Index Numbers and Time Series Analysis

Index Numbers and Time Series Analysis

Syllabus tag: KASNEB CPA | Advanced Level | CA34S1 Business Data Analytics

1. Index numbers

An index number measures the relative change in a variable (price, quantity, or value) over time, compared to a chosen base period (index = 100).

Simple price index:

Price index = (Current price / Base period price) × 100

Example: base price = KES 200, current price = KES 250 → Index = (250/200) × 100 = 125.0. This means prices have risen 25% from the base period.

2. Weighted price indices

A weighted index accounts for the relative importance of different items in the basket, preventing high-priced but low-volume items from distorting the index.

Laspeyres price index (base-period quantity weights):

Laspeyres = [Σ(P₁ × Q₀) / Σ(P₀ × Q₀)] × 100

Paasche price index (current-period quantity weights):

Paasche = [Σ(P₁ × Q₁) / Σ(P₀ × Q₁)] × 100

Fisher's ideal index: geometric mean of Laspeyres and Paasche — considered the most theoretically sound.

Percentage change between two index periods:

% change = [(Index year 2 – Index year 1) / Index year 1] × 100

3. Consumer Price Index (CPI)

The CPI measures the cost of a fixed basket of goods and services consumed by a typical household. It is the primary measure of inflation in Kenya (published monthly by the Kenya National Bureau of Statistics, KNBS). The basket is updated periodically to reflect changing consumption patterns.

Real vs nominal values: to remove the effect of inflation:

Real value = Nominal value × (Base year CPI / Current year CPI)

4. Time series components

A time series is a sequence of observations recorded at regular time intervals. Components:

• Trend (T): the long-term direction of the series (upward, downward, or flat)
• Seasonal variation (S): regular, predictable fluctuations within a year (e.g. higher retail sales in December)
• Cyclical variation (C): longer-term waves following the business cycle (expansion, peak, contraction, trough) — not always predictable
• Random/irregular variation (R): unpredictable, one-off movements

Additive model: Y = T + S + C + R (when variations are constant in absolute terms) Multiplicative model: Y = T × S × C × R (when variations grow proportionally with trend — more common in business)

5. Moving averages

A moving average smooths out short-term fluctuations to reveal the underlying trend. A 3-point moving average takes the average of three consecutive observations; a 4-point centred moving average for quarterly data is commonly used to isolate trend.

6. Seasonal adjustment

To remove seasonal effects and compare underlying trends: calculate seasonal indices (average seasonal variation for each period), then divide observed values by the seasonal index. Seasonally adjusted series are used by governments and central banks to assess the true state of the economy.