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 $a$?

  • $\begingroup$ 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? $\endgroup$ – Ryan Volpi Jul 4 at 17:02
  • $\begingroup$ You are right. It is not needed and simplifies the problem. $\endgroup$ – Celine Harumi Jul 4 at 17:06

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