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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?

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

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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, $\bar{X}^{\ast} = \frac{X_n-X_1}{n-1}$.

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