FRM Part 1 · Quantitative Analysis · Chapter QTA 11
Non-Stationary Time Series
Learning Objectives
- Describe linear and nonlinear time trends.
- Explain how regression analysis can be used to model seasonality.
- Describe a random walk and a unit root.
- Explain the challenges of modeling time series containing unit roots.
- Describe how to test if a time series contains a unit root.
- Explain how to construct an h-step-ahead point forecast for a time series with seasonality.
- Calculate the estimated trend value and construct an interval forecast for a time series.
The previous chapter assumed stationarity, a series whose statistical character stays fixed over time. Most real financial and economic series break that assumption. Prices trend upward for years, sales swing with the seasons, and asset prices wander like a drunkard, never settling on a level. A series whose mean or variance drifts is non-stationary, and applying stationary tools to it produces misleading estimates and dangerous forecasts.
This chapter covers the three most common forms of non-stationarity and, crucially, how to remove each one. Time trends and seasonalities are deterministic: they follow a fixed pattern in time, and subtracting that pattern leaves a stationary series behind. Random walks, also called unit roots, are stochastic: the non-stationarity is baked into the shocks themselves, and the fix is not subtraction but differencing. Getting the diagnosis right, deterministic trend versus unit root, decides which fix to apply, and it is the central skill of the chapter.
Key Takeaways
- A time trend makes the mean grow with time; a linear trend adds a constant each period, a log-linear trend implies a constant growth rate.
- Seasonality is modeled with dummy variables, one per season, dropping one dummy or the intercept to avoid the dummy trap.
- A random walk is the running sum of all past shocks; it never mean-reverts and its variance grows without bound. A unit root is its general form.
- Unit roots cause three problems: biased non-normal estimators (needing the Dickey-Fuller distribution), spurious regressions, and no level to forecast toward.
- Deterministic trends are removed by subtracting the fitted trend; unit roots are removed by differencing, and the ADF test tells them apart.
- A trend forecast is the fitted trend at the future date; an interval forecast surrounds it with plus and minus a critical value times the forecast standard error.
Time Trends
The simplest departure from stationarity is a time trend, a deterministic term that pushes the mean of a series steadily up or down as time passes. A linear trend adds a fixed amount each period; because the mean depends on time, the series is non-stationary, but the non-stationarity is entirely predictable.
where t is the time index (1, 2, 3, …) and δ1 is the trend slope, the average change in Y each period. The mean, δ0 + δ1t, moves with time, so the series is non-stationary even though its deviations around the line may be well behaved.
A plain linear trend has two weaknesses for financial data. If the slope is positive, the implied growth rate, the slope divided by the level, keeps falling as the level rises, which is unrealistic; and if the slope is negative, the series eventually turns negative, impossible for a price or a quantity. Constant percentage growth is the more plausible assumption for most financial and macroeconomic variables, and that is exactly what the log-linear model delivers.
Trends come in shapes. A quadratic trend adds a squared time term, letting the rate of change itself vary, useful when growth accelerates or decelerates. More common in finance is the log-linear trend, which puts a linear trend on the natural log of the series; this implies a constant percentage growth rate rather than a constant absolute change, which fits variables that grow by a roughly fixed rate each year. All of these are estimated by ordinary least squares, regressing the series (or its log) on time, and subtracting the fitted trend leaves a trend-stationary residual. Exhibit 1 lays out the family.
| Trend | Model | Implies |
|---|---|---|
| Linear | Yt = δ0 + δ1t + εt | Constant change δ1 each period |
| Quadratic | Yt = δ0 + δ1t + δ2t² + εt | Rate of change varies over time |
| Log-linear | ln Yt = δ0 + δ1t + εt | Constant growth rate δ1 |
| Log-quadratic | ln Yt = δ0 + δ1t + δ2t² + εt | Growth rate itself changes over time |
A firm’s quarterly revenue follows the fitted linear trend Yt = 100 + 5t (in millions), with a residual standard deviation of 8. Estimate revenue in quarter 24, and give a 95% interval forecast.
