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Point Optimal Testing: A Survey of the Post 1987 Literature

Listed author(s):
  • Maxwell L. King

    ()

  • Sivagowry Sriananthakumar

    ()

In the absence of uniformly most powerful (UMP) tests or uniformly most powerful invariant (UMPI) TESTS, King (1987c) suggested the use of Point Optimal (PO) tests, which are most powerful at a chosen point under the alternative hypothesis. This paper surveys the literature and major developments on point optimal testing since 1987 and suggests some areas for future research. Topics include tests for which all nuisance parameters have been eliminated and dealing with nuisance parameters via (i) a weighted average of p values, (ii) approximate point optimal tests, (iii) plugging in estimated parameter values, (iv) using asymptotics and (v) integration. Progress on using point-optimal testing principles for two-sided testing and multi-dimensional alternatives is also reviewed. The paper concludes with thoughts on how best to deal with nuisance parameters under both the null and alternative hypotheses, as well as the development of a new class of point optimal test for multi-dimensional testing.

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Paper provided by Monash University, Department of Econometrics and Business Statistics in its series Monash Econometrics and Business Statistics Working Papers with number 5/15.

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Date of creation: 2015
Handle: RePEc:msh:ebswps:2015-5
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  1. Maxwell King & Ping Wu, 1997. "Locally optimal one-sided tests for multiparameter hypotheses," Econometric Reviews, Taylor & Francis Journals, vol. 16(2), pages 131-156.
  2. Evans, Merran A. & King, Maxwell L., 1988. "A further class of tests for heteroscedasticity," Journal of Econometrics, Elsevier, vol. 37(2), pages 265-276, February.
  3. Martellosio, Federico, 2010. "Power Properties Of Invariant Tests For Spatial Autocorrelation In Linear Regression," Econometric Theory, Cambridge University Press, vol. 26(01), pages 152-186, February.
  4. Andrews, Donald W.K. & Moreira, Marcelo J. & Stock, James H., 2008. "Efficient two-sided nonsimilar invariant tests in IV regression with weak instruments," Journal of Econometrics, Elsevier, vol. 146(2), pages 241-254, October.
  5. Pierre Perron & Serena Ng, 1996. "Useful Modifications to some Unit Root Tests with Dependent Errors and their Local Asymptotic Properties," Review of Economic Studies, Oxford University Press, vol. 63(3), pages 435-463.
  6. Gao, Jiti & Gijbels, Irène, 2008. "Bandwidth Selection in Nonparametric Kernel Testing," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1584-1594.
  7. King, Maxwell L., 1985. "A Point Optimal Test for Moving Average Regression Disturbances," Econometric Theory, Cambridge University Press, vol. 1(02), pages 211-222, August.
  8. Simon Broda & Kai Carstensen & Marc Paolella, 2009. "Assessing and Improving the Performance of Nearly Efficient Unit Root Tests in Small Samples," Econometric Reviews, Taylor & Francis Journals, vol. 28(5), pages 468-494.
  9. Yixiao Sun & Peter C. B. Phillips & Sainan Jin, 2008. "Optimal Bandwidth Selection in Heteroskedasticity-Autocorrelation Robust Testing," Econometrica, Econometric Society, vol. 76(1), pages 175-194, 01.
  10. Moon, Hyungsik Roger & Perron, Benoit & Phillips, Peter C.B., 2007. "Incidental trends and the power of panel unit root tests," Journal of Econometrics, Elsevier, vol. 141(2), pages 416-459, December.
  11. King, Maxwell L., 1983. "Testing for autoregressive against moving average errors in the linear regression model," Journal of Econometrics, Elsevier, vol. 21(1), pages 35-51, January.
