Fama and French 5 Factor model as a regression model
The dataset that has been used for the analysis in this assignment is the Fama and French 5 Factor model and the main task that has been conducted is regression analysis. Various independent and dependent variables are involved with the regression analysis based on the dataset that has been provided. A function is provided by a regression model and the relationship which exists between the independent and dependent variables is investigated using the regression model. There are various types of regression models such as Linear regression model, multiple regression model, non-linear regression model, and stepwise regression model. Various kinds of predictions can be performed using regression analysis and “the effects on target variables” can be evaluated using regression models. For instance, regression models are useful while conducting studies on the reasons leading to pollution in a particular region, the effects of watching too much television on the academic results of students or other related studies.
Linear regression is one of the regression models where the relationship between the input and output variables is represented through a straight line. This is the simplest model among the different models and can be understood quickly. The linear regression model can be used to evaluate the correlation coefficient between the different variables and parameters. Estimates of the dependent variable can be obtained using the linear regression model.
There are numerous markets around the world and they have integrated with a greater number of investors. Portfolio returns are affected by firm characteristics which are considered as variables. There are various types of models that can be used in this respect such as ICAPM, CAPM, APT, and F-F-5 factor models. A lot of business and monetary transactions occur in the markets and the study is concerned with the performance of the regression model for the F-F-5 factor in developed markets like Europe or America. Data mining techniques also prove helpful in the analysis since the dataset contains extensive numerical data.
Multiple regression refers to the regression model where there are multiple input variables that might impact the outcome. In the context of multiple regression it is essential to have a conceptual clarity of the impact of the input variables and the way in which they integrate to generate the target variable outcomes. Sometimes it is easier to employ non-linear models compared to linear models and the key is to identify when non-linear models are applicable for the dataset.
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The data used in the analysis has been downloaded from an internet-based source and statistical analysis has been carried out using the data. There are portfolios of several developed countries in the dataset including countries like France, Greece, Great Britain, Hong Kong, Italy, Ireland, Norway, Australia, Canada, New Zealand and various other countries. The data that is used for the analysis consists of monthly values and includes capital gain and dividends. The assumption in the dataset is market integration and certain diversification benefits have been obtained on the Left-Hand-Side portfolios which have been used as dependent variables. One of the benefits of this has been an accurate regression that has been obtained. The inclusion of a considerable number of countries in the dataset, it can be expected to obtain a greater R-Squared and lesser error as the factors used can be termed as global factors.
The currencies of the different countries whose data has been used in the analysis are distinct. The returns in the dataset are monthly and they have been measured in US dollars. The assumption of the dataset is that there is no risk of exchange. Capital gains are what attract investors since they want to take advantage of the fluctuations of currency (Baranovskyib et al. 2019). This can be considered as a drawback of CAPM according to a researcher. Also, the researcher feels that international and national factors are important for “international relations of stock prices”. It can be stated that currencies remain in equilibrium when the goods cost the same in different countries. The exchange rates need to be taken into account in this regard. It is desirable to emulate a long-term investment policy. Currency fluctuations are exploited by investors for the purpose of gaining a large return. In some cases, currency changes can decrease the “return of a portfolio” since there is great exposure of the portfolio in international markets. Certain ways of hedging against risk of exchange include currency futures, currency forwards and swaps or currency options. Thus the dataset that has been used in the analysis is reliable and can prove to be useful for conducting the entire analysis.
The analysis that has been conducted using the data is regression analysis and various kinds of statistical measures are used in the analysis. Some of the measures that have been used include correlation coefficient, R Squared, F statistic, and others.
Descriptive statistics
The analysis that has been carried out in the current assignment is Regression Analysis and the dataset that has been used in the analysis is Fama and French 5 Factor dataset. Various measures have been obtained in the analysis and they include coefficient, standard error, t-statistic, Probability, and various other measures.
