The Regional Expressions of Secular Stagnation

This The terminology secular stagnation is suggested by Larry Summers in the last decade to demonstrate the globally economic downward trend. Although this standpoint was not paid importance at that time, the unstoppable decrease in world economy of recent years has shown this. Driven by these miscellaneous data, this report aims to search for evidence that can argue the secular stagnation. Thereby, we first analyse and select the objective tabular data from 1970s to 2010s covering all countries to apply the multiple linear regression model. Then the statistical theory is used for both preliminary measurement and subsequent regression diagnostics, also for the estimation method. In data processing, this paper explores the relationship between economic slow-down rate and (the logarithm of) affluence level with the random forest logarithm to find the threshold. Through the data visualization, the economic trend during the corresponding objective period can be observed intuitively. Finally, in the extension of the slow-down rate, we try to excavate the deep interactive terms among all countries. By the Solow growth model, it can be concluded that the gap between developing countries and developed countries is narrower due to the low original capital of poor countries. This phenomenon is also known as the catch-up effect. This report conducts the slow-down rate to verify the secular stagnation and meanwhile provides evidence for the relationship between slow-down and affluence levels.


Background
It seems that the growth rate of world economy has already gradually slowed down before the Covid-19 pandemic, which is suggested by Jacobs et al. [1]. It is noteworthy that this phenomenon is more concerning in developed countries, as majority of high-income countries are stuck in a trap that there is an insufficient demand to boost their economies. Moreover, as Younsi and Nafla [2] suppose that the monetary policies of these countries are facing invalid dangers, possibly causing financial instability and adverse effects on economic development. Therefore, people started to take notice of the reliability of a controversial hypothesis called 'secular stagnation'. In 1938, six years after Great Depression, Alvin Hansen [3], a chair professor at Harvard University, coined a new nation 'secular stagnation', which means that a diminishing growth rate could persist in long run after a great recession due to a lack of technological innovation and a slow population growth. However, his argument was subsequently proved wrong when the World War 2 started, a significant

Sources
According to Probst [4], the researcher in economic history, the OECD projections foresee the growth in all advanced economies is much slower than that a few months ago. The following Figure 1 display the predications from last May to the latest outlook, with the red arrows indicating the most dramatic downward.

Figure 1.
Economic changes in the world and some developed countries in recent years [4] Additionally, economic growth rates of developed countries have been slowed down in recent years. The most representative one is American economy, whose actual economic growth has been below the potential economic growth rate since global financial crisis [10]. Meanwhile, monetary policy becomes an invalid policy, as most interest rates of developed countries are closed to zero, it is no longer an effective method in terms of stimulating demand [11]. The statistics in Figure 2 demonstrate the business cycle of USA, and in Figure 3 the interest rates of developed countries.   Figure 3. Interest rates of developed countries [11] Lawrence Summer's research made a comparison about the difference between actual growth and economic potential trend within three economies, pointing out that the estimate of GDP potential growth of America in Figure 4 [12] was lowering down after global financial crisis while the disparity between the predict lines and actual growth has been increased. Additionally, the example of Japan and EU economy in Figure 5 and Figure 6 [12] showed that their economy went to a trap as the old monetary policies were unable to stimulate the demand any longer.

Theory derivatives
A. The R language theory such as multiple linear regression analysis and diagnostics is used. B. The statistical theory is used in three sessions, to make comparisons among measurements in data processing; in the form of OLS method in estimation part; in the form of Gauss-Markov theory to judge whether the error is reasonable. C. The approach of random forest is used for the appearance of slow down rate. D. The Solow growth model and Summer hypothesis are applied to the extension of research, including catch-up effect and leapfrogging effect.

