Time-Series Analysis for Quant Interviews: Stationarity, ARIMA, and Practical Forecasting
Time-series analysis is foundational to systematic quant work. Returns, volatility, factor exposures, macro variables, microstructure signals — everything in finance is a time series. Quant-research interviews at Two Sigma, D. E. Shaw, Citadel, Cubist, Bridgewater, and the Strats groups at Goldman Sachs, JPMorgan, and Morgan Stanley test time-series fluency directly. This guide covers what gets asked, the canonical models, and the practical issues that distinguish strong candidates from those who memorized formulas without using them.
Stationarity: The Core Concept
A stationary time series has statistical properties (mean, variance, autocorrelation) that don’t change over time. Most useful theory and methods assume stationarity. Real financial time series often aren’t stationary in raw form, so a key skill is recognizing this and transforming appropriately.
Strict vs weak stationarity
Strict stationarity: the joint distribution of any subset of values doesn’t change with time shifts. Strong condition; rarely directly testable.
Weak (covariance) stationarity: mean and variance are constant; autocovariance depends only on lag, not on time. This is what’s tested in practice (Augmented Dickey-Fuller test, KPSS test) and what most theory requires.
Common non-stationary patterns
- Trend: deterministic linear or polynomial drift. Detrending removes it.
- Random walk / unit root: stochastic non-stationarity. First differencing typically makes it stationary.
- Seasonality: periodic structure. Seasonal differencing or explicit seasonal terms.
- Time-varying volatility: heteroskedasticity. ARCH/GARCH models.
Returns vs prices
Stock prices are non-stationary (typically a random walk with drift). Returns (P_t / P_{t-1} – 1) or log returns (ln(P_t / P_{t-1})) are typically stationary. Almost every financial time-series analysis works with returns, not prices.
Autocorrelation
The correlation of a series with its own lagged values. Defined as ρ(k) = Cov(X_t, X_{t-k}) / Var(X_t).
Plot of ρ(k) vs k is the autocorrelation function (ACF). The partial autocorrelation function (PACF) is the autocorrelation after controlling for intermediate lags. ACF and PACF together help identify ARMA model orders.
Key facts:
- White noise: ACF is zero at all lags (within sampling error).
- AR(p): PACF cuts off after p; ACF decays.
- MA(q): ACF cuts off after q; PACF decays.
- ARMA(p, q): both ACF and PACF decay.
ARIMA Models
The classical workhorse of time-series modeling.
Autoregressive (AR)
X_t = φ_1 X_{t-1} + φ_2 X_{t-2} + … + φ_p X_{t-p} + ε_t. The series depends on its own past values.
Moving average (MA)
X_t = ε_t + θ_1 ε_{t-1} + … + θ_q ε_{t-q}. The series is a weighted sum of recent error terms.
ARMA(p, q)
Combines AR and MA: X_t = AR terms + MA terms + ε_t. Captures both autoregressive structure and shock persistence.
ARIMA(p, d, q)
Apply differencing of order d before fitting ARMA. This handles non-stationary series with d unit roots. ARIMA(0, 1, 0) is a random walk; ARIMA(0, 1, 1) is a random walk with MA(1) errors (used in many financial contexts).
GARCH and Volatility Modeling
Returns are typically stationary in mean but not in variance — volatility clusters (high-vol periods follow high-vol periods). GARCH(p, q) models this:
σ²_t = α_0 + α_1 ε²_{t-1} + … + α_q ε²_{t-q} + β_1 σ²_{t-1} + … + β_p σ²_{t-p}
The most common is GARCH(1, 1):
σ²_t = α_0 + α_1 ε²_{t-1} + β_1 σ²_{t-1}
This captures the empirically observed volatility clustering and is the workhorse of practitioner volatility modeling. EWMA (exponentially weighted moving average) is a degenerate case where α_0 = 0 and α_1 + β_1 = 1.
Cointegration
Two non-stationary series are cointegrated if a linear combination of them is stationary. Pairs trading is built on this: a price series is non-stationary, but the spread between two related price series may be stationary, allowing mean-reversion strategies.
Engle-Granger procedure: regress one series on the other; test residuals for stationarity. If stationary, the series are cointegrated.
Cointegration shows up at hedge funds doing statistical arbitrage; it’s a topic that distinguishes candidates with applied time-series experience from those with only textbook knowledge.
State-Space Models and Kalman Filters
State-space models represent observed time series as functions of unobserved state variables that evolve over time. The Kalman filter is the optimal linear estimator for the state given observations, under Gaussian assumptions.
