ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A Q.NO. 5

ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A

ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A

Question

In the Gauss-Markov linear model, letŷ denote the vector of fitted values andê denote the vector of residuals.

Consider the following statements:

1. The components of ŷ are pairwise uncorrelated.

2. The components of ê are pairwise uncorrelated.

Which of the above statements is/are correct?

(a) 1 only(b) 2 only(c) Both 1 and 2(d) Neither 1 nor 2

Solution

We consider the standard linear model:

Y = Xβ + ε

where E(ε) = 0 and Var(ε) = σ²I

Step 1: Fitted values (ŷ)

The fitted values are given by:

ŷ = HY

where H = X(X'X)⁻¹X' is the hat matrix

Now,

Var(ŷ) = Var(HY) = H Var(Y) H'

Since Var(Y) = σ²I,

Var(ŷ) = σ² H H'

But H is symmetric and idempotent:

H' = H and H² = H

So,

Var(ŷ) = σ² H

Important Observation:

The covariance between components of ŷ depends on off-diagonal elements of H.

Since H is not diagonal in general, its off-diagonal elements are non-zero.

• Therefore: Components of ŷ are NOT pairwise uncorrelated

• Statement 1 is FALSE

Step 2: Residuals (ê)

Residuals are:

ê = Y − ŷ = (I − H)Y

Let M = (I − H)

Then,

Var(ê) = Var(MY) = M Var(Y) M'

= σ² M M'

Since M is symmetric and idempotent:

M' = M and M² = M

So,

Var(ê) = σ² M

Important Observation:

Matrix M = (I − H) is also not diagonal in general

So, off-diagonal elements ≠ 0

Therefore: Components of residuals are NOT pairwise uncorrelated

Statement 2 is FALSE

Final Answer

Both statements are incorrect.

Correct option: (d) Neither 1 nor 2

Exam Insight

Very important concept:

• ŷ and ê are uncorrelated with each other ✔️

• But their individual components are NOT uncorrelated ❌

This is a favorite UPSC ISS concept-based trap question

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