Time-series basics — CFA L1 Quant

Time-series analysis decomposes data into trend, seasonality, and random noise. Stock prices are non-stationary (trending); stock returns are usually stationary. This distinction matters because regression and statistical inference assume stationarity. Master this reading and you understand why trying to "predict the next NIFTY value from past NIFTY values" rarely works — and what the right approach is.

Foundation

Components of a time series: • Trend (T): long-term direction • Seasonality (S): regular cycles (quarterly retail spikes, monthly housing prices, etc.) • Cyclical (C): irregular medium-term patterns • Random noise (ε): unpredictable component Stationarity: statistical properties (mean, variance, autocorrelation) don't change over time. • Strictly stationary: full distribution doesn't change • Weakly stationary: mean, variance, and autocovariance don't change Most time-series analysis assumes weak stationarity. Stock prices: NOT stationary (trending). Stock returns: USUALLY stationary. AR(1) (autoregressive) model: Y_t = α + β · Y_{t−1} + ε_t Today's value depends on yesterday's value plus noise. If |β| < 1: mean-reverting. If |β| ≥ 1: explosive (non-stationary).

Deep Dive

Stationarity tests (CFA L2 dives deeper): • Visual: plot data, look for constant mean and variance • Augmented Dickey-Fuller (ADF) test: tests for unit root (β = 1) • KPSS test: tests for stationarity directly If non-stationary: take first differences. ΔY_t = Y_t − Y_{t−1} For stock prices: P_t is non-stationary, but log returns r_t = ln(P_t / P_{t−1}) are usually stationary. This is why we model returns, not prices. AR(1) interpretation: • β > 0: positive autocorrelation (momentum) • β < 0: negative autocorrelation (mean-reversion) • β = 0: independent observations Indian RBI repo rate: high persistence (β ≈ 0.9-0.95) — past values strongly predict next. Indian inflation: moderate persistence (β ≈ 0.3-0.5). Daily stock returns: low persistence (β ≈ 0-0.1). Forecasting from AR(1): Y_{t+1} = α + β · Y_t. Long-run mean = α / (1 − β) for stationary series.

Real-world scenario

Real-world scenario — INR-USD exchange rate analysis Monthly USD-INR data, 2010-2024 (180 observations). First observation: rate is non-stationary, trending from ~46 (2010) to ~83 (2024). Cannot regress directly. First differences: month-on-month change in USD-INR. Mean ~0.18% per month (rupee depreciates ~2.2% per year). SD: 1.8% per month. AR(1) on differences: ΔS_t = α + β · ΔS_{t−1} + ε Fitted: ΔS_t = 0.001 + 0.05 · ΔS_{t−1} β = 0.05 — very low persistence. This month's INR-USD change tells you almost nothing about next month's change. Currency markets are nearly random walk on monthly data. Practical implication: forecasting USD-INR for next month from past months: not better than coin flip. Forecasting longer horizon from PPP and interest-rate parity: more reliable but still high uncertainty. For Indian exporters/importers: don't try to time currency. Use forwards or hedge mechanically. Anyone who claims "ability to forecast USD-INR" is selling, not analysing.

Advanced

Practitioner traps in time-series: 1. Spurious regression: regress two non-stationary trending variables on each other → get high R² and significant β even when no real relationship. Always check stationarity before regressing. 2. Cointegration: two non-stationary series that move together — their linear combination IS stationary. Useful for pairs trading, cross-currency analysis. 3. Look-ahead bias: using information not available at time of decision. Backtests using "today's knowledge to predict yesterday" produce inflated returns. 4. Survivorship bias: time-series of "surviving" assets understates true average. If you analyse Indian large-cap stocks today, you exclude those that delisted, merged, or went bankrupt. 5. Selection bias: choosing time period that flatters the strategy. Backtesting only the bull market makes everything look good. CFA L1 introduces stationarity; L2 tests modelling; L3 applies in portfolio construction.

Regulatory references

CFA Institute Curriculum — Level 1, Quantitative Methods, Reading 8
RBI publications on monetary-policy transmission
BIS analysis of currency-rate persistence

Common mistakes & pitfalls

Regressing two non-stationary series — produces spurious results.
Using stock prices (non-stationary) instead of returns (stationary) in regressions.
Assuming AR(1) β is constant when regimes shift.
Confusing autocorrelation with causation — AR(1) describes pattern, not mechanism.
Backtesting on entire historical sample — should reserve out-of-sample data.

Frequently asked

Why are stock returns stationary but prices not?

Prices trend (have a non-zero mean drift over time). Returns oscillate around a mean (typically positive but small) without trend. Statistical models assume stationarity; modelling returns rather than prices is the standard approach.

Should I use AR(1) for forecasting stock returns?

For developed markets, AR(1) on daily returns has β near zero — past returns don't predict next-day returns. For longer horizons or specific patterns (mean-reversion in 3-5 year windows), there's some signal but small. Don't overestimate predictive power.

What's cointegration?

Two non-stationary series whose linear combination is stationary. Stock prices of two banks may both trend, but their ratio may be stationary (mean-reverting). Pairs trading exploits this. CFA L2 introduces; L3 deepens.

Practice questions

Click each question to reveal the answer and explanation.

Q 1

In AR(1) Y_t = α + βY_{t−1} + ε, the long-run mean (assuming stationarity) is:

(a)α
(b)β
(c)α/(1−β)
(d)β/(1−α)

Correct: (c) α/(1−β)

For stationary AR(1) (|β| < 1), long-run mean = α/(1−β).

Q 2

A time series with constant mean and variance over time is:

(a)Trending
(b)Seasonal
(c)Stationary
(d)Cyclical

Correct: (c) Stationary

Stationary = constant mean, variance, autocovariance over time. The foundation assumption for time-series modelling.

Q 3

For an AR(1) with β = 0.95, the half-life of shocks is closest to:

(a)1 period
(b)5 periods
(c)14 periods
(d)50 periods

Correct: (c) 14 periods

Half-life = ln(0.5) / ln(β) = ln(0.5) / ln(0.95) ≈ 13.5 ≈ 14 periods. High β (close to 1) means slow mean-reversion.

Q 4

Stock prices vs stock returns:

(a)Both are stationary
(b)Both are non-stationary
(c)Prices are non-stationary; returns are typically stationary
(d)Prices are stationary; returns are not

Correct: (c) Prices are non-stationary; returns are typically stationary

Prices trend (non-stationary). Returns oscillate around mean (stationary). This is why we model returns, not prices.

Q 5

A "spurious regression" is most likely when:

(a)Both variables are stationary
(b)Both variables are non-stationary trending
(c)There are too many observations
(d)R² is low

Correct: (b) Both variables are non-stationary trending

Regressing two trending non-stationary series produces high R² and significant coefficients even when no real relationship exists. Always test stationarity first.

Educational purposes only. The numbers, returns, and examples used in this lesson are illustrative. Past performance does not guarantee future results. Mutual fund and securities investments are subject to market risks. This lesson is not investment advice; for advice tailored to your circumstances, consult a SEBI-registered Investment Adviser. Read our full disclaimer.