Retrospective Price Indices and Substitution Bias Retrospective - PowerPoint PPT Presentation

1 www.snb.ch Retrospective Price Indices and Substitution Bias Retrospective Price Indices and Substitution Bias W. Erwin Diewert U i University of British Columbia, Vancouver i f B i i h C l bi V Marco Huwiler Clariden Leu, Zurich Ulrich Kohli Swiss National Bank, Zurich

2 www.snb.ch Outline Outline � Introduction � A retrospective measure of the price level � A retrospective measure of the price level � Comparison with Hansen � Application to Swiss data � C � Conclusion l i

3 www.snb.ch 1. Introduction � Most countries today still compute their consumer price indices � Most countries today still compute their consumer price indices (CPI) as direct (i.e. fixed-weighted) Laspeyres indices � The weights are updated at discrete intervals, often every five or � Th i ht d t d t di t i t l ft fi ten years, at which time the old and new series are spliced together together � It is well known that, in the consumer context, the Laspeyres functional form tends to overestimate the price level due to a functional form tends to overestimate the price level due to a substitution bias

4 www.snb.ch 1. Introduction (continued) ( ) � Ideally, one should use chained, superlative indices y, , p � � However, the necessary data (especially the quantity data) are often not available often not available � Nonetheless, sooner or later the baskets will need to be updated � The question therefore arises at the time when the new information becomes available whether it can be used to assess the importance of the substitution bias, and whether it can be exploited to improve the measure of past price level behavior

5 www.snb.ch 1. Introduction (continued) ( ) � The purpose of this paper is to propose a simple way to use the � The purpose of this paper is to propose a simple way to use the new information made available at the time of updating in order to get a superlative measure of the price change over the to get a superlative measure of the price change over the corresponding period � Retroactively computed price indices then make it possible to � Retroactively computed price indices then make it possible to assess the size of the substitution bias

6 www.snb.ch 2. A retrospective measure of the price level p p Consider the following two runs of fixed-basket indices: ∑ ∑ ∑ ∑ T 1 0 1 0 2 0 − T 0 p i q p q p i q p i q i i i i i 1, 1, , , , ... , , ... , , , (1) (1) i i i i ∑ ∑ ∑ ∑ 0 0 0 0 0 0 0 0 p i q p i q p i q p i q i i i i i i i i ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ p 0 T p 1 T p 2 T T 1 T − p i q q p i q q p i q q p p q q i i i i i i i i i i i i i , , , ... , , 1 , i i i i (2) ∑ ∑ ∑ ∑ T T T T T T T T p q p q p q p q i i i i i i i i i i i i t and q i t d t denote the price and the quantity of good t N t ti Notation: p i d t th i d th tit f d i at time t , respectively; the initial period is denoted by 0, and the terminal one by T and the terminal one by T .

7 www.snb.ch 2. A retrospective measure of the price level (continued) ( ) p p Next, normalize run (2) by dividing all its elements by the first one: ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ p 0 0 T T p 1 1 T T p 2 2 T T T 1 T 1 T T − q q q p q i i i i i i i i i , i i i (2) , , ... , , 1 , ∑ ∑ ∑ ∑ T T T T T T T T p q p q p q p q i i i i i i i i i i i i ∑ ∑ ∑ ∑ 1 T 2 T T 1 T T T − p q p q p q p q i i i i i i i i . , , ... , , i i i i (3) 1, ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 0 T 0 T 0 T 0 T p p i q q p p i q q p p i q q p p i q q i i i i i i i i i i i i i i i i

8 www.snb.ch 2. A retrospective measure of the price level (continued) ( ) p p Finally, take the geometric means of the corresponding elements of (1) and (3) to get the following sequence of l f ( ) d ( ) h f ll i f pseudo Fisher indices: ∑ ∑ ∑ ∑ T 1 0 1 0 2 0 − T 0 p q p q p q p q i i i i i i i i i i , i , ... , , i (1) 1, ∑ ∑ ∑ ∑ 0 0 0 0 0 0 0 0 p q p q p q p q i i i i i i i i i i i i ∑ ∑ ∑ ∑ 2 1 T T T 1 T T T − p q p q p q p q i i i i i i i i ... , i . 1, i , , i , i (3) ∑ ∑ ∑ ∑ 0 0 T T 0 T 0 T p q p q p q p q i i i i i i i i i i i i i i i i ∑ ∑ ∑ ∑ ∑ ∑ T 1 0 T 1 T T 0 T T 1 0 1 − − T p q p q p q p q p q p q i i i i i i i i i i i i ... , . i i , i i (4) ( ) i i , 1, , ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 0 0 0 0 0 0 T T 0 0 0 0 0 0 T T 0 0 0 0 0 0 T T p q p q p q p q p q p q i i i i i i i i i i i i i i i i i i

9 www.snb.ch 3. Comparison with Hansen p H ANSEN (2007) recently proposed the following Fisher-like H ANSEN (2007) recently proposed the following Fisher like price index: 1 ⎡ ⎤ t p 2 ∑ 0 ⎢ i ⎥ s i 0 ⎢ ⎥ p 0 : i t i (5) , t = 0, 1, ..., T , P = ⎢ ⎥ H 0 ⎣ ∑ ∑ p ⎢ ⎢ ⎥ ⎥ T i i s s i ⎢ ⎥ t ⎦ p i i where: 0 0 T T p q p q 0 T . i i , i i s s = = ∑ ∑ i i 0 0 T T p q p q i i i i i i

