266 The Basics of financial economeTrics
Testing is complementary to estimation. Critical parameters are often tested
before the estimation process starts in earnest, although some tests of the
adequacy of models can be performed after estimation.
In general terms, statistics is a way to make inferences from a sample
to the entire population from which the sample is taken. This area within
the field of statistics is called inferential statistics (or inductive statistics)
and is explained in more detail in Appendix C. In financial economet-
rics, the sample is typically an empirical time series. Data may be returns,
prices, interest rates, credit spreads, default rates, company-specific finan-
cial data, or macroeconomic data. The objective of estimation techniques
is to estimate the parameters of models that describe the empirical data.
The key concept in estimation is that of estimators. An estimator is a
function of sample data whose value is close to the true value of a parameter
in a distribution. For example, the empirical average (i.e., the sum of the
sample’s values for a variable divided by the number of samples) is an esti-
mator of the mean; that is, it is a function of the empirical data that approxi-
mates the true mean. Estimators can be simple algebraic expressions; they
can also be the result of complex calculations.
Estimators must satisfy a number of properties. In particular, estimators
■ (^) Should get progressively closer to the true value of the parameter to be
estimated as the sample size becomes larger.
■ (^) Should not carry any systematic error.
■ (^) Should approach the true values of the parameter to be estimated as
rapidly as possible.
The question related to each estimation problem should be what estimator
would be best suited for the problem at hand. Estimators suitable for the
very same parameters can vary quite remarkably when it comes to quality
of their estimation. In Appendix C we explain some of the most commonly
employed quality criteria for evaluating estimators.
Being a function of sample data, an estimator is a random (i.e., stochas-
tic) variable. Therefore, the estimator has a probability distribution referred
to as the sampling distribution. In general, the probability distribution of
an estimator is difficult to compute accurately from small samples but is
simpler for large samples.^1
The sampling distribution is important. Decisions such as determining
whether a process is integrated must often be made on the basis of esti-
mators. Because estimators are random variables, decisions are based on
(^1) See Appendix C for a discussion of how the behavior of estimators changes as the
size of the sample varies.