Index Numbers & Time Series: Construction, Uses & Analysis

Core Concepts

1. Index Numbers: Definitions & Types

Index numbers measure changes in a variable (price, quantity, value) over time or between groups, expressed as a percentage of a base value (usually 100).

Key Types

classDiagram
    class IndexNumber {
        +Base Year = 100
        +Formula: (Current Value / Base Value) * 100
    }
    class PriceIndex {
        +Measures price changes
        +Types: Simple, Aggregative, Weighted
    }
    class QuantityIndex {
        +Measures volume changes
        +Types: Laspeyres, Paasche, Fisher
    }
    IndexNumber <|-- PriceIndex
    IndexNumber <|-- QuantityIndex

How They Work

• Simple Index: Uses a single commodity’s price change. Example: If a commodity costs Rs. 50 in 2022 and Rs. 60 in 2023, its simple index for 2023 is: $$\text{Index}=\left(\frac{60}{50}\right)\times 100=120‘‘‘-\ast \ast AggregativeIndex\ast \ast :Sumspricesofallcommodities(unweighted).\ast Example\ast :ForcommoditiesA–E(pricesin2022/2023:45/55,58/65,35/40,20/25,18/22),theaggregativepriceindexfor2023is:\text{[}\text{Index}=\left(\frac{55+65+40+25+22}{45+58+35+20+18}\right)\times 100=118.5‘‘‘-\ast \ast WeightedIndex\ast \ast :Usesweights(quantities/sales)toreflectimportance.\ast Example\ast :ForNEPSE’stopstocks(weights=marketcap),the\ast \ast LaspeyresPriceIndex(LPI)\ast \ast for2023(base2022)is:\text{[}LPI=\left(\frac{\sum P_{2023}\times Q_{2022}}{\sum P_{2022}\times Q_{2022}}\right)\times 100$$ Where $P_{2023}$ = 2023 prices, $Q_{2022}$ = 2022 quantities.

Comparison Table

2. Time Series Analysis

Time series data tracks a variable (e.g., sales, inflation) over time. It decomposes into:

1. Trend: Long-term movement (e.g., rising NTC electricity demand).
2. Seasonal: Repeating patterns (e.g., Daraz sales spikes in Dashain).
3. Cyclical: Economic cycles (e.g., NEPSE booms/busts).
4. Irregular: Random shocks (e.g., COVID-19 lockdowns).

Decomposition Methods

flowchart TD
    A["Time Series Data"] --> B["Trend Component"]
    A --> C["Seasonal Component"]
    A --> D["Cyclical Component"]
    A --> E["Irregular Component"]
    B --> F["Moving Averages"]
    C --> G["Seasonal Index"]
    D --> H["Economic Indicators"]
    E --> I["Residual Analysis"]

Worked Example: NTC Tariff Analysis

Data: Monthly electricity tariffs (Rs/kWh) for 2022–2023:

Steps:

1. Detrend: Apply a 12-month moving average to isolate seasonal effects. Example: For Jan 2022–Jan 2023, the moving average smooths the trend: $$\text{Trend}=\frac{6.50+6.70+\cdots +7.20}{12}=6.95\text{ (approx)}$$

2. Seasonal Index: Divide actual by trend to find seasonal factors. $$\text{Seasonal Index (Jan)}=\frac{6.50}{6.95}\times 100=93.5\%$$ Interpretation: January tariffs are 6.5% below trend (likely due to winter demand).

3. Forecast: Combine trend + seasonal adjustment. $$\text{Forecast Jan 2024}={\text{Trend}}_{2024}\times 0.935$$

In the Real World

1. NEPSE Index (Fisher Ideal Index)

   • How it’s used: NEPSE constructs a weighted index of Nepal’s top stocks (e.g., NMB, NTC, Ncell) using the Fisher formula to minimize bias. Weights are based on market capitalization.
   • Why it matters: Investors use it to track market performance. For example, if NEPSE rises from 1,800 to 1,900 (base 1,000), the index shows a 9% gain in stock prices.

2. eSewa’s Price Index for Digital Services

   • How it’s used: eSewa tracks the Laspeyres Price Index for digital transactions (e.g., Rs. 2.50 fee in 2022 vs. Rs. 3.00 in 2023, with 2022 quantities as weights).
   • Real calculation: If 80% of users paid Rs. 2.50 in 2022 and Rs. 3.00 in 2023: $$LPI=\left(\frac{3.00\times 80+2.00\times 20}{2.50\times 80+1.50\times 20}\right)\times 100=116.7$$
   • Impact: Shows a 16.7% increase in transaction costs, helping users budget.

3. NTC’s Time Series for Load Shedding

   • How it’s used: NTC analyzes monthly electricity demand (in MW) to predict shortages. Decomposing the series reveals:
     • Trend: Demand grows by 5% annually (urbanization).
     • Seasonal: Peaks in June–September (monsoon cooling) and December–January (winter heating).
     • Irregular: Sudden drops during protests (e.g., 2022 Kathmandu blackouts).
   • Action: NTC uses this to schedule load shedding (e.g., "Avoid 4–6 PM in July").

Key Formulas

1. Price Index Numbers

2. Quantity Index Numbers

3. Time Series Decomposition

$$Y_t=T_t+S_t+C_t+I_t$$

• $Y_t$ = Observed value
• $T_t$ = Trend (e.g., linear regression)
• $S_t$ = Seasonal (e.g., Jan = 0.935)
• $C_t$ = Cyclical (e.g., economic downturn)
• $I_t$ = Irregular (e.g., strikes)

Worked Example: Daraz’s Sales Forecast

Problem: Daraz wants to forecast Dashain sales (Rs. millions) using 3 years of data:

Steps:

1. Calculate Trend: Use moving averages (3-month). Trend for Nov 2023 = (120 + 130 + 140)/3 = 130 (approx).

2. Find Seasonal Index:

   • For November, divide actual by trend: $$\text{Seasonal Index (Nov)}=\frac{130}{130}\times 100=100\%$$ Interpretation: November sales are on-trend (no seasonal boost).

3. Forecast 2024:

   • Assume trend grows by 5%/year: Trend Nov 2024 = 130 × 1.05 = 136.5.
   • Apply seasonal index: 136.5 × 1.00 = Rs. 136.5 million.

Exam Tip

What Examiners Want

1. Index Numbers:

   • Always state the formula before plugging numbers.
   • Label axes in graphs (e.g., "Year" vs. "Price Index").
   • Compare Laspeyres vs. Paasche: Laspeyres uses base quantities, Paasche uses current quantities. Fisher is the geometric mean of both.

2. Time Series:

   • Decomposition steps: Trend → Seasonal → Cyclical → Irregular.
   • Moving averages: Use odd periods (e.g., 3, 5, 12 months) to center data.
   • Seasonal adjustment: If a month’s index is >100, it’s above trend; if <100, it’s below trend.

3. Real-World Links:

   • NEPSE: Use Fisher index for stock prices.
   • NTC: Decompose electricity demand for load shedding.
   • eSewa: Calculate Laspeyres index for fee hikes.

Common Mistakes to Avoid

• Forgetting weights in Laspeyres/Paasche (e.g., using quantities incorrectly).
• Ignoring the base year (always set to 100).
• Mismatched data: Ensure prices/quantities align with the year.
• Skipping units: Always label answers (e.g., "Index = 118.5%" not "118.5").

Visual Summary: Left: Laspeyres vs. Paasche curves for NEPSE stocks (2022–2023). Right: Time series decomposition of NTC’s monthly demand (trend + seasonal peaks).