The Treatment of Linear Restrictions in Regression Analysis
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Eq. (2) can be fitted subject to linear equality restrictions, which would reduce the number of parameters to be estimated.
“First, Eq. (2) can be fitted subject to linear equality restrictions, which would reduce the number of parameters to be estimated [56,57].”
From Syndemics and Health Disparities: A Methodological Note (2016) · cites this paper as [57]
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Contracting in Supply Chains: A Laboratory Investigation
Katok, E., & Wu, D. Y. (2009) · Management Science · doi:10.1287/mnsc.1090.1089 · Cited by 267 · see citing works
Inequality Constrained Least-Squares Estimation
Liew, C. K. (1976) · Journal of the American Statistical Association · doi:10.1080/01621459.1976.10481560 · Cited by 203 · see citing works
“= [1], b* and an estimate of V(b*)”
BackgroundInequality Restrictions in Regression Analysis
Judge, G. G., & Takayama, T. (1966) · Journal of the American Statistical Association · doi:10.1080/01621459.1966.10502016 · Cited by 195 · see citing works
Syndemics and Health Disparities: A Methodological Note
Tsai, A. C., & Venkataramani, A. S. (2016) · AIDS and Behavior · doi:10.1007/s10461-015-1260-2 · Cited by 176 · see citing works
“(2) can be fitted subject to linear equality restrictions, which would reduce the number of parameters to be estimated [56, 57].”
BackgroundOn Least Squares with Insufficient Observations
Chipman, J. S. (1964) · Journal of the American Statistical Association · doi:10.1080/01621459.1964.10480751 · Cited by 183 · see citing works
“Rao [9, 10], out of which the present work has evolved.”
BackgroundSelf-citationWeaker Criteria and Tests for Linear Restrictions in Regression
Wallace, T. D. (1972) · Econometrica · doi:10.2307/1912962 · Cited by 158 · see citing works
The Statistical Consequences of Preliminary Test Estimators in Regression
Bock, M. E., Yancey, T. A., & Judge, G. G. (1973) · Journal of the American Statistical Association · doi:10.1080/01621459.1973.10481349 · Cited by 92 · see citing works
Minimax linear regression estimation with symmetric parameter restrictions
Pilz, J. (1986) · Journal of Statistical Planning and Inference · doi:10.1016/0378-3758(86)90141-2 · Cited by 82 · see citing works
Miller, K. S. (1973) · SIAM Review · doi:10.1137/1015094 · Cited by 73 · see citing works
Estimation and Aggregation in Econometrics: An Application of the Theory of Generalized Inverses
Chipman, J. S. (1976) · Elsevier eBooks · doi:10.1016/b978-0-12-514250-2.50017-2 · Cited by 52 · see citing works
Self-citationAnnotated Bibliography on Generalized Inverses and Applications
Nashed, M. Z., & Rall, L. B. (1976) · Elsevier eBooks · doi:10.1016/b978-0-12-514250-2.50018-4 · Cited by 50 · see citing works
Davis‐Stober, C. P., & Dana, J. D. (2013) · Behavior Research Methods · doi:10.3758/s13428-013-0342-1 · Cited by 42 · see citing works
Projections, generalized inverses, and quadratic forms
Chipman, J. S., & Rao, M. M. (1964) · Journal of Mathematical Analysis and Applications · doi:10.1016/0022-247x(64)90002-2 · Cited by 38 · see citing works
Self-citationA mean square error test when stochastic restrictions are used in regression
Yancey, T. A., Judge, G. G., & Bock, M. E. (1974) · Communications in Statistics · doi:10.1080/03610927408827175 · Cited by 38 · see citing works
Piotrowski, T., & Yamada, I. (2008) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2008.921716 · Cited by 33 · see citing works
“3In general, in this paper, the singular values are always denoted by the same letter (in lowercase) as the matrix containing them.”
