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About text below the content, author: Wu Aoba, Australian National University Business School and School of Economics, communication mail: [email protected]
The author's previous article: Mita, 2020 Nobel Prize RDD Queen's most influential masterpiece, with data and measurement procedures
Lennox, C., Francis, J., & Wang, Z. (2012). Selection Models in Accounting Research. The Accounting Review, 87(2), 589-616. Retrieved July 11, 2020, from www.jstor.org/stable/23245616
This study explains the challenges associated with the Heckman (1979) procedure to control for selection bias, assesses the quality of its application in accounting research, and offers guidance for better implementation of selection models. A survey of 75 recent accounting articles in leading journals reveals that many researchers implement the technique in a mechanical way with relatively little appreciation of important econometric issues and problems surrounding its use. Using empirical examples motivated by prior research, we illustrate that selection models are fragile and can yield quite literally any possible outcome in response to fairly minor changes in model specification. We conclude with guidance on how researchers can better implement selection models that will provide more convincing evidence on potential selection bias, including the need to justify model specifications and careful sensitivity analyses with respect to robustness and multicollinearity.
1. The introduction
article mainly evaluates the implementation of the selection model in the accounting literature, and provides relevant suggestions for accounting scholars in using the selection model to solve problems. In view of the increasing use of selection models and the frequent comments of journal editors on controlling endogeneity and selection bias, the guidelines and suggestions listed in the article are particularly important. The article found that from 2000 to 2009, 75 of the 1,016 empirical articles published in The Accounting Review, Journal of Accounting and Economics, Journal of Accounting Research, Contemporary Accounting Research and Review of Accounting Studies used the selection model, and Only in the year from 2008 to 2009, 11% of empirical articles used the selection model.
Maddala (1991) pointed out that when the observations are distributed non-randomly between the experimental group and the control group, selectivity bias will occur, which leads to the bias of the coefficient when using the least square method to estimate the coefficient. Little (1985) proposed a more credible implementation method, requiring researchers to identify exogenous independent variables that can be deleted in the second stage in the selection model of the first stage. However, the importance of exclusion constraints is diminishing in the field of accounting research. 14 of the 75 articles did not make any exclusions, and 7 of them did not elaborate on the first stage model, so it was impossible to judge whether exclusion constraints were carried out.
The author clarified that if the exclusion constraint is not applied, then the result of the selected model is unreliable. They believe that to improve the performance of the model, careful sensitivity analysis and robustness testing are required.
Second, select the model
Generally speaking, there are two different applications for the selection model. The first is the treatment effect model—endogenous indicator variables (D) as independent variables. For example, a researcher may be interested in whether management earnings forecasts affect the cost of capital. In this case, the endogenous indicator variable (D) indicates whether the company publishes revenue forecasts, and the dependent variable is the cost of capital. The second is the sample selection model-when regression is performed on a sub-sample of observations. For example, a researcher may be interested in managing the determinants of forecast accuracy. In this case, the dependent variable is forecast accuracy, and regression is only performed on companies that have issued revenue forecasts.
In summary, the difference between the OLS model in formula (1) and the selection model in formula (3) is that the latter introduces MILLS as an independent variable. Then there are two main sources of selection bias: (1) MILLS is non-linear; (2) variable Z is excluded from formula (3). Here, the variable Z is also called the exclusion constraint. It is generally considered that Z has no direct influence on Y, and any correlation between Y and Z is conducted through MILLS.
In econometrics, the choice to exclude constraints is very important, because it can control the endogeneity of variable D so that the choice model can be realized. First of all, the variable Z must be an exogenous variable, otherwise the coefficients in the first stage are biased. Secondly, the coefficient of Z in formula (2) needs to be significant. Finally, the exclusion of variable Z from formula (3) must be valid. If the variable Z is inappropriately omitted from the model with Y as the dependent variable, it will lead to the classic missing variable problem.
In many applications, finding a good variable Z is difficult. Because, even if the exclusion constraint is not implemented, MILLS is technically identifiable because its parameters are non-linear. However, Little (1985) pointed out that econometric economists do not recommend using nonlinearity to identify selection bias for two reasons: First, if there is no variable Z, the identification of selection bias comes only from untested functional form assumptions. To illustrate this point, if Y is actually a non-linear function, but researchers mistakenly assume a linear relationship, then MILLS will have an inappropriate functional form. Second, in the absence of a constraint, the selection model will be more prone to multicollinearity problems. Multicollinearity will lead to two results: First, the standard deviation of the estimated coefficient is expanded so that the coefficient estimate will be insignificant. Second, the representation of the model is inappropriate. The risk of incorrect model setting is very high, which will affect the sensitivity of statistical judgment.
3. The selection model in accounting research
The author selected 75 articles from 2000 to 2009 that used the selection model from five accounting journals. Table 1 shows the themes of his articles, of which 16 are about research on auditing, 16 are about information disclosure, 13 are about earnings management, 11 are about corporate governance, 2 are about taxation, 1 is about management accounting, and the remaining 16 are about Other accounting and financial topics.
Panel A in Table 2 shows a significant increase in the proportion of using the selection model between 2006 and 2009, with 50 articles published between 2006 and 2009, and only 25 using the selection model between 2000 and 2005. Panel B showed that 52 of the articles used the treatment effect model, and the remaining 23 articles used the sample selection model to estimate non-random sub-samples. 32 articles used the selection model for the main analysis, and the remaining 43 articles used the selection model as an auxiliary analysis. Panel C indicated that 54 articles followed the steps of using exclusion constraints, eight studies did not have variable Z, and 6 articles both reported selection models with exclusion constraints and showed selection models without exclusion constraints. In the other 7 articles, the author did not directly express the model expression of the first stage, so it is not certain that they are subject to exclusion constraints. In general, 19% to 28% of the articles did not have exclusion constraints. 60 articles had this step, but only three articles reported that their results were robust.
