Model relationships of current and past values of a statistic or metric.

Quarterly loss ratios for a workers' compensation book wander but never explode. Last quarter's number tells you something about this quarter's, and the relationship is linear, stable, and exploitable. Time series models with constant variance turn that intuition into forecasts.

Why constant variance matters. Every model in this lesson assumes the innovation ${\epsilon }_t$ is white noise: mean zero, variance ${\sigma }^2$, uncorrelated across time. If variance drifts with the level (heteroscedasticity), you need a different toolkit. Here the spread stays flat; only the mean carries memory.

A weakly stationary series has constant mean $\mu$, constant variance ${\gamma }_0$, and autocovariance ${\gamma }_k=\text{Cov}(Y_t,Y_{t-k})$ that depends only on lag $k$, not on $t$. The autocorrelation function (ACF) is ${\rho }_k={\gamma }_k/{\gamma }_0$.

KEY: Stationarity is what lets you estimate ${\rho }_k$ by averaging across time. Without it, every $t$ is a different distribution and one sample tells you nothing.

An AR(p) regresses today on the last $p$ values plus white noise:

$$Y_t=c+{\varphi }_1{Y}_{t-1}+\cdots +{\varphi }_p{Y}_{t-p}+{\epsilon }_t$$

Common mistakes

• Confusing ACF and PACF cutoffs. AR(p) has PACF that cuts off (not ACF). MA(q) has ACF that cuts off (not PACF). Reversing them leads to fitting an MA(1) when the data want AR(1). The wrong-order forecast at $h=2$ for the Example 1 data would be 0.60 (MA(1) reverts immediately) instead of 0.7225.
• Computing AR(1) variance as ${\sigma }^2$ instead of ${\sigma }^2/(1-{\varphi }_1^2)$. With ${\varphi }_1=0.70$, the divisor 0.51 nearly doubles the variance. Skipping it gives 0.0009 when the true unconditional variance is 0.001765.
• Treating an MA(1) with $|{\theta }_1|>1$ as valid. Both $\theta$ and $1/\theta$ match the ACF, but only $|{\theta }_1|<1$ is invertible. The non-invertible root is mathematically real but software returns the invertible one; flagging the wrong root costs the question.

Bottom line

• AR(p): $Y_t=c+{\varphi }_1{Y}_{t-1}+\cdots +{\varphi }_p{Y}_{t-p}+{\epsilon }_t$. PACF cuts off at lag $p$; ACF decays.
• MA(q): $Y_t=\mu +{\epsilon }_t+{\theta }_1{\epsilon }_{t-1}+\cdots +{\theta }_q{\epsilon }_{t-q}$. ACF cuts off at lag $q$; PACF decays.
• ARMA(p,q): both ACF and PACF decay (no clean cutoff).
• Stationarity (AR): roots of $1-{\varphi }_{1}z-\cdots -{\varphi }_p{z}^p=0$ lie outside the unit circle. For AR(1): $|{\varphi }_1|<1$.

Exam shortcut

If the ACF dies geometrically and the PACF shows one clean spike, fit AR(1) and read ${\varphi }_1$ directly off lag 1. If the ACF shows one spike and the PACF tails off, fit MA(1) and back out ${\theta }_1$ from ${\rho }_1={\theta }_1/(1+{\theta }_1^2)$ (the smaller root for invertibility). If both ACF and PACF tail off and AIC/BIC favor a small mixed model, try ARMA(1,1) before adding more AR or MA terms.

The full lesson (about 2,389 words, 16 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• D1

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