Unit II :Time Series & Index Number

Time series analysis

Time Series Analysis is a statistical technique that analyzes data points collected or recorded at specific time intervals. It helps identify patterns, trends, and seasonal variations to make predictions about future data points. This method is widely used in finance, economics, business, and other fields to make informed decisions.

Concept of Time Series

A time series consists of observations measured sequentially over time. It is primarily used to:

• Understand historical data patterns.
• Forecast future values based on observed data trends.
• Identify any periodic or seasonal variations.

A time series is a sequence of data points collected or recorded at specific time intervals, usually spaced at regular intervals such as daily, monthly, or annually. Time Series Analysis is used to identify patterns, trends, and relationships within the data over time. Businesses use this analysis for sales forecasting, financial market predictions, inventory management, and more.

The primary objective of time series analysis is to understand past patterns and forecast future values based on historical data.

Application: Forecasting sales, analyzing stock prices, predicting demand for products, and monitoring production outputs.

Components of Time Series

A time series has four main components:

1. Trend (T)

A long-term movement or progression in data, showing an overall increase or decrease over time. Example: Increasing sales of smartphones over several years.

2. Seasonality (S)

A repeating pattern in data occurring at regular intervals due to seasonal factors (e.g., monthly, quarterly, annually). Example: Higher sales of ACs in summer and lower sales in winter.

3. Cyclic (C)

Long-term fluctuations occurring due to economic or business cycles. These patterns are not fixed like seasonality and can last for multiple years. Example: Economic recession causing a dip in business performance.

4. Irregular/Random (I)

Unpredictable variations caused by unusual or unforeseen events like natural disasters, strikes, or pandemics. Example: A sudden drop in tourism during the COVID-19 pandemic.

Models of Time Series

Two main models are used in time series analysis to combine these components:

Additive Model : In this model, the components are added together:

• $${\mathrm{<b>Y</b>}}_{\mathrm{<b>t</b>}}<b>=</b>{\mathrm{<b>T</b>}}_{\mathrm{<b>t</b>}}<b>+</b>{\mathrm{<b>S</b>}}_{\mathrm{<b>t</b>}}<b>+</b>{\mathrm{<b>C</b>}}_{\mathrm{<b>t</b>}}<b>+</b>{\mathrm{<b>I</b>}}_{\mathrm{<b>t</b>}}$$

Suitable when the magnitude of seasonal variations is constant over time. Example: Sales increasing by a fixed amount every year, regardless of other factors.

Multiplicative Model : 

In this model, the components are multiplied:

• $${\mathrm{<b>Y</b>}}_{\mathrm{<b>t</b>}}<b>=</b>{\mathrm{<b>T</b>}}_{\mathrm{<b>t</b>}}<b>\times </b>{\mathrm{<b>S</b>}}_{\mathrm{<b>t</b>}}<b>\times </b>{\mathrm{<b>C</b>}}_{\mathrm{<b>t</b>}}<b>\times </b>{\mathrm{<b>I</b>}}_{\mathrm{<b>t</b>}}$$

Suitable when the magnitude of seasonal variations increases or decreases with the trend. Example: Sales growing by a percentage each year due to rising demand.

Example for Better Understanding

• Additive Model Example: Monthly sales data for a retail store shows an upward trend, consistent festive sales increase in December, and minor random fluctuations.
• Multiplicative Model Example: Stock market data, where higher market indices show larger fluctuations.

Steps in Time Series Analysis

• Plotting Data: Visualize the time series to observe trends and patterns.
• Decomposition: Break down the series into its components (Trend, Seasonality, Cyclic, Irregular).
• Smoothing: Use techniques like moving averages to remove irregularities.
• Model Selection: Decide whether to use an additive or multiplicative model based on the data.
• Forecasting: Predict future data points using methods like ARIMA, exponential smoothing, or regression analysis.

Example: Let’s say a retail store wants to forecast monthly sales based on historical data.

• Trend: Sales have been increasing steadily due to new marketing strategies.
• Seasonality: Higher sales are observed during the festive seasons (Diwali, Christmas).
• Cyclic: Economic cycles like market booms or recessions impact sales over a few years.
• Irregularity: Random factors like a sudden competitor entering the market may cause unexpected dips.

