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CFA Level II Exam · Time-Series Analysis

Autoregressive (AR) Models and Covariance Stationarity

Updated 7 October 2026 · Fact-checked

An **AR(p) model** predicts a time series from its own p past values: x(t) = b0 + b1·x(t−1) + … + bp·x(t−p) + ε(t). It is valid only if the series is **covariance stationary**. For multiperiod forecasts, use the chain rule: feed each forecast into the next one.

Understand Autoregressive (AR) Models and Covariance Stationarity

A time series is a list of values recorded over time, such as quarterly sales or monthly inflation. An autoregressive (AR) model says today's value depends on its own past values. An AR(1) model uses one lag. An AR(2) model uses two lags. In general, AR(p) uses p lags.

The model only gives reliable results if the series is covariance stationary. This means three things: the expected value (mean) is constant over time, the variance is constant over time, and the covariance between the series and its own lagged values depends only on the distance between the observations, not on when they occur. If any of these fails, the regression estimates are unreliable and the usual t-tests are not valid.

For an AR(1) model, a stationary series needs a finite mean-reverting level. This level is b0 ÷ (1 − b1), and it exists only when |b1| < 1. If b1 = 1, the series is a random walk (a unit root), the mean-reverting level is undefined, and the series is not covariance stationary. You then have to transform the data, for example by first differencing, before using an AR model.

Forecasting is done step by step. A one-period-ahead forecast plugs the latest actual value into the equation. A two-period-ahead forecast uses the one-period forecast as the input. This is the chain rule of forecasting. Forecasts move toward the mean-reverting level as the horizon grows, and uncertainty grows with the horizon.

Key formulas to remember

AR(p) model
x(t) = b0 + b1·x(t−1) + b2·x(t−2) + … + bp·x(t−p) + ε(t)
AR(1) is the special case with only b1. Estimated by ordinary least squares on the lagged values.
Covariance stationarity conditions
E[x(t)] constant; Var[x(t)] constant; Cov[x(t), x(t−k)] depends only on k
All three must hold for the AR model to be valid.
Mean-reverting level (AR(1))
x* = b0 ÷ (1 − b1)
Exists only when b1 ≠ 1, and the series is stationary only if |b1| < 1.
Chain-rule forecast (AR(1))
x̂(t+1) = b0 + b1·x(t); x̂(t+2) = b0 + b1·x̂(t+1)
Use the previous forecast as the input for the next period.
Direction of mean reversion
If x(t) > x*, next value is expected to fall; if x(t) < x*, it is expected to rise
Holds for an AR(1) with 0 < b1 < 1 and finite x*.

How to solve Autoregressive (AR) Models and Covariance Stationarity questions

Use this sequence for any vignette question on AR models, stationarity or forecasts.

  1. 1Find the estimated equation in the exhibit and note b0, b1 and any further lag coefficients. Check which variable is the dependent one and what the units are.
  2. 2Check the model type: AR(1), AR(2) or a model on differences. If the data are differenced, the forecast is a change, not a level.
  3. 3Decide whether the series is covariance stationary. Look for b1 = 1 or a unit root test result. A constant mean, variance and lag covariances are required.
  4. 4For a mean-reverting level, compute b0 ÷ (1 − b1). It is undefined if b1 = 1.
  5. 5For a one-step forecast, substitute the latest observed value into the equation.
  6. 6For a multistep forecast, apply the chain rule: use the forecast from step one as the input to step two, and so on. Do not skip steps.
  7. 7Compare your forecast with the mean-reverting level to check direction, then match the answer to the three options.

Quickest way: Plug-in and mean-reversion check

When to use it: Use when the vignette gives a clean AR(1) equation and asks for a one- or two-period forecast or the mean-reverting level.

  1. Compute x* = b0 ÷ (1 − b1) first. It gives a quick sense of where forecasts will head.
  2. Compute the one-step forecast: b0 + b1 × latest value.
  3. For two steps, repeat once using that result.
  4. Sanity check: forecasts should sit between the latest value and x* when 0 < b1 < 1. Eliminate options that fall outside this range.

Common mistakes in Autoregressive (AR) Models and Covariance Stationarity

  • Using the actual latest value again in the two-period forecast instead of the first forecast.

    Students forget that the actual value two periods ahead is unknown.

