Course Overview
Course
Time Series Analysis and Forecasting
Bucharest University of Economic Studies
Duration
Lectures: 2 hours/week
Seminars: 2 hours/week
Prerequisites
Statistics, Probability Theory
Linear Algebra, Python Programming
Assessment
Final Exam: 70%
Projects & Assignments: 30%
Learning Objectives
By the end of this course, you will be able to:
- Understand stationarity, autocorrelation (ACF), and partial autocorrelation (PACF)
- Apply decomposition methods and exponential smoothing (SES, Holt, Holt-Winters)
- Build and diagnose ARMA and ARIMA models using Box-Jenkins methodology
- Model seasonal time series with SARIMA
- Model volatility clustering with GARCH, EGARCH, and GJR-GARCH
- Calculate Value at Risk (VaR) using volatility forecasts
- Analyze multivariate systems with VAR models and Granger causality tests
- Interpret Impulse Response Functions and Forecast Error Variance Decomposition
- Test for cointegration and estimate VECM models
- Implement time series analysis and forecasting in Python
Key Formulas
Decomposition
Additive: $X_t = T_t + S_t + \varepsilon_t$
Multiplicative: $X_t = T_t \times S_t \times \varepsilon_t$
Stationarity
$\mathbb{E}[X_t] = \mu$ (constant)
$\text{Var}(X_t) = \sigma^2$ (constant)
$\text{Cov}(X_t, X_{t+h}) = \gamma(h)$
AR(p) Model
$X_t = c + \sum_{i=1}^{p}\phi_i X_{t-i} + \varepsilon_t$
Stationary if roots outside unit circle
MA(q) Model
$X_t = \mu + \sum_{j=1}^{q}\theta_j\varepsilon_{t-j} + \varepsilon_t$
Always stationary
ARIMA(p,d,q)
$\phi(L)(1-L)^d X_t = \theta(L)\varepsilon_t$
$d$ = differencing order
SARIMA
$(p,d,q) \times (P,D,Q)_s$
Seasonal + non-seasonal ARIMA
VAR(p) Model
$\mathbf{Y}_t = \mathbf{c} + \sum_{i=1}^{p}\mathbf{A}_i\mathbf{Y}_{t-i} + \boldsymbol{\varepsilon}_t$
Multivariate AR extension
Granger Causality
$X \rightarrow Y$: past $X$ predicts $Y$
$H_0$: lagged $X$ coefficients = 0
GARCH(1,1)
$\sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2$
Volatility clustering model
Value at Risk
$\text{VaR}_\alpha = \mu + z_\alpha \cdot \sigma_t$
Risk measure at confidence $\alpha$
Cointegration
$Y_t = \beta X_t + u_t$, $u_t \sim I(0)$
Long-run equilibrium relationship
Model Selection
AIC $= -2\log L + 2k$
BIC $= -2\log L + k\log n$
Forecast Metrics
RMSE $= \sqrt{\frac{1}{n}\sum(y_t - \hat{y}_t)^2}$
MAPE $= \frac{100}{n}\sum|\frac{y_t - \hat{y}_t}{y_t}|$
Prophet Model
$y(t) = g(t) + s(t) + h(t) + \varepsilon_t$
Trend + Seasonality + Holidays
TBATS
Trigonometric + Box-Cox + ARMA
Multiple seasonality via Fourier
Course Chapters
Chapter 0: Fundamentals
- What is a Time Series?
- Time Series Decomposition
- Exponential Smoothing Methods
- Forecast Evaluation
- Modeling Seasonality
- Handling Trend and Seasonality
Chapter 1: Stochastic Processes & Stationarity
- Stochastic Processes
- Stationarity (Strict & Weak)
- White Noise and Random Walk
- Autocorrelation Functions (ACF/PACF)
- Lag Operator and Differencing
- Testing for Stationarity (ADF, KPSS)
Chapter 2: ARMA Models
- Autoregressive (AR) Models
- Moving Average (MA) Models
- ARMA Model Identification
- Parameter Estimation
- Model Diagnostics
- Forecasting with ARMA
Chapter 3: ARIMA Models
- Non-Stationarity & Unit Roots
- Differencing & Integration
- ARIMA(p,d,q) Models
- Unit Root Tests (ADF, KPSS)
- Box-Jenkins Methodology
- Forecasting with ARIMA
Chapter 4: SARIMA Models
- Seasonality in Time Series
- Seasonal Decomposition
- Seasonal Differencing
- SARIMA$(p,d,q) \times (P,D,Q)_s$
- Airline Model
- Seasonal Forecasting
Chapter 5: Volatility Models ARCH/GARCH
- Stylized Facts of Financial Returns
- Volatility Clustering & Heteroskedasticity
- ARCH Model (Engle, 1982)
- GARCH, IGARCH, EWMA, CGARCH, FIGARCH
- Asymmetric Models: EGARCH, GJR-GARCH, TGARCH, APARCH
- Markov-Switching GARCH
- Volatility Forecasting, Value at Risk & Expected Shortfall
Chapter 5b: Multivariate GARCH Models
- Multivariate Volatility Modeling
- VECH and BEKK Models
- DCC-GARCH (Dynamic Conditional Correlation)
- CCC-GARCH (Constant Conditional Correlation)
- Portfolio Risk & Hedging Applications
Chapter 6: VAR Models & Granger Causality
- Vector Autoregression (VAR)
- Granger Causality Testing
- Impulse Response Functions
- Forecast Error Variance Decomposition
- VAR Diagnostics & Forecasting
- Structural VAR (SVAR) Introduction
Chapter 7: Cointegration & VECM
- Spurious Regression Problem
- Cointegration Concept
- Engle-Granger Two-Step Method
- Johansen Cointegration Test
- Vector Error Correction Model (VECM)
- VECM Estimation & Interpretation
Chapter 8: Modern Extensions
- ARFIMA Models & Long Memory
- Hurst Exponent & Fractional Differencing
- Machine Learning for Time Series
- Random Forest with Lag Features
- LSTM Networks for Sequential Data
- Time Series Cross-Validation
Chapter 9: Prophet & TBATS
- Multiple Seasonality Challenge
- TBATS: Trigonometric Seasonality
- Fourier Terms & Box-Cox Transform
- Prophet: Decomposable Models
- Trend Changepoints & Holidays
- Forecasting with Uncertainty
Chapter 10: Review
- Complete Analysis Workflow
- Case Study: Bitcoin (ARIMA-GARCH Volatility)
- Case Study: Sunspots (11-Year Cycle, SARIMA)
- Case Study: US Unemployment (Structural Breaks)
- Model Selection Decision Guide
- Forecast Evaluation Metrics
Chapter 11: LLMs and Foundation Models
- Transformer Architecture and Self-Attention
- Foundation Models: Chronos, TimesFM, TimeGPT, Lag-Llama, Moirai
- Tokenization Strategies: Patching, Quantization
- Zero-Shot vs Fine-Tuning
- Use Cases: EUR/RON, Energy, Volatility
- Benchmarks, Limitations, and Future Directions
Chapter 12: Spectral Analysis
- Fourier Analysis and the Discrete Fourier Transform
- Periodogram and Spectral Density Function
- Spectral Estimation: Multitaper, Welch, Blackman-Tukey
- Cross-Spectral Analysis: Coherence, Phase, Gain
- Band-Pass Filters: Baxter-King, Christiano-Fitzgerald, HP
- Wavelet Analysis and Scalograms
Chapter 13: LPPL Models for Bubble Detection
- Financial Bubbles and Critical Phenomena
- Log-Periodic Power Law (LPPL) Model Derivation
- Crash Hazard Rate and Phase Transitions
- LPPL Estimation: Slaving and Differential Evolution
- Lomb–Scargle Confidence Indicator
- Case Studies: Dot-com, Bitcoin, Shanghai, Oil, COVID
Interactive Quizzes
Please log in with your GitHub account to access the quizzes.
Click on a chapter tab above to switch between quizzes
Chapter 0: Fundamentals
Chapter 1: Stochastic Processes & Stationarity
Question 1: Stochastic Process
What does a stochastic process $\{X_t\}$ describe?
- A) A deterministic function of time
- B) A fixed numerical value
- C) A family of random variables indexed by time
- D) A sequence of numbers in ascending order
Correct! A stochastic process is a collection $\{X_t : t \in T\}$ of random variables, one for each point in time.
Incorrect. The correct answer is C. A stochastic process is a family of random variables indexed by time.
Question 2: Autocovariance Function
What defines the autocovariance function $\gamma(t,s)$?
- A) $\mathbb{E}[X_t] \cdot \mathbb{E}[X_s]$
- B) $\text{Var}(X_t + X_s)$
- C) $\mathbb{E}[(X_t - \mu_t)(X_s - \mu_s)]$
- D) $\mathbb{E}[X_t^2]$
Correct! The autocovariance measures linear dependence between $X_t$ and $X_s$: $\gamma(t,s) = \text{Cov}(X_t, X_s)$.
Incorrect. The correct answer is C. $\gamma(t,s) = \mathbb{E}[(X_t - \mu_t)(X_s - \mu_s)]$ is the covariance between $X_t$ and $X_s$.
Question 3: Strict Stationarity
What does strict stationarity imply?
- A) Only the mean is constant
- B) Only the variance is constant
- C) All finite-dimensional distributions are invariant to time shifts
- D) The series has no visible trend
Correct! Strict stationarity requires $(X_{t_1}, \ldots, X_{t_k}) \overset{d}{=} (X_{t_1+h}, \ldots, X_{t_k+h})$ for all $h$.
Incorrect. The correct answer is C. Strict stationarity requires invariance of all distributions under time shifts.
Question 4: Strict vs. Weak Stationarity
Which statement about the relationship between strict and weak stationarity is correct?
- A) Weak stationarity implies strict stationarity
- B) The two concepts are identical
- C) Strict stationarity is not relevant in practice
- D) Strict stationarity (with finite moments) implies weak stationarity
Correct! Strict + finite moments $\Rightarrow$ weak. But NOT the reverse (exception: Gaussian processes).
Incorrect. The correct answer is D. Strict stationarity (with finite moments) implies weak, but not vice versa.
Question 5: Ergodicity
What does ergodicity allow in practice?
- A) Estimating parameters from multiple realizations of the process
- B) Estimating population parameters from a single realization
- C) Exact prediction of future values
- D) Complete elimination of uncertainty
Correct! Ergodicity guarantees the time average converges to the ensemble average: $\frac{1}{T}\sum X_t \to \mathbb{E}[X_t]$.
Incorrect. The correct answer is B. Ergodicity allows statistical inference from a single observed trajectory.
Question 6: Wold's Theorem
What does the Wold decomposition theorem state?
- A) Any time series is stationary
- B) Any weakly stationary process decomposes into MA($\infty$) + deterministic component
- C) Every time series has a trend
- D) Only AR processes are stationary
Correct! Wold: $X_t = \sum_{j=0}^{\infty}\psi_j\varepsilon_{t-j} + D_t$, where $\psi_0 = 1$ and $\sum\psi_j^2 < \infty$.
Incorrect. The correct answer is B. Wold's theorem decomposes any weakly stationary process into MA($\infty$) + deterministic.
Question 7: Lag Operator
What is $(1-L)X_t$, where $L$ is the lag operator?
- A) $X_t + X_{t-1}$
- B) $X_t - X_{t-1}$ (first difference)
- C) $X_t \cdot X_{t-1}$
- D) $X_t / X_{t-1}$
Correct! $L \cdot X_t = X_{t-1}$, so $(1-L)X_t = X_t - X_{t-1} = \Delta X_t$.
Incorrect. The correct answer is B. The operator $(1-L)$ produces the first difference: $\Delta X_t = X_t - X_{t-1}$.
Question 8: Random Walk with Drift
For $X_t = \mu + X_{t-1} + \varepsilon_t$ with $\mu > 0$ and $X_0 = 0$, which statement is correct?
- A) The series is stationary
- B) $\mathbb{E}[X_t] = \mu$ (constant)
- C) $\mathbb{E}[X_t] = \mu \cdot t$ (grows linearly)
- D) The variance is constant
Correct! Random walk with drift: $\mathbb{E}[X_t] = \mu t$ and $\text{Var}(X_t) = t\sigma^2$. Both grow, so it is non-stationary.
Incorrect. The correct answer is C. The drift $\mu$ produces a linearly growing mean: $\mathbb{E}[X_t] = \mu t$.
Question 9: ACF and Non-Stationarity
An Autocorrelation Function (ACF) that decays very slowly toward zero suggests:
- A) The series is white noise
- B) The series is stationary with short memory
- C) The series is non-stationary (possible unit root)
- D) The series has seasonality
Correct! A persistent ACF (slow decay) is the classic sign of non-stationarity. Apply the ADF test to confirm.
Incorrect. The correct answer is C. Slowly decaying ACF = non-stationarity. Solution: differencing.
Question 10: Autocorrelation Function Properties
For a weakly stationary process, $\rho(h) = \gamma(h)/\gamma(0)$ has the property:
- A) $\rho(0) = 0$
- B) $\rho(h) = \rho(-h)$ and $|\rho(h)| \leq 1$
- C) $\rho(h)$ increases with $h$
- D) $\rho(h) = 1$ for all $h$
Correct! The ACF is symmetric ($\rho(h) = \rho(-h)$), $\rho(0) = 1$, and $|\rho(h)| \leq 1$ for all $h$.
