First pass regression

What Is First-Pass Regression?

First-pass regression is a statistical technique used in quantitative finance to estimate initial parameters or coefficients for assets, typically as a preliminary step in more complex multi-stage analytical frameworks like the Fama-MacBeth procedure. It involves running a separate time series regression analysis for each individual security or portfolio against one or more explanatory factors, such as a market index. The primary output of a first-pass regression is often the estimation of an asset's beta, which measures its sensitivity to market movements. These estimated parameters are then carried forward to a subsequent stage of analysis, often a cross-sectional regression, to test theoretical relationships or price risk factors across a universe of assets.

History and Origin

The concept of regression analysis dates back to the 19th century, gaining significant traction in economics and finance with the advent of desktop computing in the 20th century.³⁶ Early applications in finance, particularly in the realm of asset pricing models, paved the way for methodologies that required sequential statistical steps.

One of the most notable historical contexts for the application of first-pass regression is in the empirical testing of the Capital Asset Pricing Model (CAPM). Pioneering work by researchers like Black, Jensen, and Scholes (1972) and later, Fama and MacBeth (1973), introduced and popularized a two-stage regression approach to test the CAPM's implications.³⁴, ³⁵ The first stage of this procedure, known as the first-pass regression, involved estimating the beta of individual securities or portfolios by regressing their historical returns against the market's returns. These estimated betas then became the input for the second stage, a cross-sectional regression, designed to examine whether average asset returns were linearly related to their estimated betas, as predicted by the CAPM.³², ³³ This two-stage methodology became a standard in empirical finance for testing financial models and estimating risk premium.³¹

Key Takeaways

• First-pass regression is the initial stage in a multi-step regression framework, commonly used in quantitative finance.
• Its main purpose is to estimate asset-specific parameters, such as beta coefficients, from time-series data.
• The parameters derived from a first-pass regression serve as inputs for subsequent analytical steps, often a cross-sectional regression.
• This technique is fundamental to methodologies like the Fama-MacBeth procedure for testing asset pricing models.
• Proper interpretation requires awareness of potential limitations, including measurement error in the estimated parameters.

Formula and Calculation

In the context of asset pricing models like the CAPM, a common setup for a first-pass regression involves regressing the excess returns of an individual asset or portfolio against the excess returns of the market portfolio. The formula for a simple first-pass regression (often called the Security Characteristic Line, or SCL) is:

$R_{i,t} - R_{f,t} = \alpha_i + \beta_i (R_{m,t} - R_{f,t}) + \epsilon_{i,t}$

Where:

• ( R_{i,t} ) = Return of asset ( i ) at time ( t )
• ( R_{f,t} ) = Risk-free rate at time ( t )
• ( R_{m,t} ) = Return of the market portfolio at time ( t )
• ( \alpha_i ) (alpha) = Intercept term, representing the asset's excess return independent of the market.
• ( \beta_i ) (beta) = Coefficient measuring the sensitivity of asset ( i )'s excess return to the market's excess return.
• ( (R_{m,t} - R_{f,t}) ) = Market risk premium at time ( t )
• ( \epsilon_{i,t} ) = The residual or error term, representing the idiosyncratic risk not explained by the market factor.

This ordinary least squares (OLS) regression is run separately for each asset or portfolio ( i ) using historical time-series data. The estimated (\hat{\beta_i}) values from these individual regressions are then used in the subsequent stage of analysis.

Interpreting the First-Pass Regression

Interpreting the results of a first-pass regression primarily focuses on the estimated coefficients, particularly beta ((\beta)). The beta coefficient indicates the sensitivity of an asset's return to changes in the market's return. A beta of 1 suggests the asset's price moves in line with the market. A beta greater than 1 implies higher volatility than the market, while a beta less than 1 suggests lower volatility.

