Unveiling the Future: Forecasting the US Debt Ceiling and Financial Market Trends

. The US debt limit has important implications for economic and financial stability. By analyzing historical data, this article finds that there is a significant correlation between the debt limit and the US financial market. The article also constructs an exponential smoothing mathematical model to successfully predict the debt limit in the future and test the accuracy of the model using historical data. The results show that the debt limit has a significant positive linear correlation with the S&P 500 Index and the Spot Gold Price Index. Although there is no linear correlation between the debt limit and the dollar index, this paper finds a hidden pattern by comparing the data of the Democratic and Republican administrations. Based on the results of the projections of the debt limit for the next five years and the correlation with the three major indices, the article makes a reasonable prediction of the future trend of the US financial market and provide insights. These results suggest that the situation in the United States may be more optimistic than expected.


Introduction
The US Debt limit refers to the maximum amount of US government bonds issued by the US government, which is used to restrict the government's borrowing behaviour, ensure financial sustainability and avoid financial crisis.Since the United States set the Debt limit in 1917, the debt scale of the United States government has been expanding, rising from the initial $11.5 billion to $31.4 trillion.The Debt limit of the United States has a certain impact on the stability of the financial market, and the dynamics of its debt market also affects global capital flows.Due to the dominant international monetary position of the US dollar, the issuance of US debt has a certain influence globally [1].
From the perspective of modern monetary theory, American debt has a certain influence on financial markets [2].For instance, within a few years, it may have serious negative impacts on science [2].In order to avoid a large amount of government debt and insolvency, the government usually reduces its borrowing behavior during economic recession, while the member countries of the OECD, just as the U.S., will increase the Debt limit to affect the depressed market [2].Therefore, the development trend forecast of financial markets is of great significance for understanding and dealing with debt problems, promoting economic development and managing financial risks.Comprehensive analysis and accurate predictions can better address debt challenges and ensure economic stability and sustainable development.This article combined the historical data of the US Debt limit and some historical data of indexes that can reflect the US financial market, namely, the S&P 500 index, the spot gold price index and the U.S. Dollar Index, and obtained the relationship between the Debt limit and these three indexes through correlation analysis.Then, use exponential smoothing to predict the future Debt limit, and use historical data to verify the model, test its fitting degree and prediction ability, and get the error zone of the model's prediction results.Finally, according to the correlation between the change of the Debt limit and the trend of the financial market, predict the future development trend of the financial market.

Data sources
Data on the US debt ceiling cited in this article come from the US Treasury, Congressional Budget Office, Federal Reserve and Wind, while data on the S&P 500, gold and US dollar indices come from Investing.com.In addition, some of the investor survey data cited in section 3.3.2 is from Bloomberg.

Pearson correlation coefficient
The Pearson correlation coefficient between two variables is defined as the quotient of the covariance and standard deviation between them.
Then the Pearson correlation coefficient: r can also be estimated from the standardized score mean of the sample points.This yields the equivalent equation: ̅ and  ̅ are mean values.  and   are mean standard deviation.

Predictive modelling
It is assumed that there is a relationship between historical values and future values.A weighted average of the actual values for the current period and the smoothed value of the index for the previous period is used to obtain the smoothing index for the current period.Let this smoothing index be the forecast value for the next period.

Correlation analysis
The debt ceiling is an important economic indicator that reflects the government's debt situation and fiscal health.At the same time, changes in the debt ceiling affect the formulation and implementation of fiscal policy, which in turn affects the bond market, interest rates, investment and consumption.
The S&P 500, spot gold and the US Dollar Index are all indicators that are closely linked to the macroeconomy.Of these, the S&P 500 is a key indicator of the performance of the US stock market, while the US Dollar Index is a measure of the relative strength of the US Dollar against a basket of foreign currencies.In times of heightened economic uncertainty, investors typically seek safe-haven assets such as gold and the US dollar.An increase in the debt ceiling could trigger market concerns about economic stability, leading to an increase in safehaven demand.This paper analyses the correlation between them to understand the potential impact of the debt ceiling on the stock market, the gold market and the US dollar exchange rate.

