ITCV / RIR表述摘录

We also ran analyses to compute the impact threshold for a confounding variable (ITCV) (Frank, 2000), which did not indicate a potential omitted variable bias. In the models presented, the ITCV is higher than 0.14 in all cases, which would be a high correlation for the variables in our table. The konfound command in Stata further suggested that even for the relatively small effect shown in Table 3, we would need to replace 26% of the sample for the effect to be zero. These relatively high bars suggest our finding is stable.

The ITCV for average experience in Model 1 in Table 3 was 0.14. In Model 2, the ITCV for experience hiring after an involuntary departure was 0.18, while that for experience with hiring after a voluntary departure was 0.16.

Board risk oversight and environmental and social performance (Amiraslani et al., 2025)

We follow Frank (2000) and Larcker and Rusticus (2010) and conduct Impact Threshold of Confounding Variable (ITCV) tests. ITCV identifies the magnitude that a correlated omitted variable would need to have to render the coefficient of interest, BRO, statistically insignificant. The results are reported in Panel B of Table 9 and indicate that, in the majority of cases, a correlated omitted variable would need to be at least as strong as (and as high as almost 14 times) the impact of the control variable with the largest impact (e.g., Size) to overturn our results. Together with the findings from the coefficient stability tests, these results help alleviate concerns about the sensitivity of our inferences to potential correlated omitted variables.

Impact of CEO's digital technology orientation and board characteristics on firm value: a signaling perspective (Filatotchev et al., 2025)

As per addressing omitted-variables concerns, we calculated the requisite partial correlations with a confounding variable to invalidate our main predictions given our analyses context (e.g., sample size, predictors included, estimate values). Specifically, we used the Busenbark, Yoon, Gamache, and Withers (2022) impact threshold of a confounding variable (ITCV) technique to calculate the correlation threshold that an omitted variable should have to invalidate our findings.

By following their suggestions (we run linear ordinary least squares models -xtreg- with fe, as the command “konfound” does not run with Bruno’s [2005] approach), we found that for an omitted variable to invalidate our findings, it would need to be correlated at r>0.08 with both Tobin’s Q and with the CEO’s relative digital technology orientation.

We found that to invalidate our main predictions, an omitted variable needs to have a pattern of partial correlations (i.e., positive with the outcome and the focal predictor) that we do not observe for any variable in our models. Specifically, the highest ITCV value is approximately 75% larger than the strongest partial correlation in our analyses. This means that the comparative thresholds of the omitted variable have to be at least 75% larger than the largest partial correlation in our data suggests, something that is very unlikely (Busenbark et al., 2022).

Balancing the radical and the incremental: CEO affiliative humor and organizational ambidexterity (Campbell et al., 2025)

Beyond controlling for relevant variables, we also assessed how robust our results were to omitted variables by conducting the impact threshold of a confounding variable (ITCV) test (Busenbark et al., 2022b). Based on the full models in Table 3, these tests indicated that it was highly unlikely that our results were susceptible to omitted variable bias that would invalidate our results. Specifically, the ITCV tests indicated that to invalidate the main effect of CEO affiliative humor on organizational ambidexterity, 45.17 % of the observations would have to be replaced with cases where there was no effect.

Public enemies? The differential effects of reputation and celebrity on corporate misconduct scandalization (Han et al., 2024)

To further rule out endogeneity concerns, we conducted a robustness of inference to replacement (RIR) analysis (Frank et al., 2021; Xu et al., 2019; Xu & Frank, 2021). The RIR analysis is equivalent to the impact threshold of a confounding variable (ITCV) analysis (Frank, 2000)—which management scholars are increasingly using to diagnose potential endogeneity issues (see Busenbark, Yoon, et al., 2022 for a review)—and is particularly useful for nonlinear models where the ITCV analysis cannot be applied (Busenbark, Yoon, et al., 2022). Our analysis was based on Model 1 in Table 2 (i.e., the main effects model), using the konfound command in Stata with the nonlinear option specified (Busenbark, Yoon, et al., 2022). The RIR analysis enables researchers to determine ‘how much of a given effect size must be biased in order to overturn an otherwise statistically significant parameter estimate’ (Busenbark, Yoon, et al., 2022, p. 44). To invalidate our current inferences for high reputation, 24.7% of the estimate (775 cases) would have to be biased. Similarly, 42.3% (1325 cases) for celebrity, 53.8% (1688 cases) for objective misconduct severity, and 72.0% (2258 cases) for perceived misconduct severity would have to be due to bias. Thus, it is highly unlikely that endogeneity is an issue in our study.

