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:


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


Подпись: L1 = -T ln(X) image596

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?


* 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.


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