Sample Size Matters: A Comparative Analysis of the I.I.D. Assumption and Risk-Return Estimations in Latin American and US Stock Markets

This study examines the impact of sample size on the independent and identically distributed (i.i.d.) assumption in financial time series and its subsequent effect on risk-return estimations. Focusing on Latin American and US stock market indices from August 2007 to December 2024, using 4,477 daily log-returns, our methodology employs moment estimations, i.i.d. tests (Ljung-Box and Augmented Dickey-Fuller), and a modified Capital Asset Pricing Model (CAPM) and Download CAPM across four rolling windows (60, 250, 500, and 1,000 days). Our findings show that financial returns consistently exhibit heavy tails and negative skewness, challenging the assumption of normality. Statistical properties and i.i.d. violations vary significantly with sample size, with tests becoming more stringent for larger samples. CAPM alpha and beta coefficients are also sample size dependent, revealing distinct risk-return profiles: the S&P 500 exhibits higher systematic risk (Beta > 1) with performance explained by the model (alpha ≈ 0), while Latin American markets are more defensive (Beta < 1) with more dispersed alphas, indicating greater influence from local factors. These results show the importance of considering sample size for robust financial modeling and informed investment decision-making, particularly in emerging markets, highlighting that no single model is universally applicable to all stock markets.