Finite Sample Econometrics: A Unified Approach

  • Aman Ullah


In the last six decades, a significant literature has developed on the finite sample econometrics. This includes approximate, analytical and simulation based research analysing the moments and distributions of various econometric estimators and test statistics. Many analytical results, however, are complicated and require the knowledge of various statistical methods.

In this paper we systematically develop unified methods for obtaining both the exact moments and the distributions of econometric estimators and test statistics. The results are derived for the normal case and extended for non-normal cases as well. A general result on the approximations is also given.


Quadratic Form Unit Root Unit Root Test American Statistical Association Exact Distribution 
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  1. Anderson, T. W. (1982), “Some Recent Developments on the Distributions of Single—Equation Estimators,” in Advances in Econometrics, Cambridge University Press.Google Scholar
  2. Anderson, T. W., and Rubin, H. (1949), “Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equation,” Annals of Mathematical Statistics 20, 46–63.CrossRefGoogle Scholar
  3. Anderson, T. W., and Sawa, T. (1973), “Distributions of Estimator of Coefficients of a Single Equation in a Simultaneous System and Their Asymptotic Expansions,” Econometrica 41, 683–714.CrossRefGoogle Scholar
  4. Anderson, T. W., and Sawa, T. (1979), “Evaluation of the Distribution Function of the Two—Stage Least Squares Estimate,” Econometrica 47, 163–182.CrossRefGoogle Scholar
  5. Banerjee, A., J. J. Dolado, D. F. Hendry and G. W. Smith (1986), “Exploring Equilibrium Relationships in Economics Through Static Models: Some Monte Carlo Evidence,” Oxford Bulletin of Economics and Statistics 48, 253–277.CrossRefGoogle Scholar
  6. Basmann, R. L. (1961), “Note on the Exact Finite Sample Frequence Functions of Generalized Classical Linear Estimators in Two Leading Overidentified Cases,” Journal of the American Statistical Association 56, 619–636.CrossRefGoogle Scholar
  7. Basmann, R. L. (1974), Exact Finite Sample Distribution for Some Econometric Estimators and Test Statistics: A Survey and Appraisal, in Frontiers of Quantitative Economics, Vol. 2, North Holland.Google Scholar
  8. Bock, M., G. G. Judge, and T. A. Yancey (1984), “A simple form for the inverse moments of non—central x2 and F random variables and for certain confluent hypergeometric functions,” Journal of Econometrics 25, 217–234.CrossRefGoogle Scholar
  9. Brown, G. F., Ramage, J. G. and V. K. Srivastava (1985), “The R-class Estimators of Disturbance Variance in Simultaneous Equation Models,” Journal of Quantitative Economics 2, 231–251.Google Scholar
  10. Carter, R. A. L. (1988), “Nonparametric Monte Carlo Estimates of t—Ratio Distributions in Dynamic Simultaneous Linear Equations Models,” manuscript, University of Western Ontario.Google Scholar
  11. Carter, R. A. L., and A. Ullah (1979), “The Finite Sample Properties of OLS and IV Estimators in Special Rational Distributed Lag Models,” Sankhya, Ser. D., 41, 1–18.Google Scholar
  12. Constantine, A. G. (1963), “Some Noncentral Distribution Problems in Multivariate Analysis,” Annals of Mathematical Statistics 34, 1270–1285.CrossRefGoogle Scholar
  13. Cramer, J. S. (1987), “Mean and Variance of R2 in Small and Moderate Samples,” Journal of Econometrics, 35, 253 266.Google Scholar
  14. Cribbett, P., J. N. Lye and A. Ullah (1989), “Evaluation of the 2SLS Distribution Function by Imhofs Procedure,” Journal of Quantitative Economics (forthcoming).Google Scholar
  15. Davidson, R. and J. G. MacKinnon (1988), “Monte Carlo Experiments,” manuscript, Queen’s University.Google Scholar
  16. Davies, R. B. (1973), “Numerical Inversion of a Characteristic Function,” Biometrika 60, 415–417.CrossRefGoogle Scholar
