# Mean of the variables in GLS estimation

I have a time series whose regression is as: $$Y_t^* = \beta_1X_t^* + e_t$$ where $$Y_t^* = Y_t - Y_{t-1}$$ and $$X_t^* =X_t-X_{t-1}$$.

So $$\hat\beta_1 = \sum(X_t^*- \bar{X}^*)(Y_t^*-\bar{Y}^*)\over \sum(X_t^*- \bar{X}^*)^2$$.

What is the value of my $$\bar{Y}^*$$ and $$\bar{X}^*$$ in that equation? Is it $$0$$ since $$X_t$$ and $$X_{t-1}$$ share the same observations?

• Surely not in general, e.g., if the series has an upward trend, the mean change of the series will be positive. Dec 20, 2021 at 14:46

The bar overhead denotes a sample mean. Now we wish to calculate the sample mean of the first differences. Clearly, these are not observed until time 2 since $$Y_1^{\ast} = Y_1 - Y_0$$ includes the unobserved $$Y_0$$. Therefore, $$\begin{eqnarray*} \bar{Y}^{\ast} &=& \frac{1}{n-1}\sum_{t=2}^n Y_t^{\ast} \\ &=& \frac{1}{n-1}\sum_{t=2}^n (Y_t-Y_{t-1}) \\ &=& \frac{Y_n-Y_1}{n-1}. \end{eqnarray*}$$ Likewise, $$\bar{X}^{\ast} = \frac{X_n-X_1}{n-1}$$.