Estimated study time: 60 minutes
Content:
Time series analysis at CFA Level 2 focuses on modeling and forecasting financial variables that evolve over time, such as GDP growth, earnings per share, interest rates, and asset prices. The most fundamental models are autoregressive (AR) models, where the current value of a variable depends on its own past values. An AR(1) model takes the form: X_t = b0 + b1*X_{t-1} + epsilon_t, where b0 is the intercept, b1 is the autoregressive coefficient, and epsilon_t is a white noise error term. An AR(p) model includes p lagged values. Candidates must determine the appropriate lag length (typically using the Akaike Information Criterion, AIC, or the Bayesian Information Criterion, BIC), test whether the model is well-specified (using autocorrelation of residuals), and interpret the mean-reverting level of the series.
The mean-reverting level of an AR(1) model is the long-run equilibrium value toward which the series gravitates: mean-reversion level = b0 / (1 - b1). This formula assumes |b1| < 1, which is the stationarity condition for an AR(1) model. If b1 = 1 exactly, the series is a random walk (a unit root process): X_t = X_{t-1} + epsilon_t. Random walk processes are non-stationary — their variance grows without bound over time and standard regression inference is invalid. The Dickey-Fuller test is used to test for the presence of a unit root. In the Dickey-Fuller regression, X_t - X_{t-1} = b0 + (b1-1)*X_{t-1} + epsilon_t, the null hypothesis is that a unit root exists (b1 = 1, coefficient on X_{t-1} equals zero). Rejection of the null means the series is stationary. Failure to reject means it is a unit root process.
If two non-stationary time series are individually unit root processes but a linear combination of them is stationary, they are said to be cointegrated. Cointegration implies a long-run equilibrium relationship between the series, and the OLS regression of one on the other produces valid (superconsistent) estimates of the long-run relationship, despite the non-stationarity of the individual series. This is the economic basis for pairs trading strategies: two cointegrated asset prices move together in the long run, and deviations from the cointegrating relationship are temporary mean-reverting opportunities. The Engle-Granger test and Johansen test are used to test for cointegration.
Seasonality is a systematic, calendar-based pattern in time series data (e.g., retail sales peaking in December, earnings announcements following quarterly cycles). In regression models, seasonality can be addressed by including seasonal dummy variables or by transforming the data to remove seasonal effects (seasonal differencing). Detecting seasonality involves examining the autocorrelation function (ACF) of the series — significant autocorrelation at regular seasonal lags (e.g., lag 4 for quarterly data, lag 12 for monthly data) indicates seasonality. The autocorrelation at lag k is: r_k = correlation(X_t, X_{t-k}). Residual autocorrelation in a fitted model indicates model misspecification — the model has not captured all systematic variation.
ARCH (Autoregressive Conditional Heteroskedasticity) models address the stylized fact that financial return volatility clusters — periods of high volatility tend to follow periods of high volatility. The ARCH(1) model specifies that the conditional variance at time t depends on the squared error from the previous period: variance_t = a0 + a1*(epsilon_{t-1})^2. GARCH(1,1) extends this by including the lagged conditional variance as well: variance_t = a0 + a1*(epsilon_{t-1})^2 + b1*variance_{t-1}. GARCH models are widely used for options pricing, Value-at-Risk estimation, and dynamic hedge ratio construction because they capture the time-varying nature of financial volatility. Testing for ARCH effects involves regressing squared residuals on lagged squared residuals and testing the joint significance of lagged terms (the Engle LM test).
Key Terms:
Quiz Questions:
Q1. An analyst estimates the following AR(1) model for quarterly GDP growth: GDP_t = 0.40 + 0.65*GDP_{t-1} + epsilon_t. The current quarter's GDP growth is 2.8%. What is the forecast for next quarter, and what is the long-run mean-reverting level of GDP growth implied by the model?
A) Forecast = 2.22%; Mean-reversion level = 1.14%. B) Forecast = 2.22%; Mean-reversion level = 1.14% is wrong — the series has a unit root. C) Forecast = 2.22%; Mean-reversion level = 0.40/(1-0.65) = 1.14%. D) Forecast = 0.40 + 0.65*(0.40) = 0.66%; Mean-reversion level = 2.8%.
Answer: C — Next quarter forecast = 0.40 + 0.65*(2.8%) = 0.40 + 1.82 = 2.22%. The mean-reversion level = b0/(1 - b1) = 0.40/(1 - 0.65) = 0.40/0.35 = 1.14%. Since |b1| = 0.65 < 1, the series is stationary and mean-reverting. Shocks will decay and the series will gravitate toward 1.14% over time.
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Q2. A researcher is modeling the monthly price level of a commodity. The Dickey-Fuller test on the price level yields a test statistic of -1.2 (critical value at 5% significance is -2.89). However, when the test is applied to first differences of the price level (i.e., monthly price changes), the test statistic is -8.4. What do these results indicate?
