Analysis of the Sector of Software & Computer Services with a New Carhart 4-Factor Model

In this paper, we analyze the sector of Software and Computer Services with a new Carhart four-factor model. The US is the leading world market for this sector and this sector is a source of significant economic opportunity in US. We compare this sector in US, UK and China to find out whether the US phenomenon has been replicated by other industrialized countries. LR, KS and AIC are used for testing parameter restrictions, residual check and model comparison, respectively. MLE is used to estimate parameters via Matlab. Empirical results show the Carhart 4 factors are still alive! The new 4-factor model fits the data well and has better in-sample fit than that of Carhart (1997) [1] and Fama-French (1993) [2]. This sector in these 3 countries can not earn statistically significant extra Alpha returns. And the Beta value in this sector of US is close to the market.


Introduction
add two more factors such as Size factor and Book-to-market factor into the CAPM model and create a 3-factor model, which is capable to explain the stock returns better than the CAPM.Carhart (1997) finds the momentum factor has great effect on stock returns.
Our research falls into the 2nd group and tries to extend the 4-factor model in Carhart (1997).But different from previous researches, instead of introducing different factors, we use a non-normal error of SSAEPD proposed by Zhu and Zinde-Walsh (2009) [11], and the EGARCH-type volatility of Nelson (1991) [12].We denote our new model as C-SSAEPD-EGARCH.SSAEPD is capable to show the skewness, fat tails and asymmetric kurtosis of data.Based on the new Carhart 4-factor model, we try to test following hypotheses: 1) With EGARCH-type volatilities in Nelson (1991) The organization of this paper is as follows.The model and methodology are discussed in section 2. Empirical results and the model comparisons will be presented in section 3. Section 4 is the conclusions and future extensions.( )
t R is the rate of return for US, UK and China indices of software & computer services industry at time t .ft R is the rate of return for the risk-free asset at time t .mt R is the rate of return for the market at time t .SMB t is the size factor, and stands for small market capitalization minus big market capitalization.HML t is the value factor, and stands for high book-to-market ratio minus low book-to-market ratio.WML t is the momentum factor, and stands for high prior return portfolios minus low prior return portfolios at time t .

SSAEPD
In the new model, the error term t z is distributed as the Standardized Standard Asymmetric Exponential Power Distribution (SSAEPD) proposed in Zhu and Zinde-Walsh (2009).The probability density function (PDF) of t z is: ( ) ( ) ( ) ( ) ( ) x is distributed as the standard AEPD(SAEPD).And ( ) Γ ⋅ is the gamma function.

MLE
We estimate the parameters in above models with the method of Maximum Likelihood Estimation (MLE).The maximum likelihood function of the model is where ( )

Empirical Analysis
where , i t p , , 1 i t p − are the prices of these indices i at time t and 1 t − , respectively.
Table 3 lists the descriptive statistics by Matlab 6 .The values of skewness are not equal to 0 and those of Kurtosis are not 3. Especially, kurtosis values are all greater than 3. P-values of JB tests are 0, which are smaller than 0.05.That mean, under 5% significance level, we can reject the null hypothesis and conclude that data do not follow Normal distribution and non-Normal error of SSAEPD may be proper.

Estimation Results
The estimates are listed in Table 4.For the new model, the Alpha returns for all   p p are smaller than 2 and that of α are not equal to 0.5.Furthermore, we find out the EGARCH term can better capture the excess kurtosis than non-Normal error.For 4 factor models, their estimates are very close to those of 3 factor models.

Carhart 4-Factor Still Alive
• Significant Tests for Parameter Restrictions Likelihood Ratio test (LR) 7 is used to test the significance of regressors in these models.The P-values for LR tests are listed in Table 5.
We find out with non-Normal errors such as SSAEPD and EGARCH-type β is statistically signif- icant.That is, market returns have significant effect on this sector returns of UK  and US.This sector in these 3 countries all don't have a statistically significant coefficient 0 β under 5% significance level which means they cannot earn sta- tistically significant Alpha returns.Non-Normality is confirmed (see column T7).ARCH and GARCH terms should be added into Carhart 4-factor model since they are all statistically significant (see column T11).
• Kolmogorov-Smirnov Test for Residuals We check the residuals for models with Kolmogorov-Smirnov test (KS).The P-values of KS test are listed in Table 6, which shows only the residuals of the new Carhart 4-factor model passes the residual diagnostics.

Model Comparison
We compare the models with AIC (see

Conclusion and Future Extensions
In this paper, sector of the software and computer services is studied.A new We choose 0 ( ) .

L
. L. Li et al.Notes: "-" means that no information is available in this paper.Mkt = Market Premium.SMB = Small Size Minus Big Size.HML = High Book-to-market Minus Low Book-to-market.WML = Past Winner portfolios Minus Past Loser portfolios.FF = Fama Frech 3-factor model.C = Carhart 4 factor model.FIML = Information Maximum Likelihood.Four Regions= North America, Europe, Japan, and Asia Pacific.IML = Illiquid portfolios Minus Liquid portfolios.Hybrid C is adding U.S. or global factors to the local model.IV = Idiosyncratic Volatility.
, investment, profitability, risk spending, quality, process, staffing, employees, revenue, net income growth of export and domestic revenues, salaries export, back-office services, annual PC sales sales, employees, place, year of start-up, type financial position, cost, VaR, ES --" means that no information is available in this paper.VaR = Value-at Risk.ES = Expected Shortfall.CMMI = Capability Maturity Model Integration.ROA = Return on Assets.

