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Time Series / Regression Models for General Time Series Data

Regression Models for General Time Series Data

Notes 5 Time Series Regression Model
Quick concept

A regression model for general time series data explains a series using a mix of time-based structure and external predictors: a trend in time, seasonal indicators, explanatory series such as temperature, and sometimes lagged y or lagged x. The same residual discipline as the rest of this unit still applies: fit, check leftovers, then choose OLS, GLS or WLS as the error pattern suggests.

Notes

Regression Models for General Time Series Data

Definition

A regression model for general time series data explains a series using a mix of time-based structure and external predictors: a trend in time, seasonal indicators, explanatory series such as temperature, and sometimes lagged y or lagged x. The same residual discipline as the rest of this unit still applies: fit, check leftovers, then choose OLS, GLS or WLS as the error pattern suggests.

Example

Conceptual electricity demand: Demand = trend effect + seasonal effect + temperature effect + error. The trend captures slow growth, the seasonal term captures winter vs summer (or weekday vs weekend), and temperature captures weather beyond the calendar average.

Typical Predictors

  • Time / trend: a time index t or a smooth trend term.
  • Seasonal indicators: month, quarter or weekday dummies (period m as in Unit 3).
  • External explanatory variables: temperature, price, holidays, promotions.
  • Lagged information: yt−1 or a lagged x, when yesterday genuinely feeds today.

Time-Series Issues that Remain

Issue Why it matters in regression
Autocorrelation Ordinary SEs and prediction intervals can be wrong; consider GLS-type errors or extra lags
Seasonality A trend-only regression leaves a repeating leftover wave
Non-stationarity A drifting level may need a trend, differencing, or another structure; do not ignore a wandering y
Changing variance Fan-shaped residuals may suggest WLS or a variance transformation

Ordinary OLS regression is often a first fit, not the last word. If residuals are still autocorrelated, additional treatment is needed before treating t-tests and forecast bands as trustworthy. Multivariate systems of several series together belong to a later unit; this topic stays with one response and its predictors.

From Fit to Forecast

Regression Model ↓ Fit ↓ Check Residuals ↓ Check Autocorrelation / Variance ↓ Choose OLS / GLS / WLS as Appropriate ↓ Predict ↓ Evaluate

Exam-Oriented Key Points

  1. General time series regression can include trend, season, external x and lags.
  2. Autocorrelation, seasonality, non-stationarity and changing variance still need checking.
  3. OLS, GLS and WLS are chosen after looking at the error pattern.
  4. Evaluate predictions on later dates, not only on the fitted sample.

Unit 4 Quick Revision

Short Exam Definitions

Topic Short definition / exam point
Time series regression Explain y using predictors, keeping time order; errors may be correlated.
Least squares / OLS Choose coefficients to minimize Σ ei2 with equal weights.
Statistical inference SE, t / p-value, confidence interval; significance ≠ practical importance.
Prediction New ŷ at new x; prediction interval is generally wider than CI for the mean.
Residual vs error e = y − ŷ (sample leftover); ε is the unobservable model shock.
Model adequacy Residual plots, Q-Q (intro), residual ACF; fitting ≠ checking.
Variable selection Forward / backward / stepwise; adjusted R2, AIC, BIC; more x is not always better.
GLS Uses an error covariance structure (correlation and/or unequal variance).
WLS Σ wiei2; higher weight for more precise observations.
General TS regression Trend + season + external x (+ lags); then diagnose and forecast.

OLS / WLS / GLS

OLS: simple independent equal-variance errors. WLS: independent errors with different variances (weights). GLS: general covariance, including autocorrelation. WLS is a special case of the GLS idea when correlation is ignored and only variances differ.

Compact Decision Intuition

Regression Model ↓ Fit ↓ Check Residuals ↓ Check Autocorrelation / Variance ↓ Choose OLS / GLS / WLS as Appropriate ↓ Predict ↓ Evaluate
  1. OLS vs GLS: equal independent errors vs structured covariance.
  2. OLS vs WLS: equal weights vs precision weights.
  3. GLS vs WLS: full covariance vs variance weights only.
  4. Confidence interval vs prediction interval: mean vs one new y (wider).
  5. Statistical vs practical significance: test rejection vs useful size.