The store can use the multiplicative model if seasonal variations grow with increasing sales or the additive model if seasonal variations remain consistent.

This structured approach allows businesses to make informed decisions like inventory planning, marketing campaigns, and financial projections.

Trend Analysis: Least Square Method (Linear and Non-Linear Equations)

The Least Square Method is a statistical tool used to determine the line of best fit for a dataset. It minimizes the sum of the squares of the vertical distances between the actual data points and the predicted values provided by a trend line. This method helps identify trends and forecast future values.

1. Linear Equation:

A linear trend assumes a straight-line relationship between the independent variable (typically time) and the dependent variable (such as sales or profits).
The equation for a straight line is:

$$\mathrm{<b>Y</b>}<b>=</b>\mathrm{<b>a</b>}<b>+</b>\mathrm{<b>b</b>}\mathrm{<b>X</b>}$$

Where:

• $Y$ = Dependent variable (forecasted value)
• $X$ = Independent variable (time period)
• $a$ = Intercept (value of $Y$ when $X = 0$)
• $b$ = Slope (rate of change in $Y$ for a unit change in $X$)

Steps to Compute Linear Trend using Least Squares

1. Compute the sums of $X$, $Y$, $X^2$, and $XY$.
2. Use the following formulas to find $a$ and $b$:

$$b = \frac{n \sum XY - \sum X \sum Y}{n \sum X^2 - (\sum X)^2}$$
$$\mathrm{<b>a</b>}<b>=</b>\frac{<b>\sum </b>\mathrm{<b>Y</b>}<b>-</b>\mathrm{<b>b</b>}<b>\sum </b>\mathrm{<b>X</b>}}{\mathrm{<b>n</b>}}$$

Where $n$ is the number of data points.

Example: If a company’s sales over five years are 10, 12, 15, 17, and 20 units, use the least squares method to predict sales for the 6th year.

2. Non-Linear Equations:

The relationship between variables does not follow a straight line. It can take forms such as:

• Exponential: $Y = ae^{bX}$
  
• Quadratic: $Y = a + bX + cX^2$
  

Steps to Compute Non-Linear Trend

1. Transform the non-linear equation into a linear form using logarithms (for exponential equations) or regression techniques (for polynomial equations).
2. Apply the least squares method to fit the transformed data.

Example in Business: A retail store uses trend analysis to forecast monthly sales based on historical data:

• Linear Trend: Sales are consistently increasing due to market growth.
• Non-Linear Trend: Sales initially rise sharply, then stabilize as the market saturates.

By identifying these trends, the store can adjust its inventory, optimize marketing campaigns, and allocate resources more efficiently.

Advantages of the Least Square Method

• Provides accurate trend forecasting by minimizing errors.
• Helps businesses make data-driven decisions.
• Suitable for both linear and non-linear trends.

Limitations

• Sensitive to outliers, which can distort the results.
• Assumes the future will follow historical patterns, which may not always be true.
• Complex for non-linear models requiring transformation techniques.

Index Numbers

An Index Number is a statistical measure used to compare changes in variables such as prices, quantities, or values over time. It provides a way to track trends and measure changes in the level of economic activities or business indicators.

Key Characteristics:

• Expressed in percentage form without the percentage sign.
• The base period value is typically set at 100.
• Used for comparing changes between different periods or locations.

Example: If the Consumer Price Index (CPI) is 120 in the current year compared to a base year of 100, it means prices have increased by 20% since the base year.

Types of Index Numbers

Index numbers can be categorized based on their purpose and method of computation. Below are the main types:

1. Price Index Numbers

These measure changes in the price level of goods and services over time.
Types:

• Wholesale Price Index (WPI): Measures the changes in wholesale prices of goods.
• Consumer Price Index (CPI): Measures changes in the retail prices of goods and services consumed by households.

