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How Much Does the Cardinal Treatment of Ordinal Variables Matter? An Empirical Investigation

Published online by Cambridge University Press:  26 February 2021


Many researchers use an ordinal scale to quantitatively measure and analyze concepts. Theoretically valid empirical estimates are robust in sign to any monotonic increasing transformation of the ordinal scale. This presents challenges for the point-identification of important parameters of interest. I develop a partial identification method for testing the robustness of empirical estimates to a range of plausible monotonic increasing transformations of the ordinal scale. This method allows for the calculation of plausible bounds around effect estimates. I illustrate this method by revisiting analysis by Nunn and Wantchekon (2011, American Economic Review, 101, 3221–3252) on the slave trade and trust in sub-Saharan Africa. Supplemental illustrations examine results from (i) Aghion et al. (2016, American Economic Review, 106, 3869–3897) on creative destruction and subjective well-being and (ii) Bond and Lang (2013, The Review of Economics and Statistics, 95, 1468–1479) on the fragility of the black–white test score gap.

© The Author(s) 2021. Published by Cambridge University Press on behalf of the Society for Political Methodology

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Edited by Jeff Gill


Acemoglu, D., Johnson, S., and Robinson, J.. 2001. “The Colonial Origins of Comparative Development: An Empirical Investigation.” American Economic Review 91(5):13691401.CrossRefGoogle Scholar
Aghion, P., Akcigit, U., Deaton, A., and Roulet, A.. 2016. “Creative Destruction and Subjective Well-Being.” American Economic Review 106(12):38693897.CrossRefGoogle ScholarPubMed
Aitchison, J., and Silvey, S. D.. 1957. “The Generalization of Probit Analysis to the Case of Multiple Responses.” Biometrika 44:131140.CrossRefGoogle Scholar
Baird, S., de Hoop, J., and Oxler, B.. 2013. “Income Shocks and Adolescent Mental Health.” Journal of Human Resources 48(2):370403.CrossRefGoogle Scholar
Baker, F. 2001. The Basics of Item Response Theory. College Park: ERIC Clearinghouse on Assessment and Evaluation, University of Maryland.Google Scholar
Banks, W. P., and Coleman, M. J.. 1981. “Two Subjective Scales of Number.” Perception & Psychophysics 29(2):95105.CrossRefGoogle Scholar
Becker, W., and Kennedy, P.. 1992. “A Graphical Exposition of the Ordered Probit.” Econometric Theory 8(1):127131.CrossRefGoogle Scholar
Bloem, J. 2020a. “Replication Data for: How Much Does the Cardinal Treatment of Ordinal Variables Matter? An Empirical Investigation.”, Code Ocean.CrossRefGoogle Scholar
Bloem, J. 2020b. “Replication Data for: How Much Does the Cardinal Treatment of Ordinal Variables Matter? An Empirical Investigation.”, Harvard Dataverse, V1, UNF:6:xZYCRzbDR5XIVSnKwo3GCg== [fileUNF].Google Scholar
Bond, T., and Lang, K.. 2013. “The Evolution of the Black–White Test Score Gap in Grades K-3: The Fragility of Results.” The Review of Economics and Statistics 95(5):14681479.CrossRefGoogle Scholar
Bond, T., and Lang, K.. 2019. “The Sad Truth About Happiness Scales.” Journal of Political Economy 127(4):16291640.CrossRefGoogle Scholar
Borghans, L., Duckworth, A. L., Heckman, J., and ter Weel, B.. 2008. “The Economics and Psychology of Personality Traits.” Journal of Human Resources 43(4):9721059.CrossRefGoogle Scholar
Bryson, A., and MacKerron, G.. 2017. “Are You Happy While You Work?The Economic Journal 127(599):106125.CrossRefGoogle Scholar
Clark, A., and Oswald, A.. 1996. “Satisfaction and Comparison Income.” Journal of Public Economics 61(3):359381.CrossRefGoogle Scholar
Conley, T. G., Hansen, C. B., and Rossi, P. E.. 2012. “Plausibly Exogenous.” The Review of Economics and Statistics 94(1):260272.CrossRefGoogle Scholar
Coombs, C. 1965. Theory of Data. New York: Wiley & Sons.Google Scholar
Cornaglia, F., Feldman, N. E., and Leigh, A.. 2014. “Crime and Mental Well-Being.” Journal of Human Resources 49(1):110140.CrossRefGoogle Scholar
Daly, M. C., and Wilson, D. J.. 2009. “Happiness, Unhappiness, and Suicide: An Empirical Assessment.” Journal of the European Economic Association 7(2–3):539549.CrossRefGoogle Scholar
Deaton, A. 2018. “What Do Self-Reports of Wellbeing Say About Life-Cycle Theory and Policy.” Journal of Public Economics 162:1825.CrossRefGoogle ScholarPubMed
Di Tella, R., MacCulloch, R., and Oswald, A.. 2003. “The Macroeconomics of Happiness.” The Review of Economics and Statistics 85(4):809827.CrossRefGoogle Scholar
Ferrer-i-Carbonell, A., and Frijters, P.. 2004. “How Important Is Methodology for the Estimates of the Determinants of Happiness?Economic Journal 114:641659.CrossRefGoogle Scholar
Frijters, P., Haisken-DeNew, J. P., and Shields, M. A.. 2004. “Money Does Matter! Evidence from Increasing Real Income and Life Satisfaction in East Germany Following Reunification.” American Economic Review 94(3):730740.CrossRefGoogle Scholar
