Transition Modeling and Econometric Convergence Tests

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1 Transition Modeling and Econometric Convergence Tests Donggyu Sul University of Auckland With Peter C.B. Phillips

2 Time Varying Common Factor Representation Panel Data: Yes, no index No. I am not covering this person A person B person C person D

3 PWT 6.1 Countries log per capita real income

4 PWT 6.1 Countries log per capita real income

5 The first convergence club

6 The second convergence club

7 The third convergence club

8 The fourth group

9 PWT 6.1 Countries log per capita real income data

10 log t Regression log H 1 H t 2logL t a blogt u t for t rt, rt 1,...,T with r 1/3 where L t logt H t 1 N h N i 1 it 1 2 h it log X it 1 N N i 1 log Xit

11 Contents 1. Panel Data: Modeling Time Varying Individual Heterogeneity: Time varying common factor representation 2. Examples and Comparison 3. Convergence Concept: Relative v.s Absolute 4. Convergence Test; logt test. 5. Clustering Method 6. Empirical Examples

12 Time Varying Common Factor Representation lnx it it t i 1,...,N; individuals t 1,...,T; time

13 Time Varying Common Factor Representation lnx it it t Common Factor: Common behavior, Representative economic agent s behavior

14 Time Varying Common Factor Representation lnx it it t Factor loading coefficients -idiosyncratic behavior -Economic distance between an individual and representative economic agent

15 Time Varying Common Factor Representation lnx it it t Parameterofinterest it Especially convergent behavior of it

16 Time Varying Common Factor Representation: Comparison Conventional Common Factor Model lnx it i t it Time Varying Common Factor Model lnx it it t

17 Time Varying Common Factor Representation: Comparison Conventional Common Factor Model lnx it i t it Time Varying Common Factor Model lnx it it t Two un-identified Idiosyncratic components One un-identified Idiosyncratic component

18 Time Varying Common Factor Representation: Comparison lnx it i t it i it t t it t it p i as t since it / t o p 1

19 Time Varying Common Factor Representation lnx it blnz 1t c lnz 2t u it lnx it b lnz 1t u it, i G 1 lnx it c lnz 2t u it, i G 2 lnx it b 1 lnz 1t u it, i G 1 lnx it b 2 lnz 1t u it, i G 2 lnx it it t it 1 if i G 1 it 2 if i G 1

20 Time Varying Common Factor Representation lnx it blnz 1t c lnz 2t u it lnx it b lnz 1t u it, i G 1 lnx it clnz 2t u it, i G 2 lnx it b 1 lnz 1t u it, i G 1 lnx it b 2 lnz 1t u it, i G 2 lnx it it t it 1 if i G 1 it 2 if i G 2

21 Time Varying Common Factor Representation: Examples Ex 1: Transitional Growth Path under Heterogeneous Technology Progress: Phillips and Sul (2006) logy it logy i logy i0 logy i e it loga it a it loga it a it log A it t t it t logy it logy jt it jt t

22 Time Varying Common Factor Representation: Examples Ex 2. Heterogeneous economic agents model 1 U C 1 r t E t it 1 it U C it it X it it C it C it r t C it C it 1 Rt C Rt C Rt 1 t, C it C it 1 Rt it C Rt C Rt 1 it t

23 Time Varying Common Factor Representation: Examples Ex 2. Heterogeneous economic agents model X it C it C it 1 Rt it C Rt C Rt 1 it t it? Relationship b.t it and Q t,tb t

24 Time Varying Common Factor Representation: Examples Ex 3. Stock Return X it X it 1i 1t 2i 2t 3i 3t it X it 1,it 1t 2,it 2t 3,it 3t it jt m j t t s 1 jt 1,it 2,it 2t 1t 1t m jt s 1 t m 1 t s 1 3,it 3t 1t it 1t 1t e js,forj 1,2,3, with m 1 0, t js 1s m j m 1 o p 1, it 1,it 2,it m 2 m 1 3,it m 3 m 1 1 o p 1, t 1t

25 Time Varying Common Factor Representation: Examples Ex 3. Stock Return X it 1,it 1t 2,it 2t 3,it 3t it X it it t it 1,it 2,it m 2 m 1 3,it m 3 m 1 1 o p 1 t 1t

26 Time Varying Common Factor Representation: Examples Ex 3. Earning and Wage in Labor Economics logy it 1,it 1t logy it 2,it 2t if i male if i female logy it 1,it 1t 2,it 2t it t 1,it 1, 2,it 2 it 1 if i male 2 if i femal

27 Time Varying Common Factor Representation: Examples Ex 3. Earning and Wage in Labor Economics logy it k s 1 sit st if i s it t sit s it s if i s

28 Convergence Concept Absolute Convergence p lim k lnx it k lnx jt k 0 Relative Convergence p lim k ln X it k ln X jt k 1 RC nests AC. When AC holds, RC should hold. Even when RC holds, AC may not hold.

