Regression Line Through Origin

Suppose we have the regression model: $y_i$ = $\beta_0$ + $\beta_1x_i$ + $\epsilon_i$

where $y_i$ = ($Y_i$ - $\bar{Y}$) and $x_i$ = ($X_i$ - $\bar{X}$).

I need to determine if the regression line is guaranteed to pass through the origin.

This will be true iff $\beta_0$ = 0. We immediately see that $\beta_0$ = ($Y_i$ - $\bar{Y})$ - $\beta_1$($X_i$ - $\bar{X})$. Where $\beta_1$ is given by $\frac{COV(X, Y)}{VAR(X)}$. I don't believe this quantity is guaranteed to be 0, so would the answer be that we are unable to determine if the regression line passes through the origin?

-- Thanks

In general, if you are using the least squares coefficients and the regression model $y_i=\beta_0+\beta_1 x_i+\epsilon_i$, where $E(\epsilon_i)=0$ for all $i$, then the coefficient $\beta_0$ is given by $$\hat{\beta}_0 = \overline{y}-\hat{\beta_1}\overline{x}.$$ If you also have $y_i = Y_i-\overline{Y}$ and $x_i=X_i-\overline{X}$ then you get $\overline{y} = \overline{Y}-\overline{Y}=0$ and similarly $\overline{x}=0$. All in all, $\hat{\beta}_0=0$, and as you say this implies that the regression line passes through the origin.