Glewwe, P. 1997. “Estimating the Impact of Peer Group Effects on Socioeconomic Outcomes: Does the Distribution of Peer Group Characteristics Matter?Economics of Education Review 16(1):3943.CrossRefGoogle Scholar
Greene, W. 2012. Econometric Analysis . 7th edn. Boston: Pearson Education.Google Scholar
Hadar, J., and Russell, W. R.. 1969. “Rules for Ordering Uncertain Prospects.” American Economic Review 49(1):2534.Google Scholar
Heckman, J. J. 1981. “The Incidental Parameters Problem and the Problem of Initial Conditions in Estimating a Discrete Time-Discrete Data Stochastic Process.” In Structural Analysis of Discrete Data with Econometric Applications, edited by Manski, C. and McFadden, D., 179195. Cambridge, MA: MIT Press.Google Scholar
Jacob, B., and Rothstein, J.. 2016. “The Measurement of Student Ability in Modern Assessment Systems.” Journal of Economic Perspectives 30(3):85108.CrossRefGoogle Scholar
Kaiser, C., and Vendrik, C. M.. 2019. “How Threatening Are Transformations of Reported Happiness to Subjective Wellbeing Research?” SocArXiv Working Paper.CrossRefGoogle Scholar
Krueger, A. B. 2017. “Where Have All the Workers Gone? An Inquiry into the Decline of the U.S. Labor Force Participation Rate.” Brookings Papers on Economic Activity 2017:187.CrossRefGoogle ScholarPubMed
Krueger, A. B., Kahneman, D., Schkade, D., Schwarz, N., and Stone, A.. 2009. “National Time Accounting: The Currency of Life.” In Measuring the Subjective Well-Being of Nations: National Accounts of Time Use and Well-Being, edited by Krueger, A. B.. Chicago: University of Chicago Press.CrossRefGoogle Scholar
Lancaster. 2000. “The Incidental Parameter Problem Since 1948.” Journal of Econometrics 95:391413.CrossRefGoogle Scholar
Lang, K. 2010. “Measurement Matters: Perspectives on Education Policy from an Economist and School Board Member.” The Journal of Economic Perspectives 24(3):167181.CrossRefGoogle Scholar
Lord, F. M. 1975. “The ‘Ability’ Scale in Item Characteristic Curve Theory.” Psychometrika 40(2):205217.CrossRefGoogle Scholar
Manski, C. 2003. Partial Identification of Probability Distributions. Springer Series in Statistics. New York: Springer.Google Scholar
McKelvey, R., and Zavoina, W.. 1975. “A Statistical Model for the Analysis of Ordered Level Dependent Variables.” Journal of Mathematical Sociology 4:103120.CrossRefGoogle Scholar
Neyman, J., and Scott, E.. 1948. “Consistent Estimation from Partially Consistent Observations.” Econometrica 16:132.CrossRefGoogle Scholar
Nielsen, E. 2017. “How Sensitive Are Standard Statistics to the Choice of Scale?” Unpublished Working Paper.Google Scholar
Nunn, N., and Wantchekon, L.. 2011. “The Slave Trade and the Origins of Mistrust in Africa.” American Economic Review 101(7):32213252.CrossRefGoogle Scholar
Oswald, A. J. 2008. “On the Curvature of the Reporting Function from Objective Reality to Subjective Feelings.” Economics Letters 100:369372.CrossRefGoogle Scholar
Putnam, R. 2001. Bowling Alone: The Collapse and Revival of American Community. 1st edn. New York: Simon & Schuster.Google Scholar
Reardon, S. F. 2008. “Thirteen Ways of Looking at the Black-White Test Score Gap.” Sanford University Working Paper.Google Scholar
Riedl, M., and Geishecker, I.. 2014. “Keep It Simple: Estimation Strategies for Ordered Response Models with Fixed Effects.” Journal of Applied Statistics 41(11):23582374.CrossRefGoogle Scholar
Schaffer, M. E. 2015. “GINIREG: Stata Module for Gini Regression.” Scholar
Schröder, C., and Yitzhaki, S.. 2016. “PISA Country Rankings and the Difficulty of Exams.” Working Paper.CrossRefGoogle Scholar
Schröder, C., and Yitzhaki, S.. 2017. “Revisiting the Evidence for Cardinal Treatment of Ordinal Variables.” European Economic Review 92:337358.CrossRefGoogle Scholar
Snell. 1964. “A Scaling Procedure for Ordered Categorical Data.” Biometrics 20:592607.CrossRefGoogle Scholar
Stevens, S. 1946. “On the Theory of Scales of Measurement.” Science 10:667680.Google Scholar
Stevenson, B., and Wolfers, J.. 2013. “Subjective Well-Being and Income: Is There Any Evidence of Satiation?American Economic Review: Papers and Proceedings 103(3):598604.CrossRefGoogle Scholar
Tamer, E. 2010. “Partial Identification in Econometrics.” The Annual Review of Economics 2:167195.CrossRefGoogle Scholar
Thorndike, R. 1966. “Intellectual Status and Intellectual Growth.” Journal of Educational Psychology 57:121127.CrossRefGoogle ScholarPubMed
Van Praag, B. 1991. “Ordinal and Cardinal Utility: An Integration of the Two Dimension of the Welfare Concept.” Journal of Econometrics 50(1–2):6989.CrossRefGoogle Scholar
Yitzhaki, S., and Schechteman, E.. 2012. “Identifying Monotonic and Non-Monotonic Relationships.” Economics Letters 116(1):2325.CrossRefGoogle Scholar
Yitzhaki, S., and Schechteman, E.. 2013. The Gini Methodology: A Primer on a Statistical Methodology. New York: Springer.CrossRefGoogle Scholar
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