BackgroundTian, Y. (2009) · Metrika · doi:10.1007/s00184-009-0255-2 · Cited by 25 · see citing works
On comparing restricted least squares estimators
Guilkey, D. K., & Price, J. M. (1981) · Journal of Econometrics · doi:10.1016/0304-4076(81)90102-0 · Cited by 23 · see citing works
Some norm inequalities concerning generalized inverses, 2
Maher, P. J. (2006) · Linear Algebra and its Applications · doi:10.1016/j.laa.2006.08.024 · Cited by 21 · see citing works
On estimation of an econometric model of short-run bank behaviour
Aigner, D. J. (1973) · Journal of Econometrics · doi:10.1016/0304-4076(73)90008-0 · Cited by 19 · see citing works
Influence diagnostic analysis in the possibly heteroskedastic linear model with exact restrictions
Liu, S., Leiva, V., Ma, T., et al. (2015) · Statistical Methods & Applications · doi:10.1007/s10260-015-0329-4 · Cited by 18 · see citing works
“Basic distributional properties of the RLS estimator, efficiency comparisons, hypothesis tests and realworld applications can be found in Chipman and Rao (1964), Trenkler (1987), Ramanathan (1993), Greene (2007) and Wooldridge (2013).”
BackgroundSome norm inequalities concerning generalized inverses
Maher, P. J. (1992) · Linear Algebra and its Applications · doi:10.1016/0024-3795(92)90045-c · Cited by 17 · see citing works
Equalities for estimators of partial parameters under linear model with restrictions
Tian, Y., & Jiang, B. (2015) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2015.09.007 · Cited by 15 · see citing works
Tian, Y., & Wang, J. (2018) · Communication in Statistics- Theory and Methods · doi:10.1080/03610926.2018.1554138 · Cited by 14 · see citing works
“…estimators, best linear unbiased estimators, etc., can widely be found in the statistical literature; see e.g., Baksalary and Kala (1979); Chipman and Rao (1964); Cysneiros and Paula (2005); Greene and Seaks (1991); Kloberdanz and Schmidt (2008); Liu et al. (2016); Mantell (1973); Paula…”
BackgroundRao, M. M. (1963) · Annals of the Institute of Statistical Mathematics · doi:10.1007/bf02865898 · Cited by 14 · see citing works
Self-citationKrafft, O. (1983) · Linear Algebra and its Applications · doi:10.1016/0024-3795(83)90154-4 · Cited by 13 · see citing works
Linear restrictions, rank reduction, and biased estimation in linear regression
Chipman, J. S. (1999) · Linear Algebra and its Applications · doi:10.1016/s0024-3795(98)10138-6 · Cited by 12 · see citing works
Self-citationBayesian inference for generalized linear model with linear inequality constraints
Ghosal, R., & Ghosh, S. K. (2021) · Computational Statistics & Data Analysis · doi:10.1016/j.csda.2021.107335 · Cited by 11 · see citing works
Replicated measurement error model under exact linear restrictions
Singh, S., Jain, K., & Sharma, S. (2012) · Statistical Papers · doi:10.1007/s00362-012-0469-7 · Cited by 11 · see citing works
“In the error free case, use of such prior information leads to more efficient estimators (Chipman and Rao 1964; Rao et al. 2008).”
MethodLinear Programming Techniques in Regression Analysis: The Hidden Danger
Crocker, D. C. (1969) · A I I E Transactions · doi:10.1080/05695556908974422 · Cited by 10 · see citing works
Attenuated blood corticosterone rhythm in rats with jejunal resection
Kato, H., Saito, M., & Shimazu, T. (1984) · Life Sciences · doi:10.1016/0024-3205(84)90620-9 · Cited by 9 · see citing works
Further remarks on constrained over-parameterized linear models
Güler, N., & Büyükkaya, M. E. (2023) · Statistical Papers · doi:10.1007/s00362-023-01426-z · Cited by 8 · see citing works
Tian, Y. (2012) · Communication in Statistics- Theory and Methods · doi:10.1080/03610926.2011.594537 · Cited by 8 · see citing works