A) The price level is stationary; monthly changes are non-stationary. B) Both the price level and monthly changes are non-stationary. C) The price level is non-stationary (contains a unit root); monthly price changes are stationary. The price series is integrated of order 1, written I(1). D) The Dickey-Fuller test is inconclusive because the statistics differ.
Answer: C — In the Dickey-Fuller test, the null hypothesis is that a unit root exists. Failing to reject (test statistic > critical value in absolute terms) means the series has a unit root. The price level has DF statistic -1.2, which is less negative than -2.89, so we fail to reject — the price level has a unit root. The first differences have DF statistic -8.4, far more negative than -2.89, so we reject the null — price changes are stationary. A series that is non-stationary in levels but stationary in first differences is integrated of order 1, or I(1).
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Q3. Two commodity price series — crude oil and natural gas — are individually found to be I(1) (unit root processes). A researcher regresses crude oil prices on natural gas prices and obtains R-squared = 0.78, with residuals that pass the Dickey-Fuller test (stationary residuals). What can the researcher conclude?
A) The regression is spurious because both series have unit roots; the R-squared is meaningless. B) Crude oil and natural gas prices are cointegrated, suggesting a long-run equilibrium relationship; the regression coefficients are valid and superconsistent. C) Stationarity of residuals indicates the OLS regression is biased. D) The result implies the two series are causally linked, not merely correlated.
Answer: B — Cointegration between two I(1) series is defined precisely as the existence of a linear combination of them that is I(0) (stationary). The stationarity of the regression residuals is that test — if the residuals from regressing oil on gas are stationary, the series are cointegrated. In this case, OLS provides valid (and superconsistent) estimates of the long-run relationship, unlike the spurious regression scenario where both series are I(1) but not cointegrated. Cointegration implies correlation but not causation (Option D is a misinterpretation).
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Q4. An analyst fits a GARCH(1,1) model to daily equity returns and obtains: variance_t = 0.00001 + 0.08*(epsilon_{t-1})^2 + 0.90*variance_{t-1}. Yesterday's return shock was -2.0% and yesterday's variance estimate was 0.0004. What is today's conditional variance estimate, and what does the sum of ARCH and GARCH coefficients (0.08 + 0.90 = 0.98) imply?
A) Variance_t = 0.00001 + 0.08*(0.02)^2 + 0.90*(0.0004) = 0.000042 + 0.000032 = 0.000074; a coefficient sum of 0.98 implies high persistence of volatility shocks. B) Variance_t = 0.00001 + 0.08*(-0.02) + 0.90*(0.0004) = 0.000349; coefficient sum > 0.5 means the series is non-stationary. C) Variance_t = 0.00001 + 0.08*(0.0004) + 0.90*(0.02)^2 = 0.000393; coefficient sum of 0.98 implies rapid mean reversion. D) Variance_t = 0.00001 + 0.08*(0.02)^2 + 0.90*(0.0004) = 0.000394; coefficient sum of 0.98 implies the series is an integrated GARCH.
Answer: D — Variance_t = 0.00001 + 0.08*(0.02)^2 + 0.90*(0.0004) = 0.00001 + 0.08*0.0004 + 0.90*0.0004 = 0.00001 + 0.000032 + 0.000360 = 0.000402. Note: (0.02)^2 = 0.0004. When the sum of ARCH (alpha) and GARCH (beta) coefficients approaches 1, volatility shocks are highly persistent and the variance process approaches non-stationarity (IGARCH). A sum of exactly 1 would mean past variance shocks never fully decay, which is the IGARCH model. A sum of 0.98 indicates very high persistence — consistent with the empirically observed slow decay of volatility in financial markets.
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Q5. Monthly retail sales data shows significant autocorrelation at lags 1, 12, and 24 in the ACF. An analyst fits an AR(1) model and finds the residuals still have significant autocorrelation at lag 12. What is the most likely explanation and the appropriate next step?
A) The AR(1) residuals are always autocorrelated; this is expected and no action is needed. B) The data likely has a seasonal pattern with a 12-month cycle; the analyst should add seasonal dummy variables (or apply seasonal differencing) to capture the annual seasonal effect. C) The residual autocorrelation at lag 12 indicates a unit root; the analyst should first-difference the series. D) The AR(1) model should be replaced with an AR(24) model to capture all lags showing autocorrelation.
Answer: B — Significant autocorrelation in residuals at lag 12 for monthly data is a classic sign of annual seasonality that the AR(1) model has not captured. Adding 11 seasonal dummy variables (one for each month, with one month as the base) or taking a seasonal difference (subtracting the same month's value from the prior year, i.e., X_t - X_{t-12}) addresses this. An AR(24) model (Option D) would be overparameterized; the structured seasonal approach is more parsimonious and interpretable.
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