Data Processing
3.3.1. GDP per capital and productivity, which is a better measure? Since either GDP per capital or productivity will be used to measure the slow down rate in the final stage, this research begins with finding out the relationship between GDP per capital and time, as well as productivity and time, respectively. Therefore, the multiple lineal regression is applied to these two relationships.
In both experiments, three dummy variables are uniformly constructed, namely 1 , 2 , 3 , which represent for progressive levels of affluence, to divide all countries in the world into three parts. Next, breakdown countries secondarily in the light of the data from World Bank [13]. Additionally, is used to describe the interactive terms that has impacts on GDP per capital or productivity. Firstly, for GDP per capital, we have the following regression function: where i is serial numbers of countries, i=1,2,…,182 x is year, is the intercept, is the slope, 1 , 2 , 3 are coefficients of dummy variables It is demonstrated by Dynan and Sheiner [14] that the expression for GDP can be conducted by expenditure-side approach as GDP = private consumption + gross investment + government investment + government spending + (exports -imports). According to the formula of GDP calculation, select the following data respectively,  Real consumption of households and government (ccon), which means the personal consumption of households and the consumption of government;  Capital stock at Constant 2011 National Prices (rnna), which means gross growth in investment and government investment, "gross" refers to the fact that GDP measures production and does not consider the various uses of the product. Production can be consumed, invested in fixed assets or inventories, or used to replace depreciated fixed assets [15];  Price level of exports (pl-x), Price level of imports (pl-m). Since the price level of the market is affected by the rate of inflation, our exports and imports are measured by the price level. Then the GDP per capital is Use R to do the regression analysis, the adjusted-2 can be worked out to quantify the relationship between GDP per capital and time as in Figure 7.  Besides, the conformity of total GDP is also be studied. However, as the Figure 8 shows below, it seems to be a subordinate choice. Secondly, for productivity, the regression function is: where i is serial numbers of countries, i=1,2,…,182 x is year, is the intercept, is the slope, 1 , 2 , 3 are coefficients of dummy variables Scheryer and Pilat [16] pointed out that the productivity can be measured by gross output, that is, the relation is presented as a production function H with gross output Q, A is a parameter of technical change Here use emp*avh in excel to stand for the expression of productivity. That is, Productivity= Number of persons engaged (in millions) * Average annual hours worked by persons engaged By R, the adjusted-2 can be observed in Figure 9.  ( 2 ), the median of the lowest half ( 1 ), and the median of the highest half ( 3 ), also with the minimum and the maximum. Likewise, the tabulation of coefficients covers four characteristics of both the intercept and the explanatory variables. 'Estimate' means the estimated value calculated by R, the second column represents for the standard error, 't value' is the test statistic for t-test, the last column is the probability of rejection in hypothesis test. Below are the significance codes, which help to judge what extent of evidence should be rejected. Finally, the multiple R-squared is the initial value and the adjusted R-squared is corrected value. The p-value can also be viewed as the test statistic.
2 correlation system test method is used to judge the fitting degree of the regression equation. The value of 2 is between 0 and 1, and the closer to 1, the better the fitting degree is. We want to assess how useful the model is in explaining the data. This can be assessed by the value of adjusted-2 . Let SSR denote the regression sum of squares, which measures how much of the variation in the y values is explained by the regression; SST denote the total sum of squares, which measures the total amount of variation in the y values. The proportion of SST explained by SSR is 2 .
By looking at the data result of the model, it can be observed that the p-value is less than 0.01. Therefore, it is known from t-test that the independent variable X is very significant in Y, and it is judged from F-test that the independent variable of the whole model is very significant. Meanwhile, 2 =0.4001 in Figure 7 means that the independent variable and the dependent variable are highly correlated. According to the R output in Figure 8, on the whole, independent variable has a significant correlation with Y, but 2 =0.05083. Therefore, it is judged that the independent variable and the dependent variable have a low correlation and a low goodness of fit. Hence, in these experiments, it can be obtained that the GDP per capital is a better measure as its adjusted-2 is closer to 1.