Incorrect. The correct answer is B. The ACF is symmetric and bounded: $|\rho(h)| \leq 1$.
Question 11: Log Returns
How do we obtain a stationary series from financial prices $P_t$?
- A) Apply the transformation $\sqrt{P_t}$
- B) Multiply by a constant
- C) Compute log returns: $r_t = \ln(P_t / P_{t-1})$
- D) Apply a moving average
Correct! Log returns $r_t = \ln P_t - \ln P_{t-1} = \Delta \ln P_t$ are typically stationary, unlike prices.
Incorrect. The correct answer is C. Log returns transform non-stationary prices into stationary series.
Question 12: Weak Stationarity
Which condition is NOT required for weak stationarity?
- A) Constant mean: $\mathbb{E}[X_t] = \mu$
- B) Constant variance: $\text{Var}(X_t) = \sigma^2$
- C) Normal distribution
- D) Covariance depends only on lag
Correct! Weak stationarity only requires constant mean, constant variance, and lag-dependent covariance. No distributional assumption!
Incorrect. The correct answer is C. Normality is NOT required for weak stationarity.
Question 13: Random Walk
For a random walk $X_t = X_{t-1} + \varepsilon_t$ with $X_0 = 100$ and $\sigma^2 = 4$, what is $\text{Var}(X_{25})$?
- A) 4
- B) 25
- C) 100
- D) 625
Correct! $\text{Var}(X_t) = t \cdot \sigma^2 = 25 \times 4 = 100$
Incorrect. The correct answer is C. For random walk: $\text{Var}(X_t) = t \cdot \sigma^2 = 25 \times 4 = 100$.
Question 14: White Noise
Which statement about white noise is FALSE?
- A) White noise has zero mean
- B) White noise has constant variance
- C) White noise must be normally distributed
- D) White noise has no autocorrelation
Correct! White noise only requires zero mean, constant variance, and no autocorrelation. Gaussian white noise is a special case.
Incorrect. The correct answer is C. White noise does NOT require normality.
Question 15: ACF Patterns
For an AR(1) process with $\phi = 0.8$, the ACF:
- A) Cuts off after lag 1
- B) Decays exponentially
- C) Oscillates around zero
- D) Is zero for all lags
Correct! For AR(1): $\rho(h) = \phi^h = 0.8^h$, which decays exponentially toward zero.
Incorrect. The correct answer is B. AR processes have exponentially decaying ACF; MA processes cut off.
Question 16: PACF Interpretation
The PACF of an MA(1) process:
- A) Cuts off after lag 1
- B) Decays exponentially
- C) Is zero for all lags
- D) Shows a seasonal pattern
Correct! For MA processes, PACF decays; ACF cuts off. For AR processes, it's the opposite.
Incorrect. The correct answer is B. MA: ACF cuts off, PACF decays. AR: ACF decays, PACF cuts off.
Question 17: Differencing
If $X_t$ is a random walk, then $\Delta X_t = X_t - X_{t-1}$ is:
- A) Still a random walk
- B) White noise (stationary)
- C) An AR(1) process
- D) A trending series
Correct! If $X_t = X_{t-1} + \varepsilon_t$, then $\Delta X_t = \varepsilon_t$, which is white noise!
Incorrect. The correct answer is B. Differencing a random walk gives white noise.
Question 18: ADF Test
In the ADF test, the null hypothesis is:
- A) The series is stationary
- B) The series has a unit root
- C) The series is white noise
- D) The series has no trend
Correct! ADF: $H_0$: unit root (non-stationary). Reject $H_0$ $\Rightarrow$ stationary.
Incorrect. The correct answer is B. ADF tests for unit root; KPSS tests for stationarity.
Question 19: KPSS Test
If KPSS rejects $H_0$, we conclude:
- A) The series is stationary
- B) The series is non-stationary
- C) The series is white noise
- D) We need more data
Correct! KPSS: $H_0$: stationary. Reject $H_0$ $\Rightarrow$ evidence of non-stationarity.
Incorrect. The correct answer is B. KPSS has opposite hypotheses to ADF.
Question 20: Deterministic vs Stochastic Trend
How should you remove a stochastic trend (unit root)?
- A) Fit a linear regression on time
- B) Apply differencing
- C) Use a moving average
- D) Apply seasonal adjustment
Correct! Stochastic trend $\Rightarrow$ differencing. Deterministic trend $\Rightarrow$ regression.
Incorrect. The correct answer is B. Use differencing for stochastic trends, regression for deterministic trends.
Chapter 2: ARMA Models
Question 1: AR Stationarity
For which value of $\phi$ is the AR(1) process $X_t = c + \phi X_{t-1} + \varepsilon_t$ stationary?
- A) $\phi = 1.2$
- B) $\phi = 1.0$
- C) $\phi = -0.8$
- D) $\phi = -1.5$
Correct! AR(1) is stationary iff $|\phi| < 1$. Only $|-0.8| = 0.8 < 1$.
Incorrect. The correct answer is C. AR(1) requires $|\phi| < 1$ for stationarity.
Question 2: ACF/PACF Pattern Recognition
You observe: ACF has a spike at lag 1, then cuts off. PACF decays gradually. What model is suggested?
- A) AR(1)
- B) MA(1)
- C) ARMA(1,1)
- D) White noise
Correct! ACF cuts off → MA process; PACF decays → confirms MA(1).
Incorrect. The correct answer is B. MA: ACF cuts off, PACF decays. AR: ACF decays, PACF cuts off.
Question 3: MA Invertibility
Is the MA(1) process $X_t = \varepsilon_t + 1.5\varepsilon_{t-1}$ invertible?
- A) Yes, MA processes are always invertible
- B) Yes, because $\theta = 1.5 > 0$
- C) No, because $|\theta| = 1.5 > 1$
- D) No, MA processes are never invertible
Correct! Invertibility requires $|\theta| < 1$. Here $|\theta| = 1.5 > 1$, so NOT invertible.
Incorrect. The correct answer is C. Invertibility condition for MA(1): $|\theta| < 1$.
Question 4: ARMA Representation
The compact form $\phi(L)X_t = \theta(L)\varepsilon_t$ represents which model?
- A) Pure AR model
- B) Pure MA model
- C) ARMA model
- D) None of the above
Correct! $\phi(L)$ is the AR polynomial, $\theta(L)$ is the MA polynomial → ARMA(p,q).
Incorrect. The correct answer is C. This notation combines AR and MA into ARMA.
Question 5: Lag Operator
What is $(1-L)^2 X_t$?
- A) $X_t - X_{t-1}$
- B) $X_t - 2X_{t-1} + X_{t-2}$
- C) $X_t + X_{t-1} + X_{t-2}$
- D) $X_t - X_{t-2}$
Correct! $(1-L)^2 = 1 - 2L + L^2$, so $(1-L)^2 X_t = X_t - 2X_{t-1} + X_{t-2}$ (second difference).
Incorrect. The correct answer is B. Expand $(1-L)^2 = 1 - 2L + L^2$.
Question 6: Information Criteria
Comparing ARMA(1,1) vs ARMA(2,1) using BIC, which statement is correct?
- A) Lower BIC always means better forecasts
- B) BIC penalizes complexity less than AIC
- C) The model with lower BIC is preferred
- D) BIC can only compare models with same number of parameters
Correct! Lower BIC indicates better fit-complexity trade-off. BIC penalizes complexity MORE than AIC.
Incorrect. The correct answer is C. BIC balances fit and complexity; lower is better.
Question 7: Ljung-Box Test
After fitting an ARMA model, you run the Ljung-Box test on residuals and get p-value = 0.03. What does this mean?
- A) Model is adequate, residuals are white noise
- B) Model is inadequate, residuals have autocorrelation
- C) Need to increase sample size
- D) Test is inconclusive
Correct! p-value < 0.05 rejects $H_0$ (white noise), indicating remaining autocorrelation → model inadequate.
Incorrect. The correct answer is B. Rejecting $H_0$ means residuals are NOT white noise.
Question 8: Forecast Properties
For a stationary AR(1) model, what happens to forecasts as horizon $h \to \infty$?
- A) Forecasts grow without bound
- B) Forecasts oscillate forever
- C) Forecasts converge to the unconditional mean $\mu$
- D) Forecasts become more accurate
Correct! $\hat{X}_{n+h|n} = \mu + \phi^h(X_n - \mu) \to \mu$ as $h \to \infty$ (since $|\phi|<1$). This is mean reversion.
Incorrect. The correct answer is C. Stationary AR forecasts revert to the unconditional mean.
Question 9: AR(1) Variance
For the AR(1) process $X_t = \phi X_{t-1} + \varepsilon_t$ with $\text{Var}(\varepsilon_t) = \sigma^2$, what is $\text{Var}(X_t)$?
- A) $\sigma^2$
- B) $\sigma^2 / (1 - \phi)$
- C) $\sigma^2 / (1 - \phi^2)$
- D) $\sigma^2 (1 + \phi^2)$
Correct! For stationary AR(1): $\gamma(0) = \text{Var}(X_t) = \sigma^2 / (1 - \phi^2)$.
Incorrect. The correct answer is C. Use $\gamma(0) = \phi^2 \gamma(0) + \sigma^2$ and solve.
Question 10: MA(1) Autocorrelation
For an MA(1) process $X_t = \varepsilon_t + \theta \varepsilon_{t-1}$, what is $\rho(2)$?
- A) $\theta / (1 + \theta^2)$
- B) $\theta^2 / (1 + \theta^2)$
- C) $\theta^2$
- D) 0
Correct! MA(q) has $\rho(h) = 0$ for $h > q$. Since MA(1) has $q=1$, $\rho(2) = 0$.
Incorrect. The correct answer is D. MA(1) ACF cuts off after lag 1.
Question 11: AR(1) Mean
For the AR(1) process $X_t = c + \phi X_{t-1} + \varepsilon_t$ with $c = 2$ and $\phi = 0.6$, what is $E[X_t]$?
- A) 2
- B) 3.33
- C) 5
- D) 0.8
Correct! $\mu = c / (1 - \phi) = 2 / (1 - 0.6) = 2 / 0.4 = 5$.
Incorrect. The correct answer is C. Use formula $\mu = c / (1 - \phi)$.
Question 12: Wold's Theorem
According to Wold's decomposition theorem, any stationary process can be written as:
- A) A finite AR process
- B) An infinite MA process plus a deterministic component
- C) A random walk
- D) White noise only
Correct! Wold: $X_t = \sum_{j=0}^{\infty} \psi_j \varepsilon_{t-j} + D_t$ where $D_t$ is deterministic.
Incorrect. The correct answer is B. Wold's theorem gives an MA($\infty$) representation.
Question 13: AR(2) Stationarity
For AR(2): $X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + \varepsilon_t$, which conditions ensure stationarity?
- A) $|\phi_1| < 1$ and $|\phi_2| < 1$
- B) $\phi_1 + \phi_2 < 1$, $\phi_2 - \phi_1 < 1$, $|\phi_2| < 1$
- C) $\phi_1^2 + \phi_2^2 < 1$
- D) $|\phi_1 + \phi_2| < 1$
Correct! AR(2) stationarity requires all three conditions in B (the stationarity triangle).
Incorrect. The correct answer is B. These define the AR(2) stationarity region.
Question 14: Yule-Walker Equations
The Yule-Walker equations are used to:
- A) Test for stationarity
- B) Estimate AR parameters from autocorrelations
- C) Estimate MA parameters
- D) Compute forecasts
Correct! Yule-Walker: $\boldsymbol{\phi} = \mathbf{R}^{-1} \boldsymbol{\rho}$ relates AR parameters to ACF.
Incorrect. The correct answer is B. Yule-Walker provides method-of-moments AR estimation.
Question 15: PACF Interpretation
The PACF at lag $k$ measures:
- A) Total correlation between $X_t$ and $X_{t-k}$
- B) Correlation between $X_t$ and $X_{t-k}$ after removing effects of intermediate lags
- C) The MA coefficient at lag $k$
- D) The variance at lag $k$
Correct! PACF measures direct correlation after controlling for $X_{t-1}, ..., X_{t-k+1}$.
Incorrect. The correct answer is B. PACF removes indirect correlations through intermediate lags.
Question 16: Parsimony Principle
The parsimony principle in ARMA modeling suggests:
- A) Always use the highest order model
- B) Choose the simplest adequate model
- C) Only use AR models
- D) Always include seasonal terms
Correct! Parsimony: prefer simpler models to avoid overfitting and improve generalization.
Incorrect. The correct answer is B. Simpler models often forecast better.
Question 17: Characteristic Roots
For AR(1) with $\phi = 0.9$, the characteristic root is:
- A) 0.9
- B) 1/0.9 ≈ 1.11
- C) -0.9
- D) 0.81
Correct! Characteristic equation: $1 - 0.9z = 0 \Rightarrow z = 1/0.9 \approx 1.11 > 1$ (stationary).
Incorrect. The correct answer is B. Root = $1/\phi$, must be outside unit circle.
Question 18: Estimation Methods
Which estimation method is generally preferred for ARMA models?
- A) Ordinary Least Squares (OLS)
- B) Maximum Likelihood Estimation (MLE)
- C) Method of Moments only
- D) Simple averaging
Correct! MLE is efficient and handles both AR and MA components properly.
Incorrect. The correct answer is B. MLE is the standard for ARMA estimation.
