Index Numbers & Time Series: Construction, Uses & Analysis

Core Concepts

1. Index Numbers: Measuring Change Over Time

Definition: An index number is a statistical measure that shows changes in a variable (or group of variables) over time or between different groups, expressed as a percentage of a base value (usually 100).

Key Types of Index Numbers

graph TD
    A["Index Numbers"] --> B["Price Index"]
    A --> C["Quantity Index"]
    A --> D["Value Index"]
    A --> E["Composite Index"]
    B --> F["Consumer Price Index (CPI)"]
    B --> G["Wholesale Price Index (WPI)"]
    C --> H["Production Index"]
    D --> I["Gross Domestic Product (GDP) Deflator"]
    E --> J["Cost of Living Index"]

Methods of Construction

Three widely used methods are compared below:

Worked Example 1: Laspeyres Price Index for NEPSE Stocks Assume the following data for NEPSE stocks (base year 2020):

Step-by-Step Calculation:

1. Compute the total expenditure in the base year (2020): $$\sum p_0{q}_0=(200\times 500)+(150\times 300)+(120\times 400)=100,000+45,000+48,000=193,000$$
2. Compute the total expenditure in the current year (2023) using base quantities: $$\sum p_n{q}_0=(280\times 500)+(210\times 300)+(180\times 400)=140,000+63,000+72,000=275,000$$
3. Calculate the Laspeyres Price Index: $$P_L=\left(\frac{275,000}{193,000}\right)\times 100=142.5$$ Interpretation: NEPSE stocks increased by 42.5% from 2020 to 2023.

2. Time Series Analysis: Decomposing Trends

Definition: A time series is a collection of observations made sequentially over time. It helps in forecasting future values by analyzing patterns like trend, seasonality, cyclicality, and irregular fluctuations.

Components of a Time Series

graph TD
    A["Time Series"] --> B["Trend (T)"]
    A --> C["Seasonal (S)"]
    A --> D["Cyclical (C)"]
    A --> E["Irregular (I)"]
    A --> F["Additive Model: Y = T + S + C + I"]
    A --> G["Multiplicative Model: Y = T × S × C × I"]

Worked Example 2: Decomposing NTC Electricity Consumption Assume the following monthly electricity consumption (in million kWh) for NTC from 2022 to 2023:

Step 1: Calculate the Trend (Moving Average)

• Use a 12-month centered moving average to smooth out seasonal fluctuations.
• For simplicity, we’ll use a 4-month moving average (since seasonal patterns in electricity are often quarterly).

Step 2: Identify Seasonal Patterns

• Compare actual values to the trend to find seasonal indices.
• For December, consumption is consistently higher (180 vs. trend ~150), indicating a winter peak.

3. Applications in Real-World Business Scenarios

In the Real World

1. eSewa & Khalti (Inflation Measurement)

   • Idea Used: Consumer Price Index (CPI) (a type of Laspeyres index).
   • How? eSewa and Khalti track transaction volumes and prices of essential goods (e.g., mobile recharge, electricity bills, grocery) to estimate inflation in Nepal. For example, if the CPI rises from 100 to 115, it signals a 15% increase in the cost of living, prompting Khalti to adjust transaction fees or eSewa to revise service charges.

2. NTC (Electricity Tariff Adjustments)

   • Idea Used: Time Series Decomposition (trend + seasonality).
   • How? NTC uses historical consumption data to predict peak demand (e.g., December-January) and adjust tariffs. If the trend shows a 5% annual increase in summer AC usage, NTC may introduce dynamic pricing to manage load.

3. NEPSE (Stock Market Index)

   • Idea Used: Composite Price Index (Fisher’s Ideal).
   • How? NEPSE’s index (e.g., NEPSE Index) combines price changes of top stocks (like Nabil Bank, Himalayan Bank) using Fisher’s formula to give a balanced view of market performance. Investors use this to decide whether to buy/sell stocks.

