The Bierens test for certain nonstationary models. (English) Zbl 1431.62391
Summary: We adapt the H. J. Bierens [Econometrica 58, No. 6, 1443–1458 (1990; Zbl 0737.62058)] test to the \(I\)-regular models of J. Y. Park and P. C. B. Phillips [Econometrica 69, No. 1, 117–161 (2001; Zbl 0999.62050)]. H. J. Bierens [loc. cit.] defines the test hypothesis in terms of a conditional moment condition. Under the null hypothesis, the moment condition holds with probability one. The probability measure used is that induced by the variables in the model, that are assumed to be strictly stationary. Our framework is nonstationary and this approach is not always applicable. We show that the Lebesgue measure can be used instead in a meaningful way. The resultant test is consistent against all \(I\)-regular alternatives.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62F03 | Parametric hypothesis testing |
| 62F05 | Asymptotic properties of parametric tests |
| 62P20 | Applications of statistics to economics |
Keywords:
Bierens test; consistent test; functional form misspecification; integrable models; local time; nonlinear cointegration; test of functional form; predictability of stock returns; unit rootReferences:
| [1] | Bierens, H. J., Consistent model specification testing, Journal of Econometrics, 20, 3105-3134 (1982) |
| [2] | Bierens, H. J., Model specification testing of time series regressions, Journal of Econometrics, 26, 323-353 (1984) · Zbl 0563.62064 |
| [3] | Bierens, H. J., ARMAX model specification testing, with an application to unemployment in Netherlands, Journal of Econometrics, 35, 161-190 (1987) · Zbl 0621.62106 |
| [4] | Bierens, H. J., ARMA memory index modeling of economic time series (with discussion), Econometric Theory, 4, 35-59 (1988) |
| [5] | Bierens, H. J., A consistent conditional moment test of functional form, Econometrica, 67, 1341-1383 (1990) |
| [6] | Bierens, H. J.; Ploberger, W., Asymptotic theory of conditional moment integrated test, Econometrica, 65, 1129-1151 (1997) · Zbl 0927.62085 |
| [7] | Billingsley, P., Convergence of Probability Measures (1968), Wiley · Zbl 0172.21201 |
| [8] | Billingsley, P., Probability and Measure (1979), Wiley · Zbl 0411.60001 |
| [9] | Chang, Y.; Park, J. Y.; Phillips, P. C.B., Nonlinear econometric models with cointegrated and deterministically trending regressors, Econometrics Journal, 4, 1-36 (2001) · Zbl 1007.62100 |
| [10] | de Jong, R. M., The Bierens test under data dependence, Journal of Econometrics, 72, 1-32 (1996) · Zbl 0855.62073 |
| [11] | de Jong, R. M., Addendum to “Asymptotics for nonlinear transformations of integrated time series”, Econometric Theory, 20, 627-635 (2004) · Zbl 1060.62100 |
| [12] | de Jong, R. M.; Wang, C.-H., Further results on the asymptotics for nonlinear transformations of integrated time series, Econometric Theory, 21, 413-430 (2005) · Zbl 1062.62210 |
| [13] | Escanciano, J. C., Goodness-of-fit statistics for linear and nonlinear time series models, Journal of the American Statistical Association, 101, 531-541 (2006) · Zbl 1119.62359 |
| [14] | Goyal, A., Welch, I., 2003. A note on predicting returns with financial ratios. Mimeo.; Goyal, A., Welch, I., 2003. A note on predicting returns with financial ratios. Mimeo. · Zbl 1232.91720 |
| [15] | Halmos, P. R., Measure Theory (1950), Springer-Verlag · Zbl 0117.10502 |
| [16] | Hong, S.H., Phillips, P.C.B., 2005. Testing linearity in cointegrating relations with an application to PPP. Cowles Foundation Discussion Paper 1541.; Hong, S.H., Phillips, P.C.B., 2005. Testing linearity in cointegrating relations with an application to PPP. Cowles Foundation Discussion Paper 1541. |
| [17] | Jeganathan, P., 2003. Second order limits of functionals of sums of linear processes that converge to fractional stable motions. Unpublished manuscript, Indian Statistical Institute.; Jeganathan, P., 2003. Second order limits of functionals of sums of linear processes that converge to fractional stable motions. Unpublished manuscript, Indian Statistical Institute. |
| [18] | Kasparis, I., 2004. Testing for functional form under non-stationarity. Unpublished manuscript, University of Cyprus.; Kasparis, I., 2004. Testing for functional form under non-stationarity. Unpublished manuscript, University of Cyprus. |
| [19] | Kasparis, I., Detection of functional form misspecification in cointegrating relations, Econometric Theory, 24, 1373-1403 (2008) · Zbl 1284.62562 |
| [20] | Lewellen, J., Predicting returns with financial ratios, Journal of Financial Economics, 74, 209-235 (2004) |
| [21] | Marmer, V., Nonlinearity, nonstationarity and spurious forecasts, Journal of Econometrics, 142, 1-27 (2007) · Zbl 1418.62514 |
| [22] | Newey, W. K., Maximum likelihood specification testing and conditional moment tests, Econometrica, 53, 1047-1070 (1985) · Zbl 0629.62107 |
| [23] | Park, J.Y., Phillips, P.C.B., 1998. Unit roots in nonlinear transformations of integrated time series. Unpublished manuscript, Yale University.; Park, J.Y., Phillips, P.C.B., 1998. Unit roots in nonlinear transformations of integrated time series. Unpublished manuscript, Yale University. |
| [24] | Park, J. Y.; Phillips, P. C.B., Asymptotics for nonlinear transformations of integrated time series, Econometric Theory, 15, 269-298 (1999) · Zbl 0964.62092 |
| [25] | Park, J. Y.; Phillips, P. C.B., Nonstationary binary choice, Econometrica, 68, 1249-1280 (2000) · Zbl 1056.62530 |
| [26] | Park, J. Y.; Phillips, P. C.B., Nonlinear regressions with integrated time series, Econometrica, 69, 117-161 (2001) · Zbl 0999.62050 |
| [27] | Pötscher, B. M., Nonlinear functions and convergence to Brownian motion: Beyond the continuous mapping theorem, Econometric Theory, 20, 1-22 (2004) · Zbl 1055.60031 |
| [28] | Stambaugh, R., Predicting regressions, Journal of Financial Economics, 54, 375-421 (1999) |
| [29] | Stinchcombe, M. B.; White, H., Consistent specification testing with nuisance parameters present only under the alternative, Econometric Theory, 14, 295-325 (1998) · Zbl 1419.62105 |
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