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CFA Level II Exam · Time-Series Analysis

Testing AR Models: Residual Autocorrelation and Mean Reversion

Updated 7 October 2026 · Fact-checked

To test an AR model, compute the autocorrelations of its residuals and test each with t = autocorrelation ÷ (1 ÷ √T). If none is significant, the model is correctly specified. The mean-reverting level of an AR(1) is b0 ÷ (1 − b1), valid when b1 is not 1.

Understand Testing AR Models: Serial Correlation and Mean Reversion

An autoregressive (AR) model forecasts a variable from its own past values. An AR(1) model is x(t) = b0 + b1·x(t−1) + ε(t). Before you use it to forecast, you must check that it is correctly specified.

The check is about the residuals. If the model captures all the dependence in the series, the residuals should look like random noise. That means the autocorrelations of the residuals, at lags 1, 2, 3 and so on, should all be zero. If a residual autocorrelation is significantly different from zero, the model has missed some pattern. The fix is usually to add more lags or consider seasonality.

The test is a t-test. The standard error of each residual autocorrelation is 1 ÷ √T, where T is the number of observations. So t = residual autocorrelation at lag k ÷ (1 ÷ √T). You compare it with the critical t-value with T − 2 degrees of freedom. For large samples at 5% significance, the critical value is about 2. Do not use the Durbin-Watson test here. It is not valid for AR models, because the lagged dependent variable is a regressor.

Mean reversion is a separate idea. A series is mean reverting if it tends to move toward a long-run level. For an AR(1), set x(t) = x(t−1) = x* and solve: x* = b0 ÷ (1 − b1). If the current value is above x*, the model forecasts a fall. If it is below, it forecasts a rise. A covariance stationary AR(1) has a finite mean-reverting level, which requires |b1| < 1. If b1 = 1 (a unit root), the level is undefined.

Key formulas to remember

AR(1) model
x(t) = b0 + b1·x(t−1) + ε(t)
Forecast the next value from the current one.
t-statistic for residual autocorrelation
t = ρ(k) ÷ (1 ÷ √T) = ρ(k) × √T
ρ(k) is the residual autocorrelation at lag k. T is the number of observations.
Degrees of freedom for the test
df = T − 2
Compare |t| with the critical value. About 2 for large samples at 5%.
Mean-reverting level (AR(1))
x* = b0 ÷ (1 − b1)
Valid when b1 ≠ 1. Needs |b1| < 1 for covariance stationarity.
Mean-reverting level (AR(2))
x* = b0 ÷ (1 − b1 − b2)
Valid when b1 + b2 ≠ 1.
Correct specification rule
No significant residual autocorrelation at any lag
If any lag is significant, the model is misspecified.

How to solve Testing AR Models: Serial Correlation and Mean Reversion questions

Use this method for any item-set question on testing an AR model or finding its mean-reverting level.

  1. 1Find the estimated AR equation and the sample size T in the vignette or exhibit.
  2. 2If asked about specification, locate the table of residual autocorrelations by lag and the t-statistics, if given.
  3. 3If t-statistics are not given, compute t = autocorrelation × √T for each lag.
  4. 4Compare each |t| with the critical value (about 2 for large T at 5%, or the value given). Any lag beyond it means significant autocorrelation.
  5. 5Conclude: no significant lags means the model is correctly specified. Any significant lag means misspecified, so add lags or revisit the model.
  6. 6For mean reversion, compute b0 ÷ (1 − b1) using the coefficients. Check that b1 is not 1 and |b1| < 1.
  7. 7Compare the current value with the mean-reverting level to state whether the forecast moves up or down.

Quickest way: Rule-of-thumb screening of residual autocorrelations

When to use it: When an exhibit lists residual autocorrelations and you are short on time.

  1. Compute the cutoff for the autocorrelation itself: critical value ÷ √T. For T = 100 and critical value 2, the cutoff is 0.20.
  2. Scan the autocorrelation column for any value larger than the cutoff in absolute terms.
  3. If none exceeds it, answer 'correctly specified'. If one does, answer 'not correctly specified'.
  4. For mean reversion, compute b0 ÷ (1 − b1) directly and compare with the current value.

