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Chapter 2
Literature Review
2.1 Temperature Model
Lau and Weng (1995) used Morlet (1983)’s wavelet transform to depict the path of the DAT. This method is usually employed in the physics. The purpose of the wavelet transform is that the time series can be decomposed into several wavelet. Differing from the Fourier transform which only analyses the average characteristic of a time series and is affected by the extreme value easily, the wavelet transform uses the basis function which is generalized by the contraction and translation of the prototype function to catch the features of a time series. It can measure the continuous (stable state) and uncontentious change (jump state) path. Lau and Weng (1995) used the monthly North Hemisphere surface temperature (NHST) over period 1945/01 to 1993/07 as sample, by the wavelet transform, the path of the DAT can be decomposed into three frequency branches: inter-annual band (2 to 5 years), inter-decade bands (10 to 60 years), century scale (90 to 180 years). They argued that the results can measure the path of the DAT efficiently.
Cao and Wei (1999, 2000, 2004) pointed out that the ”net” DAT (the original DAT is subtracted by average level in current month) exists the feature of the autocorrelation.
Thus, they employed the autoregression (AR) model to depict the path and used the sine function to catch the seasonal volatility. Cao and Wei (1999, 2000) adopted the DAT for Atlanta, Chicago, Dallas, Philadelphia over the period 1979 to 1998. The data are obtained from the National Climate Data Center (NCDC) of the National Oceanic Atmospheric Administration (NOAA). The parameters of the AR model are fitted by the maximum likelihood estimation, and the optimal lag period is four period. It shows that
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the temperature do exist the autocorrelation, and it cannot be ignored for pricing the temperature derivatives.
Alaton, Djehince, and Stillberg (2002) assumed that the long-run mean level of the DAT follows the linear regression model with the time trend and seasonal factor. The time trend measures the global warming caused by the greenhouse effect. The seasonal factor (the DAT is lower in the winter and higher in the summer) is depicted by the sine and the cosine functions. The volatility of the DAT is constant level in one month in their assumption. Moreover, the DAT is along the long-run mean level and it exists the feature of the mean-reversion (MR). According to above characteristics of the DAT, Alaton, Djehince, and Stillberg (2002) supposed the Ornstein-Uhlenbeck (OU) process to depict the path. The OU process is also called as Vasicek model (1977) or Hull-White model (1990). They used the DAT of Bromma airport meteorological station in Sweden over 40 years as sample, and estimated the parameters through three steps: (i) For the long-run mean level, they employed a linear regression model whose dependent variable is DAT and the independent variables are the time trend and the seasonal factor. Through the least squares (LS) method, the parameters of the linear regression model are estimated.
(ii) The monthly volatility is measured by the standard deviation of the ”net” DAT (the residual of the linear regression model) in the specific month. (iii) The parameter of the speed of mean-reversion is computed by the martingale estimation function method developed by Bibby and Sørensen (1995). The empirical results report that the DAT follows the global warming path, the seasonal volatility (higher in the winter and lower in the summer), and the mean-revering effect.
Brody, Syroka, and Zervos (2002) also pointed out the autocorrelation of the DAT.
The standard Brownian motion or the Gaussian white noise process cannot capture the long memory property. Thus, they introduced the fractional Brownian motion (FBM) into the mean-reverting model, and assume the deterministic processes for the speed of mean-reversion, the long-run mean level of DAT, and the seasonal volatility. In the FBM, the Hurst index (0 ≤ H ≤ 1) measures the correlation between the pre- and post increments of the DAT. If 0 ≤ H < 0.5, the correlation is negative. If 0.5 < H ≤ 1, the correlation is positive. If H = 0.5, the correlation is zero and the FBM reduces to the standard Brownian motion. Differing to Cao and Wei (1999, 2000), Brody, Syroka, and Zervos (2002) used the FBM to depict the risk premium of the autocorrelation directly.
