Time Series Analysis & Index Numbers: Trends, Cycles, Seasonality & Price Indices

TAKEAWAYS:

• Time series decomposition breaks data into T+C+S+I components (trend, cycle, seasonality, irregular) using moving averages and centred moving averages to isolate patterns.
• Index numbers measure relative changes (e.g., inflation) using Laspeyres (fixed base), Paasche (current base), or Fisher’s ideal index—each has trade-offs in bias and accuracy.
• Seasonal adjustment uses ratio-to-moving-average or deseasonalization to remove periodic fluctuations (e.g., Daraz’s monthly sales spikes during Dashain).
• Trend analysis employs linear regression or exponential smoothing to forecast future values (e.g., NTC’s monthly electricity demand).
• Index number limitations: Base year selection, price vs. quantity bias, and Fisher’s index resolves these via geometric mean.
• Real-world tie-ins: NEPSE’s price index tracks stock market trends; NTC’s consumer price index (CPI) adjusts tariffs; Daraz’s sales index optimizes inventory.

1. Time Series Analysis: The Big Picture

Time series data is ordered observations over time (e.g., monthly sales, quarterly GDP, daily temperatures). Businesses use it to:

• Predict future values (e.g., NTC forecasting peak-hour electricity demand).
• Identify patterns (e.g., Kathmandu traffic congestion during Dashain).
• Remove noise for clearer decision-making.

Components of a Time Series

Every time series is a mix of four components:

1. Trend (T): Long-term movement (e.g., rising smartphone sales in Nepal).
2. Cyclical (C): Repeating ups/downs over years (e.g., economic booms/busts).
3. Seasonal (S): Short-term, periodic fluctuations (e.g., eSewa transactions spiking on paydays).
4. Irregular (I): Random shocks (e.g., a sudden Ncell network outage).

Formula: $$Y=T+C+S+I$$ (Where $Y$ = observed value)

Caption: Daraz’s sales show a rising trend, seasonal spikes in Nov/Dec, and a cyclical uptick in Q4 2023.

2. Decomposing Time Series: Step-by-Step

Step 1: Smooth the Data with Moving Averages

To isolate trend-cycle (T+C), we use moving averages (MA). For seasonal data, use:

• Odd-period MA (e.g., 3-month MA) for quarterly data.
• Even-period MA (e.g., 12-month MA) for monthly data, then centre the MA to align with midpoints.

Example: NTC’s monthly electricity consumption (2022–2023):

Calculation for Jan 2023 (12-month MA): $${\text{MA}}_{Jan2023}=\frac{850+880+\cdots +1020}{12}=950$$

Centred MA for Feb 2023: $$\text{Centred MA}=\frac{950+970}{2}=960$$

Caption: The 12-month MA smooths out seasonal fluctuations, revealing the underlying trend.

Step 2: Extract Seasonal Component (S)

For monthly data, divide the original value (Y) by the centred MA (T+C): $$S=\frac{Y}{T+C}$$ Average these ratios for each month across years to get seasonal indices.

Example: Daraz’s monthly sales seasonal indices (2021–2022):

Interpretation:

• Nov/Dec: 50–60% seasonal boost (Dashain/Tihar sales).
• Feb: 15% dip (post-holiday lull).

Step 3: Deseasonalize the Data

Subtract the seasonal effect to isolate trend-cycle-irregular (T+C+I): $$Y_{\text{deseasonalized}}=\frac{Y}{S}$$

Example: Adjust Daraz’s Jan 2023 sales (1200 units) for seasonality: $$Y_{\text{adj}}=\frac{1200}{0.9}=1333$$ (True trend-cycle-irregular value is 1333, not 1200.)

Caption: Daraz’s sales peak in Dec (1.6x baseline) and dip in Feb (0.85x).

Step 4: Isolate Irregular Component (I)

Subtract the trend-cycle-seasonal (T+C+S) from the deseasonalized data: $$I=Y_{\text{deseasonalized}}-(T+C)$$

3. Forecasting with Time Series

Method 1: Naïve Forecasting

Assume next period = last period’s value.

• Pros: Simple, no data needed.
• Cons: Ignores trends/seasonality.
• Example: If NTC’s Jan 2023 demand = 1050 kWh, forecast Feb 2023 = 1050 kWh.

Method 2: Linear Trend Extrapolation

Fit a straight line $Y=a+bX$ to the trend-cycle (T+C) component. Example: NTC’s trend (2022–2023): $$Y=800+50X$$ (where $X$ = month number)

• Forecast for Jan 2024 ($X=25$): $800+50(25)=2050$ kWh.

Caption: The linear trend predicts a 50 kWh/month increase in demand.

