Study guide
3.3 Time-Series Analysis: AR Models, Nonstationarity & Cointegration
CFA Level 2 Chartered Financial Analyst
Track your progress as you read
Finished sections get a check mark.
1 Introduction & CFA Level II Exam Strategy
1.1 CFA Level II Exam Overview & Item-Set Vignette Strategy 1.2 2026 Curriculum Structure & 10 Topic Weight Bands 1.3 300-Hour Study Plan, Calculator Mastery & Practical Skills Modules (PSM)
2 Ethical and Professional Standards
2.1 Code of Ethics & The Seven Standards of Professional Conduct 2.2 Vignette Analysis: Conflicts of Interest & Disclosure (Standard VI) 2.3 Duties to Clients (Standard III) & Duties to Employers (Standard IV) 2.4 Standard V: Investment Analysis, Recommendations & Actions and Standard VII: Responsibilities as a CFA Member or Candidate 2.5 Application of the Code and Standards: Level II Case Method 2.6 Recommended Procedures: Compliance Programs, Supervisory Systems & Firm Policies
3 Quantitative Methods
3.1 Multiple Linear Regression, ANOVA & Hypothesis Testing 3.2 Regression Diagnostics: Multicollinearity, Heteroskedasticity & Serial Correlation 3.3 Time-Series Analysis: AR Models, Nonstationarity & Cointegration 3.4 Machine Learning, Deep Learning & Big Data Projects 3.5 Extensions of Multiple Regression: Influence Analysis, Qualitative Variables & Logistic Regression 3.6 Big Data Projects: Wrangling, Exploration, Feature Engineering & Model Training
4 Economics for Valuation
4.1 Foreign Exchange Parity Relations & Equilibrium Exchange Rates 4.2 Economic Growth Theories & Production Function Analysis 4.3 Forward Points, the Carry Trade & Policy Effects on Exchange Rates 4.4 Growth Determinants: Natural Resources, Demographics, Convergence & Trade Openness
5 Financial Statement Analysis
5.1 Intercorporate Investments: Equity Method & Business Combinations 5.2 Employee Compensation: Defined Benefit Pension Plans & Share-Based Payments 5.3 Multinational Operations: Current Rate vs Temporal Translation 5.4 Analysis of Financial Institutions & The CAMELS Framework 5.5 Evaluating the Quality of Financial Reports & Accounting Adjustments 5.6 Integration of Financial Statement Analysis Techniques
6 Corporate Issuers
6.1 Analysis of Dividends: Policy Theories, Signaling & Taxation Systems 6.2 Share Repurchases: Methods, EPS and Book Value Effects & Payout Sustainability 6.3 Cost of Capital: Advanced Topics 6.4 Corporate Governance, Board Oversight & ESG Integration 6.5 Corporate Restructuring: Types, Motivations, Evaluation & Valuation 6.6 Restructuring Financial Effects: EPS, Net Debt to EBITDA, WACC & Divestitures
7 Equity Valuation
7.1 Equity Valuation Framework & Required Rate of Return Estimation 7.2 Discounted Dividend Valuation: Gordon Growth, Multistage & The H-Model 7.3 Free Cash Flow Valuation: FCFF and FCFE Modeling 7.4 Market-Based Valuation: Price and Enterprise Value Multiples 7.5 Residual Income Valuation 7.6 Private Company Valuation: Income, Market & Asset-Based Approaches 7.7 Choosing and Combining Models: Definitions of Value, Sum-of-the-Parts & Cross-Border Comparability
8 Fixed Income Analysis
8.1 The Term Structure and Interest Rate Dynamics: Spot, Forward & Swap Curves 8.2 The Arbitrage-Free Valuation Framework & Binomial Interest Rate Trees 8.3 Valuation and Analysis of Bonds with Embedded Options: OAS & Duration 8.4 Credit Analysis Models: Structural vs Reduced-Form Approaches 8.5 Credit Default Swaps (CDS): Mechanics, Pricing & Trading Strategies 8.6 Managing Yield-Curve Risk: Key Rate Durations, One-Sided Durations & Yield Volatility Structure
9 Derivatives & Risk Management
9.1 Pricing and Valuation of Forward Commitments: Equity, Fixed Income & FX 9.2 Swap Pricing and Valuation: Interest Rate, Currency & Equity Swaps 9.3 Contingent Claims: Binomial Option Pricing Model & Arbitrage Replication 9.4 The Black-Scholes-Merton (BSM) Model, Option Greeks & Dynamic Hedging 9.5 Interest Rate Options, Swaptions & the Black Model
10 Alternative Investments
