2.5 ARIMA models and forecasting

Components of ARIMA models

ARIMA stands for Autoregressive Integrated Moving Average. These models forecast future values of a time series by combining three distinct components, each capturing a different aspect of the data's behavior. The notation ARIMA($p, d, q$) specifies the order of each component.

Choosing the right combination of $p$, $d$, and $q$ is what makes or breaks your model's accuracy.

Autoregressive (AR) component

The AR component models how the current observation depends on its own past values. The order $p$ tells you how many lagged observations feed into the model.

• An AR(1) model says the current value depends on the immediately preceding value
• An AR(2) model says the current value depends on the two most recent values

This component captures persistence in the series. If today's claim count is high, an AR component can encode the tendency for tomorrow's count to also be elevated.

The general form for an AR($p$) process is:

$X_t = c + \phi_1 X_{t-1} + \phi_2 X_{t-2} + \cdots + \phi_p X_{t-p} + \varepsilon_t$

where $\phi_1, \ldots, \phi_p$ are the autoregressive coefficients and $\varepsilon_t$ is white noise.

Integrated (I) component

The integrated component represents the degree of differencing needed to make the series stationary. The order $d$ is the number of times you difference the series before fitting the ARMA model.

• $d = 0$: the series is already stationary (you're just fitting an ARMA model)
• $d = 1$: you take first differences to remove a linear trend
• $d = 2$: you difference twice, which handles quadratic trends (rare in practice)

Most real-world actuarial time series need at most $d = 1$ or $d = 2$.

Moving average (MA) component

The MA component models the current observation as a function of past forecast errors (also called innovations or shocks). The order $q$ specifies how many lagged error terms are included.

• An MA(1) model relates the current value to the most recent forecast error
• An MA(2) model uses the two most recent forecast errors

The general form for an MA($q$) process is:

$X_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2} + \cdots + \theta_q \varepsilon_{t-q}$

where $\theta_1, \ldots, \theta_q$ are the moving average coefficients. This component captures the impact of random shocks that propagate through the series for a limited number of periods.

Stationarity in time series

Stationarity is the foundational assumption behind ARIMA modeling. A stationary time series has statistical properties that don't change over time, which is what allows you to use past behavior to predict the future. If the series isn't stationary, the patterns you estimate from historical data won't generalize forward.

A time series is (weakly) stationary when three conditions hold simultaneously:

Constant mean

The expected value of the series doesn't drift over time. If you see the series trending upward or downward, the mean isn't constant, and the series is non-stationary. This is the most visually obvious violation.

Constant variance

The spread of the data around the mean stays the same across the entire series. When variance changes over time (called heteroscedasticity), your forecast intervals become unreliable because they're calibrated to a variance that no longer applies.

Constant autocovariance

The autocovariance between $X_t$ and $X_{t+k}$ depends only on the lag $k$, not on the time $t$. If the dependence structure shifts over time, the correlation patterns you estimated from one part of the series won't hold in another.

Differencing for stationarity

When a series violates stationarity, differencing is the standard remedy. You compute differences between observations to strip out trends and stabilize the mean. The "I" in ARIMA is precisely this differencing step.

First-order differencing

$\nabla X_t = X_t - X_{t-1}$

This removes linear trends. If your claims data grows by roughly the same amount each period, first-order differencing will flatten it into a stationary series. This is by far the most common type of differencing you'll apply.

Second-order differencing

$\nabla^2 X_t = \nabla X_t - \nabla X_{t-1} = (X_t - X_{t-1}) - (X_{t-1} - X_{t-2})$

This removes quadratic trends, where the rate of change itself is changing. You'll rarely need this. If you find yourself differencing more than twice, reconsider whether ARIMA is the right model for your data.

Seasonal differencing

$\nabla_s X_t = X_t - X_{t-s}$

Here $s$ is the seasonal period (e.g., $s = 12$ for monthly data with annual seasonality, $s = 4$ for quarterly data). This removes repeating seasonal patterns by comparing each observation to the same point in the previous cycle.

Seasonal differencing is often combined with regular differencing. For example, you might apply both $\nabla$ and $\nabla_{12}$ to monthly data that has both a trend and annual seasonality.

Autocorrelation and partial autocorrelation

The ACF and PACF are your primary diagnostic tools for choosing the AR and MA orders. They reveal the temporal dependence structure of the (differenced, stationary) series.

Autocorrelation function (ACF)

The ACF at lag $k$ measures the correlation between $X_t$ and $X_{t-k}$. The ACF plot displays these correlations for a range of lags, with confidence bands (typically at $\pm 1.96 / \sqrt{n}$) to flag statistically significant values.

The ACF is the primary tool for identifying the MA order $q$.

Partial autocorrelation function (PACF)

The PACF at lag $k$ measures the correlation between $X_t$ and $X_{t-k}$ after removing the linear effects of all intermediate lags $X_{t-1}, \ldots, X_{t-k+1}$. It isolates the direct relationship at each lag.

