Non-parametric estimation of a time varying GARCH model

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(1993) and references therein) models are estimated using SAS, while MATLAB is used for the estimation of FIGARCH (1, d0, 1) model, where d0 is the fractional differencing parameter to be estimated from the data (Baillie (1996)). The definitions of these models are provided in Appendix C. R codes have been written for the estimation of tvGARCH (with d = 3, 1 and p = log n) and tvARCH models using Epanechnikov kernel. All the codes can be made available on from authors. The choices of d = 3, 1 facilitate the optimal rate of convergence of the order of n −8/9 and n −4/5 respectively and p = log n requires lesser number of parameters to be estimated in Step 1 as compared to other choices of p such as √ n. The bandwidth is selected using the cross-validation method as described in Section 3.1. Estimation of the tvARCH model has been carried out using Step 1 methodology of Section 3 with bandwidth chosen using cross validation, minimizing the mean-squared prediction error for tvARCH (Hart (1994)). EGARCH model could not be estimated for the CNY/USD data due to convergence problems. Superiority of the tvGARCH model is evident from the Table 2. The non-stationary models have clearly outperformed stationary as well as long memory models. The AMSEs of tvGARCH with d = 3 are smaller than that with d = 1 in most of the cases. However, the difference between the two is not very high. An illustrative comparison of tvGARCH (d = 3) model is also shown in Figure 3 for BRL/EURO data. The faint plot depicts the squared returns and the dark plot is the predicted volatility with the corresponding model. Clearly, the tvGARCH model has captured the ups and downs in the volatility more accurately. In Figure 4, we plot the the estimators ˆω(u), ˆα(u), ˆ β(u) and ˆα(u) + ˆ β(u) against u ∈ (0, 1] for the BSE data. Notice that similar to the least squares estimators of Fryzlewicz, Sapatinas and Subba Rao (2008), the local polynomial estimators are not guaranteed to be non-negative. Although, the estimators satisfy ˆα(u) + ˆ β(u) < 1 for this data, this may not be the case in general depending on the behaviour of the data. To compare the performance of the tvGARCH model further, in Table 3, we report the AMSE for the in-sample monthly volatility (of 22 trading days) forecasts for the same data sets, based on the monthly returns. The monthly returns are calculated as rmt = log(Pt/Pt−1), t = 1, 2,...,T, where Pt denotes the closing price on the last day of t th month and T is the total number of complete months in the data. All the datasets are of size around 125 except NSE dataset which has the size 95. This analysis 16
provides insight into the nature of the tvGARCH model for small data sets. Our numerical evidences indicated that the asymptotic properties derived in Section 4 regarding the bandwidth selection also hold for these moderate sized monthly datasets. We did not multiply the returns with 100 to avoid large values. This, together with small data size has resulted in very small AMSEs. However, for comparative purposes, this does not make any difference. Clearly, the tvGARCH is performing better than other models even for small sample sizes. One interesting conclusion that can be drawn from the above analyses is that the global crisis and specially the currency wars have vehemently turned the exchange rates volatility towards non-stationarity and short memory. This is quite possible as the fre- quent manipulation of the currencies may lead the currency rates to lose its widespread notion of the long memory behaviour. The ‘out of sample forecasting’ performance of the tvGARCH model has been judged using 50 daily forecasts computed by a rolling-window scheme. The out of sample forecasts of the tvGARCH model are computed as follows. Use the n1 = n − 50 observations for the in-sample estimation. Then, forecast into the future using the ‘last’ estimated coefficient values, that is, the estimate of coefficient functions at t = n1. Forecasts into the future are computed in the same way as in a stationary GARCH model using these last coefficient estimates. Similar method has also been used by Fryzlewicz et al. (2008) for the future forecasts using the tvARCH model. Let σ 2 t+1|t , t = n1,n1 + 1,...,n − 1 denote the one-step ahead out of sample forecasts using the previous n1 observations. We compare σ 2 t+1|t with ǫ2 t+1, t = n − 50,n − 49,...,n − 1 to get the AMSEs, which are reported in Table 4. The out of sample forecasts using tvGARCH model are better than those of the other models. The tvGARCH attains the lowest AMSE for 7 data sets, while tvARCH (2) is better in 1 case. The FIGARCH and EGARCH models have shown good forecasts for two data sets each, while GARCH and GJR models are performing abysmally. It is noticeable that the tvGARCH model with d = 1 performs better than the tv- GARCH with d = 3 in the out of sample forecasting. However, there is not much of a difference between AMSEs of tvGARCH with d = 3 and d = 1. The better performance of tvGARCH (d = 3) than tvGARCH (d = 1) in the in-sample forecasting can be ex- plained to some extent by the fact that bigger d yields a higher convergence rate of MSE. 17
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: This problem is under investigation
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 and 32: 31 Table 1: Summary statistics of t
Page 33 and 34: 33 Table 3: Aggregated mean squared
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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