Correlation analysis results
First, the debt ceiling and the S&P 500, the debt ceiling and the US dollar, and the debt ceiling and spot gold are visualized in three data sets.As Fig. 1 and Fig. 2 can initially be observed that there is a clear linear relationship, while Fig. 3 initially do not determine the linear characteristics, however, the trend reveals a hierarchy of changes in the Dollar Index in the face of the debt ceiling increase.Then, using SPSS, we analyzed the correlation between four indicators: the debt ceiling, the S&P 500, spot gold and the USD index.According to Table 1, we can see that: Debt Ceiling has a significant p-value of <0.01 and a correlation coefficient r of 0.951 with A. There is a significant correlation between the two variables.
Debt Ceiling has a significant p-value of <0.01 and a correlation coefficient of 0.927 with B. There is a significant correlation between the two variables.
Debt Ceiling has a significant p-value of 0.194 and a correlation coefficient r of -0.303 with C.There is no linear correlation.

Analysis of results
Overall, there is a strong correlation between the US calendar year debt limit and both the US Standard & Poor's index (A) and spot gold historical data (B).The US calendar year debt limit is not linearly correlated with the US dollar index (C).

The US debt ceiling and the Standard & Poor's 500 Index (A)
Investor confidence was shaken in 2011 by the US debt crisis, which led to the downgrade of the S&P 500 credit rating from AAA to AA+ and caused the S&P 500 index to fall by almost 20%.However, correlation analysis shows that the US equity market has maintained an optimistic trend, albeit with occasional volatility and corrections, despite the rising US debt ceiling.This suggests that the positive factors brought about by the raising of the US debt ceiling have to some extent dominated the performance of the US equity market.For example, the Federal Reserve's loose monetary policy has increased the amount of money in circulation in society, indirectly stimulating liquidity in the stock market and driving the overall rise.In addition, the US government's tax cuts and deregulation measures have stimulated business investment and innovation, thereby boosting economic growth.Despite the vagaries of investor confidence, they tend to go with the flow in the face of a long-term rising US stock market.

The US debt ceiling and Spot Gold Index (B)
As the US debt ceiling continues to rise, so does the spot gold index, signaling concerns about economic uncertainty and increased demand for safe-haven assets.A rising US debt ceiling means an increase in government debt, which could lead to financial distress and economic instability.In such a scenario, investors tend to seek relatively safe assets.According to the latest Markets Live Pulse investor survey (Fig. 4), gold continues to be investors' first choice for wealth preservation [5].They are concerned that the US government's debt problems could lead to a debt collapse, and as a result more than half of investors specializing in the financial sector said they would buy large amounts of gold if the US government defaulted on its national debt.Gold is seen as a safe haven asset because of its value-preserving and inflation-proofing properties in times of economic uncertainty.The continued rise in the Spot Gold Index therefore reflects investors' ongoing concerns about the economic outlook and suggests that they are seeking safe havens to protect their wealth.Although there is no clear relationship between the overall trend in the US debt ceiling and the US Dollar Index, we do find a pattern of influence on the US Dollar Index during periods of control by different political parties in the US Congress (Fig 5 and 6).Democrats and Republicans have different views on the national debt.Democrats tend to support increasing the national debt to promote economic development and social welfare, while Republicans tend to reduce the national debt to maintain fiscal stability and reduce the tax burden.We find that the US Dollar Index shows a steady and rising trend before and after the resolution of the national debt ceiling issue during the period when the Democrats control Congress.And in early 2021, after the Democrats won control of the Senate, gold fell by $50 to near 1,900, ushering in an

