Linear Filter

A linear filter transforms an input time series into an output time series via a weighted sum.

1. Definition

Given an input series $\{Y_t\}$ and coefficients $\{a_j\}_{j \in \mathbb{Z}}$, the output is

$$X_t = \sum_{j=-\infty}^{\infty} a_j Y_{t-j}$$

The sequence $\{a_j\}$ defines the filter.

2. Causal Filter

A filter is causal if $a_j = 0$ for all $j < 0$:

$$X_t = \sum_{j=0}^{\infty} a_j Y_{t-j}$$

The output depends only on present and past values of the input. Equivalently, expressed as a piecewise function:

$$a_j = \begin{cases} a_j & j \geq 0 \\ 0 & j < 0 \end{cases}$$

3. Stationarity Preservation

Proposition: If $\{Y_t\}$ is zero-mean and stationary with ACVF $\gamma_Y(h)$, and $X_t = \sum_j a_j Y_{t-j}$ with $\sum_j a_j^2 < \infty$, then:

  1. $\mathbb{E}(X_t) = 0$
  2. $\{X_t\}$ is stationary
  3. $\text{Var}(X_t) = \sigma_X^2 = \sigma_Y^2 \sum_j a_j^2$ (when $Y_t$ is white noise)

4. Filter Passing Trend

A linear filter passes a polynomial trend $m_t$ of degree $k$ exactly (i.e., $\sum_j a_j m_{t-j} = m_t$) if and only if:

  1. $\sum_{j=-\infty}^{\infty} a_j = 1$
  2. $\sum_{j=-\infty}^{\infty} j^r a_j = 0$ for $r = 1, 2, \dots, k$

The moving average filter satisfies condition 1 (passes constant trends) but not higher conditions unless modified.

5. Connection to Operators

The difference operator $\nabla = 1 - B$ is a linear filter with coefficients $a_0 = 1, a_1 = -1$.