Index Numbers and Time Series: formula sheet

Key formulas

Simple price index

Index = (Current price ÷ Base price) × 100

Use for a single item. Base period index is always 100.

Price relative

P = (p₁ ÷ p₀) × 100

p₀ is the base year price and p₁ is the current year price.

Value index

V = (Σp₁q₁ ÷ Σp₀q₀) × 100

Value = price × quantity. It combines price and quantity changes.

Percentage change from index

Change % = Index − 100

Valid when the base index is 100. A positive result is a rise, a negative result is a fall.

Quantity index (simple form)

Q = (q₁ ÷ q₀) × 100

Use for a single item's quantity.

Simple aggregative index

P01 = (ΣP1 ÷ ΣP0) × 100

Unweighted. Use only when all items are in the same unit.

Average of price relatives (arithmetic mean)

P01 = Σ[(P1 ÷ P0) × 100] ÷ n

Find each relative first, then divide the sum by the number of items n. If the question asks for the geometric mean, use the antilog of Σlog(relative) ÷ n.

Laspeyres index

L = (ΣP1Q0 ÷ ΣP0Q0) × 100

Base-year quantities as weights.

Paasche index

P = (ΣP1Q1 ÷ ΣP0Q1) × 100

Current-year quantities as weights.

Fisher ideal index

F = √(L × P)

Geometric mean of Laspeyres and Paasche.

Dorbish-Bowley index

DB = (L + P) ÷ 2

Arithmetic mean of Laspeyres and Paasche.

Marshall-Edgeworth index

ME = [ΣP1(Q0 + Q1) ÷ ΣP0(Q0 + Q1)] × 100

Uses the sum of base and current quantities as weights.

Weighted average of price relatives

Index = ΣIV ÷ ΣV, where I = (P1 ÷ P0) × 100

With V = P0Q0 this gives the Laspeyres index.

Time reversal test

P01 × P10 = 1

P10 is the same formula with base and current year swapped. If each index is written with ×100, the product is 10,000.

Factor reversal test

P01 × Q01 = Σp1q1 ÷ Σp0q0

Q01 is the quantity index formed by swapping p and q in the price index formula. The right side is the value ratio.

Circular test

P01 × P12 × P20 = 1

Extension of time reversal to three periods. Fisher's index does not satisfy it in general.

Fisher's price index

P01 = √[(Σp1q0 ÷ Σp0q0) × (Σp1q1 ÷ Σp0q1)] × 100

Geometric mean of Laspeyres and Paasche. Satisfies time reversal and factor reversal.

Fisher's quantity index

Q01 = √[(Σq1p0 ÷ Σq0p0) × (Σq1p1 ÷ Σq0p1)] × 100

Obtained from the price index by interchanging p and q.

Fisher's reversed index

P10 = √[(Σp0q1 ÷ Σp1q1) × (Σp0q0 ÷ Σp1q0)]

Swap the subscripts 0 and 1 everywhere in P01.

Marshall-Edgeworth price index

P01 = [Σp1(q0 + q1) ÷ Σp0(q0 + q1)] × 100

Satisfies the time reversal test but does not satisfy the factor reversal test.

Cost of living index (family budget or aggregate expenditure method)

CLI = ΣPW ÷ ΣW, where P = (p1 ÷ p0) × 100 and W = p0 × q0

P is the price relative of each item. W is the base-year expenditure weight. Also called the weighted average of price relatives.

Aggregate expenditure method

CLI = (Σp1q0 ÷ Σp0q0) × 100

Same result as the family budget method when weights are W = p0q0. This is the Laspeyres index.

Base shifting

New base index = (Old index ÷ Index of the new base year on the old base) × 100

Divide every value of the old series by the old index of the year chosen as the new base.

Splicing (forward, old series to new base)

Spliced index = Old index × (Overlap-year index of new series ÷ 100)

Use this to bring the old series onto the base of the new series. Overlap year is the year common to both series.

Splicing (backward, new series to old base)

Spliced index = New index × (100 ÷ Overlap-year index of the old series)

Use this to extend the new series back onto the old base, with the overlap year taken from the old series.

Deflating (real value)

Real value = (Nominal value ÷ Price index) × 100

Real wage = (Money wage ÷ CPI) × 100. Use the price index of the same year as the nominal value.

Purchasing power of money

Purchasing power = (1 ÷ Price index) × 100

It falls when prices rise.

Additive model

Y = T + S + C + I

Components are assumed independent. All terms are in the units of Y.

Multiplicative model

Y = T × S × C × I

Components are assumed to be interdependent. S, C and I are ratios, not amounts.

Trend by elimination (additive)

S + C + I = Y − T

Subtract the trend value from the actual value to get the combined other effects.

Trend by elimination (multiplicative)

S × C × I = Y ÷ T

Divide the actual value by the trend value.

Short-term and long-term split

Long-term: Trend. Short-term: Seasonal, Cyclical, Irregular

Seasonal repeats within a year. Cyclical lasts more than a year. Irregular is random.

Straight-line trend

Y = a + bX

Y is the trend value, X is time measured from the chosen origin.

Normal equations

ΣY = na + bΣX and ΣXY = aΣX + bΣX²

Use these when the origin is not in the middle.

