OBSERVABLEINDEX.RPF

OBSERVABLEINDEX.RPF demonstrates estimation of an observable index model from Sargent & Sims(1977). It differs somewhat from the model actually used in the paper as we allow for contemporaneous loadings on the indexes. The model is

${{\bf{Y}}_t} = A(L)C(L){{\bf{Y}}_t} + D(L){{\bf{Y}}_{t - 1}} + {{\bf{v}}_t}$

where ${{\bf{Z}}_t} = C(L){{\bf{Y}}_t}$ are the "indexes", $D(L)$ is diagonal and ${{\bf{v}}_t}$ is assumed to be uncorrelated across variables. Thus, all interactions among the variables are captured in the $A(L){{\bf{Z}}_t}$ term. The model takes the form of a restricted VAR. With $\dim ({\bf{Z}}) \ll \dim ({\bf{Y}})$, it will generally have many fewer free parameters than the corresponding full VAR.

The $A(L)$ and $C(L)$ aren't separately identifiable without restrictions. They obviously need to be normalized for scale and rotation since $A(L)C(L) = A(L){\bf{G}}{{\bf{G}}^{ - 1}}C(L)$. It's also possible to have a dynamic invertible factor that cancels, that is, $A(L)C(L) = A(L)G(L)G{(L)^{ - 1}}C(L)$, so a normalization which fixes the lags of $C$ is also necessary.