Figure 1: Variables in the dataset of the Fama/French 5 Research Factors excel file
The above figure shows the different variables that have been used in the analysis and they include c, cma, hml, mkt_rf, resid, rf, rff, rmw and smb. These are the parameters using which the analysis has been carried out and regression analysis has been performed.
Figure 2: Variables in the dataset of the 10 momentum-sorted portfolios csv file
Excel file
Most of the concerned variables which have been considered are shown by way of this particular image in general. These are “cma”, “hml”, “hi_prior”, “lo_prior”, “mkt_rf”, “prior_2”, “prior_4”, “prior_5”, “prior_7”, “resid”, “rf”, “rff’, “smb”, and many more. The total quantity of the observations which have been conducted here are falling into the range of “114962 to 14962” in particular.
Figure 3: Image view of the dataset from the Fama/French 5 Research Factors excel file
The above figure shows the variables and corresponding measures like coefficient, Standard Error, t-statistic, Prob and others.
Estimation of Fama and French 5 Factor model
Figure 4: Image view of the estimation analysis RF variable
According to the above figure the image view of the analysis done using EVIEWS has been conducted where the Fama and French 5 factor has been analysed. The dependent variable is taken as RF according to the above figure.
Figure 5: Interpretation of data
Here the data variables and the observations of the data after adjustment can be observed. Vector auto generation estimation along with RF coefficients are obtained from the estimation. 14901 samples are obtained after the adjustments and from that 4889 observations are included in the analysis process.
Figure 6: Image view of the histogram from the 10 momentum-sorted portfolios csv file
Excel file
As per the above figure, the image view of the histogram developed from the 10 momentum-sorted portfolios has been shown.
a) Comparison of sign and magnitude of estimates
The comparison of sign and magnitude has been conducted on the basis of the chosen dataset.
Figure 7: Estimate of the data variables from the Fama/French 5 Research Factors excel file
As per the above figure, the different variables which have been estimated to perform the analysis is shown in the figure.
b) Comparison of the tests of individual significance of the regressors
Figure 8: Graphical presentation of the variables from the Fama/French 5 Research Factors excel file
This picture in this section pertains to the graphical presentation of all of the concerned variables from the “fama/french 5 research factors file” in particular (Huy et al. 2021). It can easily be observed from the graph above that three different colours have been taken into account for showcasing different sets of observations. These are blue, green, and red representing the factors like residual, fitted, and actual respectively.
c) Comparison of R2 or other statistic
The test result suggested that R-squared statistics in this case has been -0.57. Similarly, it can also be seen that the adjusted r squared value for this model was -0.58. Therefore, it can be said that there is negative adjustivity within the R square statistics. It should be mentioned that having r square value close to 1 should be considered as good statistical competency within a model. However, in this case, r square statistics has negative attributes. Still, having 0.57 as R squared statistics is suggesting that 57% of the changes in the test variable can be explained by the changes within the response factors. Thus, it can be said that R square statistics has suggested the statistical comparison between the response and test variable.
d) Testing the statistical adequacy of the each regression
Statistical adequacy testing is done through the help of plotting the regression model with the data in terms of inspecting the trends. In order to justify a good statistical adequacy model it is important to test the best fit model with the observed regression model. Similarly, the regression model has suggested that F statistics value is 28.3 whereas mean dependent variance is 0.67. Therefore, it can be said that data accordingly plots with the regression model. Henceforth, it is evident that there is enough statistical adequacy within this regression model.
e) Testing linear restrictions on parameters of the model
Figure 9: Forecast of the dataset used in the software interface
The picture in the section above has represented the aspect of forecasting of this particular dataset utilised within this software interface. The parameter that has been forecasted in this case is “HMLF” and the actual is “HML” in this regard. The total number of forecasted samples is “114962” and the quantity of adjusted samples are “1774” respectively. The values of several parameters such as the “root-mean squared error”, “mean-absolute percent error”, “mean-absolute error”, are “0.227759”, “NA”, “0.179483” respectively.
