Core Information

Instructor: Paulo Fagandini

📧 pfagandini@iscal.ipl.pt

Main source for course material: Moodle@ISCAL

Official communication channel: 📧 Institutional Email

Bibliography

Topic 1:

📖 Gancho Custódio, S. et al. (2022) Números Índices, Edições Sílabo

Topic 2:

📖 Murteira, B.; Silva Ribeiro, C.; Andrade e Silva, J. & Pimenta, C., Introdução à Estatística, Escolar Editora, McGraw-Hill, 2010

Topic 3:

📖 Ferreira, T., Custódio, S.G. (2023) Modelos Probabilísticos, Edições Sílabo

Bibliography (EN)

Topic 1:

📖 The making of index numbers. 1^st Ed 1922. Irving Fisher.
📖 A Practical Introduction to Index Numbers. 1^st Ed 2015. Jeff Ralph, Rob O’Neill, Joe Winton.

Topics 2 and 3:

📖 Statistics for Business and Economics Global Edition. 10^th Ed 2022. Paul Newbold, William Carlson, and Betty Thorne.

This last book is a bit expensive, but it can be rented for less money here.

Continuous Assessment

Each test has a minimum grade of 7.00 (out of 20), not 6.9, not 6.95.

Comprehensive Exam

Students may opt, on the day of the Exam, to take an Exam worth 100% of the grade. This Exam will cover Topics 1, 2, and 3.

If you score less than 7.00 in the Midterm, or if you cannot attend the midterm for whatever reason, you will have to do the Comprehensive Exam.

Disclaimer

These slides are a free translation and adaptation from the slide deck for Estatística I by Prof. Sandra Custódio and Prof. Teresa Ferreira from the Lisbon Accounting and Business School, Polytechnic University of Lisbon.

Example: Apple Stock

Example: Apple Stock

Let the close price for Apple be represented by the variable \(y_t\), so \(y_{2020}\) is the close price we got for 2020 (i.e. 132.69).

How much did \(y\) grow between 2020 and 2021?

Example: Apple Stock

\[y_{2021}=y_{2020}+\Delta y_{2021}\]

\[y_{2021}=y_{2020}\left(1+\frac{\Delta y_{2021}}{y_{2020}}\right)\]

\[y_{2021}=y_{2020}\left(1+\delta^y_{2021}\right)\]

This, \(\delta_{2021}^y\), is the growth rate of \(y\) at year 2021.

Growth Rate

Growth Rate

You can write the growth rate as a decimal or as a percentage. If you use decimal notation, consider at least 4 places, if you use percentage you use at least 2. Example: 0.0123 or 1.23%.

Interpretation: What is the percentage change for the variable \(y\) between \(t-1\) and \(t\).

Example

\[177.57 = 132.69+(177.57 - 132.69)\]

\[177.57 =132.69\left(1+\frac{44.88}{132.69}\right)\]

\[177.57=132.69\left(1+0.3382\right)\]

Growth Rates

More generally \[\delta_{t+k|t}=\frac{y_{t+k}-y_t}{y_t}\]

Or \[y_{t+k}=y_t(1+\delta_{t+k|t})\]

(homework: Show step by step how you go from one to the other.)

Note: In this case \(\delta_{t+k|t}\) is what percentage the variable changed from \(t\) to \(t+k\), and \(k\) is the number of periods.

Example

We had \(y_{2020}=132.69\), and \(y_{2024} =250.42\). We will try to find \(\delta_{2024|2020}^y\)

• What is the value of \(k\) in this case?

Growth Rates

Growth Rates

Growth Rates

Let’s start with \(y_t\)… and the traditional growth rates \(\delta_t\):

\[y_{t+1} = y_t(1+\delta_{t+1})\]

\[y_{t+2} = y_{t+1}(1+\delta_{t+2})\]

\[y_{t+3} = y_{t+2}(1+\delta_{t+3})\]

Can we write \(y_{t+2}\) as a function of \(y_t\)?

Growth Rates

Generalizing

\[y_{t+k}=y_t (1+\delta_{t+1})...(1+\delta_{t+k-1})(1+\delta_{t+k})\]

When computing the average growth rate, we are trying to find \(\overline{\delta}\) such that if the growth rate was the same every period, it would have taken \(y_t\) to the value of \(y_{t+k}\) all the same:

\[y_{t+k}=y_t(1+\overline{\delta})(1+\overline{\delta})...(1+\overline{\delta})\]

Growth Rates

\[y_{t+k} = y_t \left(1+\overline{\delta}\right)^k\]

Example

We had \[\delta_{2024|2020}^y = \frac{250.42 - 132.69}{132.69} = 0.8873 = 88.73\%\]

But what now is the average growth rate between 2020 and 2024?

