###### The Carry Trade

The uncovered interest rate parity suggests that with time, high-yielding currencies will depreciate... **Read More**

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:

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

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:

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

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.

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

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

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:

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

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

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

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

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:

- 0.64.
- 0.80.
- 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.*