10.1 Commodities and Commodity Derivatives: Sectors, Participants, Futures Returns & Indexes 10.2 Private Real Estate Investments: Direct Capitalization, DCF & Cost Approach 10.3 Public Real Estate Investments: REIT Structures, NAV, FFO & AFFO 10.4 Hedge Fund Strategies: Classification, Implementation & Portfolio Role 10.5 Real Estate Investment Features, Due Diligence & Index Construction Biases
11 Portfolio Management
11.1 Using Multifactor Models: Macroeconomic, Fundamental & Statistical Risk 11.2 Measuring and Managing Market Risk: VaR, CVaR & Stress Testing 11.3 Economics & Investment Markets: Business Cycles, Term Structure & Asset Allocation 11.4 Analysis of Active Portfolio Management & The Fundamental Law 11.5 Exchange-Traded Funds (ETFs): Creation, Arbitrage, Tracking & Applications 11.6 Backtesting and Simulation: Procedures, Metrics, Biases & Scenario Analysis
3.3 Time-Series Analysis: AR Models, Nonstationarity & Cointegration
Key Takeaways Explain This Topic AI
A time series is covariance stationary if its mean, finite variance, and lag covariances are constant over time; an AR(1) model x_t = b_0 + b_1 x_{t-1} + e_t is stationary if |b_1| < 1, yielding mean-reverting level x_L = b_0 / (1 - b_1).
Correctly specified AR models produce serially uncorrelated white-noise residuals, evaluated via sample autocorrelations and information criteria (BIC penalizes extra parameters more heavily than AIC).
Random walk processes contain a unit root (b_1 = 1) with non-stationary, time-growing variance; they are detected using Augmented Dickey-Fuller (ADF) tests and transformed via first differencing (y_t = x_t - x_{t-1}).
Regressing two non-stationary I(1) series causes spurious regression unless they are cointegrated (tested via Engle-Granger ADF on residuals), which validates regression in levels and Error Correction Models (ECM), while ARCH models capture time-varying volatility clustering.
Last updated: August 2026
Time-Series Analysis in Investment Valuation
Financial and macroeconomic data are inherently sequential. Applying standard cross-sectional OLS regression to sequential time series without testing for stationarity leads to invalid statistical inference, parameter instability, and spurious regressions .
1. Covariance Stationarity
A time series { x t } \{x_t\} { x t } is defined as covariance stationary (or weakly stationary) if it satisfies three strict statistical conditions:
Constant Expected Value (Mean) : The mean of the series is constant and finite over all time periods: E ( x t ) = μ for all t E(x_t) = \mu \quad \text{for all } t E ( x t ) = μ for all t
Constant Finite Variance : The variance of the series is constant and finite over all time periods: V a r ( x t ) = E [ ( x t − μ ) 2 ] = σ 2 < ∞ for all t Var(x_t) = E[(x_t - \mu)^2] = \sigma^2 < \infty \quad \text{for all } t Va r ( x t ) = E [( x t − μ ) 2 ] = σ 2 < ∞ for all t
Constant Covariance across Lags : The covariance between observations separated by s s s periods depends solely on the displacement (lag s s s ) and not on the time index t t t : C o v ( x t , x t − s ) = E [ ( x t − μ ) ( x t − s − μ ) ] = γ s for all t , s Cov(x_t, x_{t-s}) = E[(x_t - \mu)(x_{t-s} - \mu)] = \gamma_s \quad \text{for all } t, s C o v ( x t , x t − s ) = E [( x t − μ ) ( x t − s − μ )] = γ s for all t , s
2. Autoregressive (AR) Models
An autoregressive model of order p p p , denoted AR( p p p ) , models the current value x t x_t x t as a function of its own p p p lagged values and a white noise disturbance term ϵ t \epsilon_t ϵ t : x t = b 0 + b 1 x t − 1 + b 2 x t − 2 + ⋯ + b p x t − p + ϵ t , ϵ t ∼ W N ( 0 , σ 2 ) x_t = b_0 + b_1 x_{t-1} + b_2 x_{t-2} + \dots + b_p x_{t-p} + \epsilon_t, \quad \epsilon_t \sim WN(0, \sigma^2) x t = b 0 + b 1 x t − 1 + b 2 x t − 2 + ⋯ + b p x t − p + ϵ t , ϵ t ∼ W N ( 0 , σ 2 )
The AR(1) Model & The Mean-Reverting Level
For a first-order autoregressive process, AR(1) : x t = b 0 + b 1 x t − 1 + ϵ t x_t = b_0 + b_1 x_{t-1} + \epsilon_t x t = b 0 + b 1 x t − 1 + ϵ t
Stationarity Condition : An AR(1) model is covariance stationary if and only if ∣ b 1 ∣ < 1 |b_1| < 1 ∣ b 1 ∣ < 1 .
