⏳Intro to Time Series Unit 10 – Multivariate Time Series Analysis

Key Concepts and Definitions

• Multivariate time series analysis involves studying multiple time-dependent variables simultaneously to understand their relationships and dynamics
• Time series data consists of observations recorded at regular intervals over time (hourly, daily, monthly)
• Stationarity assumes the statistical properties of a time series remain constant over time
  • Weak stationarity requires constant mean and variance
  • Strong stationarity requires the entire probability distribution to be time-invariant
• Cointegration occurs when two or more non-stationary time series have a linear combination that is stationary
• Granger causality determines if one time series is useful in forecasting another
• Impulse response functions measure the impact of a shock in one variable on the future values of other variables
• Variance decomposition quantifies the proportion of the forecast error variance in one variable explained by shocks in other variables

Multivariate Time Series Models

• Vector Autoregressive (VAR) models extend univariate autoregressive models to capture the dynamic relationships among multiple variables
  • Each variable is modeled as a linear function of its own past values and the past values of other variables
  • VAR models are useful for forecasting and analyzing the impact of shocks
• Vector Error Correction Models (VECM) are used when the time series are cointegrated
  • VECM incorporates both short-term dynamics and long-term equilibrium relationships
  • The error correction term represents the deviation from the long-run equilibrium
• Dynamic Factor Models (DFM) assume that a small number of unobserved factors drive the common dynamics of multiple time series
• State Space Models (SSM) represent the time series as a combination of unobserved state variables and observed measurements
  • Kalman filter is used for state estimation and forecasting in SSMs
• Multivariate GARCH models capture the time-varying volatility and covariance structure of multiple financial time series
• Structural VAR (SVAR) models impose economic theory-based restrictions on the relationships among variables

Data Preparation and Preprocessing

• Handling missing values through interpolation, imputation, or removal
• Dealing with outliers using robust statistical methods or intervention analysis
• Transforming variables to achieve stationarity (differencing, logarithmic transformation)
• Standardizing or normalizing variables to ensure comparability
• Selecting appropriate lag lengths based on information criteria (AIC, BIC)
• Testing for stationarity using unit root tests (Augmented Dickey-Fuller, Phillips-Perron)
• Checking for cointegration using Engle-Granger or Johansen tests
• Identifying seasonality and applying seasonal adjustment techniques (X-13ARIMA-SEATS, STL decomposition)

Model Estimation Techniques

• Ordinary Least Squares (OLS) estimation for VAR models
  • Minimizes the sum of squared residuals
  • Provides consistent and efficient estimates under certain assumptions
• Maximum Likelihood Estimation (MLE) for more complex models (VECM, SSM)
  • Maximizes the likelihood function to estimate model parameters
  • Requires distributional assumptions about the error terms
• Bayesian estimation using Markov Chain Monte Carlo (MCMC) methods
  • Incorporates prior information and updates it with observed data
  • Useful for high-dimensional models and parameter uncertainty assessment
• Generalized Method of Moments (GMM) estimation for models with endogenous variables or heteroscedasticity
• Nonlinear least squares estimation for models with nonlinear parameters
• Kalman filter and smoother for state estimation in SSMs

Forecasting Methods

• Recursive forecasting uses the estimated model to generate multi-step ahead forecasts
  • The forecasted values are used as inputs for subsequent forecasts
  • Suitable for short to medium-term forecasting horizons
• Direct forecasting estimates a separate model for each forecast horizon
  • Avoids the accumulation of errors in recursive forecasting
  • Useful for long-term forecasting and forecast comparison
• Forecast combination takes a weighted average of forecasts from multiple models
  • Improves forecast accuracy by leveraging the strengths of different models
  • Weights can be determined based on past performance or Bayesian model averaging
• Scenario analysis generates forecasts under different assumptions or policy interventions
• Probabilistic forecasting provides a range of possible future values with associated probabilities
• Rolling window forecasting updates the model estimates as new data becomes available

Model Diagnostics and Validation

• Residual analysis checks if the model assumptions are satisfied
  • Residuals should be uncorrelated, homoscedastic, and normally distributed
  • Tests for serial correlation (Durbin-Watson, Ljung-Box)
  • Tests for heteroscedasticity (White, Breusch-Pagan)
  • Tests for normality (Jarque-Bera, Shapiro-Wilk)
• Stability tests assess if the model parameters are constant over time
  • Chow test for structural breaks
  • CUSUM and CUSUMSQ tests for parameter stability
• Forecast evaluation measures the accuracy of out-of-sample forecasts
  • Mean Absolute Error (MAE), Root Mean Squared Error (RMSE)
  • Mean Absolute Percentage Error (MAPE), Theil's U statistic
• Cross-validation techniques (rolling window, k-fold) assess model performance on unseen data
• Diebold-Mariano test compares the forecast accuracy of two competing models

Applications and Case Studies

• Macroeconomic forecasting predicts key economic variables (GDP, inflation, unemployment)
  • Helps policymakers in decision-making and assessing the impact of shocks
  • Example: Forecasting the impact of monetary policy changes on economic growth
• Financial market analysis studies the interactions among stock prices, exchange rates, and interest rates
  • Useful for portfolio management and risk assessment
  • Example: Analyzing the spillover effects of stock market shocks across countries
• Energy demand forecasting predicts future energy consumption based on economic and demographic factors
  • Helps in planning energy production and infrastructure investments
  • Example: Forecasting the impact of electric vehicle adoption on electricity demand
• Environmental modeling analyzes the relationships among air pollution, weather conditions, and health outcomes
  • Supports the development of air quality management strategies
  • Example: Studying the impact of traffic emissions on respiratory diseases
• Marketing research investigates the dynamic effects of advertising and promotions on sales
  • Helps in optimizing marketing strategies and resource allocation
  • Example: Analyzing the effectiveness of multi-channel marketing campaigns

Advanced Topics and Extensions

• Bayesian Vector Autoregressive (BVAR) models incorporate prior information to improve forecasting performance
• Factor-Augmented VAR (FAVAR) models combine factor analysis with VAR to handle high-dimensional data
• Time-Varying Parameter VAR (TVP-VAR) models allow for time-varying coefficients and stochastic volatility
• Smooth Transition VAR (STVAR) models capture nonlinear regime-switching behavior
• Global VAR (GVAR) models analyze the interdependencies among multiple countries or regions
• Functional time series models deal with time series of curves or functions
• Copula-based models capture nonlinear dependence structures among variables
• Machine learning techniques (neural networks, random forests) for nonlinear modeling and forecasting