  12. A. Bhargava & L. Franzini & W. Narendranathan, 2006. "Serial Correlation and the Fixed Effects Model," World Scientific Book Chapters,in: Econometrics, Statistics And Computational Approaches In Food And Health Sciences, chapter 4, pages 61-77 World Scientific Publishing Co. Pte. Ltd..
  13. Maxwell L. King & Michael McAleer, 1987. "Further Results on Testing AR (1) Against MA (1) Disturbances in the Linear Regression Model," Review of Economic Studies, Oxford University Press, vol. 54(4), pages 649-663.
  14. Hillier, Grant H., 1987. "Classes of Similar Regions and Their Power Properties for Some Econometric Testing Problems," Econometric Theory, Cambridge University Press, vol. 3(01), pages 1-44, February.
  15. Rahman, S. & King, M.L., 1994. "A Comparison of Marginal Likelihood Based and Approximate Point Optimal Tests for Random Regression Coefficient in the Presence of Autocorrelation," Monash Econometrics and Business Statistics Working Papers 4/94, Monash University, Department of Econometrics and Business Statistics.
  16. King, Maxwell L., 1985. "A point optimal test for autoregressive disturbances," Journal of Econometrics, Elsevier, vol. 27(1), pages 21-37, January.
  17. Rothenberg, Thomas J. & Stock, James H., 1997. "Inference in a nearly integrated autoregressive model with nonnormal innovations," Journal of Econometrics, Elsevier, vol. 80(2), pages 269-286, October.
  18. Jansson, Michael, 2005. "Point optimal tests of the null hypothesis of cointegration," Journal of Econometrics, Elsevier, vol. 124(1), pages 187-201, January.
  19. King, M.L. & Harris, D.C., 1995. "The Applications of the Durbin-Watson Test to the Dynamic Regression Model Under Normal and Non-Normal Errors," Monash Econometrics and Business Statistics Working Papers 6/95, Monash University, Department of Econometrics and Business Statistics.
  20. Hwang, Jaeyoun & Schmidt, Peter, 1993. "On the power of point optimal tests of the trend stationarity hypothesis," Economics Letters, Elsevier, vol. 43(2), pages 143-147.
  21. Dimitrios Vougas, 2009. "Modification of the point optimal unit root test," Applied Economics Letters, Taylor & Francis Journals, vol. 16(4), pages 349-352.
  22. King, M.L. & Giles, D.E.A., 1984. "Autocorrelation pre-testing in the linear model: Estimation, testing and prediction," Journal of Econometrics, Elsevier, vol. 25(1-2), pages 35-48.
  23. Saikkonen, Pentti & Luukkonen, Ritva, 1993. "Point Optimal Tests for Testing the Order of Differencing in ARIMA Models," Econometric Theory, Cambridge University Press, vol. 9(03), pages 343-362, June.
  24. Rodrigues, Paulo M.M. & Taylor, A.M. Robert, 2007. "Efficient tests of the seasonal unit root hypothesis," Journal of Econometrics, Elsevier, vol. 141(2), pages 548-573, December.
  25. Elliott, Graham & Jansson, Michael, 2003. "Testing for unit roots with stationary covariates," Journal of Econometrics, Elsevier, vol. 115(1), pages 75-89, July.
  26. Brooks, Robert D., 1993. "Alternative point-optimal tests for regression coefficient stability," Journal of Econometrics, Elsevier, vol. 57(1-3), pages 365-376.
  27. Perron, Pierre & Rodriguez, Gabriel, 2003. "GLS detrending, efficient unit root tests and structural change," Journal of Econometrics, Elsevier, vol. 115(1), pages 1-27, July.
  28. Graham Elliott & Michael Jansson & Elena Pesavento, 2005. "Optimal Power for Testing Potential Cointegrating Vectors With Known Parameters for Nonstationarity," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 34-48, January.
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  30. Goffe, William L. & Ferrier, Gary D. & Rogers, John, 1994. "Global optimization of statistical functions with simulated annealing," Journal of Econometrics, Elsevier, vol. 60(1-2), pages 65-99.
  31. Donald W. K. Andrews & Marcelo J. Moreira & James H. Stock, 2006. "Optimal Two-Sided Invariant Similar Tests for Instrumental Variables Regression," Econometrica, Econometric Society, vol. 74(3), pages 715-752, 05.
