ARIMA and GARCH Explained: Forecasting a Series and Its Volatility

Understanding ARIMA and GARCH — simply put

Many economic series have memory. This month's inflation says something about next month's, and a turbulent day in the currency market tends to be followed by another. ARIMA models describe that memory in the level of a series and use it to forecast. GARCH models describe it in the size of the shocks, the volatility, and use it to forecast risk. Central banks use the first kind to project inflation, and banks and asset managers use the second to measure how much they could lose.

How does it actually help?

Consider monthly inflation in a European economy. Prices trend upward, so the model works with the monthly change in the log price index, π_t, and asks how much of last month's surprise carries into this month. The simplest useful model is π_t = μ(1 − φ) + φπ_t−1 + ε_t, where μ is average monthly inflation, φ the share of last month's deviation from that average that persists, and ε_t the new surprise. Because π_t is the first difference of the log price level, this is an ARIMA(1,1,0) model of prices: one AR term, one difference, no MA term.

In the currency market the question changes from where the exchange rate goes to how much it moves. A GARCH(1,1) model sets today's variance of returns as σ²_t = ω + αε²_t−1 + βσ²_t−1: a constant ω, plus a reaction α to yesterday's squared surprise ε²_t−1, plus a share β of yesterday's variance σ²_t−1. A large move yesterday raises the expected size of today's move, which is exactly the clustering seen in markets.

Why does this matter?

Inflation forecasts feed directly into interest-rate decisions, wage settlements and budget plans, and ARIMA models remain the benchmark that more elaborate forecasting models must beat. Volatility forecasts set the capital banks hold under value-at-risk rules, the prices of options and the margins on futures contracts. Treating volatility as constant understates risk exactly when markets turn turbulent, which is when an accurate risk measure matters most.

What makes these models reliable?

ARIMA needs a series that is stationary after differencing, with a stable mean, variance and correlation pattern, and residuals that are white noise, with nothing left to forecast. GARCH needs the mean model to be right first and needs α + β < 1, so that a burst of volatility eventually fades. Both are only as good as the stability of the past. A structural break, such as an energy shock or a new monetary regime, can make a well-fitted model forecast badly.

The takeaway

ARIMA forecasts the level of a series from its own past; GARCH forecasts how volatile it will be. The rest of this guide fits both to example data, then sets out the assumptions and the full mechanics.

Worked example 1: forecasting monthly inflation with ARIMA

The first example uses a monthly consumer price index from January 2014 to December 2024, built to resemble a low-inflation European economy with a 2% target. It is a teaching sample, and every statistic below is computed exactly from it. The table shows annual inflation each December.

Step 1: stationarity. An augmented Dickey–Fuller test on the log price level gives a p-value of 0.98: the level has a unit root and must be differenced. On the monthly change the p-value is below 0.001, so one difference is enough, and d = 1.

Step 2: identification. The autocorrelation function of the monthly change starts at 0.46 and dies out gradually, to 0.18 and 0.14 at lags 2 and 3. The partial autocorrelation is 0.46 at lag 1 and close to zero after it, at −0.04 and 0.09. A partial autocorrelation that cuts off after one lag while the autocorrelation decays is the signature of an AR(1) process.

Step 3: estimation and selection. Four candidate models are fitted to January 2014 to December 2023, holding 2024 back to test the forecast.

ARIMA(1,1,0) has the lowest AIC and BIC, and its residuals pass the Ljung–Box test. ARIMA(0,1,1) fails it: a single moving-average term leaves autocorrelation behind. The extra terms in the two larger models do not improve the fit enough to pay for themselves. The estimated model is π_t = 0.187(1 − 0.476) + 0.476π_t−1 + ε_t, with a standard error of 0.077 on φ̂. Average inflation is 0.187% a month, about 2.3% a year, and just under half of any month's surprise carries into the next.

Step 4: forecasting. Made in December 2023 for the next 12 months, the model projects cumulative inflation of 2.38% over 2024, with a 95% interval from 0.28% to 4.52%. The outcome was 2.05%, inside the interval. Month by month, though, the forecast errors were no smaller than those from assuming average inflation every month: a root mean squared error of 0.152 percentage points against 0.149. Persistence of this size helps for a month or two, after which the forecast converges to the average. Beating a naive benchmark is the test any forecasting model has to pass, and the margin is often small.

Worked example 2: modeling exchange-rate volatility with GARCH

The second example uses 2,000 daily returns, about eight years of trading days, built to resemble movements in the euro–dollar exchange rate. Returns are percentage changes from one day to the next. Their standard deviation is 0.50% a day, and calm stretches alternate with turbulent ones. Squared returns are strongly autocorrelated: an ARCH Lagrange multiplier test with five lags gives a p-value far below 0.001. The distribution also has fat tails, with excess kurtosis of 1.95, meaning more extreme days than a normal distribution allows. The model is therefore a GARCH(1,1) with Student-t errors.

α + β = 0.979, so volatility is highly persistent. A shock to volatility decays by about 2% a day, giving a half-life of about 32 trading days, roughly a month and a half. The long-run daily volatility implied by the model, √(ω / (1 − α − β)), is 0.51%, or about 8.1% a year once scaled by √252. Conditional volatility ranged from 0.32% on the calmest day to 1.08% on the most turbulent, a more than threefold swing that a constant-variance model ignores.

