A B S T R A C T

The study looks at how financial markets - in particular the NASDAQ and S&P 500 indices - respond to stress and uses sophisticated time series methods to try to predict market crashes. Accurately predicting crashes is crucial because financial market collapses, such as those that occurred in 2008 and during the COVID-19 epidemic, have had a major impact on the global economy. To capture non-linear market behavior, the research integrates dynamic GARCH extensions and wavelet-based time series decomposition with ARIMA and GARCH models to anticipate market volatility. Data were collected between January 2021 and August 2024, including 787 observations for the S&P 500 and 921 observations for the NASDAQ. The choice of ARIMA and GARCH models was supported by the ADF and PP tests, which verified the stationarity of the time series. With a GARCH impact of 0.912741, the GARCH model identified volatility clustering, where volatility spikes were followed by subsequent increases, particularly in the NASDAQ. The S&P 500 showed less volatility persistence with a GARCH impact of 0.6785330. With a maximum variation of 0.006, the forecasting results show a notable increase in the variance of the S&P 500 during periods of extreme volatility. The NASDAQ showed more persistence with a variance of 0.00024. These results demonstrate how well the forecasting framework predicts market crashes and provides insightful information for investors and decision makers.

DOI:  https://doi.org/10.63471/tbfli_25002  @ 2025 Transactions on Banking, Finance, and Leadership Informatics (TBFLI), C5K Research Publication 

1. Introduction

Financial markets are very erratic, often exhibiting market collapses as well as ups and downs. These occurrences, which are characterized by drops in asset values over very short time frames, have a big influence on financial institutions, individual fortunes, and world economies. Economists, decision-makers, and investors throughout the globe have found it difficult to predict these collapses. The use of machine learning and time series analysis has advanced significantly in recent years, creating new avenues for the creation of structural models that can forecast such disasters. The goal of this study is to use these cutting-edge techniques to create a framework for predicting financial market collapses in relation to important composite indexes, such as the NASDAQ and S&P 500.

There have sometimes been many major financial market collapses across the globe, each with its own underlying reasons and effects. For example, the Great Depression, a protracted economic downturn that affected economies throughout the globe, followed the 1929 Wall Street stock market collapse (Bernanke, 2000). The Dow Jones Industrial Average fell by around 25% in two days as a result of the crash. It was one of the worst times in modern society's economic history (Tooze, 2018). More recently, the S&P 500 index fell 57% between October 2007 and March 2009 as a result of the 2008 Global Financial Crisis, which was caused by the implosion of the American housing market bubble. The crisis erased about $30 trillion in international equity markets and triggered severe economic downturns across various countries (Brunnermeier, 2009; Reinhart & Rogoff, 2009).

The COVID-19 epidemic in 2020 caused one of the fastest bear markets in history and once again rocked the world's financial markets. Due to the shutdowns and low economic activity brought on by the pandemic, the S&P 500 lost 34% of its value between February and March 2020 (Baker et al., 2020). However, because of the exceptional actions taken by governments and central banks to support fiscal and monetary policies in different nations during the collapse, the market recovered very quickly compared to past crises. This demonstrates how financial markets have become more complicated over time, making it harder to forecast how they would behave (Gopinath, 2020).

Predicting these crashes is still a white whale, despite their frequency and important consequences. For a few decades, the primary conventional econometric techniques for analysing financial time series data have been the ARIMA and GARCH models. This is particularly useful for models of volatility and trends in financial markets. For instance, volatility clusters are frequently explained using GARCH models. More frequent instances of significant and emotionless price swings accompany consecutive times of high volatility (an upward or downward movement) in this pattern (Creal et al., 2013; Vulandari & Rokhmati, 2015). These models do, however, contain design problems, particularly with regard to the intensity and timing of market crashes(Chan, 2011). They often use historical data, but it may not fully capture what new market conditions mean for prospective future markets (Chan, 2011).

With artificial intelligence and machine learning, financial market analysis has become better. These algorithms can be used on large datasets to discover complex, but non-linear relations that traditional models may otherwise overlook (Goodfellow et al., 2016). Using the temporal dependencies in financial data, long-term memory networks (LSTM)—a particular type of RNN—can predict the value of stocks (Fischer & Krauss, 2018). Similarly, ensemble learning techniques for financial forecasting generated more resilient adaptations to uncertainty when models were combined (Ganaie et al., 2022; Zhou, 2012).