基层党组织建设与企业创新——基于治理嵌入和外部关注的双重视角 (李彬 et al., 2024)

遗漏变量偏差是影响因果推断的关键问题。参考已有研究（Larcker和Rusticus，2010），本文计算了被遗漏混淆变量的影响阈值（ITCV），即要使得因果推断无效，被遗漏的混淆变量对解释变量（PartyAC）和被解释变量（INN）的影响需要达到的阈值。结果显示（限于篇幅未列示，备索），被遗漏混淆变量的影响阈值为0.0151，意味着要使得因果推断无效，混淆变量对解释变量（PartyAC1）和被解释变量（INNI）的影响需分别达到0.123。另外，可观测控制变量的影响最大值为0.0041，明显小于被遗漏混淆变量的影响阈值0.0151，即相对于这些可观测变量，被遗漏的混淆变量必须与解释变量和被解释变量有更强的相关性才能使因果推断无效。鉴于我们已经在模型中尽可能加入了对企业创新有较强影响的变量，很难找到一个能够使推断无效的被遗漏的混淆变量。上述结果表明本文的因果推断具有稳健性。

企业金融资产配置对双元创新的影响——高管激励的调节效应(万旭仙, 王虹, 何佳, 2019)

本文借鉴 Frank [30]，Larcker& Rusticus [31]的解决方法，使用ITCV 的方法考察前文分析结果的稳健性。 ITCV 方法旨在测量内生性问题是否改变了回归的结果，如果内生性问题的严重性不足以影响 OLS回归结果的方向和显著性，则可以忽视内生性问题，认为回归结果是稳健的。ITCV 被定义为被解释变量和不可观测变量之间的偏相关性乘以解释变量与不可观测变量之间的偏相关性，也是结果显著性改变的最小值。超过ITCV，说明内生性问题严重到改变了回归结果。ITCV 值越高，说明 OLS结果越稳健。

表7中(1)和(4)为长期金融资产配置对企业探索式创新和开发式创新影响的回归结果。表中(2)给出了长期金融资产和探索式创新的 ITCV 值为-0.1825，意味着探索式创新与不可观测变量之间的偏相关性以及长期金融资产与不可观测变量之间的偏相关性一旦都达到0.4272（），OLS估计的显著性就会发生变化。在没有其它对比数据时，无法确定回归的ITCV 是否足够大到保证结果的稳健性。虽然不可观测变量不可得，但是可以用其它控制变量的相关性制定判断基准。表中(3)和(6)显示的是每个控制变量对回归结果的潜在影响，计算方法同ITCV。如果ITCV 为正，说明控制变量将强化自变量的显著性，反之将降低自变量的显著性。因此在判断大小时，可以忽略ITCV 的正负方向。从表中(3)得到对回归结果影响最大的是企业资本密集度0.0263，但是仍小于阈值0.1825。同理，表中(5)反映的是长期金融资产配置和开发式创新的ITCV 值，表中(6)显示每个控制变量对回归结果的潜在影响，对回归结果影响最大的是企业资本密集度0.0078，但是始终小于0.0834。因此可以判断，长期金融资产配置与企业探索式创新投资和开发式创新投资存在的内生性问题并未影响结果的稳定性，本文研究结论具有稳健性

Influencer warmth and competence, communication mode, and stakeholder engagement on social media (Roccapriore and Pollock, 2023)

Finally, while the HT regressions we used for our main analyses alleviate concerns about endogeneity associated with a possible fixed effect (Hausman & Taylor, 1981), to rule out any further endogeneity concerns we also performed a robustness of inference to replacement (RIR) analysis, which is equivalent to an impact threshold of a confounding variable (ITCV) analysis, but is more appropriate for nonlinear models (Busenbark, Yue, Gamache, & Withers, 2022; Frank, Maroulis, Duong, & Kelcey, 2013). RIR and ITCV analyses allow researchers to determine how strong the effect of a particular variable would have to be to potentially create an endogeneity issue that overturns the findings (Busenbark et al., 2022; Frank, 2000). We employed the konfound command in Stata and assessed the effects of our four predictors on the two outcomes based on our Table 2 results.

