Lecture 10
An Introduction to Time Series
Time series analysis
• Time series data: one variable observed at different points in time.
• Time series analysis: the use of a model to predict future values based on
previously observed values.
The importance of forecasting
• Governments forecast: unemployment rates, interest rates, expected
revenues from income taxes, etc.
• Marketing executives forecast: demand, sales, consumer preferences, etc.
• Retail stores forecast: inventory levels, cost and benefits of hiring
employees, costs and benefits of training, etc.
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temperature, humidity, etc.
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be: annually, quarterly, monthly, etc.
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Types of times series
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Income
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• Continuous: Observations are recorded continuously. For example:
• Discrete: Observations are recorded in discrete times. Time intervals can
Types of variation
• The variables can show different types of variations. We may need to
transform the data:
• Seasonal variation: variation annual in period (for example sales are typically
higher during Christmas)
• Cyclic variation:
• Variation at fixed periods (for example temperature is always lower at
night than at noon)
• Oscillations: Predictable to some extent (for example business cycles)
• Trend: long-term changes. For example: climate variables exhibit cyclic
variation over long periods of time
• Irregular fluctuations: variation left when we remove the trend and cyclic
variation (random variation).
Trend component
Upward trend
Sales
Time
Seasonal component
Sales
Summer
Fall
Winter
Spring
Time
Cyclical component
Sales
Time
Irregular component
• Unpredictable random, residual fluctuations
• Noise in the time series
• The simplest model is given by: dependent variable = linear
trend + noise
• Goal: measure trend and remove seasonal variation. What
can we do to eliminate noise in our graph?
• Moving average: use for smoothing a series of arithmetic means over
time
