The endogeneity issue in longitudinal research: Sources and solutions

doi:10.3724/SP.J.1042.2023.00507

Abstract

Abstract:

Longitudinal cross-lagged models have been widely used to analyze causal relations in behavioral and psychological sciences, and the cross-lagged panel model (CLPM) is common and important. The CLPM and related panel data models usually consist of two kinds of regression relations: (a) the autoregressive effects of a variable from its prior state (an earlier time point) to its current state (present time point) and (b) the cross-lagged effects of one variable at the prior state to another variable at the current state. These effects between the two constructs provide the foundation and crucial information in deciding on their diachronic causation. Importantly, in contrast to general regression models, a CLPM consists of a complex set of regression equations, making it more susceptible to endogeneity-related problems.
Endogeneity is a critical concern in regression analyses, which refers to situations when an explanatory variable is correlated with the residual (error) of its regression equation. It will likely lead to over- or under-estimated bias with commonly used estimators. Endogeneity is a critical concern when using regression models to analyze observational data to make causal claims. The CLPM determines diachronic causation based on two kinds of regression effects (autoregressive and cross-lagged paths). Despite its vulnerability to endogeneity, this issue has received little attention and requires systematic analyses.
The current study focuses on issues related to endogeneity in the CLPM. We first clarify the main sources of endogeneity problems. Then, we systematically analyze different endogeneity issues in the CLPM. Lastly, we provide an empirical example to illustrate the use of the instrumental variables (IV) method in the CLPM.
IV originates from econometrics and refers to the predictor of a predictor. The IV method is probably the most popular approach while dealing with endogeneity. Researchers often incorporate suitable IVs in the model to provide unbiased estimates and alleviate the endogeneity concerns. The model implied IVs (MIIVs) have been frequently used in empirical studies. A MIIV is an IV identified within the model. The MIIVs offer a promising way to deal with endogeneity in longitudinal analyses. Typically, a MIIV is a chronologically prior observation of an exogenous variable in the model. Currently, applications with IV are underutilized in psychological research. This paper tries to illustrate the use of MIIV in the CLPM by an empirical example.
To our knowledge, this is the first study to discuss endogeneity issues in the CLPM and explore the performances of MIIV in the longitudinal cross-lagged model. We find that common possible sources of endogeneity in the CLPM are: omitted variables, dynamic panel, and reciprocal relation. The omitted variables are ubiquitous in all empirical research and the omitted variable problem will affect the estimation of cross-lagged analyses. For the dynamic panel, “dynamic” refers to the use of the prior outcome as a predictor. Including the effects of this lagged outcome increases the probability of the explanatory variable being related to the residual. Besides, biases could arise from the reciprocal relation, which is also known as the feedback relation, simultaneity, or simultaneous causality.
We conclude, first, there are various types of endogeneity in the CLPM, including the omitted variables, dynamic panel, and reciprocal relation. Second, endogeneity could distort the estimation of cross-lagged effects in the CLPM. Lastly, MIIV is a promising technique to tackle endogeneity issues in the CLPM. For future research, it would be interesting to explore the performance of MIIV in models extended from the CLPM. They may include the Random Intercept-CLPM, the Latent Cure Model with Structured Residuals (LCM-SR), and the Latent Change Score Model (LCS).
This paper reviews the main sources of endogeneity in the CLPM to raise applied researchers’ awareness of the endogeneity issues in longitudinal research. We recommend the MIIV-CLPM as a solution to deal with the endogeneity issue.

Key words: endogeneity, longitudinal research, cross-lagged panel model, instrumental variables

CLC Number:

B841

FANG Junyan, WEN Zhonglin. The endogeneity issue in longitudinal research: Sources and solutions[J]. Advances in Psychological Science, 2023, 31(4): 507-518.

