Compute standard errors for implied α and β in log-model regressions

Develop a method to compute valid standard errors for the implied arithmetic parameters α (risk‑adjusted return) and β (systematic risk) that are derived from the logarithmic return regression model of cybersecurity sector private‑equity returns (following Cochrane’s 2005 framework) applied to Crunchbase data, so that statistical inference on these implied parameters is possible.

Background

The paper estimates a log return model to analyze the financial performance of private firms across cybersecurity sectors using Crunchbase data from 2010–2022. From the estimated log-model parameters, the authors derive implied arithmetic parameters α (excess return over the market risk premium) and β (systematic risk).

While reporting point estimates for α and β, the authors explicitly note that they cannot compute standard errors for these implied parameters, which prevents formal statistical inference on α and β despite providing significance for the underlying log-model parameters.