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GATE 2023 ST – Question 58

Regression Analysis · Linear Models and Estimation · 2 marks · Multiple select

For t=1,…,100,consider y_t=α₀+α₁t+α₂t²+ε_t,where ε_t are iid N(μ,1) and μ is unknown. Which parameters have unbiased estimators?

  1. α₁
  2. α₂
  3. α₀
  4. μ

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Show answer and explanation

Correct answer: (A) α₁; (B) α₂

Explanation

The mean response is (α₀+μ)+α₁t+α₂t². The three polynomial design columns have full rank,so least squares estimates α₁,α₂ unbiasedly. But replacing (α₀,μ) by (α₀+c,μ−c) leaves the entire data law unchanged,so neither separate parameter is identifiable or can have an unbiased estimator for all values. **A,B**.

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