Entropy-based benchmarking methods

Transcription

1 Entropy-based benchmarking methods Umed Temurshoev Faculty of Economics and Business University of Groningen, PO Box AV Groningen, The Netherlands This project is funded by the European Commission, Research Directorate General as part of the 7th Framework Programme, Theme 8: Socio-Economic Sciences and Humanities. Grant Agreement no: , Alexandria, VA, USA, June / 22

2 Outline 1 problem 2 Denton and Causey-Trager methods 3 SemiGRAS method 4 Entropy-based methods 5 Empirical illustration 6 Conclusion 2 / 22

3 problem What is benchmarking? Series of high-frequency data: quarterly, monthly, daily Timely data, only information about the short-term movements, less reliable 3 / 22

4 problem What is benchmarking? Series of high-frequency data: quarterly, monthly, daily Timely data, only information about the short-term movements, less reliable Series of low-frequency data: annual series High precision, reliable information on the aggregate level and long-term movements 3 / 22

5 problem What is benchmarking? Series of high-frequency data: quarterly, monthly, daily Timely data, only information about the short-term movements, less reliable Series of low-frequency data: annual series High precision, reliable information on the aggregate level and long-term movements problem: combine the relative strengths of the inconsistent low- and high-frequency series 3 / 22

6 problem Figure: problem example (Source: Bloem et al. (2001), QNA Manual, IMF, Washington DC, p. 91) 4 / 22

7 problem Methods Basis of current benchmarking methods: principle of movement preservation 5 / 22

8 problem Methods Basis of current benchmarking methods: principle of movement preservation Mathematical methods: deterministic, binding annual constraints Denton (1971), Causey and Trager (1981) 5 / 22

9 problem Methods Basis of current benchmarking methods: principle of movement preservation Mathematical methods: deterministic, binding annual constraints Denton (1971), Causey and Trager (1981) Statistical methods: stochastic, binding/nonbinding constraints Regression methods: Chow and Lin (1971), ARIMA and generalized regression-based methods (Dagum and Cholette, 2006) 5 / 22

10 problem Assessment and applications Chen (2007): 60 series from the US national economic accounts the modified Denton proportional first difference method outperforms the other methods, though the Causey-Trager growth preservation model is a close competitor in certain cases. 6 / 22

11 problem Assessment and applications Chen (2007): 60 series from the US national economic accounts the modified Denton proportional first difference method outperforms the other methods, though the Causey-Trager growth preservation model is a close competitor in certain cases. Denton procedures: BEA, Statistics Netherlands Causey-Trager procedure: US Census Bureau (Brown 2010, Titova et al. 2010) 6 / 22

12 problem This paper Principle of movement and sign preservation Motivation: abundant series are volatile and/or include both positive and negative values 7 / 22

13 problem This paper Principle of movement and sign preservation Motivation: abundant series are volatile and/or include both positive and negative values Extended pro-rata distribution (semigras method) 7 / 22

14 problem This paper Principle of movement and sign preservation Motivation: abundant series are volatile and/or include both positive and negative values Extended pro-rata distribution (semigras method) Entropy-based benchmarking methods 7 / 22

15 Denton and Causey-Trager methods Notations t = 1,..., T years i = 1,..., I sub-periods for each year zt = I 1 vector of observed indicator series for year t xt = I 1 vector of estimated benchmarked series for year t z = (z01, z02,..., z0t )0 x = (x01, x02,..., x0t )0 y = T 1 vector of annual data 8 / 22

16 Denton and Causey-Trager methods Notations t = 1,..., T years i = 1,..., I sub-periods for each year zt = I 1 vector of observed indicator series for year t xt = I 1 vector of estimated benchmarked series for year t z = (z01, z02,..., z0t )0 x = (x01, x02,..., x0t )0 y = T 1 vector of annual data Consider annual constraints ı0 xt = yt for all t, or. ı ı x = Bx = y ı0 8 / 22

17 Denton and Causey-Trager methods Denton (1971) The additive first difference (AFD) and proportional first difference (PFD) variants are: fafd = (x1 z1 )2 + N X [(xj zj ) (xj 1 zj 1 )]2, (1) j=2 fpfd = x1 1 z1 2 + N X xj j=2 xj 1 zj zj 1 2, (2) subject to Bx = y. 9 / 22

18 Denton and Causey-Trager methods Causey and Trager (1981) Growth rate preservation principle: fct = N X xj zj 2. xj 1 zj 1 (3) j=2 subject to Bx = y. 10 / 22

19 Denton and Causey-Trager methods Causey and Trager (1981) Growth rate preservation principle: fct = N X xj zj 2. xj 1 zj 1 (3) j=2 subject to Bx = y. For all methods, changing sign is possible! 10 / 22

