slide.Rmd

---
title: "The Impact of Quarterly Earnings Announcement on Stock Daily Returns"
subtitle: "based on Fama-French three-factor model"
author: "Hongyu Hu"
institute: ""
date: "`r Sys.Date()`"
output:
  beamer_presentation:
    theme: "Singapore"
    fonttheme: "structurebold"
    colortheme: "beaver"
    incremental: no
    keep_tex: yes
    slide_level: 2
fontsize: 11pt

header-includes:
- \usepackage{placeins}
- \usepackage[flushleft]{threeparttable}
---


# Motivation

## Background

<!-- According to the efficient market theory,  -->
<!-- the stock market converts all material public information -->
<!-- into daily returns through trading. -->


- There are numbers of retail investors in Chinese A-share markets 
selling or buying stocks motivated by public infromation and financial report.

- Routine quarterly earnings reports issued by listed firms are highly anticipated 
and can cause investors to bid up the stock's price or else pummel it down.


## Event date, $\tau$

We use $\tau$ as the time index rather than $t$
due to the earnings revealed date among firms are different.

```{r echo=FALSE, message=FALSE, warning=FALSE}
library(tidyverse)
library(magrittr)
library(lubridate)
datdir <- '~/NutSync/MyData/QEAData/'
ReptInfo <- dir(datdir, pattern = 'Rept.csv$') %>%
            paste0(datdir, .) %>%
            read_delim(delim='\t', na = '',
                       col_types = cols(Stkcd = col_character(),
                                        Accper = col_date(format = "%Y-%m-%d"),
                                        Annodt = col_date(format = "%Y-%m-%d"),
                                        Annowk = col_integer())) %>%
            subset(select=c(Stkcd,Sctcd,Annodt,Accper))  %>% 
            arrange(Stkcd) 

EDSZ1st <- ReptInfo %>% subset((Accper == ymd('2017-03-31')) & (Sctcd == 2), select=c('Annodt'))
EDSH1st <- ReptInfo %>% subset((Accper == ymd('2017-03-31')) & (Sctcd == 1), select=c('Annodt'))
EDSZ3rd <- ReptInfo %>% subset((Accper == ymd('2017-09-30')) & (Sctcd == 2), select=c('Annodt'))
EDSH3rd <- ReptInfo %>% subset((Accper == ymd('2017-09-30')) & (Sctcd == 1), select=c('Annodt'))

ED <- cbind(EDSH1st[sample(1:nrow(EDSH1st), 10),],
            EDSZ1st[sample(1:nrow(EDSZ1st), 10),],
            EDSH3rd[sample(1:nrow(EDSH3rd), 10),],
            EDSZ3rd[sample(1:nrow(EDSZ3rd), 10),]) %>% 
  `names<-`(c('SH.1st','SZ.1st', 'SH.3rd', 'SZ.3rd'))

print(ED, row.names=FALSE)
```



