## 3 Results and Discussion  
Modeling discrete data responses with a continuous smooth function can be used as a future predictor, when fit, to anticipate similar behavior in economics, finance, public health, sociology, etc. [13]. It can also be used as a base to derive momentum oscillators and be implemented in learning algorithms crucial to profitability of technical stock trading, for instance [14; 15; 16; 17]. In this section, we apply our method on examples from the shock that hits the economy early in the onset of COVID-19 pandemic. During the early onset of COVID-19 in March and April of 2020, financial markets were very volatile given the uncertainty about the future. Financial literature showed that uncertainty about the near-term or long-term future leads to large spreads in bond markets and high volatility in the financial markets. In [18], the authors even show that stock market uncertainty has important cross-market pricing influences and that stock-bond diversification benefits increase with stock market uncertainty. In addition, Federal Reserve intervened to calm the markets by lowering the key interest rate and creating various special lending programs to provide liquidity for the financial system and for local and state governments. All these unique developments created a chaotic scenario for many of the economic variables to experience huge swings. Indeed, the effects of COVID-19 developments and policy responses on the US stock market and bond markets are without historical precedent [19]. We choose this period given its large uncertainty about the future and its erratic oscillations in all economic and financial variables during this period. We included examples from Dow Jones Index Average (DJIA), Standard and Poor's 500 (S&P 500), two treasury rates such as 3-month treasury bond rate and 10-year treasury bond rate, and unemployment rate. We collected data from publicly available sources such as Federal Reserve Economic Data, which is published by Federal Reserve Bank of St. Louis [20]. For each data sample, a graph of the fitting function versus raw observations is presented along with a pointwise error plot. The method was capable of fitting data with frequent zigzagging, especially in the DJIA and S&P 500 index examples. In the following examples the error figures represent the absolute value of the difference between the response observations \(y_{s}\) and the corresponding values of the fitting function, \(error=|f(s)-y_{s}|\). As in Equation (1) in the previous section, we denote \(N\), the order of the polynomials in the numerator, and the denominator of the fitting function \(f\). In Figures 1 and 2, we show how the stock market indices reacted to the news of the COVID-19 pandemic. DJIA is one of the oldest and most commonly used equity indices in the market, representing a price weighted index of 30 large companies. The figures show that DJIA lost more than 32% of its value within a few days in mid-March, and it was able to recover most of it back within a couple of weeks. The figures include only business days since there are no data for stock market and bond market during the weekends. In Figure 2, we show the trend of S&P 500, which tracks the performance of 500 large companies listed on the stock exchanges in the US. Even here we can see a drop of about 28% in the value of the index and a quick recovery of most of that loss within a few weeks. Our model is able to predict with low error these zigzagging trends. In Figures 3 and 4, we show the trend of the two treasury bond rates. We chose to focus on these two bonds rates because one is very short-term (3 months) and one is long-term (10 years). The short-term one is the cost of borrowing on money markets for a short period, while the long-term one is a proxy for the cost of investment. Many companies choose to borrow for investment and borrow based on the cost of borrowing for a long-term project, which is usually at a higher premium compared to that of the treasury bond.