The PACF is the primary tool for identifying the AR order $p$.

Identifying AR and MA orders

Use the ACF and PACF together according to these signature patterns:

For mixed ARMA processes, both functions tail off, which makes identification harder. In that case, you'll typically try several candidate models and compare them using information criteria.

Box-Jenkins methodology

The Box-Jenkins methodology is the standard iterative procedure for building ARIMA models. It has four stages, and you may cycle through them multiple times before settling on a final model.

Model identification

1. Plot the series and visually assess whether it's stationary
2. If non-stationary, apply differencing (regular and/or seasonal) until the series appears stationary. Confirm with a unit root test such as the Augmented Dickey-Fuller (ADF) test
3. Compute the ACF and PACF of the stationary series
4. Use the signature patterns (see the table above) to propose one or more candidate ARIMA($p, d, q$) models

Parameter estimation

Once you've identified candidate models, estimate the parameters ($\phi$'s, $\theta$'s, and the noise variance $\sigma^2$) using maximum likelihood estimation (MLE). Most statistical software handles this automatically. The estimated coefficients should be statistically significant; drop terms whose confidence intervals include zero.

Model diagnostics

After estimation, check whether the model adequately captures the data's structure:

1. Residual ACF/PACF: The residuals should resemble white noise, with no significant autocorrelations remaining
2. Ljung-Box test: A formal test for residual autocorrelation. A non-significant p-value (e.g., $> 0.05$) supports the white noise assumption
3. Normality check: Use a Q-Q plot or the Jarque-Bera test to assess whether residuals are approximately normal
4. Heteroscedasticity check: The ARCH-LM test detects whether the residual variance changes over time

If any diagnostic fails, return to the identification stage and try alternative model orders.

Forecasting

Once diagnostics pass, use the fitted model to generate forecasts. The model produces both point forecasts and forecast intervals (discussed further below).

Seasonal ARIMA (SARIMA) models

SARIMA models extend ARIMA to handle data with periodic seasonal patterns. The full notation is ARIMA($p, d, q$) $\times$ ($P, D, Q$)$_s$, where the uppercase letters denote the seasonal orders and $s$ is the seasonal period.

Seasonal differencing

Seasonal differencing of order $D$ removes seasonal non-stationarity. With $D = 1$ and $s = 12$, you compute $X_t - X_{t-12}$, comparing each month to the same month in the prior year. Most seasonal series require at most $D = 1$.

Seasonal AR and MA components

• The seasonal AR (SAR) component of order $P$ relates the current observation to observations at lags $s, 2s, \ldots, Ps$
• The seasonal MA (SMA) component of order $Q$ relates the current observation to forecast errors at lags $s, 2s, \ldots, Qs$

These seasonal terms are multiplied with the non-seasonal terms, which is why SARIMA is sometimes called a multiplicative seasonal model. For example, an ARIMA(1,1,1) $\times$ (1,1,1)$_{12}$ model for monthly data has both regular and seasonal AR, differencing, and MA components interacting multiplicatively.

Multiplicative vs. additive seasonality

• Additive seasonality: The seasonal fluctuation has a roughly constant amplitude regardless of the series level. SARIMA naturally handles this.
• Multiplicative seasonality: The seasonal amplitude grows proportionally with the series level. A common fix is to apply a log transformation before fitting the SARIMA model, converting multiplicative seasonality into additive.

You can distinguish between the two by inspecting whether the seasonal swings widen as the series increases. If they do, apply a log transform (or Box-Cox transformation) first.

Forecasting with ARIMA models

ARIMA forecasting uses the fitted model to project the series forward. The quality of these forecasts depends on how well the model captures the true data-generating process and on the forecast horizon.

One-step-ahead forecasting

A one-step-ahead forecast predicts $X_{t+1}$ given all data up to time $t$. Because you're conditioning on actual observed values, these forecasts tend to be the most accurate. They're used for short-term operational decisions, such as next-month claims projections.

Multi-step-ahead forecasting

To forecast $h$ steps ahead, the model generates $\hat{X}_{t+1}$, then uses that forecast (in place of the unknown actual value) to produce $\hat{X}_{t+2}$, and so on iteratively. Each step introduces additional uncertainty because you're conditioning on estimated rather than observed values.

As a result, forecast accuracy degrades as the horizon $h$ increases. This is a fundamental property of ARIMA models, not a flaw.

Forecast intervals and uncertainty

Point forecasts alone are insufficient for actuarial work. ARIMA models also produce forecast intervals that quantify prediction uncertainty.

