Unit 10: Time Series - Subjective Questions

Definition: A time series is a set of observations of a variable recorded in chronological order, generally at equal intervals of time.

Principal characteristics:

• Observations are arranged according to time.
• The time intervals may be hourly, daily, monthly, quarterly, or yearly.
• Successive observations are often statistically dependent.
• The data may exhibit trend, seasonal, cyclical, and irregular movements.
• Time is treated as an important independent variable.

Examples:

• Monthly sales of a company
• Annual population of a country
• Daily closing prices of a share
• Quarterly gross domestic product

Time-series analysis identifies the patterns present in historical data and uses them to understand behavior or forecast future values.

The principal objectives of time-series analysis are:

• Understanding past behavior: It describes how a variable has changed over time.
• Identifying components: It separates trend, seasonal, cyclical, and irregular movements.
• Forecasting: Historical patterns are used to estimate future values.
• Planning and budgeting: Forecasts support production, staffing, inventory, and financial decisions.
• Performance evaluation: Actual values can be compared with expected values.
• Policy formulation: Governments use time-series data for economic and social planning.
• Control: Unexpected deviations from established patterns can be detected quickly.

For example, a retailer can analyze monthly sales to identify long-term growth and seasonal demand, thereby improving inventory planning.

A time series may contain the following four components:

1. Secular trend (): The long-term general movement of the series. It may be upward, downward, or stable. For example, population may increase over several decades.
2. Seasonal variation (): A regular pattern that repeats within one year. For example, sales of woollen clothes rise during winter.
3. Cyclical variation (): Wave-like movements around the trend lasting for more than one year. Business cycles involving prosperity, recession, depression, and recovery are examples.
4. Irregular variation (): Unpredictable movements caused by unusual events such as wars, floods, strikes, or pandemics.

An observed value is commonly represented through an additive model, , or a multiplicative model, .

Secular trend and seasonal variation differ as follows:

Thus, trend describes the broad direction, whereas seasonal variation describes recurring short-period fluctuations.

Cyclical variations are recurrent, wave-like movements around a trend that generally last for more than one year. They are often associated with business-cycle stages such as prosperity, recession, depression, and recovery. Their duration and magnitude are not perfectly regular.

Irregular variations are random and unpredictable fluctuations caused by exceptional events such as natural disasters, wars, strikes, or sudden policy changes.

Key differences:

• Cyclical movements have a broad recurring pattern; irregular movements do not.
• Cyclical changes develop over relatively long periods; irregular changes may occur suddenly.
• Cyclical movements arise from general economic forces; irregular movements arise from accidental events.

Under a multiplicative model, after estimating trend and seasonal effects, the combined cyclical and irregular factor may be obtained as

Further smoothing can estimate , leaving the unexplained residual as .

In the additive model, the observed value is expressed as

This model is appropriate when seasonal, cyclical, and irregular fluctuations remain approximately constant in absolute magnitude regardless of the level of the series. Each component is measured in the same units as .

In the multiplicative model, the observed value is expressed as

It is appropriate when fluctuations increase or decrease proportionately with the level of the series. The components , , and are usually expressed as ratios or percentages, with or representing no effect.

Comparison:

• Additive effects are absolute; multiplicative effects are proportional.
• Additive seasonal deviations sum to approximately zero over a full cycle.
• Multiplicative seasonal indices average , or sum to for seasons.

Business and economic series commonly follow the multiplicative model.

A secular trend is the smooth, long-term tendency of a time series to increase, decrease, or remain stable over an extended period. It represents the underlying direction after short-term fluctuations are disregarded.

Major causes include:

• Population changes: Population growth can raise demand, production, and consumption.
• Technological progress: Innovation may increase productivity or make older products obsolete.
• Economic development: Growth in income, investment, and infrastructure affects many series.
• Changes in tastes and habits: Consumer preferences may create persistent increases or decreases in demand.
• Inflation: Rising prices can produce an upward trend in monetary series.
• Institutional and policy changes: Taxation, regulation, and government programs can have lasting effects.

A trend does not require every observation to move in the same direction; short-term variations may occur around its general path.

Procedure:

1. Plot time on the horizontal axis and observed values on the vertical axis.
2. Mark all observations on the graph.
3. Draw a smooth curve passing through or near the plotted points.
4. Keep the deviations above and below the curve reasonably balanced.
5. Read the trend values from the fitted curve.

Merits:

• It is simple and inexpensive.
• It provides a quick visual representation of the long-term movement.
• It can accommodate nonlinear patterns.
• No lengthy calculations are required.

Limitations:

• It is subjective; different persons may draw different curves.
• It lacks a precise mathematical equation.
• Forecasting from the curve may be unreliable.
• Accuracy depends on the scale and skill of the analyst.
• It is unsuitable when reproducible numerical estimates are required.

Therefore, the method is useful for preliminary analysis but not for rigorous measurement.

The semi-average method fits a straight-line trend by dividing a time series into two equal parts.