The alpha ((\alpha)) coefficient from a first-pass regression, often referred to as Jensen's alpha, represents the asset's average return that is not explained by its exposure to the market factor. In theory, according to the CAPM, alpha should be statistically insignificant from zero if the model perfectly explains returns. A statistically significant positive alpha could suggest the asset has outperformed expectations based on its market risk, while a negative alpha suggests underperformance. However, the interpretation of alpha needs careful consideration, especially regarding the choice of the market proxy and the potential for model specification issues.

The residuals ((\epsilon)) from the first-pass regression are also critical. These represent the portion of the asset's return not explained by the independent variables included in the model. Analyzing residuals can reveal patterns or systematic factors that the current model does not capture, indicating areas for further investigation or potentially suggesting the presence of omitted variable bias.²⁹, ³⁰

Hypothetical Example

Imagine an analyst wants to estimate the beta for TechCorp stock using historical monthly returns over the past five years. They also gather data for a broad market index (e.g., S&P 500) and the risk-free rate (e.g., U.S. Treasury bill yield) for the same period.

Step 1: Calculate Excess Returns
For each month, the analyst calculates the excess return for TechCorp stock and the market index by subtracting the risk-free rate.

• TechCorp Excess Return ((R_{TechCorp,t} - R_{f,t}))
• Market Excess Return ((R_{m,t} - R_{f,t}))

Step 2: Run the First-Pass Regression
Using a statistical software package, the analyst performs an ordinary least squares regression. The dependent variable is TechCorp's excess return, and the independent variable is the market's excess return.

Let's assume the regression output for TechCorp is:
$R_{TechCorp,t} - R_{f,t} = 0.002 + 1.25 (R_{m,t} - R_{f,t}) + \epsilon_{TechCorp,t}$

Step 3: Interpret the Results
From this first-pass regression, the estimated beta ((\hat{\beta}_{TechCorp})) is 1.25. This suggests that TechCorp stock is more volatile than the overall market; for every 1% change in the market's excess return, TechCorp's excess return is expected to change by 1.25%. The alpha (0.002, or 0.2%) suggests that, on average, TechCorp outperformed what would be expected given its market exposure by 0.2% per month. These estimated values, particularly the beta, could then be used in a subsequent analysis, such as a portfolio management context, to assess TechCorp's risk contribution or to test its pricing against other factors.

Practical Applications

First-pass regression is a foundational technique with various applications in finance and economics, primarily within the field of quantitative finance.

• Asset Pricing Models: It is most commonly employed as the initial step in testing multi-factor models, such as the Capital Asset Pricing Model (CAPM) or the Fama-French Three-Factor Model. In these applications, first-pass regression estimates the sensitivity (beta) of individual securities or portfolios to various risk factors (e.g., market risk, size, value). These estimated betas are then used in a second stage to determine if these factors are "priced" in the cross-section of expected returns.²⁷, ²⁸
• Portfolio Analysis: Investors and portfolio managers use first-pass regression to understand the risk characteristics of individual assets or portfolios. By estimating betas, they can assess how sensitive their investments are to broad market movements, aiding in portfolio management and diversification strategies.²⁶
• Performance Attribution: The alpha generated from a first-pass regression can be a component of performance attribution, helping to determine if a portfolio manager's returns are simply due to market exposure or if they are generating excess returns through security selection or other skills.
• Risk Management: Understanding an asset's beta derived from a first-pass regression is crucial for assessing its systematic risk, which is the non-diversifiable risk inherent in the market. This insight informs risk budgeting and hedging strategies.²⁵
• Valuation: While not directly a valuation model, the parameters (like beta) estimated from a first-pass regression are essential inputs for asset valuation techniques that incorporate market risk, such as the cost of equity calculation in corporate finance.