Formula (Simple Price Index):

$$\mathrm{<b>P</b>}\mathrm{<b>r</b>}\mathrm{<b>i</b>}\mathrm{<b>c</b>}\mathrm{<b>e</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>I</b>}\mathrm{<b>n</b>}\mathrm{<b>d</b>}\mathrm{<b>e</b>}\mathrm{<b>x</b>}\text{<b>\hspace{0.17em}</b>}<b>=</b><b>(</b>\frac{\mathrm{<b>P</b>}\mathrm{<b>r</b>}\mathrm{<b>i</b>}\mathrm{<b>c</b>}\mathrm{<b>e</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>o</b>}\mathrm{<b>f</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>c</b>}\mathrm{<b>u</b>}\mathrm{<b>r</b>}\mathrm{<b>r</b>}\mathrm{<b>e</b>}\mathrm{<b>n</b>}\mathrm{<b>t</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>y</b>}\mathrm{<b>e</b>}\mathrm{<b>a</b>}\mathrm{<b>r</b>}}{\mathrm{<b>P</b>}\mathrm{<b>r</b>}\mathrm{<b>i</b>}\mathrm{<b>c</b>}\mathrm{<b>e</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>o</b>}\mathrm{<b>f</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>b</b>}\mathrm{<b>a</b>}\mathrm{<b>s</b>}\mathrm{<b>e</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>y</b>}\mathrm{<b>e</b>}\mathrm{<b>a</b>}\mathrm{<b>r</b>}}<b>)</b><b>\times </b>\mathrm{<b>100</b>}$$

Application:

• Used to assess inflation trends.
• Helps businesses in pricing strategies.
• Guides policymakers in economic decision-making.

2. Quantity Index Numbers

These track changes in the physical quantity of goods produced, consumed, or sold over time.

Formula:

$$Quantity \, Index \, = \left( \frac{Quantity \, of \, current \, year}{Quantity \, of \, base \, year} \right) \times 100$$

Application:

• Helps in analyzing production trends.
• Useful in inventory and supply chain management.

3. Value Index Numbers

These measure changes in the total monetary value of a variable (price × quantity) over time.

Formula:

$$Value \, Index \, = \left( \frac{Value \, of \, current \, year}{Value \, of \, base \, year} \right) \times 100$$

Application:

• Used to assess the revenue performance of businesses.
• Helps analyze market trends and sales growth.

4. Special Purpose Index Numbers

These are constructed for specific purposes, such as:

• Cost of Living Index: Measures changes in the cost of living for a particular group of people.
• Stock Market Index: Tracks the performance of a group of stocks, such as the SENSEX or NIFTY.
• Human Development Index (HDI): Measures development based on factors like health, education, and income.

Methods of Constructing Index Numbers

1. Simple Aggregative Method:

1. $$\mathrm{<b>I</b>}\mathrm{<b>n</b>}\mathrm{<b>d</b>}\mathrm{<b>e</b>}\mathrm{<b>x</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>N</b>}\mathrm{<b>u</b>}\mathrm{<b>m</b>}\mathrm{<b>b</b>}\mathrm{<b>e</b>}\mathrm{<b>r</b>}<b>=</b><b>(</b>\frac{<b>\sum </b>{\mathrm{<b>P</b>}}_{\mathrm{<b>1</b>}}}{<b>\sum </b>{\mathrm{<b>P</b>}}_{\mathrm{<b>0</b>}}}<b>)</b><b>\times </b>\mathrm{<b>100</b>}$$

   Where:

   • $P_1$ = Prices in the current period
   • $P_0$ = Prices in the base period

2. Simple Average of Relatives Method:

1. $$\mathrm{<b>I</b>}\mathrm{<b>n</b>}\mathrm{<b>d</b>}\mathrm{<b>e</b>}\mathrm{<b>x</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>N</b>}\mathrm{<b>u</b>}\mathrm{<b>m</b>}\mathrm{<b>b</b>}\mathrm{<b>e</b>}\mathrm{<b>r</b>}<b>=</b>\frac{<b>\sum </b>\frac{{\mathrm{<b>P</b>}}_{\mathrm{<b>1</b>}}}{{\mathrm{<b>P</b>}}_{\mathrm{<b>0</b>}}}<b>\times </b>\mathrm{<b>10</b>0}}{\mathrm{<b>n</b>}}$$

