Index Numbers and Moving Averages

The One Thing

Index numbers measure relative change by comparing current-period values with a selected base period, usually assigned an index of 100. Moving averages smooth time-series data by averaging consecutive observations, thereby reducing short-term fluctuations and highlighting the underlying trend.

Definitions and Results

• Index Number: A statistical measure showing the relative change in one or more variables, usually with the base-period value set at 100.
• Base Period: The reference period used for comparison; its index value is generally 100. It should be normal, representative, and free from unusual events such as war, severe shortage, or extraordinary inflation.
• Current Period: The period whose value is compared with the base period.
• Price Relative:

$$\frac{p_1}{p_0}\times 100$$ where $p_0$ is the base-period price and $p_1$ is the current-period price.

• Quantity Relative:

$$\frac{q_1}{q_0}\times 100$$ where $q_0$ is the base-period quantity and $q_1$ is the current-period quantity.

• Value Relative:

$$\frac{p_1{q}_1}{p_0{q}_0}\times 100$$ where value equals price multiplied by quantity.

• Price Index: An index measuring changes in prices between two periods.
• Quantity Index: An index measuring changes in physical quantities, such as output, sales volume, or production.
• Value Index: An index measuring changes in total monetary value:

$$V=\frac{\sum p_1{q}_1}{\sum p_0{q}_0}\times 100.$$

• Simple Aggregative Price Index:

$$\frac{\sum p_1}{\sum p_0}\times 100.$$ It is unweighted and compares the sum of current-period prices with the sum of base-period prices.

• Simple Average of Price Relatives: Using the arithmetic mean,

$$\frac{\sum \left(\frac{p_1}{p_0}\times 100\right)}{n}.$$

• Weighted Index Number: An index in which items receive weights according to their relative importance. Weights may be based on quantities, expenditure, production levels, or relative importance.
• Weighted Average of Price Relatives: Using the arithmetic mean,

$$\frac{\sum WR}{\sum W},$$ where $W$ is the weight and $R$ is the price relative.

• Laspeyres Price Index: A weighted price index using base-period quantities as weights:

$$L=\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100.$$

• Paasche Price Index: A weighted price index using current-period quantities as weights:

$$P=\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\times 100.$$

• Fisher’s Ideal Price Index: The geometric mean of the Laspeyres and Paasche indices:

$$F=\sqrt{L\times P}.$$

• Time Series: Observations recorded in chronological order, such as yearly sales or monthly prices.
• Trend: The long-term general direction of movement in a time series.
• Moving Average: An average calculated over a fixed number of consecutive observations and moved forward one period at a time.
• Odd-Period Moving Average: A moving average based on an odd number of observations, such as a 3-year or 5-year average. It can be placed directly at the middle period.
• Even-Period Moving Average: A moving average based on an even number of observations, such as a 4-year or 12-month average. It usually requires centering.
• Centered Moving Average: A moving average adjusted so that it is placed exactly at a particular time period, especially when the averaging period is even.
• Weight: A numerical value representing the relative importance assigned to an item or observation.
• Interpretation of an Index: An index above 100 indicates an increase compared with the base period; an index below 100 indicates a decrease; an index equal to 100 indicates no change.
• Good Index Number: A good index should be clearly defined, use suitable and representative items, rely on reliable data, use a suitable base period, and reflect the purpose of the study.
• Dimensionless Nature of Index Numbers: Index numbers are generally dimensionless because they compare ratios, although units must remain consistent within each calculation.
• Limitations of Index Numbers: Interpretation may be affected by changes in quality, changes in consumer preferences, unsuitable weights, non-comparable goods, and an inappropriate base year.
• Moving-Average Period: A larger period produces a smoother trend but responds slowly to sudden changes. A smaller period follows actual data more closely but may retain more irregular fluctuations.
• Moving-Average Limitation: A moving average cannot normally be calculated for the first and last few periods because sufficient neighboring observations are unavailable.
• Purpose of Moving Averages: Moving averages reduce random and seasonal fluctuations, although they may remove some short-term information.
• Comparative Nature: Index numbers are comparative measures, not absolute measurements; their meaning depends on the chosen base period and data.
• Relative Realism of Weighted Indices: Weighted indices are usually more realistic than unweighted indices because important items should influence the result more strongly.
• Fisher’s Theoretical Strength: Fisher’s index is considered a strong theoretical measure because it combines the Laspeyres and Paasche methods through their geometric mean.

Worked Methods

1. Calculating a Single-Item Price Index

• Identify the base-period price $p_0$.
• Identify the current-period price $p_1$.
• Divide the current-period price by the base-period price.
• Multiply by 100:

$$\text{Price Index}=\frac{p_1}{p_0}\times 100.$$

• Interpret the result relative to 100.