Figures/Tables 5

References 44

[1]	陈云松. (2012). 逻辑、想象和诠释:工具变量在社会科学因果推断中的应用. 社会学研究, 27(6),192-216.
[2]	陈云松, 范晓光. (2010). 社会学定量分析中的内生性问题测估社会互动的因果效应研究综述. 社会, 30(4), 91-117.
[3]	方俊燕, 温忠麟, 黄国敏. (印刷中). 纵向关系的探究:基于交叉滞后结构的追踪模型. 心理科学.
[4]	黄斌, 何沛芸, 朱宇, 魏易. (2022). 基于父母教育背景的中国家庭校外教育支出分化——兼论家庭需求视角下“双减”政策实施的优化. 中国教育学刊, (4), 19-28.
[5]	解垩, 宋颜群. (2021). 收入的社区邻里效应研究. 经济管理, 43(6),190-208.
[6]	李适源, 刘爱玉. (2022). “忧郁的孩子们”:课外补习会带来负向情绪吗?基于中国教育追踪调查(CEPS)两期数据的因果推断. 社会, 42(2), 60-93.
[7]	刘立光. (2021). 精神健康能否提升老年人主观幸福感?——兼论社会互动的中介效应. 科学经济社会, 39(4), 56-68.
[8]	王宇, 李海洋. (2017). 管理学研究中的内生性问题及修正方法. 管理学季刊, 3(35), 20-47.
[9]	温忠麟. (2017). 实证研究中的因果推理与分析. 心理科学, 40(1), 200-208.
[10]	温忠麟, 陈虹熹, 叶宝娟, 方杰, 蔡保贞. (2022). 新世纪20年国内测验信度研究. 心理科学进展, 30(8), 1682-1691.
[11]	温忠麟, 侯杰泰, 马什赫伯特. (2004). 结构方程模型检验: 拟合指数与卡方准则. 心理学报, 36(2), 186-194.
[12]	熊猛, 刘若瑾, 叶一舵. (2020). 单亲家庭儿童相对剥夺感与心理适应的循环作用关系:一项追踪研究. 心理学报, 53(1), 67-80.
[13]	张晓敏, 李亚男, 徐慧. (2022). 教育如何影响流动人口的社会融入?——基于义务教育法实施的自然实验. 教育经济评论, 7(2), 20-43.
[14]	赵西亮. (2017). 教育、户籍转换与城乡教育收益率差异. 经济研究, 52(12), 164-178.
[15]	朱琪, 陈臻, 李博湛, 杨洋. (2022). 老龄化背景下人工智能对人工成本的影响: 来自中国31个省份的证据. 华南师范大学学报(社会科学版), (2), 142-158.
[16]	Aliprantis D., & Richter F. G.-C. (2020). Evidence of neighborhood effects from moving to opportunity: Lates of neighborhood quality. Review of Economics and Statistics, 102(4), 633-647. doi: 10.1162/rest_a_00933 URL
[17]	Bascle G. (2008). Controlling for endogeneity with instrumental variables in strategic management research. Strategic Organization, 6(3), 285-327. doi: 10.1177/1476127008094339 URL
[18]	Bollen K. A. (2012). Instrumental variables in sociology and the social sciences. Annual Review of Sociology, 38(1), 37-72. doi: 10.1146/soc.2012.38.issue-1 URL
[19]	Bollen K. A. (2018). Model implied instrumental variables (MIIVs): An alternative orientation to structural equation modeling. Multivariate Behavioral Research, 54(1), 31-46. doi: 10.1080/00273171.2018.1483224 URL
[20]	Burns R. A., Crisp D. A., & Burns R. B. (2019). Re-examining the reciprocal effects model of self-concept, self-efficacy, and academic achievement in a comparison of the cross-lagged panel and random-intercept cross- lagged panel frameworks. British Journal of Educational Psychology, 90(1), 77-91. doi: 10.1111/bjep.v90.1 URL
[21]	Certo S. T., Busenbark J. R., Woo H., & Semadeni M. (2016). Sample selection bias and Heckman models in strategic management research. Strategic Management Journal, 37(13), 2639-2657. doi: 10.1002/smj.2475 URL
[22]	Cole D. A., & Maxwell S. E. (2003). Testing meditational models with longitudinal data: Questions and tips in the use of structural equation modeling. Journal of Abnormal Psychology, 112(4), 558-577. doi: 10.1037/0021-843X.112.4.558 URL
[23]	Falkenström F., Ekeblad A., & Holmqvist R. (2016). Improvement of the working alliance in one treatment session predicts improvement of depressive symptoms by the next session. Journal of Consulting and Clinical Psychology, 84(8), 738-751. doi: 10.1037/ccp0000119 pmid: 27213493
[24]	Fang J. Y., & Hau K.-T. (under review). Wen, Z. L., Appropriate modeling of endogeneity in cross-lagged panel models: A comparison of auxiliary and model implied instrumental variables.
[25]	Fang J. Y., Wen Z. L., Ouyang J. Y., Huang G. M., & Liu X. Q. (2022). The reciprocal longitudinal relationship between body mass index and subjective well-being in China. Social Science & Medicine. 297, 114829. doi: 10.1016/j.socscimed.2022.114829 URL
[26]	Gates K. M., Fisher Z. F., & Bollen K. A. (2020). Latent variable GIMME using model implied instrumental variables (MIIVs). Psychological Methods, 25(2), 227-242. doi: 10.1037/met0000229 pmid: 31246041
