Uncertainty of a 'forced' slope for a linear regression

Question

Let $\mathbf{Y} = (Y_1,...,Y_n)$ and $\mathbf{X} = (X_1,...,X_n)$ be two vectors of observations from variables $Y$ and $X$ respectively.

First, a linear regression model is used to find a relationship between the variables $Y$ and $X$. Thanks to $\mathbf{Y}$ and $\mathbf{X}$ we obtain a slope $\hat{m}$ and an intercept value $\hat{b}$ such that $(\hat{m},\hat{b})$ is given by the LSE and is an estimator of $(m,b)$ in the following equation: $$ Y = mX + b \ \ \ \ \ \ \ (1). $$ In a second step, if $\hat{b}$ is lower than a certain value $b^*$, then we will suppose that the intercept term $b$ is null and that the estimated values $\hat{Y}$ of $Y$ are simply: \begin{equation} \hat{Y} = \hat{m}X \ \ \ \ \ \ \ (2). \end{equation}

The goal is to find the (relative) standard deviation of the $\hat{Y}$ of equation (2) knowing that we used the slope given by the LSE of equation (1). Does anyone has an idea how to process?

Answer 1

1. You theoretical model is $Y = b + mX + \epsilon$, where $var(\epsilon|X) = \sigma^2$. The OLS for this model is $$ \hat m = \frac{\sum (Y_i - \bar{Y}) (X_i - \bar{X})}{\sum ( X_i - \bar{X}) ^ 2} = \frac{\sum (X_i - \bar{X})Y_i}{\sum ( X_i - \bar{X}) ^ 2}, $$ thus $$ var( \hat m |X) = \frac{\sum (X_i - \bar{X}) ^ 2 var(Y_i|X)}{(\sum (X_i - \bar X)^ 2)^2} = \frac{\sigma^2}{\sum ( X_i - \bar X) ^ 2}. $$
2. Hence, $var(\hat Y|X) = var(X \hat m ) = \frac{X^2\sigma ^ 2}{\sum (X_i - \bar X)^ 2}$.