Example

Note that \[ (1+17.21\%)^4 = (1+0.1721)^4 \approx 1.8873\approx 1 + 88.73\% \]

❓ Growth Rates · Question 1

A variable grows 10% in one year and then falls 10% in the next. Over the two years together it:

A. fell 1%

B. rose 1%

C. is unchanged

D. fell 10%

❓ Growth Rates · Question 2

A variable goes from 100 to 150 over 5 years. Its average growth rate is closest to:

A. 10%

B. 8.45%

C. 50%

D. 5%

✏️ Growth Rates · Question 3

A firm sold 240 units in 2020 and 300 units in 2023.

Compute:

(i) the cumulative growth rate over the three years;

(ii) the average annual growth rate.

Motivation

Which stock would you have purchased in 2016-10-03?

Motivation

What’s the important question here?

Motivation

Now is a new day 🌄! Prices are now \(a_{t+1}\) and \(b_{t+1}\), how much is your portfolio worth today?

Motivation

Now we have a much clear picture. We can compare their evolution starting in 2016-10-03.

What can you read from the plot?

Index Numbers

Examples

Index Numbers

Importantly, an index number is no bueno to reflect the level of a variable, but its evolution or relative value against a base.

Rolling Base

Index Numbers

Index numbers allow us to easily compute the growth rate between any two periods, just like we would do with the underlying variable.

1. Simple Index Number

\[1+\delta_{t+k,t}=\frac{x_{t+k}}{x_t}=\frac{\frac{x_{t+k}}{x_0}}{\frac{x_t}{x_0}}=\frac{I_{t+k}}{I_t}\]

Index Numbers

2. Chain Index Number

\[1+\delta_{t+k,t}=\frac{x_{t+k}}{x_t}=\frac{x_{t+k}}{x_{t+k-1}}\,\frac{x_{t+k-1}}{x_{t+k-2}}\cdots\frac{x_{t+1}}{x_{t}}\]

Index Numbers

Index Numbers

Index Numbers

We can always transform a chain index number into a fixed base index number.

1. Chose a reference, for example for period 3.

2. Remember: what you want to achieve: \(\forall t, I'_t=\frac{x_t}{x_3}\)

3. Remember: that we have \(I_t=\frac{x_t}{x_{t-1}}\)

4. Use what we did just before:

Example

Suppose we want to convert this Chain Index into Fix Base index, with the base year in 2021. This will be our 100.

Example

Note that, for example in 2025, we have \(I_{2025|2021}=116.7\), which was computed as \(123.3\times\frac{94.6}{100}\). But \(123.3\) was the base 2021 index of the previous period, which in turn was computed as \(113.3\frac{108.8}{100}\), so in reality:

\[116.7=123.3\frac{94.6}{100}=113.3\frac{108.8}{100}\frac{94.6}{100}=100\frac{113.3}{100}\frac{108.8}{100}\frac{94.6}{100}\]

If you pay attention, you will see that we have there the product of the chain Index in 2022, 2023, and 2024. We divided by 100 because, to multiply, we need to divide by 100 before.

Base Change

In case we have the data, we just need to compute the Index with the base that we want, however, if we do not have the data, and we just have a Fix Base index, we can do it all the same!

Say we have an index with base 0. Maybe we want to change it to base 2.

Base Change

Base Change

But how can we transform them, if we do not know \(x_i\) \(\forall i\)?

Example

New information? Yes! For example, without any computation we know that the price in 2026 was 30% larger than in 2021, and 5.4% larger than in 2023

Tests for good Simple Index Numbers

We can test the properties for Simple Index Numbers. These represent desirable properties for good index numbers.

