Last modified: September 16, 2024
This article is written in: 🇺🇸
Backward Shift Operator
The backward shift operator (denoted by $B$) is a powerful tool in time series analysis, used to simplify the notation and manipulation of time series models. The operator shifts the time index of a time series back by one period, making it useful in autoregressive, moving average, and mixed models.
For a time series ${X_t}$, the backward shift operator is defined as:
$$ BX_t = X_{t-1} $$
This shifts the time series back by one time unit. Higher powers of the backward shift operator correspond to multiple shifts:
$$ B^2 X_t = B(BX_t) = B X_{t-1} = X_{t-2} $$
$$ B^k X_t = X_{t-k} $$
This property is central to compactly expressing time series models such as autoregressive (AR), moving average (MA), and ARMA models.
Example 1: Random Walk
Consider a simple random walk model:
$$ X_t = X_{t-1} + Z_t $$
Where $Z_t$ is white noise. Using the backward shift operator, this can be written as:
$$ X_t = BX_t + Z_t $$
Rearranging, we get:
$$ (1 - B)X_t = Z_t $$
Here, $(1 - B)$ operates on $X_t$. We define this operator as:
$$ \phi(B) = 1 - B $$
Thus, the random walk can be compactly expressed as:
$$ \phi(B)X_t = Z_t $$
Where $Z_t$ is white noise. This operator form is useful in expressing and analyzing the structure of the random walk process.
Example 2: Moving Average (MA) Process
Consider a moving average of order 2 (MA(2)) process:
$$ X_t = Z_t + 0.2 Z_{t-1} + 0.04 Z_{t-2} $$
Using the backward shift operator, this can be written as:
$$ X_t = Z_t + 0.2 B Z_t + 0.04 B^2 Z_t $$
Factoring the right-hand side:
$$ X_t = (1 + 0.2 B + 0.04 B^2) Z_t $$
Let’s define:
$$ \beta(B) = 1 + 0.2 B + 0.04 B^2 $$
Thus, the MA(2) process can be expressed as:
$$ X_t = \beta(B) Z_t $$
This operator form simplifies the analysis of MA models by capturing the entire structure of the model in the polynomial $\beta(B)$.
Example 3: Autoregressive (AR) Process
Consider an autoregressive process of order 2 (AR(2)):
$$ X_t = 0.2 X_{t-1} + 0.3 X_{t-2} + Z_t $$
Using the backward shift operator, this becomes:
$$ X_t = 0.2 B X_t + 0.3 B^2 X_t + Z_t $$
Rearranging:
$$ (1 - 0.2 B - 0.3 B^2) X_t = Z_t $$
Let’s define the AR operator as:
$$ \phi(B) = 1 - 0.2 B - 0.3 B^2 $$
Thus, the AR(2) process can be written as:
$$ \phi(B) X_t = Z_t $$
This is the standard form of an autoregressive model, where the polynomial $\phi(B)$ captures the lagged dependencies of $X_t$ on its past values.
Example 4: Moving Average (MA) Process with Drift
An MA(q) process with drift is given by:
$$ X_t = \mu + \beta_0 Z_t + \beta_1 Z_{t-1} + \dots + \beta_q Z_{t-q} $$
Using the backward shift operator:
$$ X_t = \mu + \beta_0 Z_t + \beta_1 B Z_t + \dots + \beta_q B^q Z_t $$
Factoring the right-hand side:
$$ X_t = \mu + \beta(B) Z_t $$
Where:
$$ \beta(B) = \beta_0 + \beta_1 B + \dots + \beta_q B^q $$
Subtracting the drift term $\mu$:
$$ X_t - \mu = \beta(B) Z_t $$
This form is useful for analyzing MA processes with drift, where $\beta(B)$ captures the lagged effects of the noise terms.
Example 5: Autoregressive (AR) Process of Order $p$
An autoregressive process of order $p$ (AR(p)) is given by:
$$ X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + \dots + \phi_p X_{t-p} + Z_t $$
Using the backward shift operator:
$$ X_t = \phi_1 B X_t + \phi_2 B^2 X_t + \dots + \phi_p B^p X_t + Z_t $$
Rearranging:
$$ (1 - \phi_1 B - \phi_2 B^2 - \dots - \phi_p B^p) X_t = Z_t $$
This can be written as:
$$ \phi(B) X_t = Z_t $$
Where:
$$ \phi(B) = 1 - \phi_1 B - \phi_2 B^2 - \dots - \phi_p B^p $$
This formulation represents the AR(p) process in terms of the backward shift operator, with $\phi(B)$ summarizing the autoregressive structure.