# 15 Accruals

In this chapter, we use Sloan (1996) to provide a focus for a study of accrual processes. We use simulation analysis to understand better accounting processes, with a particular focus on **accruals**, which for this chapter we define as the portion of earnings in excess of operating cash flows.^{1} We finish up the chapter with an examination of the so-called **accrual anomaly**.

The code in this chapter uses the packages listed below. For instructions on how to set up your computer to use the code found in this book, see Section 1.2. Quarto templates for the exercises below are available at https://github.com/iangow/far_templates.

## 15.1 Sloan (1996)

While we saw evidence in Chapter 14 that capital markets do not fully price earnings surprises, Sloan (1996) goes further and examines how capital markets price *components* of earnings. Sloan (1996) points out that a number of practitioners provide investment advice predicated on identifying firms whose earnings depend on accruals rather than cash flows. Such investment advice is based on a claimed tendency for capital markets to “fixate” on earnings and to fail to recognize differences in the properties of cash flow and accrual components of earnings. In some respects, Sloan (1996) provides a rigorous evaluation of such investment advice and the premises underlying it.

### 15.1.1 Discussion questions

The following discussion questions provide an approach to reading Sloan (1996). While one approach to reading a paper involves a careful reading from start to finish, a useful skill is being able to read a paper quickly with a focus on the empirical results and the hypotheses that these relate to.

- Read the material preceding the formal statement of H1. What reasons for differential persistence of earnings components does Sloan (1996) offer? How important is it for these reasons to be correct in light of the empirical support for H1 provided in Table 3? How important is the empirical support for H1 to H2(i)?
- Which hypothesis (if any) does Table 4 test? How would you interpret the results of Table 4 in words?
- Which hypothesis (if any) does Table 5 test? How would you interpret the results of Table 5 in words?
- Which hypothesis (if any) does Table 6 test? How would you interpret the results of Table 6 in words? There are similarities between the results of Table 6 of Sloan (1996) and the results in Bernard and Thomas (1989). Both involve forming portfolios of firms based on deciles of some variable Bernard and Thomas (1989) and examining how those portfolios perform subsequently. Apart from the measure used to form portfolios, what are the significant differences between the analyses in the two papers that you can think of looking at Table 6?
- With which hypothesis (if any) is Figure 2 related? What does Figure 2 show according to Sloan (1996)?
- With which hypothesis (if any) is Figure 3 related? What does Figure 3 show according to Sloan (1996)?

## 15.2 Measuring accruals

Hribar and Collins (2002) include a definition of accruals similar to that used in Sloan (1996). Referring to prior research, they state (2002, p. 10):

Specifically, accruals (\(\mathit{ACC}_{bs}\)) are typically calculated (firm and time subscripts omitted for convenience):

\[ \mathit{ACC}_{bs} = (\Delta \mathit{CA} - \Delta \mathit{CL} - \Delta \mathit{Cash} + \Delta \mathit{STDEBT} - \mathit{DEP} )\] where

- \(\Delta \mathit{CA}\) = the change in current assets during period \(t\) (Computstat #4)
- \(\Delta \mathit{CL}\) = the change in current liabilities during period \(t\) (Compustat #5)
- \(\Delta \mathit{Cash}\) = the change in cash and cash equivalents during period \(t\) (Compustat #1);
- \(\Delta \mathit{STDEBT}\) = the [change in] current maturities of long-term debt and other short-term debt included in current liabilities during period \(t\) (Computstat #34);
- and \(\Delta \mathit{DEP}\) = depreciation and amortization expense during period \(t\) (Compustat #14).
All variables are deflated by lagged total assets (\(\mathit{TA}_{t-1}\)) to control for scale differences.

The first thing you may ask is “what does (say) ‘Computstat #4’ mean?” Prior to 2006, Compustat data items were referred to using numbers such as Computstat #4 or `data4`

. So older papers may refer to such items. Fortunately, WRDS provides translation tables from these items to the current variables.

Old item | Current item | Item description |
---|---|---|

#1 | `che` |
Cash and Short-Term Investments |

#4 | `act` |
Current Assets - Total |

#5 | `lct` |
Current Liabilities - Total |

#14 | `dp` |
Depreciation and Amortization |

#34 | `dlc` |
Debt in Current Liabilities - Total |

Hribar and Collins (2002) point out that calculating current accruals for a period are obtained by subtracting the change in current liabilities from the change in noncash current assets for that period is incorrect “because other non-operating events (e.g., mergers, divestitures) impact the current asset and liability accounts with no earnings impact.”

### 15.2.1 Discussion questions

- In the equation above, why is \(\Delta \mathit{Cash}\) subtracted?
- In the equation above, why is \(\Delta \mathit{STDEBT}\) added?
- Is it true that mergers and divestitures have “no earnings impact”? Is the absence of earnings impact important to the estimation issue? Are there transactions that have no earnings impact, but do affect cash flow from operations?
- Are there any differences between the definition of accruals in Hribar and Collins (2002) (\(\mathit{ACC}_{bs}\) above) and that in Sloan (1996)? Which definition makes more sense to you? Why?