Question 19: Forecast Error Variance
For a stationary ARMA model, as forecast horizon increases, the forecast error variance:
- A) Decreases to zero
- B) Increases without bound
- C) Converges to the unconditional variance
- D) Remains constant
Correct! As $h \to \infty$, forecast error variance $\to \gamma(0)$ (unconditional variance).
Incorrect. The correct answer is C. Long-run forecast uncertainty equals process variance.
Question 20: Box-Jenkins Methodology
The correct order of steps in Box-Jenkins methodology is:
- A) Estimation → Identification → Diagnostics
- B) Identification → Estimation → Diagnostics
- C) Diagnostics → Identification → Estimation
- D) Estimation → Diagnostics → Identification
Correct! Box-Jenkins: (1) Identify model order via ACF/PACF, (2) Estimate parameters, (3) Diagnose residuals.
Incorrect. The correct answer is B. Identification comes first, then estimation, then diagnostics.
Chapter 3: ARIMA Models
Question 1: Order of Integration
A time series requires two differences to become stationary. What is its order of integration?
- A) $I(0)$
- B) $I(1)$
- C) $I(2)$
- D) Cannot be determined
Correct! By definition, $Y_t \sim I(d)$ means $d$ differences are needed for stationarity.
Incorrect. The correct answer is C. Two differences means $I(2)$.
Question 2: Random Walk Variance
For a random walk $Y_t = Y_{t-1} + \varepsilon_t$ with $\text{Var}(\varepsilon_t) = \sigma^2$, what is $\text{Var}(Y_t)$?
- A) $\sigma^2$
- B) $t \cdot \sigma^2$
- C) $\sigma^2 / t$
- D) $\sigma^2 / (1-\phi^2)$
Correct! The variance grows linearly: $\text{Var}(Y_t) = t\sigma^2$.
Incorrect. The correct answer is B. Random walk variance grows with time.
Question 3: ADF Test Hypothesis
In the Augmented Dickey-Fuller test, what is the null hypothesis?
- A) The series is stationary
- B) The series has a unit root
- C) The series has no autocorrelation
- D) The series is normally distributed
Correct! ADF tests H0: unit root vs H1: stationary.
Incorrect. The correct answer is B. ADF null hypothesis is unit root (non-stationary).
Question 4: ARIMA Notation
What does ARIMA(2,1,1) represent?
- A) AR(2) on differenced data with MA(1) errors
- B) AR(1) with 2 differences and MA(1)
- C) MA(2) with 1 difference and AR(1)
- D) 2 lags, 1 trend, 1 seasonal component
Correct! ARIMA(p,d,q): p=AR order, d=differences, q=MA order.
Incorrect. The correct answer is A. ARIMA(2,1,1) has AR(2), 1 difference, MA(1).
Question 5: Second Difference
What is $(1-L)^2 Y_t$ expanded?
- A) $Y_t - Y_{t-1}$
- B) $Y_t - 2Y_{t-1} + Y_{t-2}$
- C) $Y_t + 2Y_{t-1} + Y_{t-2}$
- D) $Y_t - Y_{t-2}$
Correct! $(1-L)^2 = 1 - 2L + L^2$.
Incorrect. The correct answer is B. Second difference: $Y_t - 2Y_{t-1} + Y_{t-2}$.
Question 6: KPSS vs ADF
How does the KPSS test differ from the ADF test?
- A) KPSS tests for seasonality, ADF tests for trends
- B) KPSS has stationarity as null, ADF has unit root as null
- C) KPSS is more powerful than ADF
- D) There is no difference
Correct! They have reversed null hypotheses.
Incorrect. The correct answer is B. KPSS: H0=stationary, ADF: H0=unit root.
Question 7: Overdifferencing
If $Y_t \sim I(1)$ and we compute $\Delta^2 Y_t$, what happens?
- A) We get a better stationary series
- B) We introduce artificial negative autocorrelation
- C) The variance decreases
- D) Nothing changes
Correct! Overdifferencing creates an MA unit root boundary.
Incorrect. The correct answer is B. Overdifferencing introduces spurious MA structure.
Question 8: Forecast Variance
For ARIMA(0,1,0) (random walk), how does forecast variance behave as horizon $h$ increases?
- A) Stays constant
- B) Decreases to zero
- C) Grows linearly with $h$
- D) Converges to a finite limit
Correct! Forecast variance = $h\sigma^2$ for random walk.
Incorrect. The correct answer is C. I(1) forecast uncertainty grows unboundedly.
Question 9: Deterministic vs Stochastic Trend
What distinguishes a stochastic trend from a deterministic trend?
- A) Stochastic trends can be removed by detrending
- B) Deterministic trends have permanent shock effects
- C) Stochastic trends have permanent shock effects
- D) They are mathematically equivalent
Correct! Stochastic trends: shocks are permanent. Deterministic: shocks are temporary.
Incorrect. The correct answer is C. Stochastic trends have permanent shock effects.
Question 10: ADF Critical Values
In the ADF test, you reject H0 (unit root) when:
- A) Test statistic > critical value
- B) Test statistic < critical value (more negative)
- C) p-value > 0.05
- D) Test statistic equals zero
Correct! ADF rejects when statistic is sufficiently negative.
Incorrect. The correct answer is B. More negative = reject unit root.
Question 11: IMA(1,1) Model
ARIMA(0,1,1) is also known as:
- A) Random walk
- B) Simple exponential smoothing
- C) Holt's method
- D) Pure AR model
Correct! IMA(1,1) is equivalent to simple exponential smoothing.
Incorrect. The correct answer is B. ARIMA(0,1,1) = IMA(1,1) = SES.
Question 12: Constant in ARIMA
In ARIMA(1,1,0) with constant $c$, what does $c$ represent?
- A) The mean of $Y_t$
- B) The drift (average change)
- C) The variance
- D) The AR coefficient
Correct! When $d>0$, the constant represents the drift.
Incorrect. The correct answer is B. For $d \geq 1$, $c$ is the drift.
Question 13: Auto-ARIMA
What does auto_arima primarily use to select the best model?
- A) Only visual inspection
- B) Information criteria (AIC/BIC)
- C) Only residual analysis
- D) Random selection
Correct! Auto-ARIMA uses AIC/BIC to compare and select models.
Incorrect. The correct answer is B. Information criteria guide selection.
Question 14: Box-Jenkins Step 1
What is the first step in Box-Jenkins methodology?
- A) Estimate parameters
- B) Identify model order (p,d,q)
- C) Diagnose residuals
- D) Generate forecasts
Correct! Identification → Estimation → Diagnostics.
Incorrect. The correct answer is B. First identify, then estimate, then diagnose.
Question 15: ACF of I(1) Series
What is characteristic of the ACF of an I(1) series?
- A) Cuts off sharply at lag 1
- B) Decays very slowly
- C) Shows no significant lags
- D) Alternates between positive and negative
Correct! Slow ACF decay is a hallmark of non-stationarity.
Incorrect. The correct answer is B. I(1) series have slowly decaying ACF.
Question 16: Ljung-Box on ARIMA Residuals
If Ljung-Box test p-value is 0.02 on ARIMA residuals, what should you conclude?
- A) The model is adequate
- B) The model is inadequate, consider re-specifying
- C) The series is stationary
- D) The series has a unit root
Correct! p < 0.05 indicates significant residual autocorrelation.
Incorrect. The correct answer is B. Residuals should be white noise (p > 0.05).
Question 17: MLE for ARIMA
What is the standard estimation method for ARIMA parameters?
- A) Ordinary Least Squares
- B) Maximum Likelihood Estimation
- C) Method of Moments
- D) Bayesian estimation only
Correct! MLE is standard for ARIMA estimation.
Incorrect. The correct answer is B. Maximum Likelihood is standard.
Question 18: Typical $d$ for Economic Data
For most macroeconomic time series (GDP, prices), what is the typical order of integration?
- A) $I(0)$ - stationary
- B) $I(1)$ - unit root
- C) $I(2)$ or higher
- D) $I(-1)$ - overdifferenced
Correct! Most economic series are I(1).
Incorrect. The correct answer is B. GDP, prices typically are I(1).
Question 19: Conflicting Test Results
If ADF does not reject but KPSS rejects, what does this suggest?
- A) The series is definitely stationary
- B) The series is definitely non-stationary
- C) Both tests agree the series has a unit root
- D) Results are inconclusive, need more analysis
Correct! Conflicting results suggest borderline cases or structural breaks.
Incorrect. The correct answer is D. Conflicting results require further investigation.
Question 20: ARIMA Limitations
Which is NOT a limitation of ARIMA models?
- A) Assumes linearity
- B) Cannot capture volatility clustering
- C) Can handle non-stationary data
- D) Assumes constant variance
Correct! Handling non-stationarity via differencing is a strength, not a limitation.
Incorrect. The correct answer is C. Handling non-stationarity is a feature, not a limitation.
Chapter 4: SARIMA Models
Question 1: Seasonal Differencing
For monthly data with annual seasonality, what does $(1-L^{12})Y_t$ compute?
- A) $Y_t - Y_{t-1}$
- B) $Y_t - Y_{t-12}$
- C) Average of 12 months
- D) $Y_t / Y_{t-12}$
Correct! $(1-L^{12})Y_t = Y_t - Y_{t-12}$ compares each observation with the same month in the previous year.
Incorrect. The correct answer is B. The seasonal differencing operator $(1-L^s)$ computes $Y_t - Y_{t-s}$.
Question 2: SARIMA Notation
In SARIMA$(1,1,1) \times (1,1,1)_{12}$, what does the subscript 12 represent?
- A) Number of parameters
- B) Seasonal period
- C) Maximum lag order
- D) Sample size requirement
Correct! The subscript $s=12$ indicates the seasonal period (12 months for monthly data with annual seasonality).
Incorrect. The correct answer is B. The subscript represents the seasonal period.
Question 3: Airline Model
The classic "airline model" SARIMA$(0,1,1) \times (0,1,1)_{12}$ has how many parameters (excluding $\sigma^2$)?
- A) 2
- B) 4
- C) 6
- D) 12
Correct! Only 2 parameters: $\theta_1$ (regular MA) and $\Theta_1$ (seasonal MA). Despite its simplicity, it captures complex seasonal dynamics.
Incorrect. The correct answer is A. The airline model has just 2 MA parameters: $\theta_1$ and $\Theta_1$.
Question 4: Seasonal ACF Pattern
For monthly data with strong annual seasonality, where would you expect significant ACF spikes?
- A) Only at lag 1
- B) At lags 12, 24, 36, ...
- C) Randomly distributed
- D) No spikes at all
Correct! Seasonal data shows significant ACF at multiples of the seasonal period (12, 24, 36, ...).
Incorrect. The correct answer is B. Seasonal ACF shows spikes at seasonal lags and their multiples.
Question 5: Multiplicative Structure
In SARIMA, "multiplicative structure" means:
- A) Seasonal amplitude grows proportionally with level
- B) Regular and seasonal polynomials are multiplied
- C) Data must be multiplied by seasonal factors
- D) Model is estimated using multiplication
Correct! The AR polynomials $\phi(L)\Phi(L^s)$ and MA polynomials $\theta(L)\Theta(L^s)$ are multiplied, creating cross-terms.
Incorrect. The correct answer is B. Multiplicative structure refers to the multiplication of regular and seasonal polynomials.
Question 6: Full Differencing
When should you apply both regular ($d=1$) and seasonal ($D=1$) differencing?
- A) When data has only a trend
- B) When data has only seasonality
- C) When data has both trend and seasonal non-stationarity
- D) Never - they cancel each other
Correct! Both differencing types are needed when the series has both a stochastic trend (unit root) and stochastic seasonality.
Incorrect. The correct answer is C. Use both when the series exhibits both regular and seasonal non-stationarity.
Question 7: Log Transformation
When should you apply log transformation before fitting SARIMA?
- A) When data contains zeros or negative values
- B) When seasonal fluctuations grow with the level
- C) Always, regardless of data characteristics
- D) Never for seasonal data
Correct! Log transformation stabilizes variance when seasonal amplitude grows with level (multiplicative seasonality).
Incorrect. The correct answer is B. Log transformation is used for multiplicative seasonality where amplitude grows with level.
Question 8: Seasonal Period
What seasonal period ($s$) would you use for quarterly GDP data with annual patterns?
- A) $s = 1$
- B) $s = 4$
- C) $s = 12$
- D) $s = 52$
Correct! For quarterly data with annual seasonality, $s=4$ (4 quarters per year).
Incorrect. The correct answer is B. Quarterly data has 4 observations per annual cycle.
Question 9: Expansion Formula
What is $(1-L)(1-L^{12})$ expanded?
- A) $1 - L - L^{12}$
- B) $1 - L - L^{12} + L^{13}$
- C) $1 - 2L + L^{12}$
- D) $1 - L^{13}$
Correct! $(1-L)(1-L^{12}) = 1 - L - L^{12} + L^{13}$, so $(1-L)(1-L^{12})Y_t = Y_t - Y_{t-1} - Y_{t-12} + Y_{t-13}$.
Incorrect. The correct answer is B. Multiply out: $(1-L)(1-L^{12}) = 1 - L - L^{12} + L^{13}$.
Question 10: Deterministic vs Stochastic
What distinguishes stochastic seasonality (SARIMA) from deterministic seasonality (seasonal dummies)?
- A) Stochastic seasonality requires more data
- B) Stochastic seasonality evolves over time
- C) Deterministic seasonality is more flexible
- D) There is no difference
Correct! Stochastic seasonality evolves: this year's December depends on last year's December. Deterministic assumes fixed, repeating patterns.
