Consider the regression through the origin model
y = bx
Assume that there are n pairs of observations of the response and the explanatory variables,
and that these observations are indexed by i. Derive the least-squares estimator for the slope
parameter b in this model.
To derive the least-squares estimator for the slope parameter b in the regression through the origin model y = bx, we will use the method of least squares.
In the regression through the origin model, we assume that the intercept term is zero (i.e., the line passes through the origin). Therefore, we only need to estimate the slope parameter b.
Let's define the observed response variable as yi and the observed explanatory variable as xi, where i = 1, 2, ..., n.
The objective of least squares is to minimize the sum of the squared differences between the observed response variables and the predicted response variables. In other words, we want to minimize the sum of the squared residuals.
The residual for each observation, denoted as ei, is defined as the difference between the observed response variable and the predicted response variable:
ei = yi - bxi.
The sum of squared residuals is given by:
S = Σ(ei)^2 = Σ(yi - bxi)^2.
Our goal is to find the value of b that minimizes S.
To find the least-squares estimator for b, we differentiate S with respect to b and set it equal to zero:
dS/db = -2Σ(xi)(yi - bxi) = 0.
Expanding the equation, we get:
-2Σ(xi)(yi - bxi) = 0,
-2Σ(xi)(yi) + 2bΣ(xi)^2 = 0.
Rearranging the equation to solve for b, we have:
bΣ(xi)^2 = Σ(xi)(yi),
b = Σ(xi)(yi) / Σ(xi)^2.
Finally, the least-squares estimator for the slope parameter b in the regression through the origin model y = bx is given by:
b_hat = Σ(xi)(yi) / Σ(xi)^2.
This is the formula that you can use to calculate the least-squares estimator for the slope parameter b in the regression through the origin model.