Time-Series Models for Financial Data › Forecasting with ARIMA Models

Time-series models are widely used for forecasting. Using observations up to time $T$, we predict $Y_{T+h}$ for horizons $h = 1, 2, \ldots$ Forecast quality determines a model's practical value.

The optimal forecast (minimizing mean squared error) is the conditional expectation:

$$\hat{Y}_{T+h|T} = E[Y_{T+h} | Y_T, Y_{T-1}, \ldots]$$

where:

• $\hat{Y}_{T+h|T}$: forecast for time $T+h$ made at time $T$
• $E[\cdot]$: conditional expectation operator
• $Y_{T+h}$: future value at horizon $h$
• $Y_T, \ldots$: information set available at time $T$

This conditional expectation uses all available information up to time $T$ to form the best prediction of the future value. For an AR(1) model $Y_t = c + \phi Y_{t-1} + \varepsilon_t$, the one-step-ahead forecast is simply:

$$\hat{Y}_{T+1|T} = c + \phi Y_T$$

where:

• $\hat{Y}_{T+1|T}$: one-step-ahead forecast
• $c$: constant term
• $\phi$: autoregressive coefficient
• $Y_T$: last observed value

For longer horizons, we iterate the forecasting equation, replacing unknown future values with their forecasts:

$$\begin{aligned} \hat{Y}_{T+2|T} &= c + \phi \hat{Y}_{T+1|T} && \text{(iterate AR relation)} \\ &= c + \phi(c + \phi Y_T) && \text{(substitute 1-step forecast)} \\ &= c(1+\phi) + \phi^2 Y_T && \text{(simplify)} \end{aligned}$$

where:

• $\hat{Y}_{T+2|T}$: two-step-ahead forecast
• $\hat{Y}_{T+1|T}$: one-step-ahead forecast
• $c$: constant term
• $\phi$: autoregressive coefficient
• $Y_T$: last observed value

More generally, the forecast formula reveals how the current observation influences predictions at any horizon:

$$\hat{Y}_{T+h|T} = \mu + \phi^h (Y_T - \mu)$$

where:

• $\hat{Y}_{T+h|T}$: forecast for $T+h$ given information at $T$
• $\mu = c/(1-\phi)$: unconditional mean
• $\phi$: AR coefficient
• $h$: forecast horizon

The forecast is a weighted average of the unconditional mean $\mu$ and the current deviation from that mean. The weight on the current deviation is $\phi^h$, which shrinks toward zero as the horizon increases. As $h \to \infty$, forecasts converge to the unconditional mean: $\hat{Y}_{T+h|T} \to \mu$. This reflects the fundamental property that for stationary series, distant future values are increasingly disconnected from current conditions.

The forecast error at horizon $h$ is the difference between the realized value and our prediction:

$$e_{T+h|T} = Y_{T+h} - \hat{Y}_{T+h|T}$$

where:

• $e_{T+h|T}$: forecast error at horizon $h$
• $Y_{T+h}$: realized value at time $T+h$
• $\hat{Y}_{T+h|T}$: forecast made at time $T$

For AR(1), the forecast error variance grows with horizon as uncertainty accumulates:

$$\text{Var}(e_{T+h|T}) = \sigma^2 \frac{1 - \phi^{2h}}{1 - \phi^2}$$

where:

• $\text{Var}(e_{T+h|T})$: variance of the forecast error at horizon $h$
• $\sigma^2$: innovation variance
• $\phi$: AR coefficient

As $h \to \infty$, this converges to the unconditional variance $\sigma^2/(1-\phi^2)$. The uncertainty approaches the unconditional variance because at very long horizons, our conditional forecast converges to the unconditional mean, and the forecast error becomes essentially the deviation of the future observation from that mean.

A 95% forecast interval is:

$$\hat{Y}_{T+h|T} \pm 1.96 \sqrt{\text{Var}(e_{T+h|T})}$$

where:

• $\hat{Y}_{T+h|T}$: forecast for time $T+h$ given information at $T$
• $1.96$: z-score for 95% confidence
• $\text{Var}(e_{T+h|T})$: forecast error variance

This widening of forecast intervals reflects increasing uncertainty at longer horizons, a fundamental feature of time-series prediction. Wider intervals make forecast uncertainty explicit for risk-management decisions.