## 4 Closer Look into PPP  
The original PPP does not clearly state whether the common error is additive or multiplicative. This ambiguity was examined in [5]. In the \"additive\" scenario case, \(y_{1}=m_{1}+b\) and \(y_{2}=m_{2}+b\), and the source of correlation may be a common background measurement \(b\). And, the situation is somewhat different from PPP, because if \(\sigma_{b}\) is 20% of \(y_{1}=1.5\), then it is 30% of \(y_{2}=1.0\). The covariance matrix for this case is  
\[\Sigma = \left( \begin{array}{cc}\sigma_{m1}^{2}&\\ &\sigma_{m2}^{2}\end{array} \right)+\sigma_{b}^{2}\left( \begin{array}{cc}1&1 \\ 1&1\end{array} \right) \tag{7}\] \[= \left( \begin{array}{cc}0.1125&0.09\\ 0.09&0.1\end{array} \right)\]  
and the GLS solution is 1.15 \(\pm\) 0.31. With the covariance matrix of Equation 7, PPP does not occur, and the GLS solution of 1.15 is a weighted average of \(1.0\pm 0.1\) and \(1.5\pm 0.15\). Although as \(\sigma_{b}\) changes, the standard deviation \(\sigma_{\hat{\mu}}\) is scaled accordingly, the GLS solution does not change. It is reasonable that the GLS solution does not change as \(\sigma_{b}\) changes, because whatever the background fluctuation is, \(y_{1}\) and \(y_{2}\) are impacted in the same manner by \(b\). If instead \(\sigma_{b}\) is 20% of \(\mu_{m}\), then its value is unknown because \(\mu_{m}\) is unknown and must be estimated. However, regardless of the value of \(\sigma_{b}>0\), PPP cannot occur in the additive case as parameterized in Equation 7 (Theorem 1 below). In [PERSON]'s [9] notation, Equation 5 can be written as  
\[\hat{\mu}=\frac{a_{1}y_{1}+a_{2}y_{2}}{a_{1}+a_{2}} \tag{8}\]  
with  
\[a_{1}=\sigma_{2}(\sigma_{2}-\rho\sigma_{1}),\quad a_{2}=\sigma_{1}(\sigma_{1} -\rho\sigma_{2}) \tag{9}\]  
where \(\rho\) is the correlation coefficient. [PERSON] demonstrated that PPP does not occur if the condition  
\[\rho\leq\min\left(\frac{\sigma_{1}}{\sigma_{2}},\frac{\sigma_{2}}{\sigma_{1}}\right) \tag{10}\]  
is fulfilled (and does occur if the condition does not hold). It is not difficult to show that the additive case of Equation 7 satisfies this condition, which we state as Theorem 1. _Theorem 1_. If \(\Sigma\) is given by  
\[\Sigma = \left( \begin{array}{cc}\sigma_{m1}^{2}&\\ &\sigma_{m2}^{2}\end{array} \right)+\sigma_{b}^{2}\left( \begin{array}{cc}1&1 \\ 1&1\end{array} \right)\]  
then PPP does not occur. Proof. The covariance matrix \(\Sigma\) can be expressed as  
\[\Sigma = \left( \begin{array}{cc}\sigma_{1}^{2}&\rho\sigma_{1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{array} \right)\]  
with \(\rho=\frac{\sigma_{2}^{2}}{(\sigma_{1}^{2}+\sigma_{b}^{2})^{1/2}(\sigma_{2}^{2 }+\sigma_{b}^{2})^{1/2}}\). If \(\sigma_{1}^{2}<\sigma_{2}^{2}\) then \(\rho\leq\min\left(\frac{\sigma_{1}}{\sigma_{2}},\frac{\sigma_{2}}{\sigma_{1}}\right)\) is equivalent to \(\sigma_{b}^{4}\leq(\sigma_{1}^{2}+\sigma_{b}^{2})\times(\sigma_{1}^{2}+\sigma _{b}^{2})\) which obviously holds. The proof is identical if \(\sigma_{1}^{2}>\sigma_{2}^{2}\). And if \(\sigma_{1}^{2}=\sigma_{2}^{2}\), then PPP can never occur. Notice that Equations 1 and 3 have the same form as assumed in Theorem 1. Therefore, PPP also cannot occur for the multiplicative error model assumed. However, as discussed, PPP can occur with the multiplicative error model case if \(\Sigma\) is estimated as in Equations 1 and 3. Several authors explored different values for the covariance matrix \(\Sigma\) to understand the relationship between the covariance matrix and the estimates. Some numerical examples are in [2]. [PERSON] _et al._[10] and [PERSON] _et al._[11] also reported linear regression of strongly correlated data and emphasized that if \(\sigma_{1}^{2}\
eq\sigma_{2}^{2}\) then contrary to common belief, positive or negative correlation can be exploited to produce an unbiased GLS estimate that has lower variance than in the zero correlation case.