    Fix: Always chain: x̂(t+2) = b0 + b1·x̂(t+1).

  • Computing the mean-reverting level as b0 ÷ b1 or b0 × (1 − b1).

    The formula is misremembered.

    Fix: Set x = b0 + b1·x and solve: x* = b0 ÷ (1 − b1).

  • Saying a series with b1 = 1 is stationary because the model fits well.

    Students confuse a high fit with stationarity.

    Fix: A unit root (b1 = 1) means a random walk with no finite mean-reverting level. It is not covariance stationary, so the model is invalid as specified.

  • Thinking a trending series is stationary if its variance looks stable.

    Only one of the three conditions is checked.

    Fix: A series with a trending mean fails the constant-mean condition. Check all three conditions.

  • Forecasting a level from a model estimated on first differences.

    The dependent variable is a change, but the answer is read as a level.

    Fix: Add the forecast change to the latest level to get the level forecast.

Worked examples

Example 1

An analyst estimates an AR(1) model for a company's quarterly operating margin (in %): x(t) = 2.0 + 0.60·x(t−1). The latest margin is 8.0%. Questions: (1) What is the mean-reverting level? (2) What is the one-quarter-ahead forecast? (3) What is the two-quarter-ahead forecast?

Show the solution
  1. Mean-reverting level: x* = 2.0 ÷ (1 − 0.60) = 2.0 ÷ 0.40 = 5.0%.
  2. One-step forecast: x̂(t+1) = 2.0 + 0.60 × 8.0 = 2.0 + 4.8 = 6.8%.
  3. Two-step forecast by the chain rule: x̂(t+2) = 2.0 + 0.60 × 6.8 = 2.0 + 4.08 = 6.08%.
  4. Check: 8.0 > 6.8 > 6.08 > 5.0, so forecasts fall toward the mean-reverting level.

Answer: (1) 5.0%; (2) 6.8%; (3) 6.08%.

Example 2

An analyst fits an AR(1) model to a commodity price index: x(t) = 1.5 + 1.00·x(t−1) + ε(t). The vignette states the error term has a constant variance. Questions: (1) Is the series covariance stationary? (2) What is the mean-reverting level? (3) If the latest value is 100, what is the one-step forecast?

Show the solution
  1. Here b1 = 1.00, so the series has a unit root and is a random walk with drift.
  2. A random walk has a variance that grows over time and no finite mean. It is not covariance stationary.
  3. Mean-reverting level: b0 ÷ (1 − b1) = 1.5 ÷ 0 is undefined, so no finite mean-reverting level exists.
  4. One-step forecast is still computable mechanically: 1.5 + 1.00 × 100 = 101.5. But the model is not reliable for inference until the series is transformed, such as by first differencing.

Answer: (1) No, not covariance stationary. (2) Undefined. (3) 101.5, though the model should be respecified before use, for example on first differences.

Exam tips

  • Write the AR equation in the margin first, then compute x* before anything else. Many questions ask for it or use it to check direction.
  • In multistep forecasts, the trap answer uses the actual last value twice. Chain the forecasts.
  • Whenever you see b1 = 1 or a unit root test result, think random walk and not covariance stationary. First differencing is the usual fix.
  • Read whether the model is on levels or on changes. The question may ask for a level.
  • There is no penalty for wrong answers, so never leave a question blank.

Autoregressive (AR) Models and Covariance Stationarity in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Autoregressive (AR) Models and Covariance Stationarity: frequently asked questions

What does covariance stationary mean in the CFA curriculum?

It means the series has a constant mean, a constant variance, and a covariance with its own lagged values that depends only on the lag distance. All three must hold. Without them, AR regression estimates are not reliable.

How do you forecast two periods ahead with an AR(1) model?

Use the chain rule. Compute the one-period forecast from the latest actual value. Then plug that forecast into the same equation to get the two-period forecast.

How do you calculate the mean-reverting level of an AR(1) model?

Divide b0 by (1 − b1). It exists only when b1 is not 1, and the series is covariance stationary only when the absolute value of b1 is below 1.

What should you do if the series has a unit root?

Transform it, most commonly by first differencing, so the new series is covariance stationary. Then estimate the AR model on the differenced data and convert forecasts back to levels if needed.