For the number of followers, 69.02% of cases (2,013 cases) for word-based cues, 5.03% of cases (147 cases) for image-based cues, 15.94% of cases (465 cases) for competence-based cues, and 76.40% of cases (2,228 cases) for warmth-based cues would have to be biased to affect our results. For positive interactions, 50.46% of cases (1,471 cases) for word-based cues, 7.38% of cases (215 cases) for image-based cues, 46.26% of cases (1,349 cases) for competence-based cues, and 51.70% of cases (1,508 cases) for warmth-based cues would have to be biased to affect our results.

Thus, only image-based cues may be a concern, but given our use of HT models and the extensive control variables we include, particularly with respect to images, it seems unlikely that an omitted variable would result in these levels of bias. As Busenbark and colleagues (2022) noted, if researchers cannot identify a plausible omitted variable, then even low percentages are not problematic. Thus, endogeneity does not appear to be an issue.

Innovation and profitability following antitrust intervention against a dominant platform: the wild, wild west? (Thatchenkery and Katila, 2023)

To investigate the sensitivity of our results to omitted variable bias, we followed Frank (2000) and calculated how much bias needs to be present to invalidate our results, in two ways. First, we calculated the impact threshold of a confounding variable (ITCV), which tells us how strongly correlated a hypothetical omitted confounding variable would need to be to invalidate our results (Frank, 2000). ITCV is 0.04 for innovation and −0.11 for profit. In other words, to overturn the results, partial correlations between the difference-in-differences estimator, the dependent variable, and the omitted confounding variable would have to be above 0.20 (=√|0.04|) for innovation and above 0.33(=√|−0.11|) for profit. Using current control variables as a yardstick (Busenbark, Yoon, Gamache, & Withers, 2022; Larcker & Rusticus, 2010), any hypothetical omitted variable would thus need to have a larger impact than any of our (highly influential, “standard”) controls to overturn the results. It seems unlikely that an omitted variable would cross these thresholds.

Second, we calculated robustness of inference to replacement (RIR) (Frank, 2000) defined as the percentage of observations for which the observed treatment effect would need to be driven entirely by an omitted variable, not by the treatment, to invalidate the findings. Interpretation of the RIRs is “grounded in logical intuition, such that scholars typically determine whether the number of requisite overturned treatment cases appears reasonable” (Busenbark et al., 2022, p. 44). In our data, 23% of the treatment cases for innovation and 35% of cases for profit would need to be entirely overturned. These numbers are much higher than thresholds accepted in prior work (e.g., Busenbark, Lange, & Certo, 2017). Overall, our ITCV and RIR analyses indicate that it is unlikely that an omitted variable is driving the results.

The effect of CEO regulatory focus on changes to investments in R&D (Scoresby et al., 2021)

Finally, we considered the possibility of an omitted confounding variable in our analyses. It is possible that an omitted variable could affect both CEO communication from which we measure regulatory focus and also influence R&D increase. To address this concern, we ran an impact threshold of a confounding variable (ITCV) analysis (Busenbark et al., 2021; Frank, 2000) in Stata using the konfound command (Xu et al., 2019). This analysis provides scholars with a sensitivity analysis of how high the estimate of an omitted variable would have to be to alter the findings of a statistical analysis (Busenbark et al., 2021; Frank, 2000). We find that for an omitted variable to invalidate an inference of a null hypothesis regarding the relationship between promotion focus and R&D increase, it would have to be correlated with both the IV and DV at 0.109, with a threshold of 0.0118 (0.109 × 0.109). We find that for an omitted variable to sustain an inference of a null hypothesis regarding the relationship between prevention focus and R&D increase, it would have to be correlated with both the IV and DV at 0.113, with a threshold of 0.0128 (0.113 × 0.113). These are stronger combined effects than any of our other covariates, as the strongest combined effect of a variable currently in our model is R&D (log), with a threshold of 0.0083 (0.090 × 0.092). Given these results, we believe it is unlikely for an omitted variable to meaningfully affect the significance of these results.