## 15.3 Simulation analysis

We now consider some **simulation analysis**. One reason for this analysis is to better understand the basis for H1 of Sloan (1996),

A second reason for conducting simulation analysis here is to illustrate the power it offers. In many contexts, derivation of the properties of estimators or understanding how phenomena interact is very complex. While many researchers rely on intuition to guide their analyses, such intuition can be unreliable. As an example, the idea that the “FM-NW” method provides standard errors robust to both time-series and cross-sectional dependence has strong intuitive appeal, but we saw in Chapter 5 that this intuition is simply wrong.

### 15.3.1 Vectors

In our simulation analysis, we make more extensive use of base R functionality than we have in prior chapters. Chapter 28 of *R for Data Science*—“A field guide to base R”—provides material that might be helpful if code in the next section is unclear. Here we are simulating the cash flows and accounting for a simple firm that buys goods for cash and sells them on account after adding a mark-up.

### 15.3.2 Simulation function

As we have seen before, it is often good coding practice to use functions liberally in analysis.^{2} In this case, we embed the core of the simulation in a function.

The simulation function `get_data()`

below generates a time-series of data for a single “firm” and accepts two arguments. The first argument to `get_data()`

is `add_perc`

, which has a default value of `0.03`

. The value of `add_perc`

drives the amount of *allowance for doubtful debts*. The second argument is `n_years`

, which has a default value of `20`

. The value of `n_years`

drives the number of years of data generated by the simulation.

The simulation generates various cash flows and the financial statements to represent them. The main driver of the model is sales, which follows an autoregressive process. Denoting sales in period \(t\) as \(S_t\), we have

\[ S_{t} - \overline{S} = \rho (S_{t-1} - \overline{S}) + \epsilon_t \] where \(\rho \in (0, 1)\) and \(\epsilon_t \sim N(0, \sigma^2)\).

Sales then drives both cost of goods sold, which are assumed to require cash outlays in the period of sale, and accounts receivable, as all sales are assumed to be on account. The model also addresses collections, write-offs, and dividends. There are no inventories in our model.

In the simulation function, we use “base R” functionality to a fair degree. Rather than using `mutate()`

to generate variables, we refer to variables using `$`

notation, which returns the variable as a vector. For example `df |> select(ni)`

returns a data frame with a single column. In contrast, `df$ni`

gets the same underlying data, but as a vector. To calculate shareholders’ equity (`se`

), we set the initial (\(t=0\)) value to `beg_se`

. Then we calculate the ending balance of shareholders’ equity as beginning shareholders’ equity plus net income minus dividends.

```
get_data <- function(add_perc = 0.03, n_years = 20) {
# Parameters
add_true <- 0.03
gross_margin <- 0.8
beg_cash <- beg_se <- 1500
div_payout <- 1
mean_sale <- 1000
sd_sale <- 100
rho <- 0.9
# Generate sales as an AR(1) process around mean_sale
sale_err <- rnorm(n_years, sd = sd_sale)
sales <- vector("double", n_years)
sales[1] <- mean_sale + sale_err[1]
for (i in 2:n_years) {
sales[i] = mean_sale + rho * (sales[i-1] - mean_sale) + sale_err[i]
}
# Combine data so far into a data frame;
# add slots for variables to come
df <- tibble(year = 1:n_years,
add_perc = add_perc,
sales,
writeoffs = NA, collect = NA,
div = NA, se = NA, ni = NA,
bde = NA, cash = NA)
# All sales at the same margin
df$cogs <- (1 - gross_margin) * df$sales
# All sales are on credit;
# collections/write-offs occur in next period
df$ar <- df$sales
# Allowance for doubtful debts
df$add <- add_perc * df$sales
# Calculate year-1 values
df$writeoffs[1] <- 0
df$collect[1] <- 0
df$bde[1] <- df$add[1]
df$ni[1] <- df$sales[1] - df$cogs[1] - df$bde[1]
df$div[1] <- df$ni[1] * div_payout
df$cash[1] <- beg_cash + df$collect[1] - df$cogs[1] - df$div[1]
df$se[1] <- beg_se + df$ni[1] - df$div[1]
# Loop through years from 2 to n_years
for (i in 2:n_years) {
df$writeoffs[i] <- add_true * df$ar[i-1]
df$collect[i] <- (1 - add_true) * df$ar[i-1]
df$bde[i] = df$add[i] - df$add[i-1] + df$writeoffs[i]
df$ni[i] <- df$sales[i] - df$cogs[i] - df$bde[i]
df$div[i] <- df$ni[i] * div_payout
df$cash[i] <- df$cash[i-1] + df$collect[i] - df$cogs[i] - df$div[i]
df$se[i] <- df$se[i-1] + df$ni[i] - df$ni[i]
}
df
}
```

To understand a function like this it can be helpful to set values for the arguments (e.g., `add_perc <- 0.03; n_years <- 20`

) and step through the lines of code one by one, intermittently inspecting the content of variables such as `df`

as you do so.^{3}

Let’s generate 1,000 years of data.

```
set.seed(2021)
df_1000 <- get_data(n_years = 1000)
```

The first 20 years are shown in Figure 15.1.

```
df_1000 |>
filter(year <= 20) |>
ggplot(aes(x = year)) +
geom_line(aes(y = sales), colour = "red") +
geom_line(aes(y = mean(sales)), colour = "blue")
```

Now, let’s generate five-hundred random values for the `add_perc`

parameter.

