Classification shifting using income-decreasing special items: measurement and valuation issues

Abstract

Research suggests that the standard model used to detect opportunistic shifting of core expenses to special items is potentially biased. Such bias has been attributed to the use of accruals, including special item related accruals, as a control for the impact of performance on core earnings in this model. This paper provides an improved classification shifting model which both tests for such accruals-related bias and controls for other sources of error in the measurement of shifting. The paper also modifies conventional market rationality tests in accounting research to examine new dimensions of rationality in relation to measurement and valuation of shifting. The main empirical findings are as follows. First, the improved classification shifting model provides strong evidence of shifting and rejects the hypothesis that inclusion of accruals in the model causes bias. Second, estimates of shifted core expenses generated by the improved model exhibit forecasting properties of shifted earnings. Third, rationality test results are broadly consistent with rationality in relation to shifted core expenses but indicate possible partial (ir)rationality in relation to adjusted special items (i.e., special items excluding shifted core expenses). Further analysis of the latter findings, however, suggests they are more likely related to risk than irrationality. Overall, the paper contributes to improved measurement of shifting and highlights the importance of considering rational expectations when examining stock returns associated with shifting.

1 Introduction

Expense misclassification is the opportunistic transfer of operating expenses to special items in order to increase core profitability. The SEC has issued several Accounting and Auditing Enforcement Releases on such classification shifting (CS), and there is considerable research into the issue. This paper addresses the issue with a model for measuring the shifting which corrects for potential bias in previous approaches. Further, there is limited evidence on how investors respond to CS, so the paper also provides such evidence with specified market valuation and rationality tests.

When operating expenses are opportunistically shifted to special items, current core earnings are higher than expected. Thus, standard CS models identify shifting in the form of a positive contemporaneous relation between unexpected core earnings and income-decreasing special items (IDSI). Our modeling of CS deals with two measurement issues in previous research: first, the usefulness of including total accruals as a control variable for performance when estimating expected core earnings in a CS model, and second, the potential bias in the measurement of shifting due to the presence of accruals related to IDSI within total accruals. These measurement issues are related in a somewhat circular fashion. Specifically, as firm performance varies with accruals, they play an important role in controlling for current underlying economic performance when measuring expected core earnings in CS models. However, as special items (including unobservable shifted core expenses) are mostly accruals, the inclusion of accruals in CS models may induce a mechanical relation between unexpected core earnings and IDSI.

Our analysis confirms that total accruals, including special item accruals, are an important control for performance in a CS model. Previous approaches to solving the aforementioned circularity problem excluded accruals from the CS model but could not document clear evidence of shifting. This is because exclusion of accruals removes an important control for firm performance, which then undermines the role of the IDSI coefficient as an indicator of shifting. In contrast, we provide empirical results supporting inclusion of accruals as a performance control and develop a simple adjustment to the IDSI coefficient that corrects for bias due to inclusion of accruals related to shifted core expenses in total accruals. Our CS model therefore avoids “throwing out the (accruals) baby with the bathwater” whilst correcting for bias in estimation of CS related to the effect of shifting-related accruals.

Empirical analysis based on our modified CS model provides strong evidence that IDSI are used to inflate current core operating earnings. In corroboration, our estimates of shifted core expenses contained in special items (i) have earnings forecasting properties statistically indistinguishable from earnings before special items, consistent with their incorrect omission from earnings before special items; and (ii) are associated with future accounting restatements linked to special items, consistent with managerial opportunism. On the other hand, estimates of adjusted special items (i.e., special items excluding shifted core expenses) based on our model (i) have much weaker earnings forecasting ability, and (ii) show no significant association with future accounting restatements. Additional validation tests, including distinguishing results for subsamples based on investor sophistication, scope of shifting, and future shifting, provide further empirical confirmation.

In relation to our market response tests, we modify conventional rationality tests with an approach that integrates and extends previous research. Specifically, our analysis synthesizes rational expectations approaches in accounting research with the abnormal earnings valuation perspective. This explicitly allows for partial rationality scenarios where investors can use correct (incorrect) valuation multipliers for shifting-related variables but respond to incorrectly (correctly) estimated shocks in these variables. In other words, the new approach breaks down rationality in previous research into dimensions reflecting whether capital markets correctly apply appropriate valuation multipliers and accurately measure accounting shocks.

Using this new approach to assess the market response to shifting, we provide empirical evidence that is broadly consistent with rationality in relation to shifted core expenses: the market responds in a timely manner based on the implications of shifted core expenses for the present value of future forecasted abnormal earnings. The tests for adjusted special items (which exclude shifted core expenses by construction), on the other hand, indicate possible partial (ir)rationality, with the market applying the appropriate valuation coefficient but appearing to assume that past adjusted special items have a larger negative impact on expected current earnings than is implied by forecast models. Additional analysis, however, shows that such apparent partial irrationality is better explained by risk related to special items than by market irrationality. Different from previous research, our empirical findings indicate that investors respond to shifting more rationally than previously thought.

Our study contributes to the literature in two important respects. First, we offer an improved CS model, refining the measurement of the key CS parameter in the traditional model. Our refinement of the CS parameter leads to a reduction in potential estimation bias and a correction for inclusion of shifted core expenses in accruals. The modeling yields reasonable measures of shifted core expenses and adjusted special items with differing properties broadly as expected. Second, we develop a new joint test of capital market rationality with several implications for shifting. It considers two dimensions of rationality related to measurement and valuation of accounting surprises which have not been combined in previous research. The modeling views rationality on a broad spectrum, offering scenarios under which types of rationality and irrationality, i.e., partial (ir)rationality, may coexist. It also distinguishes between irrationality and risk when accounting variables forecast future returns. Using this approach, we provide evidence that the market response to shifting is broadly consistent with rational pricing. The results run counter to previous research attributing an observed relation between a component of special items and future returns to mispricing. Being theoretically grounded, our rationality test offers a tool for examining sources of partial (ir)rationality, if any, and whether risk might better explain apparent irrationality.

The remainder of the paper is organized as follows. Section 2 summarizes the main approaches to measuring CS and develops our approach. Section 3 reports empirical findings on the measurement of shifting and the ability of our estimate of shifted core expenses to forecast future accounting restatements and future earnings, together with initial results on the contemporaneous stock return response to shifting. Section 4 develops our stock return model for testing value relevance and market rationality in relation to shifted core expenses and other earnings components. Section 5 presents empirical results based on this model and considers possible ways to reconcile the results with those based on other approaches. Section 6 concludes the paper.

2 Classification shifting using special items: conceptual and measurement issues

2.1 Overview

First, we summarize two approaches to identifying CS and indicate the approach adopted in this paper. Second, we review the standard CS model developed in McVay (2006) and consider how the model has been implemented in subsequent research. Third, we improve the standard model to reduce potential estimation error in the CS regression parameter and develop a new measure of shifted core expenses based on the improved model.

2.2 Previous approaches to measuring classification shifting

Two broad approaches to identifying and measuring CS have been employed in the literature. The first approach attempts to identify CS by hypothesizing that the difference between reported core earnings and expected core earnings (based on observed core earnings drivers) is equal to a core earnings prediction error plus shifted core earnings. We summarize this approach as follows:

$$\begin{array}{c}reported\;{core\;earnings}_t-f\left[{core\;earnings\;drivers}_t\right] = {shifted\;core\;earnings}_t+{error}_t\\=f\left[{special\;items}_t\right]+{error}_t\end{array}$$

(CS1 approach)

where ${error}_{t}$ refers to the prediction error from using the core earnings drivers available at date t to predict true core earnings at date t and where shifted core earnings at date t (included in reported core earnings at date t) are a function of special items at date t. Under “CS1 approach,” shifted core earnings are potentially estimated based on the ability of special items to explain the difference in reported and predicted core earnings, although most prior research has simply tested for a positive statistical association between the difference in reported and predicted core earnings and special items. Previous research using this approach includes the key study by McVay (2006) and further related studies concerned with issues such as earnings management related to quarterly earnings announcements (Fan et al. 2010), corporate governance (Haw et al. 2011), compensation shielding practices (Joo and Chamberlain 2017), and discontinued operations (Barua et al. 2010; Ji et al. 2020).

An alternative approach to the identification and measurement of CS has recently been proposed by Cain et al. (2020). This approach views shifted core expenses or “opportunistic special items” as a function of the difference between reported special items and predicted special items based on a set of economic drivers and can be expressed as follows:

$$reported\;{special\;items}_{t}-f\left[{special\;items\;drivers}_{t}\right]={opportunistic\;special\;items}_{t}$$

(CS2 approach)

where ${special items drivers}_{t}$ refers to the economic drivers of special items, and the residual in relation to the prediction of $reported {special items}_{t}$ by these variables is the measure of opportunistic special items. As discussed by Cain et al. (2020), this measure of opportunistic special items is noisy, as forecasting special items is inherently difficult and the residual from such a forecast model is likely to include some economically driven special items. Evidence provided in their study suggests that the “CS2 approach” generates estimates of opportunistically shifted core expenses which are negatively related to future earnings before special items (as hypothesized), although this relationship is substantially weaker than the relationship between current and future earnings before special items.¹

In comparison, while the CS1 approach represents, by construction, a measure of “present” shifting based on reported core earnings at date t, the CS2 approach does not identify whether opportunistic special items are used to inflate reported core earnings at date t or at earlier or later dates. Thus, in addition to the major methodological differences between the CS1 and CS2 approaches that were previously outlined, there are also differences in the time horizon over which shifting occurs under each approach.

We adopt a CS1 approach because (i) it may be less affected by measurement error than the CS2 approach based on the forecasting of volatile special items; and (ii) it provides a focused measure of “present” shifting based on reported core earnings at date t, which permits analysis of the contemporaneous association with stock returns at date t. Furthermore, while the CS2 approach potentially captures the effect of CS at a particular date on past, present, and future reported core earnings, the aggregation of this intertemporal effect limits empirical analysis of the rationality of the market response to such (noisy) estimates of opportunistic special items under this approach.² We compare empirical findings based on our CS1 approach with Cain et al.’s (2020) findings based on the CS2 approach when we discuss validation tests and stock return tests for our measure of shifted core expenses in Sects. 3 and 5.

2.3 The McVay (2006) model of classification shifting

The McVay (2006) approach to identifying CS using IDSI is based on a two-stage model. In the first stage, unexpected core operating earnings for year t are estimated using the following model:

$${coe}_{t}={\eta }_{0}+ {\eta }_{1}{coe}_{t-1}+{\eta }_{2}{ato}_{t}+{\eta }_{3}{acc}_{t-1}+{\eta }_{4}{acc}_{t}+{\eta }_{5}\Delta {sa}_{t}+{\eta }_{6}neg\Delta {sa}_{t}+{e}_{t}$$

(1)

where ${coe}_{t}$ denotes reported core operating earnings scaled by sales, ${ato}_{t}$ denotes asset turnover equal to the ratio of sales to average net operating assets, ${acc}_{t}$ denotes total reported accruals scaled by sales, $\Delta {sa}_{t}$ denotes change in sales equal to the change in sales scaled by past year sales, and $neg\Delta {sa}_{t}$ denotes negative change in sales equal to $\Delta {sa}_{t}$ if this is negative and 0 otherwise. Unexpected core operating earnings scaled by sales are then defined as ${ucoe}_{t}={coe}_{t}-E\left({coe}_{t}\right)={e}_{t}$. In the second stage, unexpected core earnings scaled by sales for year t are regressed on IDSI as follows:

$${ucoe}_{t}=\alpha +\omega {si}_{t}+{\varepsilon }_{t}$$

(2)

where ${si}_{t}$ denotes IDSI measured as reported special items scaled by sales for year t multiplied by -1 when reported special items are negative, and 0 otherwise. According to this model, $\omega >0$ indicates shifting of core expenses to IDSI. (Hereafter, for presentational economy, we suppress stating that the earnings, accruals, and special items variables in Eqs. (1) and (2) and further related equations in Sect. 2 are all scaled by sales.)

While McVay’s empirical analysis provides support for $\omega >0$ (consistent with the existence of CS), a possible weakness identified by McVay and elaborated by Fan et al. (2010) is that inclusion of contemporaneous accruals, ${acc}_{t}$, as an independent variable in Eq. (1) might induce a “mechanical” positive association between unexpected core operating earnings and ${si}_{t}$ in Eq. (2). Specifically, if ${si}_{t}$ are accrued expenses and hence have a negative impact on ${acc}_{t}$ in Eq. (1), Fan et al. (2010) argue that a positive ${\eta }_{4}$ coefficient will lower expected core operating earnings estimated from Eq. (1) and hence increase unexpected core operating earnings in Eq. (2), leading, potentially, to a mechanical positive relationship.³ In order to avoid this potential problem, Fan et al. (2010) exclude ${acc}_{t}$ from Eq. (1) and find, in their empirical analysis based on quarterly data, that this leads to a negative sign for $\omega$, consistent with IDSI signaling poor core performance (the so-called “performance effect”). Nevertheless, Fan et al. (2010) plausibly interpret their finding of a “less negative” $\omega$ in the fourth quarter relative to other quarters as evidence of CS due to the greater incentive to manage fourth quarter earnings. More generally, however, we interpret their evidence of a negative coefficient for ${si}_{t}$ in Eq. (2) when Eq. (1) is revised to exclude ${acc}_{t}$ as support for the inclusion of total accruals (including accruals related to IDSI) as an explanatory variable for contemporaneous core operating earnings, as in McVay’s (2006) analysis. This is simply because revision of Eq. (1) to exclude total accruals removes an important control for the accruals-related performance effect, which then undermines the role of $\omega$ in Eq. (2) as a useful indicator of CS.

In summary, the empirical findings in McVay (2006) and subsequent research suggest that accruals are an important explanatory variable for core operating earnings when estimating the impact of CS on reported core operating earnings. Figure 1 summarizes the measurement approaches adopted in a number of studies on issues related to CS.⁴ As reflected in Fig. 1, results can vary conditional on how accruals are treated in the CS model, but most studies include accruals as a control for the performance effect in their implementation of the McVay model. We also include accruals as a control for the performance effect but, in contrast to prior research, provide statistical evidence on whether the potential mechanical association between IDSI and core earnings highlighted by Fan et al. (2010) affects our measure of CS.

