Backshift Operator and Difference Operator
1. Backshift Operator $B$
$$BX_t = X_{t-1}, \quad B^k X_t = X_{t-k}$$$B$ shifts the time index backwards by one step.
2. Difference Operator $\nabla$
$$\nabla X_t = X_t - X_{t-1} = (1 - B)X_t$$Higher-order differencing:
$$\nabla^2 X_t = \nabla(\nabla X_t) = X_t - 2X_{t-1} + X_{t-2}$$In general, $\nabla^p = (1 - B)^p$.
3. Trend Elimination
Theorem: If the trend is a polynomial of degree $p$, i.e., $m_t = a_0 + a_1 t + a_2 t^2 + \cdots + a_p t^p$, then $p$ times differencing eliminates the trend.
Example: for linear trend $m_t = a + bt$:
$$\nabla m_t = m_t - m_{t-1} = b \quad \text{(constant)}$$The new series $\nabla m_t$ has no trend.
4. Lag-$d$ Difference Operator $\nabla_d$
$$\nabla_d X_t = X_t - X_{t-d} = (1 - B^d)X_t$$Used for seasonality elimination: if $s_t$ has period $d$, then $\nabla_d s_t = 0$.
5. Operator Algebra
Polynomials in $B$ can be multiplied and factored like ordinary polynomials:
$$\phi(B) = 1 - \phi_1 B - \phi_2 B^2 - \cdots - \phi_p B^p$$This notation is essential for defining AR, MA, and ARMA models compactly.
6. Key Exam Problem Type
Given seasonal components with periods $d_1$ and $d_2$, determine whether $\nabla_{d_1 d_2} X_t$ is constant (TRUE: since $d_i \mid d_1 d_2$ for both $i$) versus whether $\nabla_{d_1 + d_2} X_t$ is constant (FALSE in general: $d_i \nmid d_1 + d_2$).