FRM Part I · FRM Exam Part I
Nonstationary Time Series for FRM Part I
A nonstationary time series has a mean, variance or covariance structure that changes over time, so standard time-series models can mislead. To handle it, you identify the source (trend, unit root, seasonality, break), test it, for example with a Dickey-Fuller test, and then detrend, difference or adjust before modelling.
What this chapter covers
This chapter covers what happens when the statistical properties of a series change over time. Most time-series tools you learn in Quantitative Analysis assume covariance stationarity: a constant mean, a constant variance and autocovariances that depend only on the lag. Real financial data often breaks those assumptions. Prices trend, interest rates can behave like random walks, sales and credit data are seasonal, and volatility shifts between calm and stressed periods.
The chapter moves in a logical line. You first define stationary and nonstationary series. Then you meet the main causes of nonstationarity: deterministic trends and unit roots. Next you learn how to test for a unit root with the Dickey-Fuller test, and what goes wrong if you ignore the problem (spurious regression). Finally you cover seasonality and structural breaks, including time-varying volatility.
The chapter connects to the rest of the paper in several places. It builds on regression, autoregressive models and ARMA processes in Quantitative Analysis. It supports volatility modelling such as EWMA and GARCH. It also feeds into Valuation and Risk Models, where you estimate risk from historical return data and need to know when that history is a reliable guide.
Questions here are usually short and conceptual, with a little arithmetic, so they are good marks if your concepts are clean. The same ideas also help you in neighbouring chapters on time series, regression and volatility. With 100 questions in 4 hours, you can answer most items on this chapter quickly if you recognise the pattern: which type of nonstationarity is described, what test or fix applies, and what the consequence is for inference.
Nonstationary Time Series: topics in the order to study them
- 1Stationary vs Nonstationary Time SeriesStart here because every later topic is defined against the conditions for covariance stationarity.
- 2Deterministic Trends and Trend ModelsThe simplest source of nonstationarity, where the fix is to model and remove a predictable trend.
- 3Random Walks and Unit RootsThis is the stochastic alternative to a trend, and you need it to see why a trend and a unit root need different fixes.
- 4Testing for Unit Roots: Dickey-Fuller TestOnce you know what a unit root is, you learn how to test for it and how to read the hypotheses.
- 5Spurious Regression and DifferencingThis shows the cost of ignoring a unit root and the standard remedy, so it follows the test.
- 6Seasonality and Seasonal AdjustmentA separate, more mechanical source of nonstationarity that is easier once the trend and unit-root ideas are settled.
- 7Structural Breaks and Time-Varying VolatilityFinish with the broadest topic, which links the chapter back to volatility models and the stability of estimated relationships.
How to prepare Nonstationary Time Series
Aim for a clear decision framework rather than memorised definitions. For each series, you should be able to say what is nonstationary, how you would test it and what you would do about it.
- Write the three conditions of covariance stationarity from memory: constant mean, constant variance, and autocovariance that depends only on the lag.
- Compare a trend-stationary series with a unit-root series side by side. Note that the first is fixed by removing a trend and the second by differencing.
- Learn the random walk: yₜ = yₜ₋₁ + εₜ. Work out why its variance grows with time and why shocks never die out.
- Learn the Dickey-Fuller setup: the null hypothesis is a unit root, so failing to reject means the series is treated as nonstationary. Practise reading test results and note that critical values differ from the standard t-table.
- Practise short questions on spurious regression: two unrelated trending series can show a high R² and a significant slope. Then confirm that differencing, or checking residuals, is the usual response.
- Do timed mixed sets with seasonality and breaks, then review each wrong answer by naming the type of nonstationarity you missed.
Common mistakes in Nonstationary Time Series
Treating every trending series as having a unit root.
Fix: Ask whether the trend is a predictable function of time (remove it) or the result of accumulated shocks (difference it).
Reversing the Dickey-Fuller hypotheses.
Fix: Remember that the null is a unit root. Rejecting the null points to stationarity.