A new 4 -
factor model 2 is used to analyze the sector of software & computer services sector (denoted as C-SSAEPD-EGARCH) 3 .

2
This new model is first suggested inMu (2014).The EGARCH-type volatility inNelson (1991) and non-Normal error of SSAEPD in Zhu and Zinde-Walsh (2009) are considered in the new model.We first check the simulation and the empirical results inMu (2014).Then, we re-run the simulation by setting other true parameters.The results are listed in Appendix 4.
β β are the coefficient parameters in the regression model.

3. 1 .
DataIn this paper, the sector of Software and Computer Services (SCS) is analyzed.Daily data are downloaded from the Investing.com 4.4 factors are downloaded from French's Data Library 5 .Sample period is from Nov. 1st, 2012 to Sept. 30th, 2015.Three indices of Software and Computer Services (SCS) for US, UK and P. R.China are compared.To eliminate the heteroscedasticity we calculate the log returns of these indices by following formula: Notes: Med.= Median, Max.= Maximum, Min.= Minimum.St.De.= Standard Devistion, Ske.= Skewness, Kur.= Kurtosis, P = P-value of Jarque-Bera Test.ME = Market Excess Return, SMB=Small minus Big, HML = High minus Low, WML = Momentum Factor.The null hypothesis of JB test is H_{0}: Data are distributed as Normal (0,1).
as the true values of the parameters.The data generation process (DGP) has following steps.with the following formula:( close to the true values of the parameters.For robustness exam, we also change the true values of the parameters and redo the simulation and estimation.All the simulation and estimation show the estimates are very closed to the true values of the parameters, since all error are equal to or less than −25%.11 This MatLab program is written by Mu (2014).

Appendix 5 . US 4 Factors and Global 4 Factors
In this section, we compare the US 4 factors and the Global 4 factors.Data are downloaded from the French Data Library.Because of data availability, we only analyze the monthly data for UK software & computer services sector.Table 9 shows the descriptive statistics of US 4 factors are close to those of Global 4 factors, which shows the strong global effects of US stock market.Estimates in the

Table 2 .
Researches about the software sector.
and non-normal errors of SSAEPD in Zhu and Zinde-Walsh (2009), are the Carhart 4 factors still alive in the sector of Software & Computer Services? 2) Can this new 4-factor model beat that of Carhart (1997)?To answer these questions, we run simulation to test the MatLab program used in this paper.Then, the industry of the software & computer services in US, UK and China are analyzed 1 .Data are downloaded from the Investing.com,and the sample period is from Nov. 1st, 2012 to Sept. 30th, 2015.Method of Maximum Likelihood Estimation is used to estimate the parameters.Likelihood Ratio test (LR) and Kolmogorov-Smirnov test (KS) are used for model diagnostics.Akaike Information Criterion (AIC) is used for model comparison.We find out the Carhart 4 factors are still alive!The EGARCH-type volatility can capture the excess kurtosis.The new model fits the data well and has better in-sample fit than Fama-French (1993)'s 3-factor model and Carhart (1997)'s 4-factor model in most cases.The industry of software & computer services in US, UK and China all cannot earn extra Alpha returns since the constant term in this new model is not statistically significant.This industry in US is similar to the market because the Beta coefficient ( 1 T nential Power Distribution (SSAEPD) proposed in Zhu and Zinde-Walsh (2009).

Table 4 .
Estimates.are small.The Alpha return in China is 0.15, much higher than those in UK and US.And the values of Beta ( 1 β ) for US are the largest and those for China are the smallest.Especially about US, the values of 1 Fama-French (1993) the model used inCarhart (1997).FF-Normal is the model used inFama-French (1993).data

Table 6 .
P-values of KS test.
Note: The null hypothesis of KS test is H0: Data follow a specified distribution.We set the significance level of all tests at 5%.If the P-value of KS test is bigger than 5%, then we do not reject the null hypothesis.Otherwise, we reject the null hypothesis.For example, we apply KS test for the C-SSAEPD-EGARCH model residuals with the null hypothesis of H0: C-SSEAPD-EGARCH model residuals are distributed as SSAEPD ( ).For US, its P-value is 0.37, which is bigger than 0.05.That means, under 5% significance level, we cannot reject the null hypothesis and conclude that the residuals from C-SSEAPD-EGARCH model follow SSAEPD.
Carhart 4-factor model (denoted as C-SSAEPD-EGARCH) is empirically tested using data in US, UK and China.This new model uses the non-normal error term of SSAEPD of Zhu and Zinde-Walsh (2009) and EGARCH type volatility of

Table 8 .
Simulation results.: T means the true value of parameters.E means the estimates.P means the error in percentage.Hence, we conclude the MatLab program can be applied to estimate and analyze empirical data for C-SSAEPD-EGARCH. Notes