Fig. 1

Summary of CS studies where core earnings model includes/excludes accruals

2.4 Measurement of classification shifting based on a modified McVay (2006) model

Our approach to measuring shifted core earnings is based on the McVay (2006) model but introduces two measurement modifications to improve estimation of CS. A further advantage of our approach is that it allows us to explicitly test for the potential problem, highlighted by Fan et al. (2010), of including special item accruals in the measurement of total accruals.

2.4.1 Single-step regression procedure

To reduce estimation error in $\omega$ in Eq. (2), our first modification requires the use of a single-step regression procedure in place of the two-step approach used by McVay (2006). Consistent with analysis by Chen et al. (2018), we note that the two-step procedure in the McVay model may result in measurement error in the CS measure, $\omega$, if the covariance between ${si}_{t}$ and other variables assumed to explain reported core operating earnings, ${coe}_{t}$, is non-zero. Estimation error in $\omega$ in Eq. (2) (and in other coefficients in Eq. (1)) is therefore likely unless ${si}_{t}$ is orthogonal to all other explanatory variables for ${coe}_{t}$ in Eq. (1). To correct for this issue, we replace the two-step approach with a one-step regression by adding ${si}_{t}$ to Eq. (1) as follows:

$$\begin{aligned}{coe}_{t}&={\eta }_{0}+ {\eta }_{1}{coe}_{t-1}+{\eta }_{2}{ato}_{t}+{\eta }_{3}{acc}_{t-1}+{\eta }_{4}{acc}_{t}\\&\quad+{\eta }_{5}\Delta {sa}_{t}+{\eta }_{6}neg\Delta {sa}_{t}+{\eta }_{7}{si}_{t}+{e}_{t}\end{aligned}$$

(3)

Following the logic of the McVay model, we expect that ${\eta }_{7}>0$ in the presence of CS.

Our single-step regression procedure also allows us to test for the potential mechanical relationship, identified by Fan et al. (2010), between IDSI and unexpected core earnings. The key issue raised by Fan et al. (2010) concerns the impact of including special item accruals in total accruals if they are irrelevant for generating expectations of core operating earnings. If this is the case, Eq. (3) is mis-specified and should be replaced with:

$$\begin{aligned}{coe}_{t}&={\eta }_{0}+ {\eta }_{1}{coe}_{t-1}+{\eta }_{2}{ato}_{t}+{\eta }_{3}{acc}_{t-1}+{\eta }_{4}\left({acc}_{t}-{acc\_si}_{t}\right)\\&\quad+{\eta }_{5}\Delta {sa}_{t} +{\eta }_{6}neg\Delta {sa}_{t}+{\eta }_{7}{si}_{t}+{e}_{t}\end{aligned}$$

where ${acc\_si}_{t}$ denotes accruals related to IDSI. Given that most IDSI are accruals, we assume that ${acc\_si}_{t}=-{si}_{t}$. Substituting for ${acc\_si}_{t}$ and rearranging gives:

$$\begin{aligned}{coe}_{t}&={\eta }_{0}+ {\eta }_{1}{coe}_{t-1}+{\eta }_{2}{ato}_{t}+{\eta }_{3}{acc}_{t-1}+{\eta }_{4}{acc}_{t}+{\eta }_{5}\Delta {sa}_{t}\\&\quad+{\eta }_{6}neg\Delta {sa}_{t}+\left({\eta }_{4}+{\eta }_{7}\right){si}_{t}+{e}_{t}\end{aligned}$$

(3*)

Equation (3*) shows that, under the scenario where $\left({acc}_{t}-{acc\_si}_{t}\right)$ is the appropriate control for the performance effect, the ${si}_{t}$ coefficient must be greater than or equal to the ${acc}_{t}$ coefficient due to a mechanical effect caused by the ${acc\_si}_{t}$ component of ${acc}_{t}$. That is, given that ${\eta }_{7}=0$ (${\eta }_{7}>0$) in the absence (presence) of classification shifting, it follows that (${\eta }_{4}+{\eta }_{7})\ge {\eta }_{4}$.

This leads to the following testable hypothesis. If an empirical estimation of Eq. (3) shows that the ${si}_{t}$ coefficient is significantly less than the ${acc}_{t}$ coefficient, the hypothesis in Eq. (3*) that special item accruals are not relevant for predicting core operating earnings is not supported, and the coefficient for ${si}_{t}$ can be interpreted as a measure of CS as in Eq. (3).

As we show later in Sect. 3, there is strong evidence that the ${si}_{t}$ coefficient is significantly and substantially less than the ${acc}_{t}$ coefficient. We therefore interpret ${\eta }_{7}$ in Eq. (3) as an indicator of CS consistent with ${acc}_{t}$, rather than $\left({acc}_{t}-{acc\_si}_{t}\right)$, acting as the appropriate accruals-based control for the performance effect in the single-step regression model.

2.4.2 Adjustment of CS measure for potential impact of shifted core expenses on ${{\varvec{a}}{\varvec{c}}{\varvec{c}}}_{{\varvec{t}}}$

While our empirical analysis is based on Eq. (3), we now address one further measurement problem related to ${\eta }_{7}$ as an indicator of CS. Specifically, ${\eta }_{7}$ may be an upwardly biased measure of CS due to the inclusion of accruals related to shifted core expenses in ${acc}_{t}$, and a corrected measure of CS, given by $\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}$ based on Eq. (3), reverses such bias. First, we highlight that accruals related to shifted core expenses may not be relevant for predicting true core operating profit at date t. Second, we show how this assumption leads to our corrected measure of CS, $\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}$.

In relation to shifting-related accruals, we propose that shifted core expenses are likely to be non-cash accrual items, which may more easily be portrayed by managers as transitory in comparison to cash expenses. Previous research strongly suggests that accruals are contemporaneously related to core operating earnings, consistent with weaker profitability on current activities; however, accruals related to shifted core expenses may have no such association with true contemporaneous core operating earnings. This suggests that ${acc}_{t}$ in Eq. (3) should be replaced by $({acc}_{t}-{sce}_{t})$, where ${sce}_{t}$ denotes shifted core expenses expressed as a negative number, in order for the coefficient of ${si}_{t}$ to be correctly interpreted as a measure of CS.⁵

We now show how the above assumption leads to the conclusion that ${\eta }_{7}$ in Eq. (3) is an upwardly biased measure of CS and that dividing ${\eta }_{7}$ by $\left(1+{\eta }_{4}\right)$ corrects for this bias. First, given our assumption that shifted core expenses are accruals and are not relevant for forecasting current true core operating earnings, it follows that current reported core operating earnings, ${coe}_{t}$, can be expressed as:

$$\begin{aligned}{coe}_{t}&={\eta }_{0}+ {\eta }_{1}{coe}_{t-1}+{\eta }_{2}{ato}_{t}+{\eta }_{3}{acc}_{t-1}+{\eta }_{4}\left({acc}_{t}-{sce}_{t}\right)\\&\quad+{\eta }_{5}\Delta {sa}_{t}+{\eta }_{6}neg\Delta {sa}_{t}+\theta {si}_{t}+{e}_{t}\end{aligned}$$

(4)

where $\theta$ is the “correct” CS parameter based on income-decreasing special items, ${si}_{t}$, in a specification that excludes shifted core expenses from accruals. Next, as ${sce}_{t}$ is certainly unobservable in the financial statements, substituting $\theta {si}_{t}=-{sce}_{t}$ in Eq. (4) and rearranging gives:

$$\begin{aligned}{coe}_{t}&={\eta }_{0}+ {\eta }_{1}{coe}_{t-1}+{\eta }_{2}{ato}_{t}+{\eta }_{3}{acc}_{t-1}+{\eta }_{4}{acc}_{t}+{\eta }_{5}\Delta {sa}_{t}\\&\quad+{\eta }_{6}neg\Delta {sa}_{t}+\theta \left(1+{\eta }_{4}\right){si}_{t}+{e}_{t}\end{aligned}$$

(5)

Finally, we note that Eq. (5) is identical to Eq. (3), but with the additional condition that ${\eta }_{7}=\theta \left(1+{\eta }_{4}\right)$. As such, the inclusion of shifted core expenses in ${acc}_{t}$ leads to a mechanical upward bias in ${\eta }_{7}$ in Eq. (3) above the correct CS parameter, $\theta$. It follows that ${\eta }_{7}$ in Eq. (3) must be adjusted downwards by a factor of $\frac{1}{\left(1+{\eta }_{4}\right)}$ in order to provide an estimate of the correct CS parameter, $\theta$.⁶

In summary, our analysis therefore implies a CS measure, $\theta$, given by $\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}$ based on Eq. (3). Shifted core expenses for a particular firm are given by:

$${sce}_{t}=-\theta {si}_{t}=-\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}{si}_{t}$$

(6a)

and adjusted special charges, ${adjsc}_{t}$, representing special item charges excluding ${sce}_{t}$, are given by⁷:

$${adjsc}_{t}=-{si}_{t}-{sce}_{t}=-\left(1-\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}\right){si}_{t}$$

(6b)

In Sect. 3, we report the empirical results from estimating Eqs. (3) and (5), which lend statistical support to our approach to measuring CS; and we use firm-specific estimates of ${sce}_{t}$ and ${adjsc}_{t}$, based on Eqs. (6a) and (6b), for additional empirical tests to validate our approach.

3 Empirical findings for classification shifting measured using modified McVay (2006) model

3.1 Data and estimation methods

We collect annual accounting data from Compustat for the period 1989 to 2017. We follow McVay (2006) in our sample formation. The sample period starts in 1989 after Compustat begins reporting cash flow from operations, which are used to measure accruals.⁸ We exclude firm-years with sales of less than $1 million and those where the fiscal year-end has changed from t-1 (t) to t (t + 1), and include only firm-years with the data required to measure the accounting variables specified in Eq. (3). Consistent with McVay (2006), the core operating earnings and accruals variables used to estimate these models are scaled by sales and winsorized at the 1% level. The full sample comprises 72,568 firm-year observations, which are used to estimate Eq. (3) using ordinary least squares and to estimate Eq. (5) using non-linear least squares.

To estimate shifted core expenses, ${sce}_{t}$, and adjusted special charges, ${adjsc}_{t}$, for all firm-years, we carry out annual cross-sectional estimation of Eq. (3) by industry using the two-digit SIC code for industry membership with at least 15 non-zero IDSI observations. For firm-years where the relevant industry-year estimate of $\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}$ lies between zero and one, we use Eqs. (6a) and (6b) to estimate ${sce}_{t}$ and ${adjsc}_{t}$ as $-\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}{si}_{t}$ and $-\left(1-\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}\right){si}_{t}$, respectively. For firm-years where the relevant industry-year estimate of $\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}$ is less than 0, we assume that ${sce}_{t}=0$ and ${adjsc}_{t}=-{si}_{t}$; for firm-year observations where estimated $\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}$ is greater than 1, we assume that ${sce}_{t}=-{si}_{t}$ and $adj{sc}_{t}=0$.⁹ Finally, while the firm-year estimates of ${sce}_{t}$ and ${adjsc}_{t}$ based on Eqs. (3), (6a), and (6b) are measured as a percentage of sales, our forecasting and valuation analysis uses variables measured on a per-share basis and scaled by lagged stock price. We therefore rescale ${sce}_{t}$ and ${adjsc}_{t}$ by multiplying by sales in year t, dividing by shares outstanding in year t, and scaling by lagged stock price using data from CRSP. In our empirical analysis, the ${sce}_{t}$ and ${adjsc}_{t}$ variables are measured on this basis throughout.

Our empirical analysis of ${sce}_{t}$ and ${adjsc}_{t}$ uses models that require additional accounting and stock return variables. Additional data are collected from Compustat for the accounting variables which act as control variables in a model testing the usefulness of ${sce}_{t}$ and ${adjsc}_{t}$ for forecasting future accounting restatements. Data on accounting restatements is collected from Audit Analytics. We also collect data from Compustat on earnings before special items per share to test the usefulness of ${sce}_{t}$ and ${adjsc}_{t}$ for forecasting future earnings, and we use monthly stock price data from CRSP to estimate abnormal annual stock returns required for stock return analysis. Detailed variable definitions are given in Appendix A. We also obtain the firm’s percentage of institutional holdings out of total shares outstanding from the Thomson Reuters 13F filings, for use in stock return tests related to investor sophistication. The percentage in the 13F filings data is quarterly, and we average it to match with our annual data. The additional models and tests are specified in detail in Sects. 3.4 and 3.5.

3.2 Descriptive statistics

Table 1 reports summary statistics and correlations for the main variables used in our empirical models as defined in Appendix A. Focusing first on the variables used to estimate Eqs. (3) and (5), Panel A shows that the mean of core operating earnings scaled by sales is 7.2%, the mean of accruals scaled by sales is -11.9%, and the mean of IDSI scaled by sales is 2.9%. These summary statistics are similar to those in the prior literature (e.g., McVay 2006; Fan et al. 2010, 2019). The mean of shifted core expenses and the mean of adjusted special charges, estimated from Eqs. (6a) and (6b) using industry-year estimations of Eq. (3) and expressed on a per-share basis scaled by lagged stock price, are -0.1% and -1.5%, respectively. We use earnings before special items per share scaled by lagged stock price and abnormal stock returns in further prediction and value relevance tests; the means for these variables are 3.2% and 2.8%, respectively. The remaining summary statistics are for control variables used in a model testing the relevance of shifted core expenses for forecasting accounting restatement and are broadly consistent with those reported in prior research, as discussed further in Sect. 3.4.1.

Table 1 Descriptive statistics

The correlation results reported in Table 1 Panel B show a relatively high correlation between accruals and core operating earnings, suggesting that the McVay (2006) model is likely mis-specified if accruals are excluded as in Fan et al. (2010). Importantly, IDSI are strongly associated with accruals, suggesting that the omission of one of these variables in a regression model where both are relevant explanatory variables (such as in Eqs. (3) and (5)) will lead to a biased regression coefficient and incorrect standard error for the included variable. The correlations reported in Panel B do not suggest an issue with multicollinearity and therefore lend broad support to our single equation approach to estimating CS.¹⁰

The correlation results reported in Table 1 Panel C for variables used in models for testing earnings forecasting ability and value relevance of shifted core expenses and adjusted special charges are relatively modest. The highest correlation is between shifted core expenses and adjusted special charges, consistent with variation in the magnitude of IDSI between firm-year observations impacting the magnitude of both variables.¹¹

3.3 Classification shifting tests and estimation of shifted core expenses

The empirical results related to tests of CS and the estimation of shifted core expenses are provided in Table 2 and Figs. 2 and 3, respectively.

