\section{Estabilishment and solution of model}
\subsection{Question1: Analysis of the impact of economic dynamism on demographic trends and enterprise vitality.}
We choose Beijing as an example. A certain quantity dependent variable that people are concerned about is affected by another quantity (independent variable). This kind of influence is usually only related (not causal), and this kind of relationship is also interfered by many random factors, so it is difficult to find out the relationship between them by mechanism analysis method. Regression analysis is the most suitable mathematical model to describe the relationship between these variables.. We represent trend of population as changes in the resident population, and replace business vitality with the number of surviving corporate units. The economic vitality of the dependent variable region is explained by GDP per capita.

\subsubsection{Data preprocessing}
Depending on the purpose of the data, check the extent to which the data explain the problem. We found that the use of double logarithm and standardized data processing can more accurately study the factors affecting dependent variables.
\subsubsection{Data description}
\begin{figure}[htbp]
\centering
\includegraphics[width=16cm]{excel1.jpg}
\caption{Data on the number of enterprises and population in Beijing in 2008-2019}
\label{fig:myphoto}
\end{figure}
We describe the data and mainly calculate the standard error, the maximum value and other statistics of the data, as shown in the following table.
\begin{table}[htbp]
\centering
\begin{tabular}{llllll}
\toprule
Variable & Obs&Mean&Std. Dev& Min& Max \
\midrule
GDP per capita(yuan)& 12 &101199.3&28632.22& 64491& 151428 \
Enterprise quantity &12&584479.2&264974.2&339808&1183000 \
Resident population(10000)&12&7.550499&.1690831&7.209858&7.683818\
time & 12&2013.5&3.605551&2008&2019 \

\bottomrule
\end{tabular}
\end{table}

\subsubsection{Establishment of multiple regression model}
Based on the above analysis, the following will establish a regression model of Beijing’s economic vitality affected by the number of enterprises and resident population.
\begin{equation}
y=\beta{0}+\beta{1}x{1}+\beta{2}x{2}+\epsilon{0}
\end{equation}
\subsubsection{Results and analysis of the model}
We use SPSS software to do the experiment of linear regression.
\begin{equation}
R^2=0.951
\end{equation}
\begin{equation}
F=108.63
\end{equation}

At the same time, we also find out the statistics, fitting degree and F value of the regression model.
\begin{table}[htbp]
\centering
\begin{tabular}{lllll}
\toprule
Independent variable & Unstandardized $\beta$ & Standardized $\beta$ &t & Sig \
\midrule
Constant & 4.914 & 4.914 &4.198& .002 \
Enterprise quantity & 6.035E-7 & .564 &5.887&.000 \
Resident population & .824&.492&5.128&.001 \

\bottomrule
\end{tabular}
\end{table}

According to the above data, $R^2$ represents goodness of fit, which is used to measure the fitting degree of the estimated model to the observed value. The closer it is to 1, the better the model is. The adjusted $R^2$ is more accurate than the adjusted R-square. The final adjusted $R^2$ in the figure is 0.951, indicating that the independent variable can account for 95.1\% of the change of the dependent variable.\par
From the perspective of F value: F value is the significance test of regression equation, which means whether the linear relationship between the explained variables and all the explained variables in the model is significant in general. Because f=108.63 > fa (k, n-k-1)= 4.102821015, the original hypothesis is rejected, that is to say, the combination of explanatory variables in the model has a significant impact on the explanatory variables.\par
In the above table, we can also see the significance test results of the independent variables. The last column is sig of t test, which is less than 0.05 in the table, indicating that the independent variables have a significant impact on the dependent variables.\par

