# 16 Post-earnings announcement drift

In Chapter 13, we saw that the cumulative returns of “good news” firms and “bad news” firms continued to drift apart even after earnings announcements. This result was considered an anomaly as, once earnings are announced, an efficient market should quickly impound the implications of those earnings and there should be no association with subsequent excess returns. However, subsequent research expanded on Ball and Brown (1968), finding that the post-earnings announcement drift (“PEAD”) existed with more refined measures of earnings surprise.

In this chapter, we will build some foundation concepts before studying a seminal paper in the PEAD literature more closely. As most of the research on PEAD has focused on quarterly earnings, we will spend some time understanding Compustat’s database of quarterly financial statement information, comp.fundq.

A core concept in PEAD research is earnings surprise, which can be defined quite generally as actual earnings minus expected earnings. Thus to measure earnings surprise, we need a measure of expected earnings. Early research used time-series models of earnings to develop earnings expectation models. We will look closely at Foster (1977), which is an early study of the behaviour of quarterly accounting data.

## 16.1 Fiscal years

A concept frequently encountered in accounting and finance research is the fiscal year. Most US firms have financial reporting periods ending on 31 December of each year. Most Australian firms have financial reporting periods ending on 30 June of each year, perhaps because accountants don’t want to be preparing financial statements during the summer month of January.

Some US firms (often retailers) have fiscal year-ends in January. For example, Autodesk—“a global leader in 3D design, engineering, and entertainment software and services”—had a fiscal year-end on 31 January 2021. In contrast, Akamai Technologies had a fiscal year-end ending 31 December 2020. An analyst wishing to compare financial performance of Autodesk and Akamai is likely to line up these two periods as both being in “fiscal 2020”.

Note that there is no standard definition of “fiscal year” in practice. Fedex describes the year ending 31 May 2020 as “fiscal 2019” and General Mills describes the same period as “fiscal 2020”.

Compustat has the variable fyear (called fyearq on comp.fundq), which is described as “fiscal year”. Before zooming in on fyear, note that there are four variables of type Date on comp.funda and these are listed in the table below.

Variable Description
apdedate Actual period-end date
pdate Date the data are updated on a preliminary basis
fdate Date the data are finalized

One thing we see is that datadate is different from the actual date on which the fiscal year ends. Focusing on Apple (gvkey == "001690"), whose “fiscal year is the 52- or 53-week period that ends on the last Saturday of September”, we see that datadate is the last day of the month in which the period ended.

library(dplyr, warn.conflicts = FALSE)

library(stargazer)

library(farr)
pg <- dbConnect(RPostgres::Postgres(),
bigint = "integer",
check_interrupts = TRUE)

funda <- tbl(pg, sql("SELECT * FROM comp.funda"))
fundq <- tbl(pg, sql("SELECT * FROM comp.fundq"))
company <- tbl(pg, sql("SELECT * FROM comp.company"))

funda_mod <-
funda %>%
filter(indfmt == "INDL", datafmt == "STD",
consol == "C", popsrc == "D")
apple_dates <-
funda_mod %>%
filter(gvkey == "001690") %>%

select_if(is.Date)

However, it seems that data on fields other than datadate are only available for more recent periods.

apple_dates %>%
filter(!is.na(apdedate)) %>%
## # A tibble: 6 × 4
##   <date>     <date>     <date>     <date>
## 1 2003-09-30 2003-09-27 NA         NA
## 2 2004-09-30 2004-09-25 NA         NA
## 3 2005-09-30 2005-09-24 NA         NA
## 4 2006-09-30 2006-09-30 2007-01-01 2006-10-18
## 5 2007-09-30 2007-09-29 2007-11-16 2007-10-22
## 6 2008-09-30 2008-09-27 2008-11-05 2008-10-21

In its 10-K filing for 2019, Apple says “the Company’s fiscal years 2019 and 2018 spanned 52 weeks each, whereas fiscal year 2017 included 53 weeks. A 14th week was included in the first quarter of 2017, as is done every five or six years, to realign the Company’s fiscal quarters with calendar quarters.” These facts are evident in the following:

apple_dates %>%
mutate(fyear_length = apdedate - lag(apdedate)) %>%
tail()
## # A tibble: 6 × 5
##   datadate   apdedate   fdate      pdate      fyear_length
##   <date>     <date>     <date>     <date>     <drtn>
## 1 2017-09-30 2017-09-30 2017-11-03 2017-11-02 371 days
## 2 2018-09-30 2018-09-29 2018-11-05 2018-11-01 364 days
## 3 2019-09-30 2019-09-28 2019-11-04 2019-10-30 364 days
## 4 2020-09-30 2020-09-26 2020-11-02 2020-10-29 364 days
## 5 2021-09-30 2021-09-25 2021-11-01 2021-10-28 364 days
## 6 2022-09-30 2022-09-24 2022-10-31 2022-10-27 364 days

Compustat’s documentation explains how fyear is determined.