The diagnostics confirm the fit. After estimation, the squared standardized residuals pass the Ljung–Box test (p = 0.43) and the ARCH test (p = 0.19), so the volatility pattern has been captured. The Student-t version also fits clearly better than the normal one. The forecast starts at 0.38% for the next day, below the long-run level because the sample ends in a calm spell, and rises to 0.43% at one month as volatility reverts toward its average.

The assumptions

• Stationarity after differencing. The differenced series has a constant mean, variance and correlation pattern. AR and MA coefficients describe one stable pattern of dependence. If the mean or variance drifts, the estimates average across regimes and the forecast intervals are wrong. Differencing a series that is already stationary does damage too, adding a moving-average component that cannot be inverted.
• White-noise residuals. The residuals have no autocorrelation. Any pattern left in them is predictable information the model has ignored, and the standard errors and forecast intervals are calculated on the premise that none remains.
• Stability and invertibility. The AR part must be stable (|φ| < 1 in an AR(1)) and the MA part invertible (|θ| < 1 in an MA(1)). Without stability, shocks never die out and the series has no mean to revert to. Without invertibility, several different models fit the data equally well and the shocks cannot be recovered from past values.
• Positive, stationary GARCH variance. GARCH requires ω > 0, α ≥ 0, β ≥ 0 and α + β < 1. The first three keep the variance positive. The last ensures a finite long-run variance ω / (1 − α − β) to which volatility returns. When α + β = 1 (integrated GARCH), shocks to volatility never fade and volatility forecasts never revert.
• A correct mean model. GARCH models the squared residuals of the mean equation. If the mean equation leaves autocorrelation behind, it shows up in the squared residuals and is mistaken for volatility clustering.
• Parameter stability. Estimation assumes one data-generating process over the whole sample. A regime change, such as the 2022 energy shock or a switch in monetary policy strategy, breaks that assumption, and parameters estimated on the old regime forecast the new one badly.

The formula

This section sets out the models formally. Readers who need only the intuition can skip to the limitations. Written with the lag operator L, where L y_t = y_t−1, an ARIMA(p, d, q) model is:

Parameters are estimated by maximum likelihood, and competing orders are compared with information criteria that reward fit and penalize extra parameters:

The Ljung–Box statistic tests whether the first m residual autocorrelations are jointly zero:

For the AR(1) model of the monthly change, the h-step forecast made at time T converges geometrically to the mean:

The GARCH(1,1) model combines a mean equation and a variance equation:

The long-run variance, the half-life of a volatility shock and the h-step variance forecast follow directly:

Engle's ARCH Lagrange multiplier test regresses the squared residuals on m of their own lags; T times the R² of that regression is distributed χ² with m degrees of freedom under the null of no volatility clustering.

Common extensions follow the same logic. Seasonal ARIMA adds AR, difference and MA terms at the seasonal lag, written (p, d, q)(P, D, Q)_s with s = 12 for monthly data, and ARIMAX adds explanatory variables; our review of a SARIMAX inflation forecasting study shows one in use. Asymmetric GARCH models such as EGARCH and GJR-GARCH let bad news raise volatility more than good news of the same size, a pattern common in equity markets.

When ARIMA and GARCH aren't the right tools

• Structural breaks. A model fitted to calm years cannot anticipate a shock unlike anything in its sample, as the 2022 jump in euro area inflation showed. Regime-switching models, or models with explanatory variables such as energy prices, handle known breaks better.
• Questions about causes. ARIMA and GARCH describe a series using only its own past. They forecast, but they do not say what an interest-rate rise does to inflation. That needs a model with explanatory variables, starting from regression, and a design that identifies cause and effect.
• Several series moving together. When inflation, output and interest rates feed into each other, a vector autoregression (VAR) models them jointly, and trending series that share a long-run relationship call for the cointegration methods of time series analysis. Correlated volatility across assets calls for multivariate GARCH, such as the dynamic conditional correlation (DCC) model.
• Long horizons. Beyond a few periods, ARIMA forecasts converge to the mean or the trend and add little. Long-horizon forecasts need structural models and judgment about policy.

The takeaway

Difference until the series is stationary, read the autocorrelations to pick a candidate, let AIC and BIC choose, and check that the residuals are white noise. Then test the forecast against a naive benchmark. For risk, fit a GARCH model to the residuals, check that α + β < 1, and read the persistence as the speed at which turbulence fades.

Further reading

• Time Series Analysis Explained — internal
• Understanding OLS Regression — internal
• Solving Canada's Inflation ‘Price Puzzle’: SARIMAX vs Deep Learning — internal
• Eurostat: Harmonised Index of Consumer Prices (HICP) — external reference
• European Central Bank: euro foreign exchange reference rates — external reference

Data note: both worked examples use teaching samples constructed by EconoLens, the first to resemble a low-inflation European price index of the kind published by Eurostat, the second to resemble daily euro–dollar returns from the ECB's reference rates. The figures are illustrative, not the official series. The two charts show the real euro area HICP and Brent crude series, pulled live from FRED. All estimates were computed by EconoLens.