Mean-Reverting Level ( x L x_L x L ) : If a series is covariance stationary, setting E ( x t ) = E ( x t − 1 ) = x L E(x_t) = E(x_{t-1}) = x_L E ( x t ) = E ( x t − 1 ) = x L yields: x L = b 0 + b 1 x L ⟹ x L ( 1 − b 1 ) = b 0 ⟹ x L = b 0 1 − b 1 x_L = b_0 + b_1 x_L \implies x_L(1 - b_1) = b_0 \implies \mathbf{x_L = \frac{b_0}{1 - b_1}} x L = b 0 + b 1 x L ⟹ x L ( 1 − b 1 ) = b 0 ⟹ x L = 1 − b 1 b 0
Economic Meaning : If x t > x L x_t > x_L x t > x L , the expected value for the next period is less than x t x_t x t (reverting downward toward x L x_L x L ). If x t < x L x_t < x_L x t < x L , the expected value for the next period is greater than x t x_t x t (reverting upward toward x L x_L x L ).
Multi-Period Forecasting in AR(1)
Forecasting proceeds recursively using the chain rule of forecasting:
One-step ahead forecast: x ^ t + 1 = b ^ 0 + b ^ 1 x t \hat{x}_{t+1} = \hat{b}_0 + \hat{b}_1 x_t x ^ t + 1 = b ^ 0 + b ^ 1 x t
Two-step ahead forecast: x ^ t + 2 = b ^ 0 + b ^ 1 x ^ t + 1 = b ^ 0 + b ^ 1 ( b ^ 0 + b ^ 1 x t ) = b ^ 0 ( 1 + b ^ 1 ) + b ^ 1 2 x t \hat{x}_{t+2} = \hat{b}_0 + \hat{b}_1 \hat{x}_{t+1} = \hat{b}_0 + \hat{b}_1 (\hat{b}_0 + \hat{b}_1 x_t) = \hat{b}_0(1 + \hat{b}_1) + \hat{b}_1^2 x_t x ^ t + 2 = b ^ 0 + b ^ 1 x ^ t + 1 = b ^ 0 + b ^ 1 ( b ^ 0 + b ^ 1 x t ) = b ^ 0 ( 1 + b ^ 1 ) + b ^ 1 2 x t
As the forecast horizon h → ∞ h \to \infty h → ∞ , the forecast converges asymptotically to the mean-reverting level x L x_L x L .
- 3. Residual Diagnostics & Model Selection
- Testing for Residual Autocorrelation
A correctly specified AR( p p p ) model must transform the data such that the estimated residuals e t e_t e t behave as white noise (serially uncorrelated). To verify this:
Compute the sample autocorrelations of the residuals at lag k k k : r k = Corr ( e t , e t − k ) r_k = \text{Corr}(e_t, e_{t-k}) r k = Corr ( e t , e t − k ) .
The standard error of the sample autocorrelation is s r k ≈ 1 T s_{r_k} \approx \frac{1}{\sqrt{T}} s r k ≈ T 1 , where T T T is sample size.
Compute the test statistic: t = r k 1 / T = r k T t = \frac{r_k}{1/\sqrt{T}} = r_k \sqrt{T} t = 1/ T r k = r k T with d f = T − p − 1 df = T - p - 1 df = T − p − 1 .
If the residual autocorrelation at any lag is statistically significant ( ∣ t ∣ > t c r i t |t| > t_{crit} ∣ t ∣ > t cr i t ), the AR model is underspecified and additional lags must be added.
