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Testing Fundamental Factor Model comparing Time-series and Cross-Section regression[ Edit ]

As I explained before, both APT and Fama-French Model are factor based models. Factor modeling is a common way to predict future returns based on current fundamental factors or to determine whether an asset is under or overpriced. Here I will focus on two things; the first thing I want to discuss is fundamental factor modeling which is the process of determining an equity’s future value by analyzing a combination of macro and microeconomic events and company-specific characteristics. This differs from a fundamental based analysis which focuses on forecasting overall return(or future stock value) for a particular stock. The objective of the fundamental factor model is to forecast the fluctuation of a portfolio around its expected returns. This is very important for us to know because as securities are added to the portfolio, company-specific returns are diversified away, and the common factor(systematic) portion becomes an increasingly larger part of the portfolio risk and return. Since we are going to use fundamental factor model as a tool to help us build long-short equity strategy, we need to develop our ranking system based on fundamental analysis of the company.

Before we start, I need to introduce two methods here about how to evaluate fundamental factor models. The first way like what I discussed in the last article is the time-series regression approach of Fama and French: we could build matrix where each column is a time-series of economic factor returns. These factors like I mentioned (Three Factor Model) could be SMB, HML, MKT (market premium) generated from portfolio sorts. Then, we perform N time-series regression. In particular, we run the regression of each security returns on these economic factors and estimate the βs.

By doing so, the βs are constant and the factors are time-varying depending on the regression window we choose. One of the obvious advantages here is that estimation of β is diversified away across securities, so it seems to be a superior way for portfolio construction. But the βs are slower to respond to changes that change the risk profile of a firm in this way.

The second way is ‘the barra cross-sectional regression approach’. In terms of this way, we assume that fundamental factor characteristics are betas. Then creating z-scores of the fundamental factor characteristics by generating betas for each time slice for each security. We need to perform T cross-sectional regressions(one per factor). In particular, at each time slice regress the panel of security returns on the normalized betas to estimate the unobservable factor realizations. The advantage here is the betas respond instantaneously to changes in firm characteristics and the disadvantage is that there is a potential errors( coming from model misspecification doesn’t diversify away).

Here I will show the steps about how to implement these two approaches in a practical way.

First, I will use the Fama-French factor analysis method to rank all equities based on their market cap and book value yield respectively. Then with the portfolio I am using here is going to long the top 30% of assets and short the bottom 30%. By doing this step, we can then generate two return series for each factor model.( the step shown on the first two pictures)

Since we have return series representing our factors here, we can now conduct linear regression on any return stream(I just choose the asset(‘ECA’) randomly for testing). From the third graph, I start to compute the factor sensitivities for return on ‘ECA’( the outcomes shown as the number under the code):

Before I go to the next step, I need to clarify that there are two sets of coefficients here: one is based on the historical timeframe from 2014–12–11 to 2015–12–11; the other one is just the ‘ECA’ return stream sensitivities towards the former two factors based on the data of single day which is 2015–12–11.

As usual, I will show the rolling performance of these βs for identifying the movements of these coefficients over time.

Here I will use the second approach which is factor value normalization(‘the barra cross-sectional regression approach’). The formula behind the theory is(we calculate the coefficients

from this formula):

By scaling the value of the factor, we can actually make the coefficients comparable across factors.( Here is an exception for this formula, if the factor is indicator variables means they are set to 1 for True and o for False). After calculating all the coefficients, we can estimate

and

using a cross-sectional regression(i.e. at each time step, we perform a regression using the equations for all the assets).

The following graphs show this procedure, we get the cross-sectional returns on 2015–12–11, and we compute the corresponding coefficients for all assets afterward:

Through the first step, we have generated the normalized factor sensitivities, we should use a linear regression to compute the factor betas on 2015–12–11 which shown on the picture above.

After implementing these two approaches, it is hard to say which way is better. They both have their advantages and disadvantages. We should choose the proper approach based on the traits of the data or your purpose. These two approaches use different ways to process the factor. The time-series regression approach of Fama-French ranks the data based on certain fundamental variables and calculated the factor based on the returns through the long-short equity portfolio that we designed. Once we have the factors we need, we can run N time-series regressions (per security)of any asset that we want to test with the time-series returns(factors) that we generated and find the factor sensitivities for returns of the asset(i) that we choose to test.

The Barra cross-sectional regression approach normalized the fundamental variables which in turn convert the coefficients into a comparable ones across factors. Then we just need to perform T cross-sectional regression per factor on each time slice and find out the factor betas with the certain asset that we chose before.

P.S.

I did more research on the comparison of these two approaches. I would like to share with you about som details here:

First I need to clarify two terms here. The ‘time-series’ approach that I introduced previous can also be called ‘historical betas’; and ‘cross-sectional’ or ‘unobservable’ factors can be called ‘fundamental betas’.

Based on the outcome of the tests that someone did, the ‘fundamental betas’ seem to be better.

The conclusion is summarized from the simulated portfolios over the Jan. 1990 to Feb. 2007 based on the author:

- For long-only investors, fundamental betas appear to be a more consistent measure for hedging portfolio performance.(For investors who care only about ex-post beta, the advantage of using fundamental beta may not be so pronounced, but for those who care about volatility reduction and metrics that capture downside risk and tail risk, the gains would be substantial.
- For a long-short investor who is industry-neutral, market risk is minimal and may not need to be hedged.

Please feel free to comment on inefficiencies about anything I explained here as usual. I am open to the mistakes and willing to learn from them as a beginner in this area.

**Reference:**

- Quantopian: Fundamental Factor Models
- https://www.msci.com/resources/research/articles/2010/The_Fundamentals_of_Fundamental_Factor_Models_Jun2010.pdf
- http://www.mscibarra.com/research/articles/2007/To_Beta_or_Not_to_Beta_July_2007.pdf

*Originally published at* *Haohan — Master’s of Finance and Data Scientist.*

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