# How parameters formulated for Simple Regression Model

I am reading Simple Regression Model from this book, Section 6.5 (page 267 in downloaded pdf, 276 if viewed online).

The author starts with below equation for a simple linear regression model,

$$Y_i = \alpha_1 + \beta x_i + \varepsilon_i$$

And then after few lines, he lets for conveience that, $$\alpha_1 = \alpha - \beta\overline{x}$$ so that,

$$Y_i = \alpha + \beta(x_i - \overline{x}) + \varepsilon_i$$

where $$\overline{x} = \dfrac{1}{n}\sum\limits_{i=1}^nx_i$$

My questions:
1. It is not convincing to bring in $$\overline{x}$$ just for convenience sake in the equation. Can any one please explain the logic behind bring that in the equation?
2. After above equation, the author says, $$Y_i$$ is equal to a nonrandom quantity, $$\alpha + \beta(x_i - \overline{x})$$, plus a mean zero normal random variable $$\varepsilon_i$$. Does that mean, $$\alpha + \beta(x_i - \overline{x})$$ has no randomness involved in that?

Kindly help.

1. $$\alpha_1$$s in two equations are different. Let $$\alpha_2$$ be the $$\alpha$$ in the second equation, then $$\alpha_1 = \alpha_2 + \beta \bar x$$
At the time that the computer was not popular or had no computer, the line was fit by using calculators. Bringing in $$\bar x$$ is really simplified the computation.
1. From the first equation, $$\epsilon$$ is the only random component. So source of randomness of $$Y$$ is $$\epsilon$$, the other parts $$\alpha + \beta x$$ are known or unknown constant.
• I just corrected $\alpha_1$ to $\alpha$ in 2nd equation. Still the reason is not convincing that it simplified the computation. Can you kindly elaborate further? How could $\overline{x}$ suddenly enter the equation without an associated mathematical logic. Oct 20, 2018 at 18:32
• Let $z_i=x_i-\bar x$, then (1) $\sum z_i = 0$ vs calculating $\sum x_i$, (2) $\sum z_i^2$ is easier easier than $\sum x_i^2$, and (3) $\sum z_iY_i$ is easier easier than $\sum x_iY_i$. introducing $\bar x$ into equation does not change anything in equation, similar to $+ a - a$ , which we used to proof something in math. Oct 20, 2018 at 18:43
• $Y_i = \alpha_1 + \beta x_i + \varepsilon_i$ ==> $Y_i = \alpha_1 + \beta x_i + \varepsilon_i - \beta \bar x + \beta \bar x$ ==> $Y_i = (\alpha_1 +\beta \bar x) + \beta (x_i - \bar x) + \varepsilon_i$ ==> $Y_i = \alpha + \beta (x_i - \bar x) + \varepsilon_i$ Oct 20, 2018 at 18:56