Moving Average (MA) and ARMA Models
MA( q q q ) Model : x t = μ + ϵ t + θ 1 ϵ t − 1 + ⋯ + θ q ϵ t − q x_t = \mu + \epsilon_t + \theta_1 \epsilon_{t-1} + \dots + \theta_q \epsilon_{t-q} x t = μ + ϵ t + θ 1 ϵ t − 1 + ⋯ + θ q ϵ t − q . Autocorrelations drop to zero after lag q q q .
ARMA( p , q p, q p , q ) Model : Combines p p p autoregressive terms and q q q moving average terms: x t = b 0 + ∑ i = 1 p b i x t − i + ϵ t + ∑ j = 1 q θ j ϵ t − j x_t = b_0 + \sum_{i=1}^p b_i x_{t-i} + \epsilon_t + \sum_{j=1}^q \theta_j \epsilon_{t-j} x t = b 0 + ∑ i = 1 p b i x t − i + ϵ t + ∑ j = 1 q θ j ϵ t − j .
Information Criteria: AIC vs. BIC (SC)
When comparing competing ARMA models, analysts evaluate information criteria that balance goodness-of-fit against model complexity: Akaike Information Criterion (AIC) = ln ( MSE ) + 2 k T \text{Akaike Information Criterion (AIC)} = \ln(\text{MSE}) + \frac{2k}{T} Akaike Information Criterion (AIC) = ln ( MSE ) + T 2 k Schwarz Bayesian Information Criterion (BIC / SC) = ln ( MSE ) + k ln ( T ) T \text{Schwarz Bayesian Information Criterion (BIC / SC)} = \ln(\text{MSE}) + \frac{k \ln(T)}{T} Schwarz Bayesian Information Criterion (BIC / SC) = ln ( MSE ) + T k ln ( T )
Where k = p + q + 1 k = p + q + 1 k = p + q + 1 is total estimated parameters. Lower values indicate superior models. BIC imposes a larger penalty on additional parameters than AIC (since ln ( T ) > 2 \ln(T) > 2 ln ( T ) > 2 for T ≥ 8 T \ge 8 T ≥ 8 ), making BIC favor more parsimonious models.
- 4. Nonstationarity: Random Walks & Unit Roots
- Random Walk Processes
A time series is a random walk if current value equals previous value plus a random shock: x t = x t − 1 + ϵ t , ϵ t ∼ W N ( 0 , σ 2 ) x_t = x_{t-1} + \epsilon_t, \quad \epsilon_t \sim WN(0, \sigma^2) x t = x t − 1 + ϵ t , ϵ t ∼ W N ( 0 , σ 2 )
Here, b 1 = 1 b_1 = 1 b 1 = 1 . The process possesses a unit root .
Expected value: E ( x t ) = x 0 E(x_t) = x_0 E ( x t ) = x 0 (constant).
Variance: V a r ( x t ) = t σ 2 Var(x_t) = t \sigma^2 Va r ( x t ) = t σ 2 . The variance grows infinitely with time t t t , violating covariance stationarity.
Mean-reverting level: x L = 0 1 − 1 = undefined x_L = \frac{0}{1 - 1} = \text{undefined} x L = 1 − 1 0 = undefined .
Random Walk with Drift x t = b 0 + x t − 1 + ϵ t , b 0 ≠ 0 x_t = b_0 + x_{t-1} + \epsilon_t, \quad b_0 \neq 0 x t = b 0 + x t − 1 + ϵ t , b 0 = 0
Expected value: E ( x t ) = x 0 + b 0 t E(x_t) = x_0 + b_0 t E ( x t ) = x 0 + b 0 t . The series exhibits a deterministic linear time trend and infinite variance ( t σ 2 t \sigma^2 t σ 2 ).
