Consider a situation where economic theory suggests that you impose certain restrictions on your estimated multiple regression function

These may involve the equality of parameters, such as the returns to education and on the job training in earnings functions, or the sum of coefficients, such as constant returns to scale in a production function. To test the validity of your restrictions, you have your statistical package calculate the corresponding F-statistic. Find the critical value from the F-distribution at the 5% and 1% level, and comment whether or not you will reject the null hypothesis in each of the following cases.
(a) number of observations: 152; number of restrictions: 3; F-statistic: 3.21
(b) number of observations: 1,732; number of restrictions:7; F-statistic: 4.92
(c) number of observations: 63; number of restrictions: 1; F-statistic: 2.47
(d) number of observations: 4,000; number of restrictions: 5; F-statistic: 1.82
(e) Explain why you can use the Fq,? distribution to compute the critical values in (a)-(d).
What will be an ideal response?

Answer:
(a) F3,∞ = 2.60 (5% level), F3,∞ = 3.78 (1% level). Reject the null hypothesis at the 5% level, but not at the 1% level.
(b ) F7,∞ = 2.01 (5% level), F7,∞ = 2.64 (1% level). Reject the null hypothesis at the 5% level and at the 1% level.
(c) F1,∞ = 3.84 (5% level), F1,∞ = 6.63 (1% level). Cannot reject the null hypothesis at the 5% level or at the 1% level.
(d) F5,∞= 2.21 (5% level), F5,∞ = 3.02 (1% level). Cannot reject the null hypothesis at the 5% level or at the 1% level.
(e) The F-statistic is distributed Fq,∞ in large samples. Although strictly speaking this only holds for the limiting case of n = ∞, for practical purposes the approximation is close for n > 100. This is therefore problematic for (c) above, where n = 63.