For the selection model, what is important is the researcher's choice of exclusion constraints, that is, it is an independent variable in the first-stage model and is excluded in the second-stage model. From this perspective, many accounting articles did not point out that the variable Z was excluded in the second stage. Some studies have proposed the feasibility of exclusion constraints from the economic point of view, but they have not explained that the exclusion constraints they choose are effective. Finally, multicollinearity problems may arise when using the selection model, but there are only three articles that test for multicollinearity.
The author believes that many accounting studies do not have a sufficient understanding of their econometric background when they use the selection model. This conclusion is very similar to the investigation on IV by Larcker & Rusticus (2010), but the evaluation mechanism used is quite different. Compared with the conventional IV model, the selection model is different in solving the endogenous problem. Specifically, the selection model uses the MILLS variable to control the correlation of the error term, but there is no equivalent term in the IV model. Therefore, compared with the evaluation mechanism proposed by LR, the evaluation mechanism of this paper has two main differences: First, MILLS is non-linear, which shows that even if there is no exclusion constraint, it can be estimated technically. But for the IV model, there is at least one exclusion constraint. The second difference is that researchers can judge whether there is a selection bias based on the significance of the coefficients of the MILLS variable.
4. Empirical analysis
The author found that in the published accounting articles, the exclusion constraint was not specifically pointed out, and even some articles did not carry out this process. Therefore, the author showed through the empirical analysis that the selection model without exclusion constraint cannot provide reliable and robust selection. Great results. The author selected Compustat, Audit Analytics, CRSP and I/B/E/S data from 2000 to 2009 for empirical analysis.
Auditing research usually uses selection models to control the endogenous nature of companies choosing between large and non-N. The author categorizes the independent variables used to estimate audit selection. 6 articles use logarithmic total assets to represent company size, 3 articles use logarithmic turnover to represent company size, and 1 article uses logarithmic total assets to represent company size. The digitized market value represents the company's size. Research also differs on how to express the company's profitability indicators. Three articles use indicator variables to represent losses, two use continuous variables to measure profitability, and the other three use both loss indicator variables and continuous profitability variables. The author’s empirical analysis is not intended to illustrate that accounting research should use the same indicators to measure company size and profitability. On the contrary, the author's view is that imposing different exclusion constraints on these variables will have a huge impact on the results.
These ten articles also differ in the use of exclusion constraints. The two articles did not set any exclusion constraints, because all the independent variables in the first-stage model were used as regression variables in the second-stage model. Although the econometric literature opposes this approach, these studies estimate the effect of selection bias through the non-linearity of the inverse Mills ratio. Although the other eight articles have added exclusion constraints, none of them provide a clear explanation or economic significance on why exclusion constraints are effective. The author’s purpose is not to criticize these audit studies, but to illustrate the problems that may arise when the researcher does not exclude constraints or uses arbitrary exclusion constraints. To illustrate the former, the author included all the independent variables in the first-stage model in the second-stage model. To illustrate the latter, the author excludes the company size or company profitability variable from the second-stage model, because previous studies usually impose exclusion constraints on these two variables.
5.
The author believes that the empirical analysis of the fourth part is not a direct repetition of the results of published papers. People who are skeptical may have such a view, and the research published in top journals is not very good. May contain such obvious questions, because they must be reviewed by an experienced editor. Based on this, the author reproduced a published study and tested the reliability of its conclusions, and further explained whether such conclusions are limited to audit literature and whether they are applicable to more general situations.
The author chose the article of Jackson et al. (2009) (subsequent to use JLC instead). JLC studied the impact of this method on the company's capital investment decisions. They predict and report that companies that adopt accelerated depreciation methods have higher levels of capital investment. Since the company's choice of depreciation policy is endogenous, JLC first estimated a model to explain the decision, and then they constructed the inverse Mills ratio and included it as an independent variable in the second-stage model of capital expenditure. Most of the independent variables in the first-stage model are excluded from the second-stage capital expenditure model. Like many studies in Table 2, JLC did not explain the validity of their exclusion constraints, and did not formally report on the robustness of multicollinearity and sensitivity analysis.
6. Suggestions
for the use of the model The author provides four practical suggestions for the use of the selection model. First, in the absence of exclusion constraints, it is not advisable to estimate the selection model, that is, some independent variables in the first stage should be excluded in the second stage model. Because in the absence of exclusion constraints, the result of the inverse Mills ratio is completely dependent on its non-linearity. Moreover, since the problem of multicollinearity may arise, the author suggests that even if exclusion constraints have been imposed, researchers should also perform diagnostic tests for multicollinearity.
The second point is that some accounting studies did not specify which independent variables they used in the first-stage model. This lack of information prevents readers from identifying whether they are exclusionary constraints or evaluating their statistical inferences. Therefore, the author’s second suggestion is to clarify the selection model in the first stage of the report and clearly show the Z variables they choose.
The third point is that the research should explain why the Z variable in the first-stage model can be effectively excluded in the second-stage model. Accounting researchers are used to explaining why these independent variables can be included in a model, but they often cannot explain why one or more variables in the first-stage selection model can be effectively excluded from the second-stage outcome model. It is not enough to rely on past research to prove that the independent variables included in the first and second stage models are not enough.
Fourth, since the results of the selection model are usually unstable, a sensitivity analysis report must be carried out for the robustness of statistical inference. However, in empirical accounting research, sensitivity analysis reports are not common.
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