  32. Burridge, Peter & Taylor, A M Robert, 2000. " On the Power of GLS-Type Unit Root Tests," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 62(5), pages 633-645, December.
  33. Dufour, Jean-Marie & Taamouti, Abderrahim, 2010. "Exact optimal inference in regression models under heteroskedasticity and non-normality of unknown form," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2532-2553, November.
  34. Gregoir, Stephane, 2006. "Efficient tests for the presence of a pair of complex conjugate unit roots in real time series," Journal of Econometrics, Elsevier, vol. 130(1), pages 45-100, January.
  35. King, Maxwell L & Shively, Thomas S, 1993. "Locally Optimal Testing When a Nuisance Parameter Is Present Only under the Alternative," The Review of Economics and Statistics, MIT Press, vol. 75(1), pages 1-7, February.
  36. King, Maxwell L & Edwards, P M, 1989. "Transformations for an Exact Goodness-of-Fit Test of Structural Change in the Linear Regression Model," Empirical Economics, Springer, vol. 14(2), pages 113-121.
  37. Hansen, Bruce E., 1995. "Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase Power," Econometric Theory, Cambridge University Press, vol. 11(05), pages 1148-1171, October.
  38. Elliott, Graham & Müller, Ulrich K., 2014. "Pre and post break parameter inference," Journal of Econometrics, Elsevier, vol. 180(2), pages 141-157.
  39. Ploberger, Werner, 2004. "A complete class of tests when the likelihood is locally asymptotically quadratic," Journal of Econometrics, Elsevier, vol. 118(1-2), pages 67-94.
  40. Naorayex K. Dastoor & Gordon Fisher, 1987. "On Point-Optimal Cox Tests," Working Papers 678, Queen's University, Department of Economics.
  41. King, Maxwell L. & Smith, Murray D., 1986. "Joint one-sided tests of linear regression coefficients," Journal of Econometrics, Elsevier, vol. 32(3), pages 367-383, August.
  42. Martellosio, Federico, 2008. "Testing for spatial autocorrelation: the regressors that make the power disappear," MPRA Paper 10542, University Library of Munich, Germany.
  43. King, Maxwell L. & Evans, Merran A., 1988. "Locally Optimal Properties of the Durbin-Watson Test," Econometric Theory, Cambridge University Press, vol. 4(03), pages 509-516, December.
  44. Serena Ng & Pierre Perron, 2001. "LAG Length Selection and the Construction of Unit Root Tests with Good Size and Power," Econometrica, Econometric Society, vol. 69(6), pages 1519-1554, November.
  45. Hansen, Bruce E., 2000. "Testing for structural change in conditional models," Journal of Econometrics, Elsevier, vol. 97(1), pages 93-115, July.
  46. Wooldridge, Jeffrey M., 1989. "A computationally simple heteroskedasticity and serial correlation robust standard error for the linear regression model," Economics Letters, Elsevier, vol. 31(3), pages 239-243, December.
  47. Philip A. Shively, 2001. "Trend-stationary GNP: evidence from a new exact pointwise most powerful invariant unit root test," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(4), pages 537-551.
  48. Andrews, Donald W K & Ploberger, Werner, 1994. "Optimal Tests When a Nuisance Parameter Is Present Only under the Alternative," Econometrica, Econometric Society, vol. 62(6), pages 1383-1414, November.
  49. Hwang, Jaeyoun & Schmidt, Peter, 1996. "Alternative methods of detrending and the power of unit root tests," Journal of Econometrics, Elsevier, vol. 71(1-2), pages 227-248.
  50. Ping, Wu & King, Maxwell L., 1996. "Small-sample power of tests for inequality restrictions: The case of quarter-dependent regression errors," Economics Letters, Elsevier, vol. 52(2), pages 121-127, August.
  51. Sriananthakumar, Sivagowry & King, Maxwell L., 2006. "A new approximate point optimal test of a composite null hypothesis," Journal of Econometrics, Elsevier, vol. 130(1), pages 101-122, January.