However, many database providers do not adequately explain variables, so being able to infer what an item describes is a useful skill. To build this skill, we will try to deduce what fyear means here without reading the manual. We will figure out fyear by collecting data from Compustat on fyear and datadate, then doing some statistical and graphical analysis of these data. As a side effect, we will also learn a little about regression analysis using R.

fyear_data <-
funda_mod %>%
filter(!is.na(fyear)) %>%
collect()

Let’s run a couple of regression models on the data (the discussion questions below will explore these in more detail).

fms <- list()
fms[[1]] <- lm(fyear ~ factor(month) + year - 1, data = fyear_data)
fms[[2]] <- lm(fyear ~ month + year, data = fyear_data)
sg_format <- "html"

(Change sg_format to "text" if viewing the output below on screen.)

stargazer(fms, type = sg_format,
header = FALSE, omit.stat = c("ser", "f"))
 Dependent variable: fyear (1) (2) factor(month)1 -1.000*** (0.000) factor(month)2 -1.000*** (0.000) factor(month)3 -1.000*** (0.000) factor(month)4 -1.000*** (0.000) factor(month)5 -1.000*** (0.000) factor(month)6 0.000*** (0.000) factor(month)7 0.000*** (0.000) factor(month)8 0.000*** (0.000) factor(month)9 0.000*** (0.000) factor(month)10 0.000*** (0.000) factor(month)11 0.000*** (0.000) factor(month)12 0.000*** (0.000) month 0.086*** (0.0001) year 1.000*** 0.999*** (0.000) (0.00002) Constant 0.342*** (0.031) Observations 548,169 548,169 R2 1.000 1.000 Adjusted R2 1.000 1.000 Note: p<0.1; p<0.05; p<0.01

Having run the regressions, we can add the fitted values for each model to the data set.

fyear_data <-
fyear_data %>%
mutate(pred_1 = predict(fms[[1]]),
pred_2 = predict(fms[[2]]),
resid_1 = pred_1 - fyear,
resid_2 = pred_2 - fyear)

And finally we can make some plots of the data and the fitted values.

plot_sample <-
fyear_data %>%
filter(year %in% c(2001, 2002)) %>%

plot_sample %>%
geom_point(aes(y = fyear, color = "fyear")) +
geom_line(aes(y = pred_1, color = "pred_1")) +
geom_line(aes(y = pred_2, color = "pred_2")) +
scale_x_date(date_breaks = "1 month") +
theme(axis.text.x = element_text(angle = 90))

### 16.1.1 Exercises

1. What is different between fms[[1]] and fms[[2]]? What is the function factor doing here?
2. What is the inclusion of - 1 doing in fms[[1]]? Would the omission of - 1 affect the fit of fms[[1]]? Would it affect the interpretability of results? Would the inclusion of - 1 affect the fit of fms[[2]]? Would it affect the interpretability of results?
3. Does the plot above help understand what’s going on? Why did we focus on a relatively short period in the plot? (Hint: What happens if you remove the line filter(year %in% c(2001, 2002)) %>% from the code?)
4. Using year and month, add some code along the lines of mutate(fyear_calc = ...) to calculate fyear. Check that you match fyear in each case.

## 16.2 Quarterly data

In Chapters 7 and 8, we focused on annual data from Compustat. But for many purposes we will want to use quarterly data. While Ball and Brown (1968) used annual data, the PEAD literature has generally focused on quarterly data.

In Chapter 7, we saw that once we focus on the “standard” subset of observations, we had (gvkey, datadate) forming a primary key. Alas, the same is not true for quarterly data, as we can see in the output below.

fundq_mod <-
fundq %>%
filter(indfmt == "INDL", datafmt == "STD",
consol == "C", popsrc == "D")
fundq_mod %>%
summarize(num_rows = n(), .groups = "drop") %>%
count(num_rows) %>%
collect()
## # A tibble: 2 × 2
##   num_rows       n
##      <int>   <int>
## 1        2    1496
## 2        1 1927134

To dig deeper, we create the data frame fundq_probs that relates to the problem cases observed above.

fundq_probs <-
fundq_mod %>%
filter(n() > 1) %>%
select(gvkey, datadate, fyearq, fqtr, fyr, rdq) %>%

collect()

The cause of the problems is illustrated by gvkey == "001224" and can be seen by looking at funda_mod defined above:

funda_mod %>%
filter(gvkey == "001224", between(fyear, 2012, 2015)) %>%

## # A tibble: 4 × 2
##   <chr>  <date>
## 1 001224 2012-12-31
## 2 001224 2013-12-31
## 3 001224 2014-09-30
## 4 001224 2015-09-30

In this case, the firm changed its year end from December to September during 2014. Thus Q4 of the year ending 2013-12-31 became Q1 of the year ending 2014-09-30. Compustat retains the data for the quarter ending 2013-12-31 twice: once as Q4 and once as Q1.