Common mistakes in Testing AR Models: Serial Correlation and Mean Reversion

  • Using the Durbin-Watson test on an AR model.

    Durbin-Watson is the familiar serial-correlation test from regression.

    Fix: For AR models, test residual autocorrelations at each lag with the t-test above.

  • Dividing by √T instead of multiplying, or forgetting the standard error is 1 ÷ √T.

    The formula is written as a ratio and gets misread.

    Fix: Remember t = autocorrelation × √T. The standard error shrinks as T grows.

  • Testing only lag 1.

    Lag 1 is the most common problem.

    Fix: Check every lag shown. One significant lag is enough to reject correct specification.

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

    The algebra of setting x(t) = x(t−1) is skipped.

    Fix: Solve x* = b0 + b1·x*, giving x* = b0 ÷ (1 − b1).

  • Concluding that a model with significant residual autocorrelation is fine because it has a high R-squared.

    Goodness of fit is confused with specification.

    Fix: Specification is judged by residual autocorrelation, not R-squared.

Worked examples

Example 1

An analyst estimates an AR(1) model for a monthly series using 144 observations. The residual autocorrelations are: lag 1: 0.09, lag 2: 0.21, lag 3: −0.05, lag 4: 0.04. The critical t-value at 5% is 1.98. (1) Compute the t-statistic for lag 2. (2) Is the model correctly specified?

Show the solution
  1. Standard error = 1 ÷ √144 = 1 ÷ 12 = 0.0833.
  2. Lag 2 t-statistic = 0.21 ÷ 0.0833 = 2.52.
  3. Other lags: lag 1 = 0.09 × 12 = 1.08; lag 3 = −0.05 × 12 = −0.60; lag 4 = 0.04 × 12 = 0.48.
  4. Lag 2 has |t| = 2.52, which exceeds 1.98. The others do not.

Answer: (1) t = 2.52. (2) The model is not correctly specified, because the lag 2 residual autocorrelation is significant. Consider adding a second lag.

Example 2

An AR(1) model for a company's quarterly margin (in %) is x(t) = 2.4 + 0.8·x(t−1). The current margin is 14%. (1) What is the mean-reverting level? (2) What does the model forecast for next quarter? (3) Will margins rise or fall toward the long-run level?

Show the solution
  1. Mean-reverting level = 2.4 ÷ (1 − 0.8) = 2.4 ÷ 0.2 = 12.
  2. Next quarter forecast = 2.4 + 0.8 × 14 = 2.4 + 11.2 = 13.6.
  3. Current value 14 is above 12, so the forecast of 13.6 is a move down toward 12.

Answer: (1) 12%. (2) 13.6%. (3) Margins fall toward the 12% mean-reverting level.

Exam tips

  • Always compute the cutoff or t-statistic yourself if the exhibit only gives raw autocorrelations and T.
  • Read the sample size T carefully. It may differ from the number of lags shown.
  • If a question asks which test applies to an AR model's residuals, the answer is the t-test on residual autocorrelations, not Durbin-Watson.
  • For mean reversion, check the forecast direction by comparing the current value with b0 ÷ (1 − b1).
  • A significant residual autocorrelation points to misspecification, so the usual remedy is adding lags.

Testing AR Models: Serial Correlation and Mean Reversion in other exams

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

Testing AR Models: Serial Correlation and Mean Reversion: frequently asked questions

How do you test autocorrelation of residuals in an AR model?

Compute the residual autocorrelation at each lag. Divide each by its standard error, 1 ÷ √T, to get a t-statistic. Compare it with the critical value at T − 2 degrees of freedom.

What is the mean-reverting level formula for an AR(1) model?

It is b0 ÷ (1 − b1). It is the value at which the model forecasts no change. It exists only when b1 is not 1, and a stable model needs |b1| < 1.

When is an AR model correctly specified?

When none of the residual autocorrelations is significantly different from zero. If one is significant, the model has missed a pattern and should be revised.

Why not use Durbin-Watson for AR models?

An AR model has a lagged dependent variable as a regressor, which makes the Durbin-Watson test unreliable. The t-test on residual autocorrelations is the appropriate check.