Statistical inference

Statistical inference is concerned with the issue of using a sample to say something about the corresponding population. Often we would like to know if a variable is related to another variable, and in some cases we would like to know if there is a causal relationship between factors in the population. In order to find a plausible answer to these questions we need to perform statistical tests on the parameters of our statistical model. In order to carry out tests we need to have a test function and we need to know the sampling distribution of the test function.

In the previous chapter we saw that the estimators for the population parameters were nothing more than weighted averages of the observe values of the dependent variable. That is true for both the intercept and the slope coefficient. Furthermore, the distribution of the dependent variable coincides with the error term. Since the error term by assumption is normally distributed, the dependent variable will be normally distributed as well.

According to statistical theory we know that a linear combination of normally distributed variables is also normally distributed. That implies that the distribution of the two OLS estimators is normally distributed with a mean and a variance. In the previous chapter we derived the expected value and the corresponding variance for the estimators, which implies that we have all the information we need about the sampling distribution for the two estimators. That is, we know that:

Just as for a single variable, the OLS estimators works under the central limit theorem since they can be treated as means (weighted averages) calculated from a sample. When taking the square root of the estimated variances we receive the corresponding standard deviations. However, in regression analysis we call them standard errors of the estimator instead of standard deviations. That is to make it clear that we are dealing with a variation that is due to a sampling error. Since we use samples in our estimations, we will never receive estimates that exactly equal the corresponding population parameter. It will almost always deviate to some extent. The important point to recognize is that this error on average will be smaller the larger the sample become, and converge to zero when the sample size goes to infinity. When an estimator behaves in this way we say that the estimator is consistent as described in the previous chapter.

Hypothesis testing

The basic steps in hypothesis testing related to regression analysis are the same as when dealing with a single variable, described in earlier chapters. The testing procedure will therefore be described by an example.

Example 4.1

Assume the following population regression equation:

Using a sample of 200 observations we received the following regression results:

The regression results in (4.4) present the regression function with the estimated parameters together with the corresponding standard errors within parentheses. It is now time to ask the question: Has X any effect on Y? In order to answer this question we would like to know if the parameter estimate for the slope coefficient is significantly different from zero or not. We start by stating the hypothesis:

In order to test this hypothesis we need to form the test function relevant for the case. We know that the sample estimator is normally distributed with a mean and a standard error. We may therefore transform the estimated parameter according to the null hypothesis and use that transformation as a test function. Doing that we receive:

The test function follows a t-distribution with n-k degrees of freedom, where n is the number of observations and k the number of estimated parameters in the regression equation (2 in this case). It is f-distributed since the standard error of the estimated parameter is unknown and replaced by an estimate of the standard error. This replacement increases the variation of the test function compared to what had been the case otherwise. The test function would have been normal if the standard error have been known. However, since the number of observations is sufficiently large, the extra variation will not be of any major importance. If the null hypothesis is true the mean of the test function will be zero. If that is not the case the test function will receive a large value in absolute terms. Let us calculate the test value using the test function:

The final step in the test procedure is to find the critical value that the test value will be compared with. If the test value is larger than the critical value in absolute terms we reject the null hypothesis. Otherwise, we just accept the null hypothesis and say that it is possible that the population parameter is equal to zero. In order to find the critical value we need a significance level, and it is the test maker that set this level. In this example we choose the significance level to be at the 5% level. Since the degrees of freedom equals 198 the critical value found in most tables for the f-distribution will coincide with the critical value taken from the normal distribution table. In this particular case we receive:

Critical value: tc = 1.96 (4^)

Since the test value is larger than the critical value we reject the null hypothesis and claim that there is a positive relation between X and Y.

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