For Y i= 0+ 1x i2+ i, i=1,…,n≥2, find the OLS slope estimate. Sums run over i.

$[\sum x_i^2y_i-(\sum y_i)(\sum x_i^2)]/[\sum x_i^4-(\sum x_i^2)^2]$

$[\sum x_i^2y_i-n(\sum y_i)(\sum x_i^2)]/[\sum x_i^4-n(\sum x_i^2)^2]$

GATE 2020 ST – Question 22

Regression Analysis · Linear Models and Estimation · 1 mark · Multiple choice

For $Y_i=\beta_0+\beta_1x_i^2+\epsilon_i$, i=1,…,n≥2, find the OLS slope estimate. Sums run over i.

$[n\sum x_i^2y_i-(\sum y_i)(\sum x_i^2)]/[n(\sum x_i^4-(\sum x_i^2)^2)]$
$[n\sum x_i^2y_i-(\sum y_i)(\sum x_i^2)]/[n\sum x_i^4-(\sum x_i^2)^2]$
$[\sum x_i^2y_i-(\sum y_i)(\sum x_i^2)]/[\sum x_i^4-(\sum x_i^2)^2]$
$[\sum x_i^2y_i-n(\sum y_i)(\sum x_i^2)]/[\sum x_i^4-n(\sum x_i^2)^2]$

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Correct answer: (B) $[n\sum x_i^2y_i-(\sum y_i)(\sum x_i^2)]/[n\sum x_i^4-(\sum x_i^2)^2]$

Explanation

Treat zi=xi² as the regressor. With an intercept, slope is Σ(zi−z̄)(yi−ȳ)/Σ(zi−z̄)². Multiply numerator and denominator by n and substitute zi=xi² to obtain B. The fourth powers arise because zi²=xi⁴. A nonzero regressor spread is implicit for unique estimation.

GATE 2020 ST – Question 22

Explanation

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