Method 3: Exponential Smoothing

Weights recent data more heavily: $$F_{t+1}=\alpha Y_t+(1-\alpha )F_t$$ (where $\alpha$ = smoothing factor, $0<\alpha <1$)

Example: Daraz’s sales ($\alpha =0.3$):

4. Index Numbers: Measuring Relative Change

Index numbers compare values over time (e.g., inflation, stock prices). Key types:

Where:

• $p_0,p_1$ = prices in base/current year.
• $q_0,q_1$ = quantities in base/current year.

Worked Example: NTC’s Consumer Price Index (CPI)

Data:

Step 1: Laspeyres Index (Fixed Base) $$\text{Laspeyres}=\frac{(150\times 10)+(90\times 5)+(7\times 200)}{(120\times 10)+(80\times 5)+(5\times 200)}\times 100$$ $$=\frac{1500+450+1400}{1200+400+1000}\times 100=\frac{3350}{2600}\times 100=128.8\%$$

Step 2: Paasche Index (Current Base) (Assume current quantities $q_1$ are the same as $q_0$ for simplicity.) $$\text{Paasche}=\frac{(150\times 10)+(90\times 5)+(7\times 200)}{(120\times 10)+(80\times 5)+(5\times 200)}\times 100$$ (Same as Laspeyres here, but in practice $q_1$ differs.)

Step 3: Fisher’s Ideal Index $$\text{Fisher}=\sqrt{128.8\times 128.8}=128.8\%$$ (If Paasche differed, Fisher would average them.)

Interpretation:

• NTC’s CPI = 128.8 means prices rose 28.8% from 2020–2023.
• Used to adjust electricity tariffs or pension payments.

Caption: Electricity contributes most to NTC’s CPI basket.

5. Real-World Applications

Example 1: NEPSE’s Price Index

• What it uses: Laspeyres index with a fixed basket of stocks (e.g., NABIL, NMB, NTC).
• How it works:
  • Base year = 2015 (index = 100).
  • Current index = $\frac{\text{Current stock price}}{\text{Base stock price}}\times 100$.
• Why it matters: Investors use it to compare returns across years.

Example 2: Daraz’s Sales Index

• What it uses: Seasonally adjusted time series + Paasche index for dynamic inventory planning.
• How it works:
  • Tracks monthly sales vs. a moving average.
  • Adjusts for Dashain/Janai Purnima spikes.
• Why it matters: Optimizes warehouse stock for peak seasons.

Example 3: NTC’s Tariff Adjustments

• What it uses: Consumer Price Index (CPI) based on Laspeyres.
• How it works:
  • If CPI rises from 100 to 120, tariffs increase by 20%.
• Why it matters: Ensures revenue covers inflation.

6. Common Pitfalls & Exam Tips

Mistakes to Avoid

1. Ignoring seasonality: Always adjust for seasonal patterns (e.g., Daraz’s Nov/Dec spikes).
2. Wrong MA period: Use 12-month MA for monthly data, not 3-month.
3. Base year bias: Laspeyres overstates inflation; Paasche understates it.
4. Forgetting to centre MAs: Even-period MAs must be centred for alignment.

Exam Tip: Step-by-Step Approach

For time series decomposition questions:

1. Plot the data (identify trends/seasonality).
2. Apply moving averages (odd/even periods correctly).
3. Calculate seasonal indices (average ratios per month).
4. Deseasonalize and isolate irregular component.
5. Forecast using trend extrapolation or smoothing.

For index number questions:

1. Identify Laspeyres/Paasche/Fisher required.
2. Write the formula clearly.
3. Show all calculations (partial credit for steps).
4. Interpret the result (e.g., "Inflation rose by X%").

flowchart TD
    A["Time Series Data"] --> B["Step 1: Apply Moving Averages\n(Isolate Trend-Cycle)"]
    B --> C["Step 2: Calculate Seasonal Indices\n(Y / (T+C))"]
    C --> D["Step 3: Deseasonalize\n(Y_adj = Y / S)"]
    D --> E["Step 4: Isolate Irregular Component\n(I = Y_adj - (T+C))"]
    E --> F["Step 5: Forecast\n(Naïve/Trend/Exponential Smoothing)"]
    F --> G["Business Decision\n(e.g., NTC tariff adjustment)"]

classDiagram
    class IndexType {
        +Laspeyres: Fixed Base
        +Paasche: Current Base
        +Fisher: Geometric Mean
    }
    class Application {
        +NTC: CPI for Tariffs
        +NEPSE: Stock Price Index
        +Daraz: Sales Forecasting
    }
    IndexType --> Application : "Used in"