“…work can be found, e.g., in Baksalary and Kala (1979), Baksalary and Mathew (1986), Baksalary and Mathew (1986), Baksalary and Pordzik (1990), Chipman and Rao (1964), Dent (1980), Haupt and Oberhofer (2006), Kreijger and Neudecker (1977), Mathew (1983), Rao (1971, 1973), and Tian and Wiens…”
BackgroundSimplification of Economic Models
Fisher, W. D. (1966) · Econometrica · doi:10.2307/1909770 · Cited by 8 · see citing works
Statistical inference of a stochastically restricted linear mixed model
Güler, N., & Büyükkaya, M. E. (2023) · AIMS Mathematics · doi:10.3934/math.20231244 · Cited by 7 · see citing works
Yamada, I., & Yamagishi, M. (2019) · Springer eBooks · doi:10.1007/978-3-030-25939-6_16 · Cited by 7 · see citing works
On a Restricted Least Squares Estimator
Atiqullah, M. (1969) · Journal of the American Statistical Association · doi:10.1080/01621459.1969.10501028 · Cited by 6 · see citing works
The General Linear Test in the Ridge Regression
Bae, W., Kim, M., & Kim, C. (2014) · Communications for Statistical Applications and Methods · doi:10.5351/csam.2014.21.4.297 · Cited by 5 · see citing works
“One example is the CobbDouglas production function (Chipman and Rao, 1964) in the field of econometrics; however, the general linear test problem in the ridge regression has not been studied. Obenchain (1977) studied interval estimation of the general linear combination of regression coefficients in the ridge regressi…”
BackgroundMethodEstimation of regression models under linear restrictions
Srivastava, V. K., & Agnihotri, B. S. (1980) · Biometrical Journal · doi:10.1002/bimj.4710220311 · Cited by 5 · see citing works
THE USE OF PRIOR INFORMATION IN ESTIMATING THE PARAMETERS OF ECONOMIC RELATIONSHIPS
Judge, G. G., & Yancey, T. A. (1969) · Metroeconomica · doi:10.1111/j.1467-999x.1969.tb00776.x · Cited by 5 · see citing works
Influence diagnostics in constrained general linear models
Emami, H., & Emami, M. (2015) · Communication in Statistics- Theory and Methods · doi:10.1080/03610926.2014.942430 · Cited by 2 · see citing works
“Constrained general linear models (CGLMs) are used widely in the field of econometrics, for example, in the estimation of Cobb–Douglas production functions Chipman and Rao (1964).”
MethodRen, X. (2014) · Statistics & Probability Letters · doi:10.1016/j.spl.2014.03.006 · Cited by 2 · see citing works
Robust estimation in restricted linear regression
Toka, O., ÇETİN, M., & Arslan, O. (2019) · Communications in Statistics - Simulation and Computation · doi:10.1080/03610918.2019.1661475 · Cited by 2 · see citing works
“The simplest type of linear restriction is the specification that one or more regression coefficients are equal to zero or some other constant (Chipman and Rao 1964). To give an example, in econometrics, there are lots of known equality constraints. In restricted linear regression, other known equations might be inclu…”
BackgroundInfluentialQUADRATIC EQUATIONS IN HILBERTIAN OPERATORS AND APPLICATIONS
Mizel, V. J., & Rao, M. M. (2009) · International Journal of Mathematics · doi:10.1142/s0129167x0900587x · Cited by 2 · see citing works
Self-citationOn the estimation of generalized linear probability model involving discrete random variables
Roy, A. R., & Srivastava, V. K. (1968) · Annals of the Institute of Statistical Mathematics · doi:10.1007/bf02911657 · Cited by 2 · see citing works
On Two Theorems In International Trade
Kuhn, H. W. (2011) · Springer eBooks · doi:10.1007/978-3-642-11045-0_4 · Cited by 1 · see citing works
Local Influence in Constrained General Linear Models
Emami, H., & Emami, M. (2021) · Journal of Data Science · doi:10.6339/jds.201410_12(4).0008 · Cited by 1 · see citing works
“718 Local Influence in Constrained General Linear Model Constrained general linear models (CGLMs) are used widely in the field of econometrics, for example in the estimation of Cobb-Douglas production functions Chipman and Rao (1964).”
