IDEAS home Printed from
   My bibliography  Save this paper

Estimating Stochastic Differential Equations Efficiently by Minimum Chi-Square


  • Gallant, A. Ronald
  • Long, Jonathan R.


We propose a minimum chi-square estimator for the parameters of an ergodic system of stochastic differential equations with partially observed state. We prove that the efficiency of the estimator approaches that of maximum likelihood as the number of moment functions entering the chi-square criterion increases and as the number of past observations entering each moment function increases. The minimized criterion is asymptotically chi-squared and can be used to test system adequacy. When a fitted system is rejected, inspecting studentized moments suggests how the fitted system might be modified to improve the fit. The method and diagnostic tests are applied to daily observations on the U.S. dollar to Deutschmark exchange rate from 1977 to 1992. Key Words: Diffusions, efficiency, estimation, exchange rate, minimum chi-square, partially observed state, simulation, specification test, stochastic differential equation.

Suggested Citation

  • Gallant, A. Ronald & Long, Jonathan R., 1996. "Estimating Stochastic Differential Equations Efficiently by Minimum Chi-Square," Working Papers 96-32, Duke University, Department of Economics.
  • Handle: RePEc:duk:dukeec:96-32

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. Kling, Catherine L., 1988. "Comparing welfare estimates of environmental quality changes from recreation demand models," Journal of Environmental Economics and Management, Elsevier, vol. 15(3), pages 331-340, September.
    2. T.A. Cameron & D.D. Huppert, 1988. ""Referendum" Contingent Valuation Estimates: Sensitivity to the Assignment of Offered Values," UCLA Economics Working Papers 519, UCLA Department of Economics.
    3. Cummings, Ronald G, et al, 1997. "Are Hypothetical Referenda Incentive Compatible?," Journal of Political Economy, University of Chicago Press, vol. 105(3), pages 609-621, June.
    4. Joseph Cooper & John Loomis, 1992. "Sensitivity of Willingness-to-Pay Estimates to Bid Design in Dichotomous Choice Contingent Valuation Models," Land Economics, University of Wisconsin Press, vol. 68(2), pages 211-224.
    5. Cameron, Trudy Ann, 1988. "A new paradigm for valuing non-market goods using referendum data: Maximum likelihood estimation by censored logistic regression," Journal of Environmental Economics and Management, Elsevier, vol. 15(3), pages 355-379, September.
    6. Cropper, Maureen L & Deck, Leland B & McConnell, Kenneth E, 1988. "On the Choice of Functional Form for Hedonic Price Functions," The Review of Economics and Statistics, MIT Press, vol. 70(4), pages 668-675, November.
    7. Mary Jo Kealy & Robert W. Turner, 1993. "A Test of the Equality of Closed-Ended and Open-Ended Contingent Valuations," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 75(2), pages 321-331.
    8. Anna Alberini, 1995. "Testing Willingness-to-Pay Models of Discrete Choice Contingent Valuation Survey Data," Land Economics, University of Wisconsin Press, vol. 71(1), pages 83-95.
    9. Kevin J. Boyle & F. Reed Johnson & Daniel W. McCollum & William H. Desvousges & Richard W. Dunford & Sara P. Hudson, 1996. "Valuing Public Goods: Discrete versus Continuous Contingent-Valuation Responses," Land Economics, University of Wisconsin Press, vol. 72(3), pages 381-396.
    10. Richard C. Ready & Jean C. Buzby & Dayuan Hu, 1996. "Differences between Continuous and Discrete Contingent Value Estimates," Land Economics, University of Wisconsin Press, vol. 72(3), pages 397-411.
    11. Daniel McFadden, 1994. "Contingent Valuation and Social Choice," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 76(4), pages 689-708.
    12. McConnell, K. E., 1990. "Models for referendum data: The structure of discrete choice models for contingent valuation," Journal of Environmental Economics and Management, Elsevier, vol. 18(1), pages 19-34, January.
    13. Holmes Thomas P. & Kramer Randall A., 1995. "An Independent Sample Test of Yea-Saying and Starting Point Bias in Dichotomous-Choice Contingent Valuation," Journal of Environmental Economics and Management, Elsevier, vol. 29(1), pages 121-132, July.
    14. Trudy Ann Cameron, 1992. "Combining Contingent Valuation and Travel Cost Data for the Valuation of Nonmarket Goods," Land Economics, University of Wisconsin Press, vol. 68(3), pages 302-317.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Liu, Ming & Zhang, Harold H., 1998. "Overparameterization in the seminonparametric density estimation," Economics Letters, Elsevier, vol. 60(1), pages 11-18, July.
    2. Ming Liu & Harold H. Zhang, "undated". "Specification Tests in the Efficient Method of Moments Framework with Application to the Stochastic Volatility Models," Computing in Economics and Finance 1997 93, Society for Computational Economics.
    3. Liu, Ming, 2000. "Modeling long memory in stock market volatility," Journal of Econometrics, Elsevier, vol. 99(1), pages 139-171, November.

    More about this item


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:duk:dukeec:96-32. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Department of Economics Webmaster). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.