## Path of returns when earnings report are released

```{r echo=FALSE, message=FALSE, warning=FALSE}
library(ggplot2)
datdir <- '~/NutSync/MyData/QEAData/CH3/'
filecd <- dir(datdir, pattern = 'DretMean[.]csv$') %>% paste0(datdir, .)
DR <- matrix(-40:40, ncol = 1)
for (i in filecd) {
    DRS <- read_delim(i, delim='\t', na = '') 
    DR <- cbind(DR,DRS)
}

DR %<>% `names<-`(c('Timeline',
                    str_sub(dir(datdir, pattern = 'DretMean[.]csv$'), start=1L, end=10L))) %>% `[`(31:51,)
  

multiplot <- function(..., plotlist = NULL, file, cols = 1, layout = NULL) {
    library(grid)
    
    # Make a list from the ... arguments and plotlist
    plots <- c(list(...), plotlist)
    
    numPlots = length(plots)
    
    # If layout is NULL, then use 'cols' to determine layout
    if (is.null(layout)) {
        # Make the panel ncol: Number of columns of plots nrow: Number of rows
        # needed, calculated from # of cols
        layout <- matrix(seq(1, cols * ceiling(numPlots/cols)), ncol = cols, 
            nrow = ceiling(numPlots/cols))
    }
    
    if (numPlots == 1) {
        print(plots[[1]])
        
    } else {
        # Set up the page
        grid.newpage()
        pushViewport(viewport(layout = grid.layout(nrow(layout), ncol(layout))))
        
        # Make each plot, in the correct location
        for (i in 1:numPlots) {
            # Get the i,j matrix positions of the regions that contain this subplot
            matchidx <- as.data.frame(which(layout == i, arr.ind = TRUE))
            
            print(plots[[i]], vp = viewport(layout.pos.row = matchidx$row, layout.pos.col = matchidx$col))
        }
    }
}

# for (i in 1:length(filecd)) {
#   assign(paste0('P', i),
#          ggplot(DR[, c(1L, 1L+i)], aes(x=Timeline, y=DR[, 1L+i])) +
#            geom_line() + geom_point() +
#            labs(x = "Time line", y = 'Daily return', title = names(DR)[1L+i]))
# }

P1 <- ggplot(DR[, c(1,2)], aes(x=Timeline, y=DR[, 2])) +
           geom_line() + geom_point() + ylim(-0.02, +0.02) +
           labs(x = "Time line", y = 'Daily return', title = names(DR)[2])

P2 <- ggplot(DR[, c(1,3)], aes(x=Timeline, y=DR[, 3])) +
           geom_line() + geom_point() + ylim(-0.02, +0.02) +
           labs(x = "Time line", y = 'Daily return', title = names(DR)[3])

P3 <- ggplot(DR[, c(1,4)], aes(x=Timeline, y=DR[, 4])) +
           geom_line() + geom_point() + ylim(-0.02, +0.02) +
           labs(x = "Time line", y = 'Daily return', title = names(DR)[4])

P4 <- ggplot(DR[, c(1,5)], aes(x=Timeline, y=DR[, 5])) +
           geom_line() + geom_point() + ylim(-0.02, +0.02) +
           labs(x = "Time line", y = 'Daily return', title = names(DR)[5])

multiplot(P1, P2, P3, P4, cols = 2)

```


## Methodology

- The purpose of this paper is to model and explain the **abnormal returns** attributable to quarterly earnings announcement underlying framework of **event study**.

- We define abnormal return $AR$ as, \[AR_{i\tau} = R_{i\tau} - E(R_{i\tau}),\]
where $R_{i\tau}$ equals actual stock return $R_{i\tau}$ for firm $i$ and event date $\tau$ 
and $E(R_{i\tau})$ is the expected return $E(R_{i\tau})$ predicted by a model.


## Methodology

- $\tau = 0$ represents that the quarterly earnings report of a stock
was released at data $\tau$.

- The estimation window is used for **estimating the parameters** of market model,
this allows $AR_{i\tau}$ to be calculated within the event window.

![Timeline of an event study with significant dates](figure/timeline.png)


## Hypothesis

\begin{block}{Original null hypothesis}
Special event such as quarterly financial announcement
have no effects on the mean level of security returns,
$E({\it AR}_{i\tau}) = 0$.
\end{block}

- Does the publicly issued financial report having the potential of 
influencing the demand for various stocks and subsequently affecting their prices?

- Has the stock market converted all material public information
into daily returns through trading?


## Literature

*The adjustment of stock prices to new information.*  
Fama, E. F., Fisher, L., Jensen, M. C., & Roll, R. (1969).  

*Using daily stock returns: The case of event studies.*  
Brown, S. J., & Warner, J. B. (1985).  

*Security returns around earnings announcements.*  
Ball, R., & Kothari, S. P. (1991).  

