6. Summary Index Numbres (2012)Apunte Español
|Universidad||Universidad Internacional de Cataluña (UIC)|
|Grado||Administración y Dirección de Empresas (ADE) English Programme - 2º curso|
|Año del apunte||2012|
|Fecha de subida||06/06/2014|
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• • • • Summary: Index Numbers Statistics 1 Introduction Statistics relies on repeated measuring. One way of interpreting is to compare obtained values. Comparison can be done in two ways: o Values of variable are compare to each other o Variable is compared with other variables that make sense to relate: Income/Per Capita ; Km/h ; Unemployed/Population Quantitative observations comparisons can be made with: o Difference (D) D=X1 – X0 It has the units of measurement It doesn’t provide with sufficient info. About the importance of the difference magnitude. o Ratio (Q) ! o 𝑄 = ! !! • o 𝑄 = 1 (Equal) ; Q > 0 (increment) ; Q<0 (decrement) o Result if not influenced by units of measurement o Sensitive to the magnitude of the difference Ratios, Proportions and Rates of Change o Ratio: Diving a quantity by another number ! ! o Proportion: particular kind of ratio Can be expressed as percentage or otherwise 𝑋 = 𝑋! + 𝑋! + 𝑋! ! 𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 = ! ! • Rate of Change (T) o Is also a Ratio o Allows to find variations in magnitude relative to its reference value o Used in economics, demographics o Are called simple rates of change Can be positive (increment) or negative (decrement) ! !! 𝑇 = !! ! • Linked Rates of Change o Specific case of rate of change o Simple rate of change that compares two consecutive years ! !! 𝑇!!!! = !!! ! !! !! 2.
• Index Numbers Special case of ratio that allows to analyze changes that occur in a variable time relatively to a pre-‐specified reference value ! o 𝐼!! = ! ; 𝑋! is the value of the temporal variable • • o • !! Moment of time of reference variable: reference/base period Moment of time of compared variable: actual/current period Choice of a base period is arbitrary, but it should be relatively representative to not distort the comparison. If values are multiplied by an scalar the index doesn’t change From index numbers we can compute the relative variation of the value of the variable in the current period compared to the value f the variable in the base period ! !! ! ! ! o Change: 𝑇!! = ! ! = ! − ! = ! − 1 o o o And !! !! !! !! !! = 𝐼!! o Then: 𝑇!! = 𝐼!! − 1 an increase per unit of the variable at time t compared to time 0 Index number may bot be expressed in units, but in percentage ! o Connotation changes to: 𝐼!!!"" o It exactly shows the relationship between the rate of change [%] and the index number [%] ! !! ! ! ! % 𝑇!! = ! ! . 100 = ! . 100 − ! . 100 = ! . 100 − 100 !! !! !! !! ! . 100 = (%)𝐼!!!"" ! (%)𝑇!! = 𝐼!!!"" − 100 !! !! !! • Most frequently used simple indexes o Relative price index ! 𝐼!! = ! = 𝑃𝐼!! !! o Relative quantity index ! 𝐼!! = ! = 𝑄𝐼!! o Relative value index ! ! .! 𝐼!! = ! = ! ! = 𝑃𝐼!! . 𝑄. 𝐼!! = 𝑉. 𝐼!! !! !! 3.
• • !! .!! Chain indexes It’s the ratio of consecutive values of a variable The value of any given period is related to the value of its immediately preceding period ! ! o 𝐼!!! = ! o o !!!! Estimates pace of change in consecutive values of a variables The rate of change can be calculated out of it: ! !! ! ! 𝑇!!! = ! !!! = ! − 1 !!!! o !!!! ! ! Then: 𝑇!!! = 𝐼!!! − 1 Simple not consecutive indexes can be calculated out of chain indexes ! ! ! !!! ! 𝐼!!! = 𝐼!!! . 𝐼!!!! = ! . !!! = !!! !!!! 4.
• • • Properties of simple index numbers 𝐼𝑛𝑑𝑒𝑥 𝑁𝑢𝑚𝑏𝑒𝑟𝑠 𝜖 ℝ (0,∞) If reference period = actual period, I=1 ! Reversibility: 𝐼!! = ! • Circularity: if, 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝜖 0, 𝑡, 𝑡 ! , 𝑡′′ , then !! ! o 𝐼!! . 𝐼!! . 𝐼!!! . 𝐼!!!!! = 1 o ! !! ! !! 𝐼!! . 𝐼!! . 𝐼!!! . 𝐼!!!!! = 1 𝐼!! . 𝐼!! . 𝐼!!! .
! !! !! ! 5.
• • • • • !!!! !! o • !! ! • 𝐼!!! . 𝐼!!!! Chain indexes Out of this, we can calculate the rate of change between not consecutive years ! !! !! !!! !!! • Using: 𝐼!!! = 𝐼!!!! . 𝐼!!! . 𝐼!!! . ⋯ . 𝐼!!!!! . 𝐼! !!! • Then: 𝑇!!! = 𝐼!!! − 1 ! !!!!! = 1 o 𝐼!! . 𝐼!! . 𝐼 = 𝐼!!!! Proportionality: if the value of the variable in the current period is changed proportionally, the index number will be affected by this change o 𝑋!! = 𝑋! + 𝐾𝑋! = (1 + 𝐾)𝑋! !!! !!! !! o 𝐼′!! o ∗ This allows us to recover series of index number without doing again all calculations. !! = !! = 1 + 𝐾 . 𝐼!! Links and Changes of the base The farther from the base period, the less representative become the indexes We may change base period by a more current and representative Links are used to relate series of indexes The operations are based of two properties: Reversibility and Circularity If we want to change the base from period 0 to period h then: !!! o 𝐼!! = o o We divide the entire series by the 𝑇𝑒𝑐ℎ𝑛𝑖𝑐𝑎𝑙 𝑖𝑛𝑑𝑒𝑥: 𝐼!! The result has to satisfy both properties !!! index used to link two series ...