In: Statistics and Probability
In a multivariate regression like Yi = α + β1X1i + β2X2i + β3X3i + i , explain how do we test the following three null hypothesis: H1 : β2 = 0, H2 : β3 = β2, and H3 : β1 = β2 = β3 = 0 (do not forget the name of the test and distribution under the null)
1) H1 : β2 = 0,
To test this hypothesis we use t-test
under H0
t-statistic= /SE()
The t-statistic has n – k – 1 degrees of freedom where k = number of independents variable here it is 3
The calculated value of this test statistic is compared to a critical value from the t-(df= n – k – 1) distribution.
If calculated value > a critical value we reject H0
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2) H2 : β3 = β2
To test this hypothesis we use t-test
under H0
t-statistic= (-)/(SE()^2 +SE()^2)
The t-statistic has n – k – 1 degrees of freedom where k = number of independents variable here it is 3
The calculated value of this test statistic is compared to a critical value from the t-(df= n – k – 1) distribution.
If calculated value > a critical value we reject H0
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3) H3 : β1 = β2 = β3 = 0
To test this hypothesis we use F-test
through this we test overall significance of model
F=(SST-SSE)/k-1/(SSE/(T-K))
The calculated value of this test statistic is compared to a critical value from the F(K-1,T-K) distribution.
If calculated value > a critical value we reject H0