The LS and ML methods developed in previous sections allow us to obtain a point estimate of the parameters of the model. However, even if the estimator satisfies the desirable properties, there is some probability that it will be quite erroneous, because it tries to infer a population value from a sample. Thus, a point estimate does not provide any information on the likely range of error. The estimator should be as accurate as possible (i.e., the range of error as small as possible), and this accuracy can be quantified through the variance (or standard deviation) of the estimator. Nevertheless, if we know the sample distribution of the estimator, there is a more structured approach of presenting the accuracy measure which consists in constructing a confidence interval.
A confidence interval is defined as a range of values
that is likely to include the true value of the unknown parameter.
The confidence interval means that, if we have different samples,
and in each case we construct the confidence intervals, then we
expect that about
percent of them will contain
the true parameter value. The probability amount
is
known as the level of confidence.
As we have
mentioned earlier, in order to obtain the interval estimation, we must
know the sample probability distribution of the corresponding
estimator. Result (2.74) allows us to obtain such a
distribution for every element of the
vector as:
In order to obtain the interval
estimation of
we take the distribution given in
(2.78) and the expression of the OLS estimator of
given in (2.34), in such a way that,
From our
consumption function example, we now want to calculate the
interval confidence for the three coefficients (,
,
) and the dispersion parameter
. In each case, we calculate the interval for levels
of confidence of 90 and 95 percent (
=0.1 and 0.05). The
following quantlet allow us to obtain the previous information