29 Convergence Concept: Absolute Convergence lnx it it t lnx it lnx jt it jt t Let t t, it jt t, 0 it jt 0 lnx it lnx jt if 1 if 1 0if 1

30 Convergence Concept: Relative Convergence lnx it it t lnx it lnx jt it jt t Let t t, it jt t, 0 it jt 0 lnx it lnx jt if 1 if 1 0if 1 lnx it /lnx jt it / jt 1

31 Convergence Concept: Relative v.s. Absolute X 2t 0. 45t u 2t X 1t 0.5t u 1t X 3t t t u 3t Time

32 Convergence Concept: Relative v.s. Absolute lnx it lnx jt t 1 Let 1, 1/2 lnx it lnx jt t Δ lnx it Δ lnx jt t t 1 0 Relative convergence => the same growth rate in the long run

33 Comparison: RC v.s. Cointegration lnx it it t t t 1 t 1t 2t 0.9 t 0.9 3t t In finite sample lnx 1t &lnx 2t : Cointegrated lnx 1t &lnx 3t : No Co. ln long run lnx 1t lnx 2t I 1 lnx 1t lnx 3t I 0

34 Relative Transition Parameter: Approximation of factor loading coefficients lnx it it t 1. Total number of obs. = NxT 2. Total number of unknowns = NxT + Tx1 3. Parametric estimation: Requires restrictions: Example Parametric Restriction it i it 1 it t t 1 e t

35 Relative Transition Parameter: Approximation of factor loading coefficients Problem: lnx it it t Parametric Restriction it i it 1 it t t 1 e t 1. How to justify this restriction? 2. What if the common factor is I(0) or has a linear trend 3. Is it possible to examine the convergent behavior of? it

36 Relative Transition Parameter Solution: Don t estimate them but approximate them. How? h it X it 1 N Xit N i 1 it 1 N N i 1 it working with H t 1 N h N i 1 it 1 2 If the common factor is a constant, then don t need to approximate.

37 Relative Transition Parameter: Example X 2t 0. 45t u 2t X 1t 0.5t u 1t X 3t t t u 3t Time

38 Relative Transition Parameter: Example X 1t / 1/2 X 1t X 3t X 3t / 1/2 X 1t X 3t

39 Relative Transition Parameter: Example X 1t / 1/2 X 1t X 2t X 2t / 1/2 X 1t X 2t

40 Relative Transition Parameter: How to Use h it Within X it 1 N i 1 N Xit i 1,...,N

41 Relative Transition Parameter Transition Paths for Regional Groups of the Contiguous US States Mid-Atlantic Great Lakes Pacific South Atlantic East South Central h it Within New England Mountain Plains States West South Cental X it 1 N i 1 N Xit i 1,...,N

42 1.10 OECD Transition Paths: 18 Western OECD countries from NZL AUS 1.05 USA Relative Transition POR GER 0.90 SPA

43 Relative Transition Parameter: How to Use Relative 1: (Cross) h jt X jt 1 G j i j G G X jt G 1,...,G,j

44 Examples of Phase B Transitions among the World Economies 0.81 Relative Transition Curves Eastern Asia & Pacific India China

45 Relative Transition Parameter: How to Use Relative 2: (Cross) h gt X gt 1 G g j h gt 1 G G g 1 j G G Xgt G 1,...,G,j h gt

46 1.10 OECD Transition Paths: 18 Western OECD countries from NZL AUS 1.05 USA Relative Transition POR GER 0.90 SPA

47 Evidence of Phase B & C Transitions in Historical OECD Data Relative Transition Against USA UK,NZ,AUT,CAN AUS,GER,ITA 0.92 POR,SPA FIN,SWE,NOR FRA,BEL,DEN,NET,SWS

48 Modeling and Testing Convergence h i r Sub 1 1 Sub 2 r

49 Modeling and Testing Convergence lnx it it t it i it it, it i L t t, t 1, i 0foralli. L t is a slowlying varying function. L at /L T 1asT for all a 0. Ex: log T. log at /logt loga logt /logt log a log T 1 1

50 Modeling and Testing Convergence h i r Sub 1 1 Sub 2 r

51 Modeling and Testing Convergence it i it it, it i L t t H 0 : i and 0 H A : i or 0 H 0 : Convergence for all i H A : Divergence forsome i

52 Modeling and Testing Convergence h i r Sub 1 1 Sub 2 r

53 Testing Convergence Step 1 H t 1 N N h i 1 it 1 2, h it X it N 1 N i 1 Xit Step 2 log H 1 H t 2log logt a blogt u t Step 3 Test the null of H 0 : b 0 t b with HAC estimator H 0 : i and 0 H 0 : b 0

54 Asymptotic properties of the logt regression DGP: it i it it, it i L t t Reg: log H 1 H t 2log logt a b logt u t Under H 0, b 2 Under H A, b 0, but t b Why? Under H A, h it does not converge at all.