Step 1. The point forecast is the trend evaluated at t = 24 (future shocks have expected value zero).
Step 2. The 95% interval is the point plus and minus 1.96 standard deviations.
Answer: an estimated revenue of 220 million, with a 95% interval from about 204 to 236 million. The point forecast is just the trend line read at the future date; the interval reflects the scatter of actual revenue around that line, measured by the residual standard deviation.
Modeling Seasonality With Dummies
Seasonality is a repeating within-year pattern: retail sales peak in the fourth quarter, energy demand swings with the weather, some rates move on a calendar cycle. Because the mean depends on the time of year, a seasonal series is non-stationary. The standard way to model it with regression is seasonal dummy variables, one indicator for each season, which let the regression fit a separate mean for each period of the cycle.
where there are s seasons (4 for quarterly, 12 for monthly) and each D is a dummy that equals 1 in its season and 0 otherwise. One dummy is dropped, or the intercept is, to avoid the dummy variable trap, the perfect collinearity that arises from including a full set of dummies alongside an intercept.
Each dummy coefficient measures the average level of the series in that season, relative to the omitted baseline season. The power of the regression approach is that it composes: a model can carry a time trend for long-run growth, seasonal dummies for the calendar pattern, and an autoregressive term for the cyclical dynamics, all at once, capturing the three components of a series, trend, seasonality, and cycle, in a single equation. When the residuals from a trend-and-seasonal model are stationary but still autocorrelated, adding AR or MA terms cleans them up.
Random Walks and Unit Roots
The most important form of non-stationarity in finance is the random walk. In a random walk, today’s value is simply yesterday’s value plus a fresh shock, so the series carries its entire history forward, permanently.
where repeated substitution shows the level is the starting value plus the sum of every shock to date. Each shock has a permanent effect, unlike a stationary AR where old shocks fade. A random walk with drift adds a constant, Yt = δ + Yt−1 + εt, tilting the walk steadily up or down.
Two properties make the random walk non-stationary and distinctive. Its variance is t times the shock variance, so it grows without bound as time passes, the series fans out ever wider. And because every shock is permanent, the series has no tendency to revert to any level; it wanders indefinitely. This is the opposite of a stationary AR, which mean-reverts and has constant variance. A unit root is the general name for this behavior: it is an autoregressive process whose key coefficient equals exactly 1, the boundary between the stationary world (coefficient below 1) and the explosive world (coefficient above 1). Virtually all asset prices behave like random walks, which is why unit roots dominate financial time-series work.
The random walk also connects directly to a cornerstone of finance: market efficiency. If a price already reflects all available information, then the best forecast of tomorrow’s price is today’s price, and the change is unforecastable noise, exactly the structure of a random walk. This is why returns, the differences of log prices, are the natural object of study rather than price levels: the level is non-stationary and unforecastable, while the return is stationary and is where any modelable structure, such as volatility clustering, actually lives.
A price follows a random walk with a shock standard deviation of 2 per period. What is the standard deviation of the price 25 periods from the start, and why does this matter?
Step 1. The variance of a random walk after t periods is t times the shock variance.
Answer: a standard deviation of 10, up from 2 after a single period. The spread grows with the square root of time and never stops growing, which is exactly why a random walk is non-stationary: it has no fixed variance to settle at, so historical estimates of its level are meaningless for the distant future.
Why Unit Roots Are Hard
Unit roots are not just a curiosity; they break the standard toolkit in three specific ways, and knowing all three is a reliable exam target. Exhibit 2 pairs each problem with its remedy.
First, estimation goes wrong. When a model contains a unit root, the parameter estimators are biased and no longer normally distributed, so the ordinary t-statistics and their normal critical values are invalid. Inference instead requires the Dickey-Fuller distribution, which is asymmetric, depends on the sample size, and has fatter tails than the normal, so its critical values are larger in magnitude.