Model Selection
The article collects data on the U.S. debt ceiling from 1917 to 2021.If there are multiple debt ceilings for certain years, this article uses the data near the end of the year as the debt ceiling for that year.
The most widely studied financial application area is forecasting of a given financial time series, including stock price forecasting, index prediction, forex price prediction, commodity (oil, gold, etc.) [7].The grey theory can be applied in the research of prediction, decision-making and control, especially in prediction 错 误!未找到引用源。.However, the grey prediction is not suitable for the case of large and rich historical data.Meanwhile, available data do not show fixed intervals in time.Therefore, Time Series and Grey Forecasting Models are not taken.appreciation of the dollar [6].This suggests that the Democrats have taken sound policy measures to deal with the national debt ceiling issue, thereby increasing the market's confidence in the US dollar.In contrast, during the period when the Republicans controlled Congress, the US dollar index showed a clear U-shaped fluctuation, especially in 2017, when the fluctuation trend was very clear.This suggests that there may be some uncertainty and volatility in the Republican Party's handling of the national debt ceiling issue, leading to some shaking of investor confidence.From this, we can infer that investors may rely more on the governing style of the Democratic Party, and this research result is an important reference value for us to deeply understand the impact of different political parties on the US Dollar Index and the resolution of the national debt ceiling issue.
For a more intuitive view, imaging the historical data that have collected.As shown in Fig. 7, the slope of the curve gradually increases over time, exhibiting exponential growth.For the prediction of the debt ceiling, its forecasted values are related to the past known data.The debt ceiling will change with some factors such as policies, economy and so on.In order to update the forecast results in real-time, this paper adopts.It adjusting the smoothing coefficient to renovate the predicted values and make them more closely aligned with the latest data.In addition, the weighted average method is also used to reduce the influence of outliers and make the forecast results relatively stable.

Solving the model
The exponential smoothing method is divided into primary smoothing, secondary smoothing and tertiary smoothing.The smoothing coefficient lies between 0 and 1, and its magnitude determines the weighting relationship between past data and predicted values.The closer it is to 1, the more weight is given to past data.The closer it is to 0, the more weight is given to predicted values.
Using SPSSAU, the article iterates through each type of parameter case, calculates the RMSE values from different combinations of initial values, alpha values and smoothing types, and finds the parameter values with the smallest RMSE.In this paper, the root mean square error value is used to measure the prediction effect.The smaller RMSE indicates better fitting.The best alpha value and smoothing type is found out combining with RMSE value.Finally, the best prediction effect is obtained.As shown in Table 2, the optimal model parameters are: an initial value of 9351.000, an alpha value of 0.950, and a smoothing type of cubic smoothing.The RMSE value is 9771.146.Constructing with these parameters predicted values for the backward 12 periods.In this paper, only the projected values for the next five years are used as shown in Table 3.

Analysis of results
When alpha = 0.95, the fitted value is closer to the real value, so the article choses alpha = 0.95 to forecast the U.S. debt ceiling year by year.The prediction results are: in 2023, the U.S. debt ceiling is about $62.0 trillion; in 2024, the U.S. debt ceiling is about $83.6 trillion; in 2025, the U.S. debt ceiling is about $109.6 trillion; in 2026, the U.S. debt ceiling is about $139.8 trillion; and in 2027, the U.S. debt ceiling is about $174.2 trillion.Light blue solid line is the true value; Light blue dotted line is the fitted value; Dark blue dotted line is the predicted value.
As Fig. 8 shows the forecast curve of the U.S. debt ceiling for the next five years.The real value curve and the fitted value curve have a high degree of overlap.It indicates that the exponential smoothing method has a high prediction accuracy.In order to test the accuracy of the prediction results, this paper validates the accuracy of the exponential smoothing prediction method based on previous data.From Table 4, the exact values of the debt ceiling from 1917 to 1965 predicts the projected values of the debt ceiling for the five years following 1966-1971.According to the forecast results (Table 5), the next few years, the United States debt ceiling will show a year-onyear growth trend, the past five years can maintain an exponential growth.At the same time, gold as the best hedge assets in the minds of investors, the gold index is expected to continue to rise.Although the Democratic Party advocates increasing government spending can stimulate the economy and win the confidence of some investors, resulting in some of the gold back to the dollar, but it is difficult to resist the overall upward trend of the gold index.According to data from Dow Jones Market Data, the US two-year and 10-year bond yields have been in an 'inverted state' since 5 July 2022 [9], reflecting market concerns about future economic growth.Investors may be anticipating a recession or increased uncertainty, which has pushed up the price of long-term bonds and lowered their yields.This is especially true after the huge bipartisan disagreement over the US debt ceiling in 2023.
In addition, the overall trend of the S&P 500 Index, although it may experience some volatility as a result of fluctuations in market confidence, will unsurprisingly continue to rise thanks to a number of government measures to stimulate the economy.While this will keep the US in a vicious cycle of debt spirals, US Treasury Secretary Yellen recently insisted that US Treasuries remain the safest and most liquid asset in the world [10].Moreover, according to the Commerce Department, revised final real GDP growth for the first quarter of 2023 is 2%, driven by strong consumer spending, government spending, exports and nonresidential fixed investment.The US Treasury Secretary's confidence may not be misplaced if the US can skillfully navigate the challenges ahead.
This year's US debt ceiling situation in 2023 bears some similarities to 2011, with both countries facing relatively high debt levels and political controversies.Moreover, two of the world's three major credit rating agencies, Standard & Poor's and Fitch, have downgraded the US from AAA to AA+ in 2011 and 2023 respectively, in what looks like history repeating itself.However, the good news is that the US is now in its seventh cycle of interest rate hikes, and when the economy is in recession, the US can resort to interest rate cuts and expansion to stimulate investment and consumption to counter the risk of recession or deflation.Overall, the economic outlook for the US may not be so pessimistic if the country can eventually find a steady state in the current situation.