Meanwhile, Q1 of what would have been the year ending 2014-12-31 became Q2 of of the year ending 2014-09-30.115 But there is no year ending 2014-12-31 on comp.funda for this firm, so Compustat sets fqtr to NA for the second row of data for the quarter ending 2014-03-31, as can be seen in the following.

fundq_probs %>%
filter(gvkey == "001224") %>%
## # A tibble: 6 × 6
##   gvkey  datadate   fyearq  fqtr   fyr rdq
##   <chr>  <date>      <int> <int> <int> <date>
## 1 001224 2013-12-31   2014     1     9 2014-02-11
## 2 001224 2013-12-31   2013     4    12 2014-02-11
## 3 001224 2014-03-31   2014     2     9 2014-04-30
## 4 001224 2014-03-31   2014    NA    12 2014-04-30
## 5 001224 2014-06-30   2014     3     9 2014-08-11
## 6 001224 2014-06-30   2014    NA    12 2014-08-11

So the variable fyr allows us to distinguish rows and from the following analysis, we see that (gvkey, datadate, fyr) is a valid primary key for the “standard” subset of comp.fundq.

fundq_mod %>%
summarize(num_rows = n(), .groups = "drop") %>%
count(num_rows) %>%
collect()
## # A tibble: 1 × 2
##   num_rows       n
##      <int>   <int>
## 1        1 1939642
fundq_mod %>%
count() %>%
pull()
## [1] 0

Presumably the idea is to allow researchers to link data from comp.funda with data from comp.funda. Using fyr and fyear, we can recover the relevant annual datadate.

fundq_mod %>%
select(gvkey:fyr) %>%
mutate(year = if_else(fyr <= 5L, fyearq - 1L, fyearq)) %>%
mutate(month = lpad(as.character(fyr), 2L, "0")) %>%
mutate(datadate = as.Date(paste(year, month, '01', sep = "-"))) %>%
select(-month, -year, -fqtr) %>%
collect()

## # A tibble: 1,930,126 × 5
##    <chr>  <date>      <int> <int> <date>
##  1 001000 1966-03-31   1966    12 1966-12-31
##  2 001000 1966-06-30   1966    12 1966-12-31
##  3 001000 1966-09-30   1966    12 1966-12-31
##  4 001000 1966-12-31   1966    12 1966-12-31
##  5 001000 1967-03-31   1967    12 1967-12-31
##  6 001000 1967-06-30   1967    12 1967-12-31
##  7 001000 1967-09-30   1967    12 1967-12-31
##  8 001000 1967-12-31   1967    12 1967-12-31
##  9 001000 1968-03-31   1968    12 1968-12-31
## 10 001000 1968-06-30   1968    12 1968-12-31
## # … with 1,930,116 more rows

We could then link the table above with (gvkey, datadate) combinations from comp.funda and (gvkey, datadateq, fyr) combinations from comp.fundq (we rename datadate on comp.fundq to “datadateq” to avoid a clash between annual and quarter period-ends for any given quarter).

### 16.2.1 Exercises

1. Pick a couple of gvkey values from fundq_probs. Is it possible to construct a “clean” sequence of quarterly earnings announcements for each of these firms? (Here “clean” means that, at the very least, each quarter shows up just once in the series.) What challenges does one face in this task?

2. Over the last three decades, it seems that Q2 has been the most profitable on average, while in all decades, Q4 has seen the most sales. Can you speculate as to why this might be the case?

ni_annual <-
funda_mod %>%
select(gvkey, datadate, fyr, sale, ni) %>%
collect()

ni_qtrly <-
fundq_mod %>%
select(gvkey, datadate, fyr, fqtr, saleq, niq, ibq) %>%
collect()

ni_merged <-
ni_annual %>%
inner_join(ni_qtrly, by = c("gvkey", "fyr", "datadateq"))

plot_data <-
ni_merged %>%
mutate(decade = paste0(as.character(floor(fyearq/10) * 10), "s")) %>%
filter(!is.na(fqtr), fyearq < 2020) %>%
summarize(prop_ni = sum(niq, na.rm = TRUE)/
sum(ni, na.rm = TRUE),
prop_sale = sum(saleq, na.rm = TRUE)/
sum(sale, na.rm = TRUE),
.groups = "drop") %>%
mutate(fqtr = factor(fqtr)) %>%
pivot_longer(cols = c(prop_ni, prop_sale),
names_to = "metric",
values_to = "value")

plot_data %>%
ggplot(aes(x = fqtr, y = value, fill = fqtr)) +
geom_bar(stat = "identity") +
1. Create another plot using data in ni_merged that you think might be interesting? (Feel free to add variables to ni_annual or ni_qtrly before merging.)

