Index Numbers & Time Series: Trends, Forecasting & Analysis

Key Concepts & Definitions

1. Index Numbers

An index number is a statistical measure that shows changes in a variable (price, quantity, value) over time, relative to a base period (usually 100).

Types of Index Numbers

Weighted vs. Unweighted Index

Worked Example 1: Price Index Given:

• Base year (2020) prices: $P_0=[100,200,300]$ (Rs.)
• Current year (2023) prices: $P_1=[120,250,360]$ (Rs.)
• Quantities in 2020: $Q_0=[5,10,15]$

Calculate the price index for 2023 (base = 2020). $$P_{01}=\frac{8500}{10500}\times 100=80.95$$ Interpretation: Prices in 2023 are 80.95% of 2020 levels (a 19.05% decrease).

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2. Time Series Analysis

A time series is a collection of observations ordered by time (e.g., daily hotel bookings, monthly inflation).

Components of Time Series

Decomposition Methods

1. Additive Model: $Y=T+S+C+I$
   • Used when seasonality is constant (e.g., fixed monthly sales).
2. Multiplicative Model: $Y=T\times S\times C\times I$
   • Used when seasonality grows with trend (e.g., hotel bookings).

Worked Example 2: Trend Analysis Given: Quarterly sales (in Rs. lakhs) for a hotel:

Plot the trend (ignore seasonality for now): Observation: Sales are increasing by ~10 lakhs/year (trend).

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In the Real World

1. eSewa/Khalti (Nepal):

   • Uses price indices to adjust transaction fees for inflation.
   • Example: If CPI rises from 100 to 110, fees may increase by 10% to maintain revenue.

2. NEPSE (Nepal Stock Exchange):

   • Publishes the Nepal Stock Index (Nepse Index), a weighted price index tracking 50+ stocks.
   • Formula: $\text{Nepse Index}=\frac{\sum P_{\text{current}}{Q}_{\text{base}}}{\sum P_{\text{base}}{Q}_{\text{base}}}\times 100$.

3. Hotel Demand Forecasting (Pathao/Booking.com):

   • Uses time series decomposition to predict peak seasons (e.g., Dashain, Christmas).
   • Example: If Q4 sales are always 30% higher than Q1, the system auto-adjusts room prices.

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Applications in Hotel Management

Example of a hotel occupancy index used in practice (Image: Toby Hudson, CC BY-SA 3.0 au, via Wikimedia Commons)

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Exam Tip

1. Index Numbers:

   • Always define the base year clearly.
   • For weighted indices, show the weight calculation (e.g., % of total expenditure).
   • Common mistake: Forgetting to multiply by 100 in the final index formula.

2. Time Series:

   • Plot the data (even in exams) to identify trends/seasonality.
   • For decomposition, state whether additive or multiplicative is used.
   • Forecasting: Use the last observed trend if no other method is given (e.g., "Sales increased by 5 lakhs/year; next year’s forecast = current sales + 5").

3. Real-World Links:

   • NEPSE, eSewa, and hotel chains love time series for decision-making.
   • Inflation adjustments (e.g., salary hikes) use price indices.

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Practice Question: Given the following data for a hotel’s monthly electricity bills (Rs. lakhs):

1. Calculate the price index for 2023 (base = 2022).
2. Identify the trend and seasonality (if any).
3. Forecast the April 2023 bill assuming the trend continues.

(Hint: Use the additive model for decomposition.)