  17. Davies, R. B. (1980), Seg. AS 155, “The Distribution of a Linear Contribution of x2 Random Variables,” Applied Statistics 29, 323–333.CrossRefGoogle Scholar
  18. Davis, A. N. (1976), “Statistical Distributions in Univariate and Multivariate Edgeworth Populations,” Biometrika, 661–670.Google Scholar
  19. DeGooijer, J. G. (1980), “Exact Moments of the Sample Autocorrelations from Series Generated by General ARIMA Processes of Order (p,d,q), d=0 or 1,” Journal of Econometrics 14, 365–379.CrossRefGoogle Scholar
  20. Efron, B. (1979), “Bootstrap Methods: Another Look at the Jackknife,” Annals of Statistics 7, 1–26.CrossRefGoogle Scholar
  21. Ellison, G. (1988), “Finite Sample Properties of a Co—integration Model,” manuscript, Cambridge University.Google Scholar
  22. Engle, R. F. and C. W. J. Granger (1987), “Co—integration and Error Correction: Representation, Estimation and Testing,” Econometrica 55, 251–276.CrossRefGoogle Scholar
  23. Evans, G. B. A. and N. E. Savin (1981), “Testing Unit Roots: 1,” Econometrica 49, 753–779.CrossRefGoogle Scholar
  24. Evans, G. B. A. and N. E. Savin (1984), “Testing for Unit Roots: 2,” Econometrica 50, 1241–1289.CrossRefGoogle Scholar
  25. Farebrother, R. W. (1984), “A Remark of Algorithms AS 106, AS 153 and AS 155: The Distribution of a Linear Combination of x2 Random Variables,” Applied Statistics 33, 366–369.CrossRefGoogle Scholar
  26. Fan, Y. (1988), “The Finite Sample Properties of a Generalized Co—integration Model,” manuscript, University of Western Ontario.Google Scholar
  27. Fisher, R. A. (1921), “On the Probable Error of a Coefficient of Correlation Deduced From a Small Sample,” Metron 1, 1–32.Google Scholar
  28. Fisher, R. A. (1922), “The Goodness of Fit of Regression Formulae and the Distribution of Regression Coefficients,” Journal of the Royal Statistical Society 85, 597–612.CrossRefGoogle Scholar
  29. Fisher, R. A. (1935), The Mathematical Distributions Used in the Common Tests of Significance, Econometrica, 3, 353–365.CrossRefGoogle Scholar
  30. Fuller,. (1976), Introduction to Statistical Time Series, New York: John Wiley.Google Scholar
  31. Gil Pelaez, J. (1951), “Note on the Inversion Theorem,” Biometrika 38, 481–482.Google Scholar
  32. Giles, D. E. A. (1988), “The Exact Distribution of a Simple Pre—Test Estimator,” manuscript, University of Canterbury.Google Scholar
  33. Godfrey, L. G. and A. R. Tremayne (1988), “On the Finite Sample Performance of Tests for Unit Roots,” DP 88–03, University of Sydney.Google Scholar
  34. Gurland, J. (1948), “Inversion Formulae for the Distribution of Ratios,” Annals of Mathematical Statistics, 19, 228–237.CrossRefGoogle Scholar
  35. Gurland, J. (1955), “Distribution of Definite and Indefinite Quadratic Forms,” Annals of Mathematical Statistics 26, 112–127.CrossRefGoogle Scholar
  36. Haavelmo, T. (1947), “Methods of Measuring the Marginal Propensity to Consume,” Journal of the American Statistical Association 42, 105–122.CrossRefGoogle Scholar
  37. Hansen, B. E. and P. C. B. Phillips (1988), “Estimation and Inference in Models of Cointegration: A Simulation Study,” manuscript, Yale University.Google Scholar
  38. Hendry, D. F. (1984), “Monte Carlo Experimentation in Econometrics,” in Handbook of Econometrics, Vol. II, North Holland.Google Scholar
  39. Hillier, G., T. Kinal, and V. K. Srivastava (1984), “On the Moments of Ordinary Least Squares and Instrumental Variable Estimators in a General Structural Equation,” Econometrica 52, 185–202.CrossRefGoogle Scholar
  40. Hillier, G., T. Kinal, and V. K. Srivastava (1985), “On the Joint and Marginal Densities of Instrumental Variable Estimators in a General Structural Equation,” Econometric Theory 1, 53–72.CrossRefGoogle Scholar
