Worked Case: Claims Experience with Random-Walk Behavior

Why Random-Walk Behavior Matters for Claims

A claims series may show a persistent upward or downward movement because each new observation builds on the previous level. In that situation, treating the levels as independent observations around a fixed mean can produce misleading forecasts. A random-walk model instead describes changes in the series; a differenced ARIMA model allows those changes to have their own dependence pattern.

The analysis is not just a matter of noticing that a plotted series rises. A trend could reflect persistent stochastic movement, a deterministic trend, a structural change, or another feature of the data. A useful case analysis considers the dated observations, examines first differences, compares plausible models, checks residuals, and evaluates forecasts at the horizon relevant to the decision. The hypothetical claims amounts below are measured in thousands of dollars and are not actual company or industry data.

For a random walk, shocks do not fade from the level: a shock in one period remains in all later levels. Consequently, the variance of the level increases with time. This is different from a stationary series, in which the impact of a shock does not accumulate without bound. A plot and sample autocorrelations can suggest persistence, but they do not by themselves prove that a series has a unit root. The dated observations, plausible process mechanisms, and forecast performance all matter.

Worked Case: Claims Levels and First Differences

Consider hypothetical quarterly incurred claims totals, in thousands of dollars, from 2022 Q1 through 2025 Q4:

\(120, 126, 123, 131, 135, 132, 140, 145, 142, 151, 154, 152, 160, 165, 162, 171.\)

The levels generally rise, but the consecutive changes alternate between increases and decreases. The first differences, in the same units, are:

\(6,-3,8,4,-3,8,5,-3,9,3,-2,8,5,-3,9.\)

There are 15 differences. Their sum telescopes to the last level minus the first level: \(171-120=51\). Thus the sample mean difference is \(51/15=3.4\) thousand dollars per quarter. This average increase is evidence about the observed period, not proof that future claims will rise at exactly that rate.

Both candidates have first differences fluctuating around a positive average. Their forecasts differ because the ARIMA candidate allows a recent change to influence the next change, with that influence decaying geometrically. In contrast, the random walk with drift forecasts the same expected increase, \(c\), in every future quarter.

Worked Example: Fixed-Origin Forecasts at Horizons One Through Four

Why the Forecast Formulas Hold

The conditional-mean forecast is valid when future innovations have conditional expectation zero given the information available at the forecast origin. Independent, identically distributed innovations with mean zero satisfy this condition. Martingale-difference innovations with respect to that information set also satisfy it. Merely assuming that innovations are uncorrelated does not, in general, imply zero conditional expectation.

For the random walk with drift, expand the model forward \(h\) periods:

Given the data through time \(T\), \(Y_T\) and \(c\) are known for this calculation. If each future innovation has conditional mean zero, the conditional expectation of the sum is zero. Therefore:

Here \(\mathcal{F}_T\) represents the information available through time \(T\). Under the additional assumption of independent normal innovations with variance \(\sigma^2\), the forecast error is the sum of \(h\) independent innovations, with variance \(h\sigma^2\). Thus, in the example, the four forecast-error variances are \(16,32,48,64\) thousand-dollars-squared. The corresponding standard errors are \(4,\ 4\sqrt{2},\ 4\sqrt{3},\ 8\). These calculations do not include uncertainty from estimating the model parameters.

For the ARIMA candidate, the differenced series follows an AR(1) model around \(c\). Its conditional forecast for the next difference is \(c+\phi(\Delta Y_T-c)\). Repeated substitution gives \(c+\phi^h(\Delta Y_T-c)\) at horizon \(h\). The level forecast must sum all forecasted changes because \(Y_{T+h}=Y_T+\sum_{j=1}^h\Delta Y_{T+j}\). This cumulative step is essential: a forecast for a difference is not a forecast for the level.

Residual Checks and Candidate Comparison

After fitting a candidate, inspect its residuals, which are the observed values minus fitted values on the scale modeled. For a random walk, the relevant residuals are innovations in the level equation; for a differenced ARIMA model, they are innovations in the differenced equation. A useful residual series should have no systematic pattern left that the model could have captured. Residual plots can reveal changing spread, outliers, or runs; residual autocorrelations can reveal remaining serial dependence.

The Ljung–Box statistic combines residual autocorrelations over a specified number of lags. At \(m\) lags, it is:

where \(n\) is the number of residuals and \(r_k\) is the sample residual autocorrelation at lag \(k\). Under the approximate null distribution, \(Q\) is compared with a chi-square distribution. When parameters are estimated, a common degrees-of-freedom adjustment subtracts the number of fitted AR and MA terms; the exact testing convention should be stated.

A model should not be selected from one diagnostic alone. Compare plausible candidates using parsimony, fit criteria when appropriate, residual behavior, and forecast evaluation designed to match the decision. If the task is to forecast four quarters ahead from a single cutoff, evaluate four-horizon fixed-origin forecasts. If the task concerns repeated next-quarter forecasting, use successive origins and update the information set with each newly observed value. Those are different evaluation designs and produce different forecast errors.

Worked Example: A One-Step Forecast With an Updated Origin

Common Mistakes in Random-Walk Case Analysis

• Calling a rising series stationary because its differences have a stable average. Stationarity of the differences does not make the levels stationary; a random walk’s level variance grows over time.
• Using uncorrelated innovations as if they guaranteed conditional mean zero. For conditional-mean forecasts, state a sufficient assumption such as independence from the past or the martingale-difference property.
• Forecasting differences but reporting them as levels. For a differenced model, add the forecasted changes cumulatively to the latest observed level.
• Confusing fixed-origin multi-step forecasts with rolling one-step forecasts. A fixed-origin forecast holds the information set fixed; a rolling evaluation updates it after each new observation.
• Treating a residual test as a certificate of model validity. A Ljung–Box result addresses residual autocorrelation at selected lags; it does not check every model assumption or guarantee good future performance.