`add_percs <- runif(n = 5000, min = 0.01, max = 0.05)`

We will generate simulated data for each value `add_percs`

and store these data in a list called `res_list`

.

```
set.seed(2021)
res_list <-
map(add_percs, get_data) |>
system_time()
```

```
user system elapsed
16.018 0.067 16.714
```

```
plan(multisession)
res_list <-
future_map(add_percs, get_data,
.options = furrr_options(seed = 2021)) |>
system_time()
```

```
user system elapsed
0.487 0.036 5.253
```

We then make two data frames. The first data frame (`res_df`

) stores all the data in a single data frame using the field `id`

to distinguish one simulation run from another. These runs might be considered as “firms” with each run being independent of the other.

`res_df <- list_rbind(res_list, names_to = "id")`

The second data frame (`results`

) stores the estimated persistence for each simulation run. For this purpose, we create `get_coefs()`

, which calculates *persistence* as the coefficient in a regression of income on its lagged value—a specification similar to that in Sloan (1996)—and returns that value.

```
results <-
res_list |>
map(get_coefs) |>
list_rbind(names_to = "id")
```

We plot the estimated *persistence* value against the assumed value for `add_perc`

in Figure 15.2.

```
results |>
ggplot(aes(x = add_perc, y = persistence)) +
geom_point()
```

### 15.3.3 Exercises

- When generating simulated financial statement data, it is generally important to ensure that the generated data meet basic requirements. What is one fundamental relation that we expect to hold? Does it hold for the data in
`df_1000`

? - Calculate values for cash flows from operating activities and cash flows from financing activities. (Treat payment of dividends as a financing activity.
*Hint*: You may find it easier to use the**direct method**to calculate cash flows from operating activities.) Does the cash flow statement articulate as it should? - How evident is the underlying process generating the sales process from the plot of the first twenty years of data from
`df_1000`

? Does looking at more data help? (Obviously, having a thousand years of data on a firm with a stationary process is not common.) - What is the “correct” value of
`add_perc`

that should be used? Using the plot from`results`

above, what is the relation between values of departing from that value and persistence? Does this agree with your intuition? What’s going on? What aspects of the`add_perc`

-related accounting seem unrealistic? (*Hint*: It may help to use variant of the following code`set.seed(2021); get_data(0.03)`

for various values in place of`0.03`

and to examine how the earnings process is affected.) - Does the simulation analysis speak to the underlying rationale for H1 of Sloan (1996)? If so, why? If not, what might be missing from the analysis? How might we modify the simulation to incorporate the missing elements?

## 15.4 Replicating Sloan (1996)

To better understand some elements of the empirical analysis of Sloan (1996), we conduct a replication analysis.

We start with pulling the data together. First we make a subset of `comp.funda`

, add SIC data from `comp.company`

, and call the result `funda_mod`

. We construct SIC codes using `sich`

, which provides the “historical” SIC code, where available, but we use the “header” SIC code found on `comp.company`

when `sich`

is unavailable.

```
pg <- dbConnect(RPostgres::Postgres(),
bigint = "integer",
check_interrupts = TRUE)
funda <- tbl(pg, Id(schema = "comp", table = "funda"))
company <- tbl(pg, Id(schema = "comp", table = "company"))
ccmxpf_lnkhist <- tbl(pg, Id(schema = "crsp", table = "ccmxpf_lnkhist"))
msf <- tbl(pg, Id(schema = "crsp", table = "msf"))
sics <-
company |>
select(gvkey, sic)
funda_mod <-
funda |>
filter(indfmt == "INDL", datafmt == "STD",
consol == "C", popsrc == "D") |>
left_join(sics) |>
mutate(sic = coalesce(sich, as.integer(sic)))
```

We next apply the same sample selection criteria as Sloan (1996). We focus on NYSE and AMEX firm-years (i.e., ones with `exchg`

equal to 11 and 12, respectively) and years between 1962 and 1991.

Sloan (1996, p. 293) suggests that “the financial statement data required to compute operating accruals are not available … on Compustat for banks, life insurance or property and casualty companies.” However, it is not clear if these firms are explicitly excluded (e.g., by filtering on SIC codes) or implicitly excluded by simply requiring that data for calculating accruals be available. As such, we retain these firms and merely create a related indicator variable (`finance`

).

The next step is to create variables to reflect changes in key variables. We can use `lag()`

to do this. The `lag()`

function is a **window function**, meaning that it operates on a “window” of data. Here the windows will be data for each `gvkey`

ordered by `datadate`

. Within each `gvkey`

, for each row we look at the previous row (if any) and grab the value we need.

```
acc_data_raw <-
funda_mod |>
filter(!is.na(at),
pddur == 12,
exchg %in% c(11L, 12L)) |>
mutate(finance = between(sic, 6000, 6999),
across(c(che, dlc, txp), \(x) coalesce(x, 0))) |>
group_by(gvkey) |>
window_order(datadate) |>
mutate(avg_at = (at + lag(at)) / 2,
d_ca = act - lag(act),
d_cash = che - lag(che),
d_cl = lct - lag(lct),
d_std = dlc - lag(dlc),
d_tp = txp - lag(txp)) |>
select(gvkey, datadate, fyear, avg_at, at, oiadp, dp, finance,
starts_with("d_"), sic, pddur) |>
mutate(acc_raw = (d_ca - d_cash) - (d_cl - d_std - d_tp) - dp) |>
ungroup() |>
filter(between(fyear, 1962, 1991),
avg_at > 0)
```

Some key analyses of Sloan (1996) use variables coded as deciles. To create such variables, we use two functions. The `get_deciles()`

function returns the cut-off values for each decile.