1. Identity test: If the variable in the current period is the same as in the base period, the index should be 1 (or 100). Of course this implies \[I_{0|0}=1\].
2. Proportionality test: If the variable increases by \(k\), the index should increase by \(k\). \[x_t=kx_0\ \Rightarrow\ i_{t|0}=k\]

Tests for good Simple Index Numbers

3. Time Reversal test: If you exchange the base between two periods, then the index in one base, should be the inverse of the other: \[i_{t|0}=\frac{1}{i{0|t}}\ \Leftrightarrow\ i_{t|0}i_{0|t}=1\]
4. Factor Reversal test: If we have two index numbers, one for variable \(x\) and another for \(y\), then if \(z=x\times y\) we should have \[i_{t|0}^z=i_{t|0}^x\times i_{t|0}^y\]

Tests for good Simple Index Numbers

5. Circularity test: The index at \(t\) base \(0\) should be the product of all the link index numbers up to \(t\), \[i_{t|0}=i_{t|t-1}\times i_{t-1|t-2}...i_{2|1}\times i_{1|0}\] This formalizes the relationship between fixed base and chain index numbers.

6. Homogeneity test: If the variable, for every \(t\) is multiplied by a constant, the index should not be affected.

Tests for good Simple Index Numbers

7. Well-definedness: The index number must be meaningful, and as prices and quantities usually are larger than zero, index numbers must be positive. Even further, a good index number should not take the value of \(\infty\).

❓ Simple Index Numbers · Question 1

A simple price index with base 2020 reads 125 in 2024. This tells you that the price:

A. rose 125%

B. rose 25%

C. is 125 euros

D. nothing, without the base price

❓ Simple Index Numbers · Question 2

To turn a chain (link) index into a fixed base index you:

A. multiply them, in decimal notation

B. add the link indices

C. average them

D. subtract 100 from each

✏️ Simple Index Numbers · Question 3

The price of a good was 30 in year 1, 34 in year 2 and 37 in year 3.

Compute the fixed base index with base year 1 for all three years, and the link index for year 3.

Composite Index Numbers

Composite Index Numbers

This is a great opportunity to introduce another classification for our index numbers:

Composite Index Numbers

Typically, values are collected in nominal terms, i.e. current prices, which is a value.

\[v_t=p_t\times q_t\]

Composite Index Numbers

But if we let \(\delta_{t+1}^p\) be the growth rate of prices, and \(\delta_{t+1}^p\) the growth rate of quantities we can rewrite \(v_{t+1}\) as:

\[v_{t+1}=p_t\left(1+\delta_{t+1}^p\right)q_t\left(1+\delta_{t+1}^q\right)\]

Composite Index Numbers

\[(1+\delta_{t+1}^v) = (1+\delta_{t+1}^p)(1+\delta_{t+1}^q)\]

\(\delta_t^v\) is what we call nominal change rate, while \(\delta_t^q\) is what we call real change rate. We can find out the real change rate (\(\delta^q\)), if we know the nominal change rate (\(\delta^v\)), and the prices change rate (\(\delta^p\)):

\[\left(1+\delta_{t}^q\right) = \frac{\left(1+\delta_t^v\right)}{\left(1+\delta_t^p\right)}\] \[\delta_{t}^q = \frac{\left(1+\delta_t^v\right)}{\left(1+\delta_t^p\right)}-1\]

Composite Index Numbers

Suppose we have index numbers for \(v\), \(q\), and \(p\), all with base \(b\) (remember you can re-base your index numbers easily if they do not share the same base).

\[I_{t|b}^q = \frac{I_{t|b}^v}{I_{t|b}^p}\]

Example

Let’s find the quantities index, with base 2023. Immediately we can fix \(i_{t|b}^q=100\).

\[I_{2022}^q=\frac{I_{2022}^v}{I_{2022}^p}=\frac{99}{95}=1.04\]

Example

Now we can compute what was the real change rate between 2025 and 2026:

Example

Let’s check:

\[\delta_{2026}^q=\frac{(1+\delta_{2026}^v)}{(1+\delta_{2026}^p)}-1=\frac{111/110}{109/106}-1=0.9813-1\] \[\delta_{2026}^q=-0.0187=-1.87\%\]

Baskets

Aggregate Index Numbers

From the family of formulas to create weighted composite index numbers, we will focus on the two most common:

1. Laspeyres: uses base-period quantities (prices) as weights.

2. Paasche: uses current-period quantities (prices) as weights.

Kids’ Explanation

Let’s consider that the base year is past.

Kids’ Explanation - Prices

Laspeyres - Price:

Use always \(🧺_{past}\), but relevant 🏷️! Laspeyres Price Index (LPI):

Remember that 1 corresponds to 100, and 1.25 corresponds to 125.