3. Weighted Index Numbers:

• Laspeyres' Index: $$I_L = \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times 100$$
  
• Paasche’s Index: $$I_P = \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \times 100$$
  
• Fisher’s Ideal Index: $${\mathrm{<b>I</b>}}_{\mathrm{<b>F</b>}}<b>=</b>\sqrt{{\mathrm{<b>I</b>}}_{\mathrm{<b>L</b>}}<b>\times </b>{\mathrm{<b>I</b>}}_{\mathrm{<b>P</b>}}}$$

Advantages of Index Numbers

• Simplifies complex data for easy comparison.
• Helps in decision-making for businesses and policymakers.
• Measures changes in variables over time.

Limitations of Index Numbers

• Selection of base year can influence results.
• May not account for qualitative changes in products.
• Subjective weight assignment in weighted index numbers.
• Limited scope due to reliance on historical data.

Uses of Index Numbers

Index numbers are versatile tools with wide applications in economics, business, and finance. Below are key uses:

Construction of Price Index Numbers

The construction of price index numbers involves several key steps:

1. Selection of Base Year

• The base year is a reference period, typically represented by 100 in index calculations.
• It should be a normal year without extreme fluctuations.

2. Selection of Items

• Choose representative items that reflect market consumption patterns.
• Include a balanced mix of goods and services.

3. Selection of Data Sources

• Collect reliable data on prices from wholesale or retail markets, government agencies, or trade organizations.

4. Selection of Weights (for Weighted Index Numbers)

• Assign appropriate weights based on the importance of each item in total expenditure or production.

5. Choice of Formula

• Select a method for computation (Simple or Weighted).
• Common methods include Laspeyres', Paasche's, and Fisher's formulas.

Importance of Price Index Numbers in Business

• Cost Control: Helps monitor changes in raw material prices.
• Pricing Strategy: Guides decision-making on product pricing.
• Financial Planning: Aids in forecasting future costs and revenues.
• Market Analysis: Identifies inflation trends affecting purchasing power.

Quantity and Volume Indices

• Quantity Index: Measures changes in the quantity of goods produced, sold, or consumed over time while ignoring price changes.
• Volume Index: Measures changes in the value of goods, considering both price and quantity variations.

Formula (Simple Quantity Index):

$$\mathrm{<b>Q</b>}\mathrm{<b>u</b>}\mathrm{<b>a</b>}\mathrm{<b>n</b>}\mathrm{<b>t</b>}\mathrm{<b>i</b>}\mathrm{<b>t</b>}\mathrm{<b>y</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>I</b>}\mathrm{<b>n</b>}\mathrm{<b>d</b>}\mathrm{<b>e</b>}\mathrm{<b>x</b>}\text{<b>\hspace{0.17em}</b>}<b>=</b><b>(</b>\frac{\mathrm{<b>Q</b>}\mathrm{<b>u</b>}\mathrm{<b>a</b>}\mathrm{<b>n</b>}\mathrm{<b>t</b>}\mathrm{<b>i</b>}\mathrm{<b>t</b>}\mathrm{<b>y</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>o</b>}\mathrm{<b>f</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>C</b>}\mathrm{<b>u</b>}\mathrm{<b>r</b>}\mathrm{<b>r</b>}\mathrm{<b>e</b>}\mathrm{<b>n</b>}\mathrm{<b>t</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>Y</b>}\mathrm{<b>e</b>}\mathrm{<b>a</b>}\mathrm{<b>r</b>}}{\mathrm{<b>Q</b>}\mathrm{<b>u</b>}\mathrm{<b>a</b>}\mathrm{<b>n</b>}\mathrm{<b>t</b>}\mathrm{<b>i</b>}\mathrm{<b>t</b>}\mathrm{<b>y</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>o</b>}\mathrm{<b>f</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>B</b>}\mathrm{<b>a</b>}\mathrm{<b>s</b>}\mathrm{<b>e</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>Y</b>}\mathrm{<b>e</b>}\mathrm{<b>a</b>}\mathrm{<b>r</b>}}<b>)</b><b>\times </b>\mathrm{<b>100</b>}$$