  52. Shively, Thomas S., 1988. "An analysis of tests for regression coefficient stability," Journal of Econometrics, Elsevier, vol. 39(3), pages 367-386, November.
  53. Atiq-ur-Rehman, Atiq-ur-Rehman & Zaman, Asad, 2008. "Most Stringent Test for Location Parameter of a Random Number from Cauchy Density," MPRA Paper 13492, University Library of Munich, Germany.
  54. Elliott, Graham & Pesavento, Elena, 2009. "Testing The Null Of No Cointegration When Covariates Are Known To Have A Unit Root," Econometric Theory, Cambridge University Press, vol. 25(06), pages 1829-1850, December.
  55. Perron, Pierre & Rodriguez, Gabriel, 2003. "GLS detrending, efficient unit root tests and structural change," Journal of Econometrics, Elsevier, vol. 115(1), pages 1-27, July.
  56. Evans, Merran A. & King, Maxwell L., 1985. "A point optimal test for heteroscedastic disturbances," Journal of Econometrics, Elsevier, vol. 27(2), pages 163-178, February.
  57. Ulrich K. M¸ller & Graham Elliott, 2003. "Tests for Unit Roots and the Initial Condition," Econometrica, Econometric Society, vol. 71(4), pages 1269-1286, 07.
  58. Shively, Thomas S., 1993. "Testing for autoregressive disturbances in a time series regression with missing observations," Journal of Econometrics, Elsevier, vol. 57(1-3), pages 233-255.
  59. Dong Jin Lee, 2008. "Parametric and Semiparametric Efficient Tests for Parameter Instability," Working papers 2008-40, University of Connecticut, Department of Economics, revised Aug 2009.
  60. Dufour, Jean-Marie & King, Maxwell L., 1991. "Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary AR(1) errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 115-143, January.
  61. King, Maxwell L. & Wu, Ping X., 1991. "Small-disturbance asymptotics and the Durbin-Watson and related tests in the dynamic regression model," Journal of Econometrics, Elsevier, vol. 47(1), pages 145-152, January.
  62. White, Halbert, 1982. "Regularity conditions for cox's test of non-nested hypotheses," Journal of Econometrics, Elsevier, vol. 19(2-3), pages 301-318, August.
  63. Philip Shively, 2004. "Testing for a Unit Root in ARIMA Processes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 31(7), pages 785-798.
  64. Sriananthakumar, Sivagowry, 2013. "Testing linear regression model with AR(1) errors against a first-order dynamic linear regression model with white noise errors: A point optimal testing approach," Economic Modelling, Elsevier, vol. 33(C), pages 126-136.
  65. King, Maxwell L., 1984. "A new test for fourth-order autoregressive disturbances," Journal of Econometrics, Elsevier, vol. 24(3), pages 269-277, March.
  66. King, M. L., 1981. "The alternative Durbin-Watson test : An assessment of Durbin and Watson's choice of test statistic," Journal of Econometrics, Elsevier, vol. 17(1), pages 51-66, September.
  67. Inder, Brett A, 1990. "A New Test for Autocorrelation in the Disturbances of the Dynamic Linear Regression Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 31(2), pages 341-354, May.
  68. King, Maxwell L., 1989. "Testing for fourth-order autocorrelation in regression disturbances when first-order autocorrrelation is present," Journal of Econometrics, Elsevier, vol. 41(3), pages 285-301, July.
  69. Silvapulle, Paramsothy & King, Maxwell L, 1991. "Testing Moving Average against Autoregressive Disturbances in the Linear-Regression Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 9(3), pages 329-335, July.
  70. Emma Iglesias & Jean Marie Dufour, 2004. "Finite Sample and Optimal Inference in Possibly Nonstationary ARCH Models with Gaussian and Heavy-Tailed Errors," Econometric Society 2004 North American Summer Meetings 161, Econometric Society.
  71. James G. MacKinnon, 1983. "Model Specification Tests Against Non-Nested Alternatives," Working Papers 573, Queen's University, Department of Economics.
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