Applications: dynamic linear models, time-varying betas, signal extraction from noisy data, parameter tracking. Kalman filters appear in interviews more often at quant-research-heavy firms (Two Sigma, D. E. Shaw, Cubist, Renaissance).
Common Interview Questions
Identify the model from ACF/PACF
“Here’s the ACF and PACF of a residual series. What model would you fit?” Strong candidates pattern-match: AR(p) gives PACF cut-off at lag p; MA(q) gives ACF cut-off at lag q.
Test for stationarity
“Given a financial time series, how do you test whether it’s stationary?” Answer: Augmented Dickey-Fuller (null = unit root, reject = stationary), KPSS (null = stationary, reject = non-stationary). Also visual inspection (mean, variance over rolling windows). Strong candidates know the trade-offs and limitations.
Discuss returns vs prices
“Why do quants work with returns rather than prices?” Stationarity, scale invariance (% changes are comparable across asset prices), and additive properties of log returns over time. A clean, fast answer demonstrates baseline competence.
GARCH application
“How would you forecast volatility for tomorrow’s returns?” Discuss EWMA as a starting point, GARCH(1, 1) as the next step, then mention realized volatility, implied volatility, and HAR-RV models for completeness.
Backtesting pitfalls
“You backtest a strategy and get a Sharpe ratio of 3. What concerns do you raise?” Look-ahead bias, data snooping, survivorship bias, overfitting, transaction costs, capacity constraints, regime change. Strong candidates list multiple concerns and explain how to address each.
Practical Issues That Distinguish Strong Candidates
Look-ahead bias
Using information at time t that wouldn’t have been available in real-time. Easy to introduce when not careful: future price information leaking into a feature, end-of-day data used as if it were intraday, restated financials applied retroactively.
Data snooping
Testing many strategies on the same dataset without correcting for multiple comparisons. With enough strategies, you’ll find one that “works” by chance.
Stationarity in regimes
A series may be stationary within calm regimes but exhibit structural breaks at crisis points. Models fit on calm data fail spectacularly during crises (LTCM 1998 is the canonical example).
Sampling frequency
Daily, intraday, tick-level data have very different statistical properties. Microstructure noise dominates at high frequencies; signal-to-noise improves at lower frequencies but at the cost of fewer observations.
Frequently Asked Questions
How important is time-series specifically vs general stats for quant interviews?
Critical for quant-research roles at systematic hedge funds. Time-series questions appear in nearly every research interview at Two Sigma, D. E. Shaw, Cubist, Citadel quant. For quant-developer / SWE roles, less central but still present. For trader roles at market-making firms (Optiver, Jane Street, SIG), peripheral — they care more about probability and brainteasers. Match prep depth to your target firms.
What’s the most common mistake on time-series questions?
Forgetting that returns are typically stationary but prices aren’t. Candidates apply ARMA models directly to price series, which violates assumptions and gives garbage forecasts. The fix is automatic in practice: always work with returns or differenced series unless you have a specific reason to model prices directly. Verbalizing “let me first check stationarity” or “let me work with returns” early signals competence.
What books should I use for time-series prep?
Brockwell and Davis’s Introduction to Time Series and Forecasting is the standard introductory text. Tsay’s Analysis of Financial Time Series is the standard finance-focused reference and is what most quant practitioners read. Hamilton’s Time Series Analysis is the comprehensive academic reference; useful but overkill for interviews. For interview prep specifically, Tsay’s first 8 chapters cover what you need.
How does ML / deep learning compare to classical time-series models in practice?
For most financial forecasting, classical methods (AR, GARCH, simple state-space) perform comparably to or better than deep learning, primarily because financial signals have low signal-to-noise and limited training data relative to typical ML domains. Deep learning shines for some applications (alternative data processing, high-frequency microstructure modeling, large-scale feature learning) but is not a general replacement for classical time-series methods. Interviewers want to see judgment about when to use each.
What’s the difference between a quant interview at a hedge fund vs a bank Strats group?
Hedge fund quant interviews lean more toward statistical signal generation, predictive modeling, and time-series for return forecasting. Bank Strats interviews lean more toward derivatives pricing and risk management, where time series matters but stochastic calculus and PDE methods dominate. If you’re targeting both, prep both; the underlying math overlaps but the application context differs significantly.
See also: Linear Algebra for Quant Interviews • Stochastic Calculus for Quant Interviews • Expected Value and Fair-Game Reasoning