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Introduction to Time Series

  • Difference between:
    • \(X_{1}, X_{2}, \dots, X_{100}\) representing a dice roll in 100 rounds (IID1 random).
    • \(X_{1}, X_{2}, \dots, X_{100}\) representing closing prices of a stock for 100 consecutive days (autocorrelated random).
  • What exactly is a time series?
    • Classical inference deals with IID observations.
    • A time series is ordered by time and is chronological.
    • Thus, time series data is not independent!
  • Data Types
    • Summary: Cross sectional + Time Series = Panel
    • Cross-sectional data
      • This involves different individuals at a SINGLE POINT OF TIME.
      • Examples include:
        • Max humidity at 20 diff locations.
        • 20 stocks ka closing price on a particular day.
        • Heights of students, on a particular day.
      • This involves several variables at a single point of time.
    • Time series data
      • This involves a particular individual/entity (fixed) over DIFFERENT POINTS OF TIME.
      • Examples include (transitioning the time point over a month):
        • Max HUMIDITY level for a month.
        • Closing stock price of a (single) STOCK over 6 months.
        • Quarterly student enrolment in a COLLEGE over 5 years.
      • This involves the same variable over a period of time.
    • Panel/Longitudinal data
      • This involves observations on different cross-sections over a time.
      • Multiple states, for each of the years.
      • Examples include:
        • Annual cancer mortality rates of different Indian states during 2015:2023 (e.g., Delhi, Jharkhand, Maharashtra…) for each of these years.
        • Yearly sales of 10 companies over 10 years (multiple companies, multiple years).
  • Steps in Time Series Analysis
    1. Plotting the data.
    2. Studying past behavior.
    3. Identifying underlying patterns and trends2.
    4. Using past data for forecasting.
  • Application Areas
    • Retail stores monitor sales.
    • Energy companies monitor reserves, production, demand, and price.
    • Education sectors track enrollment.
    • International Financial Organizations (World Bank, IMF) track inflation and economic activity.
    • Transportation departments predict future travel.
    • Banks track new home purchases.

  1. IID means Identical observations (random variable) come from some distribution and Independent observations (doesn't depend on observations). ↩

  2. Trend = upward or downward movement of an entity. ↩