*Has the information content of quarterly earnings announcements declined in the past three decades?*  
Landsman, W. R., & Maydew, E. L. (2002).  

*Earnings announcement promotions: A Yahoo Finance field experiment. *
Lawrence, A., Ryans, J., Sun, E., & Laptev, N. (2018). 


# Modeling the daily returns


## Data and samples

The data we use, which include data on returns,
trading, financial statements, and quarterly financial reports,
are form Wind information Inc. (WIND) and 
China Stock Market & Accounting Research (CSMAR) database.
To enable reasonable precision and power,
we eliminate stocks which

- estimate window existed non-trading status

- listed less than six months

- having less than 180 trading records in the past year
or less than 15 trading records in the past months.


## Measuring the expected returns, ${\it ER}_{i\tau}$

<!-- The starting point is to calculate the **actual daily returns**, -->
<!-- $R_{i\tau}-R_{f\tau}$, -->
<!-- of stock $i$ at date $\tau$ where -->
<!-- \( -->
<!-- R_{i\tau} = \mbox{log} (P_{i\tau} / P_{i\tau-1}) -->
<!-- \) -->
<!-- and $R_{f\tau}$ is the risk-free rate. -->

We choose the Fama-French 3-Factor model
to estimate the expected daily returns $E(R_{i\tau})$,
\begin{equation}
R_{i\tau}-R_{f\tau} = \alpha_i + \beta_{i}(R_{m\tau} - R_{f\tau}) +
s_{i}\text{SMB}_\tau + v_{i}\text{VMG}_\tau +
\varepsilon_{i \tau}.
\end{equation}
Note that this model is modified according to characteristics of Chinese A-share markets,
as Liu et al. (2019) recommended.


## A three factor model in China

The Fama-French 3-Factor model states that 
the actual daily return, $R_{i\tau}-R_{f\tau}$, 
could be explained by three mimicking risk factors,

- $R_{m\tau} - R_{f\tau}$, Market risk factor, 
the return on the value-weight market portfolio minus the risk-free return.

- $\text{SMB}_\tau$, Size factor, the return on a diversified portfolio of small stocks
minus the return on a diversified portfolio of big stocks.

- $\text{VMG}_\tau$, Value factor, the difference between the returns on 
diversified portfolios of high and low ratio of earnings-to-price stocks.



## Intercepts of portfolios, $\alpha$

![](figure/2017-09-30_6_21_wekbind_FF3_table_intercept.pdf)


## The Distribution of OLS Estimator

```{r echo=FALSE, message=FALSE, warning=FALSE}
datdir <- '~/NutSync/MyData/QEAData/CH3/'
# Specifying trem information
Accprd <- as.Date('2017-09-30')
Pretype <- '6'
Markettype <- '21'
weekterm <- 'wekbind'
modeltype <- 'FF3'


## Obtain the plots of OLS estimator used above QEA-FF data ===========================

require(grid)
library(latex2exp)

cdcoef <- paste(Accprd, Pretype, Markettype, weekterm, modeltype, 
                'OLScoef', 'CH3', 'SMB', 'VMG', '3',sep='_') %>%
          paste0(datdir, ., '.csv')
OLScoef <- as.data.frame(read_delim(cdcoef, delim=',', na = ''))
OLScoef[, 3:ncol(OLScoef)] <- round(OLScoef[, 3:ncol(OLScoef)], 6)


grid.newpage()
vplayout <- function(x,y){viewport(layout.pos.row = x, layout.pos.col = y)}
if(ncol(OLScoef)==5) {
    {pushViewport(viewport(layout = grid.layout(2,2)))
        print(qplot(OLScoef$MKT, xlab=TeX('$\\beta_1$ of MKT'),
                    ylab = "Count",bins=100), vp = vplayout(1,1:2))
        print(qplot(OLScoef$SMB, xlab=TeX('$\\beta_2$ of SMB'),
                    ylab = "Count",bins=50), vp = vplayout(2,1))       
        print(qplot(OLScoef$VMG, xlab=TeX('$\\beta_3$ of VMG'),
                    ylab = "Count",bins=50), vp = vplayout(2,2))}
} else if (ncol(OLScoef)==7) {
    {pushViewport(viewport(layout = grid.layout(2,4)))
        print(qplot(OLScoef$MKT, xlab=TeX('$\\beta_1$ of MKT'),
                    ylab = "Count",bins=100), vp = vplayout(1,1:4))
        print(qplot(OLScoef$SMB, xlab=TeX('$\\beta_2$ of SMB'),
                    ylab = "Count",bins=50), vp = vplayout(2,1))       
        print(qplot(OLScoef$HML, xlab=TeX('$\\beta_3$ of VMG'),
                    ylab = "Count",bins=50), vp = vplayout(2,2))
        print(qplot(OLScoef$RMW, xlab=TeX('$\\beta_4$ of RMW'),
                    ylab = "Count",bins=50), vp = vplayout(2,3))
        print(qplot(OLScoef$CMA, xlab=TeX('$\\beta_5$ of CMA'),
                    ylab = "Count",bins=50), vp = vplayout(2,4))}
} else {
    pushViewport(viewport(layout = grid.layout(1,1)))
    print(qplot(OLScoef$MKT, xlab=TeX('$\\beta_1$ of MKT'),
                ylab = "Count",bins=100), vp = vplayout(1,1))
}


