STA2020F: Applied Statistics
Time Series Notes
Cross-sectional data: Data observed or measured at one point in time.
Time series: A sequence of observations collected at regular equally
spaced intervals over a period of time.
• Arise in virtually every application field.
• The basic assumption is that past patterns continue in the future.
• The purpose of time series analysis is to identify and isolate
influencing factors from the past, in order to better understand the
process (pattern of behaviour and components that gave rise to it)
underlying the time series, in order to forecast future values.
• Autocorrelation and stationarity are two fundamental concepts.
Autocorrelation / serial correlation: When values of a time series are
correlated with one another.
• The random errors in the model are often positively correlated over
time, such that each random error is more likely to be similar to the
previous random error, than it would be if they were independent.
• The majority of time series are dependent – a single chance event
may affect all later observations, so we cannot assume that data
constitutes a random sample.
• They exhibit significant autocorrelation at some lag (h) = the number
of time periods between observations, at which we measure
autocorrelation.
• Standard inferential techniques don’t work for dependent
observations, only independent observations.
, Stationarity: The statistical properties (mean of the data generation
process and variance of the time series) are constant over time.
• Time series with trends and seasonality are not stationary.
• A stationary time series has no predictable patterns in the long-term.
• Time plots will show the series to be roughly horizontal (some cyclical
behaviour is possible), with a constant variance.
Time series plot: A line graph of the observed data (yt) against time (t).
• Enables us to detect and describe patterns of past behaviour.
Successive changes in values are comparable because they all relate
to a common time interval between observations.
• Helps us find a suitable statistical model to describe data, and
thereby, forecast future values of the time series, assuming that past
patterns continue into the future.
Components of a non-stationary time series:
Trend/secular trend (T)
• Long term tendency of a time series.
• The pattern may move steadily upward, downward or stay the same.
• Usually the result of long-term factors (e.g. population, preferences).
• Duration of the trend is much longer than one time period.
• Can be predicted in the future.
Time Series Notes
Cross-sectional data: Data observed or measured at one point in time.
Time series: A sequence of observations collected at regular equally
spaced intervals over a period of time.
• Arise in virtually every application field.
• The basic assumption is that past patterns continue in the future.
• The purpose of time series analysis is to identify and isolate
influencing factors from the past, in order to better understand the
process (pattern of behaviour and components that gave rise to it)
underlying the time series, in order to forecast future values.
• Autocorrelation and stationarity are two fundamental concepts.
Autocorrelation / serial correlation: When values of a time series are
correlated with one another.
• The random errors in the model are often positively correlated over
time, such that each random error is more likely to be similar to the
previous random error, than it would be if they were independent.
• The majority of time series are dependent – a single chance event
may affect all later observations, so we cannot assume that data
constitutes a random sample.
• They exhibit significant autocorrelation at some lag (h) = the number
of time periods between observations, at which we measure
autocorrelation.
• Standard inferential techniques don’t work for dependent
observations, only independent observations.
, Stationarity: The statistical properties (mean of the data generation
process and variance of the time series) are constant over time.
• Time series with trends and seasonality are not stationary.
• A stationary time series has no predictable patterns in the long-term.
• Time plots will show the series to be roughly horizontal (some cyclical
behaviour is possible), with a constant variance.
Time series plot: A line graph of the observed data (yt) against time (t).
• Enables us to detect and describe patterns of past behaviour.
Successive changes in values are comparable because they all relate
to a common time interval between observations.
• Helps us find a suitable statistical model to describe data, and
thereby, forecast future values of the time series, assuming that past
patterns continue into the future.
Components of a non-stationary time series:
Trend/secular trend (T)
• Long term tendency of a time series.
• The pattern may move steadily upward, downward or stay the same.
• Usually the result of long-term factors (e.g. population, preferences).
• Duration of the trend is much longer than one time period.
• Can be predicted in the future.
