Model Estimation 267
comparing empirical estimators with test statistics or critical values com-
puted from the sampling distribution.^2 A test statistic is used in hypothesis
testing to decide whether to accept or reject a hypothesis.^3
Estimation Methods
Because estimation methods involve criteria that cannot be justified by
themselves, they are subject to some arbitrariness. The crucial point is that,
whereas an estimation process must “fit” a distribution to empirical data,
any distribution can, with a few restrictions, be fitted to any empirical data.
The choice of distributions thus includes an element of arbitrariness. Sup-
pose we want to determine the probability distribution of the faces of a
tossed coin, and in 1,000 experiments, heads comes out 950 times. We prob-
ably would conclude that the coin is highly biased and that heads has a 95%
probability of coming up. We have no objective way, however, to rule out
the possibility that the coin is fair and that we are experiencing an unlikely
event. Ultimately, whatever conclusion we draw is arbitrary.
Four estimation methods are commonly used in financial econometrics:
■ (^) Least squares method
■ (^) Maximum likelihood method
■ (^) Method of moments method
■ (^) Bayesian method
The four methods listed above are the fundamental estimation methods.
These methods have been generalized in broad estimation frameworks that,
however, overlap, making it difficult to create a simple taxonomy of estima-
tion methods. In fact, the least squares method and the maximum likelihood
method are instances of a more general approach called the M-estimator
method. The method of moments has been generalized into the generalized
method of moments. The least squares method and the maximum likeli-
hood method are also instances of the generalized method of moments. The
Bayesian estimation method is based on a different interpretation of statis-
tics and will not be discussed in this chapter.^4 The instrumental variables
(^2) Hypothesis testing is explained in Appendix C.
(^3) Test statistics or critical values of the autoregressive parameters are tabulated and
are available in all major time-series software packages.
(^4) For readers interested in learning more about Bayesian estimation, see Svetlozar T.
Rachev, John S. J. Hsu, Biliana Bagasheva, and Frank J. Fabozzi, Bayesian Methods
in Finance (Hoboken, NJ: John Wiley & Sons, 2008).