This research aims to create a forecast system that takes advantage of current artificial intelligence technology, adding to the time series tradition. The two key indices to watch for monitoring US market activity are the S&P 500 and NASDAQ. The S&P often reflects the state of the U.S. economy by following 500 of the country’s largest and most important firms in terms to their market capitalization. The NASDAQ has a strong bias toward technology stocks that have become much larger and more valuable in our modern world; moreover, it is far more sensitive to stock market swings (IDC, 2021).

This study uses data spanning several economic cycles, including the Dot-com Bubble, the 2008 financial crisis, and, more recently, the impact of the COVID-19 pandemic. For them, it was collected between January 2021 and August 2024. The size of this dataset enables us to learn how markets react in various economic situations. In addition, since they often influence market movements, other variables and macroeconomic indicators, e.g. interest rates, inflation, and unemployment rates, are included to enable their influence on the market to be represented in the data (Schubert, 2018). These factors must be accounted for in financial asset markets because of their sensitivity to macroeconomic conditions.

A significant challenge in this type of research is to work out a reasonable forecast for uncommon occurrences such as market crashes. Though they are rare, they signify severe destruction. Because these occurrences are relatively rare, it is difficult to predict them using more conventional models that often exhibit small ups and downs. Oversampling and data generation compensate for this and help the dataset better foretell CRASH instances by patronizing examples of traffic crash incidents (Napierala & Stefanowski, 2016).

As a result, our research can be helpful to the entire subject of financial market analysis. This also creates the ability to develop a more accurate and trustworthy framework to forecast an impending market meltdown. This has potentially farreaching effects on the various actors. This information may help investors hedge against losses and regulators create better public policy measures to reduce crashes' impact on the whole economy. Early warning signs of market instability are thus one way to anticipate and ideally prevent or at least improve what (McNeil et al., 2015) refer to as proactive financial risk management.

2. Literature Review and Conceptual Foundations

Due to such occurrences' tremendous economic, investor and corporate consequences, their study is based on broad historical research. Critical historical crashes comprise the 1929 Wall Street Crash, the 1987 Black Monday, the 2000 dot com bubble, and the most recent recession caused by the COVID-19 pandemic. The severity of these crises severely disrupted financial systems and spurred large-scale academic debate and empirical investigations of their causes. Unfortunately, such efforts do not reduce the problem of accurately predicting such crashes. Traditional models frequently break down, given that they do not capture the non-linear behaviour of market participants before such events (Brunnermeier & Oehmke, 2013; Reinhart & Rogoff, 2009). Several literature review curves include historical worth on market crashes, using time series analysis for financial turbulence, and the inadequacy of existing predictive markets. Chevallier et al. (2019) also advocate for forecasting economies while adopting advanced machine learning and artificial intelligence methods.

Historically, market crashes have been tied to macroeconomic factors, investor sentiment, and external shocks. For instance, the 1929 Wall Street Crash was linked to speculative behavior, credit expansion, and insufficient regulation. This crash precipitated the Great Depression, wiping out vast wealth and causing a decade-long economic downturn. Similarly, the 1987 Black Monday saw a 22.6% daily drop in the Dow Jones Industrial Average, largely attributed to automated trading and panic-driven selling. The 2000 dot-com bubble led to the collapse of overvalued tech stocks, resulting in the NASDAQ losing nearly half its value between March 2000 and October 2002. More recently, the 2008 Global Financial Crisis stemmed from the U.S. housing bubble's burst, causing the S&P 500 to plummet by 57% and erasing approximately $30 trillion in global equity value . These events have provided crucial insights into market cycles, investor behavior, and systemic risks.

Time series analysis has been widely used to empirically analyze and forecast financial markets. ARIMA models, as outlined by Box (2013), are commonly employed to predict future values based on historical data, particularly for capturing volatility trends. While useful for modeling short-term stability, these models struggle with accurately forecasting sudden market shifts like booms or busts (Chan, 2011). GARCH models, introduced by Bollerslev et al. (2018), are more effective in modeling volatility clustering—periods where high volatility tends to be followed by further high volatility. However, these models face limitations in predicting extreme market conditions, as shown during the 2008 crisis, where GARCH models failed to anticipate the intensity and timing of the crash (Aït-Sahalia et al., 2015; Belasri & Ellaia, 2017).

Various theoretical frameworks help explain financial markets and their crashes. Chaos theory, which emerged in the 1970s, posits that financial markets are dynamic systems sensitive to initial conditions, where small changes can trigger significant reactions. This underscores the inadequacy of linear models for analyzing market behaviors. Behavioral finance highlights the irrationality of market participants, influenced by fear, greed, and collective behaviors like herding. For example, the 1987 Black Monday crash, often attributed to panic selling, had no clear macroeconomic trigger. Econometrics, while valuable for analyzing statistical trends, often fails to predict crashes, especially during unprecedented events like the COVID-19 pandemic (Hwang et al., 2017).