Table 2 Estimation of core operating earning model and evidence of CS using income-decreasing special items

Fig. 2

Median annual measures of CS coefficient. This figure reports the median coefficient (median positive coefficient) of IDSI using our one-step adaptation of the McVay (2006) model adjusted for the inclusion of shifted core earnings in accruals, $\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)} (\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}>0)$, as represented by the solid (dashed) line. The regression model is estimated each year cross-sectionally within industries

Fig. 3

Estimation differences in alternative CS measures. This figure reports estimation differences in our sample from estimating the McVay (2006) model using a two-step procedure rather than a one-step procedure (measurement issue (i)) and from not adjusting for the inclusion of shifted core expenses in current accruals in our one-stage approach (measurement issue (ii)). Estimation difference due to measurement issue (i) is the median difference between the slope coefficient on IDSI from the McVay (2006) two-stage approach and our one-stage modified approach. Estimation difference due to measurement issue (ii) is the median difference between the slope coefficient on IDSI before and after the adjustment for shifting-related accruals in our one-stage approach. All regressions are estimated each year cross-sectionally within industries

Table 2 indicates that IDSI are strongly and positively related to core operating earnings, consistent with the evidence in McVay (2006) that managers use IDSI opportunistically to reclassify core expenses as special items. Full sample OLS regression results based on Eq. (3) indicate that all variables are statistically significant at the 0.1% level, adjusted R-squared is 0.75 (0.76) with (without) industry and year fixed effects, and the coefficient for IDSI, ${\eta }_{7}$, is 0.190 (0.145) with (without) industry and year fixed effects. Given that the accruals coefficient, ${\eta }_{4}$, is 0.382 (0.352) with (without) industry and year fixed effects, our CS measure given by $\theta =\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}$ is equal to 0.137 (0.107) when the model is estimated with (without) fixed effects. This implies that shifted core expenses are, on average, above 10% of reported IDSI. Our single equation approach, which avoids the statistical problem associated with the two-equation approach (represented by Eqs. (1) and (2)), therefore suggests substantial economic significance for CS after controlling for the upward impact on ${\eta }_{7}$ of shifting-related accruals. This conclusion is further supported by the non-linear least squares (NLS) estimation of Eq. (5) reported in Table 2, which directly yields an estimate of 0.107 for $\theta$ that is significant at the 0.1% level.¹²

The Wald tests reported in Table 2 are used to test the hypothesis that ${\eta }_{7}-{\eta }_{4}\ge 0$ (or $\theta \left(1+{\eta }_{4}\right)-{\eta }_{4}\ge 0$), which, as discussed in Sect. 2.4.1, is implied if inclusion of accruals in Eq. (3) (or Eq. (5)) results in a mechanical positive impact of IDSI on core operating earnings. The results reported in Table 2 show that the hypothesis is overwhelmingly rejected at the 0.1% level in all regressions, thus lending support to the efficacy of our approach to measuring CS using IDSI and to the inclusion of accruals in Eqs. (3) and (5) as a control for the performance effect, consistent with our analysis in Sect. 2.

Finally, to estimate shifted core expenses for individual firms, we carry out annual industry-based regressions based on Eq. (3) to generate industry-year estimates of $\theta =\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}$, as summarized in Fig. 2. As shown by the solid line in Fig. 2, median annual estimates of $\theta$ are generally positive and vary over the sample period. Given we assume that negative values of $\theta$ indicate no CS, we also report the median of positive $\theta$ estimates represented by the dashed line in Fig. 2. This line indicates that for industries where CS takes place, $\theta$ is typically in the range of 0.1 to 0.3.

For comparability with McVay’s (2006) two-step approach, we also estimate her model annually on an industry basis and quantify median annual differences between our approach and her two-step approach, as shown in Fig. 3. The results summarized in Fig. 3 confirm that the McVay two-step procedure tends to lead to lower estimates of CS. More specifically, differences from using the two-step procedure shown in Fig. 3 are negative in most years and are of a higher average magnitude in these years compared to years where the difference is positive, as shown by differences for measurement issue (i) in Fig. 3. Figure 3 also shows that the higher CS measure from the single-step regression procedure is partially offset by the effect of our adjustment to control for inclusion of shifting-related accruals in total accruals, as shown by positive differences for measurement issue (ii) in all years in Fig. 3.

In summary, our single-step model provides evidence consistent with CS behavior by firms and supports the use of accruals as a control for the performance effect. The single-step regression procedure generates industry-year-based measures of CS which are generally larger than those generated by the McVay two-stage procedure, consistent with reduced measurement error in our approach.¹³

3.4 Validation tests

We provide two tests to validate our estimates of shifted core expenses and adjusted special charges based on Eqs. (6a) and (6b). First, following Cain et al. (2020), we use our measure of shifted core expenses and adjusted special charges to forecast future accounting restatements, given that restatements are likely to be associated with opportunistic behavior related to managerial shifting of core operating expenses. Second, we use a simple earnings forecast model to compare the ability of shifted core expenses and adjusted special charges to forecast future earnings before special items, given that shifted core expenses are expected to be more useful than adjusted special charges for this purpose. We also investigate whether forecast performance is affected by future shifting and by the presence of special item categories that can be more readily used for CS.

3.4.1 Using shifted core expenses to forecast accounting restatements

Cain et al. (2020) use restatements as a basis for validating a measure of opportunistic special items. Given the methodological differences between our CS1 approach to estimating shifted core expenses and their CS2 approach to estimating “opportunistic special items” (discussed in Sect. 2.2), we adopt a similar validation test. This helps provide comparable evidence on the ability of our ${sce}_{t}$ measure to capture opportunistic managerial behavior related to accounting restatements. Specifically, we report results on the ability of ${sce}_{t}$ to forecast future restatements based on the following regression model:

$${Rst}_{t}={\gamma }_{0}+{\gamma }_{1}{adjsc}_{t}+{\gamma }_{2}{sce}_{t}+{\gamma }_{3}\Delta {rev}_{t}+{\gamma }_{4}{sft\_ast}_{t}+{\gamma }_{5}{\Delta empl}_{t}+{\gamma }_{6}{sa\_vol}_{t}+{\gamma }_{7}{cf\_vol}_{t}+{\gamma }_{8}{lev}_{t}+{\gamma }_{9}{ast\_gr}_{t}+{\gamma }_{10}{btm}_{t}+{e}_{t}$$

(7)

where ${Rst}_{t}$ is either the indicator variable ${Rst}_{A\&F,t}$, which is equal to 1 if a firm’s financial statements for year t contain misstatements due to accounting irregularities or fraud which are restated at a later date (as identified by Audit Analytics) and 0 otherwise; or the indicator variable ${Rst}_{SI,t}$, which is equal to 1 if a firm’s financial statements for year t contain misstatements due to accounting irregularities or fraud specifically related to special items that are restated at a later date (as identified based on a dictionary of word approach) and 0 otherwise.¹⁴ We include several control variables identified in the literature as determinants of accounting restatements (e.g., Dechow et al. 2011; Hribar et al. 2014). These variables are change in receivables, ${\Delta rev}_{t}$; percentage of soft assets, ${sft\_ast}_{t}$; percentage change in employees, ${\Delta empl}_{t}$; sales volatility, ${sa\_vol}_{t}$; cash flow volatility, ${cf\_vol}_{t}$; leverage, ${lev}_{t}$; asset growth, ${ast\_gr}_{t}$; and book to market ratio, ${btm}_{t}$. Detailed variable definitions are given in Appendix A. The analysis starts from year 2000 because this is the year when relevant restatement data are available in Audit Analytics.

Table 3 provides results based on OLS and logit regression estimates of Eq. (7). Noting that ${sce}_{t}$ and ${adjsc}_{t}$ are negative variables, we expect that ${\gamma }_{2}<0$, reflecting a positive relation between the magnitude of shifted core expenses and the probability of restatement, and that ${\gamma }_{2}<{\gamma }_{1}$, reflecting a higher probability of restatement associated with shifted core expenses relative to adjusted special charges. The results in Table 3 show that ${\gamma }_{1}$, the coefficient for ${adjsc}_{t}$, is not significantly different from zero. These results are consistent for models using either ${Rst}_{A\&F, t}$ or ${Rst}_{SI, t}$ as the dependent variable, estimated using either OLS or logit regressions, and with or without inclusion of year and industry fixed effects. By contrast, the results show that ${\gamma }_{2}$, the coefficient for ${sce}_{t}$, is significantly negative at the 1% level for all models estimated without fixed effects except for OLS regression based on ${Rst}_{SI, t}$, where ${\gamma }_{2}$ of -0.661 (p-value 0.017) is significant at the 5% level. When industry and year fixed effects are introduced, statistical significance for ${\gamma }_{2}$ is reduced but holds at the 5% level for all model specifications reported in Table 3. Finally, Wald tests show that the null hypothesis that ${\gamma }_{1}-{\gamma }_{2}=0$ is rejected in favor of the alternative hypothesis that ${\gamma }_{2}<{\gamma }_{1}$ at the 5% level for all models except the OLS model based on ${Rst}_{A\&F, t}$ with industry and year fixed effects, where ${\gamma }_{1}-{\gamma }_{2}=0.799$ (p-value 0.057) implies rejection of the null at the 10% level.

Table 3 Predicting accounting restatements and the role of shifted core expenses

Overall, the results reported in Table 3 provide validation for our approach to estimating CS, as only the shifted core expense component of special items is associated with the probability of restatements. Furthermore, the difference between the slope coefficients on adjusted special charges and shifted core expenses is substantial and statistically significant. This represents an improvement on the results in Cain et al. (2020), which do not reject the null hypothesis of equal coefficients for “predicted special items” and “opportunistic special items” based on the authors’ CS2 approach to estimating opportunistic special items (see their Table 6 Panel A).

Table 4 Forecasting future earnings and the usefulness of shifted core expenses

3.4.2 Using shifted core expenses for forecasting earnings

Our second validation test examines the usefulness of our estimates of ${sce}_{t}$ and ${adjsc}_{t}$ for predicting future earnings before special items, ${ebsi}_{t+1}$, where the latter is measured on a per-share basis and scaled by lagged stock price, consistent with the measurement of ${sce}_{t}$ and ${adjsc}_{t}$. Specifically, we estimate the following regression model:

$${ebsi}_{t+1}={\omega }_{0}+{\omega }_{1}{ebsi}_{t}+{\omega }_{2}{adjsc}_{t}+{\omega }_{3}{sce}_{t}+{\varepsilon }_{t}$$

(8)

and test the null hypotheses that ${\omega }_{1}-{\omega }_{2}=0$ and ${\omega }_{1}-{\omega }_{3}=0$. If shifted core expenses are a good measure of recurring core expenses which have been misclassified as special items, we expect ${\omega }_{3}$ to be significantly positive and not significantly different from ${\omega }_{1}$; i.e., we do not expect to reject the null that ${\omega }_{1}-{\omega }_{3}=0$. On the other hand, if adjusted special charges are a good measure of true IDSI and have less relevance for forecasting future core earnings due to their more transitory nature, we expect that ${\omega }_{1}>{\omega }_{2}$ and hence rejection of the null that ${\omega }_{1}-{\omega }_{2}=0$.

Table 4 Panel A provides evidence that strongly supports the usefulness of ${sce}_{t}$ as a measure of shifted core expenses. The coefficient for ${sce}_{t}$, ${\omega }_{3}$, is significantly positive at the 0.1% level for the model without fixed effects and at the 1% level when industry and year effects are included. Importantly, the magnitudes of ${\omega }_{1}$ and ${\omega }_{3}$ are similar when the model is estimated either without or with fixed effects, with Wald tests failing to reject the null hypothesis that ${\omega }_{1}-{\omega }_{3}=0$. The coefficient for ${adjsc}_{t}$, ${\omega }_{2}$, is positive and significant at the 1% level but much smaller than ${\omega }_{1}$ so that the null hypothesis that ${\omega }_{1}-{\omega }_{2}=0$ is clearly rejected at the 0.1% level when the model is estimated with or without fixed effects. These results suggest that shifted core expenses contained in IDSI have similar forecasting relevance for future earnings before special items as current earnings before special items, consistent with their incorrect omission from current earnings before special items due to misclassification as special items. While adjusted special charges have a small positive coefficient for forecasting future earnings, the much lower coefficient for adjusted special charges compared to current earnings before special items is consistent with their limited role in forecasting future core performance and hence their appropriate omission from current core earnings. These results also represent an improvement on previous results in Cain et al. (2020), who find that the coefficient on their measure of “opportunistic special items” based on their alternative CS2 approach is largely smaller than the coefficient on earnings before special items, and attribute this to “the noise in the measurement of (their measure of) opportunistic special items” (Cain et al. 2020, p. 2106).

Table 5 Contemporaneous relationship between stock returns, earnings, and special items and the impact of CS

In addition to the full sample tests reported in Table 4 Panel A, we also provide results in Panels B and C from estimating Eq. (8) for subsamples of firm-year observations. The aim is to highlight how our measure of shifted core earnings performs in settings where CS is more or less likely.

Table 4 Panel B is based on estimating Eq. (8) for firm-year subsamples where there are high and low scopes for shifting, due to the nature of items contained in IDSI.¹⁵ The key result is the strong statistical significance of ${sce}_{t}$ for firms with a high scope for shifting, confirmed by ${\omega }_{3}$ of 0.543 (p-value 0.001); and the lack of statistical significance of ${sce}_{t}$ for firms with a low scope for shifting, confirmed by ${\omega }_{3}$ of -0.319 (p-value 0.407). While these findings indicate that our measure of ${sce}_{t}$ is not relevant in settings where there is little or no scope for using special items for CS, they suggest that it is highly relevant in settings where managers have a high scope for reclassifying core expenses as special items.

