12.2 Abnormal returns and their exact distribution

Motivation

On 24 Feb 2022 the Nifty fell 4.78%. Was that the invasion, or an unusually bad day that would have happened anyway? This section turns the question into a test: estimate what a normal day looks like, measure how far the event days were from it, and work out exactly how far is too far to be noise.

Reading guide

The setup, the definitions of AR and CAR, the statements of Propositions 12.1–12.2 and the worked example are for every reader; if you have read the event-study primer in M06A · 6A.2 you already have the mechanics. The proofs (folded below) are advanced and can be skipped on a first reading.

Notationas in 12.1

Event time $\tau$ counts trading days, with $\tau=0$ the first session that could react to the news.
The estimation window $\tau\in[T_0+1,T_1]$ has $L_1$ days, here $[-250,-11]$ with $L_1=240$ . The event window $\tau\in[T_1+1,T_2]$ has $L_2$ days, for example $[0,+4]$ with $L_2=5$ .
$R_{i\tau}$ is the return of asset $i$ , $R_{m\tau}$ that of the market proxy. The normal-return model is the market model (12.1), $R_{i\tau}=\alpha_i+\beta_iR_{m\tau}+\varepsilon_{i\tau}$ with $\operatorname{Var}(\varepsilon_{i\tau})=\sigma_i^2$ ; the constant-mean model drops $\beta_i R_{m\tau}$ .
A star marks the event window: $R_i^*$ and $X^*$ are the event-window returns and regressors.

Interactive figure · m12-event-windows

12.2.1Normal returns and abnormal returns#

Estimate (12.1) by ordinary least squares (OLS) on the estimation window. In matrix form, let $X$ be the $L_1\times2$ matrix with rows $(1,R_{m\tau})$ and $R_i$ the $L_1$ -vector of returns:

\hat\theta_i=\begin{pmatrix}\hat\alpha_i\\\hat\beta_i\end{pmatrix}=(X'X)^{-1}X'R_i, \tag{12.2}

\hat\sigma_i^2=\frac{1}{L_1-2}(R_i-X\hat\theta_i)'(R_i-X\hat\theta_i). \tag{12.3}

The abnormal return (AR) on an event-window day is the return minus what the model predicted for that day:

AR_{i\tau}=R_{i\tau}-\hat\alpha_i-\hat\beta_iR_{m\tau}, \qquad \tau\in[T_1+1,T_2]. \tag{12.4}

Stack these as the $L_2$ -vector $AR_i^*=R_i^*-X^*\hat\theta_i$ .

12.2.2Why abnormal returns are correlated#

Key idea

An abnormal return is noisier than the day's own noise $\sigma_i$ , because $\hat\alpha_i$ and $\hat\beta_i$ are estimates. Their error is the same on every day of the event window, so it does not average out: the abnormal returns of different days are positively correlated, and the error grows on days when the market itself moves a lot, which is exactly what geopolitical news does.

Proposition 12.1distribution of abnormal returns

Assume, conditional on market returns, that the $\varepsilon_{i\tau}$ are i.i.d. (independent and identically distributed) $N(0,\sigma_i^2)$ in both windows, and that under the null hypothesis $H_0$ of no event effect, (12.1) holds in the event window too. Then

AR_i^*\mid X,X^*\sim N\!\left(0,\;V_i\right),\qquad V_i=\sigma_i^2\left[I_{L_2}+X^*(X'X)^{-1}X^{*\prime}\right]. \tag{12.5}

Proof

Under $H_0$ , $R_i^*=X^*\theta_i+\varepsilon_i^*$ and $\hat\theta_i=\theta_i+(X'X)^{-1}X'\varepsilon_i$ . So

AR_i^* = \varepsilon_i^* - X^*(X'X)^{-1}X'\varepsilon_i. \tag{12.6}

This is a linear function of jointly normal vectors, so it is normal with mean zero. The two windows do not overlap, so $\varepsilon_i^*$ and $\varepsilon_i$ are independent and

\operatorname{Var}(AR_i^*) = \sigma_i^2I + X^*(X'X)^{-1}X'(\sigma_i^2I)X(X'X)^{-1}X^{*\prime} = \sigma_i^2\left[I+X^*(X'X)^{-1}X^{*\prime}\right].

∎

In (12.5) the first term, $\sigma_i^2 I$ , is the day's own noise. The second term is estimation error: $\hat\alpha,\hat\beta$ are noisy, and that noise is common to every day in the event window. So abnormal returns are positively correlated across days within the window, even with i.i.d. disturbances. For a single day, the diagonal element is

\operatorname{Var}(AR_{i\tau})=\sigma_i^2\left[1+\frac1{L_1}+\frac{(R_{m\tau}-\bar R_m)^2}{\sum_{s}(R_{ms}-\bar R_m)^2}\right], \tag{12.7}

using the standard inverse of the $2\times2$ matrix $X'X$ . A large market move on the event day inflates the variance through the last term. This matters for geopolitical events, which move the market.