55 Asymptotic properties of the logt regression X 1t / 1/2 X 1t X 2t 1.5 X 1t / 1/2 X 1t X 3t X 2t / 1/2 X 1t X 2t 0.5 X 3t / 1/2 X 1t X 3t

56 Clustering t {1,2} = 3.6, t {1,2,3} = 0.25, t {1,2,3,4} = 0.86, t {1,2,3,4,5} = 5.23 t {1,2,4} = 3.51, t {1,2,5} = 227, t {1,2,6} =

57 Clustering Step 1 Step 2 Last observation ordering Find core members, run the logt with i 1,...,k, k arg max k t k subject to min t k 1.65, k until t k 1.65 Step 3 Run the logt reg with adding/dropping an individual i G 1 if t b,i 0 G 1 c if t b,i 0 Step 4 Run the logt test with G c 1. If t G1 c 1.65, repeat Step 2 & 3 until t Gs c 1.65

58 Clustering t {1,2} = 3.6, t {1,2,3} = 0.25, t {1,2,3,4} = 0.86, t {1,2,3,4,5} = 5.23 t {1,2,4} = 3.51, t {1,2,5} = 227, t {1,2,6} =

59 Clustering Step 1 Step 2 Last observation ordering Find core members, run the logt with i 1,...,k, k arg max k t k subject to min t k 1.65, k until t k 1.65 Step 3 Run the logt reg with adding/dropping an individual i G 1 if t b,i 0 G 1 c if t b,i 0 Step 4 Run the logt test with G c 1. If t G1 c 1.65, repeat Step 2 & 3 until t Gs c 1.65

60 Clustering Core Core 5 6 t {1,2} = 3.6, t {1,2,3} = 0.25, t {1,2,3,4} = 0.86, t {1,2,3,4,5} = 5.23 t 1,2 3.6, t 1,2,3 0.15, t 1,2,3,4 0.86, t 1,2,3,4, t 1,2,4 3.51, t 1,2,5 227, t 1,2, t {1,2,4} = 3.51, t {1,2,5} = 227, t {1,2,6} =

61 Clustering Step 1 Step 2 Last observation ordering Find core members, run the logt with i 1,...,k, k arg max k t k subject to min t k 1.65, k until t k 1.65 Step 3 Run the logt reg with adding/dropping an individual i G 1 if t b,i 0 G 1 c if t b,i 0 Step 4 Run the logt test with G c 1. If t G1 c 1.65, repeat Step 2 & 3 until t Gs c 1.65

62 Clustering Core 5 6 t {1,2} = 3.6, t {1,2,3} = 0.25, t {1,2,3,4} = 0.86, t {1,2,3,4,5} = 5.23 t 1,2 3.6, t 1,2,3 0.15, t 1,2,3,4 0.86, t 1,2,3,4, t 1,2,4 3.51, t 1,2,5 227, t 1,2, t {1,2,4} = 3.51, t {1,2,5} = 227, t {1,2,6} =

63 Clustering Step 1 Step 2 Last observation ordering Find core members k arg max k t k subject to min t k 1.65 Step 3 Run the logt reg with adding/dropping an individual at a time. i G 1 if t b,i 0 G 1c if t b,i 0 Step 4 Run the logt test with G 1c. If t G1 c 1.65, repeat Step 2 & 3 until t Gs c 1.65

64 Absolute v.s. Relative Relative H 0 : b 0 Absolute H 0 : b 2? depending on the growth rate of t 1if t is I 1 with a drift 0.5 if t is I 1 without a drift Repeat everything with t b b 2 s.e.

