A COMPANION TO Theoretical Econometrics

Non-identified nuisance parameters

The example we discuss here is the problem of testing for the significance of jumps in the context of a jump-diffusion model. For econometric applications and references, see Saphores et al. (1998). Formally, consider the following model written, for convenience, in discrete time:

nt

St - S-1 = p + oE + X ln(Yt), t = 1, ..., T,

І=1

where E ~ N(0, 1) and ln(Y) ~ N(9, 52) and nt is the number of jumps which occur in the interval [t - 1, t]; the arrival of jumps is assumed to follow a Poisson process with parameter X. The associated likelihood function is as follows:

The hypothesis of no jumps corresponds to X = 0. It is clear that in this case, the parameters 9, 52 are not identified under the null, and hence, following the results of Davies (1977, 1987), the distribution of the associated LR statistic is non-standard and quite complicated. Although this problem is well recognized by now, a x2(3) asymptotic distribution is often (inappropriately) used in empirical applications of the latter LR test. See Diebold and Chen (1996) for related arguments dealing with structural change tests.

Let {, 62 denote the MLE under the null, i. e. imposing a Geometric Brownian Motion. Here we argue that in this case, the MC p-value calculated as described above, drawing iid N({, 62) disturbances (with { and 62 taken as given) will not depend on 9 and 52. This follows immediately from the implications of non­identification. Furthermore, the invariance to location and scale (p and o) is straightforward to see. Consequently, the MC test described in the context of pivotal statistics will yield exact p-values.

The problem of unidentified nuisance parameters is prevalent in econometrics. Bernard et al. (1998) consider another illustrative example: testing for ARCH-in­mean effects, and show that the MC method works very well in terms of size and power.

2 Conclusion

In this chapter, we have demonstrated that finite sample concerns may arise in several empirically pertinent test problems. But, in many cases of interest, the MC test technique produces valid inference procedures no matter how small your sample is.

We have also emphasized that the problem of constructing a good test - although simplified - cannot be solved just using simulations. Yet in most examples we have reviewed, MC test techniques emerge as indispensable tools.

Beyond the cases covered above, it is worthwhile noting that the MC test technique may be applied to many other problems of interest. These include, for example, models where the estimators themselves are also simulation-based, e. g. estimators based on indirect inference or involving simulated maximum likeli­hood. Furthermore, the MC test technique is by no means restricted to nested hypotheses. It is therefore possible to compare nonnested models using MC LR-type tests; assessing the success of this strategy in practical problems is an interesting research avenue.

Of course, the first purpose of the MC test technique is to control the prob­ability of type I errors (below a given level) so that rejections can properly be interpreted as showing that the null hypothesis is "incompatible" with the data. However, once level is controled, we can (and should) devote more attention to finding procedures with good power properties. Indeed, by helping to put the problem of level control out of the way, we think the technique of MC tests should help econometricians devote research to power issues as opposed to level. So an indirect consequence of the implementation of the technique may well be an increased emphasis on the design of more powerful tests.

Your data are valuable, and the statistical analysis you perform is often policy oriented. Why tolerate questionable p-values and confidence intervals, when exact or improved approximations are available?

Notes

* The authors thank three anonymous referees and the Editor, Badi Baltagi, for several useful comments. This work was supported by the Bank of Canada and by grants from the Canadian Network of Centres of Excellence (program on Mathematics of Information Technology and Complex Systems (MITACS)), the Social Sciences and Humani­ties Research Council of Canada, the Natural Sciences and Engeneering Council of Canada, and the Government of Quebec (Fonds FCAR).

1 The problem is more complicated when the structural equation includes more than one endogenous variable. See Dufour and Khalaf (1998b) for a detailed discussion of this case.

2 The underlying distributional result is due to Wilks (1932).

3 For a formal treatment see Dufour (1997).

4 Bera and Jarque (1982), Breusch and Pagan (1979, 1980) have also proposed related simulation-based techniques. However, these authors do not provide finite-sample theoretical justification for the proposed procedures. In particular, in contrast with Dwass (1957) and Barnard (1963) (and similarly to many other later authors who have proposed exploiting Monte Carlo techniques), they do not observe that appropriately randomized tests allow one to exactly control the level of a test in finite samples.

5 The subscript N in the notation adopted here may be misleading. We emphasize that RN (T0) gives the rank of S0 in the N + 1 dimensional array S0, S1,. . ., SN. Throughout this section N refers to the number of MC replications.

6 See Section 2.2 for a formal presentation of the model and test statistics. Some equa­tions are redefined here for convenience.

7 Global optimization is generally considered to be (relatively) computationally de­manding. We have experimented (see Dufour and Khalaf, 1998c, 1998b) with several MMC tests where the number of nuisance parameters referred to the simulated an­nealing algorithm was up to 20. Our simulations show that the method works well. Convergence was slow in some cases (less than 5 per 1,000). Recall, however, that for the problem at hand, one just practically needs to check whether the maximized function exceeds a, which clearly reduces the computational burdens.

