It's not true that, with OLS, only the dependent variable, $Y$, is random. In fact, both can be random variables, however OLS is centered on minimizing the mean squared deviation between $Y$ and $X\beta$, conditioned on the variable $X$. That is, if your equation is $$Y = X\beta + e$$ where $e$ is a vector of IID components, then the least squares problem is to find the vector $X$ such that $$\mathbb{E}[(Y- X\beta)^2\mid X]$$ is minimized.

In this sense, we can often "treat" the random variable $X$ as being "non-random", since we can pull it out of the operator $\mathbb{E}[ \cdot \mid X]$, however it still can be a random variable.

ORM has to do with dealing with cases where both $X$ and $Y$ contain measurment error. That is, when $X = X + \epsilon$ and $Y = y + \mu$ where $x,y$ are unknown fixed constants and $\epsilon, \mu$ are independent random variables.

It's not true that, with OLS, only the dependent variable, $Y$, is random. In fact, both can be random variables, however OLS is centered on minimizing the mean squared deviation between $Y$ and $X\beta$, conditioned on the variable $X$. That is, if your equation is $$Y = X\beta + e$$ where $e$ is a vector of IID components, then the least squares problem is to find the vector $\beta$ such that $$\mathbb{E}[(Y- X\beta)^2\mid X]$$ is minimized.

In this sense, we can often "treat" the random variable $X$ as being "non-random", since we can pull it out of the operator $\mathbb{E}[ \cdot \mid X]$, however it still can be a random variable.

ORM has to do with dealing with cases where both $X$ and $Y$ contain measurment error. That is, when $X = X + \epsilon$ and $Y = y + \mu$ where $x,y$ are unknown fixed constants and $\epsilon, \mu$ are independent random variables.

It's not true that, with OLS, only the dependent variable, $Y$, is random. In fact, both can be random variables, however OLS is centered on minimizing the mean squared deviation between $Y$ and $X\beta$, conditional on the variable $X$. That is, if you're equation is $$Y = X\beta + e$$ where $e$ is a vector of IID components, then the least squares problem is to find the vector $X$ such that $$\mathbb{E}[(Y- X\beta)^2\mid X]$$ is minimized.

In this sense, we can often "treat" the random variable $X$ as being "non-random", since we can pull it out of the operator $\mathbb{E}[ \cdot \mid X]$, however it still can be a random variable.

ORM has to do with dealing with cases where both $X$ and $Y$ contain measurment error. That is, when $X = X + \epsilon$ and $Y = y + \mu$ where $x,y$ are unknown fixed constants and $\epsilon, \mu$ are independent random variables.

It's not true that, with OLS, only the dependent variable, $Y$, is random. In fact, both can be random variables, however OLS is centered on minimizing the mean squared deviation between $Y$ and $X\beta$, conditioned on the variable $X$. That is, if your equation is $$Y = X\beta + e$$ where $e$ is a vector of IID components, then the least squares problem is to find the vector $X$ such that $$\mathbb{E}[(Y- X\beta)^2\mid X]$$ is minimized.

In this sense, we can often "treat" the random variable $X$ as being "non-random", since we can pull it out of the operator $\mathbb{E}[ \cdot \mid X]$, however it still can be a random variable.

ORM has to do with dealing with cases where both $X$ and $Y$ contain measurment error. That is, when $X = X + \epsilon$ and $Y = y + \mu$ where $x,y$ are unknown fixed constants and $\epsilon, \mu$ are independent random variables.