Chain volume index formulae

6.18    Annual chain volume indexes in the ASNA are derived by compounding successive year-to-year Laspeyres indexes. A Laspeyres volume index from year \(y-1\) to year \(y\) is derived by dividing the value of the aggregate in year y at year \(y-1\) prices (i.e. using the volumes in year \(y\) but the prices of year \(y-1\)) with the current price value in year \(y-1\); that is:

\(\large{L_Q} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }},\)

where \(P_i^y\) and \(Q_i^y\) are prices and quantities of the \(i^{th}\) product in year \(y\) and there are \(n\) products.

6.19    Annual chain Laspeyres volume indexes can be formed by multiplying consecutive year-to-year indexes; that is:

\(\large L_Q^y = \frac{{\sum\limits_{i = 1}^n {P_i^0Q_i^1} }}{{\sum\limits_{i = 1}^n {P_i^0Q_i^0} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^1Q_i^2} }}{{\sum\limits_{i = 1}^n {P_i^1Q_i^1} }} \times \frac{{\sum\limits_{i = 1}^n {P_i^2Q_i^3} }}{{\sum\limits_{i = 1}^n {P_i^2Q_i^2} }} \times \ldots \times \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^y} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }}\)

6.20    The derivation of quarterly chain Laspeyres volume indexes is in concept no different to compiling annual chain volume indexes. However there is the complication of seasonality to contend with. In the ASNA, annual base years (i.e. annual weights) are used to derive quarterly volume indexes rather than having quarterly base periods. If quarterly base periods were to be used then this should only be done using seasonally adjusted data and not original data.

6.21    Consequently the Laspeyres-type³⁴ volume index from year \(y-1\) to quarter \(c\) in year \(y\) takes the form:

\(\large L_Q^{\left( {y - 1} \right) \to \left( {c,y} \right)} = \frac{{\sum\limits_{i = 1}^n {P_i^{y - 1}4q_i^{c,y}} }}{{\sum\limits_{i = 1}^n {P_i^{y - 1}Q_i^{y - 1}} }} = \sum\limits_{i = 1}^n {\frac{{4q_i^{c,y}}}{{Q_i^{y - 1}}}s_i^{y - 1},} \)

where \(q_i^{c,y}\) , is the volume of product \(i\) in the \(c^{th}\) quarter of year \(y\) and \(s\) is the share (weight) of the \(i^{th}\) item. For more detail see Annex A to this chapter.