  41. Hurwicz, L. (1950), “Least Squares Bias in Time Series,” in T. C. Koopmans (ed.), Statistical Inference in Dynamic Economic Models, New York: Wiley.Google Scholar
  42. Imhof, P. J. (1961), “Computing the Distribution of Quadratic Form in Normal Variables,” Biometrika 48, 419–426.Google Scholar
  43. Hogue, A. and T. Peters (1986), “Finite Sample Analysis of the ARMAX Models,” Sankhya B, 48, 266–283.Google Scholar
  44. Judge, G. and M. E. Bock (1978), The Statistical Implications of Pre Test and Stein—Rule Estimators in Econometrics,North Holland.Google Scholar
  45. Kabe, D. G. (1963), “A Note on the Exact Distributions of the GCL Estimators in Two—Leading Overidentified Cases,” Journal of the American Statistical Association 58, 535–537.CrossRefGoogle Scholar
  46. Kabe, D. G. (1964), “On the Exact Distributions of the GCL Estimators in a Leading Three—Equation,” Journal of the American Statistical Association 58, 881–894.CrossRefGoogle Scholar
  47. Kadane, J. (1971), “Comparison of k—Class Estimators When the Disturbances are Small,” Econometrica 39, 723–737.CrossRefGoogle Scholar
  48. Kakwani, N. C. (1967), “The Unbiasedness of Zellner’s Seemingly Unrelated Regression Equations Estimators,” Journal of the American Statistical Association.Google Scholar
  49. King, M. L. (1987), “Testing for Autocorrelation in Linear Regression Models: A Survey,” in Specification Analysis in the Linear Model, Routledge and Kegan Paul.Google Scholar
  50. Kmenta, J. (1986), Elements of Econometrics, New York: Wiley.Google Scholar
  51. Knight, J. L. (1985), “The Moments of OLS and 2SLS When the Disturbances are Non—Normal,” Journal of Econometrics 27, 39–60.CrossRefGoogle Scholar
  52. Knight, J. L. (1986), “Non—Normal Errors and the Distribution of OLS and 2SLS Structural Estimators,” Econometric Theory 2, 75–102.CrossRefGoogle Scholar
  53. Koerts, J. and A. P. J. Abrahamse (1969), On the Theory and Application of the General Linear Model, Rotterdam University Press.Google Scholar
  54. Lugannini, R. and S. O. Rice (1984), “Distribution of the Ratio of Quadratic Forms in Normal Variables —Numerical Methids, SIAM Journal on Scientific and Statistical Computing 5, 476–488.Google Scholar
  55. Lye, J. N. (1987), “Some Further Results on the Distribution of Double k—Class Estimators,” manuscript, University of Canterbury.Google Scholar
  56. Lye, J. N. (1988), “On the Exact Distribution of a Ratio of Bilinear to Quadratic Form in Normal Variables with Econometric Applications,” manuscript, University of Canterbury.Google Scholar
  57. Maasoumi, E. (1989), Contributions to Econometrics, Vol. II, Cambridge University Press.Google Scholar
  58. Maasoumi, E. and A. Ullah (1987), “Specification Analysis in Special Rational Distributed Lag and Other Dynamic Models,” Journal of Quantitative Economics.Google Scholar
  59. Maekawa, K. (1987), “Finite Sample Properties of Several Predictors from an Autoregressive Model,” Econometric Theory 3, 359–370.CrossRefGoogle Scholar
  60. Magnus, J. R. (1986), “The Exact Moments of a Ratio of Quadratic Forms in Normal Variable,” Annals d’Economie et de Statistique, 4, 95–109.Google Scholar
  61. Magee, L., A. Ullah, and V. K. Srivastava (1987), “Efficiency of Estimators in the Regression Model with First—Order Autoregressive Errors,” in Specification Analysis in the Linear Models, Routledge and Kegan Paul.Google Scholar
  62. Mariano, R. S. (1982), “Analytical Small—Sample Distribution Theory in Econometrics: The Simultaneous—Equations Case,”International Economic Review, 503–533. Google Scholar
  63. Mehta, J. S. and P. A. V. B. Swamy (1978), “The Existence of Moments of Some Simple Bayes Estimators of Coefficients in a Simultaneous Equations Model,” Journal of Econometrics 7, 1–13.CrossRefGoogle Scholar
  64. Nagar, A. L. (1959), “The Bias and Moments Matrix of the General k—Class Estimators of the Parameters in Structural Equations,” Econometrica 27, 575–595.CrossRefGoogle Scholar