On the use of instrumental variables in accounting research (Larcker and Rusticus, 2010)

Since the estimated OLS coefficient on disclosure quality is statistically significant, we examine the potential impact of unobserved confounding variables using the approach in Frank (2000). This method is based on the notion that for an unobserved variable to affect the results it needs to be correlated with both the x-variable and the y-variable (controlling for the other variables). Frank (2000) derives the minimum correlations necessary to turn a statistically significant result into a borderline insignificant result. The Impact Threshold for a Confounding Variable (denoted as ITCV) is defined as the lowest product of the partial correlation between y and the confounding variable and the partial correlation between x and the confounding variable that makes the coefficient statistically insignificant. If the ITCV is high (low), the OLS results are robust (not robust) to omitted variable concerns.

We calculate ITCV for disclosure quality in column (3). The threshold value for disclosure quality is -0.028, implying that the correlations between x and y with the unobserved confounding variable each only need to be about 0.167 (=√0.028) for the OLS result to be overturned.

Without some additional analysis, it is difficult to determine whether the ITCV is small enough to conclude that the OLS association between disclosure quality and spreads is fragile. That is, we need to develop a benchmark for the size of likely correlations involving the unobserved confounding variable. While, by definition, we do not have the unobservable confounding variable, we do have other control variables. In column (4), we show the impact of the inclusion of each independent variable on the coefficient of disclosure quality. Similar to the ITCV the impact is defined as the product of the partial correlation between the x-variable and the control variable and the correlation between the y-variable and the control variable (partialling out the effect of the other control variables). The sign of the impact measure indicates how the inclusion of the control variable affects the coefficient of disclosure quality. A positive impact score indicates that inclusion of the control variable makes the coefficient on the disclosure quality more negative (less positive) and a negative impact score has the opposite effect.

The variable with the largest impact on the coefficient for disclosure quality is market value of equity (MVE), with a value of 0.009. This suggests that we would need a confounding variable with a stronger impact than MVE to overturn the results on disclosure quality. Specifically, the unobserved confounding variable must be more highly correlated with disclosure quality and bid-ask spreads than MVE. Under the assumption that we have a good set of control variables this provides some confidence in the estimate of the effect of disclosure quality on bid-ask spreads.

The impact of each control variable is measured after inclusion of the other control variables. As such, even though the correlations between MVE and disclosure quality and MVE and bid-ask spreads are high, the partial correlations are relatively low. In comparing the ITCV to the distribution of impact scores for the control variables we implicitly assume that the confounding variable is similarly correlated with the other control variables. To the extent that the confounding variable is relatively distinct, a more fitting comparison might be to look at the product of the raw correlations instead of the partial correlation. Column (5) includes this more conservative measure of impact. Comparing these impact scores to the ITCV suggests that the effect of disclosure quality is not nearly as robust as previously implied. However, one might still argue that even though a variable with similar impact as MVE would overturn the results, it is unlikely such a variable will be found given that we already have strong controls such as MVE and the number of analysts in the model.

The assessment of confounding variables is a very useful evaluation procedure for the OLS estimates. However, absolute standards for impact threshold are difficult to establish. Researchers can make use of the impact for the selected control variables to provide a reasonable benchmark to the ITCV.

Performance Shortfalls, Response Directions, and Belief in the Effectiveness of Responses. (Cao et al. 2024)

We also addressed the possibility that our analysis is confounded by omitted variables. To investigate this issue, we calculated the impact threshold in a confounding variable analysis (Frank, 2000) for our interaction terms. The results show that, to invalidate our findings derived from Models 2, 4, and 5 and reported in Table 2, we would have to replace 45.35%, 77.51%, and 32.15% of the sample, respectively, with cases where there is no relationship between the respective interaction terms and R&D intensity. The results also suggest that, to invalidate our findings derived from Models 2–4andreported in Table 3, 73.53%, 34.23%, and 72.78% of the sample, respectively, would have to be replaced with cases where there is no relationship between the respective interaction terms and donation intensity. Inasmuch as large portions of our sample would have to be replaced with cases in which there was no effect to invalidate our estimates, it is reasonable to believe that it is unlikely that an omitted variable would invalidate our inference related to that interaction.