10 www.snb.ch 3. Comparison with Hansen (continued) : ( ) p For comparison purposes, rewrite formulae (1) and (2) in For comparison purposes, rewrite formulae (1) and (2) in terms of price relatives and expenditure share weights: ∑ t 0 p q i i t p ∑ i 0 i , t = 0, 1, ..., T (1’) s = ∑ ∑ i 0 0 0 p q p i i i i i ∑ t T p q i i t p p ∑ ∑ i i T T i i , t = 0, 1, ..., T . (2’) ( ’) s = ∑ i T T T p q p i i i i i

11 www.snb.ch 3. Comparison with Hansen (continued) : ( ) p Using (2’), it can be seen that the period t index in (3) can Using (2 ), it can be seen that the period t index in (3) can be written as follows: ∑ t T p q i i t p ∑ T (2’) i i , t = 0, 1, ..., T . s = ∑ ∑ i T T T p p q q p p i i i i i i i i i t ∑ p ∑ ∑ T t T i s p q i i i T p p i , t = 0, 1, ..., T . (3’) i i ∑ = 0 T 0 p q ∑ p T i i i s i i T p i i

12 www.snb.ch 3. Comparison with Hansen (continued) : ( ) p Finally, using (1’) and (3’), it can be seen that the period- t ( P 0 t ) i 0:t ) in the sequence defined by pseudo Fisher price index ( P F d Fi h i i d h d fi d b (4) can be written as follows: ∑ t 0 p q i i t p ∑ i 0 i , t = 0, 1, ..., T (1’) s = ∑ i 0 0 0 p q p i i i i i t ∑ p ∑ T t T i s p q i i i T p i i i , t = 0, 1, ..., T . (3’) = ∑ 0 T 0 p q ∑ ∑ p i i T i s i i i T T p i i 1 ⎡ ⎤ 1 t t 2 ∑ p ∑ p ∑ ∑ ⎡ ⎤ 0 i T i t 0 t T ⎢ ⎥ 2 s s p q p q ⋅ i i ⎢ i i i i ⎥ 0 T ⎢ ⎥ p p (4’) ( ) 0 : t i i , t = 0, 1, ..., T . i i i i P ⎢ ⎢ ⎥ ⎥ = = ∑ ∑ ∑ ∑ ⎢ ⎢ ⎥ ⎥ F F 0 0 0 0 0 0 0 0 T T p q p q p ⎢ ⎥ ∑ ⎢ ⎥ i i i i T i s ⎣ ⎦ i i i ⎢ ⎥ T ⎣ ⎦ p i i

13 www.snb.ch 3. Comparison with Hansen (continued) : ( ) p 1 ⎡ ⎤ t 2 ∑ p 0 i ⎢ ⎥ s i 0 ⎢ ⎥ p (5) 0 : t i i , t = 0, 1, ..., T , P = ⎢ ⎥ H 0 ⎣ ∑ ∑ p ⎢ ⎢ ⎥ ⎥ T i s s i i ⎢ t ⎥ ⎦ p i i 1 ⎡ ⎡ ⎤ ⎤ 1 t t 2 2 ∑ p p ∑ p p ∑ ∑ ⎡ ⎡ ⎤ ⎤ 0 i T i t 0 t T ⎢ ⎥ 2 s s p q p q ⋅ i i ⎢ i i i i ⎥ 0 T ⎢ ⎥ p p 0 : t i i i i i i , t = 0, 1, ..., T . (4’) P ⎢ ⎥ = = ∑ ∑ ⎢ ⎥ F 0 0 0 T 0 p q p q ∑ p ⎢ ⎥ ⎢ ⎥ i i i i T i s ⎣ ⎦ i i i ⎢ T ⎥ ⎣ ⎦ p i i 0:T = P F Note that P H 0:T .

14 www.snb.ch 3. Comparison with Hansen (continued) : ( ) p “Laspeyres element”: t = ∑ p 0 : t 0 0 : t , t = 0, 1, ..., T . i P s P = H ( L ) i F ( L ) 0 p i i “Paasche element”: t ∑ p T i s i T 1 1 p p 0 0 : : t t i i 0 0 : : t t i i , t = 0, 1, ..., T . t 0 1 T P P = ≠ = H ( P ) F ( P ) 0 0 ∑ p ∑ p T i T i s s i i t T p p i i i i 0:t can be interpreted as a harmonic period- T weighted P H(P) Young index It does not have the Paasche form though Young index. It does not have the Paasche form, though.

15 www.snb.ch 4. Application to Swiss data pp � The application is for the Swiss CPI data for the period 1993 to � The application is for the Swiss CPI data for the period 1993 to 2000 � At the time the Swiss CPI was computed as a direct Laspeyres � At the time the Swiss CPI was computed as a direct Laspeyres quantity index, with fixed weights using May 1993 as the reference period p � The weights were eventually revised in 2000 � The number of categories common to both surveys amounts to 192 out of 201 � In value terms, these categories represented over 99% of the CPI