The lecture is divided in two parts:
in the first part, we discuss hypothesis testing in the normal linear regression model, in which the OLS estimator of the coefficients has a normal distribution conditional on the matrix of regressors;
in the second part, we show how to carry out hypothesis tests in linear regression models where the OLS estimator can be proved to be asymptotically normal.
In both parts, the regression model iswhere is an output variable, is a vector of inputs, is a vector of coefficients and is an error term. There are observations in the sample, so that .
We also denote:
by the vector of outputs
by the matrix of inputs
by the vector of errors
Using this notation, we can write
Moreover, the OLS estimator of is
We assume that the design matrix has full-rank, so that the matrix is invertible.
Table of contents
In this section we derive tests about the coefficients of the normal linear regression model. In this model the vector of errors is assumed to have a multivariate normal distribution conditional on , with mean equal to and covariance matrix equal towhere is the identity matrix and is a positive constant.
It can be proved (see the lecture about the normal linear regression model) that the assumption of conditional normality implies that:
the OLS estimator is conditionally multivariate normal with mean and covariance matrix ;
the adjusted sample variance of the residualsis an unbiased estimator of ; furthermore, it has a Gamma distribution with parameters and ;
is conditionally independent of .
In a test of a restriction on a single coefficient, we test the null hypothesiswhere is the -th entry of the vector of coefficients and .
In other words, our null hypothesis is that the -th coefficient is equal to a specific value.
This hypothesis is usually tested with the test statisticwhere is the -th diagonal entry of the matrix .
The test statistic has a standard Student's t distribution with degrees of freedom. For this reason, it is called a t statistic and the test is called a t test.
Under the null hypothesis has a normal distribution with mean and variance . As a consequence, the ratiohas a standard normal distribution (mean and variance ). We can writeSince has a Gamma distribution with parameters and , the ratiohas a Gamma distribution with parameters and . It is also independent of because is independent of . Therefore, the ratiohas a standard Student's t distribution with degrees of freedom (see the the lecture on the Student's t distribution).
The null hypothesis is rejected if falls outside the acceptance region.
How the acceptance region is determined depends not only on the desired size of the test, but also on whether the test is two-tailed (if we think that could be both smaller or larger than ) or one-tailed (if we assume that only one of the two things, i.e., smaller or larger, is possible). For more details on how to determine the acceptance region, see the glossary entry on critical values.
When testing a set of linear restrictions, we test the null hypothesiswhere is a matrix and is a vector. is the number of restrictions.
Example Suppose that is and that we want to test the hypothesis . We can write it in the form by setting
Example Suppose that is and that we want to test whether the two restrictions and hold simultaneously. The first restriction can be written asSo we have
This hypothesis is usually tested with the test statisticwhich has an F distribution with and degrees of freedom. For this reason, it is called an F statistic and the test is called an F test.
Under the null and conditional on , the vector , being a linear transformation of the normal random vector , has a multivariate normal distribution with meanand covariance matrixThus, we can writeSince is multivariate normal, the quadratic form has a Chi-square distribution with degrees of freedom (see the lecture on quadratic forms involving normal vectors). Furthermore, since has a Gamma distribution with parameters and the statistichas a Chi-square distribution with degrees of freedom (see the lecture on the Gamma distribution). Thus, we can writeThus is a ratio between two Chi-square variables, each divided by its degrees of freedom. The two variables are independent because depends only on and depends only on , and and are independent. As a consequence, has an F distribution with and degrees of freedom (see the lecture on the F distribution).
The F test is usually one-tailed. A critical value in the right tail of the F distribution is chosen so as to achieve the desired size of the test. Then, the null hypothesis is rejected if the F statistics is larger than the critical value.
When you use a statistical package to run a linear regression, you often get a regression output that includes the value of an F statistic. Usually this is obtained by performing an F test of the null hypothesis that all the regression coefficients are equal to (except the coefficient on the intercept).
As we explained in the lecture entitled Linear regression - maximum likelihood, the maximum likelihood estimator of the vector of coefficients of a normal linear regression model is equal to the OLS estimator . As a consequence, all the usual tests based on maximum likelihood procedures (e.g., Wald, Lagrange multiplier, likelihood ratio) can be employed to conduct tests of hypothesis about .
In this section we explain how to perform hypothesis tests about the coefficients of a linear regression model when the OLS estimator is asymptotically normal.
As we have shown in the lecture entitled OLS estimator properties, in several cases (i.e., under different sets of assumptions) it can be proved that:
the OLS estimator is asymptotically normal, that is,where denotes convergence in distribution (as the sample size tends to infinity), and is a multivariate normal random vector with mean and covariance matrix ; the value of the matrix depends on the set of assumptions made about the regression model;
it is possible to derive a consistent estimator of , that is,where denotes convergence in probability (again as tends to infinity). The estimator is an easily computable function of the observed inputs and outputs .
These two properties are used to derive the asymptotic distribution of the test statistics used in hypothesis testing.
In a z test the null hypothesis is a restriction on a single coefficient:where is the -th entry of the vector of coefficients and .
The test statistic iswhere is the -th diagonal entry of the estimator of the asymptotic covariance matrix.
The test statistic converges in distribution to a standard normal distribution as the sample size increases. For this reason, it is called a z statistic (because the letter z is often used to denote a standard normal distribution) and the test is called a z test.
We can write the z statistic asBy assumption, the numerator converges in distribution to a normal random variable with mean and variance . The estimated variance converges in probability to , so that, by the Continuous Mapping theorem, the denominator converges in probability to . Thus, by Slutsky's theorem, we have that converges in distribution to the random variablewhich is normal with meanand varianceTherefore, the test statistic converges in distribution to , which is a standard normal random variable.
When is large, we approximate the actual distribution of with its asymptotic one (standard normal). We then employ the test statistic in the usual manner: based on the desired size of the test and on the distribution of , we determine the critical value(s) and the acceptance region. The null hypothesis is rejected if falls outside the acceptance region.
In a Chi-square test, the null hypothesis is a set of linear restrictionswhere is a matrix and is a vector.
The test statistic iswhich converges to a Chi-square distribution with degrees of freedom. For this reason, it is called a Chi-square statistic and the test is called a Chi-square test.
We can write the test statistic asBy the assumptions on the convergence of and , and by the Continuous Mapping theorem, we have thatBy Slutsky's theorem, we haveBut is multivariate normal with meanand varianceThus,but, by standard results on normal quadratic forms, the quadratic form on the right hand side has a Chi-square distribution with degrees of freedom ( is the dimension of the vector )
When setting up the test, the actual distribution of is approximated by the asymptotic one (Chi-square).
Like the F test, also the Chi-square test is usually one-tailed. The desired size of the test is achieved by appropriately choosing a critical value in the right tail of the Chi-square distribution. The null is rejected if the Chi-square statistics is larger than the critical value.
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