Incorrect. The correct answer is B. Stochastic seasonality allows the seasonal pattern to evolve over time.
Chapter 5: Volatility Models ARCH/GARCH
Note: Quiz content is being updated. Current quizzes cover VAR topics.
Question 1: VAR Definition
In a VAR(2) model with 3 variables, how many coefficient matrices $\mathbf{A}_i$ are there?
- A) 2
- B) 3
- C) 6
- D) 9
Correct! VAR(p) has p coefficient matrices. VAR(2) has $\mathbf{A}_1$ and $\mathbf{A}_2$, regardless of the number of variables.
Incorrect. The correct answer is A. VAR(p) has p coefficient matrices (one for each lag).
Question 2: Number of Parameters
A VAR(2) with $K=3$ variables (including constants) has how many parameters per equation?
- A) 3
- B) 6
- C) 7
- D) 9
Correct! Per equation: $1 + K \times p = 1 + 3 \times 2 = 7$ parameters (constant + 6 lagged coefficients).
Incorrect. The correct answer is C. Each equation has 1 constant + $K \times p = 3 \times 2 = 6$ lagged terms = 7 parameters.
Question 3: Granger Causality
"$X$ Granger-causes $Y$" means:
- A) $X$ is the economic cause of $Y$
- B) Past $X$ helps predict future $Y$
- C) $X$ and $Y$ are contemporaneously correlated
- D) $X$ always increases when $Y$ increases
Correct! Granger causality is about predictive content: past values of $X$ help forecast $Y$ beyond $Y$'s own history.
Incorrect. The correct answer is B. Granger causality tests whether past $X$ improves predictions of $Y$.
Question 4: Granger Causality Test
To test if $Y_2$ Granger-causes $Y_1$ in a VAR(p), we test:
- A) All coefficients in the $Y_1$ equation equal zero
- B) Coefficients on lagged $Y_2$ in the $Y_1$ equation equal zero
- C) Coefficients on lagged $Y_1$ in the $Y_2$ equation equal zero
- D) The error covariance equals zero
Correct! We test $H_0$: all coefficients on lagged $Y_2$ in the $Y_1$ equation are jointly zero.
Incorrect. The correct answer is B. Test whether lagged $Y_2$ terms in the $Y_1$ equation are jointly zero.
Question 5: VAR Stability
A VAR(1) model is stable (stationary) if:
- A) All diagonal elements of $\mathbf{A}_1$ are less than 1
- B) The determinant of $\mathbf{A}_1$ is less than 1
- C) All eigenvalues of $\mathbf{A}_1$ are less than 1 in absolute value
- D) The trace of $\mathbf{A}_1$ equals zero
Correct! Stability requires all eigenvalues inside the unit circle ($|\lambda_i| < 1$), ensuring shocks die out over time.
Incorrect. The correct answer is C. Stability requires all eigenvalues of $\mathbf{A}_1$ to have modulus less than 1.
Question 6: Impulse Response Functions
An impulse response function shows:
- A) The correlation between two variables
- B) The effect of a shock to one variable on all variables over time
- C) The forecast accuracy of the model
- D) The p-values of coefficient tests
Correct! IRF$_{ij}(h)$ shows the response of variable $i$ at horizon $h$ to a one-unit shock in variable $j$.
Incorrect. The correct answer is B. IRFs trace how shocks propagate through the system over time.
Question 7: Lag Order Selection
Which criterion typically selects the most parsimonious VAR model?
- A) AIC (Akaike Information Criterion)
- B) BIC (Bayesian Information Criterion)
- C) FPE (Final Prediction Error)
- D) Adjusted $R^2$
Correct! BIC penalizes complexity more heavily than AIC ($k \ln n$ vs $2k$), favoring simpler models.
Incorrect. The correct answer is B. BIC has a stronger penalty for parameters, selecting more parsimonious models.
Question 8: FEVD
Forecast Error Variance Decomposition (FEVD) tells us:
- A) The correlation between variables
- B) What proportion of forecast error variance comes from each shock
- C) The optimal forecast horizon
- D) Which variables to include in the model
Correct! FEVD shows how much of the forecast uncertainty for each variable is attributable to each structural shock.
Incorrect. The correct answer is B. FEVD decomposes forecast variance by shock source.
Question 9: Cholesky Ordering
Cholesky ordering in IRF analysis assumes:
- A) All variables are equally important
- B) Variables ordered first affect later variables contemporaneously, not vice versa
- C) Shocks are uncorrelated
- D) No restrictions are needed
Correct! Cholesky imposes a recursive structure: variables ordered earlier can affect later ones within the period, but not vice versa.
Incorrect. The correct answer is B. Cholesky assumes a causal ordering for contemporaneous effects.
Question 10: Cointegration
If variables are I(1) and cointegrated, you should use:
- A) VAR in levels
- B) VAR in first differences
- C) Vector Error Correction Model (VECM)
- D) Univariate ARIMA models
Correct! VECM incorporates both short-run dynamics and the long-run equilibrium relationship through the error correction term.
Incorrect. The correct answer is C. With cointegration, VECM preserves the long-run relationship that would be lost in differences.
Question 11: VAR Residuals
In a well-specified VAR, residuals should be:
- A) Autocorrelated but homoskedastic
- B) Serially uncorrelated (white noise)
- C) Perfectly normally distributed
- D) Zero for all observations
Correct! Residuals should have no autocorrelation. Cross-equation correlation is allowed and captured in the covariance matrix.
Incorrect. The correct answer is B. The model should capture all serial dependence, leaving white noise residuals.
Question 12: Structural VAR
The main difference between SVAR and reduced-form VAR is:
- A) SVAR uses more lags
- B) SVAR identifies structural shocks with economic interpretation
- C) SVAR requires more data
- D) SVAR cannot forecast
Correct! SVAR imposes identifying restrictions to recover orthogonal structural shocks with economic meaning (e.g., "monetary policy shock").
Incorrect. The correct answer is B. SVAR uses restrictions to identify economically meaningful structural shocks.
Question 13: Granger vs True Causality
Finding that X Granger-causes Y means:
- A) X definitely causes Y economically
- B) X has predictive power for Y, but may not be true causation
- C) Y causes X
- D) X and Y are unrelated
Correct! Granger causality is about prediction, not true causation. It may be spurious due to omitted variables or anticipation effects.
Incorrect. The correct answer is B. Granger causality tests predictive content, not economic causation.
Question 14: VAR Estimation
VAR models can be estimated by:
- A) OLS on each equation separately
- B) Only maximum likelihood
- C) Only Bayesian methods
- D) Weighted least squares only
Correct! With identical regressors in each equation, OLS = GLS = MLE (under Gaussian errors). This makes estimation simple and efficient.
Incorrect. The correct answer is A. VAR can be estimated equation-by-equation with OLS.
Question 15: IRF Convergence
In a stable VAR, impulse responses as $h \to \infty$:
- A) Explode to infinity
- B) Converge to zero
- C) Oscillate forever
- D) Stay constant
Correct! Stability means eigenvalues inside unit circle, so $A^h \to 0$. Shocks have transitory effects that die out.
Incorrect. The correct answer is B. In stable VARs, all impulse responses converge to zero.
Question 16: Bidirectional Causality
If both "X Granger-causes Y" and "Y Granger-causes X", this is called:
- A) No causality
- B) Unidirectional causality
- C) Bidirectional (feedback) causality
- D) Instantaneous causality
Correct! Bidirectional or feedback causality means each variable helps predict the other. Common in financial markets.
Incorrect. The correct answer is C. When causality runs both ways, it's called bidirectional or feedback.
Question 17: Companion Form
The companion matrix of a VAR(p) is used to:
- A) Reduce the number of parameters
- B) Convert VAR(p) to VAR(1) form for stability analysis
- C) Estimate the model faster
- D) Test for Granger causality
Correct! Any VAR(p) can be written as VAR(1) in companion form. Eigenvalues of the companion matrix determine stability.
Incorrect. The correct answer is B. Companion form allows stability analysis through eigenvalue computation.
Question 18: FEVD Over Horizons
In FEVD, as the horizon $h$ increases:
- A) Own shocks always dominate
- B) The proportions converge to long-run values
- C) All shocks contribute equally
- D) FEVD becomes undefined
Correct! FEVD proportions stabilize at long horizons, showing the ultimate importance of each shock source.
Incorrect. The correct answer is B. FEVD converges to long-run decomposition as horizon increases.
Question 19: Instantaneous Causality
Instantaneous causality tests whether:
- A) Lagged X predicts Y
- B) Shocks to X and Y are correlated within the same period
- C) X and Y share a common trend
- D) The VAR is stable
Correct! Instantaneous causality tests contemporaneous correlation of residuals: $Cov(\varepsilon_{1t}, \varepsilon_{2t}) = 0$.
Incorrect. The correct answer is B. Instantaneous causality is about same-period correlation, not lags.
Question 20: Pre-Estimation Checks
Before estimating a VAR, you should always check:
- A) That all variables are I(2)
- B) The stationarity of each variable
- C) That variables are perfectly correlated
- D) That the sample size is exactly 100
Correct! Standard VAR requires stationary variables. Use ADF/KPSS tests first; if I(1), consider VECM if cointegrated.
Incorrect. The correct answer is B. Always test for stationarity before VAR estimation.
Chapter 6: VAR Models & Granger Causality
Question 1: VAR Model Structure
In a bivariate VAR(1), the equation for \\( Y_{1t} \\) includes:
- A) Only its own lagged values \\( Y_{1,t-1} \\)
- B) Only the lagged values of \\( Y_{2,t-1} \\)
- C) Lagged values of all variables: \\( Y_{1,t-1} \\) and \\( Y_{2,t-1} \\)
- D) Current and lagged values of all variables
✓ Correct! VAR equations include lagged values of ALL variables in the system.
✗ Incorrect. The correct answer is C. Each equation in a VAR includes its own lags and lags of all other variables.
Question 2: Curse of Dimensionality
A VAR(4) model with K=5 variables has how many parameters?
- A) 20
- B) 100
- C) 105
- D) 125
✓ Correct! Each equation has K×p + 1 = 5×4 + 1 = 21 parameters. With K=5 equations: 5×21 = 105.
✗ Incorrect. The correct answer is C. Total parameters = K(Kp+1) = 5(5×4+1) = 105.
Question 3: Stability Condition
For a general VAR(p), the stability condition requires that all roots of \\( \\det(\\mathbf{I}_K - \\mathbf{A}_1 z - \\cdots - \\mathbf{A}_p z^p) = 0 \\) lie:
- A) Inside the unit circle
- B) On the unit circle
- C) Outside the unit circle
- D) At the origin
✓ Correct! Stability requires all roots of the characteristic polynomial to lie outside the unit circle.
✗ Incorrect. The correct answer is C. Roots outside the unit circle ensure stationarity.
Question 4: Granger Causality Test
The Granger causality test uses which type of test statistic?
- A) Durbin-Watson statistic
- B) Wald/F-test comparing restricted and unrestricted models
- C) Augmented Dickey-Fuller statistic
- D) Jarque-Bera statistic
✓ Correct! Granger causality tests use F-statistics (or Wald statistics) comparing restricted and unrestricted models.
✗ Incorrect. The correct answer is B. The test compares models with and without the lagged variables.
Question 5: Autocovariance Matrix Property
For a weakly stationary multivariate time series, the autocovariance matrix satisfies:
- A) \\( \\boldsymbol{\\Gamma}(-h) = \\boldsymbol{\\Gamma}(h) \\)
- B) \\( \\boldsymbol{\\Gamma}(-h) = \\boldsymbol{\\Gamma}(h)' \\) (transpose)
- C) \\( \\boldsymbol{\\Gamma}(-h) = -\\boldsymbol{\\Gamma}(h) \\)
- D) \\( \\boldsymbol{\\Gamma}(-h) = \\boldsymbol{\\Gamma}(h)^{-1} \\)
✓ Correct! Unlike the univariate case, the cross-covariance matrix satisfies Γ(-h) = Γ(h)' (transpose), not equality.
✗ Incorrect. The correct answer is B. The autocovariance matrix is not symmetric in h, but satisfies the transpose relation.
Question 6: MA(infinity) Representation
The impulse response matrices \\( \\boldsymbol{\\Phi}_h \\) in the MA(\\( \\infty \\)) representation of a VAR(1) are given by:
- A) \\( \\boldsymbol{\\Phi}_h = h \\cdot \\mathbf{A} \\)
- B) \\( \\boldsymbol{\\Phi}_h = \\mathbf{A}^h \\)
- C) \\( \\boldsymbol{\\Phi}_h = \\mathbf{A}^{-h} \\)
- D) \\( \\boldsymbol{\\Phi}_h = \\boldsymbol{\\Sigma}^h \\)
✓ Correct! For VAR(1), the impulse response at horizon h is simply A raised to the power h.
✗ Incorrect. The correct answer is B. Φ_h = A^h follows from recursive substitution.
Question 7: Orthogonalized IRFs
The purpose of orthogonalizing IRFs using the Cholesky decomposition is to:
- A) Increase the number of parameters in the model
- B) Make the model stationary
- C) Produce uncorrelated shocks so that the effect of each shock can be isolated
- D) Remove the constant term from the VAR
✓ Correct! Orthogonalization via Cholesky produces uncorrelated shocks, allowing isolation of individual shock effects.