Then `form_deciles()`

uses `get_deciles()`

to convert the values of the supplied vector (`x`

) to deciles ranging from 1 to 10.

```
form_deciles <- function(x) {
cuts <- get_deciles(x)
cut(x, cuts[[1]], labels = FALSE, include.lowest = TRUE)
}
```

The final step in our data preparation calculates the core variables `earn`

, `acc`

, and `cfo`

according to the definitions found in Sloan (1996), creates a variable to store the leading value of `earn`

(using the `lead()`

function, which is a window function that complements the `lag()`

function we used above), creates deciles for `acc`

, `earn`

, `cfo`

, and `lead_earn`

, creates a two-digit SIC code, and finally filters out `finance`

firms and observations without values for `acc`

.

```
acc_data <-
acc_data_raw |>
mutate(earn = oiadp / avg_at,
acc = acc_raw / avg_at,
cfo = earn - acc) |>
group_by(gvkey) |>
window_order(datadate) |>
mutate(lead_earn = lead(earn)) |>
ungroup() |>
collect() |>
mutate(acc_decile = form_deciles(acc),
earn_decile = form_deciles(earn),
cfo_decile = form_deciles(cfo),
lead_earn_decile = form_deciles(lead_earn),
sic2 = substr(as.character(sic), 1, 2)) |>
filter(!finance, !is.na(acc))
```

The next step is to collect data on stock returns for each firm-year. We use `ccm_link`

, which we saw in Chapter 8 to link GVKEYs (Compustat) to PERMNOs (CRSP).

Following Sloan (1996), we link `(gvkey, datadate)`

combinations with for the twelve-month period beginning four months after the end of the fiscal period.

```
crsp_link <-
acc_data_raw |>
select(gvkey, datadate) |>
inner_join(ccm_link, by = "gvkey") |>
filter(between(datadate, linkdt, linkenddt)) |>
select(gvkey, datadate, permno) |>
mutate(start_month = as.Date(floor_date(datadate + months(4L), "month")),
end_month = as.Date(floor_date(datadate + months(16L) - days(1L),
"month")))
```

Table 4 of Sloan (1996, p. 304) uses abnormal returns, which are “computed by taking the raw buy-hold return … and subtracting the buy-hold return of a size-matched, value-weighted portfolio of firms. The size portfolios are based on market value of equity deciles of NYSE and AMEX firms.” We obtained the returns of individual firms above, but need to collect data on size portfolios, both the returns for each portfolio and the market capitalization cut-offs.

Data for size portfolios come from Ken French’s website, as we saw in Chapter 11. Code like used in Chapter 11 is included in the `farr`

package in two functions: `get_size_rets_monthly()`

and `get_me_breakpoints()`

.

```
size_rets <- get_size_rets_monthly()
size_rets
```

```
# A tibble: 11,700 × 4
month decile ew_ret vw_ret
<date> <int> <dbl> <dbl>
1 1926-07-01 1 -0.0142 -0.0012
2 1926-07-01 2 0.0029 0.0052
3 1926-07-01 3 -0.0015 -0.0005
4 1926-07-01 4 0.0088 0.0082
5 1926-07-01 5 0.0145 0.0139
6 1926-07-01 6 0.0185 0.0189
7 1926-07-01 7 0.0163 0.0162
8 1926-07-01 8 0.0138 0.0129
9 1926-07-01 9 0.0338 0.0353
10 1926-07-01 10 0.0329 0.0371
# ℹ 11,690 more rows
```

The table returned by `get_size_rets_monthly()`

has four columns, including two measures of returns: one based on equal-weighted portfolios (`ew_ret`

) and one based on value-weighted portfolios (`vw_ret`

). Like Sloan (1996), we will use `vw_ret`

.

```
me_breakpoints <- get_me_breakpoints()
me_breakpoints
```

```
# A tibble: 11,770 × 4
month decile me_min me_max
<date> <int> <dbl> <dbl>
1 1925-12-01 1 0 2.38
2 1925-12-01 2 2.38 4.95
3 1925-12-01 3 4.95 7.4
4 1925-12-01 4 7.4 10.8
5 1925-12-01 5 10.8 15.6
6 1925-12-01 6 15.6 22.9
7 1925-12-01 7 22.9 38.4
8 1925-12-01 8 38.4 65.8
9 1925-12-01 9 65.8 142.
10 1925-12-01 10 142. Inf
# ℹ 11,760 more rows
```

The table returned by `get_me_breakpoints()`

has four columns indicating that firms with market capitalization between `me_min`

and `me_max`

in a given `month`

should be assigned to size decile `decile`

.