Table 4 Panel C provides evidence on differences in the forecasting role of shifted core expenses between firms that shift again in the following year and firms that do not. The results indicate that the predictive role of ${sce}_{t}$ for ${ebsi}_{t+1}$ is only marginally significant at the 10% level for firms that shift again in the following year when the model is estimated without fixed effects, and is insignificant for these firms when year and industry fixed effects are included. The statistical insignificance for the ${sce}_{t}$ coefficient, ${\omega }_{3}$, for these firms is consistent with shifting in the second period obscuring the true recurring nature of these expenses. For firms that do not shift in the following year, ${\omega }_{3}$ is highly positive and statistically significant at the 1% level, confirming that these expenses are strongly recurring in settings where there is no continued shifting to obscure their recurring nature.

In summary, our results provide strong evidence of the recurring nature of our measure of shifted core expenses. Full sample results indicate that the forecast coefficient for shifted core expenses is positive and highly statistically significant and that it is not possible to reject the hypothesis that this coefficient is equal to the coefficient for earnings before special items. Subsample results show that our measure of shifted core expenses is a strong predictor of future earnings before special items in settings where the scope for shifting is high and when firms do not shift again in the following year. Taken together, the evidence in Table 4 suggests that our approach to measuring shifted core expenses generates estimates which have earnings forecasting properties consistent with the recurrence expected from a component of core earnings.

3.5 Contemporaneous stock returns and classification shifting

As previously indicated, one of the main aims of this paper is to add to the very limited evidence on capital market reaction to CS in the literature. To address this gap and to motivate our analysis of capital market rationality and CS in Sects. 4 and 5, we first analyze the contemporaneous relationship between stock returns and measures of earnings before special items, shifted core expenses, and adjusted special charges. This initial analysis focuses on the key question of whether investors identify and respond to CS in a timely fashion.¹⁶

Consistent with previous research, notably Easton and Harris (1991), Ali and Zarowin (1992), and Ohlson (2005), we include contemporaneous and lagged explanatory variables in our returns-earnings regression model as shown below:

$$\begin{aligned}{aret}_{t}&={\alpha }_{0}+{\alpha }_{1}{ebsi}_{t}+{\alpha }_{2}{adjsc}_{t}+{\alpha }_{3}{sce}_{t}\\&\quad+{\alpha }_{4}{ebsi}_{t-1}+{\alpha }_{5}{adjsc}_{t-1}+{\alpha }_{6}{sce}_{t-1}+{\varepsilon }_{t}\end{aligned}$$

(9)

where ${aret}_{t}$ is size-adjusted stock returns. We interpret ${\alpha }_{1}, {\alpha }_{2},$ and ${\alpha }_{3}$ as estimates of the stock return response to a $1 shock in ${ebsi}_{t}$, ${adjsc}_{t}$, and ${sce}_{t}$, respectively. The lagged variables ${ebsi}_{t-1}, {adjsc}_{t-1}$, and ${sce}_{t-1}$ are included in Eq. (9), as they are expected to be predictor variables for ${ebsi}_{t}, { sce}_{t}$, and ${adjsc}_{t}$. As highlighted by Ohlson (2005), the inclusion of period t-1 explanatory variables which are correlated with the period t explanatory variables improves the specification of the regression model regardless of whether these lagged variables are directly related to the dependent variable$,{ aret}_{t}$. We estimate this model for all firms and for subsamples based on investor sophistication (e.g., Jiambalvo et al. 2002; Collins et al. 2003; Truong et al. 2021).

Our broad expectation is that the coefficients for the three earnings components for period t—${\alpha }_{1}, {\alpha }_{2},$ and ${\alpha }_{3}$—will be positive, reflecting the role of earnings as a value indicator. If the capital market is rational and efficient, we expect that ${\alpha }_{1}\approx {\alpha }_{3}$ and ${\alpha }_{1}>{\alpha }_{2}$ for the full sample, reflecting an equal valuation of earnings before special items and shifted core expenses and a lower valuation of adjusted special charges due to their more transitory nature. Our tests based on investor sophistication are expected to reveal that ${\alpha }_{1}\approx {\alpha }_{3}$ and ${\alpha }_{1}>{\alpha }_{2}$ hold for firms followed by sophisticated investors but may not hold for firms with less sophisticated investor following. If the market is unable to identify CS in the full sample and therefore also in the subsample of firms followed by unsophisticated investors, we would expect that ${\alpha }_{1}>{\alpha }_{2},{\alpha }_{3}$ for the full sample and unsophisticated investor regression models, reflecting investors’ incorrect valuation of shifted core expenses as (more transitory) special items.

Table 5 reports results from estimating Eq. (9). The full sample results reported in the first two columns strongly support our expectation that ${\alpha }_{1}$, ${\alpha }_{2}$, and ${\alpha }_{3}$ are significantly positive and that ${\alpha }_{1}>{\alpha }_{2}$. Our expectation that ${\alpha }_{1}\approx {\alpha }_{3}$ is not rejected at the 10% level when fixed effects are omitted but is rejected at the 10% level when fixed effects are included. Interestingly, in the latter case, ${\alpha }_{1}-{\alpha }_{3}$ is -1.056 (p-value 0.075), which, surprisingly, suggests that ${sce}_{t}$ has a larger impact on ${aret}_{t}$ than ${ebsi}_{t}$. These results provide strong evidence that the market responds to the shifted component of IDSI in a timely manner and may even react more strongly to surprises in shifted core expenses than to surprises in earnings before special items.¹⁷ To our knowledge, our findings of a strong and positive stock return response to shifted core expenses represent the first evidence in the literature that stock returns are impacted contemporaneously by news about the CS behavior of firms. The rationality of this apparent recognition of such CS behavior by the capital market is explored further in Sects. 4 and 5 of this paper.

Table 6 NLSUR estimation of ARET function and accounting information dynamics and related value relevance and rationality tests

Table 5 also reports results from estimating Eq. (9) across subsamples based on investor sophistication. To create these subsamples, we rank firms within industry-year groups based on the percentage of institutional holdings out of total shares outstanding deflated by lagged price. Firms are divided into low, medium, and high groups, representing their sophistication rank.¹⁸ The most striking differences between the high and low sophistication subsamples are (i) the highly significant coefficient for shifted core expenses in the high sophistication subsample, where ${\alpha }_{3}$ equals 2.923 (p-value 0.001), compared to the much smaller and insignificant ${\alpha }_{3}$ coefficient—equal to 0.546 (p-value 0.696)—in the low sophistication subsample; and (ii) the highly significant coefficient for adjusted special charges in the low sophistication subsample, where ${\alpha }_{2}$ equals 0.570 (p-value 0.000), compared to the smaller and more weakly significant ${\alpha }_{2}$ coefficient—equal to 0.219 (p-value 0.052)—in the high sophistication subsample. In relation to tests of ${\alpha }_{1}-{\alpha }_{3}$, the results for both medium and high sophistication subsamples are consistent with our full sample results showing that shifted core expenses are more highly valued than earnings before special items when year and industry fixed effects are included. This consistency supports the view that high market valuation of shifted core expenses is driven by relatively sophisticated investors, and underlines the need to look further into the rationality of the market response to shifted core expenses. In relation to tests of ${\alpha }_{1}-{\alpha }_{2}$, the results for all subsamples confirm that adjusted special charges are less highly valued than earnings before special items despite the higher valuation of adjusted special charges by less sophisticated investors.

Overall, our results based on Eq. (9) provide important evidence which suggests that investors can distinguish between shifted core expenses misclassified as special items and other reported special items and that stock prices respond differently to each of these components of special items. Indeed, the results reported in Table 5 provide some support for a stronger stock return reaction to shocks in shifted core expenses than to shocks in earnings before special items. A more detailed analysis of the rationality of the market response to CS is therefore called for and is the focus of the remainder of this paper.

4 Capital market rationality and classification shifting

4.1 Overview

We now develop a new integrated approach to the market pricing of CS, which allows us to jointly test for market rationality in relation to the measurement and valuation of CS surprises. First, we highlight two dimensions of capital market rationality which guide our analysis and have not, to our knowledge, been combined in prior empirical research. Second, we derive an abnormal stock return model for analyzing capital market rationality and valuation relevance in relation to shifting. Finally, we consider the main implications of this model for analyzing the valuation impact of shifting by firms.

4.2 Informational and valuation rationality

We identify two dimensions of market rationality which have been the focus of prior capital market research in accounting. The first dimension focuses on the extent to which stock returns are related to the magnitude of contemporaneous accounting shocks implied by accounting forecast models. This perspective emanates from Sloan’s (1996) analysis of the relationship between abnormal stock returns and accounting shocks using a rational expectations approach based on Mishkin (1983).¹⁹ We refer to the dimension of capital market rationality considered in this research as “informational rationality,” as the focus is on whether the market accurately observes the shocks from the relevant forecast models and responds to this new information.

The second dimension of market rationality refers to the stock return response to accounting shocks and specifically the consistency between coefficients in accounting information dynamics (AID) and stock price changes in response to accounting shocks. For example, studies by Kormendi and Lipe (1987), Ohlson (1989, 1995), and Feltham and Ohlson (1995) highlight how valuation coefficients for accounting variables in accounting-based pricing and stock return models should be related to forecast coefficients in the underlying AID in an efficient and rational capital market.²⁰ We refer to this dimension of market rationality as “valuation rationality,” as this literature provides theoretical and empirical analysis of the role of accounting-based forecast models for understanding how the market values financial statement information.

We combine these two dimensions of market rationality in Fig. 4 to provide a more complete framework for highlighting potential scenarios in empirical research. The “full market rationality” represented by quadrant 1 in Fig. 4 requires both valuation and informational rationality. However, our framework also indicates the possibility of “partial market rationality.” For example, stock prices may accurately reflect new accounting information but not apply appropriate valuation multipliers to accounting variables (i.e., the case of informational rationality but valuation irrationality in quadrant 2 of Fig. 4). Alternatively, stock prices may on average reflect the predictive ability of accounting variables for future equity payoffs but may not reflect new accounting information accurately (i.e., the case of valuation rationality but informational irrationality shown in quadrant 3 of Fig. 4). A final possibility, of course, is that stock prices neither accurately identify accounting innovations nor apply appropriate valuation multipliers to accounting variables, as indicated in quadrant 4 of Fig. 4.

Fig. 4

Capital market rationality scenarios. This figure illustrates the two related concepts of market rationality developed in this paper. Valuation rationality occurs when the market applies correct valuation coefficients to shocks in accounting variables (which may or may not be correctly estimated). Informational rationality occurs when the market correctly estimates shocks in accounting variables (and may or may not apply the correct valuation coefficients to these shocks)

In summary, while prior research has focused on either valuation or informational rationality, our framework emphasizes the importance of both informational and valuation dimensions of capital market rationality. The next subsection operationalizes these two dimensions in relation to the valuation of CS.

4.3 Abnormal stock return model for testing market rationality

Our model builds on the abnormal returns analysis in Sloan (1996), which focuses on tests for market rationality in relation to accounting shocks. In addition to using this approach to test for informational rationality in relation to earnings before special items, shifted core expenses, and other special items, our model considers the relationship between market response coefficients and implied valuation coefficients based on abnormal earnings dynamics (as pioneered in Ohlson (1995) and Feltham and Ohlson (1995)) to provide evidence on the valuation rationality of earnings components.

Our tests of informational and valuation rationality assume the following AID²¹:

$$\left[\begin{array}{c}{aebsi}_{t+1}\\ {adjsc}_{t+1}\\ {sce}_{t+1}\\ {bv}_{t+1}\end{array}\right]=\left[\begin{array}{c}{\kappa }_{10}\\ {\kappa }_{20}\\ {\kappa }_{30}\\ {\kappa }_{40}\end{array}\right]+\left[\begin{array}{cccc}{\kappa }_{11}& {\kappa }_{12}& {\kappa }_{13}& {\kappa }_{14}\\ 0& {\kappa }_{22}& 0& 0\\ 0& 0& {\kappa }_{33}& 0\\ 0& 0& 0& {\kappa }_{44}\end{array}\right]\;\;\left[\begin{array}{c}{aebsi}_{t}\\ {adjsc}_{t}\\ {sce}_{t}\\ {bv}_{t}\end{array}\right]+\left[\begin{array}{c}{\varepsilon }_{1t+1}\\ {\varepsilon }_{2t+1}\\ {\varepsilon }_{3t+1}\\ {\varepsilon }_{4t+1}\end{array}\right]$$

(10)

where abnormal earnings before special items at date t are given by ${aebsi}_{t}={ebsi}_{t}-\left(R-1\right){bv}_{t-1}$ (and R represents one plus the cost of capital); ${bv}_{t}$ represents book value of equity at date t; and ${\varepsilon }_{it+1}, i=\mathrm{1,2},\mathrm{3,4},$ are mean zero shocks. As shown in Appendix C, if we assume a rational capital market setting, the AID represented by Eq. (10) implies the following relationship between abnormal stock returns at date t + 1, ${aret}_{t+1}$ (defined as size-adjusted returns conditioned on market-wide information at date t + 1), and accounting variables:

$${aret}_{t+1}={\beta }_{1}{aebsi}_{t+1}+{\beta }_{2}{adjsc}_{t+1}+{\beta }_{3}{sce}_{t+1}+{\beta }_{4}{bv}_{t+1}-{\pi }_{0}-{\pi }_{1}{aebsi}_{t}-{\pi }_{2}{adjsc}_{t}-{\pi }_{3}{sce}_{t}- {\pi }_{4}{bv}_{t}+{\epsilon }_{t+1}$$

(11)

where the $\pi$ coefficients for ${aebsi}_{t}$, ${adjsc}_{t}$, and ${sce}_{t}$ are jointly determined by market valuation and AID coefficients as follows:

$${\pi }_{1}={{\beta }_{1}\kappa }_{11}$$

(11′a)

$${\pi }_{2}={{\beta }_{1}\kappa }_{12}+{\beta }_{2}{\kappa }_{22}$$

(11′b)

$${\pi }_{3}={{\beta }_{1}\kappa }_{13}+{\beta }_{3}{\kappa }_{33}$$

(11′c)

and the market valuation $\beta$ coefficients for these variables are given by:

$${\beta }_{1}=\frac{{\kappa }_{11}}{\left(R-{\kappa }_{11}\right)}$$

(11′d)

$${\beta }_{2}=\frac{R{\kappa }_{12}+{\kappa }_{22}\left(R-{\kappa }_{11}\right)}{\left(R-{\kappa }_{11}\right)\left(R-{\kappa }_{22}\right)}={\beta }_{1}+\frac{R\left({\kappa }_{12}+{\kappa }_{22}-{\kappa }_{11}\right)}{\left(R-{\kappa }_{11}\right)\left(R-{\kappa }_{22}\right)}$$

(11′e)

$${\beta }_{3}=\frac{R{\kappa }_{13}+{\kappa }_{33}\left(R-{\kappa }_{11}\right)}{\left(R-{\kappa }_{11}\right)\left(R-{\kappa }_{33}\right)}={\beta }_{1}+\frac{R\left({\kappa }_{13}+{\kappa }_{33}-{\kappa }_{11}\right)}{\left(R-{\kappa }_{11}\right)\left(R-{\kappa }_{33}\right)}$$

(11′f)

and ${\epsilon }_{t+1}$ is a mean-zero disturbance term. Testing for informational rationality using the Sloan (1996) approach requires estimating Eqs. (10) and (11) and then testing if the cross-equation restrictions given by expressions (11′a) – (11′c) hold. Our extension to this approach to test for valuation rationality requires testing if the additional cross-equation restrictions given by expressions (11′d) – (11′f) hold.

