Time Series Models: Smoothing, AR, and ARCH

A time series is just a regression where the predictor is the past of the response itself. Pick the wrong memory structure and the residuals leak signal you should have captured.

A series is weakly stationary when its mean, variance, and autocovariances do not depend on time. AR and ARCH models assume stationarity. If a series trends, difference it: $\Delta Y_t = Y_t - Y_{t-1}$. If variance grows with level, take logs.

KEY: "Stationary" does not mean "constant." It means the statistical properties are time-invariant. A noisy series oscillating around a fixed mean is stationary; a series drifting upward is not.

Simple exponential smoothing (SES) forecasts the next value as a weighted average of the most recent observation and the previous forecast.

$$\hat{Y}_{t+1} = \alpha Y_t + (1-\alpha) \hat{Y}_t$$

Expanding the recursion shows the geometric decay:

$$\hat{Y}_{t+1} = \alpha \sum_{k=0}^{\infty} (1-\alpha)^k Y_{t-k}$$

The weights sum to one and decay geometrically. With $\alpha = 0.3$, the most recent point gets weight 0.30, the next 0.21, then 0.147, fading fast.

Common mistakes

• Forecasting from a non-stationary series without differencing. Fitting AR(1) to a trending series produces $\hat{\phi}_1$ near 1.0 and meaningless coefficients. Difference first, check the ACF.
• Confusing $\alpha$ in exponential smoothing with $\alpha_1$ in ARCH. Same Greek letter, different roles. The smoothing constant weights past observations; the ARCH coefficient weights past squared residuals.
• Choosing AR order from ACF instead of PACF. ACF decays geometrically for any AR; it does not tell you p. The PACF cutoff identifies p.

Bottom line

• Exponential smoothing forecasts via a weighted average of past values, weights decaying geometrically by smoothing constant $\alpha \in (0,1)$. Larger $\alpha$ reacts faster; smaller $\alpha$ is more stable.
• Holt extends simple smoothing to capture trend; Holt-Winters adds seasonality (using a level, trend, and seasonal smoothing constant).
• An AR(p) model regresses $Y_t$ on its own p most recent lags plus white noise. Stationarity requires AR roots outside the unit circle ($|\phi_1| < 1$ for AR(1)).
• Identify the AR order p from the PACF cutoff, not the ACF, which decays geometrically for any AR process and cannot pin down p.

Exam shortcut

When the prompt asks "what comes next" with one smoothing constant and a starting value, it is simple exponential smoothing; iterate $\hat{Y}_{t+1} = \alpha Y_t + (1-\alpha)\hat{Y}_t$. If the constant is given as $w$ weighting the prior smoothed value (the Frees and SOA sample-question convention), set $\alpha = 1 - w$ first. When the prompt gives lag coefficients on $Y$, it is AR; compute $c + \sum \phi_i Y_{t-i}$ and use $\mu = c/(1-\sum \phi_i)$ for the long-run mean.

The full lesson (about 3,962 words, 26 min read) adds 6 worked examples, all 10 common mistakes, a self-check, free in the app.

Learning objectives

• 3b

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