65 Monte Carlo Studies: Extreme Case Finite Sample Performance of the log t test Size: 5% 0,0.5 0,0.9 T N Nominal Power 1 1 T N i U 1,

66 Monte Carlo Studies: Extreme Case (Clustering) Failure Rates with Sum of Type I & II Errors % 10% 20% 30% 40% % 10% 20% 30% 40% 50% T 50%

67 Monte Carlo Studies: Extreme Case (Clustering) Failure Rates with α=0.2 Sum of Type I & II Errors % 10% 20% 30% 40% 50% % 10% 20% 30% 40% T 50%

68 Monte Carlo Studies: Moderate Case (Clustering): Sign Test: 50% significance level sum of type I & II errors T

69 Example 1: Cost of Living (COL) Index 1. Data: 19 Metropolitan U.S. Cities CPI indices 2. CPI cannot be used to compare the COL across cities due to the base year problem. logp o it o it logp o t e it logp it log P o it /P o i1 logp o o it logp i1 o it logp o t o i1 logp o 1 e it e i1 o o it i1 log P 1 o log P t o e it e i1 log P t o it logp t o logp t o

70 Example 1: Cost of Living (COL) Index 7.5 Min, Max, Median of 19 Consumer Price Indices 7 Discard to avoid the initial effect Max Median 5.5 Min

71 Example 1: Cost of Living (COL) Index NYC CHI HOU LAX SF0 SEA ATL STL

72 Example 1: Cost of Living (COL) Index log H 1 H t 2loglogt logt,

73 Last T Name t value Order Step1 Step2/3 1 SF0 base core S 1 t S1 2 SEA 6.1 core S 1 4 NYC S Step1 Step2/3 3 CLE CLE base core S 2 t S2 5 MIN MIN 1.0 core S LAX LAX POR -2.4 POR 5.3 S 2 8 BOS -3.7 BOS 13.9 S 2 9 CHI CHI 6.1 S 2 10 BAL t S1 c BAL PHI PHI 7.6 S 2 12 PIT PIT CIN CIN STL STL t S2 c 15 DET DET WDC WDC HOU HOU KCM KCM ATL ATL -67.2

74 Relative Transition Curves across Clubs Club1 Club2 Club

75 Example 2: International Risk Sharing 1. Data: WDI, 1975 to 2002, 38 annual obs. For 66 countries. 2. Model: Cochrane(1991); Kalemli-Ozcan, et al.(2003) logc it it logc t e it H 0 : i, i, and 0 ΔlogC it ΔlogC jt Δ it Δ jt logc t it 1 jt 1 ΔlogC t Δe it Δe jt

76 Example 2: International Risk Sharing Club A 0.05(1.27) Australia, Austria, Belgium, Canada, Denmark, Finland, France, Hong Kong, Ireland, Italy, Japan, Korea, Netherlands, Norway, Singapore, Sweden, Switzerland, UK, USA Club (-0.70) China Club B 0.04(0.67) Chile, Greece, New Zealand, Portugal, Spain, Thailand Club 2: 0.20(2.51) Brazil, Colombia, Costa Rica, Dominican Rep., El Salvador, Gabon, Guatemala, Indonesia, Malaysia, Mexico, Morocco, Paraguay, Trinidad & Tobago, Tunisia, Uruguay Club 3: 0.17(4.21) Peru, Ecuador, Philippines, Algeria, Pakistan, Honduras, India, Lesotho Club 4: 0.18(3.01) Bangladesh, Benin, Burkina Faso, Cameroon, Congo, Rep., Cote d'ivoire, Ghana, Kenya, Malawi, Nicaragua, Rwanda, Senegal, Togo, Zambia, Zimbabwe Burundi, Madagascar

77 Example 3:Growth Convergence 1. Data: PWT, 1960 to 1996, 37 annual obs. For 88 countries. 2. Model: Phillips and Sul (2006)

78 : Club Convergence Club 1 Israel, Barbados, Australia, Austria, Belgium, Canada, Denmark, Finland, France, Greece, Iceland, Italy, Japan, Netherlands, NZ, Norway, Portugal, Spain, Sweden, Swiss, U.K, USA, Hong Kong, Korea, Singapore, Taiwan Chile, Thailand Club 3 Colombia, Costa Rica, Dominican Rep. El Salvador, Fiji, Guatemala, Iran, Jordan, Paraguay, Peru, Syria, Zimbabwe Botswana, Cyprus, Ireland, Malaysia, Mauritius, Romania, Trinidad &Tobago China, Indonesia C. African Rep., Congo, D.R. Malawi, Mali, Mozambique, Niger, Zambia Group 4 Club 2 Algeria, Argentina, Brazil, Ecuador, Mexico, Panama, South Africa, Turkey, Uruguay, Venezuela Egypt, India, Jamaica, Pakistan, P.N.G, Sri Lanka, Bangladesh, Benin, Bolivia, Cameroon, Gambia, Ghana, Guyana, Honduras, Lesotho, Nepal, Nicaragua, Philippines, Rwanda, Senegal, Togo, Uganda

