Non-parametric estimation of a time varying GARCH model

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It is interesting to note that the bias expressions are free of the derivatives of other parameter functions. Also, if h1 = o(h2), then δj = αj(u0) + oP(h d+1 2 ) and the variance of the estimator does not depend on the first step bandwidth. This means that when the optimal bandwidth is used, then the estimation remains unaffected for a large choice of initial step bandwidth. This makes the estimation procedure relatively easy to imple- ment. The MSE of the final estimator is OP(h 2d+2 2 +(nh2) −1 ), which is independent of the initial step bandwidth. Notice that this MSE achieves the optimal rate of convergence at an order of n −(2d+2)/(2d+3) for an optimal bandwidth h2 of order n −1/(2d+3) and h1 = o(h2). Now in the following corollary, we prove the asymptotic normality of the estimator using martingale central limit theorem. Corollary 4.3. Under the same assumptions as that of Theorem 4.2, √ � nh2 ˆβtvGARCH(u0) − βtvGARCH(u0) − btvGARCH(u0) � � D → N3 0,e ⊤ 1,d+1A −1 2 B2A −1 2 e1,d+1V ar(v2 t )S −1 2 Ω2S −1 � 2 where βtvGARCH(u0) = [ω(u0),α(u0),β(u0)] ⊤ and btvGARCH(u0) = [Bias(ˆω(u0)),Bias(ˆα(u0)), Bias( ˆ β(u0))] ⊤ . Remark 4.1. Above results have led us to the following two important issues, which need further investigation. 1. The asymptotic distributions of the estimators of the parameter functions depend on the parameters of the stationary approximation to tvGARCH defined in (3), which is unobservable. Therefore, to derive a confidence band (or point-wise confidence intervals), one can use the bootstrap methods. Fryzlewicz, Sapatinas and Subba Rao (2008) used residual bootstrap methods of Franke and Kreiss (1992) to construct point-wise confidence intervals for the least-squares estimator of the tvARCH model. To avoid instability of the generated process, they modified their estimator so that the sum of all the estimated coefficients remain less than one. However, their method does not guarantee the estimators to be non-negative. This results in some of the bootstrapped residual squares to be negative. In order to tackle this problem, one needs to carefully formulate a bootstrap procedure and establish its working. Another approach would be to modify the estimation procedure itself to satisfy these constraints, see for example Bose and Mukherjee (2009). 14
This problem is under investigation. 2. Our method assumes that all the three tvGARCH parameter functions have the same degree of smoothness and hence they can be approximated equally well in the same interval. But if the functions possess different degrees of smoothness, then the proposed method may not give the optimal estimators (see Fan and Zhang (1999)). Therefore, one has to construct an estimator that is adaptive to different degrees of smoothness in different parameter functions. 5 Modelling and forecasting volatility using tvGARCH We analyze the currency exchange rates between five major developing economies in the forefront of global economic recovery viz. Brazil (BRL), Russia (RUB), India (INR), China (CNY) and South Africa (RND) (so called ‘BRICS’) and the developed economies viz. United States (USD) and Europe (EURO). The last decade saw the ‘BRICS’ mak- ing their mark on the global economic landscape. In recent times, these economies are severely affected due to the global financial crisis and currency wars. This was our mo- tivational factor in analyzing these exchange rates data using tvGARCH. Applications of the tvGARCH model has also been discussed in four stock indices, S & P 500, Dow Jones, Bombay stock exchange (BSE, India) and National stock exchange (NSE, India). All the data sets consist of daily percent log returns ranging from the beginning of 2000 (dates varying) to December 31, 2010 except NSE data, which start from January 2002. The data are available from the websites of US Federal Reserve, European Central Bank and www.finance.yahoo.com. Figures 1 and 2 depict the plot of the return data and au- tocorrelation functions of squared returns. In Table 1, we provide the summary statistics of of the data. To compare the in-sample prediction performance of tvGARCH with several other well known existing models, we compute the aggregated mean squared error (AMSE) (see Fryzlewicz, Sapatinas and Subba Rao (2008)): AMSE = n� (ǫ t=1 2 t − ˆσ 2 t ) 2 , where ˆσ 2 t and ǫ 2 t are the predicted volatility and squared return at time t and n denotes the sample size. These are reported in Table 2. The lowest AMSEs are presented in bold letters. Here, GARCH (1,1), EGARCH (1,1) and GJR (1,1) (see Engle and Ng 15
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: Notations. bj = bj(u0) = Bias(ˆαj
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 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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