  65. Nagar, A. L. (1962), “Double k—Class Estimators of Parameters in Simultaneous Equations Models and Their small Sample Properties,” International Economic Review 3, 168–188.CrossRefGoogle Scholar
  66. Nagar, A. L., and S. N. Sahay (1978), “The Bias and Mean Squared Error of Forecasts From Partially Restricted Reduced Form,” Journal of Econometrics 7, 227–243.CrossRefGoogle Scholar
  67. Nagar, A. L., and A. Ullah (1973), “Note on Approximate SkewnessGoogle Scholar
  68. Nagar, A. L., and Kurtosis of the Two Stage Least Squares Estimator,“ Indian Economic Review,70–80.Google Scholar
  69. Nankervis, J. S. and N. E. Savin (1985), “Testing the Autoregressive Parameter with the t—Statistics,” Journal of Econometrics 27, 143–162.CrossRefGoogle Scholar
  70. Nankervis, J. S. and N. E. Savin (1987), “Finite Sample Distributions of t and F Statistics in an AR(1) Model with an Exogenous Variable,” Econometric Theory 3, 387–408.CrossRefGoogle Scholar
  71. Pagan, A. (1984), “Econometric Issues in the Analysis of Regressions with Generated Regressors,” International Economic Review 25, 221–247.CrossRefGoogle Scholar
  72. Palm, F. C. and J. M. Sneek (1984), “Significance Tests and Spurious Correlation in Regression Models with Autocorrelated Errors,” Statistische Hefte L5, 87–105.Google Scholar
  73. Peters, T. (1986), Least Squares in Dynamic Econometric Models, Ph.D. Thesis, University of Western Ontario.Google Scholar
  74. Peters, T., and W. Veloce (1988), “Robustness of Unit Root Tests in ARMA Models With Generalized ARCH Errors,” manuscript, Brock University.Google Scholar
  75. Phillips, P. C. B. (1977), “A General Theorem in the Theory of Asymptotic Expansions as Approximations to the Finite Sample Distributions of Econometric Estimators,” Econometrica 65, 1517–1534.CrossRefGoogle Scholar
  76. Phillips, P. C. B. (1980), “Finite Sample Theory and the Distributions of Alternative Estimators of the Marginal Propensity to Consume,” Review of Economic Studies, Vol. 47, 183–224.CrossRefGoogle Scholar
  77. Phillips, P. C. B. (1983a), “Exact Small Sample Theory in Simultaneous Equations Models,” in Handbook of Econometrics, Vol. 1, Amsterdam: North Holland.Google Scholar
  78. Phillips, P. C. B. (1983b), “ERA’s: A New Approach to Small Sample Theory,” Econometrica 51, 1505–1526.CrossRefGoogle Scholar
  79. Phillips, P. C. B. (1985), “The Distribution of Matrix Quotients,” Journal of Multivariate Analysis 16, 157–161.CrossRefGoogle Scholar
  80. Phillips, P. C. B. (1986), The Exact Distribution of the Wald Statistic, Econometrica, 54, 881–895.CrossRefGoogle Scholar
  81. Phillips, P. C. B. (1987), “Fractional Matrix Calculus and the Distribution of Multivariate Tests,” in Time Series and Econometric Modelling, Boston: D. Reidell Press.Google Scholar
  82. Power, S. and A. Ullah (1987), “Nonparametric Monte Carlo Density Estimation of Rational Expectations Estimators and Their t—Ratios,” in Advances in Econometrics, JAI Press.Google Scholar
  83. Racine, J. (1988), “Semi Parametric Estimation in the Presence of Heteroskedasticity of Unknown Form,” manuscript, University of Western Ontario.Google Scholar
  84. Raj, B. and A. Ullah (1981), Econometrics: A Random Coefficient Approach,Croom Helm.Google Scholar
  85. Richardson, D. H. (1968), “The Exact Distributions of a Structural Coefficient Estimator,” Journal of the American Statistical Association 63, 1214–1226.CrossRefGoogle Scholar
  86. Rosenblatt, M. (1956), “Remarks on Some Nonparametric Estimates of a Density Function,” Annals of Mathematical Statistics 27, 832–837.CrossRefGoogle Scholar
  87. Rothenberg, T. J. (1984), “Approximating the Distribution of Econometric Estimators and Test Statistics,” in M. D. Intriligator and Z. Griliches (eds.), Handbook of Econometrics, Vol. 2, Amsterdam: North—Holland.Google Scholar
