# Regression problem with “error in variables”

Suppose that there is a deterministic relation $$y_t=ax_t$$ where $$x_t,y_t$$ are real sequences or real functions and $$a$$ a constant. But only $$X_t=x_t+e_t$$ and $$Y_t+u_t$$ can be observed, with $$e_t, u_t$$ being zero mean i.i.d. random variables.

How can I estimate the parameter $$a$$ using $$X_t$$ and $$Y_t$$?

I looks simple, but when setting a linear regression model: $$Y_t=\beta X_t+v_t$$ the error is given by $$v_t=y_t-\beta x_t+u_t-\beta e_t$$. It doesn't look good for a least squares problem and may cause inconsistency. What is the simplest approach to obtain reasonable estimates for $$a$$?

• Why is $e_t$ divided by $a$? You already have to estimate the variance $e_t$, so including the additional constant $a$ overspecifies it. Am I misunderstanding? – Ryan Volpi Jul 4 '20 at 17:02
• You are right. It is not needed and simplifies the problem. – Celine Harumi Jul 4 '20 at 17:06