Single Factor Test

What is a Factor?

As previously discussed, a factor is a variable that helps explain the returns of an asset. While the concept is intuitive, it is difficult to define precisely. Consider the book-to-market (BM) ratio—often referred to as the value factor. We hypothesize that firms with a high BM ratio tend to outperform those with a low BM ratio. If BM is the only factor in our model, the expected return of asset $i$ can be expressed as:

\[E(r_i) = r_f + \beta_{i} \lambda_{BM}\]

where:

• $E(r_i)$ is the expected return of asset $i$.
• $r_f$ is the risk-free rate.
• $\beta_{i}$ is the sensitivity of the asset’s return to the BM factor.
• $\lambda_{BM}$ is the expected return of the BM factor, which is the return of a portfolio that goes long on high BM ratio firms and short on low BM ratio firms.

However, we face a challenge: we do not know the factor return $\lambda_{BM}$ (how much the factor explains the return of an asset), nor do we know the factor loading $\beta_{i}$ (how sensitive the asset’s return is to the factor).

This creates a classic chicken-and-egg problem!

To address this, there are two main approaches:

1. Estimate $\lambda_{BM}$ first, then estimate $\beta_{i}$ using regression.
2. Or, estimate $\beta_{i}$ first, then estimate $\lambda_{BM}$ using regression.

Estimating Factor Return First Using the Sorting Technique

This method uses a factor mimicking portfolio to estimate the factor return. The idea is to sort all assets based on the factor, group them into $m$ portfolios, and use the return of a long-short portfolio (long the top group, short the bottom group) as the factor return.

Using the BM ratio as an example, the steps are as follows:

1. Sort all stocks based on their BM ratio (ascending order).
2. Group them into $m$ portfolios:
   • The first portfolio contains the stocks with the lowest BM ratio, the second portfolio contains the stocks with the next lowest BM ratio, and so on.
   • If the BM ratio truly explains asset returns, these portfolios should show increasing returns: the first portfolio has the lowest return, the second has the next lowest, and so forth.
3. Use a long-short portfolio to estimate the factor return:
   • Go long the top group (highest BM ratio) and short the bottom group (lowest BM ratio).
   • The return of this long-short portfolio is the factor return $\lambda_{BM}$.
4. Rebalance the portfolios periodically (e.g., monthly or quarterly) to capture changes in the factor return by repeating steps 1 to 3.

Testing the Factor Return

If the BM ratio is a good factor, it should have the following properties:

1. The $m$ portfolios should exhibit increasing (or decreasing) returns. → Test using ICs.
2. The long-short portfolio should have returns significantly different from zero. → Test using a t-test.

We assume the factor return is i.i.d., so each timestamp provides an independent observation. A t-test can be used to determine if the mean factor return is significantly different from zero.

For the first property, we can calculate the Information Coefficient (IC) at each timestamp. Over multiple timestamps, a t-test can be used to check if the mean IC is significantly different from zero. The ICIR (Information Coefficient Information Ratio), which is the mean IC divided by its standard deviation, can also be calculated. For more details, see the Predictablity Measure - IC, ICIR section.

[!NOTE] In practice, some investors focus only on the profitability of a factor: if the top group has a significantly higher return than the bottom group, the factor is considered valid. In academic research, factors are subject to stricter requirements, which will be discussed later.

Time Series Regression to Estimate Factor Loadings

Once we have the factor return, estimating the factor loading $\beta_{i}$ is straightforward:

• A time series of asset returns $r_i$.
• A time series of factor returns $\lambda_{BM}$.

A simple linear regression can be used to estimate the factor loading $\beta_{i}$:

\[r_i = \alpha + \beta_{i} \lambda_{BM} + \epsilon_i\]

where:

• $r_i$ is the return of asset $i$.
• $\alpha$ is the intercept.
• $\beta_{i}$ is the factor loading of asset $i$.
• $\lambda_{BM}$ is the factor return.
• $\epsilon_i$ is the error term.

