Question

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)

Answer 1

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