4. Pathao & Daraz (Demand Forecasting)

   • Idea Used: Seasonal Adjustment in Time Series.
   • How? Pathao analyzes ride demand data to find that weekday evenings (6–9 PM) have 30% higher demand than weekends. Daraz uses this to optimize delivery routes and stock inventory for peak shopping seasons (e.g., Dashain, Tihar).

4. Worked Example 3: Calculating the Cost of Living Index (Paasche Method)

Assume the following data for a worker’s basket of goods in Kathmandu (base year 2022):

Step-by-Step Calculation:

1. Compute total expenditure in the base year (2022): $$\sum p_0{q}_0=(120\times 5)+(200\times 2)+(250\times 4)=600+400+1000=2000$$

2. Compute total expenditure in the current year (2023) using current quantities: $$\sum p_n{q}_n=(150\times 4)+(250\times 1.5)+(300\times 3)=600+375+900=1875$$ Wait! This seems incorrect because the Paasche index uses current-year quantities in the numerator but base-year prices in the denominator. Let’s correct this:

   The correct Paasche formula is: $$P_P=\frac{\sum p_n{q}_n}{\sum p_0{q}_n}\times 100$$ But for the Paasche Price Index, we need to compute: $$\sum p_0{q}_n=(120\times 4)+(200\times 1.5)+(250\times 3)=480+300+750=1530$$ $$P_P=\left(\frac{1875}{1530}\right)\times 100=122.5$$ Interpretation: The cost of living decreased by 27.5% (from 100 to 72.5), which is counterintuitive. This suggests a mistake in the approach.

   Correction: The Paasche index should compare current-year prices to base-year prices using current quantities: $$P_P=\frac{\sum p_n{q}_n}{\sum p_0{q}_n}\times 100=\frac{1875}{1530}\times 100=122.5$$ But this still doesn’t make sense. Let’s re-express the Paasche formula correctly:

   The Paasche Price Index is: $$P_P=\frac{\sum p_n{q}_n}{\sum p_0{q}_n}\times 100$$ Here, $\sum p_n{q}_n=1875$ (current expenditure) and $\sum p_0{q}_n=1530$ (what current quantities would cost at base prices). $$P_P=\frac{1875}{1530}\times 100=122.5$$ This implies prices increased by 22.5%, which aligns with intuition (prices rose, but quantities bought fell).

5. Comparing Index Number Methods

6. Time Series Forecasting: Moving Averages

Worked Example 4: Predicting Ncell’s Monthly Revenue Assume Ncell’s monthly revenue (in Rs. millions) for 2023:

Step 1: Compute a 3-Month Moving Average $${\text{MA}}_3=\frac{Y_{t-1}+Y_t+Y_{t+1}}{3}$$ For April: $${\text{MA}}_3=\frac{420+480+460}{3}=453.33$$

Step 2: Forecast for January 2024

• Use the last 3-month MA (Oct, Nov, Dec): $$\text{Forecast}=\frac{500+550+540}{3}=530\text{ million Rs.}$$

Exam Tip

1. Index Numbers:

   • Always clearly state whether you’re using Laspeyres, Paasche, or Fisher’s method.
   • Label axes in graphs (e.g., "Price Index (2020=100)").
   • Interpret results in plain language (e.g., "Prices increased by 25%").

2. Time Series:

   • For moving averages, specify the window (e.g., "3-month MA").
   • In decomposition, separate trend, seasonal, and cyclical components visually.
   • Forecasting: Use the last available MA for predictions.

3. Common Pitfalls:

   • Mixing up quantities: Laspeyres uses base quantities, Paasche uses current quantities.
   • Ignoring seasonal adjustments: Always check if data has seasonal patterns (e.g., NTC electricity peaks in winter).
   • Units: Ensure all values are in the same unit (e.g., Rs., kg, kWh) before calculations.

Final Note: Index numbers and time series are essential for economic analysis, business forecasting, and policy-making. Master these concepts, and you’ll ace questions on inflation, stock market trends, and demand forecasting in your exams!