## 3 Methods and Data  
### Methods  
In this study, we combine the super-SBM model with the GML index and include undesirable outputs to measure GTFP in China's manufacturing industry. First, when applying the GML index to measure GTFP, it is necessary to construct a set of global production possibilities that includes desirable outputs and undesirable outputs. If one supposes that there are \(I\) decision making units (DMU), each uses \(N\) inputs and produces \(M\) desirable outputs and \(K\) undesirable outputs. The vectors of three factors for \(DMU_{i}\) are given by \(x\in R_{+}^{N}\), \(y\in R_{+}^{M}\) and \(b\in R_{+}^{K}\), respectively. Therefore, the current production possibility set \(P^{t}(x)\) can be expressed as the following:  
\[P^{t}(x)=\{\left(y^{t},b^{t}\right):\:\sum_{i=1}^{I}z_{i}^{t}y_{im}^{t}\geq y_{ im}^{t},\:\forall\:m;\:\sum_{i=1}^{I}z_{i}^{t}b_{ik}^{t}=b_{ik}^{t},\:\forall\:k;\:\sum_{i=1}^ {I}z_{i}^{t}x_{in}^{t}\leq x_{in}^{t},\:\forall\:n;\:\sum_{i=1}^{I}z_{i}^{t}=1, \:z_{i}^{t}\geq 0,\:\forall\:i\} \tag{1}\]  
where \(z_{i}^{t}\) represents the weight of each cross-section observation value. If \(z_{i}^{t}\geq 0\) indicates constant returns to scale, then \(\sum_{i=1}^{I}z_{i}^{t}=1\), \(z_{i}^{t}\geq 0\) indicates variable returns to scale. However, because technology retrogression may occur in \(P^{t}(x)\), [PERSON] [31] constructed the global production possibilities set \(P^{G}(x)\), which is as follows:  
\[ \begin{split} P^{G}(x)=\{\left(y^{t},b^{t}\right):\:\sum_{t=1}^{T }\sum_{i=1}^{I}z_{i}^{t}y_{im}^{t}\geq y_{im}^{t},\:\forall\:m;\:\sum_{t=1}^{T }\sum_{i=1}^{I}z_{i}^{t}b_{ik}^{t}=b_{ik}^{t},\:\forall\:k;\:\sum_{t=1}^{T} \sum_{i=1}^{I}z_{i}^{t}x_{in}^{t}\\ \leq x_{in}^{t},\:\forall\:n;\:\sum_{t=1}^{T}\sum_{i=1}^{I}z_{i}^{t }=1,\:z_{i}^{t}\geq 0,\:\forall\:i\}\end{split}  \tag{2}\]  
Second, this study draws on the research of [PERSON] and [PERSON] [26] to define the current SBM directional distance function that covers undesirable outputs as follows:  
\[ \begin{split}\overset{\rightarrow}{S}_{v}^{t}\Big{(}x_{\prime}^ {t},y_{\prime}^{t},b_{\prime}^{t},g^{x},g^{y},g^{b}\Big{)}=\underset{x,s^{ \prime},s^{\prime},s^{\prime}}{\max}\frac{\frac{1}{N}\sum_{n=1}^{N}\frac{s_{v} ^{x}}{\delta s}+\frac{1}{M+K}\left(\sum_{m=1}^{M}\frac{s_{m}^{y}}{\delta s}+ \sum_{k=1}^{K}\frac{s_{k}^{b}}{\delta s}\right)}{2}\\ S.t.\hskip 28.452756 pt\sum_{i=1}^{I}z_{i}^{t}x_{in}^{t}+s_{n}^{x }=x_{\prime,n^{\prime}}^{t},\:\forall\:n;\:\sum_{i=1}^{I}z_{i}^{t}y_{im}^{t}- s_{m}^{y}=y_{\prime}^{t},m,\\ \forall\:m;\:\sum_{i=1}^{I}z_{i}^{t}b_{ik}^{t}+s_{k}^{b}=b_{i^{ \prime},k}^{t},\:\forall\:k;\:\sum_{i=1}^{I}z_{i}^{t}=1,z_{i}^{t}\geq 0,\: \forall\:i;\:s_{m}^{y}\\ \geq 0,\:\forall\:m;\:s_{k}^{b}\geq 0,\:\forall\:k\end{split}  \tag{3}\]  
where \(\Big{(}g^{x},g^{y},g^{b}\Big{)}\) represents the direction vectors for decreasing inputs, increasing desirable outputs and decreasing undesirable outputs, respectively. \(\Big{(}s_{m}^{x},s_{m}^{y},s_{k}^{b}\Big{)}\) represents the slack vectors for redundant inputs, inadequate desirable outputs and redundant undesirable outputs, respectively. The global SBM directional distance function is as follows:  
\[ \begin{split}\overset{\rightarrow}{S}_{v}^{G}\Big{(}x_{\prime}^ {t},y_{\prime}^{t},b_{\prime}^{t},g^{x},g^{y},g^{b}\Big{)}=\underset{g^{x},g^{ y},g^{b}}{\max}\frac{\frac{1}{N}\sum_{n=1}^{N}\frac{s_{v}^{x}}{\delta s}+\frac{1}{M+K} \left(\sum_{m=1}^{M}\frac{s_{v}^{y}}{\delta s}+\sum_{k=1}^{K}\frac{s_{k}^{b}}{ \delta s}\right)}{2}\\ S.t.\hskip 28.452756 pt\sum_{t=1}^{T}\sum_{i=1}^{I}z_{i}^{t}x_{in} ^{t}+s_{n}^{x}=x_{\prime,n^{\prime}}^{t},\:\forall\:n;\:\sum_{t=1}^{T}\sum_{i=1 }^{I}z_{i}^{t}y_{im}^{t}-s_{m}^{y}=y_{\prime,n^{\prime}}^{t},\\ \forall\:m;\:\sum_{t=1}^{T}\sum_{i=1}^{I}z_{i}^{t}b_{ik}^{t}+s_{k}^{b }=b_{i^{\prime},k}^{t},\:\forall\:k;\:\sum_{i=1}^{I}z_{i}^{t}=1,z_{i}^{t}\geq 0,\: \forall\:i;\:s_{m}^{y}\\ \geq 0,\:\forall\:m;\:s_{k}^{b}\geq 0,\:\forall\:k\end{split}