8 In connection, it is worth mentioning that the MC test procedure applied to the Durbin-Watson test for AR(1) disturbances solves the inconclusive region problem.

9 See Dufour et al. (1998) for the power study.

10 See Dufour and Khalaf (1998c) for the power study.

References

Anderson, T. W., and H. Rubin (1949). Estimation of the parameters of a single equation in a complete system of stochastic equations. Annals of Mathematical Statistics 20, 46-63.

Attfield, C. L.F. (1995). A Bartlett adjustment to the likelihood ratio test for a system of equations. Journal of Econometrics 66, 207-23.

Barnard, G. A. (1963). Comment on "The Spectral Analysis of Point Processes" by M. S. Bartlett. Journal of the Royal Statistical Society, Series B 25, 294.

Bartlett, M. S. (1948). A note on the statistical estimation of supply and demand relations from time series. Econometrica 16, 323-9.

Bera, A. K., and C. M. Jarque (1982). Model specification tests: A simultaneous approach. Journal of Econometrics 20, 59-82.

Bernard, J.-T., J.-M. Dufour, L. Khalaf, and I. Genest (1998). Monte Carlo tests for heteroskedasticity. Discussion paper, Departement d'economique, Universite Laval and CRDE, Universite de Montreal.

Berndt, E. R. (1991). The Practice of Econometrics: Classic and Contemporary. Reading (MA): Addison-Wesley.

Birnbaum, Z. W. (1974). Computers and unconventional test-statistics. In F. Proschan, and R. J. Serfling (eds.) Reliability and Biometry, pp. 441-58. Philadelphia, PA: SIAM.

Bound, J., D. A. Jaeger, and R. M. Baker (1995). Problems with instrumental variables estimation when the correlation between the instruments and the endogenous explan­atory variable is weak. Journal of the American Statistical Association 90, 443-50.

Breusch, T. S. (1980). Useful invariance results for generalized regression models. Journal of Econometrics 13, 327-40.

Breusch, T. S., and A. R. Pagan (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica 47, 1287-94.

Breusch, T. S., and A. R. Pagan (1980). The Lagrange multiplier test and its applications to model specification in econometrics. Review of Economic Studies 47, 239-54.

Corana, A., M. Marchesi, C. Martini, and S. Ridella (1987). Minimizing multimodal func­tions of continuous variables with the "Simulated Annealing" algorithm. ACM Transac­tions on Mathematical Software 13, 262-80.

Dagenais, M. G., and J.-M. Dufour (1991). Invariance, nonlinear models and asymptotic tests. Econometrica 59, 1601-15.

D'Agostino, R. B., and M. A. Stephens (eds.) (1986). Goodness-of-Fit Techniques. New York: Marcel Dekker.

Davies, R. B. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64, 247-54.

Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33-43.

Davison, A., and D. Hinkley (1997). Bootstrap Methods and Their Application. Cambridge: Cambridge University Press.

Diebold, F. X., and C. Chen (1996). Testing structural stability with endogenous break­point: A size comparison of analytic and bootstrap procedures. Journal of Econometrics 70, 221-41.

Dufour, J.-M. (1989). Nonlinear hypotheses, inequality restrictions, and nonnested hypo­theses: Exact simultaneous tests in linear regressions. Econometrica 57, 335-55.

Dufour, J.-M. (1990). Exact tests and confidence sets in linear regressions with autocorrelated errors. Econometrica 58, 475-94.

Dufour, J.-M. (1995). Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics in econometrics. Discussion paper, C. R.D. E., Universite de Montreal.

Dufour, J.-M. (1997). Some impossibility theorems in econometrics, with applications to structural and dynamic models. Econometrica 65, 1365-89.

Dufour, J.-M., A. Farhat, L. Gardiol, and L. Khalaf (1998). Simulation-based finite sample normality tests in linear regressions. Econometrics Journal 1, 154-73.

Dufour, J. M., and J. Jasiak (1996). Finite sample inference methods for simultaneous equa­tions and models with unobserved and generated regressors. Discussion paper, C. R.D. E., Universite de Montreal.

Dufour, J.-M., and L. Khalaf (1998a). Monte Carlo tests for contemporaneous correlation of disturbances in multiequation SURE models. Discussion paper, C. R.D. E., Universite de Montreal.

Dufour, J.-M., and L. Khalaf (1998c). Simulation based finite and large sample inference methods in multivariate regressions and seemingly unrelated regressions. Discussion paper, C. R.D. E., Universite de Montreal.

Dufour, J.-M., and L. Khalaf (1998b). Simulation-based finite and large sample infer­ence methods in simultaneous equations. Discussion paper, C. R.D. E., Universite de Montreal.

Dufour, J.-M., and J. F. Kiviet (1996). Exact tests for structural change in first-order dynamic models. Journal of Econometrics 70, 39-68.

Dufour, J.-M., and J. F. Kiviet (1998). Exact inference methods for first-order autoregressive distributed lag models. Econometrica 66, 79-104.