Second, spurious regression. Regressing one unit-root series on another unrelated unit-root series routinely produces a large coefficient, a high R-squared, and a wildly significant t-statistic, all pointing to a strong relationship that does not exist. Because both series wander, they drift together often enough to fool a naive regression. This is a notorious trap: two unconnected trending or random-walk series can look tightly linked purely by chance.
Third, no mean reversion. A stationary AR reverts to a fixed level, so its long-run forecast is that level. A random walk has no such level; the best forecast of a random walk is simply its current value, forever. A model that wrongly assumes mean reversion where there is a unit root will produce forecasts that are confidently, and increasingly, wrong.
| Challenge | What goes wrong | Consequence |
|---|---|---|
| Biased, non-normal estimators | t-statistics are invalid; need the Dickey-Fuller distribution | Ordinary inference misleads |
| Spurious regression | Unrelated series look strongly linked | False signals, poor out-of-sample results |
| No mean reversion | No fixed level to forecast toward | Long-horizon forecasts go badly wrong |
Differencing: the Fix
The single fix for all three problems is differencing: model the change in the series rather than its level. If a series has a unit root, its first difference is stationary. The reason is clean: differencing a random walk cancels the accumulated history and leaves only the fresh shock.
where the first difference of a random walk is pure white noise, which is stationary. A series that becomes stationary after one difference is called integrated of order one; most economic series are of this kind, while a few, such as price-level indices, may need differencing twice.
Differencing is versatile: it removes a unit root, and it also removes a deterministic linear trend, and differencing at the seasonal lag (subtracting the value one year earlier) removes deterministic seasonality too. This makes it a robust catch-all when a series carries several kinds of non-stationarity at once. The one caution is over-differencing: differencing an already-stationary series introduces an artificial moving-average structure and inflates estimation error, so differencing should be applied only when a unit root is actually present, which is what the test in the next section establishes.
The diagnosis dictates the fix. A deterministic trend or seasonality is removed by subtracting the fitted pattern, leaving a trend-stationary series. A stochastic trend, a unit root, is removed by differencing. Applying the wrong one is costly: detrending a unit-root series leaves it non-stationary, and differencing a trend-stationary series over-differences it. Telling the two apart is exactly what a unit root test is for.
Testing for a Unit Root: the ADF Test
The workhorse test is the augmented Dickey-Fuller (ADF) test. It regresses the change in the series on its own lagged level, plus any needed deterministic terms and a few lagged differences that soak up short-run dynamics, and it examines the coefficient on the lagged level.
where γ (gamma) is the coefficient on the lagged level. The null hypothesis is H₀: γ = 0, a unit root, against the one-sided alternative H₁: γ < 0, a stationary series. The “augmented” lagged differences are included so the residuals are white noise, which the test requires.
The critical twist is the distribution. Under the null, the test statistic does not follow a normal distribution; it follows the Dickey-Fuller distribution, whose critical values are more negative than the normal’s (around minus 2.9 at 5 percent with a constant, versus minus 1.65 for a normal one-sided test). The decision rule is one-sided: reject the unit root, concluding the series is stationary, only when the statistic is more negative than the critical value. A statistic that is not negative enough leaves the unit root in place, and the series should then be differenced.
An ADF regression of a credit index yields a coefficient on the lagged level with a t-statistic of −1.8. The Dickey-Fuller 5% critical value (with a constant) is −2.87. Does the series contain a unit root?
Step 1. Compare the statistic with the Dickey-Fuller critical value; reject only if more negative.
Answer: the statistic of −1.8 is not below (not more negative than) the critical value of −2.87, so the null of a unit root cannot be rejected. The series is treated as non-stationary, and it should be differenced before modeling. Note the trap: judged against the ordinary normal critical value of −1.65, −1.8 would look significant, which is precisely the error the Dickey-Fuller critical values exist to prevent.
Forecasting: Point, Seasonal, and Interval
Forecasting a series with deterministic components is straightforward, because future shocks have expected value zero. The point forecast of a trend model at a future date is simply the estimated trend evaluated at that date. For a series with seasonality, the h-step-ahead forecast adds the seasonal dummy that matches the season of the target period, so a forecast for a future fourth quarter uses the fourth-quarter dummy, whatever the current quarter is.
where T is the last observed period, h the forecast horizon, and γj the dummy for the season that the period T + h falls in. A model with a trend, seasonality, and an AR term combines all three, with the AR part fading as the horizon grows.