```


## Classificaation by penalized least squares

We apply the penalized least squares from Su et al. (2016) 
to **cluster** the stocks in Chinese A-share markets,
\begin{equation}
\min_{\boldsymbol \beta} \left[ \|\mathbf{y}-\mathbf{X} \boldsymbol{\beta}\|^{2} +
\frac{\rho}{N} \sum_{i=1}^N \textstyle\prod_{k=1}^{K}
\parallel \! \beta_{i} -  \alpha_{k} \! \parallel \right].
\end{equation}
The penalty term in the multiplicative expression
\textcolor{red}{shrinks the individual-level parameter vector $\beta_{i}$
to a partical unknown group-level parameter vector $\alpha_{k}$}.
Note that we push the punishment only on parameters of factor $SMB$ and $VMG$.


##

\begin{table}[!htbp] 
\centering

\caption{Penalized least squares estimation results}

\resizebox{0.9\textwidth}{!}{
\begin{threeparttable}

\begin{tabular}{crrrr}
\hline \hline 
\textbf{Slope coefficients}   &   Unclassified   &   Group A    &    Group B    &   Group C  \\
\hline
\\
MKT     &     1.045***      &   1.170***       &      0.828***     &     0.789***    \\
        &     (200.601)     &   (174.600)      &      (99.850)     &     (56.674)    \\
SMB     &     0.728***      &   0.925***       &      0.556***     &     -0.235***   \\
        &     (65.533)      &   (61.988)       &      (34.280)     &     (-8.817)    \\
VMG     &     0.409***      &   0.546***       &      0.242***     &     -0.096***   \\
        &     (45.765)      &   (47.233)       &      (16.862)     &     (-4.176)    \\
obs     &     1636          &   1051           &      451          &     134         \\
\hline
\end{tabular}

\begin{tablenotes}
  \item Note:
  \item[a] The data running in regression
  $ R_{i\tau}-R_{f\tau} = \beta_{i}(R_{m\tau} - R_{f\tau}) + s_{i}\text{SMB}_\tau + v_{i}\text{VMG}_\tau + \varepsilon_{i \tau}$ has been zero-centered;
  \item[b] *** 1\% significant; ** 5\% significant; * 10\% significant.
\end{tablenotes}

