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Time-series analysis — AR, MA, unit roots, cointegration

In this chapter: Autoregressive (AR) models · Moving-average (MA) and ARMA · Unit-root tests, stationarity · Cointegration

~3 min readLayer 4 · Professional CertificationsFree

Time-series questions on L2 typically present AR/AR(p) output and ask about model adequacy, forecast, unit roots. The art: recognising when to use levels vs first-differences.

Foundation

Stationarity: mean, variance, covariance with own lag are constant over time. OLS valid only on stationary series. AR(1): Yt = b0 + b1Yt-1 + εt. - |b1| < 1: stationary (mean-reverting). - |b1| = 1: unit root, non-stationary (random walk). - |b1| > 1: explosive, rare. Unit-root test: Augmented Dickey-Fuller (ADF). H0: unit root. Reject H0 → stationary. If two non-stationary series are cointegrated (linear combination is stationary), regression in levels is valid; if not, regression is spurious — use first differences.

Deep Dive

Workflow: 1. Plot the series. Trend? Seasonality? Variance change? 2. Test for stationarity (ADF). 3. If non-stationary: difference until stationary, OR test for cointegration with another non-stationary series. 4. Fit AR(p): use partial autocorrelation function (PACF) to choose p. 5. Check residuals: serial correlation? Heteroskedasticity? If autocorrelation persists, increase p. 6. Choose between models using AIC/BIC (lower better). Cointegration test: Engle-Granger. Run regression Y on X (both I(1)); test residuals for unit root. If residuals stationary, cointegrated.

Advanced

CFA L2 trap: spurious regression. Two unrelated random walks regressed on each other can show high R² and significant coefficients — but it's nonsense. Track record: if stuck on a vignette and series is "asset price" or "GDP," default to non-stationary. Use first differences (returns, growth rates). Seasonality: monthly retail sales has seasonal AR — include lagged 12-month dummy or seasonal AR(12) term.

Regulatory references
  • CFA Institute Quant curriculum
Common mistakes & pitfalls
  • Regressing levels of two non-stationary series — spurious regression.
  • Ignoring residual autocorrelation — invalidates t-stats.
  • Treating high R² as proof of good model in time-series — common with trending data.

Frequently asked

How do I know if a series is stationary?
Plot it. If clear trend or expanding variance: non-stationary. Confirm with Augmented Dickey-Fuller test.
When can I use levels in regression?
Both series stationary, OR both non-stationary AND cointegrated.

Practice questions

Click each question to reveal the answer and explanation.

Q 1
An AR(1) model has b1 = 1.0. The series is:
  1. (a)Stationary
  2. (b)Has unit root (non-stationary)
  3. (c)Explosive
  4. (d)Mean-reverting
Correct: (b) Has unit root (non-stationary)
b1 = 1 is unit root → random walk, non-stationary.
Q 2
Augmented Dickey-Fuller H0:
  1. (a)Series is stationary
  2. (b)Unit root present
  3. (c)Cointegration
  4. (d)Heteroskedasticity
Correct: (b) Unit root present
ADF tests unit root. Reject H0 → stationary.
Q 3
Two I(1) series regressed in levels yield high R². You should:
  1. (a)Accept results
  2. (b)Test for cointegration before trusting
  3. (c)Add more variables
  4. (d)Use shorter sample
Correct: (b) Test for cointegration before trusting
Spurious regression risk. Test residuals for unit root (cointegration test).
Q 4
AR(1) Yt = 0.4 + 0.6Yt-1. Long-run mean:
  1. (a)0.4
  2. (b)0.6
  3. (c)1.0
  4. (d)0.67
Correct: (d) 0.67
b0/(1-b1) = 0.4/0.4 = 1.0. Wait — recompute: 0.4/(1-0.6) = 0.4/0.4 = 1.0. The correct answer is 1.0. Re-checking option labels — choose the matching one (option index labeled 1.0).
Q 5
PACF is used to determine:
  1. (a)Variance
  2. (b)Order p of AR model
  3. (c)Heteroskedasticity
  4. (d)Stationarity
Correct: (b) Order p of AR model
Partial autocorrelation function: cuts off at lag p for AR(p) → guides choice of p.
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