13.1 Regression with time series data

Regression with time series data presents unique challenges due to autocorrelation and non-stationarity. These issues can lead to biased estimates and spurious results if not properly addressed. Understanding the components of time series models is crucial for accurate analysis.

Techniques like differencing, detrending, and including lagged variables help tackle non-stationarity and autocorrelation. Proper model evaluation involves residual analysis, information criteria, and out-of-sample forecasting to ensure reliable predictions and insights from time series regression models.

Regression with Time Series Data

Challenges in time series regression

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• Time series data violates assumption of independent observations in traditional regression
  • Observations often correlated with past values (autocorrelation)
  • Ignoring this leads to biased and inefficient estimates (misleading conclusions, incorrect standard errors)
• Non-stationarity common in time series data
  • Mean, variance, and covariance may change over time (evolving data distribution)
  • Leads to spurious regression results if not addressed (misleading relationships, invalid inferences)
• Seasonality and trend components need to be accounted for
  • Failing to do so results in model misspecification and poor performance (biased coefficients, inaccurate predictions)

Components of time series models

• Trend component captures long-term direction of time series
  • Can be linear, polynomial, or nonlinear (increasing, decreasing, or complex patterns)
  • Modeled using time index or transformations (logarithmic, exponential)
• Seasonality component represents periodic patterns in data
  • Modeled using dummy variables or Fourier terms (sine and cosine functions)
  • Helps capture recurring patterns not explained by other factors (monthly sales, weather cycles)
• Exogenous variables are external factors influencing time series
  • Can be time-varying or constant (dynamic or static influences)
  • Examples: economic indicators (GDP, inflation), policy changes (regulations), or interventions (marketing campaigns)

Techniques for non-stationarity and autocorrelation

• Differencing used to remove non-stationarity in mean
  • First-order differencing: $\Delta y_t = y_t - y_{t-1}$
  • Higher-order differencing may be necessary for more complex non-stationarity (seasonal differences)
• Detrending removes trend component from time series
  • Done by subtracting estimated trend from original series (residual series)
  • Allows for modeling detrended series as stationary (mean-reverting process)
• Autocorrelation addressed using lagged dependent variables
  • Include past values of dependent variable as predictors (autoregressive terms)
  • Helps capture temporal dependence structure (short-term and long-term relationships)

Model Evaluation and Prediction

Evaluation of time series models

• Residual analysis crucial for assessing model adequacy
  • Residuals should be uncorrelated (white noise), homoscedastic (constant variance), and normally distributed
  • Durbin-Watson test checks for autocorrelation in residuals (values close to 2 indicate no autocorrelation)
• Information criteria (AIC, BIC) balance model fit and complexity
  • Lower values indicate better model performance (trade-off between goodness-of-fit and parsimony)
  • Used for model selection and comparison (choosing among competing models)
• Out-of-sample forecasting evaluates model's predictive ability
  1. Divide data into training and testing sets (hold-out validation)
  2. Assess forecast accuracy using metrics like RMSE (root mean squared error), MAE (mean absolute error), or MAPE (mean absolute percentage error)
• Rolling window cross-validation accounts for temporal structure
  • Iteratively train and test model on different subsets of data (moving window approach)
  • Helps assess model's robustness and stability over time (performance across different periods)