  88. Rothenberg, T. J.(1988), “Approximate Power Functions for Some Robust Tests of Regression Coefficients,”Econometrica, 997–1019.Google Scholar
  89. Sawa, T. (1972), “Finite Sample Properties of k—Class Estimators,” Econometrica 40, 653–680.CrossRefGoogle Scholar
  90. Sawa, T. (1978), “The Exact Moments of the Least Squares Estimator for the Autoregressive Model,” Journal of Econometrics 8, 159–172.CrossRefGoogle Scholar
  91. Silverman, B. W. (1986), Density Estimation for Statistics and Data Analysis, New York: Chapman and Hall.Google Scholar
  92. Smith, M. D. (1988) Convergent Series Expressions for Inverse Moments of Quadratic forms in Normal Variables, The Australian Journal of Statistics 30, 245–246.Google Scholar
  93. Smith, M. D. (1988a), “On the Expectations of a Ratio of Forms in Normal Variables, DP #88–02, University of Sydney.Google Scholar
  94. Srivastava, M. S. and C. G. Khatri (1979), An Introduction to Multivariate Statistics, New York: North—Holland.Google Scholar
  95. Srivastava, V.K. and R. Tiwari (1976), “Evaluation of Expectation of Products of Stochastic Matrices,” Scandinavian Journal of Statistics 3, 135–138.Google Scholar
  96. Srivastava, V.K., J. L. Knight, and A. Agnihotri (1986), “Properties of k—Class Estimators in Simultaneous Equations Models with Not Necessarily Normal Disburbances,” manuscript, University of Western Ontario.Google Scholar
  97. Takeuchi, K. (1970), “Exact Sampling Moments of Ordinary Least Squares, Instrumental Variable and Two—Stage Least Squares Estimators, International Economic Review 11, 1–12.Google Scholar
  98. Taneja, V. S. (1976), “Approximation to the Distribution of Indefinite Quadratic Forms in Noncentral Normal Variables,” Metron 34, 255–268.Google Scholar
  99. Taylor, W. E. (1983), “On the Relevance of Finite Sample Distribution Theory,” Econometric Reviews 2, 1–39.CrossRefGoogle Scholar
  100. Ullah, A. (1971), Statistical Estimation of Economic Relations in the Presence of Errors in Variables and In Equations,Ph.D. Thesis, Delhi School of Economics.Google Scholar
  101. Ullah, A. (1974), “On the Sampling Distribution of Improved Estimators for Coefficients in Linear Regression,” Journal of Econometrics 2, 143–150.CrossRefGoogle Scholar
  102. Ullah, A., and A. Nagar (1974), The Exact Mean of the Two State Least Squares Estimator of the Strutural Parameters in an Equation Having Three Endogenous Variables,“ Econometrica 42, 749–758.CrossRefGoogle Scholar
  103. Ullah, A., and A. Nagar (1987), “Unanticipated Macro Model Estimator,” Econometric Theory 3, 163–167.CrossRefGoogle Scholar
  104. Ullah, A., and A. Nagar (1988), “On the Inverse Moments of Noncentral Wishart Matrix,” manuscript, University of Western Ontario.Google Scholar
  105. Ullah, A., and R.S. Singh (1985), “The Estimation of Probability Density Functions and Its Applications in Econometrics,” Technical Report 6, University of Western Ontario.Google Scholar
  106. Ullah, A., V. K. Srivastava and R. Chandra (1983), “Properties of Shrinkage Estimators in Linear Regression When Disturbances are Not Normal,” Journal of Econometrics 21, 389–402.CrossRefGoogle Scholar
  107. Vinod, H. D. and A. Ullah (1981), Recent Advances in Regression Methods,Marcel Dekkar.Google Scholar
  108. Wegge, L. (1971), “The Finite Sample Distribution of Least Squares Estimators with Stochastic Regressors,” Econometrica 38, 241–251.CrossRefGoogle Scholar
  109. Zellner, A. (1962), “An Efficient Method of Estimating Seemingly Unrelated Regression and Test of Aggregation Bias,” Journal of the American Statistical Association.Google Scholar

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© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Aman Ullah
    • 1
  1. 1.Department of EconomicsUniversity of Western OntarioCanada

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