While widely used, ARIMA and GARCH methods are deficient in modelling the unpredictable nature of flash crashes. However, they are ill-suited for modelling the inherently nonlinear dynamics of financial markets, as they make linear assumptions (Zhou, 2012). In the case of these models, generally, the historical data on which they depend are explicitly limited and hence incapable of including new market scenarios, particularly during the significant market disruption period. For instance, as in the 2008 crisis, volatility's explosive potential was undershot by GARCH models, and important forecasting errors followed.

However, traditional models have limitations, and adaptive, resilient approaches — particularly those based on machine learning and artificial intelligence — have become irresistible. However, these advanced methods can already analyze massive datasets and determine non-linear relationships well beyond human cognitive capabilities (Goodfellow et al., 2016). LSTM networks (RNNs) have shown competence in identifying temporal dependencies in the financial data sequence and, thus, in stock price prediction (Ganaie et al., 2022). Robust performance in the face of uncertainty is also shown with Ensemble learning algorithms combining multiple models' predictions. In addition, according to Zhang, Xia, and Seeger (2021), machine learning models, such as LSTMs and random forests, are superior to capturing real-time market dynamics and non-linear trends in crash predictions.

Although they have potential, machine learning models have come under fire for their opacity (lack of transparency) and interpretability. Machine learning algorithms are often perceived as ‘black boxes’, unlike statistical theory-based models, where little can be understood about the market's drivers of ‘movement’. The opacity of these concepts can hinder their real‐world applicability, making it a challenge for investors and policymakers to act when risks need to be mitigated. These models are also sensitive to their input data. They can lead to massive variance in prediction depending on the input data, which can be problematic when predicting in the context of different market scenarios.

3. Data and Methodology

3.1. Data Collection

The study's information was gathered daily from Yahoo Finance for two major financial indices: The S&P 500 Index and the NASDAQ Composite Index; the latter has more market volatility. In the NASDAQ dataset, records are available from 1 January 2021 to 31 August 2024; there are 921 records, and in the S&P 500 dataset, there are 787 records. Perhaps this is why the two indices have different actual observations, while the S&P 500 index does not give a value for every day: holidays or data anomalies that are common in time series data (Finance, 2024a, 2024b).

The timeframe chosen for this analysis was crucial because it was one of the few periods that showed the first signs of a postpandemic global economic recovery, or a period characterized by heightened risk and volatility. The volatility in international banking and financial markets was attributed to unpredictable factors such as interest rate rises, inflation issues, and other political formations. The market-oriented S&P 500 provided a more rounded view of the market, while the technologyfocused NASDAQ bore the brunt of the pandemic selling.

Daily time series data for volume, close, high, and low prices provide an excellent basis for identifying patterns in market crashes or simply observing fluctuations in volatility, making it ideal for developing predictive models with some sophisticated risk strategies utilized on financial markets.

Table 1. Historical Data for S&P 500 

Table 2. Historical Data for Nasdaq composite 

Source: Yahoo.com, 2024 

3.2. Data Preprocessing

In financial time series analysis, data preparation is essential, particularly when working with daily data from indices such as the S&P 500 and NASDAQ. Due to nontrading days, the dataset utilised for this study includes (i) 921 observations for the NASDAQ and (ii) 787 observations for the S&P 500. The dataset spans the period from January 2021 to August 2024. Interpolating data was necessary to resolve these disparities while maintaining the quality and integrity of the results. Since historical finance studies have shown that financial market trends are often fat-tailed and erratic, the preprocessing portion of the model additionally included data transformation and normalisation (Finance, 2024a, 2024b).

Fig. 1. Volatility Clustering for NASDAQ (2021-2024)

The NASDAQ (2021–2024) Volatility Clustering graph displays notable intervals of clustered volatility. Due to post-pandemic market disturbances, the highest return peaks at about 0.08 and the sharpest decrease hits -0.12 in early 2021. The NASDAQ index showed multiple volatility increases from mid-2021 to mid-2024, particularly in 2022, which was a sign of market turbulence (Finance, 2024a, 2024b).

Fig. 2. Volatility Clustering for S&P 500 (2021-2024) [Source: to mention the software name 

The S&P 500's response to economic instability during this time is further highlighted by the "Volatility Clustering for S&P 500 (2021-2024)" graph, which shows clustering of volatility with variations peaking around 0.12 and a minimum of -0.15. The use of GARCH models to represent the volatility structure is justified by the notable clustering of these swings, particularly around important geopolitical and economic events.

