Skip to content

Risk Modelling and Survival Analysis · Applications of time series models

Actuarial Applications of Time Series and Multivariate Models

Updated 11 October 2026 · Fact-checked

Actuaries use time series to model inflation, interest rates and equity returns. Multivariate models such as VAR capture links between series. Cointegration links non-stationary series that share a stable long-run relationship. ARCH/GARCH model changing volatility. The Wilkie model is a cascade where inflation drives other series. To solve questions, state the model, check stationarity, then fit and interpret.

Understand Actuarial Applications and Multivariate Models

Economic and investment series are not independent. Inflation affects interest rates, and both affect share prices and dividends. A time series model lets you simulate these series for pricing, reserving, capital and asset-liability work.

A multivariate model describes several series together. A vector autoregressive (VAR) model of order p makes each series depend on past values of all the series. For two series, X(t) and Y(t) each depend on lags of both. This lets one series help forecast another. Stationarity conditions are checked on the whole system.

Many financial series are non-stationary, often integrated of order 1, written I(1). Differencing them once makes them stationary. But two I(1) series can be cointegrated: some linear combination of them is stationary. Think of a drunk walking a dog on a lead. Each wanders, but the gap between them stays bounded. Cointegration means a long-run equilibrium exists, so differencing both series would throw that information away. An error correction model uses the gap to pull the series back toward equilibrium.

Standard time series assume constant variance. Real returns show volatility clustering: calm periods and turbulent periods. An ARCH model lets the conditional variance depend on past squared errors. GARCH adds past conditional variances, giving a more parsimonious model. These capture fat tails in returns without fixing a constant variance.

The Wilkie model is a cascade structure. Price inflation is the driving series, modelled as an autoregressive process. Other series, such as share dividends, yields and long-term interest rates, depend on inflation, directly or through earlier series. Each has its own random error. It is a stochastic model used for long-term simulation. Know the structure and its limits: it is calibrated to historical data, so it may miss regime changes.

Key rules to remember

Integrated of order d
X(t) is I(d) if ∇^d X(t) is stationary and ∇^(d−1) X(t) is not
∇X(t) = X(t) − X(t−1). A random walk is I(1).
Cointegration
X(t), Y(t) both I(1), and Z(t) = X(t) − αY(t) is I(0) for some α
The vector (1, −α) is the cointegrating vector. A common test approach is to check whether the residuals of a regression are stationary.
VAR(1), two series
X(t) = a₁ + b₁₁X(t−1) + b₁₂Y(t−1) + e₁(t); Y(t) = a₂ + b₂₁X(t−1) + b₂₂Y(t−1) + e₂(t)
Errors may be correlated across equations. Stationarity requires the system to satisfy conditions on the coefficient matrix.
ARCH(1)
ε(t) = σ(t)·z(t), σ²(t) = α₀ + α₁ε²(t−1)
z(t) is white noise with mean 0 and variance 1. Need α₀ > 0 and α₁ ≥ 0. Stationary variance = α₀ ÷ (1 − α₁) when α₁ < 1.
GARCH(1,1)
σ²(t) = α₀ + α₁ε²(t−1) + β σ²(t−1)
Need α₀ > 0, α₁ ≥ 0, β ≥ 0. Finite stationary variance if α₁ + β < 1, equal to α₀ ÷ (1 − α₁ − β).
Wilkie inflation (structure)
ΔQ-type series: ln-inflation I(t) = μ + a·(I(t−1) − μ) + σ·z(t)
Inflation is an AR(1) process around a mean μ. Other series in the cascade depend on it. Exact parameter values are not memorised.

How to solve Actuarial Applications and Multivariate Models questions

Use this approach for any question on applications of time series and multivariate models.

  1. 1Identify the series and what the question asks: modelling, testing, forecasting or interpreting.
  2. 2Check stationarity. Look at the plot, ACF and any unit root evidence. State whether each series is I(0) or I(1).
  3. 3If series are I(1) and several are involved, ask whether a linear combination is stationary. If yes, name cointegration and use an error correction or levels model, not separate differencing.
  4. 4If the variance changes with time, or squared residuals are autocorrelated, choose ARCH or GARCH. Write the variance equation and its parameter conditions.
  5. 5For a cascade model such as Wilkie, work in order: the driving series first, then the series that depend on it.
  6. 6Write the model in standard notation, with assumptions stated.
  7. 7Compute what is asked, such as a forecast, a stationary variance or a conditional variance, showing each substitution.
  8. 8Comment on suitability and limits: parameter uncertainty, structural change, and use in the actuarial context.

Quickest way: Pick the model from the symptom

When to use it: Use in MCQs and short written parts where you must choose or describe a model quickly.

