To test a single parameter of a regression function we assume it satisfies the linear model assumptions, and seek to reject the null hypothesis.
Testing hypotheses about a single population parameter and testing multiple linear restrictions.
A statistical hypothesis is an assumption about a population parameter. The population mode
y = β0+ β1×1 + …. β ᴋXᴋ + μ
To test the hypothesis about any single parameter in the population regression function, we have to assume that it satisfies the classical linear model assumptions. βj are unknown parameters and will never be known with certainty; Nevertheless, we can hypothesize about the value of βj and then use statistical inference to test our hypothesis (Wooldridge, 2015). In most applications our interest lies in testing the null hypothesis denoted as:
H0: βj = 0
Since Bj measures the partial effect of of Xj on the expected value of y, after controlling for all other variables, H0: βj = 0 means that, once x1, x2, …. Xj-1, Xj+1….,Xk have been accounted for, Xj has no effect on the expected value of y. The statistic used to test the null hypothesis is called the “t-statistic or the t ratio.” (Wooldridge, 2015)
The rival of the null hypothesis is the alternative hypothesis. To test for multiple linear restrictions, we use the F-test and the test is known as multiple hypothesis test.
Murad, A. (1964). What Keynes Means, New Haven: Bookman Associates Inc and College University Press.
Smith, G. (2011). Essential Statistics, Regression, and Econometrics, California: Elsevier
Wooldridge, J., M. (2015) Introductory Econometrics: A modern Approach,