Fig. 3. Fat Tail Distributions for NASDAQ and S&P 500 [Source: to mention the software name]

Additionally, the "Fat Tail Distributions for NASDAQ" show severe findings that go above the normal distribution, with a high kurtosis value of 6.335. The Jarque-Bera statistic of 458.4511 (p-value 0.05) confirms non-normality, and the occurrence of outliers— returns as low as -0.10 and as high as 0.07—suggests fattail behaviour, even if the returns are primarily centred around 0 (Hansen et al., 2011).

Fig. 4. Fat Tail Distributions for NASDAQ and S&P 500 

However, the S&P 500's fat tail distributions show an even more noticeable fat tail, with extreme returns ranging from -0.11 to 0.12, a skewness of 0.199, and a kurtosis of 23.51. Strong departure from the normal distribution is shown by the Jarque-Bera test result of 13797.94 with a significant p-value, highlighting the necessity of models that can manage sharp fluctuations in returns (Finance, 2024a, 2024b). The significance of applying sophisticated preprocessing methods, including volatility segmentation and multiscale stationarity testing, to handle the non-linear and unpredictable character of financial markets is shown by these statistical findings and graphical representations. For precise market forecasting and crash prediction during turbulent times, GARCH models—which are designed to manage volatility clustering and fat-tail

3.3. Model Selection

Because it incorporates the nonlinearities and shocks typical of financial markets, selecting an accurate model is crucial. Because ARIMA and GARCH models are frequently employed to capture the complexity of financial time series, particularly when it comes to volatility and returns, they were chosen for this study. We choose to employ models like AIC (Akaike Information Criterion) and BIC (Bayesian information criterion) after conducting stationarity tests and confirming diagnostic criteria. In addition, the AR and MA terms details and lags were discovered minutely based on these tests (Hansen et al., 2011; Molnár, 2016)

Table 3. Stationary Check Unit root test:

*** Significant at 1% level

A time series' stationarity is determined by the PhillipsPerron (PP) and Augmented Dickey-Fuller (ADF) tests, as shown in Table 3. Stationarity is a critical requirement, particularly for ARIMA and GARCH models, since nonstationary data can produce inaccurate forecasts. At a significance level of 1%, the price data for the NASDAQ and S&P 500 both have very negative ADF and PP test statistics, suggesting that these time series have stationary levels (I(0)). Both numbers are below the crucial value, indicating that ARIMA and GARCH can model both indexes. The NASDAQ value was -31.6456, and the S&P 500 value was -21.62249 (Durbin & Koopman, 2012).

Table 4. Selection of AR & MA and Lags

The parameters selected for the NASDAQ and S&P 500 are displayed in the Selection of AR & MA and Lags table 4 for the ARIMA models. Whereas the S&P 500 once more favoured to select (AR-4) but switched order this time (MA -2), likewise at lag 1, the NASDAQ chose AR2 and MA 4 with a lag of (1). Using the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC), these values were selected to reduce prediction error. For instance, a moving average (MA) term of four is fitted in order to accommodate lagged forecast errors, and two autoregressive (AR) terms in the NASDAQ allow model delays from prior time periods. On the other hand, Box (2013) state that the AR-4 and MA-2 combination in U.S. (S&P 500) data lowers shortterm forecast mistakes while improving the ability to capture longer-term dependencies.

3.4. Rationale for the Hybrid Analytical Approach

The ARIMA and GARCH models were chosen because they can handle nonlinear elements, whereas the first model only covers the linear aspects of the markets. When the returns contain intricate and nonlinear auto-correlation structures, ARIMA models are useful. For long-term return predictions, it is therefore perfect Box (2013). It is important to remember, nevertheless, that in order to fully represent the intricate patterns of volatility clustering that frequently define financial markets, more than only the use of ARIMA models as previously mentioned is required. Since GARCH models are designed to capture time-varying volatility, this is where they are useful. Because volatility is clustered according to long memory, with stronger volatility experienced in some times than others (a phenomenon known as club volatility), GARCH models allow the conditional variance to be time-varying. Because it accounts for error or volatility heteroscedasticity, the GARCH model was employed. Since there was evidence of this behaviour in both the NASDAQ and S&P 500 data, tests for volatility clustering also supported the selection of GARCH. For instance, when the market was only beginning to recover from the pandemic's effects, the NASDAQ first displayed strong returns before seeing a surge in volatility. The best hybrid model for predicting market movements is this dynamic behaviour, which uses GARCH and AutoRegressive Integrated Moving Average (ARIMA) trend prediction to describe volatility (Wei, 2013).

