Forecasting Methods

• italic: Definition / About
• Bold: Property / Theorem
• Standard: Procedure / Example / Cheatsheet

Cheatsheets

• Forecasting Methods Cheatsheet

Basics

• Cross-section Data
• Time Series Data
• Pooled Data
• Stochastic Process
• Time Series Components
• Lag
• Univariate vs Multivariate Models

Stationarity and Moments

• Autocovariance and Autocorrelation
• Weakly Stationary
  • Theorem: Strict Stationary implies Weakly Stationary
• Strictly Stationary
  • Theorem: Strict Stationary implies Weakly Stationary
• Sample Autocorrelation (ACF)
• Correlogram

Parametric Models

• General Linear Process
  • Property: Stationarity Condition for General Linear Process

Autoregressive (AR) Processes

• AR(p) Process Model
  • Property: Stationarity Condition for AR(p)
  • Property: Yule-Walker Equations for AR(p)
  • Property: Variance of AR(p)
  • AR(1) Process Model
    • Property: Stationarity Condition for AR(1)
    • Property: ACF of AR(1)
    • Example: Explosive AR(1) Process
  • AR Characteristic Equation
• Random Walk
• Backshift Operator (B)

Moving Average (MA) Processes

• White Noise

• Invertible MA(q) Process

• MA(q) Process Model

  • MA(1) Process Model
    • Property: Bounds of MA(1) Autocorrelation
    • Property: Non-uniqueness of MA(1) Model
    • Property: Invertibility Condition for MA(1)
  • MA(2) Process Model

Mixed and Integrated Models (ARMA/ARIMA)

• Rationale for Non-Stationary Models
• ARMA(p,q) Process Model
  • ARMA(1,1) Process Model
    • Property: Variance and ACF of ARMA(1,1)
• ARIMA(p,d,q) Model
  • ARIMA(p,1,q) Formulation
  • Property: Characteristic Polynomial of ARIMA(p,1,q)
  • Constant Term in ARIMA
  • IMA(d,q) Model
    • IMA(1,1) Model
    • IMA(2,2) Model
  • ARI(p,d) Model
    • ARI(1,1) Model
    • Procedure: Determining Weights for ARI(1,1)

Smoothing Methods

• Smoothing Methods Overview
• Naive Method
• Averaging Method
• Moving Average Process (Stochastic)
• Single Moving Average (SMA)
• Double Moving Average (DMA)
• Single Exponential Smoothing (SES)
• Double Exponential Smoothing (Holt’s Linear Trend)
• Holt-Winter Seasonal Method

Strategy and Estimation

• Forecasting Model Accuracy Measures
• Model Selection via Data Splitting
• Box-Jenkins Strategy
• Estimating Constant Mean
• Least Squares for Linear Trend Estimation
• Least Squares for Quadratic Trend Estimation

Trends and Transformations

• Stochastic vs Deterministic Trend
• Linear and Quadratic Deterministic Trends
• Seasonal Average Model
• Procedure: Differencing to Achieve Stationarity
• Procedure: Log Transformation for Variance Stabilization
• Procedure: Percentage Changes Transformation

Forecasting

• Minimum Mean Square Error Forecast
• Deterministic Trends Forecast
• ARIMA Forecasting
• Prediction Limits
• Updating ARIMA Forecasts
• Forecasting Transformed Series

Examples

• Economic and Natural Time Series Examples

Stationarity Tests

• Procedure: Visual Stationarity Test
• Bartlett’s Test for ACF
• Box-Pierce Test
• Ljung-Box Test
• Box-Pierce vs Ljung-Box Comparison
• Dickey-Fuller Test
• Augmented Dickey-Fuller Test

Model Specification

• Partial Autocorrelation (PACF)
• PACF
• Extended Autocorrelation Function (EACF)
• AIC Criterion
• BIC Criterion

Parameter Estimation

• Method of Moments
• Parameter Estimation by Method of Moments
• Noise Variance Estimation
• Conditional Sum of Squares Function
• Conditional Least Squares
• Least Square Method
• Unconditional Sum-of-Squares Function
• Unconditional Least Squares
• Maximum Likelihood Method
• Property: Large Sample Properties of Parameter Estimates
• Canadian Hare Abundance Series

Volatility Models (ARCH/GARCH)

• Volatility Clustering
• Leverage Effect
• Principle of Parsimony in Econometric Modeling
• GARCH Relationship
• Log-Returns
• Continuously Compounded Return
• Conditional Variance
• ARCH(m) Model
  • Property: Unconditional Mean of ARCH
  • Property: Unconditional Variance of ARCH
  • Property: Kurtosis of ARCH (Heavy Tails)
• GARCH(m,s) Model
  • Property: Unconditional Variance of GARCH
  • Property: Kurtosis of GARCH (Heavy Tails)
  • Property: Volatility Persistence
• Standardized Residuals
• Building a Volatility Model
• Testing for ARCH Effects
• GARCH
• GARCH
• Forecasting with ARCH(m)
• Forecasting with GARCH(m,s)

Seasonal Models (SARIMA)

• Seasonal MA(Q) Model
• Seasonal AR(P) Model
• Multiplicative Seasonal ARMA Model
• Seasonal Differencing
• Seasonal ARIMA Model
• SARIMA Modeling for CO2 Levels