In summary, informational rationality requires consistency between $\pi$ coefficients directly estimated from Eq. (11) (ARET function hereafter) and indirect $\pi$ coefficients estimated using expressions (11′a) – (11′c) based on the $\kappa$ and $\beta$ coefficients. Valuation rationality requires consistency between $\beta$ coefficients directly estimated from Eq. (11) and indirect $\beta$ coefficients estimated from expressions (11′d) – (11′f) using $\kappa$ coefficients. From an empirical perspective, this implies that informational and valuation rationality based on Eqs. (10) and (11) can be summarized in terms of the following null hypotheses:

Informational rationality: ${{\pi }_{i}-\pi }_{i}^{AID}=0\; \mathrm{for }\;i=\mathrm{1,2},3$, where ${\pi }_{i}$ refers to the coefficients in the direct association between abnormal stock returns and accounting variables given by Eq. (11), and ${\pi }_{i}^{AID}$ refers to the coefficients calculated indirectly using expressions (11′a) – (11′c) based on AID parameters ${\kappa }_{11},{\kappa }_{12},{\kappa }_{13},{\kappa }_{22}$, and ${\kappa }_{33}$ and market valuation coefficients ${\beta }_{1},{\beta }_{2}$, and ${\beta }_{3}$.

Valuation rationality: ${{\beta }_{i}-\beta }_{i}^{AID}=0\; \mathrm{for }\;i=\mathrm{1,2},3$, where ${\beta }_{i}$ refers to the market valuation coefficients in the direct association between abnormal stock returns and accounting variables given by Eq. (11) and ${\beta }_{i}^{AID}$ refers to the coefficients calculated indirectly using expressions (11′d) – (11′f) based on AID parameters ${\kappa }_{11},{\kappa }_{12},{\kappa }_{13},{\kappa }_{22},$ and ${\kappa }_{33}$.

4.4 Implications for analysing the valuation impact of shifting by firms

By explicitly introducing an abnormal earnings based AID into the Sloan (1996) abnormal returns framework, our model not only facilitates the analysis of both informational and valuation rationality in relation to CS, but also allows us to examine the fundamental value relevance of key accounting variables explicitly, which was not possible in our market-based empirical analysis in Sect. 3.5. In this section, we first outline the implications of our model for the fundamental value relevance of earnings- and CS-related variables. We then consider alternative scenarios for market rationality in relation to these variables based on the perspective summarized in Fig. 4.

4.4.1 Implications for fundamental value relevance

Assuming the validity of the AID given by Eq. (10), the following points based on expressions (11′d) – (11′f) can be made in relation to the fundamental value relevance of our key accounting variables:

• a) If “true” special items are largely transitory and ${adjsc}_{t}$ represents a good measure of “true” special items (special items excluding shifted core expenses), then ${\kappa }_{12}$, ${\kappa }_{22}$, and hence ${\beta }_{2}$ should be close to zero. This is because completely transitory items have no forecasting ability for future abnormal earnings and only affect the market value of equity through their impact on book value of equity. While it is likely that adjusted special charges will have some forecasting ability for future total abnormal earnings (i.e., ${\kappa }_{12}>0$ and/or ${\kappa }_{22}>0$), we expect this ability to be less than that of abnormal earnings before special items, i.e., ${\kappa }_{12}+{\kappa }_{22}<{\kappa }_{11}$, and hence, from expression (11′e), that ${\beta }_{2}<{\beta }_{1}$.

• b) If shifted core expenses have a similar predictive role in relation to future total abnormal earnings as reported abnormal earnings before special items, and if the capital market can identify these shifted expenses, our model suggests that ${\kappa }_{13}+{\kappa }_{33}\approx {\kappa }_{11}$ and hence, from expression (11′f), that ${\beta }_{3}\approx {\beta }_{1}$.

• c) If the shifting behavior is carried out by poorly managed firms, it is possible that their abnormal earnings prospects may be worse than other firms that do not shift. If so, shifted core expenses must have greater persistence in relation to future abnormal earnings than reported core expenses, resulting in ${\kappa }_{13}+{\kappa }_{33}>{\kappa }_{11}$ and hence, from expression (11′f), ${\beta }_{3}>{\beta }_{1}$. In other words, shifting core expenses into special items may carry an additional “bad news” signal in relation to future abnormal earnings.

In summary, assuming accurate estimates of shifted core expenses using IDSI, we might expect shifted core expenses to be of similar fundamental value relevance to reported core earnings represented by abnormal earnings before special items. Alternatively, however, shifting behavior by firms may also provide “bad news” signals of future abnormal earnings prospects (if, for example, shifting behavior is associated with inferior management quality or weaker future performance), which could lead to a higher fundamental valuation of shifted expenses than of reported core earnings. The results in Table 5 provide some support for a higher valuation multiplier for shifted core expenses than earnings before special items. We explore further, in Sect. 5, whether this result holds when the forecasting and returns models given by Eqs. (10) and (11) are jointly estimated, and the extent to which any such evidence can be explained in terms of a “bad news” effect associated with shifting behavior.

4.4.2 Implications for informational and valuation rationality

Informational rationality in our model requires that the market correctly partitions special items into adjusted special charges and shifted core expenses and makes use of the matrix of forecast coefficients in Eq. (10) to make accurate forecasts of ${aebsi}_{t+1}$, ${adjsc}_{t+1}$, and ${sce}_{t+1}$. While previous research, notably Cain et al. (2020), suggests that the market may be unable to make such forecasts accurately due to a misunderstanding of the impact of opportunistic shifting behavior on reported core earnings and special items, we consider the alternative possibility that investors forecast and measure such shifting accurately and hence stock prices respond to correctly estimated forecast errors, as indicated by quadrants 1 and 2 in Fig. 4. Indeed, the assumption that shifted core expenses can be estimated using relatively simple approaches based on information available to investors at the time of reporting, as in both Cain et al. (2020) and the current study, itself implies that informed investors should be able to estimate shifting with some accuracy.²²

In terms of our concept of informational rationality, rejection of the null hypothesis of informational rationality due to failure to discern or measure shifting with accuracy is supported by evidence that the market uses incorrect forecast parameters for shifted core expenses to forecast future abnormal earnings before special items and shifted core expenses. Specifically, rejection of the null requires that ${{\pi }_{3}-\pi }_{3}^{AID}\ne 0$, where ${\pi }_{3}^{AID}={{\beta }_{1}\kappa }_{13}+{\beta }_{3}{\kappa }_{33}$ is based on market $\beta$ valuation coefficients from estimating the abnormal return function Eq. (11). In other words, informational irrationality is due to inaccurate estimates of shocks to abnormal earnings before special items and shifted core expenses, resulting from market failure to use ${\kappa }_{13}$ and ${\kappa }_{33}$ to forecast abnormal earnings before special items and shifted core expenses, respectively. Given that, as in Sloan (1996), our analysis tests for rejection of the null hypothesis of informational rationality, a failure to reject this hypothesis clearly does not represent acceptance of the null hypothesis. However, tests which fail to reject informational rationality with a high p-value (e.g., tests which clearly fail to reject the null hypothesis that ${{\pi }_{3}-\pi }_{3}^{AID}=0$ with a p-value substantially greater than 0.100) may be interpreted as providing a measure of support for informational rationality. Finally, it should be noted that because tests of informational rationality (i.e., tests of ${{\pi }_{i}-\pi }_{i}^{AID}=0, i=\mathrm{1,2},3$) make use of market $\beta$ valuation coefficients from the abnormal returns function Eq. (11) to estimate ${\pi }_{i}^{AID}$, the market’s use of incorrect $\beta$ valuations coefficients (which do not reflect the fundamental value relevance of accounting variables) will result in valuation irrationality even if there is informational rationality, as indicated by quadrant 2 in Fig. 4.

Valuation rationality, the second dimension of market rationality in our framework, requires that the market applies the correct valuation coefficients (based on fundamental value relevance to accounting variables) when determining the value of the firm. Previous research by McVay (2006) and Cain et al. (2020) highlights the possibility that investors apply irrationally low valuation coefficients to shifted core expenses, leading to higher current abnormal returns but lower future abnormal stock returns. It is also plausible, however, that investors who can accurately measure and forecast shifted core expenses will be able to correctly value shifted core expenses, consistent with valuation rationality. Indeed, our initial results based on the empirical model represented by Eq. (9) provide evidence that investors value shifted core expenses similarly to earnings before special items. By focusing on the association of shifted core expenses with contemporaneous abnormal returns as indicated by Eq. (11) and testing the null hypothesis that ${{\beta }_{3}-\beta }_{3}^{AID}=0$, where ${\beta }_{3}^{AID}$ is estimated using equation (11′f), our framework provides insights on the consistency of market valuation coefficients with fundamental valuation coefficients. As with informational rationality, a failure to reject the null that ${{\beta }_{3}-\beta }_{3}^{AID}=0$ cannot be interpreted as acceptance of valuation rationality. However, we interpret tests which fail to reject valuation rationality with a high p-value as providing a measure of support for valuation rationality. Finally, given that investors may use AID parameters to correctly estimate valuation coefficients but not to correctly estimate forecast errors, evidence in support of valuation rationality may also be associated with evidence of informational irrationality, as indicated by quadrant 3 of Fig. 4.

In summary, our modeling of the capital market response to CS based on extending the Sloan (1996) abnormal returns framework to consider both informational and valuation rationality fills a significant gap in the research literature. We now focus on the empirical implementation of this model using robust estimates of shifted core expenses based on our approach to CS measurement in Sect. 2.

5 Stock returns and classification shifting: empirical analysis

5.1 Estimation methods

We carry out our abnormal stock return analysis of CS by estimating Eqs. (10) and (11) using non-linear seemingly unrelated regression (NLSUR), consistent with the Mishkin test approach. Detailed variable definitions for empirical estimation of Eqs. (10) and (11) are provided in Appendix A. Consistent with our empirical analysis in Sect. 3, all accounting variables are expressed on a per-share basis and scaled by lagged stock price.

We test our cross-equation hypotheses using Wald tests for nonlinear (or linear) restrictions. Specifically, the informational rationality tests require joint estimation of Eqs. (10) and (11) to estimate Wald statistics for testing equality between the directly estimated $\pi$ coefficients from Eq. (11) and the corresponding rational ${\pi }^{AID}$ coefficients calculated using expressions (11′a) – (11′c), where the $\kappa$ coefficients are from estimating Eq. (10) and the $\beta$ coefficients are from estimating Eq. (11). The valuation rationality tests are based on testing for equality between the $\beta$ coefficients from estimation of Eq. (11)—i.e., the “market’s $\beta$ coefficients”—and the corresponding rational ${\beta }^{AID}$ coefficients calculated using expressions (11′d) – (11′f), where the $\kappa$ coefficients are from estimating Eq. (10) and the assumed cost of capital is 12% (Dechow et al. 1999; Hand and Landsman 2005).

The valuation relevance tests are simply based on testing the null hypothesis of zero difference between the valuation coefficients for each of our three earnings components, using both directly estimated $\beta$ coefficients from Eq. (11) and rational ${\beta }^{AID}$ coefficients calculated from expressions (11′d) – (11′f) using $\kappa$ coefficients from the AID given by Eq. (10). For example, we test for differences between abnormal earnings before special items and shifted core earnings coefficients by testing both null hypotheses, i.e., that ${\beta }_{1}-{\beta }_{3}=0$ and ${\beta }_{1}^{AID}-{\beta }_{3}^{AID}=0$, with the former focusing on differences in the estimated market response and the latter focusing on differences in estimated fundamental value relevance based on AID coefficients.

5.2 Tests of value relevance and market rationality

Table 6 reports results from NLSUR estimation of Eqs. (10) and (11). Panel A provides results from the joint estimation of the AID and ARET functions, as well as indirect rational estimates of coefficients in Eq. (11) based on expressions (11′a) – (11′f). Panel B provides results on tests of the relative value relevance of the earnings and special items variables and tests of the null hypotheses of valuation rationality and informational rationality outlined in Sect. 4.

The results in Table 6 Panel A provide strong evidence of statistical significance for variables in both the AID and ARET functions, together with strongly significant rationality-based estimates of relevant parameters. The signs of the AID coefficients are generally positive and highly significant (as expected), the only exception being the book value coefficient, ${\kappa }_{14}$, in the abnormal earnings dynamic, which is statistically significant and negative.²³ In relation to shifted core expenses and adjusted special charges, the coefficients are positive and statistically significant at the 0.1% level in the AID, with the minor exception of the adjusted special charge coefficient, ${\kappa }_{12}$, in the abnormal earning dynamic, which is significant at the 1% level. The regression coefficients for the ARET function are positive and statistically significant at the 0.1% level for all variables except ${\pi }_{3}$ (the coefficient for lagged shifted core expenses in the ARET function), which is significant at the 10% level. Importantly, the ${\beta }_{1}$ and ${\beta }_{3}$ valuation coefficients for abnormal earnings before special items and shifted core expenses—0.979 and 1.743, respectively—are significant at the 0.1% level, providing evidence that the market strongly responds to both reported and shifted core earnings.

