## Appendix D Derivation of the Smoothing Performance Index
\begin{split}&(\alpha+2\beta-2\alpha\beta)\sigma_{\text{sxv}}^{2}-( \alpha-1)(\beta-1)T\sigma_{\text{sxa}}^{2}-\frac{(\alpha-1)(2\beta-3)T^{2}}{6} \sigma_{\text{sxj}}^{2}\\ &+\frac{\beta(1-\alpha)}{T}\sigma_{\text{sx}}^{2}-(1-\alpha) \bigg{(}1-\frac{\beta}{3}\bigg{)}T\sigma_{\text{sxv}}^{2}-\frac{(\alpha-1)(2 \beta-3)T^{2}}{2}\sigma_{\text{sva}}^{2}\\ &-\frac{(\alpha-1)(\beta-2)T^{3}}{3}\sigma_{\text{svj}}^{2}- \frac{(1-\alpha)\bigg{(}1-\frac{\beta}{2}\bigg{)}T^{3}}{2}\sigma_{\text{sva}}^ {2}-\frac{(\alpha-1)(2\beta-5)T^{4}}{12}\sigma_{\text{saj}}^{2}\\ &-\frac{(1-\alpha)\bigg{(}1-\frac{\beta}{3}\bigg{)}T^{5}}{12} \sigma_{\text{sj}}^{2}\\ &=\frac{\alpha\beta}{T}B_{\text{x}},\\ &(\alpha+\gamma-\alpha\gamma)\sigma_{\text{sxa}}^{2}-\frac{( \alpha-1)(\gamma-3)T}{3}\sigma_{\text{sxj}}^{2}+\frac{2\gamma(1-\alpha)}{T} \sigma_{\text{sxv}}^{2}+\frac{\gamma(1-\alpha)}{T^{2}}\sigma_{\text{sx}}^{2} \\ &-(\alpha-1)(\gamma-1)T\sigma_{\text{sva}}^{2}-\frac{(\alpha-1)( \gamma-3)T^{2}}{3}\sigma_{\text{svj}}^{2}+\gamma(1-\alpha)\sigma_{\text{sv}}^ {2}\\ &-\frac{(1-\alpha)\bigg{(}1-\frac{\gamma}{2}\bigg{)}T^{2}}{2} \sigma_{\text{sa}}^{2}-\frac{(\alpha-1)(\gamma-4)T^{3}}{6}\sigma_{\text{saj}} ^{2}-\frac{(1-\alpha)\bigg{(}1-\frac{\gamma}{6}\bigg{)}T^{4}}{6}\sigma_{\text{ sj}}^{2}\\ &=\frac{\alpha\gamma}{T^{2}}B_{\text{x}},\\ &\frac{3\alpha+\delta-\alpha\delta}{3}\sigma_{\text{sxj}}^{2}+ \frac{\delta(1-\alpha)}{T^{3}}\sigma_{\text{sx}}^{2}+\frac{2\delta(1-\alpha)}{ T^{2}}\sigma_{\text{sxv}}^{2}+\frac{\delta(1-\alpha)}{T}\sigma_{\text{sxa}}^{2} \\ &-\frac{(1-\alpha)(\delta-3)T}{6}\sigma_{\text{svj}}^{2}+\frac{ \delta(1-\alpha)}{T}\sigma_{\text{sv}}^{2}+\delta(1-\alpha)\sigma_{\text{sva}} ^{2}-\frac{(\alpha-1)(\delta-3)T^{2}}{6}\sigma_{\text{saj}}^{2}\\ &+\frac{\delta(1-\alpha)T}{4}\sigma_{\text{sa}}^{2}-\frac{(1- \alpha)\bigg{(}1-\frac{\delta}{6}\bigg{)}T^{3}}{6}\sigma_{\text{sj}}^{2}\\ &=\frac{\alpha\delta}{T^{3}}B_{\text{x}},\\ &\frac{2\beta+3\gamma-2\beta\gamma}{2}\sigma_{\text{sva}}^{2}- \frac{((\beta-2)\gamma-3(\beta-1))T}{3}\sigma_{\text{svj}}^{2}+\frac{\gamma(1 -\beta)}{T}\sigma_{\text{sv}}^{2}\\ &-\bigg{(}1-\frac{\gamma}{2}\bigg{)}\bigg{(}1-\frac{\beta}{2} \bigg{)}T\sigma_{\text{sa}}^{2}-\frac{((2\beta-5)\gamma-2(4\beta-9))T^{2}}{12 }\sigma_{\text{saj}}^{2}\\ &-\frac{((\beta-1)\gamma-\beta)}{T}\sigma_{\text{sxa}}^{2}- \frac{(2\beta-3)\gamma-6\beta}{6}\sigma_{\text{ssaj}}^{2}-\frac{(2\beta-1) \gamma}{T^{2}}\sigma_{\text{sav}}^{2}-\frac{\beta\gamma}{T^{3}}\sigma_{\text{ sx}}^{2}\\ &-\frac{\bigg{(}1-\frac{\beta}{3}\bigg{)}\bigg{(}1-\frac{\gamma}{ 6}\bigg{)}T^{3}}{2}\sigma_{\text{sj}}^{2}\\ &=\frac{\beta\gamma}{T^{3}}B_{\text{x}},\\ &\frac{3\beta+2\delta-\beta\delta}{3}\sigma_{\text{svj}}^{2}- \frac{(2\beta-3)\delta}{2T}\sigma_{\text{sva}}^{2}+\frac{\delta(1-\beta)}{T^ {2}}\sigma_{\text{sv}}^{2}-\frac{((2\beta-5)\delta-6(\beta-2))T}{12}\sigma_{ \text{saj}}^{2}\\ &+\frac{\delta\bigg{(}1-\frac{\beta}{2}\bigg{)}}{2}\sigma_{ \text{sa}}^{2}-\frac{\bigg{(}1-\frac{\beta}{3}\bigg{)}\bigg{(}1-\frac{\beta}{6 }\bigg{)}T^{2}}{2}\sigma_{\text{sj}}^{2}-\frac{(2\beta-3)\delta-6\delta}{6T} \sigma_{\text{ssj}}^{2}-\frac{\beta\delta}{T^{4}}\sigma_{\text{sx}}^{2}\\ &-\frac{\delta(2\beta-1)}{T^{3}}\sigma_{\text{ssv}}^{2}-\frac{ \delta(\beta-1)}{T^{2}}\sigma_{\text{sxa}}^{2}\\ &=\frac{\beta\delta}{T^{4}}B_{\text{x}},\\ \end{split}