[27]	Hamaker E. L., Kuiper R. M., & Grasman R. P. P. P. (2015). A critique of the cross-lagged panel model. Psychological Methods, 20(1), 102-116. doi: 10.1037/a0038889 pmid: 25822208
[28]	Heckman J. J. (1976). The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models. Annals of Economic and Social Measurements, 5(4), 475-592.
[29]	Heckman J. J. (1979). Sample selection bias as a specification error. Applied Econometrics, 47(1), 153-161.
[30]	Hill A. D., Johnson S. G., Greco L. M., O’Boyle E. H., & Walter S. L. (2020). Endogeneity: A review and agenda for the methodology-practice divide affecting micro and macro research. Journal of Management, 47(1), 105-143. doi: 10.1177/0149206320960533 URL
[31]	Kazuma S. (2021). Unhappy and happy obesity: A comparative study on the United States and China. Journal of Happiness Studies, 22(3), 1259-1285. doi: 10.1007/s10902-020-00272-2
[32]	Lebenbaum M., Laporte A., & de Oliveira C. (2021). The effect of mental health on social capital: An instrumental variable analysis. Social Science & Medicine, 272, 113693. doi: 10.1016/j.socscimed.2021.113693 URL
[33]	Lu H., Tong P., & Zhu R. (2019). Longitudinal evidence on social trust and happiness in China: Causal effects and mechanisms. Journal of Happiness Studies, 21(5), 1841-1858. doi: 10.1007/s10902-019-00159-x
[34]	Maydeu-Olivares A., Shi D., & Fairchild A. (2020). Estimating causal effects in linear regression models with observational data: The instrumental variables regression model. Psychological Methods, 25(2), 243-258. doi: 10.1037/met0000226 pmid: 31294588
[35]	Patel P. C., & Cooper D. (2014). Structural power equality between family and non-family TMT members and the performance of family firms. Academy of Management Journal, 57(6), 1624-1649. doi: 10.5465/amj.2012.0681 URL
[36]	Rosseel Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1-36.
[37]	Semadeni M., Withers M. C., & Certo S. T. (2014). The perils of endogeneity and instrumental variables in strategy research: Understanding through simulations. Strategic Management Journal, 35(7), 1070-1079. doi: 10.1002/smj.2014.35.issue-7 URL
[38]	Shaver J. M. (1998). Accounting for endogeneity when assessing strategy performance: Does entry mode choice affect FDI survival? Management Science, 44(4), 571-585. doi: 10.1287/mnsc.44.4.571 URL
[39]	Shea J. (1997). Instrument relevance in multivariate linear models: A simple measure. Review of Economics and Statistics, 79(2), 348-352.
[40]	Streeter A. J., Lin N. X., Crathorne L., Haasova M., Hyde C., Melzer D., & Henley W. E. (2017). Adjusting for unmeasured confounding in nonrandomized longitudinal studies: A methodological review. Journal of Clinical Epidemiology, 87, 23-34. doi: S0895-4356(16)30334-1 pmid: 28460857
[41]	Tong E. M. W., Lum D. J. K., Sasaki E., & Yu Z. (2019). Concurrent and temporal relationships between humility and emotional and psychological well-being. Journal of Happiness Studies, 20(5), 1343-1358. doi: 10.1007/s10902-018-0002-3
[42]	Usami S., Murayama K., & Hamaker E. L. (2019). A unified framework of longitudinal models to examine reciprocal relation. Psychological Methods, 24(5), 637-657. doi: 10.1037/met0000210 URL
[43]	Wiedermann W., & von Eye A. (2020). Reciprocal relations in categorical variables. Psychological Methods, 25(6), 708-725. doi: 10.1037/met0000257 URL
[44]	Wooldridge J. M. (2010). Econometric analysis of cross section and panel data (2nd Edition). Cambridge, MA: MIT Press.