\end{threeparttable}
}

\end{table}


<!-- ```{r eval=FALSE, warning=FALSE, include=FALSE, results='asis'} -->
<!-- options(digits=3) -->
<!-- library(tidyverse) -->
<!-- library(magrittr) -->
<!-- PLS <- paste(datdir, "2017-09-30_6_21_wekbind_FF3_PLScoef_CH3_SMB_VMG_3.csv", sep = '') %>% -->
<!--  read.csv(header = T, as.is = TRUE) %>% as.data.frame() -->
<!-- PLS %<>% `[`(,c(1,3,4,6,7,9)) -->
<!-- table <- matrix(0, nrow = 6, ncol = 3) -->
<!-- table[c(1,3,5), 1] <- PLS[, 1] -->
<!-- table[c(1,3,5), 2] <- PLS[, 3] -->
<!-- table[c(1,3,5), 3] <- PLS[, 5] -->
<!-- table[c(2,4,6), 1] <- PLS[, 2] -->
<!-- table[c(2,4,6), 2] <- PLS[, 4] -->
<!-- table[c(2,4,6), 3] <- PLS[, 6] -->
<!--  -->
<!-- library(knitr) -->
<!-- library(kableExtra) -->
<!--  -->
<!-- kable(as.data.frame(table), "latex", caption = "title") %>%  -->
<!-- kable_styling(latex_options = c("striped"), full_width = F) %>% cat() -->
<!-- ``` -->

##

### Abnormal return, ${\it AR}_{i\tau}$

After we obtain the parameters from estimation window, 
for firm $i$ and event date $\tau$, 
we can decompose returns in event window as
\[
R_{i\tau}^{*} =  E(R_{i\tau}^{*} | X_\tau) + AR_{i\tau}^{*}.
\]

### Cumulative abnormal return, ${\it CAR}_\tau$

Define ${\it CAR}_i(\tau_1,\tau_2)$ as the sample sum of included ${\it AR}_{i\tau}$ from $\tau_1$ to $\tau_2$, we have 
\(
{\it CAR}_i(\tau_1, \tau_2) = \sum_{\tau=\tau_1}^{\tau_2} {\widehat{AR}_{i\tau}}.
\)
Given $N$ securities, the sample average $AR_\tau$ is
\(
\overline{AR}_\tau = \frac{1}{N} \sum_{i=1}^N \widehat{AR}_{i\tau},
\)
so the average cumulative abnormal return is
\(
\overline{CAR}_i(\tau_1, \tau_2) = \sum_{\tau=\tau_1}^{\tau_2} {\overline{AR}_{\tau}}.
\)


