Figures (1)  Tables (7)

    • Figure 1. 

      Realized volatility of oil prices and the S&P 500 index from 1990 to 2022.

    • Grid search for the BAR's threshold parameters
      Initialization: Set threshold parameters $ {\eta }_{1} $ and $ {\eta }_{2} $ such that $ {\eta }_{1}< {\eta }_{2} $, with a defined empirical search range $ [{\mathrm{a}}_{1},{\mathrm{a}}_{2}] $.
      Step 1: Define the buffer region width as $ b={\eta }_{2}-{\eta }_{1} $.
      Step 2: Constrain the buffer size by setting $ b\in (0,\dfrac{{a}_{2}-{a}_{1}}{\mathrm{c}}] $, where the empirically optimized tuning constant $ c\in \{5,\;6,\;7,\;8,\;9,\;10\} $.
      Step 3: Iterate over all possible combinations of $ {\eta }_{1} $, b, and c within their defined constraints to estimate the parameters via conditional least squares (CLS).
      Step 4: Compute the optimal parameters as
      $ \{ {\eta }_{1},\;b,\;c\}={\mathrm{argmax}}\{R^{2}\;{\text{of the BAR(p) model }}-{R}^{2}\;{\text{of benchmark model}} \} $
      Output: The statistically justified optimal threshold values $ {\eta }_{1} $ and $ {\eta }_{2} $.

      Table 1. 

      Improved grid search algorithm for the BAR model's threshold parameters.

    • StockOil
      PriceReturnVolatilityPriceReturnVolatility
      Mean1,502.6450.0000.00349.8250.0010.019
      Median1,252.8000.0010.00145.4900.0010.008
      Minimum295.500−0.1280.00010.010−1.3240.001
      Maximum4,796.5600.1100.075145.2900.7222.691
      SD997.5880.0110.00629.4070.0310.136
      Skewness1.301−0.3948.3600.603−8.43519.065
      Kurtosis4.25213.61591.2792.272472.150373.271
      ADF2.372−20.459***−5.649***2.561−18.736***−6.960***
      Notes: ADF, augmented Dickey–Fuller test statistic. The symbols *, **, and *** indicate rejection of the null hypothesis with statistical significance at 10%, 5%, and 1% levels, respectively.

      Table 2. 

      Descriptive statistics and tests for a unit root.

    • Coefficient t-statistic Coefficient t-statistic
      AR(6) Reference ARX(6)
      Parameter estimation results
      $ \omega $ −1.214*** −4.996 −1.014*** −3.431
      $ {\beta }_{1} $ 0.544*** 10.668 0.349*** 5.206
      $ {\beta }_{2} $ 0.125** 2.166 0.310*** 4.521
      $ {\beta }_{3} $ 0.002 0.036 0.232** 3.254
      $ {\beta }_{4} $ 0.016 0.283 −0.074 −1.047
      $ {\beta }_{5} $ 0.049 0.849 −0.015 −0.230
      $ {\beta }_{6} $ 0.009 0.183 −0.117 −1.861
      $ \beta $ − − 0.078 1.663
      Percentage of increase in R2
      $ {\Delta \mathrm{R}}^{2} $ − 7.71
      TAR(6) $ {V}_{t,stock}\leq -6.999 $ $ {V}_{t,stock}> -6.999 $
      Parameter estimation results
      $ {\beta }_{0} $ −0.763 −1.189 −1.566*** −4.681
      $ {\beta }_{1} $ 0.248* 1.905 0.399*** 5.256
      $ {\beta }_{2} $ 0.357** 2.286 0.304*** 3.867
      $ {\beta }_{3} $ 0.187 1.145 0.215*** 2.652
      $ {\beta }_{4} $ 0.104* 2.675 −0.111** −1.873
      $ {\beta }_{5} $ 0.081 0.580 −0.047 −0.588
      $ {\beta }_{6} $ −0.127 −0.911 −0.099 −1.342
      Percentage increase of R2
      $ {\Delta \mathrm{R}}^{2} $ − 8.72
      Notes: The table reports slope parameter estimates together with the heteroskedasticity- and autocorrelation-consistent t-statistics based on the Newey–West method. The percentage increase in $ {\text{R}}^{2} $ relative to the baseline AR(6) model in Eq. (2) is also reported. The symbols *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively.