## 16.3 Time-series properties of earnings

Foster (1977) studies the properties of quarterly accounting variables (sales, net income, and expenses) and considers the predictive ability of six models. Models 1 and 2 relate values in quarter $$Q_t$$ to values in quarter $$Q_{t-4}$$ and is therefore a seasonal model. Models 3 and 4 relate values in quarter $$Q_t$$ to values in the adjacent quarter ($$Q_{t-1}$$). Model 5 builds on Model 2 by including a component related to $$(Q_{t-1} - Q_{t-5})$$. Finally, Foster (1977) considers the “Box-Jenkins time-series methodology” as Model 6.

In this section, we will do a loose replication of Foster (1977). We say “loose” because we will not try to match the sample composition or even all the details of the approach, but the basic ideas and some of the details will be the same. We replace the Box-Jenkins approach with an alternative model, as Box-Jenkins forecasting methods are more complex and not often used in accounting research today.

n_qtrs <- 20
n_firms <- 70
focus_years <- c(1974L, 2019L)
# See Table 1 of Foster (1977) for SICs
sic2s <- as.character(c(29, 49, 28, 35, 32, 33, 37, 20, 26, 10, 36, 59))

Foster (1977) focuses on 69 firms meeting sample-selection criteria, including the availability of quarter sales and earnings information over the period 1946-1974. To keep our analysis to a comparable number of firms, we will choose 70 firms based on criteria detailed below. While Foster (1977) uses what he calls an “adaptive forecasting” approach whereby “all data available at the time the forecast was made were used to forecast”, we will limit ourselves to 20 quarters of data available at the time the model is developed. As a substantial majority of observations in Foster (1977) come from 12 two-digit SIC industries, we limit our sample to firms in the industries listed in Table 1 of that paper. Finally, while Foster (1977) evaluates the forecasting performance over the 1962-74 period, we will focus on just 2 years: 1974 and 2019.

companies <-
company %>%
mutate(sic2 = substr(sic, 1, 2)) %>%
filter(sic2 %in% sic2s) %>%
select(gvkey, sic2) %>%
collect()

Our first step is to collect the data from comp.fundq that we need into a data frame that we give the name fundq_local.

fundq_local <-
fundq_mod %>%
filter(saleq > 0 & !is.na(saleq)) %>%
select(gvkey, datadate, fyr, fqtr, fyearq, rdq, niq, saleq, ibq) %>%
collect()

Next we select firm-years available on comp.funda and link to fundq_local using the link_table we created earlier.

firm_years <-
funda_mod %>%
collect()

merged_data <-
firm_years %>%
inner_join(fundq_local,
by = c("gvkey", "datadateq", "fyearq", "fyr"))

We identify “regular” fiscal years, which we define as fiscal years comprising four quarters and extending over exactly a year (i.e., either 365 or 366 days).

qtr_num <-
merged_data %>%
count(name= "num_quarters") %>%
ungroup()

regular_fyears <-
firm_years %>%
inner_join(qtr_num, by = c("gvkey", "datadate")) %>%
group_by(gvkey) %>%

mutate(regular_year = num_quarters == 4 &
(is.na(fyear_length) | fyear_length %in% c(365, 366))) %>%
filter(regular_year) %>%

We next limit the sample to the regular fiscal years that we defined above and calculate the variables we need to estimate and apply the models.

reg_data <-
merged_data %>%
semi_join(companies, by = "gvkey") %>%
semi_join(regular_fyears, by = c("gvkey", "datadate")) %>%
select(gvkey, datadateq, fyearq, rdq, niq, saleq) %>%
group_by(gvkey) %>%
mutate(sale_lag_1 = lag(saleq, 1L),
sale_lag_4 = lag(saleq, 4L),
sale_lag_5 = lag(saleq, 5L),
sale_diff = saleq - sale_lag_1,
sale_seas_diff = saleq - sale_lag_4,
lag_sale_seas_diff  = lag(sale_seas_diff, 1L),
ni_lag_1 = lag(niq, 1L),
ni_lag_4 = lag(niq, 4L),
ni_lag_5 = lag(niq, 5L),
ni_diff = niq - ni_lag_1,
ni_seas_diff = niq - ni_lag_4,
lag_ni_seas_diff  = lag(ni_seas_diff, 1L)) %>%
ungroup()

We next create a fit_model function that estimates all six models using data from the given gvkey and the 20 quarters prior to datadateq and predicts values for datadateq for all models.

The first thing done in the function is a filter for observations that relate to the firm with the gvkey supplied as an argument to the function. Note that gvkey has two meanings in the context of the code at the point where filter is called. First, it refers to the column of reg_data. Second, it refers to the value of the gvkey supplied as an argument to the function. It is for this reason that we use gvkey to get the first meaning and use !!gvkey as a way to get the second meaning. (We do a similar thing for the same reason when we filter on dates.)