✗ Incorrect. The correct answer is C. The Cholesky decomposition orthogonalizes the error covariance matrix.
Question 8: FEVD Interpretation
If FEVD shows that 70% of the forecast error variance of unemployment at horizon 8 is due to GDP shocks, this means:
- A) GDP directly causes 70% of unemployment
- B) 70% of the uncertainty in forecasting unemployment 8 periods ahead is attributable to GDP shocks
- C) Unemployment will increase by 70% after 8 periods
- D) The correlation between GDP and unemployment is 0.70
✓ Correct! FEVD measures how much of the forecast uncertainty is attributable to each structural shock.
✗ Incorrect. The correct answer is B. FEVD decomposes forecast error variance by source.
Question 9: Granger Causality Numerical Example
Given T=100, RSS_U=45.2, RSS_R=52.8, and p=2 lags, the F-statistic for the Granger causality test is approximately:
- A) 3.09
- B) 5.42
- C) 7.98
- D) 12.35
✓ Correct! F = ((52.8-45.2)/2) / (45.2/95) = 3.8/0.4758 ≈ 7.98.
✗ Incorrect. The correct answer is C. F = ((RSS_R - RSS_U)/p) / (RSS_U/(T-2p-1)) ≈ 7.98.
Question 10: SVAR Identification
Which of the following is NOT a common identification scheme for Structural VAR models?
- A) Short-run (Cholesky) restrictions
- B) Long-run (Blanchard-Quah) restrictions
- C) OLS residual minimization
- D) Sign restrictions
✓ Correct! OLS residual minimization is not an identification scheme. Common schemes are short-run, long-run, and sign restrictions.
✗ Incorrect. The correct answer is C. Standard identification uses Cholesky, Blanchard-Quah, or sign restrictions.
Question 11: Companion Form Purpose
The companion form is useful because it:
- A) Reduces the number of parameters to estimate
- B) Converts any VAR(p) into a VAR(1), simplifying stationarity analysis, forecasting, and IRF computation
- C) Eliminates the need for the Cholesky decomposition
- D) Makes the error covariance matrix diagonal
✓ Correct! The companion form converts a VAR(p) into a VAR(1), simplifying stability analysis and forecasting.
✗ Incorrect. The correct answer is B. The companion form stacks lagged variables to create a first-order system.
Question 12: Diagnostic Tests
If the Portmanteau test for a VAR model's residuals is rejected, the most appropriate action is to:
- A) Apply a logarithmic transformation to the data
- B) Increase the lag order p or add additional variables
- C) Switch to a univariate ARIMA model
- D) Remove the constant term from the model
✓ Correct! Rejecting the Portmanteau test means residual autocorrelation remains, suggesting more lags are needed.
✗ Incorrect. The correct answer is B. Increase p or add variables to capture remaining dynamics.
Question 13: Cross-Covariance Properties
Unlike the univariate case, the cross-covariance function satisfies:
- A) \\( \\gamma_{ij}(h) = \\gamma_{ij}(-h) \\) always
- B) \\( \\gamma_{ij}(h) = \\gamma_{ji}(-h) \\), but generally \\( \\gamma_{ij}(h) \ eq \\gamma_{ij}(-h) \\)
- C) \\( \\gamma_{ij}(h) = 0 \\) for all \\( h \ eq 0 \\)
- D) \\( \\gamma_{ij}(h) = \\gamma_{ij}(h+1) \\) for all h
✓ Correct! Cross-covariance satisfies γ_ij(h) = γ_ji(-h), but generally γ_ij(h) ≠ γ_ij(-h).
✗ Incorrect. The correct answer is B. Unlike the univariate case, the cross-covariance is not symmetric in h.
Question 14: Forecast Error MSE
For a stable VAR(1), as the forecast horizon h increases, the mean squared error matrix converges to:
- A) The zero matrix
- B) The innovation covariance matrix \\( \\boldsymbol{\\Sigma} \\)
- C) The unconditional variance \\( \\boldsymbol{\\Gamma}(0) \\)
- D) Infinity
✓ Correct! For a stable VAR(1), the MSE converges to the unconditional variance Γ(0) as h → ∞.
✗ Incorrect. The correct answer is C. Long-horizon forecast uncertainty converges to the process variance.
Question 15: Restricted VAR
The likelihood ratio test for restrictions in a VAR model follows which distribution under the null?
- A) Normal distribution
- B) F-distribution
- C) Chi-squared distribution with r degrees of freedom
- D) Student's t-distribution
✓ Correct! The LR test for VAR restrictions follows a chi-squared distribution with r degrees of freedom.
✗ Incorrect. The correct answer is C. The degrees of freedom equal the number of restrictions imposed.
Question 16: Eigenvalue Example
For the matrix \\( \\mathbf{A} = \\begin{pmatrix} 0.7 & 0.2 \\\\ -0.1 & 0.6 \\end{pmatrix} \\), the eigenvalues are \\( \\lambda = 0.65 \\pm 0.132i \\). The modulus \\( |\\lambda| \\) equals:
- A) 0.782
- B) 0.663
- C) 1.30
- D) 0.44
✓ Correct! |λ| = √(0.65² + 0.132²) = √(0.4225 + 0.0174) = √0.4399 ≈ 0.663.
✗ Incorrect. The correct answer is B. The modulus of a complex eigenvalue a±bi is √(a²+b²).
Question 17: Granger Causality Pitfalls
Which of the following is a common pitfall when applying Granger causality tests?
- A) Using too many variables in the VAR
- B) A third omitted variable Z may cause both X and Y, creating a spurious Granger causal relationship
- C) The test cannot be applied to quarterly data
- D) Granger causality always implies true economic causation
✓ Correct! An omitted variable Z that causes both X and Y can create spurious Granger causality between them.
✗ Incorrect. The correct answer is B. Omitted variable bias is a well-known pitfall of Granger causality.
Question 18: Bootstrap Confidence Intervals
The bootstrap procedure for IRF confidence intervals involves:
- A) Estimating the model only once and using analytical formulas
- B) Resampling residuals with replacement, re-estimating the VAR, and computing IRFs repeatedly
- C) Increasing the sample size by adding artificial data points
- D) Removing outliers from the dataset and re-estimating
✓ Correct! Bootstrap for IRFs involves resampling residuals, re-estimating, and computing IRFs many times.
✗ Incorrect. The correct answer is B. This provides empirical confidence intervals for IRFs.
Question 19: Diebold-Mariano Test
The Diebold-Mariano test is used to:
- A) Test for Granger causality between two variables
- B) Test for the presence of unit roots
- C) Test whether one forecasting model is significantly more accurate than another
- D) Test for normality of VAR residuals
✓ Correct! The Diebold-Mariano test compares the predictive accuracy of two competing forecast models.
✗ Incorrect. The correct answer is C. It tests whether loss differentials are significantly different from zero.
Question 20: Cumulative IRF
The long-run multiplier (cumulative IRF as \\( H \\to \\infty \\)) for a stable VAR is given by:
- A) \\( \\boldsymbol{\\Psi}_\\infty = \\mathbf{A}_1 + \\mathbf{A}_2 + \\cdots + \\mathbf{A}_p \\)
- B) \\( \\boldsymbol{\\Psi}_\\infty = (\\mathbf{I}_K - \\mathbf{A}_1 - \\mathbf{A}_2 - \\cdots - \\mathbf{A}_p)^{-1} \\)
- C) \\( \\boldsymbol{\\Psi}_\\infty = \\mathbf{I}_K \\)
- D) \\( \\boldsymbol{\\Psi}_\\infty = \\mathbf{0} \\)
✓ Correct! The long-run multiplier is (I_K - A_1 - ... - A_p)^{-1}, which exists when the VAR is stable.
✗ Incorrect. The correct answer is B. This sums all impulse responses to infinity.
Chapter 7: Cointegration & VECM
Quiz content coming soon...
Chapter 8: Modern Extensions (ARFIMA, RF, LSTM)
Question 1: Hurst Exponent Interpretation
A time series has Hurst exponent $H = 0.8$. What does this indicate?
- A) The series is a pure random walk
- B) The series has long memory and is persistent (trend-following)
- C) The series is anti-persistent (mean-reverting)
- D) The series is stationary I(0)
Correct! $H > 0.5$ indicates persistence/long memory. With $H = 0.8$, high values tend to follow high values (trend-following behavior).
Incorrect. The correct answer is B. $H = 0.5$ means random walk, $H > 0.5$ means persistence, $H < 0.5$ means anti-persistence.
Question 2: ARFIMA Parameter d
In the ARFIMA(p, d, q) model, the parameter $d$ can take values:
- A) Only integer values (0, 1, 2, ...)
- B) Only $d = 0$ or $d = 1$
- C) Any real value, including fractional
- D) Only negative values
Correct! ARFIMA allows fractional differencing: $0 < d < 0.5$ gives stationary long memory, $0.5 \leq d < 1$ gives non-stationary long memory. Relation to Hurst: $d = H - 0.5$.
Incorrect. The correct answer is C. Unlike ARIMA which uses integer $d$, ARFIMA allows any real value for fractional differencing.
Question 3: Long Memory in Finance
In which financial series is long memory most commonly documented?
- A) Stock prices
- B) Daily returns
- C) Volatility (squared returns)
- D) Trading volume
Correct! Returns are approximately memoryless ($H \approx 0.5$), but volatility shows strong long memory ($H \approx 0.7-0.9$). This is a key stylized fact in finance.
Incorrect. The correct answer is C. Volatility clustering creates long memory in squared returns, which is the basis for FIGARCH models.
Question 4: Feature Engineering for ML
To apply Random Forest to time series forecasting, we must create:
- A) Dummy variables for each observation
- B) Lag features and rolling statistics
- C) Fourier transforms of the series
- D) Only the first difference of the series
Correct! Feature engineering transforms time series into supervised learning: lag features ($y_{t-1}, y_{t-2}, ...$), rolling mean/std, and calendar features (day of week, etc.).
Incorrect. The correct answer is B. We need to create features that capture temporal patterns: lags, rolling statistics, and calendar variables.
Question 5: Time Series Cross-Validation
Why can't we use standard k-fold cross-validation for time series?
- A) It's too slow for long series
- B) It violates temporal order and causes data leakage
- C) It only works for classification
- D) It requires too much data
Correct! Standard k-fold shuffles data randomly, training on future data and testing on past. This creates data leakage and overestimates performance. Use TimeSeriesSplit instead!
Incorrect. The correct answer is B. Time series must preserve temporal order; use walk-forward validation (TimeSeriesSplit).
Question 6: LSTM Advantage
What is the main advantage of LSTM over simple RNNs?
- A) It's faster to train
- B) It solves the vanishing/exploding gradient problem
- C) It requires less data
- D) It's easier to interpret
Correct! LSTM's cell state and gates (forget, input, output) allow gradients to flow through long sequences without vanishing, enabling learning of long-range dependencies.
Incorrect. The correct answer is B. The cell state acts as an "information highway" that prevents gradient degradation.
Question 7: Data Normalization
Before training an LSTM, data should be:
- A) Log-transformed
- B) Normalized/scaled to [0,1] or [-1,1]
- C) Differenced twice
- D) Converted to integers
Correct! Activation functions (sigmoid, tanh) work in limited ranges. Normalization ensures faster convergence and numerical stability. Important: fit scaler on train data only!
Incorrect. The correct answer is B. Use MinMaxScaler or StandardScaler before training neural networks.
Question 8: LSTM Hyperparameters
Which is NOT a typical LSTM hyperparameter?
- A) Number of units (neurons) per layer
- B) Input sequence length
- C) Learning rate
- D) Differencing parameter $d$
Correct! The differencing parameter $d$ is specific to ARFIMA, not LSTM. LSTM hyperparameters include architecture (layers, units), sequence length, learning rate, batch size, dropout.
Incorrect. The correct answer is D. Parameter $d$ belongs to ARFIMA; it's not used in neural network models.
Question 9: Feature Importance
Feature importance in Random Forest for time series helps us:
- A) Eliminate all low-importance variables
- B) Identify which lags and features are most predictive
- C) Determine Granger causality
- D) Calculate confidence intervals
Correct! Feature importance reveals which lags and engineered features matter most. Caution: it shows predictive power, not causality. Correlated features may share importance.
Incorrect. The correct answer is B. Feature importance identifies predictive features but does not prove causality.
Question 10: Model Selection
When comparing ARFIMA, Random Forest, and LSTM for forecasting:
- A) LSTM always wins because it's deep learning
- B) ARFIMA is always best for financial data
- C) The best model depends on data characteristics and requirements
- D) Random Forest can't be used for time series
Correct! ARFIMA: long memory, interpretability. RF: nonlinear relations, feature importance. LSTM: complex patterns, long sequences. Always validate with time series CV!
Incorrect. The correct answer is C. No single model dominates; choose based on data characteristics, interpretability needs, and available data.
Question 11: ACF Decay Pattern
Long memory processes have ACF that decays:
- A) Exponentially fast
- B) Hyperbolically (slowly)
- C) Linearly
- D) Immediately to zero
Correct! Long memory: $\rho(k) \sim k^{2d-1}$ (hyperbolic decay). Short memory (ARMA): exponential decay. This slow decay is the defining characteristic.
Incorrect. The correct answer is B. Long memory processes have slowly decaying (hyperbolic) autocorrelations, unlike ARMA which decays exponentially.
