Introduction

In this chapter, we will delve into the process of generating forecasts from Autoregressive Moving Average (ARMA) models, a fundamental aspect of time series analysis. ARMA models combine autoregressive and moving average components to capture the temporal dependencies and fluctuations in a time series. By understanding how forecasts are generated from ARMA models, we gain insights into predicting future values based on historical data patterns.

Forecasts play a crucial role in decision-making across various fields, including finance, economics, and operations. ARMA models provide a robust framework for making accurate predictions by incorporating both the past values of the series and the random fluctuations that contribute to its behavior. The key steps involved in generating forecasts from ARMA models are estimation, model selection, and prediction.

Estimation and Model Selection

The first step in generating forecasts from ARMA models is to estimate the model parameters. This involves identifying the order of autoregressive (p) and moving average (q) terms that best capture the characteristics of the time series. Estimation methods like Maximum Likelihood Estimation (MLE) or the Yule-Walker equations are employed to determine these parameters.

Forecast Calculation

Once the ARMA model parameters are estimated, the next step is to calculate forecasts for future time periods. The general formula for forecasting in an ARMA(p, q) model is given by:

$\hat{Y}_{t+h}=c+ \phi_1 Y_t+\phi_2 Y_{t-1}$ + $\ldots+\phi_p Y_{t-p+1} + \theta_1 \epsilon_t+\theta_2 \epsilon_{t-1}$+$\ldots+\theta_q \epsilon_{t-q+1}$

Where:

• $\hat{Y}_{t+h}$ is the forecasted value at time $t+h$.
• $c$ is a constant term.
• $\phi_i$ are the autoregressive coefficients.
• $Y_{t-i}$ are the observed values at time $t-i$.
• $\theta_j$ are the moving average coefficients.
• $\epsilon_{t-j}$ are the residuals at time $t-j$.

Example: Let’s consider a stock price time series data that follows an ARMA $(2,1)$ model. We’ve estimated the autoregressive coefficients as $\phi_1=0.6$ and $\phi_2=-0.3$, and the moving average coefficient as $\theta_1=0.4$.

For forecasting the stock price at time $t+1$, we use the formula: $$\hat{Y}_{t+1}=c+\phi_1 Y_t+\phi_2 Y_{t-1}+\theta_1 \epsilon_t$$