MA and ARMA — shocks with echoes, and an honest scoreboard

MA(1): a shock with an echo

The moving-average model of order one (take the mean as zero for simplicity):

$x_t = \varepsilon_t + \theta\, \varepsilon_{t-1}$

Read it as news dynamics: a fresh shock $\varepsilon_t$ hits today, and yesterday's shock echoes exactly once, scaled by $\theta$ — then it is gone forever. Contrast this with AR(1) from D1-04, where a shock enters the level and echoes forever with geometric fading. The trading question MA answers: what does a market look like where events matter briefly and then fully stop mattering? One concrete habitat: bid-ask bounce induces MA(1)-like negative lag-1 autocorrelation in high-frequency returns — trades ping-ponging between bid and ask create one-period echoes with no real price change behind them.

The MA(1) autocorrelation function can be worked out directly: only adjacent terms share a shock, so

$\rho_1 = \frac{\theta}{1 + \theta^2}, \qquad \rho_k = 0 \;\; \text{for } k \ge 2$

For $\theta = 0.5$, $\rho_1 = 0.5/1.25 = 0.4$; the largest lag-1 autocorrelation any MA(1) can produce is $0.5$, at $\theta = 1$. Note the hard cutoff: beyond lag 1, exactly zero.

Reading ACF signatures

That cutoff is the identification heuristic of the whole Box–Jenkins tradition. Given a stationary series, look at its ACF (D1-01):

• AR($p$) signature: the ACF decays geometrically — shocks echo forever, so autocorrelation fades but never cleanly stops.
• MA($q$) signature: the ACF cuts off — significant out to lag $q$, then drops inside the noise band.

(The partial ACF, which statsmodels plots alongside, does the mirror image: it cuts off at $p$ for AR models. The pair of plots is the classic order-picking tool.) The $\pm 2/\sqrt{N}$ band from D1-01 is what "drops to zero" means in practice — this is where that eyeball rule earns its keep.

ARMA: both kinds of memory

Real series often carry both a persistent component and short echoes, so combine them — ARMA($p, q$):

$x_t = c + \sum_{i=1}^{p} \phi_i\, x_{t-i} + \varepsilon_t + \sum_{j=1}^{q} \theta_j\, \varepsilon_{t-j}$

The practical selling point is parsimony: an ARMA(1,1) often fits what a pure AR would need five lags to approximate, and fewer parameters means less overfitting (the D4-01 lesson, in model-order clothing). Fitting is no longer plain OLS — the past shocks are unobserved — so use a library (display only; the exercise runtime here is numpy-only):

from statsmodels.tsa.arima.model import ARIMA
fit = ARIMA(x, order=(1, 0, 1)).fit()   # ARMA(1,1); middle 0 = no differencing

The middle number is $d$, the number of differences to take first — set it to 1 and the model differences a non-stationary series for you, which is exactly the D1-03 recipe automated (the "I" in ARIMA is "integrated").

The honest scoreboard

Where does this machinery actually make money? Time for the track's most valuable table:

• Daily returns: approximately nothing. Any stable linear pattern in returns is visible to everyone with a laptop and gets arbitraged toward zero — the efficient-markets logic from D1-01 and D1-02. Fitted coefficients come out tiny and unstable, and what little survives is usually eaten by costs (D4-01). Fit one anyway, once, on real data: watching a converged, statistically respectable model forecast the mean every single day teaches you — viscerally, permanently — how weak linear predictability in returns is. Every quant has this scar; it is load-bearing.
• Spreads: genuinely useful. A stationary spread has real, slow linear structure — the AR side of ARMA is exactly the D1-04 pairs-trading model, and the half-life logic carries over unchanged.
• Volatility: genuinely useful. Realized-volatility series have strong positive autocorrelation (D1-01 Fact 2) — persistent, forecastable memory. ARMA-type structure on volatility is the gateway to the GARCH family, and volatility forecasts feed position sizing and options work directly.

Same tool, three habitats, one lesson: the model does not create predictability — it can only harvest autocorrelation that is already there. Check the ACF first; it tells you whether there is anything to model before you model it.

Try it

Summarize it

Close the linear-models arc. In six sentences, written or spoken, summarize D1-01 through D1-05: what autocorrelation measures, what white noise and the random walk are null models for, why stationarity is the license to estimate anything, what kind of memory AR captures versus MA, and — final sentence — where in actual trading each piece of this machinery earns money and where it finds nothing.

Predict before the next lesson

Next: model selection and forecasting (D1-06). You now have a whole family of models — AR(1), AR(2), MA(1), ARMA(1,1), and upward. Before then, predict: if you fit every one of them to the same series, which will have the best in-sample fit, and why is that answer useless for picking the model you should actually trade?