Cross-sectional Regression
Cross-sectional regression is a statistical technique used to analyze the relationship between variables across multiple observations at a specific point in time. In the context of event studies, cross-sectional regression can be employed to examine the relationship between abnormal returns and other factors, such as firm characteristics or event-related variables, for a sample of firms experiencing the same event. For details see, e.g. “Introductory Econometrics for Finance” (2019) (Amazon).
Methodology
To conduct a cross-sectional regression, follow these steps:
Data Collection: Collect data on the abnormal returns and the relevant independent variables for each firm in the sample at the specific point in time or over the event window of interest.
Model Specification: Specify a linear regression model that relates the abnormal returns (dependent variable) to the independent variables of interest. The model can be expressed as:
\[ AR_i = \alpha + \beta_1\cdot X_{1,i} + \beta_2 \cdot X_{2,i} + ... + \beta_n \cdot X_{n,i} + \varepsilon_i, \] where:
\(AR_i\) represents the abnormal return (or cumulative abnormal return) of firm i
\(\alpha\) is the intercept term
\(\beta_i\) represents the regression coefficient for the j-th independent variable
\(X_{j, i}\) represents the value of the j-th independent variable for firm i
\(\varepsilon_i\) is the error term for firm i
Estimation: Estimate the model parameters using ordinary least squares (OLS) or other appropriate estimation techniques. This step will provide the regression coefficients and their standard errors, which can be used to assess the significance of the relationships between the abnormal returns and the independent variables.
Hypothesis Testing: Formulate and test hypotheses about the relationships between the abnormal returns and the independent variables. For example, you might test whether the abnormal returns are significantly related to firm size or the magnitude of the event’s impact on the firm.
Interpretation: Interpret the results of the regression analysis, focusing on the magnitude, direction, and statistical significance of the estimated relationships. This will provide insights into the factors that drive the abnormal returns in response to the event of interest.
Advantages
Insights into Determinants of Abnormal Returns: Cross-sectional regression can help identify the factors that influence abnormal returns, providing valuable insights into the drivers of the event’s impact on security returns.
Flexibility: This method allows for the inclusion of multiple independent variables, enabling researchers to control for various factors that might influence the relationship between the event and the abnormal returns.
Applicability: Cross-sectional regression can be applied to various research settings, making it a versatile tool for event study analysis.
Disadvantages
Assumption of Linearity: Cross-sectional regression assumes a linear relationship between the dependent and independent variables, which might not always hold true.
Model Specification Issues: The accuracy of cross-sectional regression depends on the correct specification of the model, including the selection of appropriate independent variables and functional forms.
Endogeneity and Multicollinearity: The presence of endogeneity or multicollinearity among the independent variables can lead to biased and unreliable estimates.
In conclusion, cross-sectional regression is a valuable statistical tool for analyzing the determinants of abnormal returns in event studies. By identifying the factors that influence the event’s impact on security returns, researchers can gain a deeper understanding of the underlying mechanisms and improve their analysis of the event’s consequences. However, it is essential to consider the method’s limitations and ensure proper model specification and estimation techniques are used.