediate and Brent), where our sample period ranged from 1996Q1 to 2020Q4. Given that, on the one hand, our sample period was relatively short and, on the other hand, our data comprised measures of uncertainty for eight countries and eight measures of international spillovers, we used the Lasso estimator to estimate our forecasting models. Taken together, our empirical results demonstrated that, depending on the model specification, uncertainty and international spillovers had predictive value for the realized variance (and its "good" and "bad" counterparts) at an intermediate (two quarters) and a long (one year) forecasting horizon.

Compared to the current literature, which has relied only on the role of US uncertainty in predicting oil market volatility, our paper extends this line of research by highlighting the importance of not only the uncertainty of the G7 countries and China but also their respective spillovers of uncertainty to the rest of the world. This being the first study of its kind, it is impossible to provide comparative quantitative assessment of our results with the existing papers in this related area; however, its academic value in terms of depicting the pertinent role of uncertainty and its spillovers beyond the US in forecasting oil-market volatility cannot be overlooked.

In addition, our results can be used by policy authorities to obtain information on the future path of the volatility of oil prices due to uncertainty of G7 countries and China, as well as the associated global spillovers of uncertainty from these economies. This knowledge, in turn, could be useful to predict economic activity, given that oil-price volatility is known to lead business cycles. Our results, therefore, may help policymakers to reach appropriate policy decisions in the wake of the movements in the uncertainties of major global economies and the spillovers. Moreover, with volatility being a key input in portfolio decisions, the forecastability of oil-price volatility due to the uncertainties of G7 and China, as well as the associated spillovers, should be of vital importance to traders in the oil market.

Having indicated the important implications of our results, it is also necessary to acknowledge one limitation of our study in terms of the low-frequency of our data. Ideally, we would have preferred to have conducted the forecasting exercise of realized variance of oil at a higher frequency, as it is of grea<sup>t</sup> importance for policymakers and investors to make timely policy and portfolio decisions; however, the uncertainty spillover indexes

were available only at a quarterly frequency and, hence, constrained us in our ability to provide higher-frequency (say, for example, daily or monthly) results.

As a part of future research, it would be interesting to extend our analysis to other commodity markets, in particular gold, which is a well-established safe haven in the wake of heightened uncertainty [38,39].

**Author Contributions:** R.G. and C.P. contributed equally to all parts of the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data used in this research can be obtained from the authors upon request.

**Acknowledgments:** The authors thank the anonymous reviewers for their helpful comments. The usual disclaimer applies.

**Conflicts of Interest:** The authors declare no conflict of interest.
