The previous sections have developed, through point and interval estimation, a method to infer a population value from a sample. Hypothesis testing constitutes another method of inference which consists of formulating some assumptions about the probability distribution of the population from which the sample was extracted, and then trying to verify these assumptions for them to be considered adequate. In this sense, hypothesis testing can refer to the systematic component of the model as well as its random component. Some of these procedures will be studied in the following chapter of this book, whilst in this section we only focus on linear hypotheses about the coefficients and the parameter of dispersion of the MLRM.
In order to present how to compute hypothesis testing about the coefficients, we begin by considering the general statistic which allows us to test any linear restrictions on . Afterwards, we will apply this method to particular cases of interest, such as the hypotheses about the value of a coefficient, or about all the coefficients excepting the intercept.
In order to test any linear hypothesis about the coefficient, the problem is formulated as follows:
The matrix and the vector can be considered as artificial instruments which allow us to express any linear restrictions in matrix form. To illustrate the role of these instruments, consider an MLRM with 4 coefficients. For example, if we want to test
Expression (2.138) includes the unknown parameter , so in order to obtain a value for the statistic, we have to use the independence between the quadratic form given in (2.138), and the distribution (2.125) is (see Hayashi (2000)), in such a way that:
A way of solving this question consists of employing the so-called p-value provided by a sample in a specific test. It can be defined as the lowest significance level which allows us to reject , with the available sample:
It we use the p-value, the decision rule is modified in stages and as follows: to calculate the p-value, and if , is accepted. Otherwise, it is rejected.
Econometric softwar does not usually contain the general F-statistic, except for certain particular cases which we will discuss later. So, we must obtain it step by step, and it will not always be easy, because we have to calculate the inverses and products of matrices. Fortunately, there is a convenient alternative way involving two different residual sum of squares (): that obtained from the estimation of the MLRM, now denoted (unrestricted residual sum of squares), and that called restricted residual sum of squares, denoted . The latter is expressed as:
If we use (2.140) to test a linear hypothesis about , we only need to obtain the corresponding to both the estimation of the specified MLRM, and the estimation once we have substituted the linear restriction into the model. The decision rule does not vary: if is true, should not be much different from , and consequently, small values of the statistic provide evidence in favour of .
Having established the general F statistic, we now analyze the most useful particular cases.
To obtain (2.142) we must note that, under , the matrix becomes a row vector with zero value for each element, except for the element which has 1 value, and . Thus, the term becomes . Element becomes .
Moreover, we know that the squared root of the F random variable expressed in (2.142) follows a t-student whose degrees of freedom are those of the denominator of the F distribution, that is to say,
It must be noted that, given the form of in (2.141), (2.143) is a two-tailed test, so once we have calculated the statistic value , is rejected if .
An interesting particular case of the t-statistic consists of testing , which simplifies (2.143), yielding:
The statistic given in (2.143) is the same as (2.122), which was derived in order to obtain the interval estimation for a coefficient. This leads us to conclude that there is an equivalence between creating a confidence interval and carrying out a two-tailed test of the hypothesis (2.141). In this sense, the confidence interval can be considered as an alternative way of testing (2.141). The decision rule will be: given a fixed level of significance and calculating a percent confidence interval, if the value in ( ) belongs to the interval, we accept the null hypothesis, at a level of significance . Otherwise, should be rejected. Obviously, this equivalence holds if the significance level in the interval is the same as that of the test.
Nevertheless, the F statistic (2.148) has an alternative form as a function of the explained sum of squares . To prove it, we begin by considering:
Furthermore, from the definition of given in (2.130) we can deduce that:
We must note that the equivalence between (2.148) and (2.151) is only given when the MLRM has a constant term.
The earlier mentioned relationship between the confidence interval and hypothesis testing, allows us to derive the test of the following hypothesis easily:
Now, we present the quantlet linreg in the stats quantlib which allows us to obtain the main measures of fit and testing hypothesis that we have just described in both this section and the previous section.
For the example of the consumption function which we presented in previous sections, the quantlet XEGmlrm05.xpl obtains the statistical information
The column represents the squared sum of the regression (ESS), the squared sum of the residuals (RSS) and the total squared sum (TSS). The column represents the means of calculated by dividing by the corresponding degrees of freedom(df). The F-test is the statistic to test , which is followed by the corresponding p-value. Afterwards, we have the measures of fit we presented in the previous section, that is to say, , adjusted-( ), and Standard Error (SER). Moreover, multiple R represents the squared root of .
Finally, the output presents the columns of the values of the estimated coefficients (beta) and their corresponding standard deviations (SE). It also presents the t-ratios (t-test) together with their corresponding p-values. By observing the p-values, we see that all the p-values are very low, so we reject , whatever the significance level (usually 1, 5 or 10 percent), which means that all the coefficients are statistically significant. Moreover, the p-value of the F-tests also allows us to conclude that we reject , or in other words, the overall regression explains the variable. Finally, with this quntlet it is also possible to illustrate the computation of the F statistic to test the hypothesis .