Unit 10: Time Series

I. Orientation — Meaning and Basic Framework

A time series is a set of quantitative observations recorded in chronological order, usually at equal intervals such as hours, months, quarters, or years. Its governing principle is that observations close together in time may be related; therefore, their order contains information that an unordered dataset would lose.

A. Introduction to time series

The study of a time series aims to identify systematic patterns in historical observations and use them for description, comparison, control, or forecasting.

• Definition: If a variable is observed at successive times (t=1,2,\ldots,n), the ordered values (Y_1,Y_2,\ldots,Y_n) form a time series.
  • (t) denotes the time period.
  • (Y_t) denotes the observed value in period (t).
  • Examples include annual population, monthly sales, daily temperature, and hourly electricity demand.
• Chronological order: Time order must be preserved because (Yt) may depend on earlier observations such as (Y{t-1}). Rearranging monthly sales from smallest to largest would destroy seasonal and trend information.
• Time intervals: Observations may be annual, quarterly, monthly, weekly, daily, or more frequent. Equal spacing is generally preferred because many measurement methods compare corresponding periods or calculate centered averages.
• Stock and flow series:
  1. Stock series: Measured at a particular instant, such as bank deposits on 31 March or inventory at month-end.
  2. Flow series: Measured over an interval, such as monthly income or annual rainfall.
• Discrete and continuous recording:
  1. Discrete series: Values are observed only at specified times, such as quarterly profits.
  2. Continuous process: The variable exists continuously, but readings may be sampled, as with temperature measured every hour.
• Time-series graph: Time is placed on the horizontal axis and the observed variable on the vertical axis. A line joining successive observations can reveal direction, recurring peaks, turning points, and unusual fluctuations.
• Principal objectives:
  • Description: Summarize the historical movement of a variable.
  • Decomposition: Separate trend, seasonal, cyclical, and irregular movements.
  • Forecasting: Estimate future values from established patterns.
  • Control: Detect deviations from expected behavior, such as an unexpected fall in production.
• Dependence rather than independence: Ordinary samples often assume independent observations, whereas time-series observations may exhibit autocorrelation—the correlation of a series with its own lagged values.
• Comparability requirement: Values should be measured using consistent definitions, units, geographical coverage, and accounting rules. A change from rupees to thousands of rupees or from one price base to another must be adjusted before comparison.
• Limitations of historical patterns: Forecasts assume some continuity in the forces generating the series. Structural changes, policy shifts, wars, technological innovations, or pandemics may make earlier patterns unreliable.

II. Components of Time Series — Sources of Temporal Movement

A time series is commonly analyzed as the combined effect of four components: secular trend, seasonal variation, cyclical variation, and irregular variation. Decomposition makes the underlying movement easier to interpret.

A. Components of time series

The components distinguish long-run direction from periodic, business-cycle, and unpredictable movements.

• Secular trend ((T)): Trend is the smooth, persistent movement of a series over a long period.
  • An upward population trend may result from births exceeding deaths.
  • A downward trend in the cost of data storage may reflect technological progress.
  • Trend does not require every observation to rise or fall; short-term fluctuations can occur around its general direction.
• Seasonal variation ((S)): Seasonal movement is a regular pattern that repeats within a year.
  • Ice-cream sales may peak in summer, while retail sales may increase during recurring festivals.
  • Causes include climate, customs, holidays, institutional calendars, and payment schedules.
  • Quarterly data have four seasonal positions, while monthly data have twelve.
• Cyclical variation ((C)): Cyclical movement consists of wave-like fluctuations around the trend lasting longer than one year.
  • Economic series may pass through expansion, peak, contraction, and trough.
  • Unlike seasonality, cycles do not have a fixed length or perfectly regular amplitude.
  • Investment, employment, prices, and output commonly respond to business cycles.
• Irregular variation ((I)): Irregular movement is the residual effect of exceptional or unpredictable events.
  • Floods, strikes, earthquakes, sudden policy changes, and supply disruptions may produce irregular observations.
  • Random measurement error may also enter this component.
  • It is usually isolated after the systematic components have been estimated.
• Additive model: Components are added when their absolute magnitudes are approximately independent of the level of the series.

Y_t = T_t + S_t + C_t + I_t

• (Y_t) is the observed value at time (t).
• (T_t), (S_t), (C_t), and (I_t) are respectively the trend, seasonal, cyclical, and irregular components at time (t).
• Seasonal effects in this model are expressed in the original unit, such as “20 units above trend.”
  • Multiplicative model: Components are multiplied when fluctuations change proportionally with the level of the series.

Y_t = T_t × S_t × C_t × I_t

• (Y_t) and (T_t) retain the original measurement scale.
• (S_t), (C_t), and (I_t) are ratios or index factors, commonly centered around (1) or (100\%).
• A seasonal index of (1.20) means that the relevant period is typically (20\%) above its nonseasonal level.
  • Choice between models:
    1. Additive behavior: Seasonal changes remain near a constant amount—for example, quarterly demand repeatedly differs from trend by about 500 units.
    2. Multiplicative behavior: Seasonal changes expand as the series grows—for example, December sales remain about (30\%) above normal.
  • Interaction of components: A single observation may embody all four forces. High sales could reflect long-run growth, a festival season, an economic expansion, and an unexpected competitor shortage simultaneously.

