Time-Series Modeling in Finance: ARIMA, GARCH, and Volatility Forecasting

Introduction

Time-series modeling in finance uses statistical tools to describe and forecast how prices and volatility evolve over time. You can use ARIMA models to capture trends and short-term autocorrelation in price series, and GARCH models to estimate conditional volatility that changes with market shocks.

Why does this matter to you as an investor or risk manager? Forecasts from these models feed position sizing, risk metrics, option pricing inputs, and algorithmic strategies. How precise can these forecasts be, and how should you interpret model outputs when markets surprise you?

In this article you will learn the assumptions behind ARIMA and GARCH, how to set up and validate them, how to interpret parameter estimates, and how to combine forecasts for practical trading and risk management. Real-world examples using $AAPL and $NVDA show calculations and decision-relevant outputs.

• ARIMA models target predictable structure in levels or returns after enforcing stationarity, useful for short-horizon price forecasts.
• GARCH models estimate time-varying conditional variance, crucial for volatility forecasting and risk metrics like Value at Risk.
• Stationarity, residual diagnostics, and parsimony are essential to avoid overfitting and misleading signals.
• Combine ARIMA expected returns with GARCH forecasted volatility to build probabilistic price paths and size positions consistently.
• Watch for heavy tails, leverage effects, and structural breaks; extend baseline models with EGARCH, TGARCH, or t-distributed errors when appropriate.

Modeling price trends with ARIMA

ARIMA stands for AutoRegressive Integrated Moving Average. It models a time series by combining autoregression, differencing for integration, and a moving average of past shocks. For financial prices you usually model log prices or returns to address nonstationarity and multiplicative effects.

When to use ARIMA

Use ARIMA when the series shows autocorrelation that is stable through time and when you believe past values contain information about short-term future movements. Stocks often require differencing because price levels are nonstationary. Daily log returns are more likely to be stationary and are the common input for ARIMA on returns.

Specification and interpretation

ARIMA(p,d,q) means p autoregressive lags, d differences, and q moving average lags. A simple ARIMA(1,1,1) on log prices yields a model for first differences, which is equivalent to modeling returns. The AR term captures momentum or mean reversion in returns, while the MA term captures serial correlation due to market microstructure or news clustering.

Practical ARIMA example

Suppose you model daily log prices for $AAPL with ARIMA(1,1,1) and estimate phi=0.20 for the AR coefficient, theta=-0.30 for the MA coefficient. If yesterday's log return was 0.002 and the last residual estimate was 0.001, the one-step ahead expected return is phi times last return plus theta times last residual. That gives 0.20 times 0.002 plus -0.30 times 0.001, or 0.0001 in log return units.

Convert that to a price forecast by exponentiating the log return over today's price. The forecast is small; ARIMA typically produces small short-horizon expected returns for liquid equities, but the sign and confidence interval matter for strategy design.

Modeling volatility with GARCH

GARCH stands for Generalized AutoRegressive Conditional Heteroskedasticity. It models conditional variance as a function of past squared shocks and past variance. GARCH captures volatility clustering where large moves tend to be followed by large moves and calm periods follow calm periods.

GARCH(1,1) mechanics

The common specification is sigma_t squared equals omega plus alpha times epsilon_{t-1} squared plus beta times sigma_{t-1} squared. Here epsilon is the model residual and sigma squared is the conditional variance. Alpha measures the reaction to recent shocks and beta measures persistence of volatility.

Numeric GARCH example

Estimate a GARCH(1,1) on $NVDA daily returns and find omega=1.0e-6, alpha=0.07, beta=0.90. If yesterday's residual squared was 0.0004 and yesterday's sigma squared was 0.00025, then today's sigma squared equals 1.0e-6 plus 0.07 times 0.0004 plus 0.90 times 0.00025. That computes to about 0.000307. The daily volatility in percent is the square root times 100 so sqrt(0.000307) times 100 is about 1.75 percent daily volatility.

Use that conditional volatility to scale position sizes, compute expected squared returns for risk forecasts, or feed option pricing models with time-varying sigma. Note how alpha plus beta near one implies high persistence and slow mean reversion of volatility.

Combining ARIMA and GARCH for forecasting

ARIMA models the conditional mean, while GARCH models the conditional variance. Fitting them jointly or iteratively gives a richer picture than either alone. You can first model the mean with ARIMA and then estimate GARCH on residuals to capture volatility dynamics.

Step-by-step workflow

1. Transform data to log prices or returns to achieve stationarity.
2. Use ACF and PACF plots and information criteria such as AIC or BIC to select ARIMA orders.
3. Estimate ARIMA and save residuals for volatility modeling.
4. Fit GARCH to residuals, test alternative error distributions like Student t to capture heavy tails.
5. Validate in-sample and perform rolling out-of-sample forecasts to assess stability.

After you have forecasts for the conditional mean and variance, construct probabilistic price paths. For one-step ahead forecasts, expected log price equals last log price plus ARIMA forecast. Variance equals GARCH forecasted sigma squared. A 95 percent predictive interval is expected log price plus or minus 1.96 times forecast sigma, assuming normal innovations. If you use t-distributed errors, adjust the critical quantiles accordingly.