Deviant versus Aspirational Risk Taking: The Effects of Performance Feedback on Bribery Expenditure and R&D Intensity (Xu et al. 2019)

To further address any concern over the omitted-variable bias, we followed Frank, Maroulis, Duong, and Kelcey’s (2013) analysis and calculated the impact threshold of a confounding variable to quantify the degree of bias necessary to invalidate our results. In our sample (n 5 9,633), the estimated effect was 20.73 (Model 2, Table 2); the threshold for making an inference was calculated to be 20.24, based on a standard error of 0.12 and critical t value of 21.96. Thus, bias must account for 20.49 (20.73 2 (20.24)), or about 67% (20.49/20.73) of the estimated effect to invalidate our results, which is highly improbable (Frank et al., 2013).

参考文献

Amiraslani, H., Deller, C., Ittner, C.D., Keusch, T. (2025) Board risk oversight and environmental and social performance. Journal of Accounting and Economics, 79, 101754. https://doi.org/10.1016/j.jacceco.2024.101754

Boivie, S., Gee, I.H., Gentry, R.J., Graffin, S.D. (2025) Do boards learn to hire? The effect of board experience with CEO replacement on CEO performance. Strategic Management Journal, 46, 2467–2491. https://doi.org/10.1002/smj.3725

Campbell, R.J., Short, C.E., Graffin, S.D. (2025) Balancing the radical and the incremental: CEO affiliative humor and organizational ambidexterity. Research Policy, 54, 105131. https://doi.org/10.1016/j.respol.2024.105131

Filatotchev, I., Lanzolla, G., Syrigos, E. (2025) Impact of CEO’s Digital Technology Orientation and Board Characteristics on Firm Value: A Signaling Perspective. Journal of Management, 51, 875–912. https://doi.org/10.1177/01492063231200819

Han, J.-H., Pollock, T.G., Paruchuri, S. (2024) Public enemies? The differential effects of reputation and celebrity on corporate misconduct scandalization. Strategic Management Journal, 45, 2727–2762. https://doi.org/10.1002/smj.3638

Larcker, D.F., Rusticus, T.O. (2010) On the use of instrumental variables in accounting research. Journal of Accounting and Economics, 49, 186–205. https://doi.org/10.1016/j.jacceco.2009.11.004

Roccapriore, A.Y., Pollock, T.G. (2023) I Don’t Need a Degree, I’ve Got Abs: Influencer Warmth and Competence, Communication Mode, and Stakeholder Engagement on Social Media. Academy of Management Journal, 66, 979–1006. https://doi.org/10.5465/amj.2020.1546

Scoresby, R.B., Withers, M.C., Ireland, R.D. (2021) The effect of CEO regulatory focus on changes to investments in R&D. Journal of Product Innovation Management, 38, 401–420. https://doi.org/10.1111/jpim.12591

Thatchenkery, S., Katila, R. (2023) Innovation and profitability following antitrust intervention against a dominant platform: the wild, wild west? Strategic Management Journal, 44, 943–976. https://doi.org/10.1002/smj.3470

李彬, 姚瑶, 李海霞 (2024) 基层党组织建设与企业创新——基于治理嵌入和外部关注的双重视角. 外国经济与管理, 46, 18–32. https://doi.org/10.16538/j.cnki.fem.20230409.204

万旭仙, 王虹, 何佳 (2019) 企业金融资产配置对双元创新的影响——高管激励的调节效应. 科技进步与对策, 36, 124–132

Cao, Z., Jiang, F., Wang, D. (2024) Performance Shortfalls, Response Directions, and Belief in the Effectiveness of Responses. Academy of Management Journal, 67, 178–207. https://doi.org/10.5465/amj.2021.0241

Xu, D., Zhou, K.Z., Du, F. (2019) Deviant versus Aspirational Risk Taking: The Effects of Performance Feedback on Bribery Expenditure and R&D Intensity. Academy of Management Journal, 62, 1226–1251. https://doi.org/10.5465/amj.2016.0749

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