变量	M (SD)	幸福感(T1)	幸福感(T2)	幸福感(T3)	BMI (T1)	BMI (T2)	BMI (T3)
幸福感(T1)	3.51 (0.83)	1
幸福感(T2)	3.63 (0.87)	0.34^**	1
幸福感(T3)	3.62 (0.83)	0.23^**	0.30^**	1
BMI (T1)	23.53 (6.10)	0.10^**	0.17^**	0.13^**	1
BMI (T2)	23.80 (5.72)	0.10^**	0.18^**	0.14^**	0.95^**	1
BMI (T3)	24.02 (5.58)	0.10^**	0.15^**	0.11^**	0.89^**	0.89^**	1

变量	M (SD)	幸福感(T1)	幸福感(T2)	幸福感(T3)	BMI (T1)	BMI (T2)	BMI (T3)
幸福感(T1)	3.51 (0.83)	1
幸福感(T2)	3.63 (0.87)	0.34^**	1
幸福感(T3)	3.62 (0.83)	0.23^**	0.30^**	1
BMI (T1)	23.53 (6.10)	0.10^**	0.17^**	0.13^**	1
BMI (T2)	23.80 (5.72)	0.10^**	0.18^**	0.14^**	0.95^**	1
BMI (T3)	24.02 (5.58)	0.10^**	0.15^**	0.11^**	0.89^**	0.89^**	1

系数/拟合指标	CLPM			CLPM-cov			MIIV-CLPM
系数/拟合指标	Est.	SE	p	Est.	SE	p	Est.	SE	p
${{\beta }_{x1}}$	0.71	0.09	<0.001	0.73	0.09	<0.001	0.59	0.09	<0.001
${{\beta }_{x2}}$	0.70	0.09	<0.001	0.72	0.09	<0.001	0.69	0.09	<0.001
${{\beta }_{y1}}$	0.97	0.01	<0.001	0.97	0.01	<0.001	1.00	0.02	<0.001
${{\beta }_{y2}}$	0.97	0.01	<0.001	0.97	0.01	<0.001	0.94	0.02	<0.001
${{\gamma }_{x1}}$	0.00	0.12	0.898	0.00	0.12	0.810	0.03	0.24	0.291
${{\gamma }_{x2}}$	0.00	0.12	0.898	0.00	0.12	0.810	0.04	0.24	0.291
${{\gamma }_{y1}}$	0.10	0.00	0.005	0.10	0.00	0.007	0.02	0.01	0.661
${{\gamma }_{y2}}$	0.09	0.00	0.005	0.09	0.00	0.007	0.02	0.01	0.661
${{\theta }_{x}}$	—	—	—	-0.03	0.05	0.523	—	—	—
${{\theta }_{y}}$	—	—	—	0.10	0.39	0.003	—	—	—
${{\eta }_{x}}$	—	—	—	-0.03	0.00	0.548	—	—	—
${{\eta }_{y}}$	—	—	—	0.05	0.01	0.132	—	—	—
${{z}_{x1}}$	—	—	—	—	—	—	0.03	0.01	0.001
${{z}_{x2}}$	—	—	—	—	—	—	0.21	0.01	0.001
${{z}_{y1}}$	—	—	—	—	—	—	0.05	0.03	<0.001
${{z}_{y2}}$	—	—	—	—	—	—	0.37	0.03	<0.001
χ²	130.371			152.757			0.212
df	9			17			2
CFI	0.971			0.967			1.000
RMSEA	0.117			0.090			0.000

系数/拟合指标	CLPM			CLPM-cov			MIIV-CLPM
系数/拟合指标	Est.	SE	p	Est.	SE	p	Est.	SE	p
${{\beta }_{x1}}$	0.71	0.09	<0.001	0.73	0.09	<0.001	0.59	0.09	<0.001
${{\beta }_{x2}}$	0.70	0.09	<0.001	0.72	0.09	<0.001	0.69	0.09	<0.001
${{\beta }_{y1}}$	0.97	0.01	<0.001	0.97	0.01	<0.001	1.00	0.02	<0.001
${{\beta }_{y2}}$	0.97	0.01	<0.001	0.97	0.01	<0.001	0.94	0.02	<0.001
${{\gamma }_{x1}}$	0.00	0.12	0.898	0.00	0.12	0.810	0.03	0.24	0.291
${{\gamma }_{x2}}$	0.00	0.12	0.898	0.00	0.12	0.810	0.04	0.24	0.291
${{\gamma }_{y1}}$	0.10	0.00	0.005	0.10	0.00	0.007	0.02	0.01	0.661
${{\gamma }_{y2}}$	0.09	0.00	0.005	0.09	0.00	0.007	0.02	0.01	0.661
${{\theta }_{x}}$	—	—	—	-0.03	0.05	0.523	—	—	—
${{\theta }_{y}}$	—	—	—	0.10	0.39	0.003	—	—	—
${{\eta }_{x}}$	—	—	—	-0.03	0.00	0.548	—	—	—
${{\eta }_{y}}$	—	—	—	0.05	0.01	0.132	—	—	—
${{z}_{x1}}$	—	—	—	—	—	—	0.03	0.01	0.001
${{z}_{x2}}$	—	—	—	—	—	—	0.21	0.01	0.001
${{z}_{y1}}$	—	—	—	—	—	—	0.05	0.03	<0.001
${{z}_{y2}}$	—	—	—	—	—	—	0.37	0.03	<0.001
χ²	130.371			152.757			0.212
df	9			17			2
CFI	0.971			0.967			1.000
RMSEA	0.117			0.090			0.000