Cross-Sectional and Time Series Data

Structure of Economic Data

• Cross-sectional data provides a snapshot of information at one specific point in time.

• It is gathered by observing multiple subjects like individuals, firms, or countries simultaneously.

• Cross-sectional data analysis compares differences between the subjects.

• Notation for cross-sectional data: Yᵢ for i = 1, 2, 3, ..., N individuals.

• Examples: microeconomics, labor economics, business economics, health economics, state and local public finance, demographic economics.

• Example application: measuring obesity levels in South Africa using a random sample of 1000 people.

• Their height and weight are measured to calculate the percentage categorized as obese, for example, 32%.

• Time series data consists of observations on one or more variables over a period of time

• Time series data is arranged in chronological order, which may have different time frequencies.

• Notation for time series data: Yₜ for t = 1, 2, 3, ..., T or t = 1990, 1991,...2002.

• Examples of time series data: stock prices, GDP, and money supply.

• Analyzing time series is more difficult because observations are dependent across time.

• Past events can influence future events

• Lags in behavior are expected.

• A variable lagged one period is denoted as Yₜ₋₁, or two periods as Yₜ₋₂.

• Leading periods are denoted as Yₜ₊ₖ.

• Time series seasonal patterns occur weekly, monthly, and quarterly.

• Panel data consists of a time series for each cross-sectional member.

• The time series aspect (t) is depicted vertically.

• The cross-sectional aspect (i) is depicted horizontally.

• Examples of panel data include sales for 50 firms over 5 years, GDP for 20 countries over 20 years, and exports of 150 countries over 10 years.

Basic Data Handling

• Looking at raw data and getting a 'feel' helps to understand and better interpret results before econometric analysis.

• Observing the number of observations, start and end dates, and spotting outliers/discontinuities/structural breaks is crucial.

• Graphical analysis provides a 'big picture' view of the data.

• Graphical analysis makes it easy to check for outliers/structural breaks

• Time series data is represented using line graphs.

• Cross-sectional data is represented with bar graphs/histograms.

• Main graphical tools include histograms, scatter plots, line graphs, bar graphs, and pie charts.

• Histograms show distribution.

• Scatter plots show the relationship between two series.

• Line graphs show series comparisons.

• Summary statistics available in EViews include mean, variance, standard deviation, covariance, and correlation.

Components of a Time Series

• Trend: smooth upward/downward movement.

• Cycle: rise and fall over long periods.

• Seasonal: patterns within a year.

• Irregular: random component that is either episodic (unpredictable, but identifiable) or residual (unpredictable and unidentifiable).

• An index is a number expressing relative change in value from one period to another.

• An example of an index is the CPI

• Change is measured relative to a base value, which should always be 100

• Two indices can be compared directly only if their base years are the same.

Data Transformations

• Use caution when changing the frequency of time series data.
• When reducing frequency:
  • For stock variables (e.g., CPI), choose specific dates or averaging.
  • For flow variables (e.g., GDP), sum the values.
• Increasing frequency can be achieved through extrapolation.

Nominal vs. Real Data

• Nominal series has a price component which obscures fundamental features.
• Two variables may show a "false" correlation due to price components
• To convert to real terms, use an appropriate price deflator.
• Use CPI for consumption expenditure.
• Use PPI for manufacturing production.

Log Transformations

• Time series data sometimes exhibits a strong trend caused by underlying growth, resulting in an exponential curve.
• Exponential growth dominates other series features and obscures relationships.
• A log transformation, inverse of exponential function, linearizes the exponential trend.
• Allow regression coefficients to be interpreted as elasticities
• Cob-Douglas function is a non-linear function example.

Differencing

• Differencing is used to remove a trend from a series.
• Differencing provides the absolute changes from one period to the next.
• Start with first-order differencing
• Use second-order if a trend still exists.

Growth Rates

• Growth rates are used especially in economics and investigate how changes in one variable relate to changes in another.

• Discrete compounding (Annual data): Growth rate of Y = (Y – Yₜ₋₁) / Yₜ₋₁

• Continuous compounding (Annual data): ln (Y/ Yₜ₋₁) = ln(Y) – ln (Yₜ₋₁)

• Monthly data often uses month-on-previous-month growth rates.

• Alternatively, year-on-year growth uses monthly data (month-on-same-month in the previous year) to reduce volatility

• To annualize monthly growth: Y = [(Y - Yₜ₋₁)¹²/Yₜ₋₁] − 1 {Discrete} OR = 12 x (Y/ Yₜ₋₁) {Continuous}

• To annualize quarterly data: Y = [(Y - Yₜ₋₁)⁴/Yₜ₋₁] − 1 {Discrete} OR = 4 x (Y/Yₜ₋₁) {Continuous}