INTERVAL ESTIMATION AND HYPOTHESIS TESTING

 

2.51  Interval Estimation

Instead of relying on the point estimate alone, we may construct an interval around the point estimator, say within two or three standard errors on either side of the point estimator, such that this interval has, say, 95 percent probability of including

the true parameter value. This is roughly the idea behind interval estimation.

 

Given a model we want to find out how “close,” say, ˆ β2 is to β2. For this purpose we try to find out two positive numbers δ and α, the latter lying between 0 and 1, such that the probability that the random interval ( ˆ β2 − δ, ˆ β2 + δ) contains the true β2 is 1 − α. Symbolically,

Pr ( ˆ β2 − δ ≤ β2 ≤ ˆ β2 + δ) = 1 – α. Such an interval, if it exists, is known as a confidence interval; 1 − α is known as the confidence coefficient; and α (0 < α < 1) is known as the level of significance.  The endpoints of the confidence interval are known as the confidence limits (also known as critical values), ˆ β2 − δ being the lower confidence limit and ˆ β2 + δ the upper confidence limit.

 

An interval estimator, in contrast to a point estimator, is an interval constructed in such a manner that it has a specified probability 1 – α  of including within its limits the true value of the parameter.

 

It is very important to know the following aspects of interval estimation:

1. Since β2, although an unknown, is assumed to be some fixed number, either it lies in the interval or it does not. For the method described, the probability of constructing an interval that contains β2 is 1 − α.

2. The interval is a random interval; that is, it will vary from one sample to the next because it is based on ˆ β2, which is random.

3. Since the confidence interval is random, the probability statements attached to it

should be understood in the long-run sense, that is, repeated sampling.

 

4. The interval is random so long as ˆ β2 is not known. But once we have a specific sample and once we obtain a specific numerical value of ˆ β2, the interval is no longer random; it is fixed. In this case, we cannot say that the probability is 1 − α that a given fixed interval includes the true β2. In this situation, β2 is either in the fixed interval or outside it. Therefore, the probability is either 1 or 0.

 

 

b) Confidence Intervals for Regression Coefficients β1 and β2

Confidence Interval for β2

We can use the normal distribution to make probabilistic statements about β2 provided the true population variance σ2 is known. If σ2 is known, an important property of a normally distributed variable with mean μ and variance σ2 is that the area under the normal curve between μ ± σ is about 68 percent, that between the limits μ ± 2σ is about 95 percent, and that between μ ± 3σ is about 99.7 percent.

But σ2 is rarely known, and in practice it is determined by the unbiased estimator ˆσ 2. If we replace σ by ˆσ , the t statistic is . where  se ( ˆ β2) now refers to the estimated standard error. It can be shown that the t variable thus defined follows the t distribution with n − 2 df.

 

We can use the t distribution to establish a confidence interval for β2 as follows:

Pr (−tα/2 ≤ t ≤ tα/2) = 1 – α. where the t value in the middle of this double inequality is the t statistic value computed and where tα/2 is the value of the t variable obtained from the t distribution for α/2 level of significance and n − 2 df; it is often called the critical t value at α/2 level of significance.

 

The width of the confidence interval is proportional to the standard error of the estimator. That is, the larger the standard error, the larger is the width of the

confidence interval. Put differently, the larger the standard error of the estimator, the greater is the uncertainty of estimating the true value of the unknown parameter. Thus, the standard error of an estimator is often described as a measure of the precision of the estimator (i.e., how precisely the estimator measures the true population value).

ˆ β1 ± tα/2 se ( ˆ β1)

Pr [ ˆ β1 − tα/2 se ( ˆ β1) ≤ β1 ≤ ˆ β1 + tα/2 se ( ˆ β1)] = 1 − α

ˆ β2 ± tα/2 se ( ˆ β2)

Pr [β2 − tα/2 se ( ˆ β2) ≤ β2 ≤ ˆ β2 + tα/2 se ( ˆ β2)] = 1 – α with n-2 df.