\subsubsection{Improvement of model-based problems}
\begin{figure}[htbp]
\centering
\begin{minipage}{7cm}
\centering
\includegraphics[scale=0.3]{point1.jpg}
\caption{Scatter chart of GDP and population}
\end{minipage}
\hspace{10pt}
\begin{minipage}{7cm}
\centering
\includegraphics[scale=0.3]{point2.jpg}
\caption{Scatter chart of enterprises and GDP}
\end{minipage}
\end{figure}
Then, we found the problem from the scatter diagram. In the previous regression analysis, we all assumed that the perturbation term was spherical, but the cross-sectional data we used had the problem of heteroscedasticity. Next, we will use the white test to test the heteroscedasticity hypothesis of the model.\par

The content is that under the original assumption of the same variance:[H{0}:E(\epsilon_i^2|X)=\sigma^2] the difference between the robust covariance matrix and the common covariance matrix converges to a zero matrix
[\hat{S}-s^2S{xx}= \frac{1}{n} \sum{i=1}^n e{i}^2x{i}x{i}’-s^2\frac{1}{n} \sum{i=1}^n x{i}x{i}’=\frac{1}{n}\sum{i=1}^n(e{i}^2-s^2)x{i}x{i}’ \rightarrow{p} 0{KxK}]
After applying white test, we find that heteroscedasticity does exist.Try to OLS + robust standard error. This method still carries out OLS regression, but using robust standard error, even if heteroscedasticity exists, all parameters and tests can be carried out as usual. In other words, if there is a robust standard error, the heteroscedasticity will not affect the calculation results.\par
\begin{figure}[htbp]
\centering
\includegraphics[width=12cm]{improve2.png}
\caption{Data improved by OLS + robust standard error}
\label{fig:myphoto}
\end{figure}
From this analysis process, our experimental results are relatively ideal. We can conclude that population trends and the number of enterprises have a significant impact on economic vitality.
\subsection{Question2:Analysis of the impact of economic policies on economic vitality}
The common methods to estimate policy effect are: instrumental variable method, breakpoint regression, tendency score matching method, double difference method, composite control method, etc. We use the double difference method here to solve the second problem.
We selected GDP data from two regions of Shanghai and Jiangsu from 2009 to 2018, and analyzed the impact of the one belt policy on the two regions. As we all know, the “one belt” policy is implemented in 2014. When the time axis was before 2014, the impact of the policy on two cities was zero. After 2014, policies began to have an impact on the city.
\subsubsection{Data description}
According to the original data, we generate a series of data, which is described as follows
\begin{figure}[htbp]
\centering
\includegraphics[width=10cm]{f7.png}
\caption{The image of GDP changing with time in Hebei Province from 2009 to 2018}
\label{fig:myphoto}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=10cm]{f8.png}
\caption{The image of GDP changing with time in Hebei Province from 2009 to 2018}
\label{fig:myphoto}
\end{figure}
\subsubsection{Principle introduction}
Grey prediction is to predict the system which contains both known and uncertain information
Then, it is to predict the gray process which changes in a certain range and is related to time.
Grey prediction processes the original data to find the rule of system change, and
The data sequence with strong regularity is generated, and then the corresponding differential equation model is established
And predict the future development trend of things.\par
GM (1,1) uses the original discrete non negative data column to reduce the randomness through one accumulation generation
Then, by establishing the differential equation model, the new discrete data series with more regularity are obtained
The approximate estimates of the original data generated by the subtraction are solved to predict the subsequent development of the original data.