The following code assigns firm-years (i.e. `(permno, datadate)`

combinations) to size deciles according to market capitalization and size cut-offs applicable during the month of `datadate`

.

```
crsp_dates <-
msf |>
select(date) |>
distinct() |>
mutate(month = floor_date(date, 'month'))
me_values <-
crsp_link |>
mutate(month = floor_date(datadate, 'month')) |>
inner_join(crsp_dates, by = "month") |>
inner_join(msf, by = c("permno", "date")) |>
mutate(mktcap = abs(prc) * shrout / 1000.0) |>
select(permno, datadate, month, mktcap) |>
collect()
me_decile_assignments <-
me_breakpoints |>
inner_join(me_values,
join_by(month, me_min <= mktcap, me_max > mktcap)) |>
select(permno, datadate, decile)
```

For each `datadate`

and size decile, the following code calculates the cumulative returns over the twelve-month period beginning four months after `datadate`

.

```
cum_size_rets <-
me_decile_assignments |>
select(datadate, decile) |>
distinct() |>
mutate(start_month = datadate + months(4),
end_month = datadate + months(16)) |>
inner_join(size_rets,
join_by(decile, start_month <= month, end_month >= month)) |>
group_by(datadate, decile) |>
summarize(ew_ret = exp(sum(log(1 + ew_ret), na.rm = TRUE)) - 1,
vw_ret = exp(sum(log(1 + vw_ret), na.rm = TRUE)) - 1,
n_size_months = n(),
.groups = "drop")
```

Now we have the data we need to calculate size-adjusted returns. We simply combine `crsp_data`

with `me_decile_assignments`

and then with `cum_size_rets`

and calculate `size_adj_ret`

as a simple difference.

```
size_adj_rets <-
crsp_data |>
inner_join(me_decile_assignments, by = c("permno", "datadate")) |>
inner_join(cum_size_rets, by = c("datadate", "decile")) |>
mutate(size_adj_ret = ret - vw_ret) |>
select(gvkey, datadate, size_adj_ret, n_months, n_size_months)
```

For our regression analysis, we simply join our processed data from Compustat (`acc_data`

) with our new data on size-adjusted returns (`size_adj_rets`

).

Before running regression analyses, it is important to examine our data. One useful benchmark is the descriptive statistic reported in Sloan (1996). The following descriptive statistics are quite similar to those reported in Table 1 of Sloan (1996), providing some degree of assurance.

```
reg_data |>
group_by(acc_decile) |>
summarize(across(c(acc, earn, cfo), \(x) mean(x, na.rm = TRUE))) |>
kbl(digits = 3, booktabs = TRUE)
```

acc_decile | acc | earn | cfo |
---|---|---|---|

1 | -0.175 | 0.035 | 0.211 |

2 | -0.087 | 0.095 | 0.182 |

3 | -0.061 | 0.104 | 0.165 |

4 | -0.044 | 0.111 | 0.155 |

5 | -0.030 | 0.118 | 0.148 |

6 | -0.017 | 0.118 | 0.135 |

7 | -0.001 | 0.127 | 0.128 |

8 | 0.023 | 0.135 | 0.113 |

9 | 0.060 | 0.142 | 0.082 |

10 | 0.172 | 0.154 | -0.018 |

### 15.4.1 Table 2 of Sloan (1996)

Having done a very basic check of our data, we can create analogues of some of the regression analyses found in Sloan (1996).

The following output parallels the “pooled” results in Table 2 of Sloan (1996).

```
fms <- list()
fms[[1]] <- lm(lead_earn ~ earn, data = reg_data)
fms[[2]] <- lm(lead_earn_decile ~ earn_decile, data = reg_data)
modelsummary(fms,
estimate = "{estimate}{stars}",
gof_map = c("nobs", "r.squared"),
stars = c('*' = .1, '**' = 0.05, '***' = .01))
```

(1) | (2) | |
---|---|---|

(Intercept) | 0.030*** | 1.396*** |

(0.001) | (0.039) | |

earn | 0.704*** | |

(0.005) | ||

earn_decile | 0.749*** | |

(0.006) | ||

Num.Obs. | 12877 | 12877 |

R2 | 0.607 | 0.553 |

To produce “industry level” analysis like that in Table 2 of Sloan (1996), we create a small function (`run_table_ind()`

) to produce regression coefficients by industry and

The function `stats_for_table()`

compiles descriptive statistics.

Finally, `summ_for_table()`

calls `run_table_ind()`

and `stats_for_table()`

and produces a summary table.

```
summ_for_table <- function(lhs = "lead_earn", rhs = "earn") {
reg_data |>
distinct(sic2) |>
pull() |>
map(run_table_ind, lhs = lhs, rhs = rhs) |>
list_rbind() |>
select(-sic2) |>
map(stats_for_table) |>
list_rbind(names_to = "term") |>
kbl(digits = 3, booktabs = TRUE)
}
```

The following tables parallel the “industry level” results reported in Table 2 of Sloan (1996).

`summ_for_table(lhs = "lead_earn", rhs = "earn")`

term | mean | q1 | median | q3 |
---|---|---|---|---|

(Intercept) | 0.026 | 0.012 | 0.026 | 0.036 |

earn | 0.664 | 0.611 | 0.697 | 0.815 |

`summ_for_table(lhs = "lead_earn_decile", rhs = "earn_decile")`

term | mean | q1 | median | q3 |
---|---|---|---|---|

(Intercept) | 1.612 | 0.937 | 1.257 | 1.745 |

earn_decile | 0.670 | 0.649 | 0.739 | 0.819 |

Our results thus far might be described as “qualitatively similar” to those in Table 2 of Sloan (1996). The main difference may be in the magnitude of the pooled coefficient on `earn`

in the regression with `lead_earn`

as the dependent variable. Table 2 of Sloan (1996) reports a coefficient of 0.841, notably higher than the mean coefficient from the industry-level regressions (0.773). In contrast, the mean coefficients in our pooled and industry-level analyses are much closer to each other.