Having obtained data related to the firm, we split the data into train_data, the data from previous periods that we will use to estimate the models, and test_data, the data for the period that we will forecast. (Note that we exit the function and return NULL if we don’t have 20 quarters of data to train the model.)

We first estimate models 2 and 4, which require no more than estimation of a drift term ($$\delta$$ in the notation of Foster, 1977), which we estimate as the mean of the respective changes over the training period.

We then create predicted values for models 1–4.

The next thing we do is fit models 5 and 6. Model 6 in Foster (1977) was based on the Box-Jenkins method. Our model 6 can be viewed as an “unconstrained” version of model 5. In model 6, we regress (in the notation of Foster, 1977) $$Q_t$$ on $$Q_{t-1}$$, $$Q_{t-4}$$ and $$Q_{t-5}$$.

In the final step, we add predicted values for models 5 and 6 to the same for models 1–4, then calculate the prediction errors and return the results of our analysis.

firm_data <-
reg_data %>%
filter(gvkey == !!gvkey)

train_data <-
firm_data %>%

if (nrow(train_data) < n_qtrs) return(NULL)

test_data <-
firm_data %>%

# Estimate models 2 & 4
model_24 <-
train_data %>%
group_by(gvkey) %>%
summarize(sale_diff = mean(sale_diff, na.rm = TRUE),
ni_diff = mean(ni_diff, na.rm = TRUE),
sale_seas_diff = mean(sale_seas_diff, na.rm = TRUE),
ni_seas_diff = mean(ni_seas_diff, na.rm = TRUE))

# Fit models 1, 2, 3 & 4
df_model_1234 <-
test_data %>%
# We drop these variables because we will replace them with
# their means from model_24
select(-sale_diff, -ni_diff, -sale_seas_diff, -ni_seas_diff) %>%
inner_join(model_24, by = "gvkey") %>%
mutate(ni_m1 = ni_lag_4,
sale_m1 = sale_lag_4,
ni_m2 = ni_lag_4 + ni_seas_diff,
sale_m2 = sale_lag_4 + sale_seas_diff,
ni_m3 = ni_lag_1,
sale_m3 = sale_lag_1,
ni_m4 = ni_lag_1 + ni_diff,
sale_m4 = sale_lag_1 + sale_diff)

# Fit model 5
sale_fm5 <- tryCatch(lm(sale_seas_diff ~ lag_sale_seas_diff,
data = train_data, model = FALSE),
error = function(e) NULL)

ni_fm5 <- tryCatch(lm(ni_seas_diff ~ lag_ni_seas_diff,
data = train_data, model = FALSE),
error = function(e) NULL)

# Fit model 6
sale_fm6 <- tryCatch(lm(saleq ~ sale_lag_1 + sale_lag_4 + sale_lag_5,
data = train_data, model = FALSE),
error = function(e) NULL)

ni_fm6 <- tryCatch(lm(niq ~ ni_lag_1 + ni_lag_4 + ni_lag_5,
data = train_data, model = FALSE),
error = function(e) NULL)

if (!is.null(sale_fm5) & !is.null(ni_fm5)) {
results <-
df_model_1234 %>%
mutate(ni_m5 = ni_lag_4 + predict(ni_fm5, newdata = test_data)) %>%
mutate(sale_m5 = sale_lag_4 + predict(sale_fm5, newdata = test_data)) %>%
mutate(ni_m6 = predict(ni_fm6, newdata = test_data)) %>%
mutate(sale_m6 = predict(sale_fm6, newdata = test_data))%>%
select(gvkey, datadateq, fyearq, niq, saleq, matches("(ni|sale)_m[0-9]")) %>%
pivot_longer(cols = ni_m1:sale_m6,
names_to = "item", values_to = "value") %>%
mutate(abe = if_else(grepl("^ni", item), abs(value - niq)/value,
abs(value - saleq)/value),
se = abe^2) %>%
separate(item, into = c("item", "model"), sep = "_m") %>%
select(-niq, -saleq)

results
}
}

To keep the analysis relatively small, we will focus on 70 firms. In each test period, we choose the firms within our larger sample that have the largest sales. We then use the Map function to apply the fit_model above to each firm-year in test_years and store the results in results.

top_firms <-
reg_data %>%
filter(fyearq %in% focus_years) %>%
group_by(gvkey, fyearq) %>%
summarize(total_sales = sum(saleq),
.groups = "drop") %>%
group_by(fyearq) %>%
arrange(desc(total_sales)) %>%
mutate(rank = row_number()) %>%
filter(rank <= n_firms)

test_years <-
reg_data %>%
semi_join(top_firms, by = c("gvkey", "fyearq")) %>%

results <-
Map(fit_model, test_years$gvkey, test_years$datadateq) %>%

Before graphing results comparable to Table 3 in Foster (1977), let’s look at the results in a more raw form. We address outliers with a fix_outlier function and plot a histogram of abe for net income for each of the 2 for all six models.

