Question

Consider the regression model $y_{i}=\beta_{0}+i\beta_{1}+\epsilon_{i} (i=1,2,...,n>2)$ where $\beta_{0}$ and $\beta_{1}$ are unknown parameters and $\epsilon_{i}$ are random errors. Let $y_{i}$ be the observed value of $Y_{i}(i=1,2,...,n)$. Using the method of ordinary least squares, the estimate of $\beta_{1}$ is

• A. $\frac{1}{n^{2}-1}[\frac{12}{n}\sum_{i=1}^{n}iy_{i}-6(n+1)\overline{y}]$
• B. $\frac{1}{(n^{2}-1)n}[12\sum_{i=1}^{n}iy_{i}-6(n+1)\overline{y}]$
• C. $\frac{1}{n(n^{2}-1)}[12\sum_{i=1}^{n}iy_{i}-6n\overline{y}]$
• D. $\frac{1}{n(n^{2}-1)}[12\sum_{i=1}^{n}iy_{i}-6\overline{y}]$

Hint:

For regression where $x$ is just an index $1 \dots n$, the variance of $x$ is always $\frac{n^2-1}{12}$. Using this standard result makes calculating least-squares parameters for time-series or indexed data much faster.

Answer 1

The Correct Option is A