Last modified: June 14, 2023

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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:

$$B{X}_t=X_{t-1}$$

This shifts the time series back by one time unit. Higher powers of the backward shift operator correspond to multiple shifts:

$$B^2{X}_t=B\left(B{X}_t\right)=B{X}_{t-1}=X_{t-2}$$

$$B^k{X}_t=X_{t-k}$$

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:

$$X_t=B{X}_t+Z_t$$

Rearranging, we get:

$$(1-B)X_t=Z_t$$

Here, $(1-B)$ operates on $X_t$. We define this operator as:

$$\varphi \left(B\right)=1-B$$

Thus, the random walk can be compactly expressed as:

$$\varphi \left(B\right)X_t=Z_t$$

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:

$$X_t=Z_t+0.2Z_{t-1}+0.04Z_{t-2}$$

Using the backward shift operator, this can be written as:

$$X_t=Z_t+0.2B{Z}_t+0.04B^2{Z}_t$$

Factoring the right-hand side:

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

Let’s define:

$$\beta \left(B\right)=1+0.2B+0.04B^2$$

Thus, the MA(2) process can be expressed as:

$$X_t=\beta \left(B\right)Z_t$$

This operator form simplifies the analysis of MA models by capturing the entire structure of the model in the polynomial $\beta \left(B\right)$.

Example 3: Autoregressive (AR) Process

Consider an autoregressive process of order 2 (AR(2)):

$$X_t=0.2X_{t-1}+0.3X_{t-2}+Z_t$$

Using the backward shift operator, this becomes:

$$X_t=0.2B{X}_t+0.3B^2{X}_t+Z_t$$

Rearranging:

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

Let’s define the AR operator as:

$$\varphi \left(B\right)=1-0.2B-0.3B^2$$

Thus, the AR(2) process can be written as:

$$\varphi \left(B\right)X_t=Z_t$$

This is the standard form of an autoregressive model, where the polynomial $\varphi \left(B\right)$ 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:

$$X_t=\mu +{\beta }_0{Z}_t+{\beta }_1{Z}_{t-1}+\cdots +{\beta }_q{Z}_{t-q}$$

Using the backward shift operator:

$$X_t=\mu +{\beta }_0{Z}_t+{\beta }_{1}B{Z}_t+\cdots +{\beta }_q{B}^q{Z}_t$$

Factoring the right-hand side:

$$X_t=\mu +\beta \left(B\right)Z_t$$

Where:

$$\beta \left(B\right)={\beta }_0+{\beta }_{1}B+\cdots +{\beta }_q{B}^q$$

Subtracting the drift term $\mu$:

$$X_t-\mu =\beta \left(B\right)Z_t$$

This form is useful for analyzing MA processes with drift, where $\beta \left(B\right)$ 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:

$$X_t={\varphi }_1{X}_{t-1}+{\varphi }_2{X}_{t-2}+\cdots +{\varphi }_p{X}_{t-p}+Z_t$$

Using the backward shift operator:

$$X_t={\varphi }_{1}B{X}_t+{\varphi }_2{B}^2{X}_t+\cdots +{\varphi }_p{B}^p{X}_t+Z_t$$

Rearranging:

$$(1-{\varphi }_{1}B-{\varphi }_2{B}^2-\cdots -{\varphi }_p{B}^p)X_t=Z_t$$

This can be written as:

$$\varphi \left(B\right)X_t=Z_t$$

Where:

$$\varphi \left(B\right)=1-{\varphi }_{1}B-{\varphi }_2{B}^2-\cdots -{\varphi }_p{B}^p$$

This formulation represents the AR(p) process in terms of the backward shift operator, with $\varphi \left(B\right)$ summarizing the autoregressive structure.