Confidence interval

An alternative approach to the test of significance approach described by the example in the previous section is the so called confidence interval approach. The two approaches are very much related to each other. The idea is to create an interval estimate for the population parameter instead of working with a point estimate.

The basic steps are about the same. All tests need to start from some hypothesis and we will use (4.5) in this example. By choosing a 5% significance level and the corresponding critical values from the ^-distribution table we may form the following interval:

This expression says that there is a 95% chance that a f-value with 198 degrees of freedom lies between the limits given by the interval. In order to form a confidence interval for our case we substitute the test function (4.6) into (4.9). That will result in the following expression:

which may be transformed in the following way:

which provides a 95% confidence interval for B . Hence, there is a 95% chance that the given interval will cover the true population parameter value. Alternatively, in repeated sampling the interval will cover the population parameter in 95 cases out of 100 on average.

If the confidence interval does not cover the value given by the null hypothesis (zero in this case) we will be able to reject the null hypothesis. By plugging in the values, we receive a confidence interval that may be expressed in the following way:

which in this case equals

0.469 ± 1.96 x 0.213

Remember that, since both the parameter and the corresponding standard errors are estimates based on sample information, the interval is random. One should therefore not forget that it is the interval that with a certain probability will cover the true population parameter, and not the other way around. Two important concepts to remember and distinguish in these circumstances are the confidence level and significance level. They are defined in the following way:

Confidence level

The percent of the times that the constructed confidence interval will include the population parameter. When it is expressed as a percent, it is sometimes called the confidence coefficient.

Significance level

The probability of rejecting a true null hypothesis.

Hence, before being able to construct a confidence interval we have to pick a significance level, which is usually set to 5 percent. Given the significance level we know that the confidence level of our test or corresponding interval will be 95 percent. The significance level is often denoted with the Greek letter a, which implies that the confidence level equals 1-a.

P-value in hypothesis testing

Most econometric software that produces regression output report p-values related to each estimated parameter. To investigate the p-value is a fast way to reach the conclusion that we otherwise would receive by carrying out all the steps in the test of significance approach or the confidence interval approach. By looking at the p-value we can directly say if the parameter is significantly different from zero or not.

The P-value for sample outcome

The P-value for a sample outcome is the probability that the sample outcome could have been more extreme than the observed one.

If the P-value equals or is greater then the specified significance level: H0 is concluded. If the P-value is less than the specified significance level: H1 is concluded.

Regression output from Excel

Table 4.1 Regression output from Excel

In Table 4.1 we have an example of how regression output could look like. This particular output is generated using MS Excel, but most statistical software offers this information in their output. The example is based on a random sample of 10 observations.

Observe that regression outputs always assumes a two sided test. That has implications on the P-value. The P-Value in this particular case can therefore be calculated as

The P-value from a one-sided hypothesis would therefore be

Since the P-value for the intercept is larger than any conventional significance levels, say 5 percent, we can not reject the null hypothesis that the intercept is different from zero. For the slope coefficient on the other hand the P-value is much smaller than 5 percent and therefore we can reject the null hypothesis and say that it is significantly different from zero.

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