Question

 

3. The model

y_t = β₀ + β₁x_1t + β₂x_2t + β₃x_3t + u_t

was estimated by ordinary least squares from 26 observations. The results were

yˆ_t = 2 + 3.5x_1t − 0.7x_2t + 2.0x_3t

(1.9) (2.2)    (1.5)

t-ratios are in parentheses and R² = 0.982. The same model was estimated with the restriction β1 = β2. Estimates are

y_ˆt = 1.5 + 3(x_1t + x_2t) − 0.6x_3t R² = 0.876

(2.7) (2.4)

(a) Test the significance of the restriction β₁= β₂. State the assumptions under which the test is valid.

(b) Suppose that x_2t were dropped from the equation: would the R² rise or fall?

(c) Would the R² rise or fall if x_2t were dropped?

Answer 1

(a)

H0: β₁= β₂

H1: β₁ β₂

We will use F test to test for linear restrictions in linear model.

F Statistic is given as,

F = [(R²_U - R²_R )/q ] / [(1 - R²_U) / (n-k-1)]

where

R²_U , R²_R  are R² value of the unrestricted model (model 1 with  β₁ β₂ ) and the restricted model (model 2 with  β₁ = β₂ )

q is number of restrictions in the null hypothesis. Here q = 1

n is number of observations. n = 26

k is number of explanatory variables in unrestricted model. Here k = 3

F = [(0.982- 0.876)/1 ] / [(1 - 0.982) / (26 -3 -1)] = 129.56

Degree of freedom = q, n-k-1 = 1, 26-3-1 = 1, 22

Critical value of F at 5% significance level and df = 1, 24 is 4.30

Since the observed F (129.56) is greater than the critical value, we reject the null hypothesis H0 and conclude that there is strong evidence that β₁ β_2.

Assumptions - For the F test all of the assumptions of the Gaussian-noise simple linear regression model hold: the true model must be linear, the errors around it must be Gaussian, the error variance must be constant, the error must be independent of explanatory variables and independent across measurements.

(b)

If any variable is dropped from the model, R² of the model will always decrease/fall.

(c)

R² will fall if variable x_2t were dropped from the model.