The results on the relative value relevance of earnings and special items variables in Table 6 Panel B indicate the following. First, consistent with the largely transitory nature of adjusted special items, ${adjsc}_{t}$ has a weaker impact on future abnormal earnings than ${aebsi}_{t}$, with ${\kappa }_{11}-({\kappa }_{12}+{\kappa }_{22})$ equal to 0.298 (p-value 0.000). Second, this difference is reflected in a weaker association between ${adjsc}_{t+1}$ and ${aret}_{t+1}$ than between ${aebsi}_{t+1}$ and ${aret}_{t+1}$; the results confirming ${\beta }_{1}-{\beta }_{2}$ of 0.720 (p-value 0.000) and ${\beta }_{1}^{AID}-{\beta }_{2}^{AID}$ of 0.604 (p-value 0.000) are highly significant. Third, in marked contrast to the results for adjusted special charges, there is no evidence of a statistically significant difference between the impact of ${aebsi}_{t}$ and ${sce}_{t}$ on future abnormal earnings, with ${\kappa }_{11}-({\kappa }_{13}+{\kappa }_{33})$ equal to -0.145 (p-value 0.134). Fourth, consistent with the role of both ${aebsi}_{t}$ and ${sce}_{t}$ as highly significant forecast variables for future abnormal earnings, the results do not support the hypothesis that stock returns react less strongly to shifted core earnings than to core earnings represented by abnormal earnings before special items. Indeed, ${\beta }_{1}-{\beta }_{3}$ of -0.764 (p-value 0.104) and ${\beta }_{1}^{AID}-{\beta }_{3}^{AID}$ of -0.270 (p-value 0.134) are both negative, implying that shifted core expenses have a larger impact on both market and fundamental value than if they were reported as part of earnings before special items, although these differences are not significant at the 10% level. We therefore conclude that while a potential “bad news” effect associated with the shifting behavior is not supported at standard statistical significance levels, there is strong evidence of comparable value relevance between core earnings and shifted core expenses.²⁴

Table 6 Panel B also provides evidence on market rationality in relation to earnings and special items components. Focusing first on informational rationality, there is support for informational rationality in relation to ${aebsi}_{t}$ and ${sce}_{t}$. Specifically, failure to reject the null that ${\pi }_{1}-{\pi }_{1}^{AID}=0$ based on ${\pi }_{1}-{\pi }_{1}^{AID}=$ -0.027 (p-value 0.295) implies that we cannot reject the hypothesis that the market’s estimation of ${\kappa }_{11}$ is equal to the correct forecast coefficient. Similarly, failure to reject the null that ${\pi }_{3}-{\pi }_{3}^{AID}=0$ based on ${\pi }_{3}-{\pi }_{3}^{AID}=$ 0.089 (p-value 0.859) implies that we cannot reject the hypothesis that the market correctly estimates ${\kappa }_{13}$ and ${\kappa }_{33}$ and hence correctly assesses the expected impact of shifted core expenses (equal to ${\kappa }_{13}+{\kappa }_{33}$) on next period abnormal earnings. In relation to ${adjsc}_{t}$, on the other hand, rejection of the null that ${\pi }_{2}-{\pi }_{2}^{AID}=0$ based on ${\pi }_{2}-{\pi }_{2}^{AID}=$ 0.525 (p-value 0.000) implies that the market may be informationally irrational in relation to adjusted special charges. In particular, when forecasting next period abnormal earnings, the market appears to assume that ${\kappa }_{12}$ and/or ${\kappa }_{22}$ are substantially higher than reported in Table 6 Panel A (or, in other words, that the market appears to assume that adjusted special charges have a larger negative impact on expected future earnings than ${\kappa }_{12}+{\kappa }_{22}$ implied by the AID). Overall, therefore, while we have evidence which supports informational rationality in relation to abnormal earnings before special items and shifted core expenses, there is also some initial evidence of market irrationality in relation to the predictive role of adjusted special charges for future abnormal earnings. We return to this issue in Sect. 5.4 when we carry out additional market rationality tests.

Finally, turning to the valuation rationality tests reported in Table 6 Panel B, there is evidence of valuation irrationality in relation to ${aebsi}_{t+1}$ based on ${\beta }_{1}-{\beta }_{1}^{AID}=$ 0.100 (p value 0.001), with this result indicating a small but statistically significant divergence between market valuation of earnings before special items and valuation based on fundamentals. On the other hand, the results for ${adjsc}_{t+1}$ and ${sce}_{t+1}$—based on ${\beta }_{2}-{\beta }_{2}^{AID}=$ -0.016 (p-value 0.786) and ${\beta }_{3}-{\beta }_{3}^{AID}=$ 0.594 (p-value 0.263), respectively—fail to reject the null hypothesis of valuation rationality for each of these variables, although the magnitude of the market response to shifted core expenses is higher than that implied by fundamentals based on the AID. Putting the results on informational and valuation rationality together suggests that the market response to abnormal earnings before special items and shifted core expenses is reasonably rational, with rejection only of the null hypothesis of valuation rationality for abnormal earnings before special items based on a relatively small divergence between market and fundamental valuation coefficients. In the case of adjusted special charges, the market valuation coefficient closely approximates the fundamental coefficient based on the AID consistent with valuation rationality, although informational rationality for this variable appears to be rejected by the large and statistically significant divergence of ${\pi }_{2}$ from ${\pi }_{2}^{AID}$.

To summarize, our analysis provides strong support for the market’s ability to identify CS and to respond rationally to estimates of shifted core expenses. Our findings therefore challenge the “implicit” assumption, in previous research, that the market fails to contemporaneously respond to shifted core earnings. Indeed, estimates of both market and fundamental valuation differences between earnings before special items and shifted core earnings suggest a higher valuation of shifted core expenses. Thus, the results suggest some “bad news” effect linked to CS, although these differences are not significant at conventional confidence levels. Finally, while there is broad support for informational and valuation rationality in relation to shifted core expenses and for valuation rationality in relation to adjusted special charges, there is surprising evidence of informational irrationality in relation to adjusted special charges. In Sect. 5.3, we explore (i) a possible “bad news” effect in relation to shifted core expenses, and (ii) informational irrationality in relation to adjusted special charges, in further analysis considering firms’ shifting history.

5.3 Multiperiod classification shifting, value relevance, and market rationality

To explore potential differences between firms which have no recent prior history of CS and firms which have engaged repeatedly in CS in prior years, we estimate Eqs. (10) and (11) using NLSUR for two subsamples. The first subsample consists of firms which have not shifted in any of the prior three years. The second subsample consists of firms which have shifted in each of the prior three years. Results for these two subsamples are reported in Table 7.

Table 7 Estimation of ARET function and accounting information dynamics using NLSUR for firms partitioned according to prior year shifting

Table 7 Panel A provides results from the joint estimation of the ARET function and associated AID for each of our two subsamples based on CS history. A striking feature of the reported results for the AID is the large increase in ${\kappa }_{13}$—the coefficient for ${sce}_{t}$ in the abnormal earnings dynamic—from 0.595 (p-value 0.002) for firms with no CS in the prior three years to 0.980 (p-value 0.000) for firms with three years of prior CS. This increase is also reflected in ${\beta }_{3}$, the valuation coefficient for ${sce}_{t+1}$ in the ARET function, which changes from 1.843 (p-value 0.046) for firms with no CS in the prior three years to 4.079 (p-value 0.000) for firms with three years of prior CS. Given that the results reported in Table 7 Panel A for the ${\kappa }_{11}$ and ${\beta }_{1}$ coefficients show no such increase, prior multiperiod shifting therefore increases the relative importance of shifted core expenses as an indicator of future earnings and abnormal stock returns in comparison to reported abnormal earnings before special items.

The value relevance tests reported in Table 7 Panel B provide evidence on the statistical significance of differences in the value relevance of abnormal earnings before special items and shifted core expenses related to firms’ prior multiperiod shifting behavior. These tests provide strong evidence of a statistically significant “bad news” effect for firms which engage in prior year CS. Thus, while our previous finding, for all firms, that ${\kappa }_{11}-({\kappa }_{13}+{\kappa }_{33})$ is not significantly different from zero is repeated for firms with no CS in the prior three years, this difference in the forecasting role of ${aebsi}_{t}$ and ${sce}_{t}$ in relation to future abnormal earnings is equal to -0.579 (p-value 0.006) for firms with CS in the prior three years. The same pattern is displayed for the difference in the market valuation coefficients, ${\beta }_{1}-{\beta }_{3}$, which moves from a statistically insignificant -0.861 (p-value 0.351) for firms with no CS in the prior three years to a highly significant -3.164 (p-value 0.001) for firms with CS in the prior three years. More modest increases in the difference in fundamental valuation coefficients, ${\beta }_{1}^{AID}-{\beta }_{3}^{AID}$, are reported for firms with differing CS histories, but once again this difference is statistically insignificant for firms with no CS in the prior three years, -0.384 (p-value 0.343), and highly statistically significant for firms with CS in the prior three years, -1.030 (p-value 0.006).

Finally, Table 7 Panel B also provides tests of valuation and informational rationality for these subsamples. While valuation rationality is supported by the failure to reject the null hypothesis that ${\beta }_{3}-{\beta }_{3}^{AID}=0$ for firms with no CS in the prior three years (as in our full sample results), the null hypothesis is rejected for firms engaged in CS in the prior three years, where ${\beta }_{3}-{\beta }_{3}^{AID}$ is equal to 2.237 (p-value 0.032). This evidence therefore suggests that valuation irrationality in relation to ${sce}_{t+1}$ occurs through a market overreaction to firms with prior multiperiod shifting when compared to the fundamental valuation coefficient implied by the AID. Hence, far from failing to respond contemporaneously to CS (as implied in previous research), our results suggest that the market overreacts to shifting by “repeat offenders.”²⁵ In relation to informational rationality, all subsample results support rationality in relation to abnormal earnings before special items and shifted core earnings but again reject rationality in relation to adjusted special charges, where ${\pi }_{2}-{\pi }_{2}^{AID}$ is equal to 0.699 (p-value 0.000) for firms with no CS in the prior three years and 0.501 (p-value 0.000) for firms that have shifted in each of the prior three years.

In summary, our results for subsamples based on prior CS history, reported in Table 7, suggest that our full sample results in Table 6 are broadly consistent with results for the subsample of firm years where there has been no CS in the prior three years. Further, evidence of a statistically significant “bad news” effect and valuation irrationality in relation to shifted core expenses is only provided for firms with a history of repeated shifting in the prior three years. Previous findings in relation to the apparent rejection of rationality for adjusted special charges, on the other hand, apply similarly to both subsamples, suggesting that this effect is prevalent across the full sample of firms. Further analysis and findings in relation to market rationality are now considered.

5.4 Robustness tests in relation to market rationality

Our full sample results provide no evidence of market irrationality in relation to shifted core earnings (Table 6 Panel B), and our subsample results in Table 7 only provide evidence of valuation irrationality in relation to shifted core expenses for firms that have repeatedly shifted in the prior three years. These findings contrast strongly with recent findings in Cain et al. (2020), which suggest that the market is initially deceived into treating opportunistic special items as non-recurring, only to be “disappointed” by the recurrence of these expenses in the future (i.e., the market initially appears to value such expenses as transitory and only corrects this valuation error when they recur). In comparison to our research design, Cain et al. (2020), however, do not provide a direct test of valuation rationality, as they do not consider the market’s contemporaneous response to their measure of opportunistic special items. Instead, they provide indirect evidence of valuation irrationality, based on results showing that opportunistic special items are negatively related to future returns. They argue that this is due to eventual market recognition (and “disappointment”) that shifted expenses recur. Given the major difference between our evidence suggesting that the market broadly values shifted core expenses correctly (and even overreacts to these expenses in the case of repeat shifters) and Cain et al.’s (2020) evidence suggesting that the market undervalues such expenses, we carry out further robustness tests on the market rationality in relation to shifted and adjusted components of IDSI.

Our further empirical analysis is based on a lagged abnormal returns regression model where a measure of abnormal stock returns is regressed on lagged shifted core expenses and lagged adjusted special charges, similar to the analysis in Cain et al. (2020) (see their Table 5). However, whereas Cain et al.’s analysis is entirely empirical, we develop a theoretical model in Appendix D based on the assumption that market irrationality reverses in the following year. The model shows that either informational or valuation irrationality in relation to a particular accounting variable will result in statistically significant coefficients of opposite sign for the prior two periods. Given these implications from our model in Appendix D, our additional empirical analysis is based on the following regression model:

$$\begin{aligned}{aret}_{t+1}&={\varphi }_{0}+{\varphi }_{a1}{adjsc}_{t}+{\varphi }_{s1}{sce}_{t}+{\varphi }_{a2}{adjsc}_{t-1}\\&\quad+{\varphi }_{s2}{sce}_{t-1}+{\varepsilon }_{t+1}\end{aligned}$$

(12)

where the null hypotheses of market rationality in relation to adjusted special charges and shifted core expenses are expressed as $\left\{{\varphi }_{a1}=0, { \varphi }_{a2}=0\right\}$ and $\left\{{\varphi }_{s1}=0, { \varphi }_{s2}=0\right\}$ respectively.²⁶ Based on the previous results in Table 6 Panel B where there was no evidence of market irrationality in relation to shifted core expenses, we do not expect to reject the null for shifted core expenses $\left\{{\varphi }_{s1}=0, { \varphi }_{s2}=0\right\}$. On the other hand, given the previous evidence in Table 6 Panel B that ${\pi }_{2}-{\pi }_{2}^{AID}$ is significantly greater than zero, we expect to reject the null for adjusted special charges $\left\{{\varphi }_{a1}=0, { \varphi }_{a2}=0\right\}$ in favor of the alternative hypothesis $\left\{{\varphi }_{a1}<0, { \varphi }_{a2}>0\right\}$.²⁷ In contrast, consistency with the previous results of Cain et al. (2020) implies rejection of $\left\{{\varphi }_{s1}=0, { \varphi }_{s2}=0\right\}$ in relation to shifted core expenses, but no rejection of $\left\{{\varphi }_{a1}=0, { \varphi }_{a2}=0\right\}$ in relation to adjusted special charges. More specifically, market underreaction to shifted core expense (due to investors incorrectly treating them as transitory items) in line with Cain et al.’s findings would imply that ${\varphi }_{s1}>0$ if market underreaction at date t is corrected at date t + 1 and/or that ${\varphi }_{s2}>0$ if the correction takes longer than one year.²⁸

The regression results based on estimating Eq. (12) and a simpler one-period lag model are reported in Table 8 Panel A. In addition to using a size-adjusted returns measure, ${aret}_{t+1}$, as the dependent variable (as in Eq. (12)), we also report results for an alternative measure of abnormal returns, denoted ${aret}_{t+1}^{4\mathrm{F}}$, based on the Fama and French (1993) / Carhart (1997) four-factor model in Table 8 Panel B. The use of ${aret}_{t+1}^{4\mathrm{F}}$ as an alternative dependent variable to ${aret}_{t+1}$ provides a check that any inferences in relation to market rationality are robust to the inclusion of further controls for risk.