  1. Trend that does not settle, differencing needed: think I(1).
  2. Two I(1) series moving together with a stable gap: cointegration.
  3. Several series affecting each other's future: VAR.
  4. Variance clusters, residuals look uncorrelated but squared residuals do not: ARCH/GARCH.
  5. Inflation drives other economic series in a chain: Wilkie cascade.
  6. For GARCH(1,1) variance: α₀ ÷ (1 − α₁ − β), only if α₁ + β < 1.

Common mistakes in Actuarial Applications and Multivariate Models

  • Differencing cointegrated series separately and ignoring the long-run relationship.

    Students learn that I(1) means difference it.

    Fix: If a combination is stationary, keep the levels relationship and use an error correction model.

  • Saying cointegrated series must be stationary.

    Confusing the series with the combination.

    Fix: Each series is I(1). Only the linear combination is I(0).

  • Using the GARCH stationary variance formula when α₁ + β ≥ 1.

    Applying the formula without checking the condition.

    Fix: Check α₁ + β < 1 first. Otherwise there is no finite stationary variance.

  • Saying ARCH models change the mean of the series.

    Mixing up mean and variance equations.

    Fix: ARCH/GARCH model the conditional variance. The mean equation is separate.

  • Describing the Wilkie model as a set of independent models.

    Listing the series without the links.

    Fix: State that inflation is the driver and other series depend on it in a cascade.

  • Treating VAR as just several separate AR models.

    Ignoring the cross-lag terms.

    Fix: Each equation includes lags of all series, and errors can be correlated across equations.

Worked examples

Example 1

A GARCH(1,1) model has σ²(t) = 0.00002 + 0.10ε²(t−1) + 0.85σ²(t−1). (a) Is the stationary variance finite? Find it. (b) Given ε(t−1) = 0.02 and σ²(t−1) = 0.0004, find σ²(t).

Show the solution
  1. (a) α₁ + β = 0.10 + 0.85 = 0.95, which is less than 1, so the stationary variance is finite.
  2. Variance = α₀ ÷ (1 − α₁ − β) = 0.00002 ÷ 0.05 = 0.0004.
  3. (b) ε²(t−1) = 0.02² = 0.0004.
  4. σ²(t) = 0.00002 + 0.10 × 0.0004 + 0.85 × 0.0004.
  5. = 0.00002 + 0.00004 + 0.00034 = 0.0004.

Answer: (a) Yes; the stationary variance is 0.0004. (b) σ²(t) = 0.0004.

Example 2

X(t) and Y(t) are both I(1). Regressing X on Y gives X(t) = 2Y(t) + Z(t), where Z(t) is stationary. Explain what this means and how it affects modelling.

Show the solution
  1. Both series are I(1), so each is non-stationary and needs differencing once to be stationary.
  2. Z(t) = X(t) − 2Y(t) is a linear combination that is I(0). So X and Y are cointegrated, with cointegrating vector (1, −2).
  3. Interpretation: X and Y can wander, but X − 2Y stays around a stable level. There is a long-run equilibrium.
  4. Modelling: differencing both series separately would lose the long-run information. Use an error correction model, where changes in X and Y depend on past values of Z(t) as well as past changes.
  5. Actuarial use: this can link, for example, two asset series or prices and dividends, so simulations keep them from drifting apart without limit.

Answer: X and Y are cointegrated with vector (1, −2). Model them with an error correction model using Z(t) = X(t) − 2Y(t), not by differencing separately.

Exam tips

  • Write the stationarity condition for the model before calculating anything.
  • In discussion parts, link the model to a purpose such as asset-liability modelling or capital, and name one limitation.
  • For GARCH, always check α₁ + β < 1 and show the check.
  • Use standard notation: ∇ for differencing, I(d) for integration order, and name the cointegrating vector.
  • In Paper B, state the model, the R function or step, and interpret the output, not just the numbers.

Practice questions from Applications of time series models

Actuarial Applications and Multivariate Models: frequently asked questions

What is cointegration in simple terms?

Two or more non-stationary I(1) series are cointegrated if some linear combination of them is stationary. They can drift, but not apart from each other. This shows a long-run equilibrium.

What is the difference between ARCH and GARCH?

ARCH makes conditional variance depend on past squared errors. GARCH also includes past conditional variances. GARCH usually needs fewer parameters to capture the same volatility clustering.

How does the Wilkie model work?

It is a cascade of stochastic series. Price inflation is the driving series, usually an autoregressive process. Other series, such as dividends and interest yields, depend on inflation and on each other in a set order, each with its own random error.

What is a VAR model used for in CS2?

A VAR model describes several series together, so each depends on lagged values of all of them. It is used to forecast and to study how series affect each other, for example inflation and interest rates.