2. Calculating a Single-Item Quantity Index

• Identify the base-period quantity $q_0$.
• Identify the current-period quantity $q_1$.
• Divide the current-period quantity by the base-period quantity.
• Multiply by 100:

$$\text{Quantity Index}=\frac{q_1}{q_0}\times 100.$$

3. Calculating a Single-Item Value Index

• Calculate current-period value as $p_1{q}_1$.
• Calculate base-period value as $p_0{q}_0$.
• Divide current-period value by base-period value.
• Multiply by 100:

$$\text{Value Index}=\frac{p_1{q}_1}{p_0{q}_0}\times 100.$$

4. Simple Aggregative Price Index

• Add all current-period prices to obtain $\sum p_1$.
• Add all base-period prices to obtain $\sum p_0$.
• Divide the current-period total by the base-period total.
• Multiply by 100:

$$\text{Simple Aggregative Price Index}=\frac{\sum p_1}{\sum p_0}\times 100.$$

• This method gives every item equal influence.

5. Simple Average of Price Relatives

• For each item, calculate its price relative:

$$R=\frac{p_1}{p_0}\times 100.$$

• Add all price relatives.
• Divide by the number of items $n$:

$$\text{Index}=\frac{\sum R}{n}.$$

• This is the arithmetic mean of the individual price relatives.

6. Weighted Average of Price Relatives

• Calculate each item’s price relative $R$.
• Assign each item an appropriate weight $W$, based on quantities, expenditure, production levels, or relative importance.
• Multiply each relative by its weight to obtain $WR$.
• Add the weighted relatives.
• Add the weights.
• Divide:

$$\text{Weighted Index}=\frac{\sum WR}{\sum W}.$$

• Weighted indices are generally more realistic because more important items exert greater influence.

7. Laspeyres Price Index

• Use base-period quantities $q_0$ as weights.
• For each item, calculate current-period price multiplied by base-period quantity, $p_1{q}_0$.
• Calculate base-period price multiplied by base-period quantity, $p_0{q}_0$.
• Sum both products.
• Apply:

$$L=\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100.$$

8. Paasche Price Index

• Use current-period quantities $q_1$ as weights.
• For each item, calculate $p_1{q}_1$.
• Calculate $p_0{q}_1$.
• Sum both products.
• Apply:

$$P=\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\times 100.$$

9. Fisher’s Ideal Price Index

• Calculate the Laspeyres price index $L$.
• Calculate the Paasche price index $P$.
• Multiply $L$ by $P$.
• Take the square root:

$$F=\sqrt{L\times P}.$$

• Fisher’s index combines base-period and current-period quantity weighting through the geometric mean.

10. Three-Year Moving Average

• Select the first three consecutive observations.
• Add them together.
• Divide by 3:

$$\text{Three-year moving average}=\frac{\text{sum of three consecutive observations}}{3}.$$

• Place the result at the middle year.
• Move forward one year and repeat using the next three observations.
• Continue across the series where three observations are available.

11. Five-Year Moving Average

• Select the first five consecutive observations.
• Add them together.
• Divide by 5:

$$\text{Five-year moving average}=\frac{\text{sum of five consecutive observations}}{5}.$$

• Place the result at the middle year.
• Move forward one year and repeat using the next five observations.

12. Four-Year Centered Moving Average

• Calculate the average for each successive group of four observations.
• Because four is an even number, each average lies between two periods.
• Take two adjacent four-year moving averages.
• Average them to center the result:

$$\text{Centered moving average}=\frac{\text{first four-year moving average}+\text{next four-year moving average}}{2}.$$

• Place the centered value at the appropriate time period.

Where It Goes Wrong

• Forgetting that $p_0$ and $q_0$ refer to the base period, while $p_1$ and $q_1$ refer to the current period.
• Using current-period quantities in the Laspeyres formula or base-period quantities in the Paasche formula; Laspeyres uses $q_0$, whereas Paasche uses $q_1$.
• Treating an index as an absolute measurement rather than a comparison whose interpretation depends on the base period.
• Selecting a base period affected by war, severe shortage, or extraordinary inflation instead of a normal and representative period.
• Placing an even-period moving average directly on one observation without centering it.
• Calculating moving averages for the first or last few periods when sufficient neighboring observations are unavailable.

What Gets Asked

• Calculate and interpret a single-item price, quantity, or value index.
• Calculate a simple aggregative price index.
• Calculate a simple arithmetic average of price relatives.
• Calculate a weighted average of price relatives using $\sum WR/\sum W$.
• Calculate a Laspeyres price index using base-period quantities.
• Calculate a Paasche price index using current-period quantities.
• Calculate Fisher’s Ideal Price Index from the Laspeyres and Paasche indices.
• Explain the meaning of an index above, below, or equal to 100.
• Identify suitable characteristics of a base period and a good index number.
• Explain the limitations of index numbers, including changes in quality, consumer preferences, weights, comparability, and base-year selection.
• Calculate and position a 3-year moving average.
• Calculate and position a 5-year moving average.
• Calculate and center a 4-year moving average.
• Explain how the choice of moving-average period affects smoothness, responsiveness, and the retention of short-term information.
• Explain how moving averages reduce random and seasonal fluctuations and reveal the trend in a time series.