# Analysis of Potential Factors Influencing China's Regional Sustainable Economic Growth
## 3 Material and Methods
### Empirical Methods
The Lasso-Lars model was also used to develop a PM2.5 concentration time series [96]. The Lasso variable selection method and multi-factor model were applied to study the construction of enhanced index funds [97]. Ridge regression is a biased estimation regression method dedicated to collinearity data analysis. In essence it is an improved OLS method. The disadvantage is that the unbiasedness of the least squares method needs to be abandoned and some information is lost. It also reduces the accuracy but can obtain more realisticregression coefficients, making it a more reliable regression method and more suitable for the fitting process of ill-conditioned data. Compared with ordinary regression, although the square value of the correlation coefficient is reduced, the significance of the regression coefficient is significantly higher than that of ordinary regression. Therefore, it is more practical for collinearity problems and ill-conditioned data [95]. It was pointed out that ridge regression is a promising alternative method for deleting related variables to reduce multicollinearity. The study found that ridge regression will give better results for some economic models (such as production functions) than other economic models [98]. The shortcomings of multiple regression were described, and ridge regression was introduced to show how to obtain biased estimates with minor mean squared errors [99]. Ridge regression and geographically weighted regression models were applied to test the impacts of geographic location, transportation, and policy on the regional economy [100]. Various methods such as ridge regression were used to forecast vital macroeconomic variables for Australia [101]. A pre-test and contraction method based on generalized ridge regression estimation was proposed, which was applied to multicollinearity and high-dimensional problems [102]. Lasso regression is a compressed estimation regression method. A more refined model can be obtained by constructing a penalty function. Compared with ridge regression, Lasso regression can better solve overfitting [95]. If the sum of the absolute values of the coefficients is less than a constant, \"Lasso\" minimizes the sum of the squared residuals. Due to the nature of this constraint, it tends to produce some cases where the coefficients are precisely zero, thereby enabling interpretable models to be obtained [103]. In addition, the authors of a previous study pointed out that the Lasso idea is quite general and can be applied to various statistical models. An extension to generalized regression models and tree models was briefly described in the paper [104]. Subsequently, an improved Lasso-based principal component analysis technique was proposed [105]. Finally, the adaptive Lasso estimator was used to identify the critical variables affecting economic growth [106]. It is noted that the use of the Lasso regression model solved the problems of multicollinearity and overfitting, and its ability to minimize the number of influences on the forecast also allowed more flexibility in policy design [107]. The models used for ridge regression and Lasso variable selection are highlighted. Assuming that \(\big{(}x_{i1},x_{i2},\ldots,x_{ip};y_{i}\big{)},i=1,2,\ldots n\), \(n\) is the sets of observations of the explanatory variables, and assuming that the data are standardized, note that \(\hat{\beta}=\big{(}\hat{\beta_{1}},\hat{\beta_{2}},\ldots,\hat{\beta_{p}}\big{)} ^{T}\), while the estimate of \(\big{(}\hat{\beta_{0}},\hat{\beta}\big{)}\) is defined as:  
\[\big{(}\hat{\beta_{0}},\hat{\beta}\big{)}=\text{argmin}\Bigg{\{}\sum_{i=1}^{n} \Bigg{(}y_{i}-\beta_{0}-\sum_{j=1}^{p}\beta_{j}x_{ij}\Bigg{)}^{2}\Bigg{\}},s.t.\sum_{j=1}^{p}\beta_{j}^{2}\leq\lambda \tag{1}\]  
\[\big{(}\hat{\beta_{0}},\hat{\beta}\big{)}=\text{argmin}\Bigg{\{}\sum_{i=1}^{n} \Bigg{(}y_{i}-\beta_{0}-\sum_{j=1}^{p}\beta_{j}x_{ij}\Bigg{)}^{2}\Bigg{\}},s.t.\sum_{j=1}^{p}\lvert\beta_{j}\rvert\leq\lambda \tag{2}\]  
Equation (1) represents ridge regression and Equation (2) represents the Lasso method.