Practical application: position sizing and VaR

Combine ARIMA expected return and GARCH volatility to compute a Sharpe-like signal for sizing. For example, if ARIMA predicts a 0.1 percent daily return and GARCH forecasts 1.75 percent daily volatility, the expected return divided by volatility is about 0.057 daily. Use that to scale allocation with risk budget constraints.

For Value at Risk, use the conditional mean and variance to estimate the loss percentile. For a 1-day 99 percent VaR, compute conditional mean minus z times sigma where z corresponds to the 99 percent quantile. If you model heavy tails, use empirical quantiles or t-distribution critical values to avoid underestimating tail risk.

Model validation and implementation

Validation is the linchpin of reliable forecasts. You must inspect residuals for autocorrelation and heteroskedasticity, test for normality, and conduct out-of-sample rolling window tests. Backtests should replicate realistic trading costs and lookahead constraints so you avoid optimistic performance estimates.

Key diagnostic checks

• ACF and PACF of residuals to check autocorrelation.
• Ljung Box test to assess serial correlation in residuals.
• ARCH tests on residuals to confirm volatility clustering remains after model fitting.
• QQ plots and jarque bera test to check for heavy tails and skewness.

If residual diagnostics fail, revisit model orders, try alternative distributions such as Student t, or add asymmetric volatility models like EGARCH or GJR-GARCH to capture leverage effects where negative shocks increase volatility more than positive shocks.

Implementation tips

Use rolling estimation with a fixed window and re-estimate parameters regularly to adapt to regime changes. Keep models parsimonious when you implement in production to reduce computational overhead and parameter instability. Automate residual monitoring and set thresholds for model retraining when diagnostics degrade.

Real-World Examples and Scenario Walkthroughs

Below are concrete examples tying the concepts together using realistic numbers and steps you can repeat with your data.

Example 1: Short-term trade signal for $AAPL

You fit ARIMA(1,1,1) on daily log prices for $AAPL over a year of data. The AR coefficient is positive but small at 0.12 and the MA coefficient is -0.20. ARIMA yields a one-day expected return of 0.05 percent, with a standard error from residuals of 1.2 percent. Fit GARCH(1,1) on ARIMA residuals and get sigma forecast of 1.1 percent daily.

Combine these to compute an expectation over risk. The expected return over volatility is 0.045, weak but actionable when combined with portfolio constraints and transaction costs. You might restrict execution to days with additional confirmations, such as momentum in sector ETFs or low implied volatility in options market.

Example 2: Volatility forecasting for risk limits on $NVDA

For a concentrated $NVDA exposure, you estimate a GARCH(1,1) that produces alpha 0.09 beta 0.88, indicating high persistence. The unconditional variance equals omega divided by 1 minus alpha minus beta. If omega is 1.2e-6 and alpha plus beta is 0.97, the long-run variance is roughly 4.0e-5 which annualizes to about 40 percent volatility. Use the one-day sigma forecast in margin calculations and overnight position limits to prevent breach of risk limits during earnings or macro events.

Common Mistakes to Avoid

• Ignoring stationarity. If you model price levels without differencing, you get spurious regressions that mislead forecasts. Always test and transform data to stationarity.
• Overfitting with high order models. Complex ARIMA or GARCH orders can fit noise. Favor parsimonious models and cross-validate with rolling windows.
• Assuming normal errors. Financial returns have heavy tails and skew. Use Student t or empirical residual bootstrapping for tail risk estimates.
• Neglecting structural breaks. Economic regimes change and parameters can shift suddenly. Monitor parameter stability and use change-point detection when needed.
• Misusing significance tests. A statistically significant coefficient can still be economically negligible. Assess economic significance and transaction costs before acting on signals.

FAQ

Q: When should I model prices versus returns?

A: Model returns or log returns when you need stationarity. Prices are typically nonstationary and require differencing. For long-term level forecasts such as valuation, modelling price levels with cointegration approaches may be appropriate, but short-term forecasting usually works on returns.

Q: Can ARIMA and GARCH predict large crashes?

A: They capture conditional dynamics and clustering but they do not predict rare, exogenous shocks reliably. Use stress testing, scenario analysis, and tail-robust methods to supplement these models for crash risk.

Q: Should I use t-distributed errors in GARCH models?

A: Yes, Student t-errors often better capture heavy tails in financial returns. They produce wider predictive intervals and more realistic tail risk estimates than Gaussian errors.

Q: How do I choose the model re-estimation frequency?

A: Base it on data frequency and market regime stability. For daily data, monthly or weekly rolling re-estimation is common. Re-estimate more often around known regime shifts or when diagnostics indicate degradation.

Bottom Line

ARIMA and GARCH are complementary tools for modeling the conditional mean and variance of financial time series. You can use ARIMA for short-term return forecasts and GARCH for time-varying volatility estimates that drive risk metrics and option inputs.

Always validate models thoroughly, prefer parsimony, and account for heavy tails and structural breaks. Combine mean and variance forecasts to build probabilistic price paths and to size positions sensibly. Keep monitoring and adapt your models as market conditions change, and remember no model replaces sound risk management practices.