Table 8 Shifted core expenses, adjusted special charges, and future stock returns

The results on the rationality of the market in relation to shifted core expenses fail to reject the null $\left\{{\varphi }_{s1}=0, { \varphi }_{s2}=0\right\}$, as these coefficients are statistically insignificant when either ${aret}_{t+1}$ or ${aret}_{t+1}^{4\mathrm{F}}$ is the dependent variable. These results therefore support our previous evidence in Table 6 that it is not possible to reject the view that the market is rational both in valuation of shifted core expenses and in forecasting the magnitude of shifting by firms. These findings, however, are inconsistent with Cain et al. (2020), who report that opportunistic special items (measured as a positive number) are negatively related to future stock returns.

As expected from our previous analysis in Table 6, informational rationality in relation to adjusted special charges appears to be challenged due to the rejection of $\left\{{\varphi }_{a1}=0, { \varphi }_{a2}=0\right\}$ in Table 8 Panel A. The results show that ${\varphi }_{a1}=-0.687$ (p-value 0.000) without fixed effects and ${\varphi }_{a1}=-0.704$ (p-value 0.000) with fixed effects. Nevertheless, the finding that ${\varphi }_{a2}=-0$.095 (p-value 0.277) without fixed effects and ${\varphi }_{a2}=-0$.122 (p-value 0.171) with fixed effects is statistically insignificant, is inconsistent with the reversal of apparent irrationality indicated by ${\varphi }_{a1}<0$. As discussed in Appendix D, a possible explanation for these results (and our previous finding that ${\pi }_{2}-{\pi }_{2}^{AID}>0$ in Table 6 Panel B) is that adjusted special charges proxy for risk which is not controlled for in the measurement of ${aret}_{t+1}$. Alternatively stated, ${\varphi }_{a1}<0$ may be due to the higher required rate of return for firms with high levels of adjusted special charges. The results reported in Table 8 Panel B, where ${aret}_{t+1}^{4\mathrm{F}}$ is used to measure abnormal stock returns, lend at least partial support to a risk-related interpretation of the relationship between adjusted special charges and future stock returns, with the magnitude and statistical significance of ${\varphi }_{a1}$ falling to -0.211 (p-value 0.020) without fixed effects and to -0.216 (p-value 0.016) with fixed effects. In other words, controlling more fully for risk in the measurement of abnormal stock returns greatly reduces the impact of adjusted special charges on future returns and suggests that our previous finding that ${\pi }_{2}-{\pi }_{2}^{AID}>0$ may be better explained by risk than by irrational investors overestimating the negative impact of adjusted special charges on expected future abnormal earnings.

Finally, given that Table 8 provides no evidence that either ${sce}_{t}$ or ${sce}_{t-1}$ are related to stock returns at date t + 1 (consistent with our main findings that the market responds in a broadly rational manner to news about shifted core expenses), we conclude by considering how this might be reconciled with the findings in Cain et al. (2020). Specifically, their results show a significant negative relation between their (absolute) measure of opportunistic special items measured at date t and future stock returns measured over a two-year period comprising years t + 2 and t + 3.²⁹ On reflection, given that our measure of shifted core expenses represents present shifting (i.e., contemporaneous shifting of core expenses to special items), their finding of market irrationality may be primarily attributable to intertemporal shifting, risk aside. In line with this, while investors may be adept at detecting shifting between core earnings and special items in the same year as suggested by our results, they may also be slow to recognize the role of IDSI reported in year t in artificially inflating prior year core earnings, leading to a negative stock return correction after year t in response to this component of opportunistic special items in year t, consistent with Cain et al. (2020). Despite major differences in both methodology and empirical results, our study and Cain et al. (2020) may therefore provide complementary insights on investor understanding of CS using IDSI, which can be explored in future research.

In summary, the alternative empirical tests of market rationality reported in Table 8 are broadly consistent with our prior results in Table 6. Most importantly, the results providing support for market rationality in relation to shifted core earnings in Table 6 are supported by results in Table 8. Similarly, the apparent results of informational irrationality in relation to adjusted special charges in Table 6 are consistent with the results reported in Table 8, although we suggest that priced risk related to adjusted special charges may be a more plausible explanation for their positive impact on future returns than informational irrationality. Overall, we conclude that market rationality is relatively high in relation to both shifted core expenses and adjusted special charges, and that the market responds to CS using IDSI in a timely and broadly rational manner.

6 Summary and conclusions

This paper provides an improved model for measuring classification shifting and develops a novel approach for assessing market rationality, which is applied to shifted and adjusted components of special items.

Our CS model is based on a development of the model in McVay (2006) which reduces potential measurement error in the CS regression parameter and provides explicit measures of shifted core expenses and adjusted special charges. Our measure of shifted core expenses is significantly related to the probability of future restatements linked to special items, reflecting the role of managerial opportunism in shifting. It is also significantly related to future earnings before special items, similar to current earnings before special items and in marked contrast to much weaker forecasting relevance of adjusted special charges, reflecting its incorrect omission from core earnings. Contemporaneous returns-earnings regressions show a significantly positive stock return response to shifted core expenses, which is higher for firms with higher institutional ownership, suggesting high valuation of these expenses by sophisticated investors.

Combining insights from Sloan’s (1996) analysis of stock returns and accounting information with insights from Ohlson’s (1995) abnormal earnings valuation perspective, we also develop a novel approach to assessing the rationality of the market response to shifting. Empirical results based on this approach indicate that stock returns respond in a broadly rational manner to shifted core expenses and that an apparent irrationality result associated with other adjusted special items is most likely attributable to risk rather than market anomaly.

Overall, our analysis suggests that information on opportunistic shifting can be obtained from published financial reports and that market participants make use of this information in the valuation of stocks. More broadly, our results reinforce the important role of rational expectations in interpreting stock returns results and corroborate the notion, in Penman and Zhu (2014), that anomalous returns associated with accounting variables that forecast earnings may be attributed to risk.

Appendices

Appendix A

Variable definitions

Variable	Definition
${coe}_{t}$	Core operating earnings equal to operating income before depreciation (oibdp) scaled by sales (sale).
${ato}_{t}$	Assets turnover ratio equal to sales / [(noat + noat-1)/2], where noat is net operating assets measured as operating assets minus operating liabilities [total assets (at) - cash (che) - short term investments (ivao)] - [ total assets (at) - total debt (dltt and dlc) - book value of common and preferred equity (ceq and pstk) – minority interest (mib)]. Average net operating assets is required to be positive.
${acc}_{t}$	Operating accruals equal to [net income before extraordinary items (ibc) - cash from operations (oancf-xidoc)] scaled by sales.
${\Delta sa}_{t}$	Percentage change in sales equal to change in sales between date t and t-1 divided by sales at date t-1.
${neg\Delta sa}_{t}$	Percentage change in sales ${\Delta sa}_{t}$ if ${\Delta sa}_{t}$ is less than zero, and 0 otherwise.
${si}_{t}$	Income-decreasing special items equal to special items (spi) scaled by sales when special items are negative, and 0 otherwise. Income-decreasing special items are measured in absolute terms.
${sce}_{t}$	Shifted core earnings, per share, and scaled by lagged price as estimated per Eq. (6a) (method used to estimate shifted core earnings is described in the text).
${adjsc}_{t}$	Adjusted special charges, per share, and scaled by lagged price, estimated as the difference between income-decreasing special items and shifted core earnings as per Eq. (6b).
${ebsi}_{t}$	Earnings before special items measured as [income before extraordinary items and discontinued operations (ibc) - special items (spi)], per share, and scaled by lagged price.
${aebsi}_{t}$	Abnormal earnings before special items measured as [income before extraordinary items and discontinued operations (ibc) - special items (spi) – 0.12 × common equity at date t-1 (ceq)], per share, and scaled by lagged price. We use a cost of equity of 12%, consistent with previous studies by Dechow et al. (1999) and Hand and Landsman (2005).
${bv}_{t}$	Common equity (ceq), per share, and scaled by lagged price.
${aret}_{t}$	Annual abnormal stock returns are calculated from monthly returns recorded from the beginning of April to the end of March using the raw buy-hold returns and subtracting the buy-hold returns on CRSP sized matched portfolio (Ali and Zarowin 1992). Consistent with Campbell et al. (2009), we use the delisting return when the return is missing. Otherwise, we assume a zero return. We also consider annual abnormal stock returns, ${aret}_{t+1}^{4\mathrm{f}}$, measured relative to the Fama and French (1993) / Carhart (1997) four-factor model. Betas are estimated at the stock level using a rolling 20-month estimation window and retrieved from the Beta Suite by WRDS.
${Rst}_{A\&F/SI,t}$	${Rst}_{AF, t}$(${Rst}_{SI, t}$) is an indicator variable equal to 1 if the firm-year contains misstatements due to accounting irregularities or fraud (misstatements related to special items) as defined by Audit Analytics (a dictionary of words linked to special items) which are subsequently restated and 0 otherwise. Accounting irregularities and fraud related to special items are identified if a word search within the restatement descriptions in Audit Analytics retains any of the following words: restruct, reorg, impair, write, loss, integration, onetime, transitory, special, severance, year2000, settle, nonrecurring, flood, fire, assetretire, disaster, expense, and classification of income.
${\Delta rev}_{t}$	Change in receivables, equal to change in receivables (rect) divided by average total assets (at).
${sft\_ast}_{t}$	Percentage of soft assets, equal to [total assets (at) - property, plant and equipment (ppent) – cash (che)]/ total assets.
${\Delta empl}_{t}$	Percentage change in employees, equal to change in the number of employees (emp) between date t and t-1 divided by the number of employees at date t-1.
${sa\_vol}_{t}$	Sales volatility, equal to the standard deviation of the firm’s rolling five-year sales divided by average total assets.
${cf\_vol}_{t}$	Cash flow volatility, equal to the standard deviation of the firm’s rolling five-year cash flow from operations (oancf) divided by average total assets.
${lev}_{t}$	Leverage, equal to long term debt (dltt) divided by total assets.
${ast\_gr}_{t}$	Asset growth, equal to change in total assets between date t and t-1 divided by total assets at date t-1.
${btm}_{t}$	Book to market ratio, equal to equity (ceq)/ market value (csho $\times$ prc).

Appendix B: Numerical illustration of ${sce}_{t}$ and ${adjsc}_{t}$ estimation for the ‘mean non-zero IDSI firm’

This appendix provides an illustration of our methodology for estimating ${sce}_{t}$ and ${adjsc}_{t}$ using parameter estimates for Eqs. (3)-(6) in Sect. 2.4.2 and means of variables for a sample of 31,707 firm-year observations with non-zero IDSI. This illustration aims both to illuminate our measurement procedure and to highlight the significant economic impact of CS on the core operating earnings margin.

Estimation of CS for a firm where explanatory variables in Eq. (3) are equal to the mean of 31,707 firm years with non-zero IDSI (i.e., ${si}_{t}\ne 0$)

Panel A: Impact of explanatory variables on ${coe}_{t}$
Variable	Parameter	Parameter Estimate	Mean of variable	Impact on ${coe}_{t}$
Intercept	${\eta }_{0}$	0.049		4.90%
${coe}_{t-1}$	${\eta }_{1}$	0.756	5.50%	4.16%
${ato}_{t}$	${\eta }_{2}$	-0.001	2.702	-0.27%
${acc}_{t-1}$	${\eta }_{3}$	-0.181	-16.80%	3.04%
${acc}_{t}$	${\eta }_{4}$	0.382	-16.80%	-6.42%
$\Delta {sa}_{t}$	${\eta }_{5}$	0.100	10.00%	1.00%
$neg\Delta {sa}_{t}$	${\eta }_{6}$	0.466	-4.90%	-2.28%
${si}_{t}$	${\eta }_{7}$	0.190	6.50%	1.24%
Expected ${coe}_{t}$ for mean non-zero IDSI firm				5.36%
Panel B: Calculation of ${sce}_{t}$ and ${adjsc}_{t}$
Uncorrected ${sce}_{t}$ $\left(-{\eta }_{7}{si}_{t}=-0.190*6.50\%\right)$			-1.24%
Uncorrected ${adjsc}_{t}$ $\left(-{si}_{t}-{sce}_{t}=-6.50\%+1.24\%\right)$			-5.27%
Corrected ${sce}_{t}$ $\left(-\theta {si}_{t}=-\left(\frac{{\eta }_{7}}{\left(1+{\eta }_{4}\right)}\right){si}_{t}=-0.137*6.50\%\right)$			-0.89%
Corrected ${adjsc}_{t}$ $\left(-{si}_{t}-{sce}_{t}=-6.50\%+0.89\%\right)$			-5.61%

1. Panel A shows parameter estimates (from Tables 2), means of each of the explanatory variables in Eq. (3) for non-zero IDSI sample firm-years, and the mean impact of each of these variables on reported core earnings at date t, ${coe}_{t}$. The final column also shows expected ${coe}_{t}$ of 5.36% for the “mean non-zero IDSI firm” equal to the sum of all variable impacts and the intercept term based on Eq. (3). Panel B shows estimates of ${sce}_{t}$ and ${adjsc}_{t}$ uncorrected for upward bias in CS parameter ${\eta }_{7}$ of 1.24% and 5.27% of sales, respectively, and estimates of ${sce}_{t}$ and ${adjsc}_{t}$ corrected for upward bias in CS parameter ${\eta }_{7}$ based on the corrected CS parameter $\theta ={\eta }_{7}/\left(1+{\eta }_{4}\right)$ of 0.89% and 5.61%, respectively.

Appendix C: Abnormal stock returns model

We first derive the equity valuation function, then use it to derive the abnormal stock returns model used in our empirical analysis.