Shortcut when ΣX = 0

a = ΣY ÷ n and b = ΣXY ÷ ΣX²

Works when X is measured from the middle year (odd n) or from the centre of the series (even n, in years or half-year units).

Odd-period moving average

3-year MA for year t = (Yt-1 + Yt + Yt+1) ÷ 3

The average is placed against the middle year.

Even-period moving average

Take totals of 4 values, add two consecutive totals, divide by 8

This centres the average against a year. It is called centring.

Semi-average slope

b = (Average of 2nd half − Average of 1st half) ÷ gap between the two middle years

Each average is placed at the middle of its half.

Values lost in moving average

Odd period m: (m − 1) ÷ 2 lost at each end

A 3-year MA loses one value at each end; a 5-year MA loses two.

Simple average method

Seasonal index = (Average of the season ÷ Grand average) × 100

Grand average is the average of all the seasonal averages. Use it when there is no clear trend.

Sum of indices check

Sum of indices = 100 × number of seasons

400 for quarters and 1200 for months. Use this to check your answer.

Centred moving average for quarterly data

CMA = (Sum of two consecutive 4-quarter totals) ÷ 8

It falls against a quarter, not between two quarters. For monthly data, the 12-month version divides by 24.

Ratio to moving average

Ratio = (Actual value ÷ CMA) × 100

Average these ratios for each season across the years.

Adjusted index, ratio to moving average

Adjusted index = Average ratio of the season × (100 ÷ Average of all the seasonal average ratios)

Do this only when the average of the seasonal ratios is not exactly 100.

Link relative

Link relative = (Current season value ÷ Previous season value) × 100

The first season of the first year has no previous season, so it has no link relative.

Chain relative

Chain relative = (Average link relative of the season × Chain relative of the previous season) ÷ 100

The chain relative of the first season is taken as 100.

Correction for trend in link relative method

L = (Average link relative of first season × Last chain relative) ÷ 100; d = (L − 100) ÷ number of seasons; corrected chain of the k-th season = chain − (k − 1) × d

The first season stays at 100. Then divide each corrected chain by the average of all corrected chains and multiply by 100 to get the index.

Quick revision

• Index number: a measure of the relative change in a variable between a base period and a current period, with the base usually set at 100.
• Simple aggregative price index = (ΣP1 ÷ ΣP0) × 100.
• Laspeyres uses base year quantities as weights: (ΣP1Q0 ÷ ΣP0Q0) × 100.
• Paasche uses current year quantities as weights: (ΣP1Q1 ÷ ΣP0Q1) × 100.
• Fisher's index is the geometric mean of Laspeyres and Paasche: √(L × P).
• Time reversal test: P01 × P10 = 1. Factor reversal test: P01 × Q01 = ΣP1Q1 ÷ ΣP0Q0.
• Deflated (real) value = current value ÷ price index × 100.
• Time series components: trend, seasonal, cyclical and irregular. Additive model: Y = T + S + C + I. Multiplicative model: Y = T × S × C × I.
• An even-period moving average must be centred by taking a second two-term average.
• Least squares trend line: Y = a + bX. With Σx = 0, a = ΣY ÷ n and b = ΣxY ÷ Σx².
• Seasonal variation repeats within a year, while cyclical variation spans longer periods.

Common mistakes

• Putting the base value in the numerator. Fix: Remember that the current year always goes on top. Check that a price rise gives an index above 100.
• Forgetting to multiply by 100. Fix: An index is a percentage figure. Always multiply the ratio by 100 before choosing an option.
• Using current-year quantities in the Laspeyres formula (or base quantities in Paasche). Fix: Remember: Laspeyres = Q0 (base-year basket). Paasche = Q1 (current-year basket). Mnemonic: 'L for Legacy basket'.
• Forgetting to multiply by 100. Fix: Index numbers are always shown with base = 100. Multiply the final ratio by 100 every time.
• Swapping only the prices in P10 and leaving the quantities as they were. Fix: In P10, swap every 0 and 1 in the whole formula. Base year becomes current year for both p and q.
• Getting Q01 by just reading off a different column without swapping p and q. Fix: Take the price formula and replace every p with q and every q with p. Write Q01 in full before computing.
• Dividing ΣPW by the number of items instead of ΣW. Fix: Always divide by the sum of weights, ΣW, never by n.
• Using current-year quantities or prices as weights in the family budget method. Fix: Weights are base-year expenditure p0q0 unless the question says the weights are given directly.
• Calling a yearly festival sales rise cyclical. Fix: Anything that repeats within a year with a fixed period is seasonal. Cyclical needs more than a year and has no fixed length.
• Treating a flood or strike effect as seasonal. Fix: If it is unpredictable and not repeating, it is irregular.

Exam tips

• Learn the three types by definition. Questions often ask which type a given example represents.
• Remember that the base year index is 100 and use it to check every answer.
• Revise the limitations as a short list: approximate, base year dependent, sample bias, formula choice, and poor long-term comparison.
• Practise one-line numerical questions on simple price and value indices, as they are quick marks.
• Since there is no negative marking, attempt every question and eliminate wrong options first.
• Write the formula name and its Q subscript at the top of your rough work before filling the table.
• Questions often say only 'price index using weights'. Check whether the weights given are base-year (Laspeyres) or current-year (Paasche) values.
• If both L and P are given, Fisher and Dorbish-Bowley need no table work at all. Do these first to save time.