Unit Root Testing: Dickey-Fuller & Augmented Dickey-Fuller (ADF)
We cannot test b 1 = 1 b_1 = 1 b 1 = 1 using standard t t t -tests because nonstationary series violate the assumptions underlying the Student's t t t -distribution. The Dickey-Fuller (DF) test transforms the AR(1) equation by subtracting x t − 1 x_{t-1} x t − 1 from both sides: x t − x t − 1 = b 0 + ( b 1 − 1 ) x t − 1 + ϵ t ⟹ Δ x t = b 0 + g x t − 1 + ϵ t x_t - x_{t-1} = b_0 + (b_1 - 1) x_{t-1} + \epsilon_t \implies \Delta x_t = b_0 + g x_{t-1} + \epsilon_t x t − x t − 1 = b 0 + ( b 1 − 1 ) x t − 1 + ϵ t ⟹ Δ x t = b 0 + g x t − 1 + ϵ t
Where g = b 1 − 1 g = b_1 - 1 g = b 1 − 1 . The hypotheses are:
H 0 : g = 0 H_0: g = 0 H 0 : g = 0 ( b 1 = 1 ⟹ b_1 = 1 \implies b 1 = 1 ⟹ Unit root present / Nonstationary)
H a : g < 0 H_a: g < 0 H a : g < 0 ( b 1 < 1 ⟹ b_1 < 1 \implies b 1 < 1 ⟹ Covariance stationary)
The test uses specialized Dickey-Fuller critical τ \tau τ -values (which are substantially more negative than standard normal critical values, e.g., approx − 2.89 -2.89 − 2.89 at α = 0.05 \alpha = 0.05 α = 0.05 with an intercept). If t s t a t < τ c r i t t_{stat} < \tau_{crit} t s t a t < τ cr i t , reject H 0 H_0 H 0 and conclude the series is stationary.
Augmented Dickey-Fuller (ADF) : Adds lagged changes ∑ i = 1 k c i Δ x t − i \sum_{i=1}^k c_i \Delta x_{t-i} ∑ i = 1 k c i Δ x t − i to eliminate higher-order residual autocorrelation: Δ x t = b 0 + g x t − 1 + ∑ i = 1 k c i Δ x t − i + ϵ t \Delta x_t = b_0 + g x_{t-1} + \sum_{i=1}^k c_i \Delta x_{t-i} + \epsilon_t Δ x t = b 0 + g x t − 1 + i = 1 ∑ k c i Δ x t − i + ϵ t
Remedy for Unit Roots: First Differencing
If a time series contains a unit root (integrated of order 1, denoted x t ∼ I ( 1 ) x_t \sim I(1) x t ∼ I ( 1 ) ), taking the first difference produces a stationary series ( y t ∼ I ( 0 ) y_t \sim I(0) y t ∼ I ( 0 ) ): y t = Δ x t = x t − x t − 1 y_t = \Delta x_t = x_t - x_{t-1} y t = Δ x t = x t − x t − 1
5. Seasonality in Time Series
Quarterly or monthly financial data often exhibit regular seasonal swings (e.g., retail holiday revenue).
Detection : Seasonality is detected when the sample autocorrelations of residuals spike significantly at the seasonal lag (lag 4 for quarterly data, lag 12 for monthly data).
Remedy : Add a seasonal lag term directly into the autoregressive equation: x t = b 0 + b 1 x t − 1 + b 4 x t − 4 + ϵ t ( for quarterly seasonality ) x_t = b_0 + b_1 x_{t-1} + b_4 x_{t-4} + \epsilon_t \quad (\text{for quarterly seasonality}) x t = b 0 + b 1 x t − 1 + b 4 x t − 4 + ϵ t ( for quarterly seasonality )
6. Autoregressive Conditional Heteroskedasticity (ARCH)
In asset return series, periods of high volatility cluster together, as do periods of low volatility ( volatility clustering ). While return errors ϵ t \epsilon_t ϵ t may be serially uncorrelated ( C o v ( ϵ t , ϵ t − s ) = 0 Cov(\epsilon_t, \epsilon_{t-s}) = 0 C o v ( ϵ t , ϵ t − s ) = 0 ), their variance is conditional on past squared shocks .