### 15.4.2 Table 3 of Sloan (1996)

In Table 3, Sloan (1996) decomposes the right-hand side variable from above into accrual and cash-flow components. We replicate these analyses in Table 15.7.

```
modelsummary(fms,
estimate = "{estimate}{stars}",
gof_map = c("nobs", "r.squared"),
stars = c('*' = .1, '**' = 0.05, '***' = .01))
```

(1) | (2) | |
---|---|---|

(Intercept) | 0.028*** | −1.311*** |

(0.001) | (0.073) | |

acc | 0.640*** | |

(0.007) | ||

cfo | 0.723*** | |

(0.005) | ||

acc_decile | 0.534*** | |

(0.007) | ||

cfo_decile | 0.760*** | |

(0.007) | ||

Num.Obs. | 12877 | 12877 |

R2 | 0.613 | 0.470 |

`summ_for_table(lhs = "lead_earn", rhs = "acc + cfo")`

term | mean | q1 | median | q3 |
---|---|---|---|---|

(Intercept) | 0.025 | 0.010 | 0.024 | 0.031 |

acc | 0.671 | 0.589 | 0.667 | 0.766 |

cfo | 0.714 | 0.651 | 0.723 | 0.828 |

`summ_for_table(lhs = "lead_earn_decile", rhs = "acc_decile + cfo_decile")`

term | mean | q1 | median | q3 |
---|---|---|---|---|

(Intercept) | -0.880 | -2.029 | -1.284 | -0.587 |

acc_decile | 0.485 | 0.438 | 0.520 | 0.598 |

cfo_decile | 0.696 | 0.672 | 0.735 | 0.845 |

Again we have “qualitatively similar” results to those found in Sloan (1996).

### 15.4.3 Pricing of earnings components

An element of the analysis reported in Table 5 of Sloan (1996) regresses abnormal returns on contemporaneous earnings and components of lagged earnings and we provide results from such a regression here.

```
mms <- list(lm(size_adj_ret ~ lead_earn + acc + cfo,
data = reg_data),
lm(size_adj_ret ~ lead_earn_decile + acc_decile + cfo_decile,
data = reg_data))
modelsummary(mms,
estimate = "{estimate}{stars}",
gof_map = c("nobs", "r.squared"),
stars = c('*' = .1, '**' = 0.05, '***' = .01))
```

(1) | (2) | |
---|---|---|

(Intercept) | −0.027*** | 0.186*** |

(0.006) | (0.018) | |

lead_earn | 2.538*** | |

(0.061) | ||

acc | −1.961*** | |

(0.062) | ||

cfo | −1.826*** | |

(0.057) | ||

lead_earn_decile | 0.082*** | |

(0.002) | ||

acc_decile | −0.055*** | |

(0.002) | ||

cfo_decile | −0.056*** | |

(0.002) | ||

Num.Obs. | 12867 | 12867 |

R2 | 0.124 | 0.109 |

In the notation of Sloan (1996), the coefficient on `acc`

can be expressed as \(- \beta \gamma^{*}_1\), which is minus one times the product of \(\beta\), the coefficient on `lead_earn`

(i.e., earnings roughly contemporaneous with `size_adj_ret`

), and \(\gamma^{*}_1\), the implied market coefficient on accruals.

With estimates of \(\hat{\beta} = 2.538\) and \(\widehat{\beta \gamma^{*}_1} = 1.961\), we have an implied estimate of \(\hat{\gamma}^{*}_1 = 0.7726.\)

This estimate \(\hat{\gamma}^{*}_1 = 0.7726\) is higher than the estimate of \(\hat{\gamma}_1 = 0.6399\). But can we conclude that the difference between these two coefficients is statistically significant?

One approach to this questions would be to evaluate whether \(\widehat{\beta \gamma^{*}_1} = 1.961\) as estimated from the market regression is statistically different from the value implied by \(\hat{\beta} \times \hat{\gamma_1} = 2.538 \times 0.6399 = 1.624\).

However, as pointed out by Mishkin (1983) , this procedure “implicitly assumes that there is no uncertainty in the estimate of \(\hat{\gamma_1}\). This results in inconsistent estimates of the standard errors of the parameters and hence test statistics that do not have the assumed F distribution. This can lead to inappropriate inference ….”^{4}

Given the issue of “inappropriate inference” described above, Mishkin (1983) uses “iterative weighted non-linear least squares” (Sloan (1996), p. 302) to estimate a system of equations and then calculates an F-statistic based on comparison of goodness-of-fit of an unconstrained system of equations with that of a constrained system of equations (i.e., one in which \(\gamma\) is constrained equal in both equations). While Sloan (1996) uses this “Mishkin (1983)” test in his analysis reported in Tables 4 and 5, this approach involves significant complexity.^{5}

Fortunately, Abel and Mishkin (1983) suggest a simpler approach that they show is asymptotically equivalent to the Mishkin test. The intuition for this approach is that if components of lagged earnings (accruals and cash flows) are mispriced in a way that predicts stock returns, then this should be apparent from a regression of stock returns on those lagged earnings components. Kraft et al. (2007) provide additional discussion of the Mishkin test and the approach used in Abel and Mishkin (1983).^{6} In effect, this allows us to skip the “middleman” of contemporaneous earnings in the regression analysis.