fix_outlier <- function(x) {
if_else(x < 0 | x > 1, 1, x)
}

results %>%
filter(item == "ni") %>%
filter(!is.na(abe)) %>%
mutate(abe = fix_outlier(abe)) %>%
ggplot(aes(x = abe)) +
geom_histogram(bins = 40) +
facet_grid(model ~ fyearq)

Next, we rank the models for each observation. produce a graphical version of our analogue of Table 3 of Foster (1977).

model_ranks <-
results %>%
mutate(rank = row_number()) %>%
group_by(fyearq, item, model) %>%
summarize(avg_rank = mean(rank, na.rm = TRUE),
.groups = "drop") %>%
pivot_wider(names_from = c("model"), values_from = "avg_rank")

model_ranks %>% knitr::kable(digits = 2)
fyearq item 1 2 3 4 5 6
1974 ni 4.25 3.38 3.49 3.45 2.93 3.51
1974 sale 5.52 4.11 3.28 2.67 2.57 2.86
2019 ni 3.24 3.33 3.50 3.78 3.34 3.81
2019 sale 3.48 3.51 3.71 3.71 3.43 3.17

Finally, we produces results analogous to Table 3 of Foster (1977).

results_summ <-
results %>%
mutate(abe = fix_outlier(abe),
se = fix_outlier(se)) %>%
group_by(fyearq, item, model) %>%
summarize(mabe = mean(abe, na.rm=TRUE),
mse = mean(se, na.rm=TRUE),
.groups = "drop")

results_summ %>%
pivot_longer(cols = mabe:mse, names_to = "metric", values_to = "value") %>%
ggplot(aes(x = model, y = value, fill = model)) +
geom_bar(stat = "identity") +
facet_grid(fyearq ~ item + metric)

### 16.3.1 Exercises

1. What does the fix_outlier function do? Does Foster (1977) do anything to address outliers? If so, how does the approach in Foster (1977) compare to that of fix_outlier? Do you agree with the approach taken in fix_outlier? What would you do differently?

2. How do the results in Figure @ref(fig:table_3) compare with those in Foster (1977)?

3. What do you make of the significantly “worse” performance of models predicting ni than those predicting sale? Does this imply that ni is simply more difficult to forecast? Can you suggest an alternative approach to measuring performance that might place these models on a more “level playing field”?

## 16.4 Post-earnings announcement drift

Bernard and Thomas (1989) is a careful and persuasive paper that rewards close reading if you have time (now or later). Bernard and Thomas (1989) builds on an earlier paper . Foster et al. (1984) measured earnings surprise relative to expectations from Model 5 of Foster (1977), but consider two alternative denominators.

$FE_i^1 = \frac{Q_{i,t} - E[Q_{i,t}]}{|Q_{i,t}|}$ $FE_i^2 = \frac{Q_{i,t} - E[Q_{i,t}]}{\sigma \left(Q_{i,t} - E[Q_{i,t}]\right)}$ In the first model, $$FE_i^1$$ is calculated with the absolute value of the forecast item as the denominator. In the second model, $$FE_i^2$$ is calculated with the standard deviation of the forecast error as the denominator.

Bernard and Thomas (1989) require at least 10 quarters of data to produce a forecast and use up to 24 quarters of data. If there are fewer than 16 observations, then Bernard and Thomas (1989) use the simpler Model 1 of Foster (1977). The following function is adapted from fit_model above, but adapted to reflect the features just discussed.

Rather than net income, Bernard and Thomas (1989) use “Income Before Extraordinary Items” (ibq) and we follow them in this regard. While Bernard and Thomas (1989) focus on NYSE/AMEX-listed firms, we just use all firms that we can match from Compustat to CRSP as outlined below.

reg_data_fos <-
merged_data %>%
semi_join(regular_fyears, by = c("gvkey", "datadate")) %>%
select(gvkey, datadateq, fyearq, rdq, ibq) %>%
group_by(gvkey) %>%
mutate(ib_lag_4 = lag(ibq, 4L),
ib_seas_diff = ibq - ib_lag_4,
lag_ib_seas_diff  = lag(ib_seas_diff, 1L),
qtr = quarter(datadateq, with_year = TRUE)) %>%
ungroup()

As before, we split the data into train_data, which we use to fit the data and test_data, which is the holdout period for the forecast. The calculation denom_m2 provides the denominator for $$FE_i^2$$.