Estimation of the F-F-5M for the 10 portfolios
The statistical programme used for time series-focused econometric analysis is called Econometric Views, or EViews. The collaboration with economic specialists who are also trained analysts and offer targeted support for the use of EViews with the goal of providing the finest assistance to academics, policy makers, government agencies, and academicians for econometrics analysis. E-views had tools for frequency filtering and data interpolation, which can be used to generate the missing data (Burkhanov, 2020). E-views software is useful for time series analysis as a result. Other programmes, including STATA and SPSS, can only provide the data missing values. The database occasionally required high-frequency data with hours and seconds frequencies. A multi-year, bimonthly, fortnightly, ten-day, and daily frequency are also available. With a large database, the E-views package can support this kind of data. Databases frequently call for long-run variance and covariance calculations, where errors are randomly distributed and signify a specific kind of sequence. Using the nonparametric kernel, E-views enabled the calculation of symmetric or one-sided long-run variances.
Test of F-F-5M estimation
Although the E-views programme is beneficial for time series analysis, panel data analysis with long-term data sets cannot be done with it. This is due to the E-views package's limitations on the matrices of various observations and variables. Therefore, it might not be possible to acquire the findings for panel data sets (Huy et al. 2020). The task has been possible like that of the algorithm employed by E-views to estimate the parameters within the log probability is not the best choice for obtaining the outcomes for arbitrary maximising or minimization. This is thus because the algorithms' foundation has been the sum of the derivatives of the likelihood contribution's outer product. As a result, it might not be possible to present a good estimate using the general configuration. The functional form and the statistical features have been properly described.
Figure 10: Correlogram of the Residuals
According to the above figure, the analysis called correlogram of residuals has been obtained using specific functionality of EVIEWS software application. As per the figure there are a total of 771 observations that have been used in the above analysis. The different techniques that have been used with respect to the data are Autocorrelation and Partial correlation.
Estimation test
The estimation has been performed upon the concerned sets of data for that matter. The above image showcases all of the parameters which have been obtained after conducting the aforementioned process. In this case, RF is the dependent variable and the utilized method is named as least squares (Radivojevi? et al. 2019). The total number of variables that have been taken into consideration are “5” in general. The respective parameters that have also been acquired are standard error, coefficient, prob., and t-statistic respectively. The corresponding values of several parameters have also been fetched and that are “mean dependent variable”, "schwarz criterion”, “durbin-watson stat”, “akaike info criterion” amongst others.
Figure 11: Variables for estimation testing of the dataset
Equation Estimation is performed as per the above figure using EViews and the method of Least Quares has been employed. The NLS and ARMA model has been employed to evaluate the least square errors that has been obtained from the above figure. For Sample 1, the value of Least squares according to the ARMA model has been obtained as 14962. The different variables used for testing of the dataset includes hml, cma, rf, rff, rmw and smb.
Figure 12: Estimation command from the excel file in the interface
In this context, the estimation process has been comprehensively performed with respect to the data set in question. The above image showcases the outcomes of the aforementioned process in particular (Liu et al. 2019). Here, the name of the dependent variable is “RFF” which has been considered while performing the least squares method. The four respective variables that have been taken into consideration in this case are “HML”, “CMA”, “MKT_RF”, and “SMB” respectively.
The corresponding parameters like the “coefficient”, “t-statistic”, “prob.”, and “Std. Error” have been garnered along with their values (Dell’Anna et al. 2022). These values are obtained in relation to the previously mentioned four parameters on the whole (M?XON et al. 2021). The results of the primary parameters are obtained by way of performing the software operations in the platform of software known as EVIEWS. The parameters that have been obtained are “r-squared”, “adjusted r-squared”, “long likelihood”, “s.e. Of regression”, “mean dependent var”, “s.d dependent var”, and many others. The values in this regard are “-0.577130”, “-0.583298”, “-764.2770”, “0.653732”, “0.676615”, “0.519539” respectively.