Linear regression analysis
Next stage, it is necessary to work out the trend of slow down rate to visualise the secular stagnation by studying the relationship between slow down rate and affluence level. Firstly, a scatterplot should be drawn to explore the distribution of slow down rate in different countries and the relationship between slow down rate and the affluence level. Choose GDP per capital as a measure for these two variables according to the last stage, because it is more appropriate than the productivity. For the y-axis, that is, the slow down rate, which is defined to be equal to the average growth rate from 1994 to 2017 minus the average growth rate from 1970 to 1993. That is, Slow down rate = average growth rate from 1994 to 2017 -average growth rate from 1970 to 1993 If the slope is positive, this means that there exists an acceleration in the economic growth, while the negative value means the downward economic growth rate. However, if the GDP per capital is used as x-axis, there will be a strong influential outlier, compelling all scatters clustering to the left. Hence, a transformation is conducted by adding logarithm to the GDP per capital. The influential observation is weakened in this case. Likewise, rank the values of GDP per capital from the smallest to the largest through the time period 1983-1993, where the assumed slow-down has not happened. Therefore, a scatterplot can be drawn as shown in Figure 10 to indicate the distribution of slow-down rate in all countries. By R, the scatterplot is runed out. It can be obviously observed that there are three outliers, all should be checked. One is on the right-hand-side, this country has the largest GDP but its slow down rate is negative. It can be found from the data table that this country is Qatar, which is an oil produced country [16]. Thus, its export level is much higher than other countries, explaining the high GDP value. Another two outliers are outside the interval [-0.1, 0.1], they are Georgia and Equatorial Guinea, the former [17] accelerated the economic development in the 1990s and the later [18] implemented the economic restructuring in 1987. From the scatterplot, the tendency can be roughly observed. Next, a fitted linear regression can be worked out, which appears to imply the secular stagnation as its slope is negative. However, the practical meaning of this graph may be illogical because of the constant drop. This tendency derives from the laws of the real world. Therefore, there exist a threshold point that divide the tendency of slow down rate into two parts, with a gradient descent.

Data transformation
To find the threshold, position it relative to the middle of the scatterplot at first according to the distribution of scatters, i.e., within the interval (-5, -4). By the approach of random forest from Prasetiyowati, Maulidevi and Surendro [19], traverse all points in this interval to find a point where 10 the residual sum of squares is the smallest among them. Hence, after calculating all possible points, the country code with the smallest SSR is CHN. Next, take the means of the two segments, one has the logarithm of GDP per capital smaller than CHN and the other has the logarithm of GDP per capital larger than CHN. Then, a transformed fitted line plot with a gradient decent at the threshold can be displayed in Figure 11. The average value of the former segment is 0.03477754 and the average value of the later segment is -0.007733709. Figure 11. The transformed slow-down rate It can be summarised from Figure 10 that the world undertakes the economic downward pressure as the global economy grows steadily. Moreover, there's a relationship between slow down rate and affluence level, a richer country suffers from a more severe economic downward. Since the slow down rate is below zero, it can be concluded that the world economy is experiencing secular stagnation.

Regression diagnostics
To examine the fitness of the linear regression, run R and we obtain four regression diagnostic diagrams as shown in Figure 12.  Figure 12, the two graphs on the left-hand side are residuals and fitting diagrams, they reveal the relationship between fitted values and true values by residuals and standardized residuals. Both suggest that the points are reasonably close to a straight line, with the scatters evenly distributed on both sides of the horizontal line. Additionally, the Normal Q-Q figure is used to judge whether the residuals satisfy normal distribution. It shows three outliers and a non-distinct tail that away from the straight line, but most of points are within the interval [-1.96, 1.96]. However, the problem is that the points don't fall on the line at an angle of 45 degrees. This will be explained later. Finally, the residual versus leverage diagram is displayed. In this Scale-Location graph, several influential observations exist, meaning that they make strong influence on the regression.  Figure 13 is the normal Q-Q plot of the transformed tendency, the degree of the slope of the straight line is about 45 0 , which is more appropriate than before despite of some points do not fall on the line. To sum up, the fitted line is reasonable to assumption although some deviations exist.