## Path of group-CAR


```{r echo=FALSE, message=FALSE, warning=FALSE}
library(ggplot2)
timeline <- c(-20:+40) 
windlen <- length(timeline)

## Read the result from MATLAB ========
library(R.matlab)
PLSpath <- "~/NutSync/QEA/Matlab_PLS"

# Group information
PLSclus <- readMat(file.path(PLSpath, paste('group', Accprd, Pretype, Markettype, weekterm, 
                                            'CH3', 'SMB', 'VMG', '3', sep = '_') %>% 
                                      paste0('.mat'))) %>% `[[`(1) 
colnames(PLSclus) <- c("group")
grpnum <- length(unique(PLSclus))

cdcar <- paste(Accprd, Pretype, Markettype, weekterm, modeltype, 
                'CAR', 'CH3', 'SMB', 'VMG', '3',sep='_') %>%
          paste0(datdir, ., '.csv')
QEAcar <- as.data.frame(read_delim(cdcar, delim=',', na = ''))

## Path of CAR ========================================================================

titchar <- paste0('Paths of grouped cumulative abnormal return (CAR) ', 
                  '\nattributed to quarterly earnings announcement ',
                  '\narround accounting period ', Accprd)
ggcar <- data.frame(matrix(0, windlen*(grpnum+1),3))
for (i in 1:(grpnum+1)) {
    ifelse(i==1, ggcar[(1:windlen),] <- cbind(timeline,QEAcar[,i],c(i)),
           ggcar[(i-1)*windlen + (1:windlen), ] <- cbind(timeline,QEAcar[,i],c(i)))
}
colnames(ggcar) <- c('timeline','CAR', 'group')
ggcar$group <- as.factor(ggcar$group)


if (grpnum+1==4) {
  ggplot(ggcar, aes(timeline, CAR, 
                    linetype = group, colour = group)) + 
    geom_line() + geom_point() +
    scale_linetype_manual(name='Group',values=c("solid", 'solid', 'solid', "dotted"),
                          labels=c('PLS_G1', 'PLS_G2', 'PLS_G3','Unclassified')) +
    scale_colour_manual(name="Group", values = c("blue", "red", 'green', "black"),
                        labels=c('PLS_G1', 'PLS_G2', 'PLS_G3','Unclassified')) +
    labs(title = titchar, x = "Time line", y = 'Cumulative Abnormal Return') + 
    theme(plot.title = element_text(size=11), 
          axis.ticks.y = element_blank()) 
  
} else if (grpnum+1==3) {
    ggplot(ggcar, aes(timeline, CAR, 
                      linetype = group, colour = group)) + 
        geom_line() + geom_point() + coord_fixed() +
        scale_linetype_manual(name='Group',values=c("solid", 'solid', "dotted"),
                              labels=c('PLS_G1', 'PLS_G2', 'Unclassified')) +
        scale_colour_manual(name="Group", values = c("blue", "red", "black"),
                        labels=c('PLS_G1', 'PLS_G2', 'Unclassified')) +
        labs(title = titchar, x = "Time line", y = 'Cumulative Abnormal Return') + 
        theme(plot.title = element_text(size=11), 
              axis.ticks.y = element_blank())
} else print('Group number is error')

```


<!-- ## Power test  -->

<!-- Under the null hypothesis that  -->
<!-- the event has no influence on the mean and variance of returns, -->
<!-- we can obtain the distribution of AR, -->
<!-- \(AR_{i} \sim N(0, V_i),\) -->
<!-- where  -->
<!-- \( -->
<!-- V_i = \sigma_{\varepsilon_{i}^{*}}^{2}+X_{i}^{*}\left(X_{i}^{*'} X_{i}\right)^{-1} X_{i}^{*^{\prime}} \sigma_{\varepsilon_{i}}^{2}. -->
<!-- \) -->

<!-- \[ -->
<!-- \begin{aligned} -->
<!-- V_i & = {\it Var}( AR_{i}) = E\left[\hat{\varepsilon}_{i}^{*} \hat\varepsilon_{i}^{*^{\prime}} \right] \\ -->
<!-- & =E\left[\left[\varepsilon_{i}^{*}-X_{i}^{*}\left(\hat{\beta}_{i}-\beta_{i}\right)\right] -->
<!-- \left[\varepsilon_{i}^{*}-X_{i}^{*}\left(\hat{\beta}_{i}-\beta_{i}\right)\right]^{\prime} \right] \\ -->
<!-- & = E\left[\varepsilon_{i}^{*} \varepsilon_{i}^{*^{\prime}} - \varepsilon_{i}^{*} \left(\hat{\beta}_{i}-\beta_{i}\right)^{\prime} X_{i}^{*} -->
<!-- - X_{i}^{*}\left(\hat{\beta}_{i}-\beta_{i}\right) \varepsilon_{i}^{*^{\prime}} \right. \\  -->
<!-- & \quad \left. + X_{i}^{*}\left(\hat{\beta}_{i}-\beta_{i}\right) \left(\hat{\beta}_{i}-\beta_{i}\right) X_{i}^{*^{\prime}} \right] \\ -->
<!-- & = I \sigma_{\varepsilon_{i}^{*}}^{2}+X_{i}^{*}\left(X_{i}^{*'} X_{i}\right)^{-1} X_{i}^{*^{\prime}} \sigma_{\varepsilon_{i}}^{2}. -->
<!-- \end{aligned} -->
<!-- \] -->

