BBA 1st Semester - Quantitative Techniques

–I
UNIT – 5

📈 Time Series and Index Numbers 📊

🔄 TIME SERIES
📋 Introduction to Time Series
A time series is a sequence of observations or data points collected and
recorded at regular intervals over a period of time. These observations are
arranged chronologically to show how a particular variable changes over
time. Time series analysis is crucial in business for forecasting, planning,
and decision-making processes.

Key Characteristics:

Data points are collected at regular intervals (daily, weekly, monthly,

quarterly, annually)
Observations are dependent on time

Used for analyzing patterns and trends

Essential for forecasting future values

🧩 Components of Time Series

Time series data can be decomposed into four main components:

1. 📈 Trend (T)
The trend represents the long-term movement or direction in the data
over an extended period.

Characteristics:

Shows the general pattern of increase, decrease, or stability

Eliminates short-term fluctuations

Indicates the underlying growth or decline pattern
Can be linear, exponential, or curvilinear

Types of Trends:

Upward Trend: Values increase over time

Downward Trend: Values decrease over time

Horizontal/Static Trend: Values remain relatively constant

2. 🌊 Seasonal Variations (S)

Seasonal variations are regular, predictable patterns that repeat over

fixed periods (usually within a year).

Characteristics:

Occur due to seasonal factors like weather, festivals, holidays

Repeat in a systematic manner
Duration is usually one year or less

Predictable and regular in nature

Common Seasonal Patterns:

 Ice cream sales peak in summer

Umbrella sales increase during monsoon

Gift sales surge during festivals

3. 🔄 Cyclical Variations (C)

Cyclical variations are long-term fluctuations that occur over periods

longer than one year, often related to business cycles.

Characteristics:

Duration varies and is unpredictable

Related to economic conditions

Periods of prosperity and recession

Irregular in timing and amplitude

Examples:

Economic boom and recession cycles

Real estate market cycles

Stock market cycles

4. ⚡ Irregular/Random Variations (I)

Irregular variations are unpredictable, random fluctuations that cannot be

attributed to trend, seasonal, or cyclical factors.

Characteristics:

Sudden, unexpected changes

 Caused by unforeseen events
Non-recurring and unpredictable

Also called random or erratic movements

Causes:

Natural disasters
Political events

Strikes and labor disputes

Sudden policy changes

📊 Time Series Models

Model Type Formula Description

Additive Model Y=T+S+C+I Components are added together

Multiplicative Model Y=T×S×C×I Components are multiplied together

Mixed Model Y=T×S×C+I Combination of both approaches

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📈 Methods of Measuring Trend

1. 🔄 Moving Averages Method
The moving averages method is a technique used to smooth out
fluctuations in time series data to identify the underlying trend.

Simple Moving Average

Definition: The arithmetic mean of a fixed number of consecutive

observations that moves through the data series.
Process:

1. Select the number of periods (n) for averaging

2. Calculate the average of first n observations

3. Drop the first observation and add the next one

4. Continue this process throughout the series

Formula: Moving Average = (Sum of n consecutive values) / n

Weighted Moving Average

Assigns different weights to different periods, giving more importance to

recent observations.

Advantages:

Simple to calculate and understand

Smooths out irregular fluctuations

Useful for short-term forecasting

Reduces the impact of random variations

Disadvantages:

Loses information at the beginning and end of series

May not be suitable for all types of trends

Sensitive to the choice of period length

2. 📐 Least Squares Method

The least squares method is a mathematical approach to find the best-
fitting trend line through the data points.
Linear Trend

For a straight-line trend: Y = a + bX

Where:

Y = Dependent variable (observed values)

X = Independent variable (time)

a = Y-intercept

b = Slope of the line

Calculation Process:

1. Normal Equations:
ΣY = na + bΣX
ΣXY = aΣX + bΣX²

2. Solving for coefficients:

b = (nΣXY - ΣXΣY) / (nΣX² - (ΣX)²)

a = (ΣY - bΣX) / n

Advantages:

Provides objective mathematical solution

Minimizes the sum of squared deviations

Suitable for long-term forecasting

Gives precise trend equation

Disadvantages:
 Assumes linear relationship
Sensitive to extreme values

May not capture non-linear trends

Requires mathematical computation

📊 INDEX NUMBERS
📋 Introduction to Index Numbers
Index numbers are statistical measures that show the relative change in a
variable or group of variables over time, compared to a base period.

