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CFA Level II · CFA Level II Exam

Time-Series Analysis for CFA Level II: Chapter Guide

Time-series analysis models a variable observed over time to forecast it. At Level II you fit trend and autoregressive models, test residuals for serial correlation, check covariance stationarity and unit roots, and judge forecasts by out-of-sample error. In the vignette, read the regression output, then apply the rule it points to.

What this chapter covers

Time-series analysis is about data indexed by time, such as sales, inflation or an index level. You learn to pick a model, check that it is valid, and use it to forecast. The chapter starts with simple trend models, moves to autoregressive (AR) models, and then asks whether the model is trustworthy.

The flow is a chain of tests. A trend model can leave serial correlation in its errors. An AR model needs covariance stationarity. Durbin-Watson applies to trend models, while t-tests on residual autocorrelations apply to AR models. You also check for a unit root, and then look at mean reversion, seasonality and ARCH effects. Each test leads to a fix: add a lag, first-difference the series, add a seasonal lag, or model the variance.

This chapter builds on regression and its violations in Quantitative Methods, so heteroskedasticity and serial correlation should already feel familiar. Questions give you output tables, so you must read them quickly and apply the right rule.

Time-series analysis sits within Quantitative Methods, which carries a modest topic weight, so you cannot afford to spend too long on it. But the questions are predictable and rule-based. Once you know which test applies to which problem, most points are mechanical: compute a forecast, read a t-statistic, spot a unit root, name the fix. Because every question comes from a vignette with exhibits, practising reading regression output is the main skill. There is no penalty for wrong answers, so always answer every question.

Time-Series Analysis: topics in the order to study them

  1. 1Trend Models: Linear and Log-Linear TrendsIt is the simplest model and introduces the forecast, the time index and the problem of serially correlated errors.
  2. 2Autoregressive (AR) Models and Covariance StationarityAR models are the core of the chapter, and stationarity is the condition that makes them valid.
  3. 3Testing AR Models: Serial Correlation and Mean ReversionOnce you can fit an AR model, you learn how to check its residuals and compute the mean-reverting level.
  4. 4Unit Roots and Random WalksThis explains why some series fail stationarity and how first differencing fixes them, building on the mean-reversion idea.
  5. 5Seasonality, ARCH and Forecast AccuracyThese are the final refinements, and they only make sense once the basic model and its tests are clear.

How to prepare Time-Series Analysis

Plan for focused sessions that mix a short rule review with output-reading practice. The volume is small, so depth in applying the rules matters more than extra reading.

  1. Read each topic once and write the model equation, the test used and the fix for each problem on a single page.
  2. Practise forecasting by hand: plug values into a linear trend, a log-linear trend and an AR(1) model, and compute a two-step-ahead forecast by feeding the first forecast back in.
  3. Learn the mean-reverting level formula for AR(1), b0 ÷ (1 − b1), and how it compares with the current value to tell you the direction of the forecast.
  4. Work through regression output tables and practise finding the relevant t-statistic, then compare it with the critical value given in the vignette.
  5. Build a decision flow: check residual autocorrelations, then stationarity, then unit root, then seasonality, then ARCH, and note the fix at each step.
  6. Do full item sets under time pressure, giving yourself about 2 to 3 minutes per question, and review each miss by naming the rule you skipped.
  7. In the last days, reread your one-page summary and redo only the questions you got wrong.

Common mistakes in Time-Series Analysis

  • Using the Durbin-Watson statistic to test an AR model.

    Fix: For AR models, test residual autocorrelations with t-statistics. Keep Durbin-Watson for trend models and ordinary regressions.

  • Forecasting two steps ahead with the actual lagged value instead of the first forecast.

    Fix: Compute the one-step forecast first, then use that number as the lagged input for the next forecast.

  • Applying the mean-reverting formula to a series with a unit root.

    Fix: Check b1 first. If b1 = 1, the denominator is zero and the series has a unit root (a random walk, with or without drift), so there is no mean-reverting level.

  • Concluding the model is fine because the coefficients are significant.

    Fix: Significance of coefficients says nothing about the residuals. Always check serial correlation, stationarity and ARCH separately.

  • Choosing a model by in-sample fit instead of out-of-sample error.

    Fix: Judge forecasting ability by out-of-sample RMSE. The model with the lower value is the more accurate forecaster.

  • Confusing a log-linear trend with a linear one.

    Fix: Use log-linear when the series grows at a roughly constant percentage rate. Remember that the forecast needs the exponential of the fitted value to return to the original units.

Last-day revision: Time-Series Analysis

  • Linear trend: yt = b0 + b1t + εt. Log-linear trend: ln(yt) = b0 + b1t + εt, used when growth is a constant rate.
  • A trend model with serially correlated errors is misspecified, so consider an AR model instead.
  • Covariance stationarity needs a constant mean, a constant variance and constant covariance with lagged values.
  • AR(1): xt = b0 + b1xt−1 + εt. Multi-step forecasts use earlier forecasts as inputs.
  • Mean-reverting level of AR(1) = b0 ÷ (1 − b1). It is defined only when b1 ≠ 1.
  • To test for serial correlation, compare each residual autocorrelation's t-statistic with the critical value. Use t = autocorrelation ÷ (1/√T), where T is the number of observations.
  • The Durbin-Watson test is not valid for AR models, so use residual autocorrelations.
  • A random walk has b0 = 0 and b1 = 1 (without drift) and has a unit root. A random walk with drift has b0 ≠ 0 and b1 = 1. Neither has a finite mean-reverting level.
  • Fix a unit root by first differencing, then fit an AR model to the differenced series.
  • Seasonality shows as a significant residual autocorrelation at the seasonal lag. Add the seasonal lag as a variable.
  • ARCH exists if the coefficient a1 in the regression of squared residuals on their lagged value is significantly different from zero. If it does, correct it with generalized least squares or by modeling the variance with ARCH-type models.
  • Compare models by out-of-sample root mean squared error, where lower is better.

Time-Series Analysis in other exams

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

Time-Series Analysis: frequently asked questions

How do I decide between a linear and a log-linear trend model?

Look at the pattern. If the series rises by a roughly constant amount each period, use a linear trend. If it grows by a roughly constant percentage, use a log-linear trend. If the residuals of either remain serially correlated, move to an AR model.

What is the quickest way to spot a unit root in an exam question?

Check whether the AR coefficient is 1 or whether the vignette reports a Dickey-Fuller test that fails to reject the null. Either result means a unit root, so the series is not covariance stationary. The fix is first differencing.

Do I need to memorise critical values for the t-tests?

No. The vignette normally gives you the critical value or the information needed to judge significance. Your job is to compute the t-statistic correctly and compare it.

How much time should I give this chapter?

Time-series analysis is a small part of Quantitative Methods, so keep it compact. A few focused sessions on the formulas and output-reading practice are usually enough, with the most time spent on item sets.