The following regression results come from applying the approach suggested by Abel and Mishkin (1983).

```
eff <- list(lm(size_adj_ret ~ acc + cfo, data = reg_data),
lm(size_adj_ret ~ acc_decile + cfo_decile, data = reg_data))
modelsummary(eff,
estimate = "{estimate}{stars}",
gof_map = c("nobs", "r.squared"),
stars = c('*' = .1, '**' = 0.05, '***' = .01))
```

(1) | (2) | |
---|---|---|

(Intercept) | 0.041*** | 0.078*** |

(0.006) | (0.018) | |

acc | −0.327*** | |

(0.049) | ||

cfo | 0.038 | |

(0.037) | ||

acc_decile | −0.011*** | |

(0.002) | ||

cfo_decile | 0.006*** | |

(0.002) | ||

Num.Obs. | 13798 | 13798 |

R2 | 0.005 | 0.007 |

Here we can see (in the first regression) that lagged accruals are negatively associated with abnormal returns, consistent with the market overpricing these accruals due to an assumed level of persistence that is higher than warranted.

### 15.4.4 Exercises

In the code creating

`acc_data_raw`

, we used`coalesce()`

to set the value of certain variables to zero when missing on Compustat. Does this seem appropriate here? Are the issues similar to those observed with regard to R&D in Chapter 7? It may be helpful to find some observations from recent years where this use of the`coalesce()`

function has bite and think about the issues in context of financial statements for those firm-years.Can you reconcile the results from the Abel and Mishkin (1983) test with those from the previous regressions? (

*Hint*: Pay attention to sample composition; you may need to tweak these regressions.)The equations estimated in Table 5 of Sloan (1996) could be viewed as a structural (causal) model. Can you represent this model using a causal diagram? In light of the apparent econometric equivalence between that structural model and the estimation approach used in Abel and Mishkin (1983), how might the structural model using the latter approach?

A critique of Sloan (1996) made by Kraft et al. (2007) is that the coefficients may be biased due to omitted variables. This critique implies a causal interpretation of the coefficients in Sloan (1996). How might the critique of Kraft et al. (2007) be represented on the causal diagrams above? How persuasive do you find the the critique of Kraft et al. (2007) to be?

Apart from the different data sources used, another difference between the simulation analysis earlier in this chapter and the regression analysis in Table 3 of Sloan (1996) is the regression model used. Modify the code below to incorporate the appropriate formulas for cash flow from operating activities (

`cfo`

) and accruals (`acc`

). Then replicate the pooled analysis of Panel A of Table 3 of Sloan (1996) using the resulting`sim_reg_data`

data frame. What do you observe?

```
<-
sim_reg_data |>
res_df mutate(cfo = [PUT CALC HERE], acc = [PUT CALC HERE]) |>
group_by(id) |>
arrange(id, year) |>
mutate(lag_cfo = lag(cfo),
lag_acc = lag(acc)) |>
ungroup()
```

- Which hypothesis does Figure 1 of Sloan (1996) relate to? What aspects of the plot make it easier or more difficult to interpret the results? The following code replicates a version of Figure 1 from Sloan (1996) using our simulated data. On the basis of Figure 15.3, Figure 15.4, and Figure 15.5, and the arguments given in Sloan (1996), is H1 true in our simulated data? Given the other analysis above, is H1 true in our simulated data?

```
year_of_event <- 10
decile_data <-
sim_reg_data |>
filter(year == year_of_event) |>
mutate(cfo_decile = form_deciles(cfo),
ni_decile = form_deciles(ni),
acc_decile = form_deciles(acc)) |>
select(id, ends_with("decile"))
reg_data_deciles <-
sim_reg_data |>
inner_join(decile_data, by = "id")
```

```
reg_data_deciles |>
filter(ni_decile %in% c(1, 10)) |>
mutate(ni_decile = as.factor(ni_decile),
event_year = year - year_of_event) |>
group_by(ni_decile, year) |>
summarize(ni = mean(ni, na.rm = TRUE), .groups = "drop") |>
ggplot(aes(x = year, y = ni, group= ni_decile, color = ni_decile)) +
geom_line()
```

```
reg_data_deciles |>
filter(cfo_decile %in% c(1, 10)) |>
mutate(cfo_decile = as.factor(cfo_decile),
event_year = year - year_of_event) |>
group_by(cfo_decile, year) |>
summarize(ni = mean(ni, na.rm = TRUE), .groups = "drop") |>
ggplot(aes(x = year, y = ni, group = cfo_decile, color = cfo_decile)) +
geom_line()
```

```
reg_data_deciles |>
filter(acc_decile %in% c(1, 10)) |>
mutate(acc_decile = as.factor(acc_decile),
event_year = year - year_of_event) |>
group_by(acc_decile, year) |>
summarize(ni = mean(ni, na.rm = TRUE), .groups = "drop") |>
ggplot(aes(x = year, y = ni, group = acc_decile, color = acc_decile)) +
geom_line()
```