Question 12: ARFIMA Stationarity
An ARFIMA process with $d = 0.3$ is:
- A) Non-stationary
- B) Stationary with long memory
- C) Stationary with short memory
- D) A unit root process
Correct! For $0 < d < 0.5$, the process is stationary but has long memory. For $d \geq 0.5$, it becomes non-stationary.
Incorrect. The correct answer is B. With $d = 0.3$ (between 0 and 0.5), the process is stationary but exhibits long memory.
Question 13: Random Forest Ensemble
Random Forest reduces overfitting compared to a single decision tree by:
- A) Using deeper trees
- B) Averaging predictions from many trees trained on bootstrap samples
- C) Using fewer features
- D) Training on the full dataset only
Correct! Bagging (bootstrap aggregating) + random feature selection creates diversity. Averaging many trees reduces variance without increasing bias.
Incorrect. The correct answer is B. Random Forest uses ensemble averaging of trees trained on bootstrap samples with random feature subsets.
Question 14: Data Leakage
Which is an example of data leakage in time series ML?
- A) Using lag features
- B) Scaling data using statistics from the entire dataset (train+test)
- C) Using rolling window features
- D) Splitting data chronologically
Correct! Scaling with test data statistics leaks future information. Always fit scaler on training data only, then transform both train and test.
Incorrect. The correct answer is B. Using full dataset statistics for scaling incorporates future information into training.
Question 15: LSTM Gates
Which gate in LSTM decides what information to discard from the cell state?
- A) Input gate
- B) Forget gate
- C) Output gate
- D) Update gate
Correct! Forget gate: $f_t = \sigma(W_f[h_{t-1}, x_t] + b_f)$. Output close to 0 means "forget", close to 1 means "keep".
Incorrect. The correct answer is B. The forget gate controls what information from the previous cell state to discard.
Question 16: Dropout Regularization
Dropout in LSTM helps to:
- A) Speed up training
- B) Prevent overfitting by randomly zeroing neurons
- C) Increase model capacity
- D) Handle missing data
Correct! Dropout randomly sets neurons to zero during training, forcing the network to learn redundant representations. This reduces overfitting.
Incorrect. The correct answer is B. Dropout is a regularization technique that prevents overfitting.
Question 17: Early Stopping
Early stopping monitors validation loss to:
- A) Speed up computation
- B) Stop training when the model starts overfitting
- C) Select the best features
- D) Adjust the learning rate
Correct! When validation loss stops improving (or starts increasing), training stops. This prevents overfitting to training data.
Incorrect. The correct answer is B. Early stopping halts training when validation performance degrades, indicating overfitting.
Question 18: Sequence Length
Choosing the LSTM input sequence length (lookback) involves:
- A) Always using the maximum available data
- B) Balancing memory capacity vs. computational cost
- C) Using exactly 30 time steps
- D) Matching the seasonal period only
Correct! Longer sequences capture more history but increase computation and may not help if dependencies are short. Use domain knowledge and experimentation.
Incorrect. The correct answer is B. Sequence length is a hyperparameter that should be tuned based on the data's temporal dependencies.
Question 19: R/S Analysis
The R/S (Rescaled Range) statistic is used to estimate:
- A) The ARIMA order
- B) The Hurst exponent
- C) The GARCH parameters
- D) The seasonal period
Correct! R/S analysis: $E[R/S] \sim cn^H$. Plot $\log(R/S)$ vs $\log(n)$; slope estimates $H$. Developed by Hurst for Nile River floods.
Incorrect. The correct answer is B. The R/S statistic is a classical method to estimate the Hurst exponent.
Question 20: Direction Accuracy
Direction accuracy measures:
- A) The magnitude of forecast errors
- B) How often the model correctly predicts up/down movements
- C) The correlation between forecast and actual
- D) The variance of forecast errors
Correct! Direction accuracy = % of times sign($\Delta \hat{y}$) = sign($\Delta y$). Important for trading strategies even if RMSE is high.
Incorrect. The correct answer is B. Direction accuracy measures the proportion of correctly predicted directional moves.
Chapter 9: Prophet & TBATS for Multiple Seasonality
Question 1: Multiple Seasonality Challenge
Why can't standard SARIMA handle hourly electricity demand data?
- A) SARIMA can only handle monthly data
- B) SARIMA allows only one seasonal period ($m$ parameter)
- C) SARIMA doesn't support trend components
- D) SARIMA requires normally distributed data
Correct! Hourly data has daily (24h), weekly (168h), and annual (8760h) patterns. SARIMA's single $m$ parameter cannot capture all these simultaneously.
Incorrect. The correct answer is B. SARIMA only allows one seasonal period, making it unsuitable for multiple seasonality.
Question 2: TBATS Acronym
What does TBATS stand for?
- A) Trend, Baseline, ARMA, Transform, Seasonal
- B) Trigonometric, Box-Cox, ARMA, Trend, Seasonal
- C) Time-Based Automatic Time Series
- D) Temporal Bayesian Adaptive Trend System
Correct! TBATS: Trigonometric (Fourier terms), Box-Cox (variance stabilization), ARMA (errors), Trend (damped local), Seasonal (multiple periods).
Incorrect. The correct answer is B. Each letter represents a component of the model.
Question 3: Fourier Terms
In TBATS, increasing the number of Fourier harmonics ($K$) for a seasonal pattern:
- A) Always improves forecast accuracy
- B) Allows more flexible (complex) seasonal shapes
- C) Reduces the model complexity
- D) Eliminates the need for Box-Cox transformation
Correct! More harmonics = more flexibility but also more parameters (risk of overfitting). Maximum useful $K = m/2$.
Incorrect. The correct answer is B. More Fourier terms allow more complex seasonal patterns.
Question 4: Prophet Decomposition
Prophet decomposes a time series into which components?
- A) AR, MA, and seasonal components
- B) Trend, seasonality, holidays, and error
- C) Mean, variance, and autocorrelation
- D) Level, slope, and curvature
Correct! Prophet: $y(t) = g(t) + s(t) + h(t) + \varepsilon_t$ where $g$ = trend, $s$ = seasonality, $h$ = holidays.
Incorrect. The correct answer is B. Prophet uses an additive model with trend, seasonality, holidays, and error.
Question 5: Prophet vs TBATS
When would you choose Prophet over TBATS?
- A) When you need automatic model selection
- B) When you have known holidays and changepoints to incorporate
- C) When you need the most parsimonious model
- D) When your data has no trend
Correct! Prophet excels at incorporating domain knowledge: holidays, events, known changepoints. TBATS is better for automatic modeling.
Incorrect. The correct answer is B. Prophet's strength is easy integration of domain knowledge.
Question 6: Seasonality Mode
For retail sales where December sales are 3x the monthly average, which seasonality mode is more appropriate in Prophet?
- A) Additive seasonality
- B) Multiplicative seasonality
- C) Both work equally well
- D) Neither---use ARIMA instead
Correct! When seasonal amplitude scales with level, use multiplicative. Additive: constant effect. Multiplicative: proportional effect.
Incorrect. The correct answer is B. Multiplicative seasonality is appropriate when seasonal effects are proportional to the level.
Question 7: Prophet Changepoints
In Prophet, changepoints allow the model to:
- A) Change the seasonal period automatically
- B) Adjust the trend slope at specific points in time
- C) Switch between additive and multiplicative modes
- D) Detect and remove outliers
Correct! Changepoints allow piecewise linear trends with different slopes. Prophet can detect them automatically or you can specify manually.
Incorrect. The correct answer is B. Changepoints adjust trend slopes at specific time points.
Question 8: Model Selection
You have daily call center data with weekly seasonality only. Which model is most appropriate?
- A) TBATS (designed for multiple seasonality)
- B) Prophet (handles any seasonality well)
- C) Standard SARIMA (simpler and sufficient)
- D) LSTM neural network (most flexible)
Correct! Principle of parsimony: use the simplest model that fits. With only weekly seasonality ($m=7$), SARIMA works fine.
Incorrect. The correct answer is C. For single seasonality, SARIMA is often sufficient.
Question 9: Prophet Uncertainty
Prophet generates prediction intervals by:
- A) Assuming normally distributed residuals
- B) Sampling from the posterior distribution of parameters
- C) Using bootstrap resampling of historical errors
- D) Applying a fixed multiplier to point forecasts
Correct! Prophet uses Bayesian estimation: MAP for point forecasts, MCMC/simulation for uncertainty intervals from both trend and observation noise.
Incorrect. The correct answer is B. Prophet's Bayesian approach samples from the posterior distribution.
Question 10: Energy Demand Forecasting
For hourly energy demand with daily, weekly, and annual patterns plus holidays, which approach is best?
- A) SARIMA with $m=24$
- B) TBATS with three seasonal periods
- C) Prophet with custom holidays
- D) Either TBATS or Prophet, depending on holiday importance
Correct! Both handle multiple seasonality. Prophet if holidays are crucial, TBATS for automatic selection. Often try both and compare via CV.
Incorrect. The correct answer is D. Both can work; choice depends on specific requirements.
Question 11: Box-Cox Transformation
In TBATS, the Box-Cox transformation is used to:
- A) Remove the trend
- B) Stabilize variance
- C) Remove seasonality
- D) Speed up computation
Correct! Box-Cox: $y^{(\lambda)} = (y^\lambda - 1)/\lambda$. Stabilizes variance when it increases with level. $\lambda=0$ gives log transform.
Incorrect. The correct answer is B. Box-Cox transformation stabilizes variance.
Question 12: Prophet Default Trend
In Prophet, the default trend type is:
- A) Logistic growth
- B) Piecewise linear
- C) Exponential
- D) Polynomial
Correct! Default is piecewise linear with automatic changepoint detection. Logistic growth must be explicitly specified for saturating trends.
Incorrect. The correct answer is B. Prophet's default trend is piecewise linear.
Question 13: Fourier Parameters
For weekly seasonality ($m=7$) with $K=3$ Fourier terms, how many parameters are added?
- A) 3
- B) 6
- C) 7
- D) 14
Correct! Each harmonic adds 2 parameters (sine and cosine coefficients): $2 \times K = 2 \times 3 = 6$.
Incorrect. The correct answer is B. Each Fourier term has both sine and cosine components.
Question 14: Missing Data
Which statement about missing data handling is correct?
- A) SARIMA handles missing data better than Prophet
- B) Prophet handles missing data and irregular timestamps gracefully
- C) TBATS automatically imputes missing values
- D) All methods require complete data
Correct! Prophet can handle missing data and irregular timestamps. SARIMA/TBATS typically require equally spaced complete data.
Incorrect. The correct answer is B. Prophet's design handles missing data well.
Question 15: Holiday Effects
Prophet's holiday effects are modeled as:
- A) Multiplicative adjustments to trend
- B) Additive effects with optional windows
- C) Separate ARIMA components
- D) Fourier terms at holiday frequencies
Correct! Holidays are modeled as indicator variables with additive effects. Lower/upper windows capture effects on adjacent days.
Incorrect. The correct answer is B. Holiday effects are additive with configurable windows.
Question 16: Overfitting with Fourier
A symptom of overfitting with too many Fourier terms is:
- A) Smooth, realistic seasonal patterns
- B) Jagged seasonality and poor out-of-sample performance
- C) Faster model fitting
- D) Better handling of outliers
Correct! Too many harmonics cause jagged/unrealistic seasonal shapes and oscillating forecasts. Use CV to select optimal $K$.
Incorrect. The correct answer is B. Overfitting shows as jagged patterns and poor out-of-sample accuracy.
Question 17: TBATS Selection
TBATS automatically selects the number of Fourier terms using:
- A) Cross-validation
- B) AIC (Akaike Information Criterion)
- C) Visual inspection
- D) Fixed rules based on frequency
Correct! TBATS uses AIC to automatically select the optimal number of Fourier terms for each seasonal period.
Incorrect. The correct answer is B. AIC balances fit and complexity in TBATS.
Question 18: Prophet Cross-Validation
Prophet's built-in cross_validation function uses:
- A) Standard k-fold CV
- B) Rolling origin (time series) CV
- C) Leave-one-out CV
- D) Random holdout
Correct! Prophet uses rolling origin (simulated historical forecasts) with configurable initial, period, and horizon parameters.
Incorrect. The correct answer is B. Prophet properly handles temporal ordering in CV.
Question 19: Changepoint Prior
In Prophet, increasing changepoint_prior_scale makes the trend:
- A) Smoother and more stable
- B) More flexible and responsive to changes
- C) Linear without changepoints
- D) Logistic instead of linear
Correct! Higher prior scale = more flexibility = more changepoints detected. Lower = smoother trend. Default is 0.05.
Incorrect. The correct answer is B. Higher changepoint_prior_scale allows more trend flexibility.
Question 20: Model Interpretability
A key advantage of Prophet over black-box ML models is:
- A) Always higher accuracy
- B) Interpretable component decomposition (trend, seasonality, holidays)
- C) Faster training time
- D) No hyperparameters to tune
Correct! Prophet's decomposition lets analysts understand and explain forecasts: which component drives predictions. This is valuable for business decisions.
Incorrect. The correct answer is B. Prophet provides interpretable component decomposition.
Chapter 10: Review - From Data to Forecast
Question 1: Analysis Workflow
What is the correct order of the time series analysis workflow?