B. Significance and limitations

Component analysis provides an interpretive framework, but the components are estimated rather than directly observed.

• Seasonal adjustment: Removing (S_t) permits fair comparison between adjacent periods. In a multiplicative model, a basic adjustment is (Y_t/S_t).
• Trend assessment: Managers can distinguish sustained growth from temporary seasonal peaks or cyclical recovery.
• Forecast construction: A trend projection may be combined with expected seasonal indices to obtain period-specific forecasts.
• Non-unique decomposition: Different trend methods or time spans can produce different estimates of (T_t), leaving different residual values for (C_t) and (I_t).
• Changing patterns: Seasonal timing and magnitude may evolve as consumer behavior, climate, regulations, or technology changes.
• Component overlap: A long irregular disturbance can resemble a cycle, while a short cycle may be confused with changing seasonality.

III. Measurement of Time-Series Variations — Decomposition Methods

Measurement converts visually observed movements into trend values, seasonal indices, cyclical ratios, and residuals. The chosen procedure should match the frequency, length, and behavior of the data.

A. Measurement of variations of time series

Time-series variations are measured by estimating the components sequentially, usually beginning with trend and seasonality.

• Graphical or freehand trend: Plot (Y_t) against time and draw a smooth line through the observations so that deviations are reasonably balanced.
  • It is simple and shows turning points clearly.
  • It is subjective: two analysts may draw different trend lines, so it is unsuitable when reproducible numerical estimates are required.
• Semi-average method: Divide the series into two equal time groups, calculate each group’s mean, and join the means at their respective midpoint dates.
  • For an odd number of observations, the central observation is commonly omitted.
  • The resulting straight trend is easy to calculate but cannot represent curvature.
• Moving-average method: Replace each value with the mean of a fixed number of neighboring observations.

M_t = (Y_t + Y_(t-1) + ... + Y_(t-k+1)) / k

• (M_t) is the (k)-period moving average associated with period (t).
• (Y_t) is the observation at time (t).
• (k) is the number of observations averaged.
• A 12-month or four-quarter average suppresses a stable annual seasonal pattern.
• If (k) is even, averages fall between periods and must be centered by averaging two consecutive moving averages.
  • Least-squares trend: Fit the line that minimizes the sum of squared differences between observations and fitted values.

T_t = a + bt

• (T_t) is the fitted trend at time (t).
• (a) is the estimated trend value when (t=0).
• (b) is the average change in trend per time period.
• (t) is a coded time variable, often centered so that (\sum t=0).

a = ΣY / n
b = Σ(tY) / Σ(t²)

• (\Sigma) denotes summation over all observations.
• (Y) is an observed value, and (n) is the number of observations.
• These simplified formulas apply when coded time values satisfy (\sum t=0).
  • Seasonal indices by simple averages: Average all January values, all February values, and so on; divide each period average by the grand average and multiply by (100).
• Indices should average (100).
• Monthly indices should total (1200), and quarterly indices should total (400).
• This method is straightforward but can confound trend with seasonality.
  • Ratio-to-trend method: Express each observation as a percentage of its estimated trend, group ratios by season, and average them.

Seasonal relative = (Y_t / T_t) × 100

• (Y_t) is the actual observation and (T_t) is its fitted trend.
• The method is effective when a suitable trend equation represents long-run movement.
  • Ratio-to-moving-average method: Divide each observation by its centered moving average, average the resulting ratios for corresponding seasons, and normalize the indices.
• The centered moving average approximates the combined trend-cycle component.
• The ratios primarily contain seasonal and irregular effects; averaging corresponding periods reduces irregular variation.
  • Link-relative method: Express each period as a percentage of the immediately preceding period, average link relatives by season, and convert them into chain relatives.
• This method tracks sequential change but is sensitive to extreme observations and requires correction for drift.
  • Cyclical measurement: After removing trend and seasonality under a multiplicative model, calculate residual percentages.

Cycle-irregular ratio = (Y_t / (T_t × S_t)) × 100

• (Y_t) is the actual value, (T_t) is the trend estimate, and (S_t) is the seasonal factor expressed as a ratio.
• Smoothing these ratios over several periods can reveal cyclical movement; remaining erratic deviations represent (I_t).
  • Worked measurement example: If actual quarterly sales are (240), the fitted trend is (200), and the seasonal index is (120), then (S_t=1.20).

Cycle-irregular ratio = [240 / (200 × 1.20)] × 100 = 100

• The observation is fully explained by trend and seasonality at this stage, leaving no measured cycle-irregular departure from the expected level.
  • Method selection: Moving averages suit flexible smoothing, least squares provides an objective equation and projection, and seasonal-index methods support recurring within-year comparisons.
  • Measurement constraints: Missing observations, short series, changing seasonal patterns, outliers, and structural breaks can distort every decomposition method; estimates should therefore be interpreted in their historical context.