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Generalized autocontours: Evaluation of multivariate density models

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  • González-Rivera, Gloria
  • Sun, Yingying

Abstract

We propose a new tool, the Generalized Autocontour (G-ACR), as the basis for a battery of dynamic specification tests that are applicable (in-sample or out-of-sample) to univariate or multivariate random processes. We apply this methodology to the modeling of a multivariate system by specifying the dynamics of the marginal distributions of each process in the system, together with a copula that ties up the marginals to produce their multivariate distribution. We work with the probability integral transforms (PIT) of the system that, under a correct specification of the conditional model, should be i.i.d. U[0,1]. The dimensionality of the system is not a constraint, because the information contained in the vector of PITs is condensed into an indicator, which is the basis of the proposed tests. We construct hyper-cubes of different sizes within the maximum hyper-cube formed by a multidimensional uniform density [0,1]n, and assess the locations of the empirical PITs (duplex, triplex, n-plex of observations) within the corresponding population hyper-cubes. If the conditional model is correct, the volumes of the population hyper-cubes must be the same as those in their empirical counterparts. This approach allows the researcher to focus on different areas of the conditional density model, so as to assess the regions of interest. We estimate a trivariate model for a very large number of trades on the stocks of three large U.S. banks and find that the contemporaneous dependence among institutions is asymmetric, which implies that when liquidity drains (due to a lack of trading) in one institution, we should expect a concurrent effect among similar institutions. On the other hand, when liquidity is plentiful (due to dense trading), the trades on the stocks of the institutions are not correlated. We assess the models’ performances by evaluating their one-step-ahead density forecasts of trades.

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  • González-Rivera, Gloria & Sun, Yingying, 2015. "Generalized autocontours: Evaluation of multivariate density models," International Journal of Forecasting, Elsevier, vol. 31(3), pages 799-814.
    • Handle: RePEc:eee:intfor:v:31:y:2015:i:3:p:799-814
    DOI: 10.1016/j.ijforecast.2014.03.019
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    1. Madhavan, Ananth, 2000. "Market microstructure: A survey," Journal of Financial Markets, Elsevier, vol. 3(3), pages 205-258, August.
    2. Gloria González-Rivera & Zeynep Senyuz & Emre Yoldas, 2011. "Autocontours: Dynamic Specification Testing," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(1), pages 186-200, January.
    3. Heinen, Andreas & Rengifo, Erick, 2007. "Multivariate autoregressive modeling of time series count data using copulas," Journal of Empirical Finance, Elsevier, vol. 14(4), pages 564-583, September.
    4. Francis X. Diebold & Jinyong Hahn & Anthony S. Tay, 1999. "Multivariate Density Forecast Evaluation And Calibration In Financial Risk Management: High-Frequency Returns On Foreign Exchange," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 661-673, November.
    5. Andrew J. Patton, 2006. "Estimation of multivariate models for time series of possibly different lengths," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(2), pages 147-173, March.
    6. Diebold, Francis X & Gunther, Todd A & Tay, Anthony S, 1998. "Evaluating Density Forecasts with Applications to Financial Risk Management," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 863-883, November.
    7. González-Rivera, Gloria & Yoldas, Emre, 2012. "Autocontour-based evaluation of multivariate predictive densities," International Journal of Forecasting, Elsevier, vol. 28(2), pages 328-342.
    8. Cameron,A. Colin & Trivedi,Pravin K., 2005. "Microeconometrics," Cambridge Books, Cambridge University Press, number 9780521848053, Enero-Abr.
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    Citations

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    Cited by:

    1. Jonas Dovern & Hans Manner, 2020. "Order‐invariant tests for proper calibration of multivariate density forecasts," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 35(4), pages 440-456, June.
    2. Gloria Gonzalez‐Rivera & Yun Luo & Esther Ruiz, 2020. "Prediction regions for interval‐valued time series," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 35(4), pages 373-390, June.
    3. Rossi, Barbara & Sekhposyan, Tatevik, 2019. "Alternative tests for correct specification of conditional predictive densities," Journal of Econometrics, Elsevier, vol. 208(2), pages 638-657.
    4. João Henrique G. Mazzeu & Gloria González-Rivera & Esther Ruiz & Helena Veiga, 2020. "A bootstrap approach for generalized Autocontour testing Implications for VIX forecast densities," Econometric Reviews, Taylor & Francis Journals, vol. 39(10), pages 971-990, November.
    5. González-Rivera, Gloria & Sun, Yingying, 2017. "Density forecast evaluation in unstable environments," International Journal of Forecasting, Elsevier, vol. 33(2), pages 416-432.
    6. Yun Luo & Gloria González-Rivera, 2024. "A Truncated Mixture Transition Model for Interval-Valued Time Series," Journal of Financial Econometrics, Oxford University Press, vol. 22(4), pages 1130-1169.
    7. Lenza, Michele & Moutachaker, Inès & Paredes, Joan, 2023. "Density forecasts of inflation: a quantile regression forest approach," CEPR Discussion Papers 18298, C.E.P.R. Discussion Papers.
    8. Gloria Gonzalez-Rivera & Yun Luo & Esther Ruiz, 2018. "Prediction Regions for Interval-valued Time Series," Working Papers 201817, University of California at Riverside, Department of Economics.
    9. Gonçalves Mazzeu, Joao Henrique & González-Rivera, Gloria & Ruiz Ortega, Esther & Veiga, Helena, 2016. "A Bootstrap Approach for Generalized Autocontour Testing," DES - Working Papers. Statistics and Econometrics. WS 23457, Universidad Carlos III de Madrid. Departamento de Estadística.
    10. Kolassa, Stephan, 2016. "Evaluating predictive count data distributions in retail sales forecasting," International Journal of Forecasting, Elsevier, vol. 32(3), pages 788-803.
    11. Anatolyev, Stanislav & Baruník, Jozef, 2019. "Forecasting dynamic return distributions based on ordered binary choice," International Journal of Forecasting, Elsevier, vol. 35(3), pages 823-835.
    12. Gloria Gonzalez-Rivera & Yun Luo, 2020. "A Truncated Mixture Transition Model for Interval-valued Time Series," Working Papers 202005, University of California at Riverside, Department of Economics.

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