fit_model_fos <- function(gvkey, quarter) {

n_qtrs <- 24
min_qtrs_fos <- 16
min_qtrs <- 10

firm_data <-
reg_data_fos %>%
filter(gvkey == !!gvkey)

train_data <-
firm_data %>%
filter(qtr < !!quarter) %>%

test_data <-
firm_data %>%
filter(qtr == !!quarter)

if (nrow(train_data) < min_qtrs) return(NULL)
if (nrow(train_data) >= min_qtrs_fos) {
# Fit model 5
ib_fm <- tryCatch(lm(ib_seas_diff ~ lag_ib_seas_diff,
data = train_data, na.action = na.exclude,
model = FALSE),
error = function(e) NULL)
} else {
ib_fm <- NULL
}

if (!is.null(ib_fm)) {
train_results <-
train_data %>%
mutate(fib = ib_lag_4 + predict(ib_fm))
} else {
train_results <-
train_data %>%
mutate(fib = ib_lag_4)
}

denom_m2 <-
train_results %>%
mutate(fe = ibq - fib) %>%
pull() %>%
sd()

if (is.null(ib_fm)) {
results <-
test_data %>%
mutate(fib = ib_lag_4)
} else {
results <-
test_data %>%
mutate(fib = ib_lag_4 + predict(ib_fm, newdata = test_data))
}

results %>%
mutate(fe1 = (ibq - fib)/abs(ibq),
fe2 = (ibq - fib)/denom_m2)
}

So far, we have not worried about calendar time very much. But in this analysis, we are going to form portfolios each quarter based on the earnings surprise of each firm in that quarter. For this purpose we want to measure earnings surprise in each calendar quarter. In constructing reg_data_fos above, we calculated qtr as quarter(datadateq, with_year = TRUE). The quarter function comes from the lubridate package and, with with_year = TRUE, will return (say) the number 2014.2 if given the data 2014-06-30.

Bernard and Thomas (1989) focus on quarters from 1974 through 1986 and we follow this. The following code gets a list of all quarters on the data set in that range. For each quarter, the function get_results is called. This function calls fit_model_fos, a function that produces data for a given gvkey-quarter combination, for each gvkey, then assembles the results into a data frame and returns them.

Note that running the following code chunk takes several minutes to run.116 So only run it if you want to play around with the output yourself.

quarters <-
reg_data_fos %>%
filter(qtr >= 1974, qtr < 1987) %>%
select(qtr) %>%

arrange(qtr) %>%
pull()

get_results <- function(quarter) {

gvkeys <-
reg_data_fos %>%
filter(qtr == quarter) %>%
select(gvkey) %>%

pull()

Map(fit_model_fos, gvkeys, quarter) %>%

}
results <- bind_rows(lapply(quarters, get_results))

Bernard and Thomas (1989) form portfolios based on deciles of earnings surprise. Decile 1 will have the 10% of firms with the lowest (most negative) earnings surprise. Decile 10 will have the 10% of firms with the highest (most positive) earnings surprise. But to avoid lookahead bias, the cutoffs for assigning firms to deciles will be based on the distribution of earnings surprises for the previous quarter. The get_deciles function takes a vector of data and returns the decile cut-offs for that vector. Note that we set the highest limit to Inf, as we want to put any firms whose earnings surprise is greater than the maximum in Decile 10. Note also that we return the breaks inside a list so that the returned value can be stored in a column of a data frame, which we do in creating decile_cuts. The code creating decile_cuts is fairly self-explanatory, but note that we will be most interested in fe1_deciles_lag and fe2_deciles_lag, which are the decile cut-offs based on data from the previous quarter.

get_deciles <- function(x) {
breaks <- quantile(x, probs = seq(from = 0, to = 1, by = 0.1),
na.rm = TRUE)
breaks[length(breaks)] <- Inf
list(breaks)
}

decile_cuts <-
results %>%
group_by(qtr) %>%
summarize(fe1_deciles = get_deciles(fe1),
fe2_deciles = get_deciles(fe2),
.groups = "drop") %>%
arrange(qtr) %>%
mutate(fe1_deciles_lag = lag(fe1_deciles),
fe2_deciles_lag = lag(fe2_deciles))

We add the decile cut-offs to the results data frame in the following code. Note that we need to use rowwise here to calculate fe1_decile and fe2_decile row by row. While the cut function is a vectorized function, it accepts just one value for the second argument breaks. However, we want to apply different breaks to each row (based on which quarter it is in). Using rowwise allows us to do this. Once we have applied the breaks, we no longer need them, so we drop them in the last line below.

results_deciles <-
results %>%
inner_join(decile_cuts, by = "qtr") %>%

mutate(fe1_decile = cut(fe1, fe1_deciles_lag, labels = FALSE),
fe2_decile = cut(fe2, fe2_deciles_lag, labels = FALSE)) %>%
filter(!is.na(fe1_decile) | !is.na(fe2_decile)) %>%
select(-matches("^fe[12]_deciles"))