Procedure:

1. Divide the observations into two equal groups.
2. If the number of observations is odd, usually omit the middle observation before dividing the series.
3. Calculate the arithmetic mean of each group.
4. Place each mean at the midpoint of its corresponding time period.
5. Join the two midpoint values to obtain the linear trend.

If the semi-averages are and , located at coded times and , the slope is

The intercept is

and the trend equation is

The method is objective and easy, but it ignores much of the information within each half and assumes a linear trend.

Divide the data into two groups:

• First group:
• Second group:

Their semi-averages are

These averages correspond to the midpoints of years and . Therefore, the annual slope is

Taking year as , the trend equation is

where is the number of years after year .

For the seventh year, . Hence,

Thus, the estimated trend value for the seventh year is approximately .

A moving average is obtained by successively averaging a fixed number of consecutive observations. It smooths short-term variations and reveals the underlying trend-cycle movement.

For an -period moving average,

Procedure:

• Select a period equal to the seasonal cycle, such as quarters or months.
• Calculate averages for successive groups of observations.
• Place each average at the center of its group.

When is odd, the average corresponds to an actual central period. When is even, its center lies between two periods. Therefore, two adjacent moving averages are averaged again:

This is called centering. It aligns the smoothed value with an actual time period and makes comparison with the original observation possible.

For a three-year moving average, each value is the average of three consecutive observations:

Thus, the three-year moving averages, assigned to years through , are

Interpretation: The smoothed values show a clear upward trend. The temporary decline from to in the original observations is moderated by averaging, allowing the underlying growth pattern to be seen more clearly.

For observations , assume the fitted trend is

The residual is . The least-squares method minimizes

Differentiate with respect to and set the result equal to zero:

Therefore,

Similarly, differentiating with respect to gives

so

These are the normal equations. If time is coded so that , they simplify to

and

Hence, the fitted trend is .

Code the five successive years as . The calculations are:

Thus,

Because ,

and

The fitted trend equation is

The sixth year corresponds to . Therefore,

The forecast for the sixth year is .

The preferred method depends on the purpose and data. Moving averages are useful for smoothing, while least squares are generally preferable when an equation and forecasts are required.

Seasonal variation refers to regular and recurring movements in a time series that are completed within one year. These patterns may occur monthly, quarterly, weekly, or daily.

A seasonal index measures the relative level of a particular season compared with the average period. In percentage form, an index of means that the season is normally above average, while an index of means it is below average.

Major causes:

• Climate and weather
• Festivals and customs
• Holidays and vacation periods
• Institutional schedules
• Production and consumption habits

Uses:

• Planning production and inventories
• Scheduling labor and transport
• Preparing seasonal budgets
• Comparing data after seasonal adjustment
• Improving short-term forecasts

For quarterly data, properly adjusted indices sum to ; for monthly data, they sum to .

Under the simple-average method, seasonal indices are calculated without separately estimating the trend.

Procedure:

1. Arrange the observations by year in rows and seasons in columns.
2. Compute the average for each month or quarter.
3. Compute the grand average of all observations or the average of the seasonal averages.
4. Calculate each seasonal index as

5. Adjust the indices if their total differs from the required value.

For seasons, the correction factor is

Each adjusted index is

Thus, quarterly indices must sum to , and monthly indices must sum to .

The method is simple, but it may be distorted when a strong trend or cyclical movement is present.

The grand average of the four quarterly averages is

The seasonal indices are:

Their sum is

so no adjustment is required.

Interpretation:

• First-quarter sales are approximately below the normal quarterly level.
• Second-quarter sales are approximately below normal.
• Third-quarter sales are approximately above normal.
• Fourth-quarter sales are approximately above normal.

Hence, the strongest seasonal demand occurs in the fourth quarter.

Ratio-to-trend method:

1. Fit a trend line and calculate trend values .
2. Compute the ratio for each period:

3. Group ratios by month or quarter and find their averages.
4. Adjust the averages so that their total is .

This method removes the estimated trend, but cyclical and irregular effects may remain in the ratios.

Ratio-to-moving-average method:

1. Calculate moving averages with a period equal to the seasonal cycle.
2. Center them when the period is even.
3. Compute

4. Group the ratios by season and average them, often after removing extreme values.
5. Adjust the resulting indices to total .

Because the centered moving average represents the trend-cycle component, dividing by it largely isolates seasonal and irregular effects. This method is widely used when a stable seasonal pattern exists.

The link-relative method compares each observation with the observation in the immediately preceding season.

Steps:

1. Calculate link relatives:

2. Arrange the link relatives according to months or quarters.
3. Find the average or median link relative for each season.
4. Convert the average link relatives into chain relatives. Assign to the first season and compute successive values using

5. Use the last season's link relative to obtain the chained value for the first season of the next cycle.
6. Correct for drift if this value differs from .
7. Convert the corrected chain relatives into seasonal indices whose average is .

The method uses comparisons between consecutive periods and can accommodate a trend, but it is relatively lengthy and may be affected by irregular observations.