Time Series Interview Questions: Stationarity, ARMA & Autocorrelation Traps

Q: What time series topics come up most in quant interviews?

The most frequent topics are the definition of weak stationarity, AR(1) algebra (unconditional variance, autocorrelation function, shock half-life), ACF vs PACF signatures for identifying AR and MA models, and unit roots. Researcher interviews at systematic funds also probe spurious regression, volatility clustering, and how to validate models on serially dependent data without lookahead bias.

Q: Is a random walk stationary?

No. A random walk X_t = X_{t-1} + e_t has variance t*sigma^2, which grows linearly with time, violating the constant-variance requirement of weak stationarity. This is why quants typically model returns, which are approximately stationary, rather than price levels.

Q: How do you compute the half-life of an AR(1) process?

A shock to an AR(1) process with coefficient phi decays as phi^k after k periods, so the half-life solves phi^k = 1/2, giving k = ln(0.5)/ln(phi). For example, with phi = 0.6 the half-life is about 1.36 periods. Trading desks use this to measure how quickly a signal loses its predictive power.

Q: What is spurious regression and why do interviewers ask about it?

Spurious regression occurs when you regress one non-stationary series on another and get high R-squared and significant t-statistics even though the series are unrelated; Granger and Newbold documented this for independent random walks. Interviewers ask it because it tests whether a candidate will blindly trust regression output on financial data. The correct responses are to difference the series to stationarity or to test for cointegration.

The time-series toolkit quant researcher interviews actually test — with a fully worked AR(1) example and the traps that catch prepared candidates.

Updated July 3, 2026 · QuantVault

Why quant interviews test time series

Almost everything a quant researcher touches is a time series: prices, returns, order flow, factor exposures. So while trader interviews lean on probability and mental math, researcher and ML-researcher loops at systematic funds — think Two Sigma, D.E. Shaw, or Citadel — test whether you can reason about serially dependent data without fooling yourself. The interviewer is rarely checking whether you memorized the ARIMA acronym. They are checking three things: do you know what stationarity actually means, can you do the AR(1) algebra cold, and do you recognize the classic ways time-series analysis goes wrong on financial data.

The core toolkit

Stationarity. A process is (weakly) stationary if its mean and variance are constant in time and $\mathrm{Cov}(X_t, X_{t+k})$ depends only on the lag $k$. This is the single most-asked definition. The canonical follow-up: is a random walk stationary? No — $X_t = X_{t-1} + \varepsilon_t$ has $\mathrm{Var}(X_t) = t\sigma^2$, which grows without bound. Prices behave like random walks; returns are closer to stationary. That is why we model returns.

The AR(1) model. The workhorse of interview questions:

$$X_t = \phi X_{t-1} + \varepsilon_t, \qquad \varepsilon_t \sim \text{i.i.d.}(0, \sigma^2)$$

It is stationary iff $|\phi| < 1$. Under stationarity, taking variances of both sides gives $\mathrm{Var}(X) = \phi^2 \mathrm{Var}(X) + \sigma^2$, so

$$\mathrm{Var}(X) = \frac{\sigma^2}{1 - \phi^2}, \qquad \rho_k = \phi^k.$$

The geometrically decaying autocorrelation function (ACF) is the fingerprint of an AR process. Know it alongside its mirror image for MA processes:

Model	ACF	PACF
AR(p)	Decays geometrically	Cuts off after lag $p$
MA(q)	Cuts off after lag $q$	Decays geometrically
ARMA(p,q)	Decays (after lag $q$)	Decays (after lag $p$)

Forecasting. For AR(1), the $h$-step conditional forecast is $E[X_{t+h} \mid X_t] = \phi^h X_t$: your best guess decays toward the unconditional mean. This connects directly to regression to the mean — an AR(1) with $\phi < 1$ is mean reversion in equation form, and it underlies most pairs-trading and signal-decay questions.

Worked example: the signal half-life question

“A trading signal follows an AR(1) with $\phi = 0.6$ and $\sigma^2 = 1$. What is its unconditional variance? How many periods until a shock decays by half? If the signal reads 2 today, what is your 2-step-ahead forecast and its uncertainty?”

Variance. Since $|\phi| < 1$ the process is stationary, so

$$\mathrm{Var}(X) = \frac{1}{1 - 0.36} = \frac{1}{0.64} = 1.5625.$$

Half-life. A shock decays as $\phi^k$, so solve $\phi^k = \tfrac{1}{2}$:

$$k = \frac{\ln 0.5}{\ln 0.6} = \frac{-0.693}{-0.511} \approx 1.36 \text{ periods}.$$

This is the number every systematic desk cares about — it tells you how fast you must trade before the signal is gone.

Forecast. $E[X_{t+2} \mid X_t = 2] = \phi^2 \cdot 2 = 0.36 \times 2 = 0.72$. The forecast error variance is

$$\sigma^2\,\frac{1 - \phi^{2h}}{1 - \phi^2} = \frac{1 - 0.6^4}{0.64} = \frac{0.8704}{0.64} = 1.36,$$

already 87% of the unconditional variance of 1.5625. Interviewers love the closing observation: after just two steps, knowing today’s signal tells you very little — with $\phi = 0.6$, predictability dies fast.

The traps that separate candidates

Spurious regression. Regress one random walk on an independent random walk and you will routinely see high $R^2$ and huge t-statistics. Granger and Newbold demonstrated this in 1974, and it remains the most common trap question. The fix is to difference to stationarity or test for cointegration — never regress levels on levels blindly.
“Returns are uncorrelated, so they’re independent.” False, and a favorite follow-up. Daily returns have near-zero autocorrelation, but squared returns are strongly autocorrelated — volatility clusters. Zero autocorrelation does not imply independence; this is the motivation for GARCH and links to the martingale view of prices, where returns are unpredictable in mean but not in variance.
Unit-root tests have low power. An ADF test often cannot distinguish $\phi = 1$ from $\phi = 0.98$ in realistic samples. Saying “the test failed to reject, therefore unit root” is exactly the kind of overclaim interviewers probe.
Over-differencing. Differencing a series that is already stationary introduces an MA(1) with negative autocorrelation and destroys the signal you wanted to model.
Lookahead in validation. Random train/test splits leak future information in serially dependent data; you must split in time. This trap reappears constantly in ML interview questions about backtesting.

How to prepare

Get the AR(1) derivations to the point where you can produce them in under two minutes, memorize the ACF/PACF table, and rehearse a one-line answer for each trap above. If your target role is heavier on theory, the continuous-time analogues (Ornstein-Uhlenbeck as the limit of AR(1)) come up too — our stochastic processes bank covers that bridge.

Then drill under time pressure: work through our time series interview questions with full worked solutions, shore up the foundations in the statistics bank, and browse the full problem bank to mix time-series questions into a realistic interview set.

Frequently asked questions

What time series topics come up most in quant interviews?

The most frequent topics are the definition of weak stationarity, AR(1) algebra (unconditional variance, autocorrelation function, shock half-life), ACF vs PACF signatures for identifying AR and MA models, and unit roots. Researcher interviews at systematic funds also probe spurious regression, volatility clustering, and how to validate models on serially dependent data without lookahead bias.

Is a random walk stationary?