- A) Model fitting, Data exploration, Diagnostics, Forecasting
- B) Data exploration, Stationarity testing, Model selection, Diagnostics, Forecasting
- C) Forecasting, Data exploration, Model fitting, Diagnostics
- D) Diagnostics, Model selection, Data exploration, Forecasting
Correct! The systematic workflow: Explore data → Test stationarity → Select model → Fit → Diagnose → Forecast.
Incorrect. The correct answer is B. Always start with data exploration before model fitting.
Question 2: RMSE vs MAE
If RMSE is much larger than MAE, this suggests:
- A) The model is overfitting
- B) There are some large outlying errors
- C) The model is underfitting
- D) The data is stationary
Correct! RMSE penalizes large errors more (squared). A large RMSE/MAE ratio indicates some very large forecast errors.
Incorrect. The correct answer is B. RMSE is more sensitive to large errors than MAE.
Question 3: MAPE Limitation
MAPE is problematic when:
- A) The data has a trend
- B) Actual values are close to or equal to zero
- C) The forecast horizon is long
- D) Multiple models are being compared
Correct! MAPE = $\frac{100}{n}\sum|\frac{y_t - \hat{y}_t}{y_t}|$. Division by small $y_t$ causes extreme or undefined values.
Incorrect. The correct answer is B. MAPE divides by actual values, causing issues near zero.
Question 4: Volatility Clustering
If S&P 500 returns show periods of high volatility followed by high volatility, you should consider:
- A) ARIMA with higher order
- B) Exponential smoothing
- C) GARCH model for variance
- D) Seasonal differencing
Correct! Volatility clustering is a key feature of financial returns. GARCH: $\sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2$.
Incorrect. The correct answer is C. GARCH models capture time-varying volatility.
Question 5: ADF vs KPSS
If ADF test fails to reject and KPSS rejects, the series is likely:
- A) Stationary
- B) Non-stationary
- C) Trend-stationary
- D) White noise
Correct! ADF H0: unit root. KPSS H0: stationary. Both suggest non-stationarity when ADF doesn't reject and KPSS rejects.
Incorrect. The correct answer is B. Use both tests together for confirmation.
Question 6: Multiplicative Seasonality
Air passengers data shows increasing seasonal amplitude over time. Which decomposition is appropriate?
- A) Additive: $Y_t = T_t + S_t + R_t$
- B) Multiplicative: $Y_t = T_t \times S_t \times R_t$
- C) Both work equally well
- D) Neither---use differencing instead
Correct! When seasonal amplitude scales with level, use multiplicative. Log transform converts to additive.
Incorrect. The correct answer is B. Multiplicative for proportional seasonal effects.
Question 7: SARIMA for Monthly Data
For monthly data with yearly seasonality, the seasonal period $m$ in SARIMA is:
- A) 4
- B) 7
- C) 12
- D) 52
Correct! Monthly data with yearly pattern: $m=12$. Quarterly: $m=4$. Weekly: $m=52$. Daily with weekly pattern: $m=7$.
Incorrect. The correct answer is C. Monthly data has 12 observations per year.
Question 8: Model Comparison
When comparing SARIMA and Prophet forecasts, which metric is scale-independent?
- A) RMSE
- B) MAE
- C) MAPE
- D) MSE
Correct! MAPE expresses error as percentage. Useful for comparing across different scales/datasets.
Incorrect. The correct answer is C. MAPE is percentage-based and scale-independent.
Question 9: Structural Breaks
US Retail Sales experienced a structural break during COVID-19. Prophet handles this via:
- A) Automatic differencing
- B) Changepoint detection in the trend
- C) Seasonal adjustment
- D) GARCH modeling
Correct! Prophet's changepoints allow the trend slope to change at specific points, capturing structural breaks like COVID-19.
Incorrect. The correct answer is B. Changepoints model regime changes in trend.
Question 10: Ljung-Box Test
After fitting an ARIMA model, the Ljung-Box test on residuals tests for:
- A) Normality
- B) Remaining autocorrelation
- C) Heteroscedasticity
- D) Stationarity
Correct! Ljung-Box tests if residuals are white noise (no autocorrelation). Reject = model inadequate.
Incorrect. The correct answer is B. It tests if residual autocorrelations are jointly zero.
Question 11: ACF/PACF Interpretation
ACF decays exponentially and PACF cuts off after lag 2. This suggests:
- A) MA(2)
- B) AR(2)
- C) ARMA(2,2)
- D) Random walk
Correct! PACF cutoff → AR order. ACF cutoff → MA order. Decaying ACF + PACF cutoff at 2 = AR(2).
Incorrect. The correct answer is B. PACF determines AR order.
Question 12: Log Returns
For S&P 500, we use log returns $r_t = \ln(P_t/P_{t-1})$ instead of prices because:
- A) Prices are always stationary
- B) Returns are approximately stationary; prices are not
- C) Log returns are easier to compute
- D) Returns have stronger autocorrelation
Correct! Prices follow random walk (non-stationary). Returns are (approximately) stationary and suitable for ARIMA/GARCH.
Incorrect. The correct answer is B. Stationarity is required for most time series models.
Question 13: AIC vs BIC
Compared to AIC, BIC typically selects:
- A) More complex models
- B) Simpler (more parsimonious) models
- C) Identical models to AIC
- D) Models with better in-sample fit
Correct! BIC penalizes parameters more heavily: $\text{BIC} = -2\ln L + k\ln n$ vs $\text{AIC} = -2\ln L + 2k$.
Incorrect. The correct answer is B. BIC's stronger penalty favors simpler models.
Question 14: Cross-Validation
For time series cross-validation, we use:
- A) Random k-fold CV
- B) Leave-one-out CV
- C) Rolling origin (expanding window) CV
- D) Stratified CV
Correct! Time series CV must respect temporal order. Train on past, validate on future. Never use future data to predict past.
Incorrect. The correct answer is C. Random splits violate temporal ordering.
Question 15: Multiple Seasonality
Hourly data with daily, weekly, and yearly patterns is best handled by:
- A) SARIMA with $m=24$
- B) Simple exponential smoothing
- C) TBATS or Prophet
- D) ARIMA with differencing
Correct! SARIMA allows only one seasonal period. TBATS and Prophet can handle multiple seasonalities.
Incorrect. The correct answer is C. Multiple seasonality requires specialized models.
Question 16: GARCH Persistence
In GARCH(1,1), high volatility persistence means $\alpha + \beta$ is:
- A) Close to 0
- B) Close to 1
- C) Greater than 1
- D) Negative
Correct! $\alpha + \beta$ measures persistence. Close to 1 = shocks persist long. Must be $< 1$ for stationarity.
Incorrect. The correct answer is B. High persistence means volatility shocks decay slowly.
Question 17: Forecast Uncertainty
As forecast horizon increases, prediction intervals typically:
- A) Narrow
- B) Stay constant
- C) Widen
- D) Oscillate
Correct! Uncertainty accumulates over time. Forecast variance increases with horizon. Long-term forecasts are inherently less precise.
Incorrect. The correct answer is C. Uncertainty grows with forecast horizon.
Question 18: Direction Accuracy
A model with 45% direction accuracy for stock returns is:
- A) Better than random guessing
- B) Worse than random guessing
- C) Optimal for trading
- D) Statistically significant
Correct! Random guessing = 50%. 45% is worse than flipping a coin. Financial returns are notoriously difficult to predict.
Incorrect. The correct answer is B. Below 50% is worse than random.
Question 19: Model Selection Principle
The principle of parsimony suggests choosing:
- A) The model with the best in-sample fit
- B) The simplest model that adequately fits the data
- C) The most complex model available
- D) The model with the most parameters
Correct! Occam's razor: simpler models generalize better. Complex models overfit. Balance fit vs complexity.
Incorrect. The correct answer is B. Parsimony prefers simpler adequate models.
Question 20: Complete Workflow Summary
Which statement best summarizes the time series analysis workflow?
- A) Fit the most complex model first, then simplify
- B) Start with visualization, test stationarity, fit models, validate on held-out data
- C) Use the same model for all datasets
- D) Skip diagnostics if in-sample fit is good
Correct! Systematic workflow: Explore → Transform → Model → Diagnose → Validate → Forecast. Data characteristics guide model choice.
Incorrect. The correct answer is B. A systematic, data-driven approach is essential.
Chapter 11: LLMs and Foundation Models for Time Series
Question 1: Self-Attention Mechanism
What is the formula for the Scaled Dot-Product Attention in the Transformer (Vaswani et al., 2017)?
- A) $\text{Attention}(Q,K,V) = \text{softmax}(QK^T) \cdot V$
- B) $\text{Attention}(Q,K,V) = \text{softmax}\!\left(\frac{QK^T}{\sqrt{d_k}}\right) V$
- C) $\text{Attention}(Q,K,V) = \sigma(QK^T + b) \cdot V$
- D) $\text{Attention}(Q,K,V) = Q \cdot \text{softmax}(K^T V)$
Correct! The scaling factor $\sqrt{d_k}$ prevents softmax saturation for large key dimensions, ensuring stable gradients during training.
Incorrect. The correct answer is B. Scaling by $\sqrt{d_k}$ is essential — without it, the dot product grows with dimension and softmax saturates.
Question 2: Architectural Paradigms
What type of Transformer architecture does Chronos (Amazon) use?
- A) Decoder-only (GPT-style)
- B) Encoder-decoder (T5)
- C) Encoder-only (BERT-style)
- D) Recurrent network (LSTM)
Correct! Chronos uses the T5 encoder-decoder backbone with no architectural changes — the innovation is the quantization-based tokenization of time series values.
Incorrect. The correct answer is B. Chronos uses the T5 encoder-decoder architecture, treating forecasting as language modeling over quantized tokens.
Question 3: Patching Tokenization
If a time series has 512 time steps and we use patch size $P = 32$, how many tokens result?
- A) 512 tokens
- B) 16 tokens
- C) 32 tokens
- D) 256 tokens
Correct! Patching groups $P$ consecutive points into a single token: $512 / 32 = 16$ tokens. This reduces attention complexity from $O(T^2)$ to $O((T/P)^2)$.
Incorrect. The correct answer is B. $512 \div 32 = 16$ tokens. Patching dramatically reduces sequence length.
Question 4: Zero-Shot vs Fine-Tuning
When is zero-shot forecasting preferable to fine-tuning?
- A) When you have millions of observations from the target domain
- B) When data is limited, rapid prototyping is needed, or cross-domain deployment is required
- C) When causal interpretability is needed
- D) When the series has fewer than 10 observations
Correct! Zero-shot is ideal for cold-start, rapid prototyping, and multi-series deployment without individual training. Surprising finding: Chronos zero-shot outperforms task-specific models (DeepAR, N-BEATS) trained on the target data.
Incorrect. The correct answer is B. Zero-shot excels with limited data, rapid prototyping, and cross-domain deployment.
Question 5: Chronos — Tokenization
How does Chronos convert continuous time series values into discrete tokens?
- A) Direct embedding of raw values
- B) Mean-scale normalization followed by quantization into 4096 bins via normal CDF
- C) Differencing the series and one-hot encoding
- D) Converting the series to text and using the standard GPT tokenizer
Correct! Chronos pipeline: normalize $\tilde{x}_t = x_t / |\bar{x}|$, quantize into 4096 bins via $\Phi$ (normal CDF), then model as a sequence of categorical tokens with T5.
Incorrect. The correct answer is B. Chronos normalizes via mean-scale and quantizes into 4096 bins, then treats forecasting as language modeling.
Question 6: LoRA (Low-Rank Adaptation)
What percentage of parameters are trainable in LoRA fine-tuning?
- A) 100% — all parameters are updated
- B) 0.1–1% — only small low-rank matrices ($W' = W + BA$, $r \ll d$)
- C) 50% — half of the network layers
- D) 10–20% — only the last layers
Correct! LoRA (Hu et al., 2022) freezes base weights and adds small trainable matrices: $W' = W + \Delta W = W + BA$, where $B \in \mathbb{R}^{d \times r}$, $A \in \mathbb{R}^{r \times d}$, $r \ll d$. Similar performance to full fine-tuning at much lower cost.
Incorrect. The correct answer is B. LoRA trains only 0.1–1% of parameters by adding low-rank matrices.
Question 7: Moirai — Universal Design
What makes Moirai (Salesforce) unique among time series foundation models?
- A) It is the only model with API access
- B) It handles any number of variates, any frequency, and any prediction length
- C) It has the most parameters (710M)
- D) It uses exclusively synthetic data for pre-training
Correct! Moirai uses a BERT-style architecture with patching, variate IDs, and frequency embeddings — the closest to a "universal" time series model. Pre-trained on LOTSA (27B observations across 9 domains).
Incorrect. The correct answer is B. Moirai is the most flexible — handling any-variate, any-frequency, any-length.
Question 8: Foundation Model Limitations
Which of the following is NOT a major limitation of current time series foundation models?
- A) Most are univariate only
- B) They cannot produce probabilistic forecasts
- C) They lack causal reasoning
- D) They are black boxes that are hard to interpret
Correct! Foundation models actually excel at probabilistic forecasting: Chronos generates categorical distributions, Lag-Llama produces Student-t parameters, and Moirai outputs mixture distributions. The real limitations are: univariate-only, no causality, and black-box nature.
Incorrect. The correct answer is B. Foundation models actually produce excellent probabilistic forecasts — this is not a limitation.
Question 9: Lag-Llama — Output Distribution
What output distribution does Lag-Llama use instead of discrete token probabilities?