An interval forecast puts a band around the point forecast to express its uncertainty. The width depends on the standard deviation of the forecast error; for a linear trend model with normal shocks, the forecast error is just the future shock, so a 95 percent interval is the point forecast plus and minus 1.96 times the residual standard deviation, exactly as computed in Worked Example 1. Models with AR or MA components have a forecast-error variance that grows with the horizon, so their intervals widen the further out they reach, which is the honest picture of forecasting: the further ahead, the less certain.
A quarterly sales model is Yt = 50 + 2t + 12 D4t (in millions), where D4 is the fourth-quarter dummy and the current period is t = 40, a third quarter. Forecast sales two quarters ahead.
Step 1. Two quarters ahead is t = 42, which is a first quarter, so the Q4 dummy is 0.
Answer: 134 million. Because the target quarter is not a fourth quarter, the seasonal bump does not apply, only the trend does. Had the forecast landed on a fourth quarter, the extra 12 million would be added. The key is to track the season of the period being forecast, not the season you are standing in now.
Two series both look non-stationary. One is a stock price that never returns to any level; the other is a company’s revenue that rises by a steady amount each year around a clear line. Which fix, detrending or differencing, applies to each?
Show answer
The stock price behaves like a random walk, a stochastic trend, so it is made stationary by differencing: model the change (the return) rather than the level. The revenue has a deterministic linear trend, so it is made stationary by detrending: subtract the fitted trend line and model the residual. Matching the fix to the type of non-stationarity is essential; differencing the revenue would over-difference it, and detrending the price would leave it non-stationary.
An ADF test gives a test statistic of −3.5, and the Dickey-Fuller 5% critical value is −2.87. What do you conclude, and what would you have wrongly concluded using a normal table?
Show answer
Since −3.5 is more negative than the critical value of −2.87, the null of a unit root is rejected: the series is stationary and does not need differencing. Using a normal table would not change this particular conclusion, but the general danger runs the other way. A statistic like −2.0 looks significant against the normal one-sided value of −1.65 yet is not significant against the Dickey-Fuller value of −2.87, so relying on the normal table would falsely declare a unit-root series stationary.
A researcher regresses one stock’s price level on an unrelated stock’s price level and finds a t-statistic of 8 and an R² of 0.9. Why is this not evidence of a real relationship?
Show answer
Because both price levels are non-stationary (they behave like random walks), this is a textbook spurious regression. Two unrelated wandering series drift together often enough to produce a large coefficient, a high R², and a hugely significant t-statistic, none of which reflect a genuine link. The correct approach is to difference both series first, regressing one return on the other; on the stationary differences the apparent relationship almost always vanishes.
A monthly sales model with a trend and monthly dummies is used to forecast six months ahead. Why does it matter which calendar month the target falls in?
Show answer
Because the seasonal dummy applied must match the season of the period being forecast, not the current month. The h-step forecast is the trend value at the target date plus the dummy coefficient for that target month. Forecasting into December uses the December dummy and its seasonal bump; forecasting into a low month uses that month’s coefficient instead. Tracking the target period’s season is the whole point of an h-step seasonal forecast.
Chapter Summary
- A time trend makes the mean grow with time. Linear trends add a constant each period; log-linear trends imply a constant percentage growth rate; quadratic trends let the rate vary.
- Seasonality is modeled with dummy variables, one per season, dropping one dummy or the intercept to avoid the dummy trap; each coefficient is that season’s average level.
- Trends, seasonal dummies, and an AR term can be combined to capture a series’ trend, seasonality, and cycle in one model.
- A random walk is the running sum of all past shocks: no mean reversion, and a variance that grows as t times the shock variance. A unit root is its general form, an AR coefficient of exactly 1.