Kids’ Explanation - Prices

Paasche - Price:

Use always \(🧺_{today}\)! Paasche Price Index (PPI):

Remember that 1 corresponds to 100, and 1.27 corresponds to 127.

Conclusions

1. In both index numbers, for prices, quantities are constant: All variation comes from prices.

2. In both index numbers, for quantities, prices are constant: All variation comes from quantities.

Fisher Index

Some authors argue, however, that the true measure for cost of life is in between both metrics. This is captured by the Fisher Index:

\[FPI_{t|0}=\sqrt{LPI_{t|0}\times PPI_{t|0}}\]

\[FQI_{t|0}=\sqrt{LQI_{t|0}\times PQI_{t|0}}\]

Example: the data

A basket of 15 goods, observed in 2024, 2025 and 2026. Each cell is what the basket costs with the row year’s prices and the column year’s quantities, \(\sum_{k=1}^{15} p_{row}^k\, q_{col}^k\).

Example: LPI, PPI and FPI

What are the three indices if we choose 2024 as the base year?

❓ Composite Index Numbers · Question 1

The Laspeyres price index weights prices using quantities from:

A. the current period

B. the base period

C. the average of both

D. neither, it uses no quantities

❓ Composite Index Numbers · Question 2

The Fisher index is:

A. the arithmetic mean of Laspeyres and Paasche

B. the geometric mean of Laspeyres and Paasche

C. always between 100 and 200

D. always equal to Laspeyres

✏️ Composite Index Numbers · Question 3

For a basket you computed \(LPI_{t|0}=110\) and \(PPI_{t|0}=104\).

Compute Fisher’s price index, and explain why it must always fall between the other two.

Consumer Price Index

To measure inflation, we use a Laspeyres Price Index, usually known as CPI (Consumer Price Index).

Consumer Price Index

To create a price index, you need to follow at least 3 steps.

1. Determine representative basket: Survey families 👩‍👩‍👧‍👦 👨‍👩‍👦‍👦 👨‍👨‍👧‍👦
2. Survey retail stores to find out sale prices. 🏪
3. Compute the index with the collected data. 📈
   1. Compute the cost of the 🧺 with the prices at the base period.
   2. Compute the cost of the same 🧺 with the prices of every successive period.

Application

A Labor union leader is complaining that the average salary in 2022 was 1500, and in 2025 it is 1650. This increase, they say, is not enough to maintain the workers’ purchasing power, because of inflation. The data for inflation is the following:

1. The year with the largest increase in prices was 2024?
2. The average growth rate for the wages between 2022 and 2025 was _______?
3. Is the union leader right?

Application

For the first question, we can compute the chain or link index:

Application

We know salary in 2022 was 1500, and in 2025 1650. If the growth rate was constant, say \(\delta_w\) we would have:

Summarizing

Remember this relationship we saw before? \[I^q_{t|b}=\frac{I^v_{t|b}}{I^p_{t|b}}\]

Real equals nominal over CPI.

This, dividing by CPI a nominal quantity, is known as to deflate by the CPI.

About the Test for Factor Reversal for Composite Index Numbers

Note that neither Laspeyres nor Paasche pass the test for factor reversal (see here), \[VI_{t|0}\neq LPI_{t|0}\times LQI_{t|0}\quad\quad VI_{t|0}\neq PPI_{t|0}\times PQI_{t|0}\]

But we can verify that: \[LPI_{t|0}\times PQI_{t|0}=IV_{t|0} \quad \quad PPI_{t|0}\times LQI_{t|0}=IV_{t|0}\]

About the Test for Factor Reversal for Composite Index Numbers

(… continuation)

Example (do the rest as homework): \[LPI_{t|0}\times PQI_{t|0}=\frac{\sum_{k=1}^m p_t^kq_0^k}{\sum_{k=1}^m p_0^kq_0^k}\frac{\sum_{k=1}^m p_t^kq_t^k}{\sum_{k=1}^m p_t^kq_0^k}=\frac{\sum_{k=1}^m p_t^kq_t^k}{\sum_{k=1}^m p_0^kq_0^k}\]

Note that this last item is exactly \(IV_{t|0}\).

