*Last modified: September 16, 2024*

*This article is written in: ðŸ‡ºðŸ‡¸*

## 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:

$$ X_t = (1 + 0.2 B + 0.04 B^2) Z_t $$

Letâ€™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:

$$ (1 - 0.2 B - 0.3 B^2) X_t = Z_t $$

Letâ€™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.