Formula (Value Index):

$$\mathrm{<b>V</b>}\mathrm{<b>a</b>}\mathrm{<b>l</b>}\mathrm{<b>u</b>}\mathrm{<b>e</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>I</b>}\mathrm{<b>n</b>}\mathrm{<b>d</b>}\mathrm{<b>e</b>}\mathrm{<b>x</b>}\text{<b>\hspace{0.17em}</b>}<b>=</b><b>(</b>\frac{\mathrm{<b>V</b>}\mathrm{<b>a</b>}\mathrm{<b>l</b>}\mathrm{<b>u</b>}\mathrm{<b>e</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>o</b>}\mathrm{<b>f</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>C</b>}\mathrm{<b>u</b>}\mathrm{<b>r</b>}\mathrm{<b>r</b>}\mathrm{<b>e</b>}\mathrm{<b>n</b>}\mathrm{<b>t</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>Y</b>}\mathrm{<b>e</b>}\mathrm{<b>a</b>}\mathrm{<b>r</b>}}{\mathrm{<b>V</b>}\mathrm{<b>a</b>}\mathrm{<b>l</b>}\mathrm{<b>u</b>}\mathrm{<b>e</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>o</b>}\mathrm{<b>f</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>B</b>}\mathrm{<b>a</b>}\mathrm{<b>s</b>}\mathrm{<b>e</b>}\text{<b>\hspace{0.17em}</b>}\mathrm{<b>Y</b>}\mathrm{<b>e</b>}\mathrm{<b>a</b>}\mathrm{<b>r</b>}}<b>)</b><b>\times </b>\mathrm{<b>100</b>}$$

Where:

• Value = Price × Quantity

Methods for Measuring Indices

1. Fixed Base Method

In the fixed base method, the quantities or volumes of a specific base year are used as a constant reference point to calculate index numbers for subsequent years.

Formula for Quantity Index (Laspeyres' Method):

$${\mathrm{<b>I</b>}}_{\mathrm{<b>Q</b>}}<b>=</b>\frac{<b>\sum </b>{\mathrm{<b>Q</b>}}_{\mathrm{<b>1</b>}}<b>\times </b>{\mathrm{<b>W</b>}}_{\mathrm{<b>0</b>}}}{<b>\sum </b>{\mathrm{<b>Q</b>}}_{\mathrm{<b>0</b>}}<b>\times </b>{\mathrm{<b>W</b>}}_{\mathrm{<b>0</b>}}}<b>\times </b>\mathrm{<b>100</b>}$$

Where:

• $Q_1$ = Quantity in the current year
• $Q_0$ = Quantity in the base year
• $W_0$ = Weight based on the base year

Steps to Calculate:

1. Select the base year.
2. Gather data on quantities or volumes.
3. Calculate the index for each year using the base year data.

Advantages:

• Simple to compute and compare over time.
• Easy to interpret trends relative to a single base year.

Disadvantages:

• Becomes less relevant if the base year data becomes outdated.

2. Chain Base Method

In the chain base method, each year serves as the base year for the next year. This provides a dynamic approach by linking successive years.

Formula for Chain Base Quantity Index:

$${\mathrm{<b>I</b>}}_{\mathrm{<b>Q</b>}}<b>=</b>\frac{{\mathrm{<b>Q</b>}}_{\mathrm{<b>n</b>}<b>+</b>\mathrm{<b>1</b>}}}{{\mathrm{<b>Q</b>}}_{\mathrm{<b>n</b>}}}<b>\times </b>\mathrm{<b>100</b>}$$

Steps to Calculate:

• Use the previous year as the base year.
• Calculate the index number for each year.
• Multiply successive indices to create a continuous series.

Advantages:

• Keeps the index relevant by updating the base year.
• Suitable for rapidly changing environments.

Disadvantages:

• Complex to interpret due to changing base years.
• Errors can accumulate when linking indices.