      Table 3. 

      Within-sample estimation results of the AR, ARX, and TAR models for monthly oil volatility.

    • Coefficient t−statistic Coefficient t−statistic
      AR(6) Reference ARX(6)
      Parameter estimation results
      $ \omega $ −1.214*** −4.996 −1.014*** −3.431
      $ {\beta }_{1} $ 0.544*** 10.668 0.349*** 5.206
      $ {\beta }_{2} $ 0.125** 2.166 0.310*** 4.521
      $ {\beta }_{3} $ 0.002 0.036 0.232** 3.254
      $ {\beta }_{4} $ 0.016 0.283 −0.074 −1.047
      $ {\beta }_{5} $ 0.049 0.849 −0.015 −0.230
      $ {\beta }_{6} $ 0.009 0.183 −0.117 −1.861
      $ \beta $ − − 0.078 1.663
      Percentage of increase in R2
      $ {\Delta \mathrm{R}}^{2} $ − 7.71
      BAR(6) $ {V}_{t,stock} $ ≤ −7.104 $ {V}_{t,stock} $ > −6.999
      Parameter estimation results
      $ {\gamma }_{0} $ −1.149* −1.786 −1.549*** −4.743
      $ {\gamma }_{1} $ 0.154 0.971 0.394*** 5.505
      $ {\gamma }_{2} $ 0.317** 2.014 0.296*** 3.838
      $ {\gamma }_{3} $ 0.143* 1.932 0.248*** 3.048
      $ {\gamma }_{4} $ 0.115 0.778 −0.125 −1.542
      $ {\gamma }_{5} $ 0.126 0.719 −0.022 −0.294
      $ {\gamma }_{6} $ −0.068 −0.490 −0.129* −1.811
      Percentage increase of R2
      $ {\Delta \mathrm{R}}^{2} $ − 9.21
      Notes: The table reports the slope parameter estimates and Newey–West heteroskedasticity- and autocorrelation-consistent t−statistics. In addition, it shows the percentage of the increase in R2 relative to the baseline AR(6) model in Eq. (2). For the BAR(6) model, two thresholds are reported (−7.104 and −6.999), which define a buffer zone between the regimes and capture the hysteresis effect. The symbols *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively.

      Table 4. 

      Within-sample estimation results of the AR, ARX, and BAR models for monthly oil volatility.

    • Model: ARX(6) Model: TAR(6) Model: BAR(6)
      Recursive Rolling Recursive Rolling Recursive Rolling
      2011−2022 $ \Delta R_{oss}^{2} $ −4.635** −1.168** 6.319*** 5.001*** 8.176*** 6.855***
      p−Value 1.63 × 10−2 1.4 × 10−2 6.76 × 10−4 1.66 × 10−3 3.71 × 10−4 2.57 × 10−4
      2013−2022 $ \Delta R_{oss}^{2} $ −2.301*** 1.586*** 9.674*** 7.225** 11.405*** 8.633***
      p−Value 2.06 × 10−2 1.27 × 10−3 2.06 × 10−3 1.05 × 10−2 1.94 × 10−3 2.08 × 10−3
      2015−2022 $ \Delta R_{oss}^{2} $ −6.542** −2.519** 11.392** 9.830** 13.237** 11.328***
      p−Value 1.52 × 10−2 1.04 × 10−2 2.66 × 10−2 2.83 × 10−2 3.07 × 10−2 2.67 × 10−4
      2017−2022 $ \Delta R_{oss}^{2} $ −4.142** −1.089*** 12.792** 11.609** 14.531* 13.212***
      p−Value 2.41 × 10−2 7.64 × 10−3 4.2 × 10−2 3.74 × 10−2 1.0 × 10−4 1.2 × 10−2
      Notes: The $ \Delta R_{oss}^{2} $ measure represents the percentage of reduction in the MSPE of the focal model relative to the MSPE of the reference model in Eq. (2). The p−values correspond to the Clark–West[50] test (CW test), which evaluates whether the MSPE of the equity market-implied volatility differs significantly from that of the reference crude oil volatility model.

