Analysis of 48 US Industry Portfolios with a New Fama-French 5-Factor Model

In this paper, we analyze US stock market with a new 5-factor model in Zhou and Li (2016) [1]. Data we use are 48 industry portfolios (Jul. 1963-Jan. 2017). Parameters are estimated by MLE. LR and KS are used for model diagnostics. Model comparison is done with AIC. The results show Fama-French 5 factors are still alive. This new model in Zhou and Li (2016) [1] fits the data better than the one in Fama and French (2015) [2].


Introduction
In 2015, Fama and French suggest a 5-factor model (denoted as FF5-Normal) 1 to capture the market, size, value, profitability and investment patterns in stock returns, which is found better than their 3-factor model in [3]. Since then, many researches about the 5-factor model are developed (see Table 1). These researches can be divided into following 2 groups. The 1st group of researches empirically tests the FF5-Normal model using different data. For example, [4] [5] find out that the FF5-Normal model works well in India.
They find out their new model has better in-sample fit than that of [2] and the non-normal error assumption of SSAEPD is capable of capturing many stylized facts in financial time series such as skewness and asymmetric fat-tailedness 2 .
Based on the new model of [1], in this paper, we try to test following hypothesis: If different data such as 48 industry portfolios 3 are considered, can the new model of [1] still beat the 5-factor model in [2]? To find answers for above question, simulation is used to check the validity of [1]'s MatLab program 4  Model comparison is done with Akaike Information Criterion (AIC).
Simulation results show the MatLab program is valid and can be used for empirical analysis. Empirical results show the 5 factors in [2] are still alive! The GARCH-type volatility and SSAEPD can successfully capture the excess kurtosis.
The new model of [1] fits the data well and has better in-sample fit than the 5-factors model of Fama and French.
The organization of this paper is as follows. Section 2 is the model and methodology. Section 3 presents the empirical results. Section 4 provides the conclusions and future extensions. The appendices contain additional information that may be helpful to understand our paper. 2 The history of SSAEPD is displayed in Appendix 3.
3 [1] analyze 25 Fama-French portfolios, which is different from the dataset we use. 4 Simulation results are listed in Appendix 4. [1] extend Fama-French's 5-factor model based on the GARCH-type volatility in [13] and non-Normal error distribution of SSAEPD in [10], and show their new model is better for 25 Fama-French portfolios. This new model in [1] is listed as follows (denoted as FF5-SSAEPD-GARCH).

The FF5-SSAEPD-GARCH Model
( ) the FF5-SSAEPD-GARCH model is reduced to Fama-French's 5-factor model and in the following section we will compare these two models.

MLE
Maximum Likelihood Estimation (MLE) is used to estimate previous model. The likelihood function is  portfolio, the "Ships'' industry) is not 0 and the kurtosis is more than 3. The p-value of Jarque-Bera test for each portfolio is 0, which is smaller than 5% significance level. Hence, we can reject the null hypothesis and conclude that the asset returns do not follow Normal distribution. Thus, non-Normal error assumption of SSAEPD might be able to fit the data better.

Estimation Results
The estimates for our new model are displayed in Table 3. We find out that our

Model Diagnostics
To test the significance of coefficients in FF5-SSAEPD-EGARCH, Likelihood Ratio test (LR) is applied 6 , which is calculated using Equation (9).

Tests for Parameter Restrictions • Tests for Parameters in the Mean Equation
The P-values of LR are listed in Table 4. The null hypothesis of the joint significance test is     T0  T1  T2  T3  T4  T5  TJ  T0  T1  T2  T3  T4 T8  T9  T10 T11 T12 T13 T14 T15 T16  T8  T9  T10 T11 T12 T13 T14 T15  5-factor model. For instance, we do the joint significance test for hypothesis For 46 out of the 48 portfolios, the p-value of the LR are smaller than the significance level 5%, which means our GARCH-type volatilities are quite necessary. As for individual hypotheses, we discover that most P-values of LR are smaller than the significance level 5%. And to be specific, ARCH term • Tests for Parameters in SSAEPD We also run significance tests for the parameters in the SSAEPD and the results of parameter restrictions show strong non-Normality. For example, for the Applied Mathematics

Residual Check
In this subsection, the residuals for previous models are checked with both Kolmogorov-Smirnov test and graphs. Our results show 41 out of the 48 portfolios have residuals which do follow SSAEPD. That means, the FF5-SSAEPD-GARCH is adequate for the 48 industry portfolios. But the FF5-Normal model is not adequate for the data, since all the 48 portfolios have residuals which do not follow the Normal error distribution.
• p-values of the KS test are also listed in Table 6. All of the 48 portfolios have smaller p-values than 0.05, which means these 48 industry portfolios reject the nulls. Hence, the error terms of the portfolios do not follow Normal distribution. And the FF-Normal model is not adequate for the data.
• PDFs of Residuals By method of "eye-rolling'', the PDF of residuals is compared with theoretical PDFs. Taking the portfolio of Agriculture industry for example, in Figure 1(a), the probability density function (PDF) for the estimated residuals ˆt z in FF5-SSAEPD-EGARCH and that of ( ) 1 2ˆ, , SSAEPD p p α are plotted. These curves are very close to each other, which means the residuals are distributed as SSAEPD. Hence, the FF5-SSAEPD-GARCH model fits the data well.
Similarly, the probability density function (PDF) for the estimated residuals ˆt u in FF5-Normal and that of ( )

2,
Normal µ σ are shown in Figure 1(b). And there are big differences between these two curves, which means the residuals do not follow Normal distribution. 7 The null hypothesis of KS test is 0 H : Data follows a specified distribution. If the P-value of KS test is bigger than 5% significance level, the null hypothesis is not rejected. Otherwise, the null hypothesis is rejected. 8 The null hypothesis is

Model Comparison
In this subsection, we compare the model in [1] with the 5-factor model of Fama and French. The Akaike Information Criterion (AIC) is used as the model selection criterion. Table 7 lists the AIC values. We find that 47 out of 48 AIC values of the FF5-SSAEPD-GARCH model are smaller than those of the FF5-Normal model. Hence, we conclude that the new model we used (FF5-SSAEPD-GARCH) is better than the 5-factor model of Fama and French.

Conclusions
In this paper, we empirically test the new 5-factor model suggested in [1]. Their new model generalizes the 5-factor model in [2] by introducing a non-normal

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