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GRANGER CAUSALITY & VAR ANALYSIS
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--- A. GRANGER CAUSALITY TESTS ---
  ATM_IV → RV              : F=  62.381, p=0.0000, 100.0% sig  ***
  RV → ATM_IV              : F=  17.933, p=0.1501, 60.9% sig  
  Skew → RV                : F=  20.792, p=0.0177, 94.2% sig  **
  RV → Skew                : F=   6.589, p=0.0628, 75.4% sig  *
  ATM_IV → Return          : F=   1.799, p=0.2688, 24.6% sig  
  Return → ATM_IV          : F=   3.977, p=0.1741, 58.0% sig  
  Skew → Return            : F=   1.412, p=0.3817, 13.0% sig  
  Return → Skew            : F=  13.576, p=0.0133, 95.7% sig  **
  PC_Ratio → Return        : F=   1.010, p=0.4999, 5.8% sig  
  Return → PC_Ratio        : F=   2.819, p=0.1638, 46.4% sig  
  Impl_Skew → RV           : F=   1.343, p=0.3856, 17.4% sig  
  Impl_Kurt → RV           : F=   5.619, p=0.0861, 75.4% sig  *

--- B. BIVARIATE VAR: IV_ATM ↔ RV (pooled) ---

  VAR estimated for 20 tickers
  Avg R² (IV equation): 0.9363
  Avg R² (RV equation): 0.3977

  Average Impulse Response Function:
    h    IV→IV    RV→IV    IV→RV    RV→RV
    0   1.0000   0.0000   0.0000   1.0000
    2   0.8923  -0.0485   0.6683   0.1410
    4   0.8046  -0.0417   0.5105   0.1038
    6   0.7625  -0.0415   0.4224   0.0597
    8   0.7212  -0.0397   0.3869   0.0295
   10   0.6819  -0.0375   0.3534   0.0127
   12   0.6455  -0.0354   0.3296   0.0018
   14   0.6114  -0.0332   0.3088  -0.0040
   16   0.5795  -0.0312   0.2907  -0.0073
   18   0.5496  -0.0292   0.2747  -0.0090

--- C. FORECAST ERROR VARIANCE DECOMPOSITION ---
   Horizon   % RV by IV   % RV by RV
         1         0.00       100.00
         2        28.01        71.99
         3        44.32        55.68
         4        51.28        48.72
         5        56.10        43.90
         6        58.76        41.24
         7        61.24        38.76
         8        63.29        36.71
         9        64.98        35.02
        10        66.41        33.59
        11        67.61        32.39
        12        68.66        31.34
        13        69.58        30.42
        14        70.40        29.60
        15        71.12        28.88
        16        71.77        28.23
        17        72.36        27.64
        18        72.89        27.11
        19        73.37        26.63
        20        73.82        26.18

GRANGER CAUSALITY & VAR ANALYSIS COMPLETE.