The two-stage regression approach, of which the first-pass regression is the initial step, has become a "staple in applied finance" due to its simplicity and adaptability for incorporating additional risk measures.²³, ²⁴

Limitations and Criticisms

Despite its widespread use, first-pass regression, particularly as part of a multi-stage methodology, carries several limitations and criticisms:

• Measurement Error in Betas: One of the most significant concerns is that the betas estimated in the first-pass regression are themselves estimates and thus contain measurement error. If these estimated betas are then used as independent variables in a second-stage regression, this measurement error can lead to biased and inconsistent results in the second stage, affecting the reliability of conclusions about risk premium and factor pricing.²⁰, ²¹, ²² While methods exist to adjust standard errors for this, it remains a challenge.¹⁹
• Omitted Variable Bias: If the first-pass regression model does not include all relevant explanatory variables that influence the dependent variable (e.g., asset returns), the estimated coefficients, including beta, can be biased.¹⁷, ¹⁸ This occurs when an omitted variable is correlated with both the dependent variable and an included independent variable, leading the included variable's coefficient to absorb some of the omitted variable's effect.¹⁶
• Model Specification: The choice of factors in the first-pass regression is crucial. If the chosen factors do not adequately capture the true systematic risks, the estimated betas and alphas may not accurately reflect the asset's true characteristics. For instance, early tests of the CAPM using simple first-pass regressions often found significant alphas, suggesting the market factor alone might not fully explain returns.¹⁵
• Assumptions of OLS: First-pass regressions typically rely on Ordinary Least Squares (OLS) assumptions (e.g., linearity, homoscedasticity, no autocorrelation in residuals). Violations of these assumptions can lead to inefficient or biased estimates.¹⁴
• Data Frequency and Period: The choice of data frequency (e.g., daily, monthly) and the specific time period used for the first-pass regression can influence the stability and accuracy of the estimated parameters. Betas can vary over time, and a static estimate from a single historical period might not be representative of future risk.

Researchers continually refine methodologies to address these challenges, for example, by using rolling regressions or more advanced econometric techniques to account for time-varying betas and measurement error.¹², ¹³

First-Pass Regression vs. Two-Pass Regression

The term "first-pass regression" inherently implies that it is part of a multi-stage process, most notably the "two-pass regression" methodology. The distinction lies in their roles within the analytical framework.

First-Pass Regression is the initial stage, focusing on estimating asset-specific sensitivities to predetermined risk factors using time series data. For example, in testing the CAPM, a first-pass regression calculates an individual stock's beta by regressing its historical returns against the market's historical returns. The outcome of this stage is a set of estimated betas (one for each asset).¹⁰, ¹¹

Two-Pass Regression refers to the entire two-stage process. The first stage is the first-pass regression, which yields the asset-specific parameters. The second stage, often called the cross-sectional regression or second-pass regression, then uses these estimated parameters from the first pass as explanatory variables to explain the average returns across a group of assets.⁹ This second stage aims to determine if the estimated risk exposures (from the first pass) are indeed priced in the market, meaning whether investors are compensated for bearing those risks. The Fama-MacBeth procedure is a prominent example of a two-pass regression approach, where monthly cross-sectional regressions are performed using the betas estimated in the first pass, and the resulting coefficients are then averaged over time.⁸ The key difference is that first-pass regression is a component, while two-pass regression describes the complete methodological framework.

FAQs

What is the primary goal of a first-pass regression in finance?

The primary goal of a first-pass regression in finance is to estimate the exposure of individual assets or portfolios to various risk factors, most commonly their beta to the market. These estimated exposures then serve as inputs for a subsequent, second-stage analysis, often a cross-sectional regression, to test financial theories or determine risk premia.⁶, ⁷

How does first-pass regression relate to the Capital Asset Pricing Model (CAPM)?

First-pass regression is a crucial step in empirically testing the Capital Asset Pricing Model (CAPM). It is used to estimate the beta for individual securities or portfolios by regressing their returns against market returns. This beta is then used in the second stage of the CAPM's empirical test to see if higher beta (higher market risk) is associated with higher expected returns.³, ⁴, ⁵

What are residuals in the context of first-pass regression?

In a first-pass regression, residuals represent the portion of an asset's actual return that is not explained by the explanatory factors included in the regression model. They signify the unpredictable, idiosyncratic component of the asset's return that is unique to that asset and not systematically related to the factors.¹, ² Analyzing residuals can help in evaluating the model specification and identifying potential issues.