Further Readings

There is a large number of books available that provide further in-depth discus­sions of the material (or parts of the material) presented in this article. The list of such books includes texts by Billingsley (1968, 1979), Davidson (1994), Serfling (1980) and Shiryayev (1984), to mention a few. Hall and Heyde (1980) give a thorough discussion of martingale limit theory.

Recent books on asymptotic theory for least mean distance estimators (includ­ing maximum likelihood estimators) and generalized method of moments esti­mators for general classes of nonlinear models include texts by Bierens (1994), Gallant (1987), Gallant and White (1988), Potscher and Prucha (1997), and White (1994). For recent survey articles see, e. g., Newey and McFadden (1994), and Wooldridge (1994).

Notes

1 Strictly speaking, h(.) has to be Borel measurable; for a definition of Borel measur­ability see Billingsley (1979), section 13.

2 For an understanding of the example it proves helpful to plot Z 1(ю), Z2(ro),.. . for 0 < ю < 1.

3 The space of all random variables with finite rth moment is often referred to as the space Lr. On this space convergence in rth mean is often referred to as Lr-convergence.

4 For a proof see, e. g., Serfling (1980, p. 15).

5 See Billingsley (1979, p. 180 and Example 21.21).

6 We note that the rescaled quantities like 4n (0n – 0) typically do not converge a. s. or i. p., and thus a new notion of convergence is needed for these quantities.

7 See, e. g., Billingsley (1968, p. 12), Billingsley (1979, p. 345), and Serfling (1980, p. 16).

8 See, e. g., Serfling (1980, pp. 13-14).

9 See, e. g., Serfling (1980, p. 24).

10 That is, let A C Rk denote the set of continuity points of g, then PZ(A) = P(Z Є A) = 1. Of course, if g is continuous on Rk, then A = Rk and the condition PZ(A) = 1 is trivially satisfied.

11 The event {Vn = 0} has probability approaching zero, and hence it is irrelevant which value is assigned to Wn/ Vn on this event.

12 See, e. g., Shiryayev (1984, p. 366).

13 See, e. g., Shiryayev (1984, p. 364).

14 See, e. g., Shiryayev (1984, p. 487), or Davidson (1994, p. 314).

15 See, e. g., Stout (1974, p. 180).

16 See, e. g., Stout (1974, p. 181).

17 More precisely, Qn is a function from Q x 0 into R such that Q(., 0) is у-measurable for each 0 Є 0.

18 The same line of argument holds if (uniform) convergence a. s. is replaced by (uni­form) convergence i. p.

19 For a proof see Jennrich (1969, Theorem 2).

20 See, e. g., Billingsley (1979, p. 308).

21 Of course, the case о2 = 0 is trivial since in this case Zt = 0 a. s.

22 See, e. g., Billingsley (1979, p. 310).

23 See, e. g., Billingsley (1979, p. 312).

24 The theorem is given as Problem 27.6 in Billingsley (1979, p. 319).

25 The proof given in Amemiya seems not to be entirely rigorous in that it does not take into account that the elements of Sn and hence those of Wn depend on the sample size n.

26 See, e. g., Ganssler and Stute (1977, p. 372).

27 See footnote 21.

28 See, e. g., Ganssler and Stute (1977, p. 365 and 370).

References

Amemiya, T. (1985). Advanced Econometrics. Cambridge: Harvard University Press. Anderson, T. W. (1971). The Statistical Analysis of Time Series. New York: Wiley.

Andrews, D. W.K. (1987). Consistency in nonlinear econometric models: A generic uni­form law of large numbers. Econometrica 55, 1465-71.

Bierens, H. J. (1994). Topics in Advanced Econometrics. Cambridge: Cambridge University Press.

Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.

Billingsley, P. (1979). Probability and Measure. New York: Wiley.

Davidson, J. (1994). Stochastic Limit Theory. Oxford: Oxford University Press.

Davidson, J., and R. de Jong (1997). Strong laws of large numbers for dependent heteroge­neous processes: A synthesis of recent new results. Econometric Reviews 16, 251-79.

Fuller, W. A. (1976). Introduction to Statistical Time Series. New York: Wiley.

Gallant, A. R. (1987). Nonlinear Statistical Models. New York: Wiley.

Gallant, A. R., and H. White (1988). A Unified Theory of Estimation and Inference in Nonlinear Dynamic Models. New York: Basil Blackwell.

Ganssler, P., and W. Stute (1977). Wahrscheinlichkeitstheorie. New York: Springer Verlag.

Hall, P., and C. C. Heyde (1980). Martingale Limit Theory and Its Application. New York: Academic Press.

Hannan, E. J. (1970). Multiple Time Series. New York: Wiley.

Jennrich, R. I. (1969). Asymptotic properties of nonlinear least squares estimators. Annals of Mathematical Statistics 40, 633-43.

McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Annals of Probability 2, 620-8.

McLeish, D. L. (1975). Invariance principles for dependent variables. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 32, 165-78.

Newey, W. K., and D. L. McFadden (1994). Large sample estimation and hypothesis testing. In R. F. Engle, and D. L. McFadden (eds.) Handbook of Econometrics, Volume 4. pp. 2113­245, New York: Elsevier Science B. V.

Phillips, P. C.B., and V. Solo (1992). Asymptotics for linear processes. Annals of Statistics 20, 971-1001.

Potscher, B. M., and I. R. Prucha (1986). A class of partially adaptive one-step M-estimators for the non-linear regression model with dependent observations. Journal of Econometrics 32, 219-51.

Potscher, B. M., and I. R. Prucha (1989). A uniform law of large numbers for dependent and heterogeneous data processes. Econometrica 57, 675-83.

Potscher, B. M., and I. R. Prucha (1997). Dynamic Nonlinear Econometric Models, Asymptotic Theory. New York: Springer Verlag.

Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. New York: Wiley.

Shiryayev, A. N. (1984). Probability. New York: Springer Verlag.

Stout, W. F. (1974). Almost Sure Convergence. New York: Academic Press.

Theil, H. (1971). Principles of Econometrics. New York: Wiley.

White, H. (1994). Estimation, Inference and Specification Analysis. Cambridge: Cambridge University Press.

Wooldridge, J. M. (1994). Estimation and inference for dependent processes. In R. F. Engle and D. L. McFadden (eds.) Handbook of Econometrics, Volume 4. pp. 2641-738, New York: Elsevier Science B. V.

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