The estimated $\beta_{i}$ measures the sensitivity of the asset’s return to the factor return.

If there are multiple factors, we can use the sorting technique to estimate the factor returns for each, then use time series regression to estimate the factor loadings for each asset:

\[E(r_i) - r_f = \alpha_i + \beta_{i1} \lambda_{F1} + \beta_{i2} \lambda_{F2} + ... + \beta_{in} \lambda_{Fn}\]

If the model is valid, we should have:

\[H_0: \alpha_1 = \alpha_2 = ... = \alpha_i = ... = \alpha_n = 0\]

This test checks the validity of the model—rejecting the null hypothesis means the model is not valid.

In practice, it is common to preprocess the data before running the regression. This may include removing outliers, normalizing variables, or adjusting for autocorrelation and heteroscedasticity.

[!NOTE] Black, Jensen, and Scholes used this time-series approach to estimate the unconditional CAPM model, where the market beta is the only factor. They could not reject the null hypothesis that the intercept is zero, indicating the CAPM model was valid in their sample.

Limitations of the Sorting Technique

The sorting technique is straightforward and easy to implement. It was the first method proposed by Fama and MacBeth (1973) to estimate factor returns. However, it has several limitations:

• It only works for factors that can be sorted, such as BM ratio, size, or momentum. It cannot be used for factors that cannot be ranked, such as interest rates, inflation, or GDP growth.
• It considers only one factor at a time. Ideally, a factor-mimicking portfolio should be sensitive only to the target factor and independent of others. The univariate sorting technique cannot guarantee this. (Bivariate or multivariate sorting can help, but these approaches are more complex and have their own limitations.)
• Grouping is intended to reduce noise, but the choice of the number of groups and their sizes is subjective and can significantly affect results.

Using Firm Characteristics as Factor Loadings

The second approach is to estimate the factor loading $\beta_{i}$ first. For some style factors, such as the BM ratio, each firm has its own BM ratio—so we can use the BM ratio directly as the factor loading. Then, at each timestamp, we run a cross-sectional regression to estimate the factor return $\lambda_{BM}$. Suppose we have $n$ companies at timestamp $t$:

\[E(r_{i}) - r_{f} = \alpha + \beta_{i} \lambda_{BM}\]

where:

• $E(r_{i})$ is the expected return of asset $i$ at timestamp $t$.
• $r_{f}$ is the risk-free rate at timestamp $t$.
• $\alpha$ is the intercept at timestamp $t$.
• $\beta_{i}$ is the factor loading, equal to the BM ratio of asset $i$ at timestamp $t$.

By running this cross-sectional regression at each rebalancing period, we obtain a time series of factor returns $\lambda_{BM}$.

Evaluating the Factor Return Using Firm Characteristics

As before, we can use ICs, ICIR, and t-tests to evaluate the factor return $\lambda_{BM}$.

In this approach, ICs are calculated at the stock level (whereas the previous method used group portfolios). For each timestamp, we compute the IC, treat each as an independent observation, and use a t-test to determine whether the mean IC and the mean factor return $\lambda_{BM}$ are significantly different from zero.

Summary

As the number of factors increases, the regression results (factor returns) will change. The goal here is not to find the “true” factor return, but to determine whether a factor (or effect) is valid.

By using either the sorting technique or firm characteristics as factor loadings, we can apply statistical tests to assess the validity of a factor. Once validated, multiple factors can be combined into a factor model, and further statistical tests can be performed to evaluate the model as a whole.

For practical application, here is a typical process for constructing a factor model:

1. Propose an idea: Identify a potential factor based on economic theory or empirical evidence.
2. Single Factor Test: Use the sorting technique or firm characteristics as factor loadings to test whether the factor has predictive power.
3. Add to the existing factor model: If the factor is valid, incorporate it into the existing factor model and test the overall model.