Linear Filters of Stationary Processes

Why Filter a Stationary Process?

A time series is often transformed by combining its current value with values at nearby dates. A moving average, for example, can smooth short-run fluctuations; a difference can emphasize changes. Such transformations are linear filters. Their coefficients determine how much each time point contributes to the output.

A central question is whether filtering a weakly stationary process produces another weakly stationary process. The answer is yes under a useful regularity condition: the filter coefficients must be absolutely summable. This condition also makes the output’s mean and autocovariance well defined. We will derive the result and then calculate the covariance patterns of finite moving averages of white noise.

Definition of a Linear Filter

Let \(\{X_t\}\) be a real-valued weakly stationary process with finite second moments. Write its mean as \(\mu_X=E[X_t]\) and its autocovariance at lag \(h\) as \(\gamma_X(h)=\operatorname{Cov}(X_t,X_{t+h})\). Weak stationarity means that the mean is constant over time and the autocovariance depends only on the lag, not on the date \(t\).

A finite filter has only finitely many nonzero coefficients and is therefore absolutely summable. One-sided filters, which use the present and past values only, are included by setting \(\psi_j=0\) for \(j<0\). Allowing a two-sided filter makes the stationarity result no harder to state.

Absolute summability controls the total weight placed on the input series. In particular, it guarantees that the filtered sum converges in mean square: the centered input values all have the same finite standard deviation, and the sum of the absolute coefficient weights is finite. It also ensures that the covariance formula below is an absolutely convergent series.

How Filtering Changes the Mean and Autocovariance

The output mean is constant because it is a fixed weighted sum of the constant input mean. For the covariance, the constant \(c\) has no effect, and each pair of input terms contributes according to the lag between their dates. That lag is \(h+j-k\) when the output values are \(Y_t\) and \(Y_{t+h}\).

The inequality in Step 3 follows from the Cauchy–Schwarz inequality applied to two centered input values, each with variance \(\gamma_X(0)\). The bound permits the covariance of the finite partial filters to converge to the displayed double sum. Since the mean is constant and covariance depends only on the lag, the filtered process is weakly stationary.

This result concerns second-order properties. It does not say that filtering makes a process Gaussian or independent. For example, if the input is white noise, its values are uncorrelated at different dates, but they need not be independent. A filter can create nonzero autocovariances even when the input has none.

Worked Example: Filtering a Stationary Input

Finite Moving Averages of White Noise

Suppose \(\{\varepsilon_t\}\) is white noise with mean zero, variance \(\sigma_\varepsilon^2\), and zero autocovariance at every nonzero lag. This definition does not require independence. For a finite moving-average filter

the output mean is zero. At lag \(h\geq0\), only terms that use the same white-noise date in both output values contribute to their covariance. Matching the dates gives \(j=h+k\), so

For negative lags, symmetry gives \(\gamma_Y(-h)=\gamma_Y(h)\). In particular, \(\gamma_Y(0)=\sigma_\varepsilon^2\sum_{j=0}^{q}\psi_j^2\). The output autocorrelation is \(\rho_Y(h)=\gamma_Y(h)/\gamma_Y(0)\), provided the output variance is positive. A finite filter therefore produces zero autocovariance beyond lag \(q\) when its input is white noise.

Why Absolute Summability Matters

Absolute summability is a transparent sufficient condition for a well-behaved filter. It ensures that the weighted contributions do not accumulate without bound and supports both the mean-square definition of the output and the covariance calculation. In these examples, each filter has only finitely many coefficients, so the condition holds automatically. For an infinite filter, checking \(\sum_j|\psi_j|<\infty\) is an important step before using the stationarity result.

The covariance formula also shows that filtering reshapes dependence rather than simply copying it. For general stationary input, the output covariance combines many input autocovariances with coefficient products. For white noise, all nonzero-lag input autocovariances vanish, leaving only the matching coefficient products. This is why finite moving averages of white noise have a finite covariance cutoff.

Common Mistakes

• Assuming the output covariance equals the input covariance. A filter combines values at several dates, so coefficient pairs contribute to each output lag. Use the double-sum formula or, for white noise, the matching-coefficient formula.
• Claiming independence from zero autocovariance. Zero covariance at nonzero lags means the process is uncorrelated across dates; it does not by itself imply independence.
• Forgetting the coefficient signs. Products such as \(\psi_{k+h}\psi_k\) can be negative. Keeping those signs is essential for the correct output autocovariance.
• Assuming a finite covariance cutoff for every stationary input. The cutoff beyond lag \(q\) applies to a finite moving average of white noise. A stationary input with nonzero autocovariances at larger lags can give a filtered output with nonzero autocovariances there as well.