A Comparison of Weighted Time Dummy Hedonic and Time-Product Dummy Indexes Jan de Haan, Rens Hendriks and Michael Scholz
Background • Aizcorbe, Corrado and Doms (2003) – “When Do Matched-Model and Hedonic Techniques Yield Similar Price Measures?” • Aizcorbe and Pho (2005) – “Differences in Hedonic and Matched-Model Price Indexes: Do the Weights Matter?” • Silver and Heravi (2005) – “A Failure in the Measurement of Inflation: Results from a Hedonic and Matched Experiment Using Scanner Data” • Krsinich (2016) – “The FEWS Index: Fixed Effects with a Window Splice”
The TDH and TPD models • The Time Dummy Hedonic model: T K ∑ ∑ = δ + δ + β + ε t t t t p D z 0 ln i i k ik i = = t k 1 1 • The Time Product Dummy model: − T N 1 ∑ ∑ t = α + δ t t + γ + ε t p D D ln i i i i i = = t i 1 1
Weighted TDH and TPD Indexes t = δ t ˆ P 0 exp( ) ∏ t t s p i ( ) i ( ) K ∑ t t = β − t ˆ P ∈ z z i S 0 0 exp ∏ TDH k k k s 0 p 0 i ( ) = k 1 i ∈ i S 0 ∏ t t s p ( ) i ( ) i = ∏ t t γ − γ t P ∈ i S 0 0 ˆ ˆ exp TPD s 0 p 0 i ( ) i ∈ i S 0
Decomposition in regression residuals (1) • Weighted TDH and TPD sum to zero in each period. t s s 0 t i i p p 0 ˆ ˆ ∏ ∏ = = i i 1 t p p 0 t ∈ ∈ i S i S 0 i i • The TDH and TPD indices can be written as: t s 0 s t i i s s 0 t t t t i i p p p p ˆ ˆ 2 2 ∏ ∏ ∏ ∏ = = = t P i i i i 0 p p p p 0 0 0 0 ˆ ˆ t t ∈ ∈ ∈ ∈ i S 0 i i S i i S 0 i i S i
Decomposition in regression residuals (2) t ( ) • P s 0 0 = − TPD D u u 0 0 exp D TPD D TDH t P s ( ) ( ) 0 0 TDH M t ( ) • s • − t t N u u exp N TDH N TPD t s ( ) ( ) M [ ] ( ) ( ) • − t − − t u u u u 0 0 ( ) 0 0 ( ) exp M TPD M TPD M TDH M TDH ( ) ( ) ( ) ( )
Empirical Illustration (1) Weighted TPD & TDH Indexes 130 120 110 100 90 80 70 60 50 40 1 3 5 7 9 1113151719212325272931333537394143454749 TPD TDH
Empirical Illustration (2) Weighted Average Residuals 0,4 0,3 0,2 0,1 0 -0,1 -0,2 -0,3 -0,4 -0,5 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 Disapp. TPD Disapp. TDH New TPD New TDH
Empirical Illustration (3) Aggregate Expenditure Shares 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 Disapp. New Matched (0) Matched (t)
Empirical Illustration (4) Decomposition of TPD-TDH Ratio 1,3 1,2 1,1 1 0,9 0,8 0,7 0,6 0,5 0,4 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 Ratio Disapp. New Third term
Empirical Illustration (5) TPD-TDH Indexes – Group Level 130 120 110 100 90 80 1 3 5 7 9 1113151719212325272931333537394143454749 TPD TDH
Empirical Illustration (6) Decomposition – Group Level 1,02 1,01 1 0,99 0,98 0,97 0,96 0,95 0,94 0,93 0,92 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 Ratio Disapp. New Third term
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