20 SemiGRAS method Apply the Generalized RAS idea to benchmarking (Gu nlu k-s enesen and Bates 1988,Junius and Oosterhaven 2003) Result The solution of the benchmarking problem is X = Ps Ns 1, where the annual adjustment factors s are derived from q c s = 0.5 P ı y + y y + 4 (P ı) (N ı), where Z = (z1, z2,..., zt ), X = (x1, x2,..., xt ) and Z = P N. Level preservation principle step problem due to discontinuities between years 11 / 22

21 Entropy-based methods Proportional first difference preservation xj zj = xj 1 zj 1 + εj0, denote gj = zj zj 1 and εj = zj 1 εj0, then 12 / 22

22 Entropy-based methods Proportional first difference preservation xj zj = xj 1 zj 1 + εj0, denote gj = zj zj 1 and εj = zj 1 εj0, then xj = gj xj 1 + εj for all j = 2,... N(= IT ) 12 / 22

23 Entropy-based methods Proportional first difference preservation xj zj = xj 1 zj 1 + εj0, denote gj = zj zj 1 and εj = zj 1 εj0, then xj = gj xj 1 + εj for all j = 2,... N(= IT ) xn gn xn 1 = εn xn 1 gn 1 xn 2 = εn 1... x 2 g2 x 1 = ε 2 1 gn g 0 N g2 xn xn 1... x1 εn εn 1 =... ε1 12 / 22

24 Entropy-based methods Proportional first difference preservation xj zj = xj 1 zj 1 + εj0, denote gj = zj zj 1 and εj = zj 1 εj0, then xj = gj xj 1 + εj for all j = 2,... N(= IT ) xn gn xn 1 = εn xn 1 gn 1 xn 2 = εn 1... x 2 g2 x 1 = ε 2 1 gn g 0 N g2 PFD constraints: CP x = ε xn xn 1... x1 εn εn 1 =... ε1 12 / 22

25 Entropy-based methods Proportional first difference preservation Annual constraints with non-binding possibilities: y = Bx + τ 13 / 22

26 Entropy-based methods Proportional first difference preservation Annual constraints with non-binding possibilities: y = Bx + τ τ y B x + y = Γ x + e = ε 0 CP 13 / 22

27 Entropy-based methods Proportional first difference preservation Annual constraints with non-binding possibilities: y = Bx + τ τ y B x + y = Γ x + e = ε 0 CP OLS? 13 / 22

28 Entropy-based methods Proportional first difference preservation Annual constraints with non-binding possibilities: y = Bx + τ τ y B x + y = Γ x + e = ε 0 CP OLS? Generalized cross-entropy (Golan et al. 1996): principle of minimum discrimination information (Kullback 1959) 13 / 22

29 Entropy-based methods Proportional first difference preservation Annual constraints with non-binding possibilities: y = Bx + τ τ y B x + y = Γ x + e = ε 0 CP OLS? Generalized cross-entropy (Golan et al. 1996): principle of minimum discrimination information (Kullback 1959) Treat x j as a discrete random variable with a compact support and M possible outcomes rj = (rj1,..., rjm )0 with 2 M <, i.e., x j = M X rjm pjm = r0j pj. m=1 13 / 22

30 Entropy-based methods Proportional first difference preservation Similarly, ek = PJ j=1 vkj wkj = v0k wk 14 / 22

31 Entropy-based methods Proportional first difference preservation Similarly, ek = PJ j=1 vkj wkj = v0k wk Reparameterize: y = Γ x + e = Γ Rp + Vw 14 / 22

32 Entropy-based methods Proportional first difference preservation Similarly, ek = PJ j=1 vkj wkj = v0k wk Reparameterize: y = Γ x + e = Γ Rp + Vw Let q and u be prior weights of x and e, respectively 14 / 22

33 Entropy-based methods Proportional first difference preservation Similarly, ek = PJ j=1 vkj wkj = v0k wk Reparameterize: y = Γ x + e = Γ Rp + Vw Let q and u be prior weights of x and e, respectively The GCE estimator is min I (p, q, w, u) = p0 log(p/q) + w0 log(w/u) p,w (4) subject to y = Γ Zx + Vw, (5) 0 (6) 0 (7) ı = (I ı )p, ı = (I ı )w, 14 / 22

34 Entropy-based methods Additive first difference preservation xj zj = (xj 1 zj 1 ) + εj for all 2 = 1,... N 15 / 22

35 Entropy-based methods Additive first difference preservation xj zj = (xj 1 zj 1 ) + εj for all 2 = 1,... N zj zj 1 = xj xj 1 + εj 15 / 22

36 Entropy-based methods Additive first difference preservation xj zj = (xj 1 zj 1 ) + εj for all 2 = 1,... N zj zj 1 = xj xj 1 + εj zn zn 1 zn 1 zn 2 =... z2 z xn xn 1... x1 εn εn ε1 15 / 22

37 Entropy-based methods Additive first difference preservation xj zj = (xj 1 zj 1 ) + εj for all 2 = 1,... N zj zj 1 = xj xj 1 + εj zn zn 1 zn 1 zn 2 =... z2 z1 xn xn x1 AFD constraints: z = CA x + ε εn εn ε1 15 / 22