12.2.3The cumulative abnormal return and its exact test#

The cumulative abnormal return (CAR) adds the abnormal returns over the window. Let $\gamma$ be the $L_2$ -vector of ones (or of ones on a sub-window). Then

CAR_i=\gamma'AR_i^*,\qquad \operatorname{Var}(CAR_i)=\gamma'V_i\gamma=\sigma_i^2\left[L_2+\gamma'X^*(X'X)^{-1}X^{*\prime}\gamma\right]. \tag{12.8}

For the constant-mean model, $X$ is a column of ones, $X'X=L_1$ , and (12.8) becomes $\sigma_i^2(L_2+L_2^2/L_1)$ .

Proposition 12.2exact $t$ -test

Under the assumptions of Proposition 12.1, the standardised CAR

SCAR_i=\frac{CAR_i}{\sqrt{\hat\sigma_i^2\,\gamma'[I+X^*(X'X)^{-1}X^{*\prime}]\gamma}} \sim t_{L_1-2}. \tag{12.9}

Proof

By (12.8), $Z=CAR_i/\sqrt{\operatorname{Var}(CAR_i)}\sim N(0,1)$ .
$(L_1-2)\hat\sigma_i^2/\sigma_i^2\sim\chi^2_{L_1-2}$ , the usual OLS result: the residual vector is $M\varepsilon_i$ with $M=I-X(X'X)^{-1}X'$ idempotent ( $MM=M$ : it projects onto the residual space) of rank $L_1-2$ .
The residuals $M\varepsilon_i$ are independent of $\hat\theta_i-\theta_i=(X'X)^{-1}X'\varepsilon_i$ , because they are jointly normal with $\operatorname{Cov}=\sigma^2 M X(X'X)^{-1}=0$ . They are also independent of $\varepsilon^*_i$ , which comes from a different window.
Hence $\hat\sigma^2_i$ is independent of $Z$ , and $SCAR_i = Z/\sqrt{\chi^2_{L_1-2}/(L_1-2)}$ is $t_{L_1-2}$ by definition.∎

With $L_1=240$ the $t_{238}$ critical value at 5% two-sided is 1.970, close to the normal 1.96.

12.2.4Worked example: the Nifty on 24 Feb 2022#

Worked example 12.1single event, constant-mean model

On 24 Feb 2022, the day Russia invaded Ukraine, the Nifty 50 fell 4.78% to 16,247.95¹. The Nifty cannot be its own benchmark, so use the constant-mean model. Suppose the estimation window $[-250,-11]$ gives a mean daily return of 0.05% and a residual standard deviation of 1.00%. These two numbers are assumptions; the lab below computes them from data. The returns are the actual Nifty closes².

Day $\tau$	Date	Close	Return	$AR_\tau$ (return − 0.05%)	CAR from day 0
0	24 Feb 2022	16,247.95	−4.78%	−4.83%	−4.83%
+1	25 Feb	16,658.40	+2.53%	+2.48%	−2.35%
+2	28 Feb	16,793.90	+0.81%	+0.76%	−1.59%
+3	2 Mar	16,605.95	−1.12%	−1.17%	−2.76%
+4	3 Mar	16,498.05	−0.65%	−0.70%	−3.46%

Day 0 alone, $[0,0]$ : $\operatorname{Var}(AR_0)=0.01^2(1+1/240)$ , so $\operatorname{sd}=1.002\%$ , $t=-4.83/1.002=-4.82$ and $p\approx3\times10^{-6}$ .
Five days, $[0,+4]$ : $CAR=-3.46\%$ , and by (12.8) $\operatorname{sd}=1.00\%\times\sqrt{5+25/240}=2.26\%$ , so $t=-1.53$ and $p\approx0.13$ .

The one-day effect is unmistakable. The five-day CAR is not significant: the next day's rebound and four days of ordinary noise swamp it. Window length is a power choice. By (12.8), for a 5% one-day shock with 1.5% daily noise, a single-firm test at the $t_{238}$ critical value detects it 91% of the time with a $[0,0]$ window, 48% with $[-1,+1]$ and 16% with $[-5,+5]$ ; the timeline figure above shows each case, and exercise m12_event_study confirms it by simulation.

Practitioner notes

Fat tails (M02) mean the exact $t$ result is only approximate. A 5-sigma day in a GARCH world (volatility that clusters, M02) is less rare than the $t$ tail says. For a single event, compare the AR with the empirical distribution of estimation-window returns, a bootstrap or rank test, as well as the $t$ .
Volatility rises around geopolitical events. Using estimation-window $\sigma$ then overstates significance after the event; see the Boehmer–Musumeci–Poulsen (BMP) test in 12.3.

The lab downloads daily closes for the Nifty 50, Brent and gold futures and USDINR, sets day 0 to the first session of each market on or after the event date, estimates a constant-mean model on the 240 returns in $[-250, -11]$ , and plots cumulative abnormal returns over $[-5, +20]$ . The band is $\pm1.96$ times the standard deviation of the CAR under no event effect, $\hat\sigma_i\sqrt{L + L^2/L_1}$ , the constant-mean case of (12.8). The table reports $AR_0$ with the $t$ -ratio of (12.7), and the CARs over $[0, +4]$ and $[-5, +20]$ with their $t$ -ratios (12.9). Switch events to compare 24 Feb 2022 with 7 Apr 2025 and 7 May 2025.

What to look at Worked example 12.1 used assumed estimation-window numbers; here they come from the data. Check whether the day-0 abnormal return is significant while the 26-day CAR is not, and whether the CARs leave the band before day 0 (leakage or anticipation, 12.3).