Autoregressive Models and Multiperiod Forecasts

The current-time values of a time series are related to the previous time values. This property is termed autoregressive. Autoregressive models are abbreviated (\(AR_{p}\)) models. \(p\) is known as the order of the model. It indicates the number of lagged values of the dependent variable used. More specifically, a time series that has been regressed on past values is an autoregressive model.

In the autoregressive model, we abandon the dependent (y) and independent (x) notion and use\(x_t\) since there is no longer a difference.

A p-order autoregressive model, \(AR_{p}\), is expressed as:

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}+\text{b}_{2}\text{x}_{\text{t}-2}+…+\text{b}_{\text{p}}\text{x}_{\text{t}-\text{p}}+\epsilon_{\text{t}}$$

Example: Autoregressive Model

First-order autoregressive model: \(\text{AR}(1): \text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}+\epsilon_{\text{t}}\)

Second order autoregressive model: \(\text{AR}(2):\text{x}_{t}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}+\text{b}_{2}\text{x}_{\text{t}-2}+\epsilon_{\text{t}}\)

Longer interval differences can be used to account for seasonality:

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}+\text{b}_{2}\text{x}_{\text{t}-4}+\epsilon_{\text{t}}$$

Chain Rule of Forecasting

The chain rule of forecasting can be used to derive multiperiod forecasts using an \(AR_{p}\) model. It involves calculating a one-step-ahead forecast before a two-step ahead forecast as the independent variable is a lagged value of the dependent variable.

One-Step Ahead Forecast

The one-step-ahead forecast of an AR(1) model is given by:

$$\hat{\text{x}}_{\text{t}}=\hat{\text{b}}_{0}+\widehat{\text{b}}_{1}\text{x}_{\text{t}-1}$$

Example: Calculating the One-step Ahead Forecast

Given an AR(1) model where \(\hat{\text{b}}_0=2\) and \(\hat{\text{b}}_1=1.8\), the one-step-ahead forecast of \(\text{x}_{1}\) when \(\text{x}_0=2\) is closest to: 

Solution

$$\hat{\text{x}}_{\text{t}}=\hat{\text{b}}_{0}+\widehat{\text{b}}_{1}\text{x}_{\text{t}-1}$$

$$\hat{\text{x}}_{\text{1}}=2+1.8\times2=5.6$$

Two-Step Ahead Forecast

The two-step ahead forecast for an AR(1) model is determined as:

$$\hat{\text{x}}_{\text{t}+1}=\hat{\text{b}}_{0}+\hat{\text{b}}_{1}\hat{\text{x}}_{\text{t}}$$

Example: Calculating the Two-step Ahead Forecast

Calculate the two-step ahead forecast of \(\text{x}_{2}\)

$$\hat{\text{x}}_{2}=2+1.8\hat{\text{x}}_1$$

$$\hat{\text{x}}_{2}=2+1.8\times5.6=12.08$$

Question

Consider an AR(1) model with the following prediction equation:

$$\text{x}_{\text{t}}=0.8+0.5\text{x}_{\text{t}-1}$$

If the current value of \(x\) is 4.0, the two-step-ahead forecast is closest to:

1. 0.64.
2. 0.80.
3. 1.60.

Solution

The correct answer is A.

One-step ahead forecast:

If \(\text{x}_{\text{t}}=4,\) then \(\hat{\text{x}}_{\text{t}+1}=0.8+0.5\times4=1.6\)

Two-step ahead forecast: 

If \(\hat{\text{x}}_{\text{t}+1}=1.6,\) then \(\hat{\text{x}}_{\text{t}+2}=0.8+0.5\times1.6=0.64\)

Reading 5: Time Series Analysis

LOS 5 (d) Describe the structure of an autoregressive (AR) model of order p and calculate one- and two period-ahead forecasts given the estimated coefficients.