MethodInfluence Analysis of Constrained Regression Models
Kim, M.-G. (2007) · Communications for Statistical Applications and Methods · doi:10.5351/ckss.2007.14.2.281 · Cited by 1 · see citing works
Papakyriazis, P. A., & Papakyriazis, A. (1986) · International Journal of Systems Science · doi:10.1080/00207728608926830 · Cited by 1 · see citing works
On maximum information strategies for estimation of stochastic multiple response systems
Papakyriazis, P. A. (1986) · International Journal of Systems Science · doi:10.1080/00207728608926825 · Cited by 1 · see citing works
Estimation and testing in the restricted general linear model
Hallum, C. R. (1972) · ThinkTech (Texas Tech University) · Cited by 1 · see citing works
Zoia, M. G. (2006) · RePEc: Research Papers in Economics
A METHOD OF STRUCTURAL ESTIMATION UNDER LINEAR RESTRICTIONS
Kariya, T. (1973) · The economic studies quarterly/Economic studies quarterly/Kikan riron keizaigaku · doi:10.11398/economics1950.23.3_72
A Search Algorithm for the Estimation of Interregional Migration in Local Areas
Meier, P. (1973) · Environment and Planning A Economy and Space · doi:10.1068/a050045
The quasi-stochastically constrained least squares method for ill linear regression
Li, S., Sheng, Y., & Li, Y. (2015) · Communication in Statistics- Theory and Methods · doi:10.1080/03610926.2013.828076
“Among these, the most effective ones are the constrained methods (Chipman and Rao, 1964; Srivastava et al., 1984).”
BackgroundThe Life and Work of M. M. Rao
Green, M. L., Krinik, A. C., & Swift, R. J. (2011) · Journal of Statistical Theory and Practice · doi:10.1080/15598608.2011.10412045
Two Diagnostic Plots in Constrained Regression
Kim, M.-G. (2009) · Communications for Statistical Applications and Methods · doi:10.5351/ckss.2009.16.3.495
LINEAR POLICY MODELS AND QUADRATIC OPTIMIZATION*
Condos, A. (1970) · Metroeconomica · doi:10.1111/j.1467-999x.1970.tb00142.x
Chipman, J. S. (1983) · Physica-Verlag eBooks · doi:10.1007/978-3-662-41526-9_15
Self-citationTesting equality of regression coefficients: Some enlargements
Katzenbeisser, W. (1981) · Statistische Hefte
Test auf Gleichheit von Regressionskoeffizienten: Einige Erweiterungen
Katzenbeisser, W. (1981) · Statistische Hefte · doi:10.1007/bf02932816
Some Problems of Real and Stochastic Analysis Arising from Applications
Rao, M. M. (2020) · CRC Press eBooks · doi:10.1201/9781003067597-1
Self-citationRestricted Stein-rule estimation in ultrastructural linear measurement error models
Khademnoe, O., & Emami, H. (2022) · Communication in Statistics- Theory and Methods · doi:10.1080/03610926.2022.2135380
“In the regression analysis one of the best attempts to provide better estimators is the incorporation of prior information available in the form of exact restrictions Chipman and Rao (1964).”
MethodA matrix approach to determine optimal predictors in a constrained linear mixed model
Güler, N., & Büyükkaya, M. E. (2024) · Open Mathematics · doi:10.1515/math-2024-0114
Comparison of Optimal Predictors Under Two Competing Constrained Linear Models
Jiang, B. (2026) · Iranian Journal of Science · doi:10.1007/s40995-026-01993-4
“…literature that focus on an exact restriction on unknown parameters under GLMs from different perspectives; see, e.g., Baksalary and Kala (1979); Chipman and Rao (1964); Dent (1980); Güler and Büyükkaya (2023b, 2024); Jiang and Tian (2023); Jiang et al. (2020); Li et al. (2023); Lu et al.…”
- Authors
- Chipman, J. S., & Rao, M. M.
- Venue
- Econometrica
- Publisher
- JSTOR
- Published
- 1964-01
- Volume
- 32
- Issue
- 1/2
- Pages
- 198
- ISSN
- 0012-9682 (ISSN-L)
- OpenAlex
- W2327560351
- Cited by
- 64
- References
- 0
- Citations by year
- 2026: 1 · 2024: 1 · 2023: 2 · 2022: 1 · 2021: 2 · 2020: 1 · 2019: 2 · 2018: 1 · 2015: 5 · 2014: 2 · 2013: 1 · 2012: 2
- FWCI
- 3.29
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Econometrica (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Applied Mathematics › Statistical and numerical algorithms
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Keywords
- linear restrictions, regression analysis, hypothesis testing, constrained least squares
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