Time-Series Models for Financial Data › Financial Applications of ARIMA Models

ARIMA models find practical application across several areas of quantitative finance. This section examines three key applications: exploiting mean-reverting behavior in spreads, quantifying risk through conditional forecasts, and projecting macroeconomic variables.

Interest rate spreads and basis trades often exhibit mean-reverting behavior well-captured by AR models. If a spread $S_t$ follows AR(1) with $\phi < 1$:

$$S_t = \mu + \phi(S_{t-1} - \mu) + \varepsilon_t$$

where:

• $S_t$: spread at time $t$
• $\mu$: long-run mean spread
• $\phi$: autoregressive coefficient (determines speed of reversion; must be $< 1$)
• $\varepsilon_t$: random shock

When the spread deviates from $\mu$, the expected deviation decays geometrically. The half-life measures the time $\tau$ required for this deviation to shrink by 50% ($E[S_{t+\tau}-\mu] = \frac{1}{2}(S_t-\mu)$). Since the expected deviation at $\tau$ is $\phi^\tau(S_t - \mu)$, we solve:

$$\begin{aligned} \phi^\tau &= 0.5 && \text{(decay condition)} \\ \tau \ln(\phi) &= \ln(0.5) && \text{(take logs)} \\ \tau &= \frac{-\ln(2)}{\ln(\phi)} && \text{(solve for } \tau \text{)} \end{aligned}$$

where:

• $\tau$: half-life of the mean reversion process
• $\phi$: autoregressive coefficient

For $\phi = 0.95$, the half-life is approximately 13.5 periods.

ARIMA forecasts provide inputs for Value-at-Risk calculations. The conditional distribution of future returns:

$$r_{T+h|T} \sim N(\hat{r}_{T+h|T}, \sigma^2_{h|T})$$

where:

• $r_{T+h|T}$: distribution of returns at $T+h$ conditional on time $T$
• $\hat{r}_{T+h|T}$: conditional expected return
• $\sigma^2_{h|T}$: conditional forecast variance

allows computation of forward-looking VaR that accounts for current conditions rather than using unconditional distributions.

Central bank policy analysis uses ARIMA models for inflation, GDP growth, and employment forecasts. These series often require $d=1$ or $d=2$ and seasonal components (SARIMA) that extend the basic ARIMA framework.