38 Entropy-based methods Additive first difference preservation Annual constraints with non-binding possibilities: y = Bx + τ y B τ = x + y = Γ x + e z CA ε 16 / 22

39 Entropy-based methods Growth rate preservation xj xj 1 = zj zj 1 + ζj, denote gj = zj zj 1 then 17 / 22

40 Entropy-based methods Growth rate preservation xj xj 1 = zj zj 1 + ζj, denote gj = zj zj 1 then xj = (gj + ζj )xj 1 for all 2 = 1,... N 17 / 22

41 Entropy-based methods Growth rate preservation xj xj 1 = zj zj 1 + ζj, denote gj = zj zj 1 then xj = (gj + ζj )xj 1 for all 2 = 1,... N Assume ζj = ζ + εj xj 1, then xj = (gj + ζ)xj 1 + εj 17 / 22

42 Entropy-based methods Growth rate preservation xj xj 1 = zj zj 1 + ζj, denote gj = zj zj 1 then xj = (gj + ζj )xj 1 for all 2 = 1,... N Assume ζj = ζ + εj xj 1, then xj = (gj + ζ)xj 1 + εj Work in progress 17 / 22

43 Empirical illustration Aggregate indicators of closeness PN 1 Average absolute level difference: AALD = j=1 xj zj N 18 / 22

44 Empirical illustration Aggregate indicators of closeness PN 1 Average absolute level difference: AALD = 2 Average absolute change difference: P P AACD = N j=2 (xj xj 1 ) (zj zj 1 ) N 1 = N j=2 j=1 xj zj N (xj zj ) (xj 1 zj 1 ) N 1 18 / 22

45 Empirical illustration Aggregate indicators of closeness PN 1 Average absolute level difference: AALD = 2 Average absolute change difference: P P AACD = 3 N j=2 (xj xj 1 ) (zj zj 1 ) N 1 = N j=2 j=1 xj zj N (xj zj ) (xj 1 zj 1 ) N 1 Average absolute proportional difference: xj zj 100 PN AAPD = N 1 j=2 xj 1 zj 1 18 / 22

46 Empirical illustration Aggregate indicators of closeness PN 1 Average absolute level difference: AALD = 2 Average absolute change difference: P P AACD = N j=2 (xj xj 1 ) (zj zj 1 ) N 1 = N j=2 j=1 xj zj N (xj zj ) (xj 1 zj 1 ) N 1 3 Average absolute proportional difference: xj zj 100 PN AAPD = N 1 j=2 xj 1 zj 1 4 Average absolute relative proportional difference: xj zj zj 100 PN AARPD = N 1 j=2 xj 1 zj 1 / zj / 22

47 Empirical illustration Denton (1971) data z = ( )0 19 / 22

48 Empirical illustration Denton (1971) data z = ( Bz = 400 6= ) =y / 22

49 Empirical illustration Denton (1971) data z = ( Bz = 400 6= ) =y r0j = (0.1, 0.55, 1, 1.45, 1.9) zj with q0j = ( ) 19 / 22

50 Empirical illustration Denton (1971) data z = ( Bz = 400 6= ) =y r0j = (0.1, 0.55, 1, 1.45, 1.9) zj with q0j = ( ) v0k = ( 1.7, 0.85, 0, 0.85, 1.7) σz with u0k = ( ) except for the annual constraints with u0k = ( ) 19 / 22

51 Empirical illustration Results Table: Some aggregate indicators SemiGRAS AFD Denton AFD Entropy PFD Denton PFD Entropy CT fct AALD AACD AAPD (%) AARPD (%) / 22

52 Conclusion Conclusions 1 Advantages of entropy-based benchmarking methods (Any choice of) Binding and/or nonbinding constraints 21 / 22

53 Conclusion Conclusions 1 Advantages of entropy-based benchmarking methods (Any choice of) Binding and/or nonbinding constraints Reliability indicators for each element 21 / 22

54 Conclusion Conclusions 1 Advantages of entropy-based benchmarking methods (Any choice of) Binding and/or nonbinding constraints Reliability indicators for each element Applicable for any size of yearly observations 21 / 22

55 Conclusion Conclusions 1 Advantages of entropy-based benchmarking methods (Any choice of) Binding and/or nonbinding constraints Reliability indicators for each element Applicable for any size of yearly observations Using non-sample information 21 / 22

56 Conclusion Conclusions 1 Advantages of entropy-based benchmarking methods (Any choice of) Binding and/or nonbinding constraints Reliability indicators for each element Applicable for any size of yearly observations Using non-sample information Controlling for sign change 21 / 22

57 Conclusion Conclusions 1 Advantages of entropy-based benchmarking methods (Any choice of) Binding and/or nonbinding constraints Reliability indicators for each element Applicable for any size of yearly observations Using non-sample information Controlling for sign change 2 Plausible competitors to current benchmarking methods 21 / 22

58 Thanks for your attention! 22 / 22