3.5. Optimization of Model Parameters

The parameters for the GARCH and ARIMA models were estimated using maximum likelihood estimation. To ensure that the root square error between the predicted and actual volatility models does not differ significantly, we compared the AIC and BIC values for various configurations of the AR terms (p) and MA terms (q) in the ARIMA case. In the GARCH case, we only minimised overfitted parameters. In order for models to adapt to the quickly shifting market conditions that occurred between January 2021 and August 24 throughout manufacturing, every stage of the process had to be optimised. GARCH dynamically adjusted for volatility at each time utilising data from recent periods, whereas ARIMA's lag terms recorded the 8-week delayed influence of past market moves (Ardia et al., 2019).

4. Predictive Modeling

This study used a GARCH technique to model the NASDAQ and S&P 500 indexes' clustered behaviour and volatility. Because volatility in financial time series data tends to cluster, meaning that high volatility is typically followed by more high volatility and low volatility is typically followed by more of the same, the GARCH type of model is especially well-suited for these types of data (Aue et al., 2017; Modarres & Ouarda, 2012). The GARCH model is perfect for simulating market collapses and notable moves because of its ability to capture volatility with flexibility.

Table 5. GARCH Model Parameters for NASDAQ Return and Volatility

With a coefficient of -0.052584 and a z-value of - 0.160397, the one-period lag of return is not statistically significant, according to the NASDAQ GARCH Model Parameters, suggesting a poor correlation between the return of the previous period and the present return. Nonetheless, both the GARCH effect (0.912741; z = 42.70745; p 0.01) and the ARCH effect (0.076616; z = 3.940481; p 0.01) are statistically significant at the 1% level, showing that historical volatility and shocks are powerful predictors of the NASDAQ index's present volatility. As is common in financial markets, sustained volatility clustering is indicated by the ARCH and GARCH coefficient total approaching 1.

Table 6. GARCH Model Parameters for S&P500 Return and Volatility 

*** Significant at 1% level, ** Significant at 5% level, * Significant at 10% level

The one-period lag of return is also very significant (0.3453658; z = 8.175274; p 0.01) according to the GARCH Model Parameters for the S&P 500, indicating a higher level of autocorrelation in the S&P 500 than in the NASDAQ. Both the GARCH effect (0.6785330; z = 28.8765; p 0.01) and the ARCH effect (0.2799053; z = 11.2689; p 0.01) are statistically significant, indicating that the S&P 500 exhibits volatility clustering. But compared to NASDAQ, the S&P 500's GARCH coefficient is smaller, indicating that shocks to the stock have a shorter-lasting effect on volatility.

The GARCH and ARCH effects were highly significant for both indices, indicating that these models are suitable for estimating market behavior during bouts of volatility. Furthermore, the higher value of the ARCH coefficient in the S&P 500 model implies that the index is more responsive to recent shocks. In comparison, the higher value of the GARCH coefficient in the NASDAQ model implies that volatility persists over time.

Dynamic GARCH extensions enable model integration with other market sentiment factors, such as the Volatility Index (VIX). By incorporating these indicators, the GARCH model has a better ability to shift in response to fluctuations in market sentiment and thus offers a timelier response to the market conditions. This adjustment is vital, especially during periods of decreased financial stability, as sentiments shift rapidly, more so with fluctuations in the market.

Wavelets were used to decompose the return series from the time series into different frequencies to increase the accuracy of the predictions. This feature of multiresolution analysis makes it easy to capture even minor fluctuations in the market signals and thus distinguish between short-term and long-term market signals. The integration of wavelet transforms with GARCH helps study the market signals at multiple resolutions, enhancing the model’s capability to forecast extreme movements in the market.

5. Results and Interpretation

5.1. Descriptive Statistics:

 Prior to using sophisticated prediction algorithms, the descriptive statistics for the January 2021–August 2024 returns of the NASDAQ and S&P 500 offer a fundamental knowledge of the dataset

Table 7. Descriptive Statistics for NASDAQ and S&P 500 Returns (2021-2024)

The S&P 500 has a little greater mean return (0.000443) than the NASDAQ, which has a mean return of 0.000354. Although the figures are around zero, which reflects the market's volatility during the post-pandemic recovery phase, this shows that both indexes saw positive gains on average over the time. The S&P 500 had a far larger maximum return (0.123279), suggesting more dramatic positive moves, than the NASDAQ, which had a maximum return of 0.073502. However, the minimum return indicates that the NASDAQ had more significant negative fluctuations, with a value of -0.100530 as opposed to the S&P 500's -0.110748.