79 : Evolution of Convergence Clubs Club 1 Israel, Barbados, Australia, Austria, Belgium, Canada, Denmark, Finland, France, Greece, Iceland, Italy, Japan, Netherlands, NZ, Norway, Portugal, Spain, Sweden, Swiss, U.K, USA, Hong Kong, Korea, Singapore, Taiwan Chile, Thailand Club 3 Colombia, Costa Rica, Dominican Rep. El Salvador, Fiji, Guatemala, Iran, Jordan, Paraguay, Peru, Syria, Zimbabwe Botswana, Cyprus, Ireland, Malaysia, Mauritius, Romania, Trinidad &Tobago China, Indonesia C. African Rep., Congo, D.R. Malawi, Mali, Mozambique, Niger, Zambia Group 4 Club 2 Algeria, Argentina, Brazil, Ecuador, Mexico, Panama, South Africa, Turkey, Uruguay, Venezuela Egypt, India, Jamaica, Pakistan, P.N.G, Sri Lanka, Bangladesh, Benin, Bolivia, Cameroon, Gambia, Ghana, Guyana, Honduras, Lesotho, Nepal, Nicaragua, Philippines, Rwanda, Senegal, Togo, Uganda

80 : Evolution of Convergence Clubs Club 1 Israel, Barbados, Australia, Austria, Belgium, Canada, Denmark, Finland, France, Greece, Iceland, Italy, Japan, Netherlands, NZ, Norway, Portugal, Spain, Sweden, Swiss, U.K, USA, Hong Kong, Korea, Singapore, Taiwan Chile, Thailand Club 3 Colombia, Costa Rica, Dominican Rep. El Salvador, Fiji, Guatemala, Iran, Jordan, Paraguay, Peru, Syria, Zimbabwe Botswana, Cyprus, Ireland, Malaysia, Mauritius, Romania, Trinidad &Tobago China, Indonesia C. African Rep., Congo, D.R. Malawi, Mali, Mozambique, Niger, Zambia Group 4 Club 2 Algeria, Argentina, Brazil, Ecuador, Mexico, Panama, South Africa, Turkey, Uruguay, Venezuela Egypt, India, Jamaica, Pakistan, P.N.G, Sri Lanka, Bangladesh, Benin, Bolivia, Cameroon, Gambia, Ghana, Guyana, Honduras, Lesotho, Nepal, Nicaragua, Philippines, Rwanda, Senegal, Togo, Uganda

81 : Evolution of Convergence Clubs Club 1 Israel, Barbados, Australia, Austria, Belgium, Canada, Denmark, Finland, France, Greece, Iceland, Italy, Japan, Netherlands, NZ, Norway, Portugal, Spain, Sweden, Swiss, U.K, USA, Hong Kong, Korea, Singapore, Taiwan Chile, Thailand Club 3 Colombia, Costa Rica, Dominican Rep. El Salvador, Fiji, Guatemala, Iran, Jordan, Paraguay, Peru, Syria, Zimbabwe Botswana, Cyprus, Ireland, Malaysia, Mauritius, Romania, Trinidad &Tobago China, Indonesia C. African Rep., Congo, D.R. Malawi, Mali, Mozambique, Niger, Zambia Group 4 Club 2 Algeria, Argentina, Brazil, Ecuador, Mexico, Panama, South Africa, Turkey, Uruguay, Venezuela Egypt, India, Jamaica, Pakistan, P.N.G, Sri Lanka, Bangladesh, Benin, Bolivia, Cameroon, Gambia, Ghana, Guyana, Honduras, Lesotho, Nepal, Nicaragua, Philippines, Rwanda, Senegal, Togo, Uganda

82 Table 2: Convergence Clubs and Convergence between Clubs Club 1 (37) (0.040) Club 1 (26) 0.182(0.055) Club 2 (28) 0.144(0.052) Club 2 (17) 0.275(0.074) Club 3 (16) (0.075) Club 3 (14) 0.224(0.095) Group 4 (7) (0.087) Group 4 (31) (0.022) Club 1 2(19 14) 0.013(0.013) Club 1 2(13 8) 0.109(0.043) Club 2 3(14 8) 0.122(0.058) Club 2 3(9 7) 0.330(0.080) Notes: 1) The numbers in parentheses are the number of countries. 2) This residual group shows evidence of several tiny convergent subgroups. 3) The affix * (respectively ** ) indicates rejection of the null hypothesis of the convergence at the 5% (1%) level.

83 Summary 1. Data Dependent Clustering method: Simple but Powerful 2. Can use when the common factor has a growing component. 3. Form (AC or RC) subgroups easily. t

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