Definition: An index number is a statistical measure designed to show

changes in a variable or group of related variables with respect to time,
geographical location, or other characteristics.

🎯 Meaning and Significance

Index numbers serve as:

Economic barometers indicating economic conditions

Tools for comparison across different time periods

Basis for policy formulation by governments and organizations

Inflation indicators showing price level changes

📈 Types of Index Numbers

1. Based on Number of Variables
 Type Description Example

Simple Index Measures change in single variable Price of wheat

Composite Measures change in group of Consumer Price

Index variables Index
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2. Based on Purpose

A. Price Index Numbers

Measure changes in price levels

Compare prices over different periods

Examples: Consumer Price Index, Wholesale Price Index

B. Quantity Index Numbers

Measure changes in quantities consumed/produced

Show volume changes over time

Examples: Industrial Production Index

C. Value Index Numbers

Measure changes in total value

Combination of price and quantity changes

Value = Price × Quantity

🛠️ Uses of Index Numbers

1. Economic Analysis

Inflation Measurement: Track price level changes

 Economic Planning: Formulate economic policies

Trend Analysis: Identify economic patterns

Comparative Studies: Compare different regions/periods

2. Business Applications

Cost Control: Monitor input cost changes

Pricing Decisions: Adjust selling prices

Wage Negotiations: Base for salary adjustments
Performance Evaluation: Measure business growth

3. Government Policy

Monetary Policy: Control money supply

Fiscal Policy: Taxation and expenditure decisions

Welfare Measures: Social security adjustments

Subsidy Allocation: Determine subsidy requirements

4. Academic Research

Economic Research: Study economic phenomena

Forecasting: Predict future trends

Statistical Analysis: Data comparison and analysis

🔧 Methods of Construction of Index Numbers

1. 📊 Laspeyres' Index
Named after: Étienne Laspeyres (German economist)
Formula:

Price Index: P₀₁ = (Σp₁q₀ / Σp₀q₀) × 100

Quantity Index: Q₀₁ = (Σq₁p₀ / Σq₀p₀) × 100

Where:

p₀ = Base year prices

p₁ = Current year prices

q₀ = Base year quantities

q₁ = Current year quantities

Characteristics:

Uses base year quantities as weights

Base-weighted index
Easier to calculate as weights remain constant

Tends to overstate price increases

Advantages:

Simple to understand and calculate

Weights remain constant over time

Suitable for regular computation

Facilitates comparison across periods

Disadvantages:

Doesn't reflect current consumption patterns

 May become outdated over time
Upward bias in price measurement

Ignores substitution effects

2. 📊 Paasche's Index
Named after: Hermann Paasche (German economist)

Formula:

Price Index: P₀₁ = (Σp₁q₁ / Σp₀q₁) × 100

Quantity Index: Q₀₁ = (Σq₁p₁ / Σq₀p₁) × 100

Characteristics:

Uses current year quantities as weights

Current-weighted index

Reflects current consumption patterns

Tends to understate price increases

Advantages:

Reflects current market conditions

More relevant for current decision-making

Considers substitution effects

Updated consumption patterns

Disadvantages:

Requires current year data for weights

 More complex calculations
Weights change every period

Difficult to compare across long periods

3. 📊 Fisher's Index
Named after: Irving Fisher (American economist)

Formula: Fisher's Index = √(Laspeyres' Index × Paasche's Index)

Also known as:

Ideal Index

Geometric Mean of Laspeyres' and Paasche's indices

Characteristics:

Combines both base and current year weights

Neutralizes the biases of both methods

Satisfies most index number tests

Considered the most accurate method

Advantages:

Eliminates biases of individual methods

More accurate representation

Satisfies important mathematical tests

Balanced approach to index construction

Disadvantages:
 Complex calculations required
Requires both base and current year data

Difficult to interpret economically

More expensive to compute regularly

📈 Comparison of Methods
Aspect Laspeyres' Paasche's Fisher's

Weights Used Base year quantities Current year quantities Both periods

Bias Upward bias Downward bias No bias

Calculation Simple Moderate Complex

Data Required Base year data Current year data Both periods

Accuracy Moderate Moderate High

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💰 Cost of Living Index

📋 Definition and Purpose
The Cost of Living Index measures the relative change in the cost of
maintaining a standard of living between different time periods or
locations.

Purpose:

Measure inflation impact on consumers

Adjust wages and salaries

Compare living costs across regions

Formulate economic policies
🏗️ Construction Method
1. Selection of Items

Food Items: Essential food commodities

Clothing: Basic clothing requirements

Housing: Rent and housing costs

Fuel & Lighting: Energy expenses

Miscellaneous: Healthcare, education, transport

2. Weight Assignment

Weights are assigned based on:

Family Budget Surveys

Expenditure Patterns

Income Group Analysis

Regional Variations

3. Price Collection

Regular market surveys

Multiple source verification

Quality specifications

Seasonal adjustments

📊 Applications
Wage Adjustments: Dearness allowance calculations
 Pension Revision: Social security adjustments

Policy Making: Government welfare schemes

International Comparisons: Standard of living analysis

🏭 Wholesale Price Index (WPI)

📋 Definition and Scope
The Wholesale Price Index measures the average change in prices of
goods traded in wholesale markets.

Coverage:

Primary Articles: Agricultural products, minerals

Fuel & Power: Coal, petroleum products, electricity

Manufactured Products: Industrial goods, consumer products

🎯 Characteristics
Base Year

Periodically revised to reflect current economic structure

Current base year varies by country

Ensures relevance and accuracy

Commodity Coverage

Broad Coverage: Represents major economic sectors

Weighted System: Based on economic importance

Regular Updates: Commodity basket revision

📈 Uses and Applications
1. Economic Indicators

Inflation Measurement: Producer price inflation

Economic Monitoring: Industrial price trends

Policy Formulation: Monetary and fiscal policies

2. Business Applications

Contract Escalation: Price adjustment clauses

Cost Analysis: Input cost monitoring

Pricing Strategy: Market price benchmarking

3. Government Functions

Tax Policy: Indirect tax adjustments

Subsidy Programs: Agricultural price support
Trade Policy: Export-import decisions

🔄 Calculation Process
1. Data Collection: Wholesale market price surveys
2. Weight Assignment: Based on gross value of output

3. Index Calculation: Using appropriate formula

4. Quality Control: Data verification and validation

5. Publication: Regular release of index values
📊 Summary and Key Points
🔄 Time Series Analysis
Components: Trend, Seasonal, Cyclical, Irregular
Trend Methods: Moving averages for smoothing, Least squares for
mathematical precision
Applications: Forecasting, planning, decision-making

📈 Index Numbers
Purpose: Measure relative changes over time

Types: Simple vs. Composite, Price vs. Quantity vs. Value

Construction Methods: Laspeyres' (base-weighted), Paasche's
(current-weighted), Fisher's (ideal)

💡 Practical Applications
Economic Analysis: Inflation measurement, policy formulation

Business Planning: Cost control, pricing decisions

Social Welfare: Wage adjustments, living standard comparisons

📚 Study Tips:
Focus on understanding concepts rather than memorizing formulas
Practice calculations with different data sets

Understand the practical applications in real-world scenarios

Pay attention to advantages and disadvantages of each method

Connect theoretical knowledge with current economic indicators