Using a standard t-table to judge the Dickey-Fuller statistic.
Fix: Under the null the distribution is nonstandard, so use Dickey-Fuller critical values.
Trusting a high R² in a regression of two trending series.
Fix: Check for unit roots first. A high R² with highly autocorrelated residuals is a classic sign of spurious regression.
Confusing seasonality with a trend or a cycle.
Fix: Seasonality repeats at a fixed calendar frequency. A trend is a long-run drift, and a cycle has no fixed calendar period.
Assuming a stationary model stays valid across a structural break.
Fix: If a break is described, expect estimates from the full sample to be unreliable and consider estimating the sub-periods separately.
Last-day revision: Nonstationary Time Series
- Covariance stationarity means constant mean, constant variance and autocovariance that depends only on the lag.
- A deterministic trend is a predictable function of time; a trend-stationary series is stationary after removing it.
- A random walk, yₜ = yₜ₋₁ + εₜ, has a unit root, and its variance grows with time.
- With a unit root, shocks have permanent effects; with a stationary AR(1), shocks decay.
- Differencing a unit-root series once often makes it stationary.
- Dickey-Fuller null hypothesis: a unit root is present. Alternative: the series is stationary.
- Dickey-Fuller test statistics are compared with special critical values, not the normal t-table.
- Failing to reject the null does not prove a unit root; it only means you lack evidence against it.
- Spurious regression: unrelated nonstationary series can show a high R² and a significant slope.
- Seasonality is a regular pattern at a fixed calendar frequency, handled with seasonal dummies or seasonal adjustment.
- A structural break is a shift in parameters; estimates across the break can be misleading.
- Volatility often changes over time, so a constant-variance assumption can fail for returns.
Nonstationary Time Series practice questions
- Quarterly sales of a firm are modeled as Y_t = 100 + 2t + S_q, where S_q is a seasonal dummy effect measured relative to Q4 (the base). The …
- A risk analyst fits Y_t = a + bX_t + u_t using levels of two series that each have a unit root. The Durbin-Watson statistic is 0.15 and resi…
- An analyst fits an AR(1) model to monthly changes in a bond spread over 2005-2023 and suspects that the intercept and slope changed after a …
- A analyst models a monthly log price series as Y_t = Y_{t-1} + e_t, where e_t is white noise with variance 0.0004. Y_0 = 4.60 is known. What…
- An analyst regresses the level of one stock market index on the level of an unrelated commodity price index. Both series are random walks wi…
- A series follows Y_t = 2 + 0.6t + e_t, where e_t is white noise. An analyst first-differences this trend-stationary series. Which statement …
- A price series follows the random walk with drift P_t = 0.5 + P_(t-1) + e_t, where e_t is white noise with variance 4. After taking first di…
- A random walk without drift is defined as y_t = y_{t-1} + e_t, with y_0 = 0 and e_t i.i.d. with variance 0.25. What is the variance of y_t a…
Nonstationary Time Series in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Nonstationary Time Series: frequently asked questions
What is the difference between a trend-stationary and a unit-root series?
A trend-stationary series fluctuates around a deterministic trend, so removing the trend leaves a stationary series. A unit-root series is driven by accumulated shocks, so shocks are permanent and differencing is the usual fix.
Do I need to calculate a Dickey-Fuller statistic by hand?
Usually not. You are more likely to be asked to state the null hypothesis, interpret a reported statistic against critical values, or choose the right action. Make sure you can state the null and the conclusion for both outcomes.
Why does differencing help?
For a random walk, the first difference yₜ − yₜ₋₁ equals the shock εₜ, which is stationary if the shocks are well behaved. That removes the unit root, so standard regression methods become valid again.
How does this chapter link to volatility models?
Time-varying volatility is a form of changing variance, which breaks the constant-variance condition for stationarity in the simple sense. Models such as EWMA and GARCH allow the conditional variance to change, so this chapter sets up the reason for using them.