Figure 13: Image view of the Histogram
The above picture in the section-above also displays all of the outcomes obtained by conducting the process of estimation. Here, “HML” is the dependent variable in this context. Similarly, the method named least squares has also been taken up while conducting the aforementioned process in particular. A sum total of “771” observations have been made after taking into account the adjustments. Here, the concerned variables are “CMA”, "RF", “RMW”, “RF”, and “SMB” for that matter (Das et al. 2022). The subsequent values are also obtained in this regard in view of the parameters called “Std. Error”, “Coefficient”, “Prob.”, and “t-Statistics” respectively. Those parameters that are obtained properly in this case are “r-squared”, “s.d dependent var”, “long likelihood”, “mean dependent var”, “adjusted r-squared”, “s.e. Of regression”, and many others. The values rendered are “0.601979”, “5.222688”, “-2012.831”, “0.561012”, “0.599901”, “3.303528” respectively.
Links with results
The research work in this case focussed on the regression analysis with the help of EVIEWS software. The excel format datasets had to be taken into consideration for carrying out data analysis. The emphasis has been on using the variables, as well as parameters that were present in the software. The software analysis has been used for presenting a systematic representation in order to complete the requirements. As per the requirements, the variables have been used for forming graphs, as well as estimation testing. The testing part focussed on the dataanalysis, like, the variables that helped in presenting the outputs for histograms, as well as results.
Figure 14: Equations used for analysis
The above image in Figure 14 has been used for showing how the commands can be presented in EVIEWS software
Figure 15: HML Residuals overview
The above image in Figure 15 has been used for presenting the HML Residuals in EVIEWS software.
Model Comparison
The comparison between both the models such as CAPM and Fama-French 3 factor model is given in the following section:
Fama-French Three factor model
Market Risk Premium: The term called market risk premium is nothing but the underlying difference between the anticipated return of the market and the “risk free” rate (Susilo et al. 2020). This essentially renders the concerned investor with a very excess return in comparison to the extra volatility of the returns. The measured amount stays over as well as above the “risk free” rate in this regard.
High Minus the Low: HML refers to nothing but the “value” premium. It is essentially the spread within the returns between the organisations with a very high “book to market” value ratio along with the organisations with a very low “book to market” ratio. In relation to the “SMB” factor, as the “HML” factor gets determined, the corresponding “beta” coefficient could easily be obtained by way of linear regression (Yue et al. 2020). This “HML” beta coefficient has the possibility of taking negative or positive values. The aforementioned factor provides the crucial insight that the “long term” value stocks aim to enjoy greater return than the growth stocks.
Small Minus the Big: SMB is the size effect predicated upon the factor of market capitalization in respect of an organisation. SMB generally gauges the previous excess of the small-cap organisations over the big-cap organisations. Once the SMB gets defined, the beta coefficient also gets determined through linear regression (Chen et al. 2019). Essentially, the crucial insight obtained in this context is that the small cap organisations aim to witness much higher returns as opposed to the large cap ones.
CAPM Model
Systematic Risk: CAPM generally considers the systematic risk, that is essentially left-out of the other models of return, like the dividend-discount model (Raymond et al. 2020). The term systematic risk is also known by the name market risk. It is a crucial variable by dint of being unforeseen and can not be totally mitigated as it can not be anticipated in fullness.
Elimination of Unsystematic Risk: CAPM fundamentally assumes that the concerned investor holds a much diversified portfolio, very similar to the market portfolio for that matter. A much diversified portfolio helps in removing the unsystematic risk on the whole.
Ease of Usage: CAPM pertains to the simplistic calculator which can be stress-tested for deriving an array of the possible outcomes. These outcomes can render a certain amount of confidence around all the needed rates of the returns.
Investment Appraisal: Any investor can utilise CAPM for the purpose of investment appraisal, in comparison to the other rates. It provides a superior rate of discount in particular (Ren et al. 2020). This particular model lucidly links the systematic risk and the required return respectively.
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