An ARCH(1) model specifies the conditional variance σ t 2 \sigma_t^2 σ t 2 as: σ t 2 = a 0 + a 1 ϵ t − 1 2 , a 0 > 0 , a 1 ≥ 0 \sigma_t^2 = a_0 + a_1 \epsilon_{t-1}^2, \quad a_0 > 0, a_1 \ge 0 σ t 2 = a 0 + a 1 ϵ t − 1 2 , a 0 > 0 , a 1 ≥ 0
Testing for ARCH : Estimate the primary AR model, obtain residuals e t e_t e t , and regress squared residuals on lagged squared residuals: ≠ t 2 = a 0 + a 1 e t − 1 2 + u t \ne_t^2 = a_0 + a_1 e_{t-1}^2 + u_t = t 2 = a 0 + a 1 e t − 1 2 + u t
Test H 0 : a 1 = 0 H_0: a_1 = 0 H 0 : a 1 = 0 (no ARCH) vs H a : a 1 > 0 H_a: a_1 > 0 H a : a 1 > 0 using an individual t t t -test or the Engle LM test ( L M = T × R 2 ∼ χ 1 2 LM = T \times R^2 \sim \chi^2_1 L M = T × R 2 ∼ χ 1 2 ). If a 1 a_1 a 1 is statistically significant, ARCH effects are present.
Consequences & Forecasting : If ARCH exists, OLS standard errors are invalid. Generalized Autoregressive Conditional Heteroskedasticity ( GARCH ) models must be estimated, and one-step-ahead volatility is forecasted as σ ^ t + 1 2 = a ^ 0 + a ^ 1 e t 2 \hat{\sigma}_{t+1}^2 = \hat{a}_0 + \hat{a}_1 e_t^2 σ ^ t + 1 2 = a ^ 0 + a ^ 1 e t 2 .
7. Multi-Time-Series Regressions & Cointegration
When regressing one time series Y t Y_t Y t on another X t X_t X t , the stationarity of both series governs the validity of the regression:
Case Nature of Y t Y_t Y t and X t X_t X t Validity of Direct OLS Regression Correct Econometric Procedure
Case 1 Both Y t Y_t Y t and X t X_t X t are I ( 0 ) I(0) I ( 0 ) (Stationary) Valid Estimate standard OLS regression in levels.
Case 2 One series is I ( 1 ) I(1) I ( 1 ) and the other is I ( 0 ) I(0) I ( 0 ) Invalid Transform I ( 1 ) I(1) I ( 1 ) series via first differencing before modeling.
Case 3 Both Y t Y_t Y t and X t X_t X t are I ( 1 ) I(1) I ( 1 ) (Nonstationary) & NOT Cointegrated Invalid (Spurious Regression) First difference both series ( Δ Y t \Delta Y_t Δ Y t on Δ X t \Delta X_t Δ X t ). Direct OLS yields false high R 2 R^2 R 2 , inflated t t t -stats, and D W → 0 DW \to 0 D W → 0 .
Case 4 Both Y t Y_t Y t and X t X_t X t are I ( 1 ) I(1) I ( 1 ) & Cointegrated Valid (True Long-Run Equilibrium) Estimate OLS in levels; estimates are super-consistent . Model dynamic adjustments via Error Correction Model (ECM).
Cointegration & The Engle-Granger Test
Two nonstationary I ( 1 ) I(1) I ( 1 ) series are cointegrated if there exists a cointegrating parameter β \beta β such that their linear combination is stationary ( I ( 0 ) I(0) I ( 0 ) ): ≠ t = Y t − β 0 − β 1 X t ∼ I ( 0 ) \ne_t = Y_t - \beta_0 - \beta_1 X_t \sim I(0) = t = Y t − β 0 − β 1 X t ∼ I ( 0 )
Engle-Granger Two-Step Test :
Run OLS regression in levels: Y t = b 0 + b 1 X t + e t Y_t = b_0 + b_1 X_t + e_t Y t = b 0 + b 1 X t + e t .
Test the residual series e t e_t e t for a unit root using the Augmented Dickey-Fuller test with specialized Engle-Granger critical values .
If H 0 H_0 H 0 of a unit root in residuals is rejected , the series are cointegrated.
Error Correction Model (ECM) : If cointegrated, the short-run dynamics are modeled with an error-correction mechanism: Δ Y t = α 0 + α 1 Δ X t + λ ( Y t − 1 − b ^ 1 X t − 1 ) + u t \Delta Y_t = \alpha_0 + \alpha_1 \Delta X_t + \lambda (Y_{t-1} - \hat{b}_1 X_{t-1}) + u_t Δ Y t = α 0 + α 1 Δ X t + λ ( Y t − 1 − b ^ 1 X t − 1 ) + u t
Where λ < 0 \lambda < 0 λ < 0 is the error correction coefficient measuring the speed of adjustment back to long-run equilibrium.