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Picture Math

the "picture" is part of the demonstration of the formula of Variance of the slope coefficient in Linear Regression, my question is why the first part in circle is equal to Const?

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    $\begingroup$ I think the constant refers to it not being a random variable (containing $\epsilon$). $\endgroup$ – Jirapat Samranvedhya Aug 17 '17 at 20:06
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Note that \begin{align} \hat{\beta}_0 &= \beta_0 \sum \frac{1}{n} + \beta_1 \bar{x} + \beta_0 \bar{x} \sum w_i - \beta_1 \bar{x} \sum x_i w_i + \sum (x_i - \bar{x}w_i)\epsilon_i\\ &= \beta_0 + \beta_1 \bar{x} + 0 - \beta_1\bar{x} + \sum (x_i - \bar{x}w_i)\epsilon_i\\ &=\beta_0 + \sum (x_i - \bar{x}w_i)\epsilon_i\\ & = \beta_0 + \sum v_i\epsilon_i\\ & = \beta_0 + o_p(1). \end{align} When $\beta_0$ is parameter and the "error term" converges to $0$ in probability.

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  • $\begingroup$ Thanks a lot @V.V it was easy. you explained it very well. I voted for your answer but the system requires for me to have at least 15 reputation. Best wishes. $\endgroup$ – J.r. Paladines Aug 19 '17 at 14:14
  • $\begingroup$ I think that you can still except it. Anyway, you are welcome. $\endgroup$ – V. Vancak Aug 19 '17 at 14:31

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