• A 95% forecast interval gives the range within which the future value is expected to fall with 95% probability (under the model assumptions)
• Interval width grows with the forecast horizon, reflecting increasing uncertainty
• These intervals assume normally distributed errors; if that assumption is violated, consider bootstrap-based intervals instead

Forecast intervals are critical for reserve setting and capital adequacy calculations, where understanding the range of possible outcomes matters as much as the central estimate.

Model selection and evaluation

When you have several candidate ARIMA models, you need objective criteria to choose among them. The goal is a model that fits the data well without being unnecessarily complex (overfitting).

Akaike information criterion (AIC)

$AIC = -2 \ln(\hat{L}) + 2k$

where $\hat{L}$ is the maximized likelihood and $k$ is the number of estimated parameters. Lower AIC is better. The AIC penalizes complexity but can still favor slightly overparameterized models, especially with small samples.

Bayesian information criterion (BIC)

$BIC = -2 \ln(\hat{L}) + k \ln(n)$

where $n$ is the sample size. The BIC imposes a heavier penalty on additional parameters than the AIC (since $\ln(n) > 2$ for $n \geq 8$). It tends to select more parsimonious models, which is often preferable for forecasting.

Residual analysis and diagnostics

Even after selecting a model via information criteria, verify it with residual diagnostics:

• Ljung-Box test: Tests whether residual autocorrelations are collectively zero. Significant results (low p-value) indicate remaining structure the model hasn't captured.
• Jarque-Bera test: Tests for normality of residuals. Non-normal residuals affect the validity of forecast intervals and hypothesis tests.
• ARCH-LM test: Tests for autoregressive conditional heteroscedasticity. If significant, the residual variance is time-varying, and you may need a GARCH-type extension.

If diagnostics reveal problems, revisit your model specification rather than proceeding with unreliable forecasts.

Extensions and variations of ARIMA

The standard ARIMA framework can be extended to handle additional data features. These extensions are common in actuarial practice where real-world data rarely fits neatly into a basic ARIMA structure.

ARMAX models with exogenous variables

ARMAX models add external predictor variables to the ARMA framework. For example, when forecasting auto insurance claims, you might include unemployment rate or fuel prices as exogenous inputs. The model becomes:

$X_t = c + \sum_{i=1}^{p} \phi_i X_{t-i} + \sum_{j=1}^{q} \theta_j \varepsilon_{t-j} + \sum_{m=1}^{r} \beta_m Z_{m,t} + \varepsilon_t$

where $Z_{m,t}$ are the exogenous variables. This assumes the series is already stationary.

ARIMAX models with exogenous variables

ARIMAX extends ARMAX by allowing for non-stationary series (i.e., including the differencing step). You difference the series to achieve stationarity and then include exogenous variables in the model. This is the more general and commonly used formulation when your target series has a trend and you also want to incorporate external drivers.

Nonlinear ARIMA models (NARIMA)

Standard ARIMA assumes linear relationships. When the data exhibits asymmetric behavior, regime changes, or threshold effects, nonlinear extensions may be appropriate:

• Threshold AR (TAR): The AR coefficients switch depending on whether the series crosses a threshold value
• Smooth Transition AR (STAR): Similar to TAR but with a smooth rather than abrupt transition between regimes
• Bilinear models: Include interaction terms between lagged values and lagged errors

These models are more flexible but require larger samples for reliable estimation and are harder to interpret. Use them when diagnostic evidence clearly points to nonlinearity.

Applications of ARIMA in actuarial science

ARIMA models appear throughout actuarial practice wherever time-indexed data needs to be projected forward.

Forecasting claims frequency and severity

Claims frequency (number of claims per period) and severity (average cost per claim) both evolve over time and often exhibit trends, seasonality, and autocorrelation. ARIMA models fitted to historical claims data help actuaries:

• Set premiums that reflect expected future claims costs
• Establish adequate reserves for outstanding liabilities
• Detect emerging trends (e.g., increasing claim severity due to medical cost inflation)

For example, fitting an ARIMA(1,1,1) to quarterly claims severity data might reveal that severity grows with a trend and that recent quarters' deviations from trend persist into the next quarter.

Forecasting mortality rates

Mortality rates at various ages change over time due to medical advances, lifestyle changes, and other factors. ARIMA models (often applied to log mortality rates or as part of the Lee-Carter framework) allow actuaries to:

• Project future life expectancies for annuity and pension valuation
• Price life insurance products that reflect anticipated mortality improvements
• Assess longevity risk, the risk that policyholders live longer than expected

Forecasting economic variables

Interest rates, inflation, and GDP growth directly affect actuarial calculations. ARIMA models applied to these economic time series support:

• Asset-liability management: Matching the duration and cash flows of assets to liabilities
• Investment strategy: Anticipating interest rate movements that affect bond portfolios
• Risk assessment: Stress-testing reserves under different economic scenarios

Accurate economic forecasts are particularly important for long-tail lines of business, where claims may not be settled for years and the discount rate materially affects reserve estimates.