\subsubsection{Establishment and analysis of the model}
1)Establishment\par
Assume that $x^{(0)}=\left(x^{(0)}(1), x^{(0)}(2), \cdots, x^{(0)}(n)\right)$ is the nonnegative original,We did Accumulating Generation Operator once and generated a new sequence $x^{(1)}=\left(x^{(1)}(1), x^{(1)}(2), \cdots, x^{(1)}(n)\right)$\par
We take equation $x^{(0)}(k)+a z^{(1)}(k)=b$ as the basic form of GM (1,1) model (k = 2,3,… N).(X we define the change of GDP, Z we define the change of time series)
Where b is the amount of ash used, and -a is the development coefficient.
\begin{figure}[htbp]
\centering
\includegraphics[width=12cm]{f1.png}
\caption{The image of GDP changing with time in Hebei Province from 2009 to 2018}
\label{fig:myphoto}
\end{figure}
2)check of quasi exponential law\par
In order to ensure the preciseness of the model, we need to test the quasi exponential law of the model before building it.\par
After the test, we will get an image about the series ratio. The vertical coordinate of the image represents the smoothness of the data inflection point, as long as the value below the critical value is considered.Here are the test results.
\begin{figure}[htbp]
\centering
\includegraphics[width=10cm]{f3.png}
\caption{We can see from the figure that the smoothness of the first two years is not ideal, so we do not consider the impact of the government’s macro-control policies on the GDP of Hebei Province from 2014 to 2015}
\label{fig:myphoto}
\end{figure}
3)Selected model\par
There are three types of model selection:
\begin{itemize}
\item Traditional GM (1,1) prediction
\item New information GM (1,1) prediction
\item Metabolism GM(1,1) prediction
\end{itemize}
\begin{figure}[htbp]
\centering
\includegraphics[width=10cm]{f2.png}
\caption{We can see that the three models work well, in which the traditional information and the new information models overlap}
\label{fig:myphoto}

\subsubsection{Data evaluation and forecast}
1)Data evaluation
(1) residual inspection
It can be divided into three types: absolute residual, relative residual and average relative residual\par
If the residual is less than 20\%, it is considered that GM (1,1) meets the general requirements for the fitting of the original data.\par
If the residuals are less than 10\%, GM (1,1) is considered to have a good fitting effect on the original data.
(2) grade ratio deviation inspection
First, the order ratio $\sigma_{k}$ of the original data is calculated from $x^{(0)}(k-1)$ and $x^{(0)}(k-1)$:
According to the predicted development coefficient (a), the corresponding stage ratio deviation and average stage ratio deviation are calculated\par
If the deviation is less than 0.2, it is considered that GM (1,1) can fit the original data to the general requirements.\par
If the deviation is less than 0.1, GM (1,1) is considered to be a good fit for the original data.
\begin{figure}[htbp]
\centering
\includegraphics[width=10cm]{f5.png}
\caption{The figure above is the residual test, and the figure below is the level ratio deviation test}
\label{fig:myphoto}
\end{figure}
After the above two methods are tested, we come to the conclusion that GM (1,1) fits the original data very well.
2)Forecast future trends
Due to the requirements of the subject, we should divide it into short-term and long-term ones. For short-term ones, we set the time as two years and for long-term ones as 50 years.
\begin{figure}[htbp]
\centering
\includegraphics[width=10cm]{f4.png}
\caption{Forecast the change of GDP in Hebei Province by 2020}
\label{fig:myphoto}
\end{figure}
\begin{figure}[htbp]
\centering
\begin{minipage}{7cm}
\centering
\includegraphics[scale=0.4]{f4.png}
\caption{Forecast the change of GDP in Hebei Province by 2020}
\end{minipage}
\hspace{10pt}
\begin{minipage}{7cm}
\centering
\includegraphics[scale=0.4]{f6.png}
\caption{Forecast the change of GDP in Hebei Province by 2068}
\end{minipage}
\end{figure}

\subsection{Question3:A mathematical model for analyzing the vitality of regional economy}
When using statistical analysis method to study multi variable subjects, too many variables will increase the complexity of the subject. People naturally want to get more information with fewer variables. In many cases, there is a certain correlation between variables. When there is a certain correlation between two variables, it can be explained that the two variables reflect the information of this subject has a certain overlap. Principal component analysis is to delete redundant variables (closely related variables) and establish as few new variables as possible, so that these new variables are irrelevant, and these new variables keep the original information as much as possible in reflecting the information of the subject.\par
\subsubsection{Data description}

Here are 19 cities’ specific data on nine different indicators. We will build a model based on these nine data and select the main components to summarize the original data