We will need to grab stock returns for our earnings announcements. The following code is more or less copy-pasted from earlier chapters. Note that running the following code chunk takes several minutes to run.117

ccmxpf_lnkhist <- tbl(pg, sql("SELECT * FROM crsp.ccmxpf_lnkhist"))

ccmxpf_lnkhist %>%
collect()

results_deciles %>%
select(gvkey, rdq) %>%
mutate(permno = as.integer(lpermno)) %>%
select(gvkey, rdq, permno)

event_date = "rdq", win_start = -60, win_end = 60)

Finally, we calculate mean size-adjusted returns for each decile at each trading day relative to the earnings announcement. There are some issues implicit in these calculations that are discussed in Bernard and Thomas (1989) and alluded to in the discussion questions at the end of this section. Here we are largely following the approach described on p.7 of Bernard and Thomas (1989).

plot_data <-
results_deciles %>%
mutate(decile = fe2_decile) %>%
left_join(link_table, by = c("gvkey", "rdq")) %>%
left_join(rets, by = c("rdq", "permno")) %>%
group_by(decile, relative_td) %>%
summarize(ar = mean(ret - decret, na.rm = TRUE),
.groups = "drop")
## Warning in left_join(., link_table, by = c("gvkey", "rdq")): Detected an unexpected many-to-many relationship between x and y.
## ℹ Row 1546 of x matches multiple rows in y.
## ℹ Row 1439 of y matches multiple rows in x.
## ℹ If a many-to-many relationship is expected, set relationship = "many-to-many" to silence this warning.
## Warning in left_join(., rets, by = c("rdq", "permno")): Detected an unexpected many-to-many relationship between x and y.
## ℹ Row 1 of x matches multiple rows in y.
## ℹ Row 5507329 of y matches multiple rows in x.
## ℹ If a many-to-many relationship is expected, set relationship = "many-to-many" to silence this warning.

In making the plot below, note that we set abnormal returns for the first trading day in each plot to zero, so that the plots depict returns starting from day $$t-59$$ in the first plot and from day $$t+1$$ in the second plot. We also add a label to the last day of the series so that we don’t need a legend.

plot_data %>%
filter(relative_td <= 0) %>%
filter(!is.na(decile)) %>%
mutate(decile = as.factor(decile)) %>%
mutate(first_day = relative_td == min(relative_td),
last_day = relative_td == max(relative_td),
ar = if_else(first_day, 0, ar),
label = if_else(last_day, as.character(decile), NA_character_)) %>%
select(-first_day) %>%
group_by(decile) %>%
arrange(relative_td) %>%
mutate(car = cumsum(ar)) %>%
ggplot(aes(x = relative_td, y = car,
group = decile, color = decile)) +
geom_line() +
geom_label(aes(label = label), na.rm = TRUE) +
theme(legend.position = "none")
plot_data %>%
filter(relative_td >= 0) %>%
filter(!is.na(decile)) %>%
mutate(decile = as.factor(decile)) %>%
mutate(first_day = relative_td == min(relative_td),
last_day = relative_td == max(relative_td),
ar = if_else(first_day, 0, ar),
label = if_else(last_day, as.character(decile), NA_character_)) %>%
group_by(decile) %>%
arrange(relative_td) %>%
mutate(car = cumsum(ar)) %>%
ggplot(aes(x = relative_td, y = car, group = decile, color = decile)) +
geom_line() +
geom_label(aes(label = label), na.rm = TRUE) +
theme(legend.position = "none")

### 16.4.1 Discussion questions

1. A common feature of Bernard and Thomas (1989) and Ball and Brown (1968), is that both were addressing issues with “conventional wisdom” at their respective times. How had conventional wisdom changed in the years between 1968 and 1989?
2. Evaluate the introduction of Bernard and Thomas (1989). How clear is the research question to you from reading this? How does this introduction compare with other papers we’ve read in the course? With other papers you have seen?
3. How persuasive do you find Bernard and Thomas (1989) to be? (Obviously answering this one requires reading the paper fairly closely.)
4. The analysis above considers the 13-year period from 1974 to 1986. What changes would you need to make to the code to run the analysis for the 13-year period from 2007 to 2019? (If you choose to make this change and run the code, what do you notice about the profile of returns in the post-announcement period? Does it seem necessary to make an additional tweak to the code to address this?)
5. Considering a single stock, what trading strategy is implicit in calculating ar as ret - decret?
6. In calculating mean returns by decile and relative_td (i.e., first using group_by(decile, relative_td) and then calculating ar by aggregating mean(ret - decret, na.rm = TRUE)), are we making assumptions about the trading strategy? What issues are created by this trading strategy? Can you suggest an alternative trading strategy? What changes to the code would be needed to implement this alternative?
7. Is it appropriate to add returns to get cumulative abnormal returns as is done in car = cumsum(ar)? What would be one alternative approach?