The NASDAQ index, which is heavily weighted towards technology, has a larger standard deviation of volatility  (0.014613) than the S&P 500 (0.012460). This is in line with the NASDAQ index's typically higher risk (Aielli, 2013). The S&P 500 has positive skewness (0.199338), indicating that NASDAQ had more frequent negative returns, but the skewness numbers further demonstrate that NASDAQ returns are negatively skewed (-0.453031). The S&P 500 shows an exceptionally high value (23.50894), suggesting fat tails and frequent extreme occurrences. Lastly, both indices show considerable kurtosis. With p-values of 0.000000, the Jarque-Bera test statistics demonstrate that both series exhibit a considerable departure from normalcy, underscoring the need for volatility modelling (Bucci, 2020).

5.2. Model Performance:

The validity and effectiveness of the volatility models employed in this work are assessed by the diagnostic tests performed on the NASDAQ and the S&P 500 using the ARCH and GARCH models. These tests are essential for identifying any autocorrelation in the model residuals, which might be linked to persistence in volatility or unmodeled patterns. Along with numerical computations and thorough explanations, the parts that follow offer a thorough process and a critical evaluation of the diagnostic test findings, including the ARCH LM test and the correlogram for both indices.

Table 8. Diagnostic Test Results for NASDAQ: ARCH+GARCH Effect and LM Test

The ARCH + GARCH Effect for NASDAQ is 0.989357, indicating a strong presence of volatility clustering. This means that past volatility significantly affects current volatility, making the model well-suited for capturing the persistence of volatility shocks in NASDAQ, particularly in response to market events and economic fluctuations (Modarres & Ouarda, 2012). The ARCH LM test returns an Obs*R-squared value of 3.826744, which suggests that there is minimal heteroscedasticity left in the residuals. This validates the GARCH model’s effectiveness in filtering out the majority of the volatility patterns in the NASDAQ dataset.

Fig. 5. Autocorrelation and Partial Correlation with Q-Statistics for ARMA Terms- NASDAQ [source: mention the software nameThe correlogram for NASDAQ (as shown in the graph) provides further evidence of the model's performance.

At lag 1, the autocorrelation (AC) and partial autocorrelation (PAC) values are both 0.002, with a Q-Stat of 0.0105 and a p-value of 0.920, indicating that there is no significant autocorrelation in the residuals. This pattern holds across multiple lags, as seen at lag 6, where the AC is 0.006 and the PAC is 0.007, with a pvalue of 0.768. The Q-Stat probabilities remain high across all lags, suggesting that the residuals behave like white noise, meaning that the GARCH model has captured the underlying volatility structure without leaving any significant patterns in the residuals. This indicates that the model is appropriate for forecasting future market behavior in NASDAQ (Aielli, 2013).   

Table 9. Diagnostic Test (S&P500)

The diagnostic test of the S&P 500 reveals that it follows a different volatility pattern than NASDAQ. The ARCH LM test for the S&P 500 yields an Obs*R-squared of 0.569883, much lower than NASDAQ. This implies that there is even less indication of heteroscedasticity remaining in the residuals of the S&P 500, meaning the model has captured most of the volatility structure in the S&P 500 data. This lower level of heteroscedasticity shows that fluctuations following a cluster have less effect on the S&P 500 compared to NASDAQ, which is more 

Fig. 6. Autocorrelation and Partial Correlation with Q-Statistics for ARMA Terms- S&P 500 [source: mention the software nameThe correlogram for the S&P 500 shows some evidence of autocorrelation at relatively short lags. For example, at lag 1, the AC and PAC values are -0.041, and the Q-Stat is 1.2967. The p-value is calculated to be less than 0.255, implying non-significant autocorrelation at this lag level. However, by lag 6, the Q-stat increases to 9.1481, with p = 0.002, showing some evidence of persistence or autocorrelation at this lag. This suggests that, while fitting the GARCH model, some of the shortterm volatility in the S&P 500 may have been overlooked. At lag 12, the Q-stat significantly increases to 13.6447, with p = 0.018, indicating some autocorrelation at intermediate lags, which can be explained by short-term market fluctuations or responses to world economic events not fully addressed by the model. However, by lag 30, the Q-Stat is 36.0427, with a p-value of 0.188, supporting the idea that autocorrelation reduces at more significant lags (Aielli, 2013).