(i) Equity valuation function based on market rationality

The linear information dynamics represented by Eq. (10) can be represented as the following vector autoregressive process:

$${{\varvec{x}}}_{{\varvec{t}}+1}={\varvec{\kappa}}{{\varvec{x}}}_{{\varvec{t}}}+{{\varvec{\varepsilon}}}_{{\varvec{t}}+1}$$

(A1)

where ${{\varvec{x}}}_{{\varvec{t}}}^{\boldsymbol{^{\prime}}}=\left[1\; {aebsi}_{t}\; {adjsc}_{t}\; {sce}_{t}\; {bv}_{t}\right]$, ${\varvec{\kappa}}$ is a matrix of forecast coefficients as follows:

$$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ {\kappa }_{10}& {\kappa }_{11}& {\kappa }_{12}& {\kappa }_{13}& {\kappa }_{14}\\ {\kappa }_{20}& 0& {\kappa }_{22}& 0& 0\\ {\kappa }_{30}& 0& 0& {\kappa }_{33}& 0\\ {\kappa }_{40}& 0& 0& 0& {\kappa }_{44}\end{array}\right]$$

and ${{\varvec{\varepsilon}}}_{{\varvec{t}}+1}^{\boldsymbol{^{\prime}}}=\left[0\; {{\varvec{\varepsilon}}}_{1t+1}\; {{\varvec{\varepsilon}}}_{2t+1}\; {{\varvec{\varepsilon}}}_{3t+1}\; {{\varvec{\varepsilon}}}_{4t+1}\right]$ is a vector of mean zero disturbance terms. We assume that $\underset{s\to \infty }{\mathrm{lim}}{R}^{s}{{\varvec{\kappa}}}^{s}=0$ and that the abnormal earnings valuation model holds such that equity value is given by:

$${p}_{t}={bv}_{t}+\sum_{j=1}^{\infty }{R}^{-j}{E}_{t}\left[{aebsi}_{t+j}+{adjsc}_{t+j}+{sce}_{t+j}\right]$$

(A2)

where total abnormal earnings at date t + j is equal to $\left({aebsi}_{t}+{adjsc}_{t}+{sce}_{t}\right)$ and R is equal to one plus the cost of equity. It follows from (A1) and (A2) that:

$${p}_{t}={bv}_{t}+{\varvec{i}}{\varvec{\kappa}}{\left[R{\varvec{I}}-{\varvec{\kappa}}\right]}^{-1}{{\varvec{x}}}_{{\varvec{t}}}$$

(A3)

where ${\varvec{i}}=\left[0\; 1\; 1\; 1\; 0\right]$. Equation (A3) can be rewritten as the following linear equity valuation function:

$${p}_{t}={\varvec{\beta}}{{\varvec{x}}}_{{\varvec{t}}}={\beta }_{0}+{\beta }_{1}{aebsi}_{t}+{\beta }_{2}{adjsc}_{t}+{\beta }_{3}{sce}_{t}+{\beta }_{4}{bv}_{t}$$

(A4)

where the valuation coefficients ${\beta }_{1}, {\beta }_{2}, \mathrm{and}\; {\beta }_{3}$ are as given by Eqs. (11′d) – (11′f) in the main text. Finally, if we add a white noise “other non-accounting information” variable, ${\vartheta }_{t}$, to the right side of the ${aebsi}_{t}$ forecast equation in Eq. (10), then a mean-zero disturbance term based on “other non-accounting news” revealed at date t, ${e}_{t}={\vartheta }_{t}/\left(R-{\kappa }_{11}\right)$, is added to the equity valuation function as below:

$${p}_{t}={\varvec{\beta}}{{\varvec{x}}}_{{\varvec{t}}}+{e}_{t}={\beta }_{0}+{\beta }_{1}{aebsi}_{t}+{\beta }_{2}{adjsc}_{t}+{\beta }_{3}{sce}_{t}+{\beta }_{4}{bv}_{t}+{e}_{t}$$

(A5)

(ii) Abnormal returns model based on market rationality (Eq. (11))

The abnormal returns model is derived from the equity valuation function by defining unexpected equity returns at a date t + 1 conditional on information at date t as:

$${uret}_{t+1}={up}_{t+1}+{udv}_{t+1}$$

where ${up}_{t+1}$ denotes unexpected equity value at date t + 1 and ${udv}_{t+1}$ denotes unexpected net dividends at date t + 1. It follows, using equation (A5), that unexpected returns at date t + 1 can be re-expressed as:

$$\begin{aligned}{uret}_{t+1}&={\beta }_{1}{uaebsi}_{t+1}+{\beta }_{2}{uadjsc}_{t+1}+{\beta }_{3}{usce}_{t+1}\\&\quad+{\beta }_{4}{ubv}_{t+1}+{\epsilon }_{t+1}\end{aligned}$$

(A6)

where ${uaebsi}_{t+1}$, ${uadjsc}_{t+1}$, ${usce}_{t+1}$, and ${ubv}_{t+1}$ are unexpected variables corresponding to ${aebsi}_{t+1}$, ${adjsc}_{t+1}$, ${sce}_{t+1}$, and ${bv}_{t+1}$, respectively, and ${\epsilon }_{t+1}=\left({e}_{t+1}+{udv}_{t+1}\right)$ is a mean-zero disturbance term reflecting unexpected non-accounting and dividend information at date t + 1. Defining ${uaebsi}_{t+1}$, ${uadjsc}_{t+1}$, ${usce}_{t+1}$, and ${ubv}_{t+1}$ as the residuals ${\varepsilon }_{1t+1}$, ${\varepsilon }_{2t+1}$, ${\varepsilon }_{3t+1}$, and ${\varepsilon }_{4t+1}$, respectively, in Eq. (10), it follows that unexpected returns at date t + 1 can be written as:

$$\begin{aligned}{uret}_{t+1}&={\beta }_{1}\left({aebsi}_{t+1}-{\kappa }_{10}-{\kappa }_{11}{aebsi}_{t}-{\kappa }_{12}{adjsc}_{t}-{\kappa }_{13}{sce}_{t}-{\kappa }_{14}{bv}_{t}\right)\\&\quad+{\beta }_{2}\left({adjsc}_{t+1}-{\kappa }_{20}- {\kappa }_{22}{adjsc}_{t}\right)+{\beta }_{3} \left({sce}_{t+1}-{\kappa }_{30}-{\kappa }_{33}{sce}_{t}\right)\\&\quad+{\beta }_{4}\left({bv}_{t+1}-{\kappa }_{40}-{\kappa }_{44}{bv}_{t}\right)+{\epsilon }_{t+1}\end{aligned}$$

(A7)

Next, following Sloan (1996), we operationalize abnormal stock returns at date t + 1, ${aret}_{t+1}$, as size-adjusted returns, so that for a given stock at date t + 1:

$${aret}_{t+1}={ret}_{t+1}-{ret}_{t+1}^{s}={uret}_{t+1}+\left(R-1\right)-{ret}_{t+1}^{s}$$

(A8)

where ${ret}_{t+1}$ is the realized stock return at date t + 1 and ${ret}_{t+1}^{s}$ is mean return on the size-matched portfolio at date t + 1. Under the assumption that the mean return on the size-matched portfolio is approximately equal to the cost of equity, i.e., that $\left(R-1\right)-{ret}_{t+1}^{s}\approx 0$, it follows from (A8) that ${aret}_{t+1}\approx u{ret}_{t+1}$. Replacing $u{ret}_{t+1}$ with ${aret}_{t+1}$ in equation (A7) and rearranging the right-hand side of (A7) then gives Eq. (11), with $\pi$ coefficients as defined in Eqs. (11′a) – (11′c) in the main text.

Appendix D: Further analysis of market rationality

We derive a simple lagged abnormal returns model where the assumption of market rationality in our contemporaneous abnormal returns model (represented by Eq. (11) and expressions (11′a) – (11′f)) is violated. This additional model provides motivation for regressing abnormal returns at date t + 1 on lagged accounting variables (as in Table 8) when tests based on our main contemporaneous abnormal returns model indicate market irrationality. It also highlights the important issue of distinguishing between market irrationality and risk-related explanations when accounting variables forecast future stock returns.

We assume that informational and/or valuation irrationality in one period is reversed in the following period, i.e., any irrational component of ${aret}_{i}$ is reversed at date i + 1. Using matrix notation for ease of presentation, we write the underlying rational contemporaneous abnormal returns function as:

$${aret}_{t+1}^{AID}={{\varvec{\beta}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}\boldsymbol{ }{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}+{e}_{t+1}$$

(A9)

where ${aret}_{t+1}^{AID}$ denotes rational abnormal returns at date t + 1, ${{\varvec{\beta}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}$ denotes rational valuation coefficients based on the AID, ${{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}$ denotes rational accounting shocks based on the AID, and ${e}_{t+1}$ is a mean zero disturbance term. Next, it follows that observed abnormal returns, ${aret}_{t+1}$, under the assumption of informational and valuation irrationality (which reverses the following period), can be written as:

$$\begin{aligned}{aret}_{t+1}&={{\varvec{\beta}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}+{{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}+{\varvec{\beta}}{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}\\&\quad-{{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}-{\varvec{\beta}}{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}+{e}_{t+1}\end{aligned}$$

(A10)

where ${{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}={\varvec{\beta}}-{{\varvec{\beta}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}$ is the error in the valuation coefficients used by the market and ${{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}={{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}-{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}$ is the market error when estimating accounting shocks. Specifically, the first term on the RHS of (A10) is the rational response to correctly estimated shocks at date t + 1, the second term is the effect of valuation irrationality at date t + 1, the third term is the effect of informational irrationality at date t + 1, the fourth term is the reversal of prior valuation irrationality at date t, and the fifth term is the reversal of prior informational irrationality at date t. Finally, making use of the following definitions:

$$\begin{aligned}{\varvec{\kappa}}=\;&{{\varvec{\kappa}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}+{{\varvec{\kappa}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}},\;\;{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}={-{\varvec{\kappa}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{x}}}_{{\varvec{t}}},\;\;\\&{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}={-{\varvec{\kappa}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{x}}}_{{\varvec{t}}-1},\;\; {{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}={{\varvec{x}}}_{{\varvec{t}}}-{{\varvec{\kappa}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}{{\varvec{x}}}_{{\varvec{t}}-1}\end{aligned}$$

where ${\varvec{\kappa}}$ is the matrix of forecast coefficients used by the market to forecast accounting variables and ${{\varvec{\kappa}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}$ is the correct matrix of forecast coefficients which should be used under informational rationality, we can rewrite equation (A10) as the following lagged abnormal returns model:

$$\begin{aligned}{aret}_{t+1}&=-\left({\varvec{\beta}}{{\varvec{\kappa}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}+{{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}\right){{\varvec{x}}}_{{\varvec{t}}}\\&\quad+\left({\varvec{\beta}}{{\varvec{\kappa}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}+{{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{\kappa}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}\right){{\varvec{x}}}_{{\varvec{t}}-1}+{\vartheta }_{t+1}\end{aligned}$$

(A11)

where ${\vartheta }_{t+1}={e}_{t+1}+{\varvec{\beta}}{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}$ is a mean zero disturbance term which is independent of prior date accounting variables in ${{\varvec{x}}}_{{\varvec{t}}}$ and ${{\varvec{x}}}_{{\varvec{t}}-1}$, implying that equation (A11) can be estimated using standard regression methods. Note that if only informational irrationality occurs, i.e., ${{\varvec{\kappa}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}\ne 0$ and ${{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}=0$, then:

$$\begin{aligned}{aret}_{t+1}&=-{\varvec{\beta}}{{\varvec{\kappa}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{x}}}_{{\varvec{t}}}+{\varvec{\beta}}{{\varvec{\kappa}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{x}}}_{{\varvec{t}}-1}+{\vartheta }_{t+1}\\&={{\varvec{\beta}}{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}-{{\varvec{\beta}}{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}+{\vartheta }_{t+1}\end{aligned}$$

(A11′)

reflecting informational irrationality at date t + 1 due to use of incorrect forecast parameters and reversal of prior period information irrationality due to use of incorrect forecast parameters at date t. If only valuation irrationality occurs, i.e., ${{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}\ne 0$ and ${{\varvec{\kappa}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}=0$, then:

$$\begin{aligned}{aret}_{t+1}&=-{{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{x}}}_{{\varvec{t}}}+{{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{\kappa}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}{{\varvec{x}}}_{{\varvec{t}}-1}+{\vartheta }_{t+1}\\&=-{{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}+{\vartheta }_{t+1}\end{aligned}$$

(A11′′)

reflecting reversal of valuation irrationality at date t only (as valuation irrationality at date t + 1, ${{\varvec{\beta}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}{{\varvec{u}}{\varvec{x}}}_{{\varvec{t}}+1}^{{\varvec{A}}{\varvec{I}}{\varvec{D}}}$, is part of the disturbance term ${\vartheta }_{t+1}$ in this lagged model).

Finally, we use the model to consider the case where a single accounting variable, ${x}_{i}\in {\varvec{x}}$, is related to future abnormal stock returns. Specifically, if we observe:

$${aret}_{t+1}=\varphi {x}_{it}-\varphi {{x}_{it-1}+\varepsilon }_{t+1}$$

where ${\varepsilon }_{t+1}$ is a mean zero disturbance term, this is consistent with informational irrationality in relation to this variable as in equation (A11′); i.e., it is consistent with $\varphi =-{\varvec{\beta}}{{\varvec{\kappa}}}_{{\varvec{i}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}$, where ${{\varvec{\kappa}}}_{{\varvec{i}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}$ is a column vector of forecast coefficient errors made by the market when using ${x}_{it}$ to forecast value-relevant accounting variables, ${{\varvec{x}}}_{{\varvec{t}}+1}$. On the other hand, if we observe:

$${aret}_{t+1}=\varphi {x}_{it}+{\varepsilon }_{t+1}$$

a plausible explanation is that ${{\varvec{\kappa}}}_{{\varvec{i}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}=0$ (and hence ${\varvec{\beta}}{{\varvec{\kappa}}}_{{\varvec{i}}}^{{\varvec{E}}{\varvec{R}}{\varvec{R}}}=0$) and that $\varphi$ is a “risk premium” where the accounting variable ${x}_{it}$ proxies for sensitivity to risk not controlled for in the estimation of ${aret}_{t+1}$.