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But, they did not estimate the models parameters, and cannot compare the performance between the FBM and the standard Brownian motion.
Benth and ˇSaltyt˙e-Bench (2005) advocated that the OU process and the L´evy process are the better model for the DAT, and assumed that the long-run mean level and the volatility follow the truncated Fourier series which is the linear combination of a series of sine and cosine functions and measures the seasonality. They used the DAT of Alta, Bergen, Kristiansand, Oslo, Stavanger, Tromsø, Trondheim in Norway as sample, and pointed out that the DAT in Norway follows the leptokurtic distribution. It implies that the extreme temperature occurs with less probability. Moreover, the empirical results shows that the DAT exists the one-lag positive autocorrelation, the two-lag negative autocorrelation, and the volatility clustering in short term. They suggested that the gen-eralized autoregression conditional heteroskedasticity (GARCH) model can depict these features.
Campbell and Diebold (2005) also employed the truncated Fourier series to measure the long-run mean levels of the DAT and volatility. Apart from the seasonality, the DAT also exists the volatility clustering effect. Thus, they introduced the GARCH model with seasonal DAT and volatility to depict the path of the DAT. Campbell and Diebold (2005) used the DAT of Atlanta, Chicago, Las Vegas, and Philadelphia as sample, the empirical results show that the DAT in each city does not follow the normal distribution.
Subtracting the time trend, the seasonal factor, and autocorrelated factor, the ”net”
DAT obeys the left-skew and platykurtic distribution. Thus, the investors cannot use the standard Brownian motion or the Gaussian white noise process to describe the path of the DAT. Moreover, Campbell and Diebold (2005)’s model can predict the path of the DAT exactly in long term than in short term.
Benth and ˇSaltyt˙e-Bench (2007) also used the truncated Fourier series to measure the long-run mean levels of the DAT and volatility. Obtaining the DAT of Stockholm in Sweden over the period 1961/01/01 to 2006/05/26 as sample, the empirical results are similar to Benth and ˇSaltyt˙e-Bench (2005).
Past literatures like Benth and ˇSaltyt˙e-Bench (2007) usually employed the likelihood ratio test (LRT), the Akaike information criterion (AIC), the Bayesian information crite-rion (BIC), in-sample and out-of-sample predicting performance to determine the optimal lag period. However, Zapranis and Alexandridis (2008, 2009a, 2009b) used the concepts
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of the wavelet transform and the neural networks which is developed by Lau and Weng (1995) to choose the lag period. The sample includes the DAT of Paris over the period 1971/01/01 to 2000/12/31. The time series of the DAT can be decomposed into 1-year and 7-year wavelets.
Huang, Shiu, and Lin (2008) extended Alaton, Djehince, and Stillberg (2002)’s frame-work. They supposed the GARCH model with the time trend and seasonal factor of the long-run mean level. The sample is obtained from Taiwan Central Weather Bureau over the period 19740/01/01 to 2003/12/31. The GARCH(4,3) performs the better fitting for the DAT. It says that the DAT in Taiwan exists the four lag period unexpected shock effect and three lag period volatility clustering effect. However, under Huang, Shiu, and Lin (2008)’s model, the volatility is negative with positive probability. In the future, the parameters will be restricted against the negative volatility.
Benth and ˇSaltyt˙e-Bench (2011) introduced the continuous-time autoregressive (CAR) model which is similar to the Barndorff-Nielsen and Shephard (2006)’s stochastic volatil-ity model. Owing to the volatilvolatil-ity under the GARCH model be predictable, it does not correspond to reality. And, the volatility exists the characteristics of the autocor-relation and seasonality, so the CAR model can depict the DAT more exactly. Using the DAT of Stockholm of Sweden over the period 1961/01/01 to 2006/05/26 as sample, Benth and ˇSaltyt˙e-Bench (2011) pointed out that the DAT follows the three lag period aurocorrelation and platykurtic distribution.