<!-- Consequently we have -->
<!-- \(\overline{CAR}(\tau_1, \tau_2) \sim N(0, \bar \sigma^2\left(\tau_1, \tau_2)\right),\) -->
<!-- where $\bar \sigma^2_i(\tau_1, \tau_2)$ is unknown and can be consistently eatimated by -->
<!-- \( -->
<!-- \hat{\bar\sigma}^2(\tau_1, \tau_2) = -->
<!-- \frac{1}{N^2} \sum_{i=1}^{N} \hat \sigma^2_i (\tau_1, \tau_2). -->
<!-- \) -->

<!-- ## -->


<!-- ##  -->
<!-- <!-- Note that our original null hypothesis is event haved no effect on  -->
<!-- <!-- mean and variance of security returns. -->
<!-- A **modified null hypothesis would be**: event has effect on mean return only. -->
<!-- So the distribution of average $CAR$ could be transformed to standard normal distribution, -->
<!-- \[ -->
<!-- \frac{\overline{\it CAR}(\tau_1, \tau_2)}{\hat{\bar\sigma}(\tau_1, \tau_2) } -->
<!-- \stackrel{a}{\sim} N(0, 1) -->
<!-- \] -->
<!-- where -->
<!-- \( -->
<!-- \hat{\bar\sigma}(\tau_1, \tau_2) -->
<!-- = \frac{1}{N^2} \sum_{i=1}^{N}\left[ {\it CAR}_i(\tau_1, \tau_2) - \overline{\it CAR}(\tau_1, \tau_2)  \right]^2. -->
<!-- \) -->


<!-- ## -->


<!-- ## Review the regular steps in event studies -->

<!-- 1. Exactly define the event and identify the exact event date -->

<!-- 2. Determine the estimation and event window -->

<!-- 3. Determine the estimation method for **expected return** calculation -->

<!-- 4. Calculate **abnormal returns** (AR) and cumulative abnormal returns (CAR) -->

<!-- 5. Define null and alternative hypotheses, then test for statistical significance -->


<!-- ## -->

<!-- Through the previous section, we have learned about that -->
<!-- the publicly issued quarterly earnings announcement of listed frim -->
<!-- has the potential of affecting itself prices. -->
<!-- Next, we analyze the **origin** and **fluctuation** of abnormal return ($AR$). -->



# Identify the abnormal return


## A Perspective on Trading Strategies

When the intercept, $\hat \alpha_i$, in Fama-French 3-factor model
is highly statistical significantly,
the abnormal return, ${\it AR}$, equals
\begin{equation}
AR_{i\tau}^* = \hat\alpha_i + \epsilon_{i\tau}^*, 
\end{equation}
Obviously, only $\epsilon_{i\tau}^{*}$ is the abnormal return 
brought by quarterly earnings announcement.
There is an interesting perspective to explain the existence and variation of $AR$,

- $\alpha$, Earnings momentum (Post-earning announcement drift)

- $\epsilon$, Value or growth investing (Reaction to information about firm's value)


## The values of abnormal return

Because of the abnormal return consists of two parts, $\alpha$ and $\epsilon$, 
so its positive and negative also depends on the directions and 
absolute values of $\alpha$ and $\epsilon$. 
Assume the abnormal return $AR$ is positive, there maight be three cases included, 

- both $\alpha$ and $\epsilon$ are positive;

- $\alpha$ is positive, $\epsilon$ is negative, 
and the absolute value of $\alpha$ is greater than $\epsilon$'s;

- the third is the negative $\alpha$ and the positive $\epsilon$, 
but the absolute value of $\epsilon$ is greater than $\alpha$'s.