Dwass, M. (1957). Modified randomization tests for nonparametric hypotheses. Annals of Mathematical Statistics 28, 181-7.

Efron, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans, CBS-NSF Regional Conference Series in Applied Mathematics, Monograph No. 38. Society for Industrial and Applied Mathematics, Philadelphia, PA.

Efron, B., and R. J. Tibshirani (1993). An Introduction to the Bootstrap, vol. 57 of Monographs on Statistics and Applied Probability. New York: Chapman & Hall.

Gallant, A. R., and G. Tauchen (1996). Which moments to match? Econometric Theory 12, 657-81.

Gibbons, M. R., S. A. Ross, and J. Shanken (1989). A test of the efficiency of a given port­folio. Econometrica 57, 1121-52.

Goffe, W. L., G. D. Ferrier, and J. Rogers (1994). Global optimization of statistical functions with simulated annealing. Journal of Econometrics 60, 65-99.

Gourieroux, C., and A. Monfort (1995). Statistics and Econometric Models, Volumes One and Two. Cambridge: Cambridge University Press.

Gourieroux, C., A. Monfort, and E. Renault (1993). Indirect inference. Journal of Applied Econometrics 8S, 85-118.

Hajivassiliou, V. A. (1993). Simulation estimation methods for limited dependent variables. In G. S. Maddala, C. R. Rao, and H. D. Vinod (eds.) Handbook of Statistics, Volume 11, Econometrics. pp. 519-43. Amsterdam: North-Holland.

Hall, P. (1992). The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag.

Horowitz, J. L. (1997). Bootstrap methods in econometrics: Theory and numerical perfor­mance. In D. Kreps, and K. W. Wallis (eds.) Advances in Economics and Econometrics, vol. 3, pp. 188-222. Cambridge: Cambridge University Press.

Jarque, C. M., and A. K. Bera (1980). Efficient tests for normality, heteroscedasticity and serial independence of regression residuals. Economics Letters 6, 255-9.

Jarque, C. M., and A. K. Bera (1987). A test for normality of observations and regression residuals. International Statistical Review 55, 163-72.

Jeong, J., and G. S. Maddala (1993). A perspective on application of bootstrap methods in econometrics. In G. S. Maddala, C. R. Rao, and H. D. Vinod (eds.) Handbook of Statistics, Volume 11, Econometrics, pp. 573-610. Amsterdam: North-Holland.

Keane, M. P. (1993). Simulation estimation for panel data models with limited dependent variables. In Maddala, Rao, and Vinod (1993), pp. 545-571.

Kiviet, J. F., and J.-M. Dufour (1997). Exact tests in single equation autoregressive distri­buted lag models. Journal of Econometrics 80, 325-53.

Kolmogorov, A. N. (1933). Sulla determinazione empiricadi una legge di distribtuzione. Giorna. Ist. Attuari 4, 83-91.

Laitinen, K. (1978). Why is demand homogeneity so often rejected? Economics Letters 1, 187-91.

Lehmann, E. L. (1986). Testing Statistical Hypotheses, 2nd edn. New York: John Wiley & Sons.

Maddala, G. S., C. R. Rao, and H. D. Vinod (eds.) (1993). Handbook of Statistics, Volume 11, Econometrics. Amsterdam: North-Holland.

Mariano, R. S., and B. W. Brown (1993). Stochastic simulation for inference in nonlinear errors-in-variables models. In Maddala, Rao, and Vinod (1993), pp. 611-27.

Nelson, C. R., and R. Startz (1990a). The distribution of the instrumental variable estimator and its t-ratio when the instrument is a poor one. Journal of Business 63, 125-40.

Nelson, C. R., and R. Startz (1990b). Some further results on the exact small properties of the instrumental variable estimator. Econometrica 58, 967-76.

Saphores, J.-D., L. Khalaf, and D. Pelletier (1998). Modelling unexpected changes in stump - age prices: an application to Pacific Northwest National Forests. Discussion paper, GREEN, Universite Laval, Quebec.

Shao, S., and D. Tu (1995). The Jackknife and Bootstrap. New York: Springer-Verlag.

Smirnov, N. V. (1939). Sur les ecarts de la courbe de distribution empirique (Russian/ French Summary). Matematiceskil Sbornik N. S. 6, 3-26.

Staiger, D., and J. H. Stock (1997). Instrumental variables regression with weak instru­ments. Econometrica 65, 557-86.

Stewart, K. G. (1997). Exact testing in multivariate regression. Econometric Reviews 16, 321-52.

Vinod, H. D. (1993). Bootstrap methods: Applications in econometrics. in Maddala, Rao, and Vinod (1993), pp. 629-61.

Wilks, S. S. (1932). Certain generalizations in the analysis of variance. Biometrika 24, 471-94.

Zellner, A. (1962). An efficient method for estimating seemingly unrelated regressions and tests for aggregate bias. Journal of the American Statistical Association 57, 348-68.