For what value of alpha (α), the BLUE of beta1 (β₁) and beta2 (β₂) are uncorrelated for the model| iss previous year solution paper-2 q.no.1

ISS PREVIOUS YEAR 2016 SOLUTION PAPER-2 SET A

Question: 1

For what value of alpha (α), the BLUE of beta1 (β₁) and beta2 (β₂) are uncorrelated for the model

E(y1) = 2β₁ + β₂

E(y2) = β₁ − β₂

E(y3) = β₁ + αβ₂

Options:

(a) −1 (b) 0 (c) 1 (d) 2

Solution

The model can be written in matrix form as

Y = Xβ + ε

where

β = (β₁ , β₂)'

and the design matrix X is

X =[ 2 1 ][ 1 −1 ][ 1 α ]

The covariance matrix of the BLUE estimator is

Var(β̂) = σ² (X'X)^(-1)

First compute X'X.

X'X =

[ 2² + 1² + 1² 2×1 + 1×(−1) + 1×α ][ 2×1 + 1×(−1) + 1×α 1² + (−1)² + α² ]

Now simplify.

2² + 1² + 1² = 6

2×1 + 1×(−1) + 1×α = 2 − 1 + α = 1 + α

1² + (−1)² + α² = 1 + 1 + α² = 2 + α²

Thus,

X'X =

[ 6 1 + α ][ 1 + α 2 + α² ]

The covariance between β̂₁ and β̂₂ is proportional to the off-diagonal element of (X'X)^(-1).

For a 2×2 matrix

[ a b ][ b c ]

the inverse is

(1/(ac − b²)) ×

[ c −b ][ −b a ]

Thus,

Cov(β̂₁ , β̂₂) ∝ −(1 + α)

For β̂₁ and β̂₂ to be uncorrelated, the covariance must be zero.

Therefore,

1 + α = 0

α = −1

Final Answer

The value of α for which the BLUE of β₁ and β₂ are uncorrelated is

α = −1

Correct option: (a) −1.

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ISS PREVIOUS YEAR 2016 SOLUTION PAPER-1 SET A Q.NO-1