Advanced Time Series Models

In advanced time series we move beyond the basics to apply the Box-Jenkins methodology for ARIMA and SARIMA, systematically identifying model orders and addressing seasonal patterns. Diagnostic steps ensure residuals show no unmodeled structure. The section concludes with time series regression, where external variables enter forecasts while solutions for autocorrelated residuals are introduced. This equips learners to create more robust models that capture real-world complexities.

ARIMA MODELING

Learning Objectives

• Recall the Box-Jenkins methodology and ARIMA(\(p, d, q\)) models

• Identify AR and MA terms using autocorrelation (ACF) and partial autocorrelation (PACF)

• Estimate ARIMA parameters (manually/automated)

• Diagnose residuals and produce multi-step forecasts

Indicative Content

• Box-Jenkins (ARIMA) Overview

  • AR(\(p\)), I(\(d\)), MA(\(q\)) components

  • Iterative approach: stationarity → identification → estimation → diagnostics → forecasting

• Model Identification

  • ACF for MA(\(q\)), PACF for AR(\(p\))

• Parameter Estimation

  • Trying different (p, d, q) sets, minimizing AIC or BIC

• Diagnostic Checking

  • Analyzing residual plots for white noise

  • Checking autocorrelation in residuals

• Forecasting

  • Generating forecasts once the model is validated

SEASONAL ARIMA MODELING

Learning Objectives

• Include seasonal effects into ARIMA (SARIMA)

• Differentiate non-seasonal differencing (\(d\)) from seasonal differencing (\(D\))

• Identify seasonal AR(\(P\)) and MA(\(Q\)) from ACF/PACF

• Validate and forecast using a seasonal ARIMA model

Indicative Content

• SARIMA Notation

  • \(\text{ARIMA}(p, d, q)(P, D, Q)_S\), with \(S\) as the seasonal period

• Seasonal Differencing & Stationarity

  • Checking ACF/PACF at seasonal lags

• Model Identification

  • Determining \(p, q\) and \(P, Q\) based on correlation patterns

• Parameter Estimation

  • Choosing best (p, d, q)(P, D, Q) with minimal AIC or BIC

• Diagnostic Checks & Forecasting

  • Ensuring residuals are free from seasonal autocorrelation

  • Producing forecasts with seasonal pattern recognition

TIME SERIES REGRESSION

Learning Objectives

• Integrate external covariates in time series models

• Recognize how autocorrelated errors break standard regression assumptions

• Apply the Durbin-Watson test to detect first-order autocorrelation

• Use or adapt specialized techniques (e.g., ARIMAX) to handle autocorrelation

Indicative Content

• Multiple Linear Regression on Time Series

  • Observations with time-order plus covariates

• Autocorrelation of Errors

  • Bias in parameter estimates and standard errors

• Durbin-Watson Test

  • Checking for first-order autocorrelation in residuals

• Remedial Approaches

  • Feasible generalized least squares, maximum likelihood

  • ARIMAX or specialized regressions

• Predictive Use Cases

  • Combining domain features with time-lagged data

TOOLS AND METHODOLOGIES (ADVANCED TIME SERIES MODELS)

• Python Libraries

  • Common modules for ARIMA/SARIMA, time-series regressions (e.g., statsmodels)

  • Residual analysis and forecast functions

• ARIMA & SARIMA

  • Applying Box-Jenkins steps: stationarity, identification, estimation, diagnostics

  • Handling seasonal terms (\(P, D, Q\)) for monthly or quarterly data

• Time Series Regression

  • Managing autocorrelated errors

  • Durbin-Watson as a key diagnostic

• Model Validation & Forecasting

  • Checking leftover patterns in residuals

  • Multi-step or seasonal forecast generation

  • Integrating external factors for enriched models