## 15.5 Accrual anomaly

Table 6 of Sloan (1996) provides evidence that the market’s apparent mispricing of accruals implies trading strategies that give rise to abnormal returns. Such strategies are generally termed **anomalies** because they are inconsistent with efficient markets hypothesis. Fama and French (2008, p. 1653) define anomalies as “patterns in average stock returns that … are not explained by the Capital Asset Pricing Model (CAPM).” Implicit in the Fama and French (2008) seems to be the notion that the CAPM is the true model of market risk and a general version of the definition of Fama and French (2008) would replace the CAPM with the posited true model of market risk. Dechow et al. (2011, p. 23) argue that “the accrual anomaly is not really an anomaly at all. In fact, the original research documenting the accrual anomaly predicted that it would be there. The term anomaly is usually reserved for behavior that deviates from existing theories, but when Sloan (1996) first documented the accrual anomaly, he was testing a well-known theory and found that it was supported.”

While Table 6 provides portfolio returns for years \(t+1\), \(t+2\), and \(t+3\), we only collected returns for year \(t+1\) in the steps above.

```
modelsummary(fm,
estimate = "{estimate}{stars}",
gof_map = c("nobs", "r.squared"),
stars = c('*' = .1, '**' = 0.05, '***' = .01))
```

(1) | |
---|---|

acc_decile1 | 0.120*** |

(0.020) | |

acc_decile2 | 0.109*** |

(0.020) | |

acc_decile3 | 0.042** |

(0.020) | |

acc_decile4 | 0.058*** |

(0.020) | |

acc_decile5 | 0.067*** |

(0.020) | |

acc_decile6 | 0.042** |

(0.020) | |

acc_decile7 | 0.025 |

(0.020) | |

acc_decile8 | 0.034* |

(0.020) | |

acc_decile9 | −0.003 |

(0.020) | |

acc_decile10 | −0.012 |

(0.020) | |

Num.Obs. | 300 |

R2 | 0.255 |

```
hedge_ret <- fm$coefficients["acc_decile1"] - fm$coefficients["acc_decile10"]
p_val <- linearHypothesis(fm, "acc_decile1 = acc_decile10")$`Pr(>F)`[2]
```

The hedge portfolio return is 0.1322 with a \(p\)-value of 4.4e-06.

### 15.5.1 Discussion questions

- In estimating the hedge portfolio regression, we included a line
`summarize(size_adj_ret = mean(size_adj_ret))`

. Why is this step important? - Green et al. (2011) say “The simplicity of the accruals strategy and the size of the returns it generates have led some scholars to conclude that the anomaly is illusory. For example, Khan (2008) and Wu et al. (2010) argue that the anomaly can be explained by a mis-specified risk model and the q-theory of time-varying discount rates, respectively; Desai et al. (2004) conclude that the anomaly is deceptive because it is subsumed by a different strategy; Kraft et al. (2006) attribute it to outliers and look-ahead biases; Ng (2005) proposes that the anomaly’s abnormal returns are compensation for high exposure to bankruptcy risk; and Zach (2006) argues that there are firm characteristics correlated with accruals that cause the return pattern.” Looking at Sloan (1996), but without necessarily looking at each of the papers above, what evidence in Sloan (1996) seems inconsistent with the claims made by each paper above? Which do you think you would need to look more closely at the paper to understand? What evidence do you think Zach (2006) would need to provide to support the claim of an alternative “cause”?
- Do Green et al. (2011) address the alternative explanations advanced in the quote in Q1 above? Do you think that they need to do so?
- How persuasive do you find the evidence regarding the role of hedge funds in Green et al. (2011)?
- Xie (2001) (p. 360) says that “for firm-years prior to 1988 when Compustat item #308 is unavailable, I estimate \(\textit{CFO}_t\) as follows …”. Why would item #308 be unavailable prior to 1988? What is the equivalent to #308 in Compustat today?
- Study the empirical model on p. 361 of Xie (2001), which is labelled equation (1). (This is the much-used “Jones model” from Jones (1991).) What are the assumptions implicit in this model and the labelling of the residual as “abnormal accruals”? (Hint: Take each component of the model and identify circumstances where it would be a reasonable model of “normal” accruals.)
- What is “channel stuffing”? (Hint: Wikipedia has a decent entry on this.) What effect would channel stuffing have on abnormal accruals? (Hint: Think about this conceptually and with regard to equation (1). Do you need more information than is provided in Xie (2001) to answer this?)

This is one definition that can be tightened and vary by context.↩︎

Wickham et al. (2023, p. 441) suggest that “a good rule of thumb is to consider writing a function whenever you’ve copied and pasted a block of code more than twice.”↩︎

Indeed, this is the process often used to create a function like this in the first place.↩︎

Note that Mishkin (1983) is actually critiquing a different econometric procedure whereby residuals from the first regression are included in a version of the second, but the quoted criticism is equally applicable to the procedure we describe here.↩︎

This is apparent from inspection of the Stata

`.ado`

file provided by Judson Caskey to implement the Mishkin (1983) approach.↩︎Note Kraft et al. (2007) appear to assume that the OLS test used in Mishkin (1983) is the same as the test proposed in Abel and Mishkin (1983), but differences in these tests do not affect the substance of the discussion in Abel and Mishkin (1983).↩︎