      Table 5. 

      Out-of-sample forecast results of the ARX, TAR, and BAR models for monthly oil volatility.

    • Model: ARX(5) Model: TAR(5) Model: BAR(5)
      Recursive Rolling Recursive Rolling Recursive Rolling
      2011−2022
      $ \Delta R_{oss}^{2} $ −4.225*** −0.682* 6.263*** 5.629*** 6.994*** 6.532***
      p−value 6.70 × 10−3 6.62 × 10−2 1.59 × 10−3 8.10 × 10−4 6.09 × 10−4 5.11 × 10−4
      2013−2022
      $ \Delta R_{oss}^{2} $ −3.163*** 0.615*** 8.232*** 6.492*** 9.071** 7.695***
      p−value 8.90 × 10−3 4.34 × 10−4 4.81 × 10−3 8.70 × 10−3 1.29 × 10−2 9.56 × 10−4
      2015−2022
      $ \Delta R_{oss}^{2} $ −7.587*** −3.776*** 9.541** 8.066** 10.627** 9.162*
      p−value 8.44 × 10−3 1.05 × 10−2 2.73 × 10−3 4.48 × 10−2 4.08 × 10−2 5.03 × 10−2
      2017−2022
      $ \Delta R_{oss}^{2} $ −4.359** −1.547** 10.944** 9.807** 11.697* 11.191*
      p−value 2.46 × 10−2 1.53 × 10−2 4.27 × 10−2 3.79 × 10−2 8.47 × 10−2 5.03 × 10−2
      Notes: The $ \Delta R_{oss}^{2} $measure is calculated as the percentage of decrease in the MSPE of the focal model compared with the MSPE of a reference model in Eq. (2). Additionally, the significance levels (p−values) from Clark and West's[50] (CW) tests, which assess the equality of the MSPE between the equity market inferred volatility and the reference crude oil volatility model, are presented.

      Table 6. 

      Out-of-sample forecast performance of the ARX(5), TAR(5), and BAR(5) models.

    • Model: ARX(7) Model: TAR(7) Model: BAR(7)
      Recursive Rolling Recursive Rolling Recursive Rolling
      2011−2022
      $ \Delta R_{oss}^{2} $ −3.563*** −0.776** 6.259*** 6.026*** 7.377*** 6.574***
      p−value 3.33 × 10−3 3.98 × 10−2 9.08 × 10−4 3.96 × 10−4 4.84 × 10−2 4.91 × 10−4
      2013−2022
      $ \Delta R_{oss}^{2} $ −2.544*** 1.454*** 8.257*** 7.271* 9.161*** 8.308***
      p−value 8.6 × 10−3 1.38 × 10−4 3.54 × 10−3 5.07 × 10−2 3.25 × 10−3 1.15 × 10−3
      2015−2022
      $ \Delta R_{oss}^{2} $ −6.124*** −2.367** 9.350** 8.044* 10.625* 9.883*
      p−value 9.56 × 10−3 1.94 × 10−2 3.11 × 10−2 5.63 × 10−2 5.1 × 10−2 8.07 × 10−2
      2017−2022
      $ \Delta R_{oss}^{2} $ −3.614** −1.199*** 10.395** 9.669** 11.814** 10.918**
      p−value 3.03 × 10−2 7.04 × 10−3 4.71 × 10−2 4.5 × 10−2 4.82 × 10−2 4.42 × 10−2
      Notes: The $ \Delta R_{oss}^{2} $ measure is calculated as the percentage of decrease in the MSPE of the focal model compared with the MSPE of the reference model in Eq. (2). Additionally, the significance levels (p−values) from the CW[50] tests, which assess the equality of the MSPE between the equity market's inferred volatility and the reference crude oil volatility model, are presented.

      Table 7. 

      Out-of-sample forecast performance of the ARX(7), TAR(7), and BAR(7) models.