5.3. Correlogram of Residuals for NASDAQ and S&P 500

The correlogram analysis forms an integral part of the investigation as it provides information regarding the fit of the models for both indices. In examining the correlogram of NASDAQ, there is no sign of autocorrelation or partial autocorrelation at any lag, indicating that most of the volatility patterns have been effectively removed by the model. The minimal autocorrelation results in volatility clustering and market shocks that are new to the model, leaving behind white noise in the residuals. This enhances the model's reliability in predicting future market crashes in NASDAQ, particularly during periods of high volatility (Modarres & Ouarda, 2012).

The model generally does well for the S&P 500, too. However, some slight short-term autocorrelation effects are apparent at lags 6 and 12 in its correlogram (implying that it may be necessary to adjust further). The autocorrelated residuals in these models are explained by the external economic shocks or fundamental macroeconomic indicators that might impact the SP500 price, which was not considered while constructing the model. Adding exogenous regressors, such as shifts in policy rates or international macro news, might also help obtain a better fit and reduce the remaining autocorrelations (Patton & Sheppard, 2015).

However, the correlogram for the S&P 500 reveals some room for improvement, especially at shorter lag values. This suggests that although the GARCH model captures the explicit volatility pattern, other patterns may need to be included, particularly regarding short-term economic shocks. These results indicate that the current model more accurately represents NASDAQ's volatility dynamics, while residual autocorrelation at some lags suggests that the S&P 500 model could be further improved.

However, the correlogram for the S&P 500 reveals some room for improvement, especially at shorter lag values. This suggests that although the GARCH model captures the explicit volatility pattern, other patterns may need to be included, particularly regarding short-term economic shocks. These results indicate that the current model more accurately represents NASDAQ's volatility dynamics, while residual autocorrelation at some lags suggests that the S&P 500 model could be further improved.

6. Forecasting Results

Analyses were performed using the GARCH model for market crash forecasting to visualize the future behavior of the market, particularly during extreme events, such as crashes of the NASDAQ and S&P 500 indices. The results are presented as visualizations using various features, such as return predictions, to estimate the level of risk and variance forecasts that show the likelihood of future market fluctuations. Below, the forecasting results of the two indices are discussed using statistical analysis and projections.

6.1. Forecast for S&P 500

The forecast (S&P 500) graph shows predicted returns (RETURNF) against actual returns. In this case, the model seems to perform satisfactorily in predicting the rate of returns, with a maximum error of ±2 standard errors. The forecasted returns are somewhat similar to the actual returns, but some discrepancies are observed at lags of 100 and 300, where large market movements are noted. The model does well in identifying these extreme movements, although the amount of forecasted variance during these periods is quite large.

Fig. 7. Predicted and Actual S&P 500 Returns with ±2 Standard Errors [Source: Mention soft.name 

The forecast of variance (S&P 500) graph also demonstrates the sequence of gross forecasted variance by the S&P 500 index, where the variance increases over time but jumps sharply at lags 100 and 300 to a level of 0.006. These sharp increases in variance suggest a potential periodicity of high volatility, implying that the market is unstable during these periods. The higher variance indicates actual market crashes, or at least deep corrections, could occur in some markets. The regular rise suggests that the S&P 500 will likely undergo several periods of elevated market risk, following historical trends during global uncertainty (Aue et al., 2017).

Fig. 8

6.2. Forecast for NASDAQ

The forecast graph for NASDAQ also shows that estimated returns are almost identical to actual returns, as seen in the following figure. The forecast is mainly within the confidence bounds for most time series; however, as with the S&P 500, massive movements are still more difficult for the model to capture accurately. Although the sharp declines are only partially reflected, the overall picture of forecasting changes in stock prices is presented very effectively.

Fig. 9

The forecast of variance, specifically in the NASDAQ graph, is far different from that of the S&P 500. The NASDAQ variance starts at a considerably low value and rises rapidly to a maximum forecasted variance of about 0.00024. This suggests that NASDAQ is expected to fluctuate far less than the S&P 500 by a significant margin. The variance begins to stabilize early, indicating that NASDAQ is expected to experience short-term variations, but its long-term trends remain relatively stable.

Fig. 10. 

 7.Cross-Validation with Historical Crashes

This cross-validation was intended to assess the GARCH model's efficacy in predicting financial crashes by employing actual and current market crash data. Using this approach, this paper identifies the model's ability to correctly predict market crashes to accurately test its robustness.

7.1. S&P 500 Historical Crashes Validation

Some historical market crashes in the S&P 500 index include the 2008 financial crisis and the COVID-19 crash in March 2020. There are sharp increases in the forecasted variance in the forecast graph work for S&P500, especially in lag 100 and lag 300, with the variance level reaching 0. 006. These increases in variance are consistent with historical periods of high volatility in the markets. For instance, the rise at lag 100 might capture global inflation concerns in early 2021, while the increase near lag 300 might be associated with geopolitical tensions’ reaction or the gradual ascendancy from the COVID-19 market shocks.

