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Unformatted text preview: 0 using the model above? . 2. Let
y1 = β1 + β2 + ε1
y2 = β1 – β2 + ε2
y3 = β1 + 2β2 + ε3
y4 = 2β1 + 3β2 + ε4
y5 = 3β1 – 4β2 + ε5
where the εi’s are independent and identically distributed normal random
variables with mean 0 and variance σ2. The observed values of yi’s are given as
follows: y1 = 5.2, y2 = −2.6, y3 = 7.2, y4 = 9.6 and y5 = −10.
(i) (ii) Express the above model in the matrix form y = Xβ + ε, where y = (y1, y2, ⋯,
y5)′, β = (β1, β2)′ and ε = (ε1, ε2, ..., ε5)′. Obtain the least squares estimates for
β1 and β2.
Find the sum of squares error (SSE) and its degrees of freedom. (iii) Find estimate of
(iv) Test the hypothesis H0: β1 = 0 against H1: β1 ≠ 0 at the 5% significance level. 1 3. Let yi and εi be random variables such that
yi = β0 + β1 (−1)i+1 + εi, i = 1, 2, ⋯, 12,
where β0, β1 are unknown parameters and εi’s are independent and distributed as
Express the above model in the matrix form y = Xβ + ε, with β = (β0, β1)′.
Obtain X′X and X′y and hence find the least squares estimators of β0 and β1.
(ii) Show that the
(iii) Show that the F-test for testing H0: β1 = 0 against H1: β1 ≠ 0, rejects H0
(i) where is the least squares estimator of β1. Answers to selected problems
1. (ii) ,
, s.e. of 2. (i)
3. (i) (ii) Rewrite ; (iii) ; (iv) s.e. of = 0.04714; (vi) (a) tobs = 7.87, (b) Fobs = 62.03. ; (ii) SSE = 3.2188 with 3 d.f.; (iv) tobs = 15.99.
, , and 2...
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