28.2 Testing
28.2.1 Parametric Test
(S. J. Brown and Warner 1985) provide evidence that even in the presence of non-normality, the parametric tests still perform well. Since the proportion of positive and negative abnormal returns tends to be equal in the sample (of at least 5 securities). The excess returns will coverage to normality as the sample size increases. Hence, parametric test is advocated than non-parametric one.
Low power to detect significance (Kothari and Warner 1997)
- Power = f(sample, size, the actual size of abnormal returns, the variance of abnormal returns across firms)
28.2.1.1 T-test
Applying CLT
\[ \begin{aligned} t_{CAR} &= \frac{\bar{CAR_{it}}}{\sigma (CAR_{it})/\sqrt{n}} \\ t_{BHAR} &= \frac{\bar{BHAR_{it}}}{\sigma (BHAR_{it})/\sqrt{n}} \end{aligned} \]
Assume
Abnormal returns are normally distributed
Var(abnormal returns) are equal across firms
No cross-correlation in abnormal returns.
Hence, it will be misspecified if you suspected
Heteroskedasticity
Cross-sectional dependence
Technically, abnormal returns could follow non-normal distribution (but because of the design of abnormal returns calculation, it typically forces the distribution to be normal)
To address these concerns, Patell Standardized Residual (PSR) can sometimes help.
28.2.1.2 Patell Standardized Residual (PSR)
- Since market model uses observations outside the event window, abnormal returns contain prediction errors on top of the true residuals , and should be standardized:
\[ AR_{it} = \frac{\hat{u}_{it}}{s_i \sqrt{C_{it}}} \]
where
\(\hat{u}_{it}\) = estimated residual
\(s_i\) = standard deviation estimate of residuals (from the estimation period)
\(C_{it}\) = a correction to account for the prediction’s increased variation outside of the estimation period (Strong 1992)
\[ C_{it} = 1 + \frac{1}{T} + \frac{(R_{mt} - \bar{R}_m)^2}{\sum_t (R_{mt} - \bar{R}_m)^2} \]
where
\(T\) = number of observations (from estimation period)
\(R_{mt}\) = average rate of return of all stocks trading the the stock market at time \(t\)
\(\bar{R}_m = \frac{1}{T} \sum_{t=1}^T R_{mt}\)