Conclusion
Through our research, we conclude that the US debt limit will show an annual increase in the coming years.We find that there is a significant linear positive correlation between the debt limit and the S&P 500 and the Spot Gold Price Index.Although the S&P 500 and the Spot Gold Price Index can fluctuate in the short term for a variety of reasons, they are generally trending upwards.The bullish trend in the S&P 500 index suggests that the US government's stimulus measures are working, but the market still lacks confidence in the US dollar and investors are more inclined to put their money into gold as a safe haven.Although there is no linear correlation between the debt ceiling and the US Dollar Index, we find that the US Dollar Index has shown a steady and rising trend before and after the resolution of the national debt ceiling issue during the period when the Democrats controlled Congress.In contrast, during the period when the Republican Party controlled Congress, the US Dollar Index showed a clear U-shaped fluctuation, suggesting that the Democrats took sound policy measures in dealing with the national debt ceiling issue, thus strengthening market confidence in the US Dollar.Currently, the US is in the midst of its seventh round of interest rate hikes.When the economic situation is recessionary, the United States can stimulate investment and consumption by lowering interest rates and expanding the table to deal with the risk of recession or deflation.In the current situation, a continuation of the debt-for-debt approach may not be entirely without optimistic prospects if the US is eventually able to find a steady state.
The importance of this study is that the debt limit has far-reaching implications for the US domestic economy and financial markets.By modelling and forecasting changes in the debt limit and analyzing its correlation with key indicators, people can better understand these effects and provide decision support to relevant stakeholders, helping investors and market participants to develop more informed investment and risk management strategies.However, the current study has some shortcomings.First, the accuracy and predictive power of the models depend on the quality and reliability of the historical data.Therefore, in future studies, this paper should further improve the model to enhance its accuracy and predictive power, and expand the scope of the study to other countries and regions to compare the debt limits of different countries and regions, so as to provide more references for risk management in the global financial market.Second, the results of the study may be affected by other factors, which may lead to uncertainty in the conclusions.Therefore, this paper can suggest directions for further research to verify the reliability of the results.
Continuous updating of the index smoothing values realizes real-time forecasting.The basic formula is:   =   + (1 − )  .  and  −1 are the exponentially smoothed values of the m-th and m-1-st cycles.  is the measured value of the  ℎ cycle. (  ≤0≤1) is the smoothing coefficient.m(m=1,2,3,...) is periodic number.

Fig. 1 .
Fig. 1.Historical Closing Data Of The S&P 500 Index In The United States (Picture credit: Original).

Fig. 5 .
Fig. 5. Changes in the US dollar index during the Democratic Party's control period within one month before and after the resolution of the debt ceiling issue (Picture credit: Original).

Fig. 6 .
Fig. 6.Changes in the US Dollar Index During the Republican Party's Control Period within One Month Before and After the Debt Cap Problem was Solved (Picture credit: Original).

Table 3 .
Projections for the next 5 years of the debt ceiling

Table 5 .
Relative errors in predicted and accurate values,