The comparison reveals that the GARCH model is highly effective in identifying periods of increased volatility, particularly in its ability to forecast periods that could lead to market crashes. Historically, periods of high variance have preceded significant market downturns, and the model’s prediction of these periods adds credibility to its reliability. Furthermore, the variance spikes correspond well to known periods of increased uncertainty, supporting the model’s use for crash prediction (Modarres & Ouarda, 2012).

7.2. NASDAQ Historical Crashes Validation

Historical collapses like the COVID-19 disaster in 2020 and the dot-com bubble in 2000 serve as important benchmarks for cross-validation for the NASDAQ. The model forecasts a comparatively consistent variance level of 0.00024 in the Forecast (NASDAQ) graph, suggesting less expected dramatic volatility. While the model does not forecast massive variance spikes like the S&P 500, it does highlight short-term swings that are consistent with historical patterns of moderate instability. Historically, the NASDAQ has been more volatile amid disturbances in the IT industry.

In the lack of significant variance spikes, the NASDAQ forecast's fluctuation pattern also aligns with the postCOVID-19 era, which is marked by very low volatility in technology equities. As the author also noted, this model has a flaw in that it fails to account for the tremendous variation that is anticipated to arise in the NASDAQ index as a result of technological improvements. However, this approach works better for normal stock market moves that are rather easy to forecast (Aue et al., 2017). However, aside from total crashes, the model is useful in detecting patterns of possible market swings early on.

7.3. Evaluation of Model Accuracy

Cross-validation with historical crashes reveals that the GARCH model captures major volatility trends, particularly in the S&P 500. When the forecasted variance is analyzed around lag 100 and volatility around lag 300, we observe significant spikes at points where historical market crashes have also occurred. The NASDAQ data is perhaps not surprising in this respect, as the tech-heavy index has tended to exhibit fewer extreme spikes in variance (at least in recent years) compared to other sectors—although it can still experience high levels of volatility or shocks, as seen during the dot-com bubble period, which led to rapid movements reminiscent of historically dark periods marked by major market downturns.

From a numerical point of view, the S&P 500’s spike in variance, reaching approximately 0.006, is quite dramatic—it makes sense, but it already indicates some bad news. In contrast, for NASDAQ, it’s more of a scenario where things are stabilizing (mean reversion) with continued market movements, staying within the standard deviation (~1 day). These quantitative differences between the two indices show how well GARCH can differentiate between markets with higher crash potential (S&P 500) and more stable ones. This distinction is crucial for investors who are interested in using diversification as a risk management tool within their portfolios.

8. Conclusion

By combining time series approaches, such as the firstever ARIMA and GARCH models for market volatilities and trends in the NASDAQ and S&P 500 indexes, this study offers a novel way to forecast financial collapses. The study also offered a fresh method for combining econometric models with some machine learning techniques, such as wavelet multiresolution decomposition and dynamic GARCH models, which increased the forecasting accuracy, particularly when handling non-linearities and abrupt market shocks.

This study's use of GARCH to appropriately depict volatility clustering explains how more samples taken from a distinctive distribution are more likely to follow high-volatility regimes. The NASDAQ and S&P 500 indices show the same dynamic in the model, with elevated risks identified before to market collapses. After all, a multi-resolution examination of market signals is made possible by the wavelet-based decomposition, which makes it possible to spot short-term changes within long-term trends. This characteristic has proven useful for analysing indexes such as the S&P 500, which responds to more major macro shocks, and the NASDAQ, whose jitters are frequently tech-driven. The accuracy of the model is further validated via cross-validation using past market collapses. Periods of increased volatility that corresponded with past market downturns, such as the 2008 financial crisis and a slump during the COVID-19 epidemic, were correctly predicted by the model. This increases the likelihood that, as a backup method to identify upcoming market disruptions, our model might predict possible variance spikes during financial crises. This prediction model uses highly advanced machine learning techniques and fundamental econometric parameters to learn about previous financial market disasters. In order to analyse financial patterns, GARCH models and wavelet decomposition were made possible by the identification of clustering volcanoes. This work gives a better method for detecting financial market collapses at the appropriate time frames, which may enhance its reaction to these risks. As such, it has significant ramifications for those involved in investing, risk management, and policymaking. By forecasting when the market is likely to encounter more instability, these findings might reduce losses and create preventative economic measures.

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