## 28.2 Testing

### 28.2.1 Parametric Test

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

• 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

$C_{it} = 1 + \frac{1}{T} + \frac{(R_{mt} - \bar{R}_m)^2}{\sum_t (R_{mt} - \bar{R}_m)^2}$

where

• $$T$$ = number of observations (from estimation period)

• $$R_{mt}$$ = average rate of return of all stocks trading the the stock market at time $$t$$

• $$\bar{R}_m = \frac{1}{T} \sum_{t=1}^T R_{mt}$$

### 28.2.2 Non-parametric Test

• No assumptions about return distribution

• Sign Test (assumes symmetry in returns)

• binom.test()
• Wilcoxon Signed-Rank Test (allows for non-symmetry in returns)

• Use wilcox.test(sample)
• Gen Sign Test