Advantages
 The multiple linear regression model is used to conduct regression analysis on GDP per capital and productivity, and the optimal measurement is selected to make the results more reliable.  Dummy variables are introduced in the experiment to divide countries into three partitions, so that the data can improve the accuracy of the model, also it can reflect the relationship between the affluence level and slow down rate more accurately.  The least square method is used in curve fitting of slow-down, which can make the parameters fitting result better.

Improvements
The strategies for improvements are based on the potential threatens of the data analysis. In this study, firstly, the limited data in data processing will affect the final results to some extent. One is that the data of GDP in target countries is incomplete in the origination year, which gives rise to an inaccuracy in the process of data analysis. Considering the necessity to analyse the GDP of developed countries, developing countries and poor countries in the same time period to keep conformity, chose from 1983 to 1993, which is before the gradient decent and relatively complete in data. Another limitation is that, when filtrating the data, there exist two countries (CUW and SXM) that only have the data from 2005 to 2017. They are deleted as they break the consistency of the selected data. However, some other countries also have missing data when calculating the GDP, although these are much fewer than that in two countries. For example, data are not available for some years in selected countries in the time period 1983-1993, some countries only have data from 1990 to 1993. Thus, the GDP is ultimately measured by its definition, that is, GDP = Private Consumption + Gross Investment + Government Investment + Government Spending + (Exports -Imports). The indispensable problem is that the time periods from 1983 to 1993 or 1990 to 1993 is too short, so it may be one-sided to judge the opulence degree of a country based on the data within 10 years, let alone the economic changes. Secondly, the optimality of variables in the regression model is noncommittally remaining a problem. From the perspective of classification, the developed and developing countries can also be divided by other categories, such as exports, technology and people's living standards, etc., leading to enormous different consequences. From the perspective of interactive terms, since the price level of exports and price level of imports are used in the first stage, the price level may be affected by inflation. Adversely, this term is not added to the model as the interactive term is set to be default 0. There is another variable in the excel tabulation needed to be noticed---the purchasing power parity (PPP). If convert the expression of GDP per capital into the one that with the variable purchasing power parity (PPP), it may be far from the actual result due to the variability of theoretical exchange rate in the calculation. The only thing that ought to be ensured is that each country measures the GDP in the same way with the same monetary unit. Thirdly, the regression diagnostics exist some defects inevitably. In Residuals vs Fitted diagram, which is a scatterplot of residuals-fitted values (Y-X) for residual analysis. This graph can be used to detect nonlinearity, unequal error variances, and outliers. In the Figure 12, the distribution is basically uniform on both sides of the line, but the scatters seem to be inclined to a central distribution and the line is slightly skew to a horizontal straight line. The error distribution also does not conform to the Gauss-Markov Condition. Similarly, the Y-axis of the scale-location diagram is the extraction of the Y-axis of Residuals vs Fitted diagram and both have the same X-axis. For the scale-location diagram, it shows whether the residuals are uniformly distributed along the range of predictors to check the equivalence of variance hypothesis. Theoretically, there would be a horizontal straight line with evenly distributed data, but in reality, it seems not to be a horizontal line with the far left and right points deviate from it. The Normal Q-Q plot represents for the normality of the univariate distribution of the data set. If the data is normally distributed, the points will fall on the 45-degree reference line. Otherwise, the points will deviate from the reference line. But in fact, the 45-degree sloped line in the model does not start at the origin and the data does not fall completely in line, with a tail divorced from the precedent trajectory. Therefore, the data does not fit the Y=X line, and the error cannot follow a perfectly normal distribution as well. For the Residuals vs Leverage diagram, which can check the outliers of the regression model. It can also be used to detect the heteroscedasticity and nonlinearity of data set. The Figure 12 witnesses that some points are too far away from the regression line, which contributes to the imprecision and unpredictability of the model. Generally, the problems in the regression diagnostics stem from the precedent limited data and the variables that may not be correct. Instead of deleting or keeping the missing values, maybe the preferred method is to do missing value processing by the multiple imputation method, that is, to seek for alternative values. Furthermore, to upgrade the accuracy of the regression model, some algorithm conversion can be used and the variable adjustments can also be applied. Finally, outliers may sometimes affect the result severely, this could be amended by some mathematical transformations.