## $\alpha$

Following Carhart (1997),
We construct a PR1QR factor mimicking the momentum effect to explain the intercept,
$\alpha$, in Fama-French 3-factor model,
\begin{equation}
\begin{aligned}
R_{i\tau}-R_{f\tau} = & \mu_i + \beta_{i}(R_{m\tau} - R_{f\tau}) +
s_{i}\text{SMB}_\tau + v_{i}\text{VMG}_\tau  \\
& + p_i \text{PR1QR}_\tau  + e_{i\tau}.
\end{aligned}
\end{equation}

<!-- the equal-weight average of firms  -->
<!-- with the highest 30 percent eleven-month returns lagged three month -->
<!-- minus the equal-weight average of firms  -->
<!-- with the lowest 30 percent eleven-month returns lagged three month -->


## $\epsilon$ 

Inspired by Bamber (1987), we regress the grouped cumulative abnormal return, 
$\sum_{\tau=-k}^{k}\epsilon_{g_i\tau}^{*}$,
within event-window $\tau=(-k,k)$ on the index of standardized unexpected earnings (SUE) 
calculated by seasonal random walk (SRW),
\begin{equation}
\sum_{\tau=-k}^{k} \epsilon_{g_i\tau}^{*} = \mu + \gamma \text{SUE}_{g_i}+
e_{g_i},
\quad {\rm for} \ k=1,\cdots,k_0.
\end{equation}
Note that we classified $i \in \hat G_k$ if $\hat \beta_i = \hat \alpha_i$ for some $k=1,\cdots,K_0$.

##

Under the hypothesis that the value invest strategy 
is a significant source of return variation for the portfoloios
in both the smaller-size and higher-$EP$ quintiles,
we expect those stocks to have a more significant and larger value of $\gamma$.

<!-- We compute $SUE_{i\tau}$ using a seasonal random walk,  -->
<!-- in which $SUE{i\tau} = \Delta_{i\tau} / \sigma(\Delta_i)$,  -->
<!-- $\Delta{i\tau}$ equals the quarter-over-quarter change in stock $i$’s quarterly earnings,  -->
<!-- and $\sigma(\Delta_i)$ is the standard deviation of $\Delta_{i\tau}$ for the last eight quarters. -->


<!-- ## -->

<!-- Furthermore, We could compare the daily coefficients $\gamma$, -->
<!-- which be long to any number of days -->
<!-- before and after the release of the quarterly earnings announcement, -->
<!-- running below group panel regression, -->
<!-- \[ -->
<!-- \epsilon_{g_i\tau} = \mu_{g_i} + ( \gamma \text{SUE}_{g_i}) \times  D_\tau -->
<!-- + ( \gamma \text{SUE}_{g_i}) \times  (1 - D_\tau) -->
<!-- + e_{g_i\tau}, -->
<!-- \] -->


## The adjustment effect of event QEA on momentum

Furthermore, we regress the momentum effect $PR1QR$ on the average of grouped
abnormal return $\tilde \epsilon_{g_i}$ attributable to quarterly earnings announcement,
\begin{equation}
\text{PR1QR}_{\tau} = \upsilon +
\omega \tilde\epsilon_{g_i\tau} + \zeta^{\prime} \tilde Z_{g_i\tau} + \varepsilon_{g_i\tau},
\quad {\rm for} \ k=1,\cdots,k_0.
\end{equation}
where $\tilde Z_{\tau}$ are the value-weighted control variables 
$Turnover$, $Volatility$, and $Illiquidity$ etc.


## Conclusion

- We aims to identify the different parts of abnormal returns attributable to quarterly earnings announcement and to illustrate the pattern of its effects on stocks daily returns;

- We hold the hypothesis that, in Chinese A-share markets, the release of corporate earnings will produce significant grouped abnormal returns, which are not captured by the Fama-French three factor model;

- We expect the abnormal return could be explained by momentum effect and investing strategy. 

<!-- Keywords: Asset pricing; Cluster analysis; heterogenous coefficients;  -->
<!-- Factor models; Chinese A-share markets -->