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Financial Innovation

Table 2 Results of permutation tests

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

Benchmark/rival model

h = 1

h = 2

h = 4

HAR-RV-US vs. HAR-RV-states/q = 0.95

0.7680

0.0000

0.0000

HAR-RV-US vs. HAR-RV-states/q = 0.75

0.0980

0.0000

0.0000

HAR-RV-US vs. HAR-RV-states/q = 0.5

0.0000

0.0000

0.0000

HAR-RV-US vs. HAR-RV-states/q = 0.25

0.1220

0.0000

0.0000

HAR-RV-US vs. HAR-RV-states/q = 0.05

0.5540

0.0000

0.0000

  1. The p values reported in this table are based on 500 simulation runs. In every simulation run, the data on the state-level economic conditions are sampled without replacement. The dependent variable and the predictors of the classic HAR-RV model are not resampled. Using the simulated data, the HAR-RV-states model is then estimated by means of the quantile Lasso estimator (the intercept and the classic HAR-RV terms are not penalized) and the model prediction errors are stored. The penalty parameter is determined by tenfold cross-validation. 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 i denotes the simulation index, \(e_t\) denotes the model prediction errors, B denotes the benchmark model, and R denotes the rival model. The benchmark (rival) model is the first (second) model given in the first column of the table. The benchmark model is estimated by the quantile-regression technique. The prediction errors of the benchmark model are based on the estimates reported in Table 1. The p values are then computed as \(( \# RP_i \ge RP_{ref} ) / 500\), where the reference values, \(RP_{ref}\), of the relative performance statistic are the values reported in Table 1. 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
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