Estimation method
First, it should be checked that whether the residuals have a normal distribution, through drawing a histogram of residuals, this consumption can be verified as in Figure 14. As it is tested before, the estimates are as following ( Figure 15): Figure 15. The R output for the regression of GDP per capital The missing values in 1 is caused by the fewer number in the richest countries. The coefficient shows that there is a relation between the year and GDP per capital, also the p-value indicates the strong evidence. The standard error deviation is within a reasonable range. Have 2 = 40.01%, meaning that the regression explains 0.01% of the variation in the data. This is not a low percentage, suggesting a useful regression relationship and the relation is not causal.

The relationship between slow down rate and affluence levels
After the verification, the fitted line is a piecewise function and there is a continuous linear relationship with a discrete shift between the slow down rate and affluence level. It has an overall decrease in the slow down rate, which implies the affluence level of countries is in direct proportion to the slow down rate. Since the slow down rate of almost all countries is negative without the logarithm, it comes out that the world is confronted with secular stagnation. However, what the interactive nexuses between poorer countries and richer countries or of slow down at similar rates are needed to be explained, although the hypothesis of the subject has been verified. A correlative terminology is the Solow growth model, which demonstrates three factors --technology, capital accumulation and labour force that drive economic growth. It implies that the less capital in poor countries to start with can narrow the gap between rich and poor countries as each additional unit of capital is inclined to yield higher return than in a richer country [20]. This is also the concept of catch-up growth. Hence, the gradient decent can be explained by this model because the difference of slow down rate on both sides of the threshold is stable and will not become wider. Meanwhile, it is suggested by Steinmueller [21] that the development strategy of 'leapfrogging' is supported by information and communication technologies (ICTs), it also narrows the gaps in industrialized and developing countries. The leapfrogging indicates that small and incremental innovations can sustain the leading position of a dominant enterprise [22]. Recently, as the Summers hypothesis says, the productivity improvement for agriculture and manufacturing are close to saturation [23]. Therefore, the underlying rate of technology progress has slowed down. More productivity improvements are needed in service sector for which may require large ICT developments. Since a large amount of technology transfer comes through FDI (foreign direct investment), by searching for the data source traversing all countries from 1970 to 2019, it can be seen in Figure 16 [24] that the countries with high FDI tend to have the similar slow-down rate. But the countries with low FDI do not follow this trait owing to several special countries. Figure 16. FDI in specific countries [19]

Conclusion
This report aims to study the relationship between the slow down rate of economy and affluence level among all countries in the world to explore the forward or backward evidence for secular stagnation. At the beginning, GDP and productivity are compared to choose a better measure for economy, also namely affluence level. After some calculations and comparisons, GDP per capital is more suitable to measure the level of economic development of a country, whether from a theoretical perspective or from specific data analysis. It can be thought of as a "best fit" unit of measure to argue the tendency of economy. Next, the tendency of slow down rate is worked out through seeking the relationship between slow down rate and affluence level. The negative value of slow down rate implies the secular stagnation. Finally, to examine the accuracy of multiple linear regression model and fitted line plot, the regression diagnostics and estimation method are used. The result comes out to be reasonable and appropriate but lack of precision. However, the problem is that, the absence of data on GDP, the Capital Stock at Constant 2011 National Prices (RNNA), Price Level of Exports (PL-X), and Price Level of Imports (PL-M), which