- A) Normal distribution $N(\mu, \sigma^2)$
- B) Student-t distribution with parameters $(\mu, \sigma, \nu)$
- C) Uniform distribution over an interval
- D) Categorical distribution over 4096 bins
Correct! Lag-Llama outputs Student-t parameters $(\mu, \sigma, \nu)$ — the heavy-tailed distribution is particularly suited for financial data with frequent extremes.
Incorrect. The correct answer is B. Lag-Llama uses the Student-t distribution, which naturally captures heavy tails in financial data.
Question 10: Scaling Hypothesis
What has been found about scaling laws for time series foundation models compared to NLP?
- A) Larger models are always significantly better
- B) Evidence is mixed — data quality matters more than parameter count
- C) Scaling works identically to NLP
- D) Smaller models are always superior
Correct! Unlike NLP, returns are diminishing: Chronos-Large (710M) $\approx$ Chronos-Base (200M). Time series have less diversity than text, and many patterns are well-described by AR + seasonality. Data quality > parameter count.
Incorrect. The correct answer is B. Scaling for time series differs from NLP — data quality matters more than model size.
Chapter 12: Spectral Analysis and Frequency-Domain Methods
Question 1: Discrete Fourier Transform
What does the Discrete Fourier Transform (DFT) of a time series measure?
- A) The linear trend of the series
- B) The decomposition of the series into sinusoidal components at different frequencies
- C) The number of missing observations
- D) The correlation between different series
Correct! The DFT decomposes the series into sinusoidal components: $d(\omega_j) = \frac{1}{\sqrt{T}} \sum_{t=1}^{T} x_t e^{-2\pi i \omega_j t}$, revealing amplitude and phase at each frequency.
Incorrect. The correct answer is B. The DFT decomposes the series into sinusoidal components at different frequencies.
Question 2: Nyquist Frequency
What happens when a signal contains frequencies above the Nyquist limit ($f_N = 1/2$)?
- A) The frequencies are detected correctly
- B) Aliasing occurs — high frequencies appear as spurious low frequencies
- C) The signal is automatically filtered
- D) The amplitude becomes zero
Correct! Aliasing occurs when true frequencies exceed the Nyquist limit — they "fold back" and appear as spurious low frequencies, corrupting the spectral analysis.
Incorrect. The correct answer is B. Aliasing causes frequencies above the Nyquist limit to appear as spurious low frequencies.
Question 3: Periodogram — Inconsistency
Why is the periodogram an inconsistent estimator of the spectral density?
- A) Bias increases with sample size
- B) Variance does NOT decrease as $T \to \infty$
- C) It cannot detect low frequencies
- D) It requires stationary data
Correct! The periodogram $I(\omega_j) \xrightarrow{d} f(\omega_j) \cdot \chi^2_2/2$ — variance remains constant regardless of $T$. This motivates smoothed estimators (Welch, multitaper).
Incorrect. The correct answer is B. The periodogram's variance does not decrease with $T$ — it remains noisy regardless of the amount of data.
Question 4: Welch Method
How does Welch's method reduce spectral estimation variance?
- A) It uses only the central half of the data
- B) It averages periodograms over overlapping windowed segments
- C) It applies differencing to the series
- D) It removes frequencies above a threshold
Correct! Welch's method splits the series into overlapping segments, applies a window (e.g., Hann), computes each segment's periodogram, and averages — reducing variance at the cost of frequency resolution.
Incorrect. The correct answer is B. Welch averages periodograms over overlapping windowed segments.
Question 5: Long Memory
How does long memory manifest in the frequency domain?
- A) The spectral density is constant (flat)
- B) The spectral density diverges at frequency zero: $f(\omega) \propto |\omega|^{-2d}$
- C) The spectrum has a single peak at the Nyquist frequency
- D) All autocorrelations are zero
Correct! Long memory (ARFIMA, $0 < d < 0.5$) produces spectral power that rises toward infinity at frequency zero, reflecting hyperbolically decaying autocorrelations.
Incorrect. The correct answer is B. Long memory manifests as spectral density divergence at frequency zero.
Question 6: Spectral Coherence
What does the squared coherence $C^2_{xy}(\omega)$ between two series measure?
- A) The total correlation between the series
- B) The linear association at each frequency, analogous to frequency-specific $R^2$
- C) The amplitude difference between the series
- D) The number of common cycles
Correct! $C^2_{xy}(\omega) = |f_{xy}(\omega)|^2 / [f_x(\omega) f_y(\omega)] \in [0,1]$ — it is a frequency-specific $R^2$, measuring linear association.
Incorrect. The correct answer is B. Coherence is a frequency-specific $R^2$ — it measures linear association at each individual frequency.
Question 7: Hodrick-Prescott Filter
What is the main limitation of the HP filter ($\lambda = 1600$ for quarterly data)?
- A) It cannot be applied to quarterly data
- B) It is a high-pass filter that leaks low-frequency noise
- C) It requires stationary data
- D) It only works for series with linear trends
Correct! HP is not an ideal band-pass — its transfer function rises gradually, allowing some unwanted low-frequency components through. BK and CF filters are closer to the ideal rectangular shape.
Incorrect. The correct answer is B. HP leaks low-frequency noise, as it is not an ideal band-pass filter.
Question 8: Wavelet vs Fourier Analysis
What is the main advantage of wavelet analysis over Fourier for non-stationary series?
- A) Wavelets have better frequency resolution
- B) Wavelets provide time-frequency representation — showing WHEN frequencies occur
- C) Wavelets require less data
- D) Wavelets automatically remove the trend
Correct! Fourier shows WHAT frequencies are present but not WHEN. Wavelets (scalogram) show both — essential for series where cycles appear, disappear, or shift over time.
Incorrect. The correct answer is B. Wavelets provide simultaneous time-frequency localization, essential for non-stationary series.
Question 9: Spectral Leakage
What causes spectral leakage and how is it reduced?
- A) Signal frequency is too high; reduced by subsampling
- B) Frequency doesn't fall exactly on the DFT grid; reduced by windowing (tapering, e.g. Hann)
- C) Series has too few observations; reduced by zero-padding
- D) Data contains missing values; reduced by interpolation
Correct! Leakage occurs when the true frequency doesn't coincide with a DFT bin. Windows (Hann, DPSS) attenuate sidelobes, concentrating energy at the correct frequency.
Incorrect. The correct answer is B. Spectral leakage is caused by frequency mismatch with the DFT grid and is reduced by tapering.
Question 10: Business Cycle
What is the standard NBER band for business cycles in quarterly data?
- A) 2–4 quarters (6 months – 1 year)
- B) 6–32 quarters (1.5–8 years)
- C) 40–100 quarters (10–25 years)
- D) 1–2 quarters (3–6 months)
Correct! The NBER business cycle band is 6–32 quarters (1.5–8 years). Band-pass filters (BK, CF) extract exactly these components.
Incorrect. The correct answer is B. The standard NBER business cycle is 6–32 quarters (1.5–8 years).
Chapter 13: LPPL Models for Bubble Detection
Question 1: Super-Exponential Growth
What is the main mathematical signature of a financial bubble?
- A) Price grows linearly in time
- B) Log-price grows faster than linearly (super-exponential growth)
- C) Volatility is constant
- D) Returns are normally distributed
Correct! In a bubble, $\ln P(t) = A + B(t_c - t)^m$ with $B < 0$ and $0 < m < 1$, producing super-exponential growth — the growth rate accelerates toward $t_c$.
Incorrect. The correct answer is B. Super-exponential growth in log-price is the key signature of a speculative bubble.
Question 2: Ising Model — Ordered Phase
In the financial analogy of the Ising model, what does the regime $T < T_c$ (ordered phase) correspond to?
- A) Efficient market with random trading
- B) Strong herding — bubble regime, one group dominates
- C) High-frequency trading with no directional bias
- D) Bear market with declining prices
Correct! Below $T_c$, spins align — most agents make the same decision (buy or sell). This represents herding and bubble formation.
Incorrect. The correct answer is B. Below $T_c$, spins are aligned (strong herding), corresponding to a bubble regime.
Question 3: Susceptibility at the Critical Point
Why does the susceptibility $\chi \to \infty$ at the critical point $T_c$ matter for financial markets?
- A) The market becomes insensitive to news
- B) The market is perfectly efficient
- C) Even a small perturbation can trigger a system-wide cascade
- D) Volatility drops to zero
Correct! Diverging susceptibility means the system's response to a perturbation becomes infinite. A minor piece of news can flip large clusters of traders simultaneously.
Incorrect. The correct answer is C. At the critical point, the market is maximally fragile — any small perturbation can trigger a cascade.
Question 4: Discrete Scale Invariance
What does discrete scale invariance (DSI) produce in the LPPL model?
- A) Constant exponential growth
- B) Log-periodic oscillations with a preferred scaling ratio $\lambda$
- C) Gaussian-distributed returns
- D) Zero correlation between returns
Correct! DSI produces log-periodic oscillations with $\lambda = e^{2\pi/\omega}$. Each hesitation cycle is $\lambda$ times shorter than the previous one.
Incorrect. The correct answer is B. Discrete scale invariance generates periodic oscillations in log-time, with a preferred scaling ratio $\lambda$.
Question 5: LPPL Equation — Parameter B
Why is the condition $B < 0$ essential in the LPPL model?
- A) It ensures the price decreases over time
- B) It ensures super-exponential price growth as $t \to t_c$
- C) It eliminates log-periodic oscillations
- D) It forces $t_c$ to be in the past
Correct! With $B < 0$, the term $B(t_c - t)^m$ becomes less negative as $t \to t_c$ (since $(t_c-t)^m \to 0$), so log-price rises with acceleration.
Incorrect. The correct answer is B. $B < 0$ ensures super-exponential price growth toward $t_c$ (a bubble, not deflation).
Question 6: Partial Linearization
What advantage does partial linearization (slaving) provide in LPPL estimation?
- A) It eliminates all 7 parameters from optimization
- B) It reduces the search from 7D to 3D $(t_c, m, \omega)$ + OLS for the rest
- C) It guarantees a unique global optimum
- D) It does not require real market data
Correct! For fixed $(t_c, m, \omega)$, the parameters $(A, B, C_1, C_2)$ enter linearly and are solved exactly by OLS. Only 3D nonlinear search remains.
Incorrect. The correct answer is B. Partial linearization reduces the problem from 7 to 3 nonlinear parameters, solving the rest via OLS.
Question 7: Filter Conditions
Why is the condition $0.1 \leq m \leq 0.9$ necessary for a valid bubble signal?
- A) It ensures growth is slower than exponential
- B) It ensures growth faster than exponential but not instantaneous
- C) It eliminates oscillations from the model
- D) It forces the price to mean-revert
Correct! With $0 < m < 1$, growth is super-exponential (faster than exponential). $m \geq 1$ would mean sub-exponential growth, and $m = 0$ an instantaneous singularity.
Incorrect. The correct answer is B. The condition $m \in (0.1, 0.9)$ ensures super-exponential but physically reasonable growth.
Question 8: LPPLS Confidence Indicator
How is the LPPLS Confidence Indicator (CI) constructed?
- A) By computing daily volatility
- B) By fitting LPPL over multiple windows and counting the fraction passing all 8 filter conditions
- C) By comparing the current price to a moving average
- D) By applying the ADF test on returns
Correct! The CI is computed by fitting LPPL over multiple time windows ending at the current date, then counting how many pass all 8 filters.
Incorrect. The correct answer is B. LPPLS CI = fraction of windows producing valid LPPL fits (8/8 conditions).
Question 9: Negative Control
Why is negative control testing (e.g., the COVID-19 crash) important for validating LPPL?
- A) It shows LPPL can predict any type of crash
- B) It confirms LPPL does NOT produce false alarms for exogenous (non-bubble) shocks
- C) It shows the market was efficient in 2020
- D) It validates that COVID-19 was an endogenous bubble
Correct! LPPL should fail on data without an endogenous bubble. The COVID crash was exogenous, and LPPL correctly produces no signal — validating the model's specificity.
Incorrect. The correct answer is B. Negative control shows LPPL correctly distinguishes endogenous bubbles from exogenous crashes.
Question 10: Phase Transition
What happens to the magnetization $|M|$ at the critical temperature $T_c$ in the 2D Ising model?
- A) $|M|$ remains at 1 (perfect order persists)
- B) $|M|$ drops continuously to 0 (second-order phase transition)
- C) $|M|$ jumps discontinuously from 1 to 0 (first-order transition)
- D) $|M|$ oscillates between 0 and 1
Correct! The 2D Ising model undergoes a continuous (second-order) phase transition at $T_c = 2/\ln(1+\sqrt{2}) \approx 2.269$. Magnetization vanishes as $|M| \sim (T_c - T)^{1/8}$ (Onsager, 1944).
Incorrect. The correct answer is B. It is a continuous (second-order) transition — $|M|$ drops gradually to zero.
Resources
Recommended Textbooks
- Hyndman, R.J. & Athanasopoulos, G. (2021). Forecasting: Principles and Practice, 3rd ed. OTexts (free online)
- Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press.
- Tsay, R.S. (2010). Analysis of Financial Time Series, 3rd ed. Wiley.
- Shumway, R.H. & Stoffer, D.S. (2017). Time Series Analysis and Its Applications, 4th ed. Springer.
Online Resources
Contact Information
Instructor
Prof. dr. Daniel Traian Pele
Department of Statistics and Econometrics
Location
Bucharest University of Economic Studies
Faculty of Cybernetics, Statistics and Economic Informatics