Chapter 13 Time series and index numbers
13.1 Time series
A time series is a sequence of observations of a variable recorded at successive points in time:
\[\{x_t\}_{t=1}^T = x_1, x_2, \dots, x_T\]
Examples include annual GDP, monthly unemployment rates, daily stock prices, or yearly average cinema ticket prices.
Time series analysis focuses on: the direction and speed of change of a variable and comparisons of its value across time.
13.1.1 Flow and stock
Many economic time series variables can be classified into stocks and flows:
- Stock variables are measured at a specific point in time.
Examples: population on 31 December, money supply at the end of the year, inventory levels.
- Flow variables are measured over a period of time.
Examples: annual income, monthly exports, quarterly GDP.
The stock–flow distinction is fundamental but not universal. Many important time series – such as prices, rates, ratios, and indices – do not belong to either category.
13.2 Absolute and relative change
Let \(x_0\) be the value of a variable in the base period and \(x_t\) its value in period \(t\).
13.2.1 Absolute change
The absolute change is defined as:
\[\Delta x = x_t - x_0 \tag{13.1}\]
It measures the change in original units (e.g. PLN, euros, number of guitars).
13.2.2 Relative change (growth rate)
The relative change (or growth rate) expresses change relative to the initial level:
\[g = \frac{x_t - x_0}{x_0} \tag{13.2}\]
Often expressed as a percentage:
\[g(\%) = \frac{x_t - x_0}{x_0} \cdot 100\% \tag{13.3}\]
Relative measures are essential when one compares changes in variables with different units.
13.3 Index numbers
An index number expresses the level of a variable relative in period \(t\) to a chosen base period \(b\), usually normalized to 100.
For a simple index:
\[I_{t/b} = \frac{x_t}{x_b} \cdot 100 \tag{13.4}\]
Interpretation:
- \(I = 100\): no change in comparison with the base period
- \(I > 100\): increase
- \(I < 100\): decrease
13.3.1 Fixed-based index
A fixed-base index always compares period \(t\) to the same base period, usually denoted as \(0\):
\[I^{FB}_t = \frac{x_t}{x_0} \cdot 100 \tag{13.5}\]
Advantages:
- Easy interpretation
- Direct comparability to the base year
Disadvantage:
- Base period may become outdated over long horizons
13.3.2 Chain index
A chain index compares each period only to the previous one:
\[I^{CH}_n = \frac{x_t}{x_{t-1}} \cdot 100 \tag{13.6}\]
The fixed-base index with base period \(0\) can be recovered by chaining together the successive chain index numbers. The fixed-base index for period \(t\) (base \(0\)) is obtained as:
\[I^{FB}_t = \prod_{i=1}^{t}\left(\frac{I^{CH}_i}{100}\right) \cdot 100 \tag{13.7}\]
Chain indices are flexible but require careful interpretation when accumulated over time.
13.4 Average rate of change
When a variable changes over multiple periods, we often seek a single average rate that summarizes the entire process. Compound Annual Growth Rate (CAGR) — or its analogue for non-annual periods—is the usual answer.
The key idea of CAGR is to replace a sequence of possibly uneven growth rates with a constant rate that would generate the same total change over the whole period.
Let \(x_0\) be the initial value, \(x_n\) the final value, and \(n\) the number of periods.
The average annual rate of change is defined as:
\[\text{CAGR} = \left( \frac{x_n}{x_0} \right)^{\frac{1}{n}} - 1 \tag{13.8}\]
Expressed in percentages:
\[\text{CAGR}(\%) = \left[ \left( \frac{x_n}{x_0} \right)^{\frac{1}{n}} - 1 \right] \cdot 100\% \tag{13.9}\]
This measure accounts for compounding and is preferred over arithmetic averages of growth rates.
13.5 Aggregate price indices
When analyzing multiple goods, prices and quantities must be aggregated using weights.
Let:
- \(p_t\) – price vector in period \(t\)
- \(q_t\) – quantity vector in period \(t\)
13.5.1 Laspeyres index
The Laspeyres price index uses base-period quantities as weights:
\[I = \frac{\sum p_t q_0}{\sum p_0 q_0}\cdot 100, \tag{13.10}\]
Interpretation:
- Measures how much more (or less) it would cost in period (n) to buy the base-period basket.
- Tends to overstate inflation, as it ignores substitution effects.
13.6 Exercises
Exercise 13.1 The table below presents price dynamics in San Escobar. Using the information from the table, calculate the fixed-base price index for the year 2019.
| Year | Chain index | Fixed base index (2017 = 100) |
|---|---|---|
| 2017 | 110 | 100 |
| 2018 | 120 | 120 |
| 2019 | 110 |
Exercise 13.2 The table below presents the end-of-year prices of fund X and the indices calculated from them. Complete missing prices, as well as fixed-base and chain indices, then compute CAGRs over selected subperiods.
Assume [2017] = 100 for the fixed-base index. All prices are integers.
| 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | |
|---|---|---|---|---|---|---|---|
| price | 1695 | 1637 | |||||
| fixed-base index | 103.26 | 129.02 | |||||
| chain index | 129.09 | 108.61 | 115.36 |
Calculate the CAGR between the end of 2015 and the end of 2020: %.
Calculate the CAGR between the end of 2014 and the end of 2020: %.
Calculate the CAGR between the end of 2014 and the end of 2021: %.
Exercise 13.3 According to the Gapminder website (https://www.gapminder.org/topics/guitars-per-capita/), the number of guitars per million inhabitants of the Earth increased from 200 guitars per million in 1962 to 11,000 guitars in 2014. What was the average annual rate of change?
Exercise 13.4 Using the Central Statistical Office’s Local Data Bank (Bank Danych Lokalnych GUS), find the average ticket price for cinemas and theatres for each available year. Then compute average annual rate of change for both series over the full period covered by the data.
Exercise 13.5
Prices rise by 20% in 2024 and then fall by 20% in 2025. What is the CAGR? %
Prices rise by 20% and next year fall by 25%. What is the CAGR? %
Prices rise by 25% and next year fall by 20%. What is the CAGR? %
Exercise 13.6 Treat the basket in the table as a student consumption basket. Calculate the Laspeyres, Paasche and Fisher price indices for 2025 (2015 = 100) and discuss their interpretation as measures of student inflation.
| product | p0 | p1 | q0 | q1 |
|---|---|---|---|---|
| Cheeseburger | 3.0 | 6.0 | 5 | 6 |
| Donut | 1.5 | 3.5 | 6 | 6 |
| Fries | 4.0 | 9.0 | 3 | 2 |
| Ice cream | 2.5 | 7.5 | 4 | 2 |
| Cola | 3.0 | 5.0 | 7 | 8 |