About the Factor Reversal Test for Composite Index Numbers

\[FPI_{t|0}\times FQI_{t|0}=\] \[\sqrt{LPI_{t|0}\times PPI_{t|0}}\times\sqrt{LQI_{t|0}\times PQI_{t|0}}\] \[\sqrt{LPI_{t|0}\times PPI_{t|0}\times LQI_{t|0}\times PQI_{t|0}}\] \[\sqrt{LPI_{t|0}\times PQI_{t|0}\times LQI_{t|0}\times PPI_{t|0}}\] \[\sqrt{IV_{t|0}\times IV_{t|0}}=\sqrt{IV_{t|0}^2}=IV_{t|0}\]

About the Time Reversal Test for Composite Index Numbers

Laspeyres and Paasche do not pass the time Reversal test, let’s see an example:

\[LPI_{t|0}\times LPI_{0|t} = \frac{\sum_{k=1}^m p_t^k q_0^k}{\sum_{k=1}^m p_0^k q_0^k}\frac{\sum_{k=1}^m p_0^k q_t^k}{\sum_{k=1}^m p_t^k q_t^k}\neq 1\]

You can do the same for Paasche, and for the quantities index numbers.

About the Time Reversal Test for Composite Index Numbers

Fisher on the other hand…

\[FPI_{t|0}\times FPI_{0|t}=\sqrt{LPI_{t|0}PPI_{t|0}}\sqrt{LPI_{0|t}PPI_{0|t}}\]

About the Time Reversal Test for Composite Index Numbers

Fisher on the other hand…

\[FPI_{t|0}\times FPI_{0|t}=\sqrt{LPI_{t|0}PPI_{t|0}}\sqrt{LPI_{0|t}PPI_{0|t}}\]

\[=\sqrt{\frac{\sum_{k=1}^m p_t^k q_0^k}{\sum_{k=1}^m p_0^k q_0^k}\frac{\sum_{k=1}^m p_t^k q_t^k}{\sum_{k=1}^m p_0^k q_t^k}}\sqrt{\frac{\sum_{k=1}^m p_0^k q_t^k}{\sum_{k=1}^m p_t^k q_t^k}\frac{\sum_{k=1}^m p_0^k q_0^k}{\sum_{k=1}^m p_t^k q_0^k}}\]

\[=\sqrt{\frac{\color{red}{\sum_{k=1}^m p_t^k q_0^k}}{\color{blue}{\sum_{k=1}^m p_0^k q_0^k}}\frac{\color{green}{\sum_{k=1}^m p_t^k q_t^k}}{\sum_{k=1}^m p_0^k q_t^k}\frac{\sum_{k=1}^m p_0^k q_t^k}{\color{green}{\sum_{k=1}^m p_t^k q_t^k}}\frac{\color{blue}{\sum_{k=1}^m p_0^k q_0^k}}{\color{red}{\sum_{k=1}^m p_t^k q_0^k}}}\]

About the Circularity Test for Composite Index Numbers

None, not even Fisher’s Index, passes the circularity test.

Homework: Verify it.

Example

Example

1. According to LPI for year 2, prices for public transportation grew 11.17%.
2. Real change, according to Paasche, for public transportation was 7.35%.
3. The composite index for change in expenditures with public transportation in year 2 compared the expenditure in year 1 is what type of change? (real, nominal, quantity, prices?) How much was it?

Example

According to LPI for year 2, prices for public transportation grew 11.17%.

Example

Example

\[LPI_{2|1}=\frac{\sum_{k=1}^m p_2^k q_1^k}{\sum_{k=1}^m p_1^k q_1^k}\]

Example

The composite index for change in expenditures with public transportation in year 2 compared the expenditure in year 1 is what type of change? (real, nominal, quantity, prices?) How much was it?

The value index for public transportation is a nominal index (price times quantity).

Example

\[VI_{2|1}=LPI_{2|1}\times PQI_{2|1} = 1.1117 \times 1.0735 \approx 1.1934\]

The value of public transportation trips grew 19.34% from year 1 to year 2. This includes the effect of an increase in prices and quantities.

❓ Inflation · Question 1

The Consumer Price Index is constructed as a:

A. Paasche quantity index

B. simple index number

C. Fisher index

D. Laspeyres price index

❓ Inflation · Question 2

Your nominal wage rises 10% while the CPI rises 12%. Your purchasing power:

A. rose about 10%

B. did not change

C. rose 2%

D. fell about 1.8%

✏️ Inflation · Question 3

A worker earned 1500 three years ago and earns 1650 today. Over the same period the CPI went from 100 to 112.

Did the worker gain or lose purchasing power, and by how much?