Non-parametric estimation of a time varying GARCH model

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32 Table 2: Aggregated mean squared errors of the ‘in sample forecasts’ Series tvGARCH tvGARCH tvARCH (1) tvARCH (2) GARCH EGARCH GJR FIGARCH (d = 3) (d = 1) INR/USD 35.59 35.12 36.37 33.27 40.23 38.03 40.26 38.68 INR/EURO 2119.02 2162.22 2158.96 2137.97 2234.45 2524.83 2234.46 2249.93 CNY/USD 0.72 0.72 0.74 0.71 1.03 − 1.22 0.96 CNY/EURO 76.64 77.58 80.12 79.75 84.02 84.17 84.55 85.73 BRL/USD 1174.72 1197.15 1276.72 1117.56 1249.60 1163.88 1312.59 1221.22 BRL/EURO 3563.09 3603.22 4295.43 3844.53 4942.11 4402.06 5320.83 4861.54 RUB/EURO 65.27 65.36 68.77 68.34 73.98 72.81 74.04 69.34 RND/EURO 935.43 940.02 977.31 966.16 993.15 981.79 1016.55 989.07 S & P 500 2154.41 2620.90 2979.67 2652.07 2614.41 2476.76 2679.29 2572.90 Dow Jones 1715.59 2063.03 2330.29 2067.89 2075.91 1951.45 2125.98 2025.00 BSE 5688.85 5702.34 6170.73 6026.22 6358.63 6095.25 6539.42 6381.01 NSE 6205.13 6244.79 6764.44 6556.36 7134.17 6765.58 7398.32 7112.78
33 Table 3: Aggregated mean squared errors of the monthly volatility forecasts Series tvGARCH tvGARCH tvARCH (1) tvARCH (2) GARCH EGARCH GJR FIGARCH (d = 3) (d = 1) INR/USD (×10 −5 ) 5.9571 6.2781 6.3978 6.0262 7.4865 7.5525 7.6340 7.5385 INR/EURO (×10 −4 ) 1.4162 1.5060 1.6407 1.5872 1.8460 1.8557 1.8557 1.8806 CNY/USD (×10 −7 ) 2.5545 3.0306 3.0607 3.0216 4.3104 3.3115 4.2701 4.8166 CNY/EURO (×10 −4 ) 1.4054 1.5678 1.6280 1.5514 1.9860 1.6661 1.9820 1.9929 BRL/USD 0.0029 0.0031 0.0031 0.0030 0.0040 0.0037 0.0052 0.0048 BRL/EURO 0.0108 0.0120 0.0119 0.0118 0.0136 0.0136 0.0133 0.0135 RUB/EURO (×10 −4 ) 4.0295 4.3653 4.3969 4.4216 5.8115 6.5392 5.8298 5.5266 RND/EURO 0.0121 0.0131 0.0130 0.0128 0.0149 0.0147 0.0149 0.0149 S & P 500 0.0079 0.0085 0.0085 0.0079 0.0125 0.0151 0.0180 0.0137 Dow Jones 0.0047 0.0051 0.0052 0.0047 0.0065 0.0065 0.0148 0.0085 BSE 0.0205 0.0216 0.0217 0.0204 0.0245 0.0244 0.0245 0.0245 NSE 0.0147 0.0161 0.0161 0.0149 0.0187 0.0173 0.0187 0.0187
Page 1 and 2: Non-parametric estimation of a time
Page 3 and 4: 1 Introduction The first decade of
Page 5 and 6: have been discussed in Section 2. S
Page 7 and 8: Under Assumption 1(i), (3) is a sta
Page 9 and 10: The estimator of αi(u0) as a solut
Page 11 and 12: 4 Asymptotic results Towards provin
Page 13 and 14: Notations. bj = bj(u0) = Bias(ˆαj
Page 15 and 16: This problem is under investigation
Page 17 and 18: provides insight into the nature of
Page 19 and 20: Proof of Proposition 2.3. We can wr
Page 21 and 22: Now consider n� 1 nh k=p+1 (uk
Page 23 and 24: ⎡ X ⊤ ⎢ 1 W1 ⎢ ⎣ and usin
Page 25 and 26: Hence the result can be easily prov
Page 27 and 28: The variance expression given in Th
Page 29 and 30: (iii) a GJR process if where ω,α,
Page 31: 31 Table 1: Summary statistics of t
Page 35 and 36: −1.5 0.0 1.0 −2 −1 0 1 −1.0
Page 37 and 38: volatility volatility 0 10 20 30 0

Non-parametric estimation of a time varying GARCH model

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