- Unit roots cause biased non-normal estimators (needing the Dickey-Fuller distribution), spurious regressions, and forecasts with no level to revert to.
- Deterministic trends and seasonality are removed by subtracting the fitted pattern; a unit root is removed by differencing, which also removes trends and (at the seasonal lag) seasonality. Avoid over-differencing.
- The ADF test regresses the change on the lagged level; the null is a unit root, rejected only when the statistic is more negative than the Dickey-Fuller critical value.
- A trend point forecast is the fitted trend at the future date; a seasonal forecast adds the dummy for the target season; an interval forecast is the point plus and minus a critical value times the forecast standard error.
Frequently Asked Questions
What is a time trend, and what is the difference between a linear and a nonlinear trend?
A time trend is a deterministic component that makes the mean of a series change with time, which by itself makes the series non-stationary. A linear trend adds a constant amount each period, so the series grows by the same quantity every step. A nonlinear trend, such as a quadratic, lets the rate of change itself vary over time, while a log-linear trend implies a constant percentage growth rate rather than a constant absolute change. Trends are estimated by regressing the series on time, and once a deterministic trend is removed the remaining series can be stationary.
How is seasonality modeled with regression?
Seasonality, a repeating within-year pattern, is modeled by adding seasonal dummy variables to a regression. For a series with s periods per cycle, four for quarterly data or twelve for monthly data, one dummy is created for each season, and either the intercept is dropped or one dummy is omitted to avoid perfect collinearity, the dummy variable trap. Each dummy coefficient captures the average level of the series in that season. These seasonal dummies can be combined with a time trend and autoregressive terms to model the trend, the seasonality, and the cyclical dynamics together.
What is a random walk and a unit root?
A random walk is a series whose current value equals its previous value plus a random shock, so it is the running sum of all past shocks. Because each shock has a permanent effect, a random walk never reverts to a fixed level, and its variance grows without bound as time passes, which makes it non-stationary. A unit root is the general version of this: it is an autoregressive process whose key coefficient equals exactly one, the boundary where a process stops being stationary. Almost all asset prices behave like random walks, which is why unit roots are the most important form of non-stationarity in finance.
Why are unit roots challenging to model?
Unit roots create three problems. First, the usual parameter estimators are biased and no longer normally distributed, so standard t-statistics are invalid and the special Dickey-Fuller distribution must be used instead. Second, regressing one unit-root series on another often produces a spurious relationship, a strong and significant-looking link between series that are actually unrelated. Third, a unit root process does not mean-revert, so it has no fixed level to forecast toward, and forecasts from a mis-specified model can be badly wrong. The solution to all three is to difference the series.
How do you test whether a series contains a unit root?
The standard test is the augmented Dickey-Fuller (ADF) test. It regresses the change in the series on its own lagged level, plus any deterministic terms and a few lagged differences to clean up short-run dynamics, and examines the coefficient on the lagged level. The null hypothesis is that this coefficient is zero, meaning a unit root is present and the series is non-stationary; the alternative is that it is negative, meaning the series is stationary. Because the test statistic follows the Dickey-Fuller distribution rather than the normal, its own critical values must be used, and a statistic more negative than the critical value rejects the unit root.
What is spurious regression?
Spurious regression is a false relationship that appears when two unrelated non-stationary series are regressed on each other. Because both series wander over time, a naive regression often produces a large coefficient, a high R-squared, and a very significant t-statistic, suggesting a strong link where none exists. The problem arises specifically with unit-root or trending series and is a major pitfall in financial modeling. It is avoided by making both series stationary first, usually by differencing, before running the regression.
How do you forecast a series with a trend or seasonality, and construct an interval?
For a series with a deterministic trend, the point forecast at a future date is the estimated trend evaluated at that date, since future shocks have expected value zero. For a seasonal series, the forecast adds the seasonal dummy that matches the season of the future period being forecast. An interval forecast surrounds the point forecast with a band of plus and minus a critical value times the standard error of the forecast, so a 95 percent interval for a linear trend model with normal errors is the point forecast plus and minus 1.96 times the residual standard deviation.
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