Do U.S. economic conditions at the state level predict the realized volatility of oil-price returns? A quantile machine-learning approach

Table 4 Results of an out-of-sample analysis

Benchmark/rival model	Mean RP	p value	Mean RP	p value	Mean RP	p value
Horizon	h = 1	h = 1	h = 2	h = 2	h = 4	h = 4
HAR-RV vs. HAR-RV-US/q = 0.95	− 0.0019	0.5920	− 0.0030	0.5160	0.0040	0.3000
HAR-RV vs. HAR-RV-US/q = 0.75	− 0.0005	0.5040	− 0.0011	0.7760	− 0.0009	0.6060
HAR-RV vs. HAR-RV-US/q = 0.5	− 0.0007	0.6720	− 0.0005	0.6860	− 0.0011	0.7300
HAR-RV vs. HAR-RV-US/q = 0.25	− 0.0010	0.7480	− 0.0008	0.5980	− 0.0009	0.5700
HAR-RV vs. HAR-RV-US/q = 0.05	0.0024	0.3300	− 0.0020	0.4980	− 0.0026	0.6920
HAR-RV vs. HAR-RV-states/q = 0.95	− 0.0093	0.7380	− 0.0143	0.6560	0.0646	0.0780
HAR-RV vs. HAR-RV-states/q = 0.75	− 0.0033	0.5840	0.0252	0.0600	0.0466	0.0200
HAR-RV vs. HAR-RV-states/q = 0.5	0.0020	0.2940	0.0074	0.2220	0.0507	0.0020
HAR-RV vs. HAR-RV-states/q = 0.25	− 0.0043	0.6660	0.0132	0.1540	0.0465	0.0060
HAR-RV vs. HAR-RV-states/q = 0.05	− 0.0047	0.6520	0.0049	0.3280	0.0443	0.0660
HAR-RV-US vs. HAR-RV-states/q = 0.95	− 0.0073	0.5500	− 0.0113	0.5780	0.0608	0.0840
HAR-RV-US vs. HAR-RV-states/q = 0.75	− 0.0028	0.5120	0.0263	0.0560	0.0474	0.0160
HAR-RV-US vs. HAR-RV-states/q = 0.5	0.0027	0.2600	0.0080	0.2060	0.0517	0.0020
HAR-RV-US vs. HAR-RV-states/q = 0.25	− 0.0032	0.5100	0.0140	0.1340	0.0474	0.0060
HAR-RV-US vs. HAR-RV-states/q = 0.05	− 0.0071	0.7280	0.0068	0.3040	0.0467	0.0620

The models are estimated 500 times on bootstrapped data sampled without replacement. For every estimation, the relative-performance statistic, RP, statistic is computed as \(RP = 1 - \sum _{t=1}^T \rho _\alpha \left( e_{t, R} \right) / \sum _{t=1}^T \rho _\alpha \left( e_{t, B} \right)\), where \(e_t\) denotes the out-of-sample prediction errors and the summation, \(t,,\ldots , T\), runs over the out-of-sample data. The out-of-sample data are those data not included in the estimation. The fraction of out-of-sample data for every bootstrap sample is 30%. The benchmark (B) model is the first model given in the first column of the table, and the rival (R) model is the second model given in that column. The HAR-RV-states model includes the state-level components in the vector of potential predictors. The benchmark model is estimated by the quantile-regression technique, while the HAR-RV-states model is estimated by the quantile Lasso technique. The intercept and the classic HAR-RV terms are not penalized. The penalty parameter is determined by tenfold cross-validation and is re-optimized at the beginning of a month. A positive mean of the out-of-sample RP statistic shows that the rival model outperforms on average the benchmark model. The parameter h denotes the forecast horizon. The parameter q denotes the quantile being analyzed. The dependent variable is the natural log of the realized volatility of oil-price returns