Regression Model Estimator

[mrkb80]

Homework Statement

Assume regression model $y_i = \alpha + \beta x_i + \epsilon_i$ with $E[\epsilon_i] = 0, E[\epsilon^2] = \sigma^2, E[\epsilon_i \epsilon_j] = 0$ where $i \ne j$. Suppose that we are given data in deviations from sample means.

If we regress $(y_i-\bar{y})$ on $(x_i-\bar{x})$ without a constant term, what is the expected value of the least squares estimator of the slope coefficient?

Homework Equations

The Attempt at a Solution

I was thinking I could start with $S(\beta)=\Sigma (y_i-x_i \beta)^2$ and replace y and x with $(y_i-\bar{y})$ and $(x_i-\bar{x})$ and then take FOC to get the estimator.

 

[mighty2000]

However, I'm not sure how to proceed from there.You are on the right track. The first step would be to substitute (y_i-\bar{y}) and (x_i-\bar{x}) into the sum of squares equation, which would give you:

S(\beta) = \sum (y_i - \bar{y} - \beta(x_i - \bar{x}))^2

= \sum (y_i - \beta x_i + \beta \bar{x} - \bar{y})^2

= \sum (y_i - \beta x_i)^2 + \beta^2 \sum (x_i - \bar{x})^2 + \bar{y}^2 + 2\beta \bar{x} \sum (y_i - \beta x_i) - 2\bar{y} \sum (y_i - \beta x_i)

Next, you can take the first derivative of S(\beta) with respect to \beta and set it equal to 0 to find the FOC:

\frac{\partial S(\beta)}{\partial \beta} = -2 \sum (y_i - \beta x_i)x_i + 2\beta \sum (x_i - \bar{x})^2 + 2\bar{x} \sum (y_i - \beta x_i) = 0

Simplifying this equation, you get:

\beta \sum x_i^2 - \sum x_i y_i + \beta \sum x_i^2 - \beta \sum \bar{x}x_i + 2\beta \bar{x} \sum x_i - \bar{x} \sum y_i = 0

Now, you can use the given assumptions to simplify this equation further. Since E[\epsilon_i] = 0, we know that \bar{y} = \alpha + \beta \bar{x}. Substituting this into the equation, we get:

2\beta \sum x_i^2 - \sum x_i y_i + 2\beta \bar{x} \sum x_i - \bar{x} (n\alpha + \beta \sum x_i) = 0

Using the assumption that E[\epsilon_i \epsilon_j] = 0 for i \ne j, we know that \sum x_i \epsilon_i = 0. Substituting this into the