Moving averages

The simple moving average as a linear filter#

The trailing SMA of order $k$ is

$$ \hat{T}t = \frac{1}{k} \sum{j=0}^{k-1} y_{t-j} $$

and the centred version for odd $k = 2q+1$ is

$$ \hat{T}t = \frac{1}{2q+1} \sum{j=-q}^{q} y_{t+j} $$

• $k$ — the window length.
• $q$ — the half-width, $(k-1)/2$.
• $y_t$ — the observation at time $t$.

Both are finite impulse response filters with rectangular weights. Their behaviour is best understood in the frequency domain.

Frequency response, and why the rectangular window is poor#

For a symmetric filter with weights ${a_j}_{j=-q}^{q}$, the transfer function is $A(\omega) = \sum_j a_j e^{-i\omega j}$. For the centred SMA this evaluates to the Dirichlet kernel

$$ A(\omega) = \frac{\sin(k\omega/2)}{k \sin(\omega/2)} $$

• $\omega$ — angular frequency in radians per sample, $\omega \in [0, \pi]$.

Two consequences follow, and both are visible in practice:

• Zeros at $\omega = 2\pi j / k$. A centred SMA of order $k$ annihilates any cycle whose period divides $k$ exactly. This is precisely why an SMA of order $m$ removes seasonality of period $m$, and it is the design principle behind classical decomposition.
• Sidelobes that do not decay fast, and $A(\omega) < 0$ over some bands. A negative transfer means the filter inverts components at those frequencies. The smoothed series can show a peak where the data had a trough. This artefact is called the Slutsky-Yule effect: Slutsky (1937) and Yule (1926) showed that repeatedly averaging pure noise manufactures apparent cycles that were never in the data.

The trailing SMA has the same magnitude response but a linear phase delay of $(k-1)/2$ samples. That delay is exactly the lag measured empirically in the developer block, and it is unavoidable for any causal filter with non-negative weights.

Better kernels#

Rectangular weights minimise variance for a locally constant signal and nothing else. Alternatives that trade a little variance for far better frequency behaviour:

• Henderson filters (Henderson, 1916) minimise the sum of squared third differences of the smoothed output subject to reproducing cubic polynomials exactly. The 13-term Henderson MA is the trend filter inside X-11.
• Loess fits a locally weighted low-degree polynomial with tricube weights, reproducing local linear or quadratic structure and behaving sensibly at the endpoints. This is the smoother inside STL, discussed in trend, seasonality and noise.
• Savitzky-Golay (1964) fits a degree-$p$ polynomial by least squares in each window, which preserves peak height and width far better than an SMA — the standard choice in spectroscopy, available as scipy.signal.savgol_filter.
• Hodrick-Prescott solves $\min_\tau \sum (y_t - \tau_t)^2 + \lambda \sum (\Delta^2 \tau_t)^2$. Widely used in macroeconomics and widely criticised: Hamilton (2018), Why you should never use the Hodrick-Prescott filter, shows it induces spurious dynamics and that the conventional $\lambda$ values are arbitrary.

Variance reduction and the bias-variance statement#

For $y_t = \mu_t + \varepsilon_t$ with $\varepsilon_t$ white noise of variance $\sigma^2$,

$$ \mathrm{Var}(\hat{T}_t) = \frac{\sigma^2}{k}, \qquad \mathrm{Bias}(\hat{T}t) = \frac{1}{k}\sum{j=0}^{k-1} \mu_{t-j} - \mu_t $$

Variance falls as $1/k$. Bias grows with $k$ whenever $\mu_t$ is not locally constant; for a linear trend of slope $\beta$ the trailing bias is exactly $-\beta(k-1)/2$. Minimising mean squared error gives an optimal $k$ that scales as $\sigma^{2/3}$ against curvature — the same $O(n^{-1/5})$ bandwidth story as in kernel regression, and the reason no universal window exists.

The white-noise assumption also fails for autocorrelated errors. With AR(1) errors of parameter $\phi$, the variance of the mean of $k$ consecutive observations is inflated by roughly $(1+\phi)/(1-\phi)$, so a rolling mean over positively autocorrelated data is noisier than it looks.

The relationship to exponential smoothing#

The SMA gives weight $1/k$ to the last $k$ observations and zero beyond. Exponential smoothing gives weight $\alpha(1-\alpha)^j$ to lag $j$, never reaching zero. Equating the average age of the weights,

$$ \frac{k-1}{2} = \frac{1-\alpha}{\alpha} \quad \Longleftrightarrow \quad \alpha = \frac{2}{k+1} $$

which is the standard span convention in pandas.Series.ewm(span=k). The two smoothers are comparable in responsiveness at that setting, but exponential smoothing requires storing one number instead of $k$, and produces no leading NaN block. That is developed in exponential smoothing.

Reading#

• Yule (1926), Why do we sometimes get nonsense-correlations between time-series?, JRSS 89(1).
• Slutsky (1937), The summation of random causes as the source of cyclic processes, Econometrica 5(2).
• Savitzky and Golay (1964), Smoothing and differentiation of data by simplified least squares procedures, Analytical Chemistry 36(8).
• Hamilton (2018), Why you should never use the Hodrick-Prescott filter, Review of Economics and Statistics 100(5).
• Hyndman and Athanasopoulos, Forecasting: Principles and Practice, 3rd ed., section 3.3 — otexts.com/fpp3/moving-averages.html.

What to learn next#

• Exponential smoothing — weighting recent values more, without a hard window edge.
• Feature engineering for time series — turning rolling windows into safe model inputs.
• Evaluating a forecast — the baselines that beat a moving average above.