Full Program

*
* Observable index model
* From Sargent & Sims(1977), "Business cycle modeling without pretending
* to have too much a priori economic theory," in "New Methods in
* Business Cycle Research: Proceedings from a Conference", Federal
* Reserve Bank of Minneapolis.
*
* This differs somewhat from the model actually used in the paper above
* because we allow for contemporaneous loadings on the indexes. The model is
*
*    Y(t)=A(L)C(L)Y(t)+D(L)Y(t-1)+v(t)
*
* where Z(t)=C(L)Y(t) are the "indexes", D(L) is diagonal and v(t) is
* assumed to be uncorrelated across variables. Thus, all interactions
* among the variables are captured in the A(L)Z(t) term. The model takes
* the form of a restricted VAR. With dim(Z)<<dim(Y), it will generally
* have many fewer free parameters than the corresponding full VAR.
*
* The A(L) and C(L) aren't separately identifiable without restrictions.
* They obviously need to be normalized for scale and rotation since
* A(L)C(L)=A(L)GG^-1C(L). It's also possible to have a dynamic
* invertible factor that cancels, that is, A(L)C(L)=A(L)G(L)G(L)^-1C(L),
* so a normalization which fixes the lags of C is also necessary.
*
open data bbedata.rat
calendar(m) 1959
data(format=rats) 1959:01 2001:07 ipp ipf ipc ipcd ipcn ipe ipi ipm ipmd ipmnd ipmfg ipd ipn ipmin iput $
ip ipxmca pmi pmp gmpyq gmyxpq lhel lhelx lhem lhnag lhur lhu680 lhu5 lhu14 lhu15 lhu26 lpnag lp lpgd $
lpmi lpcc lpem lped lpen lpsp lptu lpt lpfr lps lpgov lphrm lpmosa pmemp gmcq gmcdq gmcnq gmcsq gmcanq $
hsfr hsne hsmw hssou hswst hsbr hmob pmnv pmno pmdel mocmq msondq fsncom fspcom fspin fspcap fsput fsdxp $
fspxe exrsw exrjan exruk exrcan fyff fygm3 fygm6 fygt1 fygt5 fygt10 fyaaac fybaac sfygm3 sfygm6 sfygt1 $
sfygt5 sfygt10 sfyaaac sfybaac fm1 fm2 fm3 fm2dq fmfba fmrra fmrnba fclnq fclbmc ccinrv pmcp pwfsa pwfcsa $
pwimsa pwcmsa psm99q punew pu83 pu84 pu85 puc pucd pus puxf puxhs puxm lehcc lehm hhsntn
*
* NLAGSC is the number of lags in C. It includes the 0 lag as well.
* NLAGSD is the number of lags in D.
* NLAGSA is the number of lags in A. It includes the 0 lag as well.
* NLAGSRF is the derived number of lags in the reduced form.
*
compute nlagsc=2
compute nlagsd=2
compute nlagsa=2
compute nlagsrf=%imax(nlagsc+nlagsa,nlagsd)
*
* NF is the number of factors (indexes)
*
compute nf=1
*
* Specifies the reduced form for the VAR. The variable list is taken
* from this.
*
system(model=reducedform)
variables ip ipd ipn ipp ipc ipcd ipcn ipe ipi ipm ipmd ipmnd ipmfg ipmin iput
lags 1 to nlagsrf
end(system)
*
* Determine the maximum estimation range
*
inquire(equation=%modeleqn(reducedform,1)) rstart rend
*
compute ny=%modelsize(reducedform)
*
dec vect[series] factors(nf)
*
* Set up and initialize the D's (one vector per variable)
*
dec vect[vect] d(ny)
do i=1,ny
   compute d(i)=%zeros(nlagsd,1)
end do i
*
* Set up and initialize the C's (one NF x NY matrix per lag)
*
dec vect[rect] c(nlagsc+1)
do i=1,nlagsc+1
   compute c(i)=%zeros(nf,ny)
end do i
*
* Set up and initialize the A's (one NY x NF matrix per lag). This is
* initialized with small random numbers so we'll have derivatives with
* respect to C parameters at the guess values.
*
dec vect[rect] a(nlagsa+1)
do i=1,nlagsa+1
   compute a(i)=%ranmat(ny,nf)*.1
end do i
*
* Put together a full list of variables and lags. For general
* convenience, these are blocked by lag. Also, initialize the (log)
* variable shock variances using a linear regression.
*
dec vect lsv(ny)
dec table fulltable(2,ny*(nlagsrf+1))
do i=1,ny
   compute xseries=%modeldepvars(reducedform)(i)
   linreg(noprint) xseries rstart rend
   # xseries{1 to nlagsd}
   compute lsv(i)=log(%sigmasq)
   compute d(i)=%beta
   do j=0,nlagsrf
      compute fulltable(1,j*ny+i)=xseries
      compute fulltable(2,j*ny+i)=j
   end do j
end do i
equation fulleqn *
# %rlfromtable(fulltable)
*
* These are the multiplied out coefficients in ARF(L)Y(t)=v(t), though
* this is actually organized as ARF(L)Y(t)=-v(t).
*
dec rect arf(ny,ny*(nlagsrf+1))
************************************************
function %%FillARF
local integer i j k f v
compute arf=%zeros(ny,ny*(nlagsrf+1))
do i=1,ny
   compute arf(i,i)=-1.0
   do j=1,nlagsd
      compute arf(i,j*ny+i)=d(i)(j)
   end do j
   do j=0,nlagsa
   do k=0,nlagsc
         do f=1,nf
            do v=1,ny
               compute arf(i,(j+k)*ny+v)+=a(j+1)(i,f)*c(k+1)(f,v)
            end do v
         end do f
      end do k
   end do j
end do i
end %%FillARF
************************************************
dec symm sv
dec real jacobian
*
* This is called before the main function evaluation to multiply out the
* coefficients and create the target covariance matrix from the log
* variance parameters.
*
function %%IndexStart
compute %%fillarf()
compute sv=%diag(%exp(lsv))
compute jacobian=log(abs(%det(%xsubmat(arf,1,ny,1,ny))))
end %%IndexStart
*
* This includes the normalization that C(L) has a 1 coefficient on lag
* zero for the first variable, and 0 coefficients on the other lags on
* the first variable.
*
nonlin c a d lsv c(1)(1,1)=1.0 c(2)(1,1)=0.0 c(3)(1,1)=0.0
*
* Because the model is linear in variables, non-linear in parameters,
* it's possible to dramatically speed up the calculation by forming the
* crossproduct of the data and computing the implied sigma by a
* quadratic form.
*
cmom(equation=fulleqn) rstart rend
compute rnobs=%nobs
compute %cmom=%cmom/rnobs
*
* Because the likelihood function is computed in one shot using the
* quadratic form, we need to use FIND rather than MAXIMIZE. Since this
* includes current Y's on the RHS, we need to include a Jacobian term,
* which needs to be multiplied by the number of observations, since it
* occurs in each one.
*
find(method=bfgs,iters=800,stderrs) max %logl
   compute %%IndexStart()
   compute [symm] sigmahat=arf*%cmom*tr(arf)
   compute %logl=%logdensitycv(sv,sigmahat,rnobs)+rnobs*jacobian
end find
*
* Create reduced form VAR
*
compute ainv=-1.0*inv(%xsubmat(arf,1,ny,1,ny))
compute rfcoeffs=%zeros(ny*nlagsrf,ny)
do k=1,nlagsrf
   compute phi=ainv*%xsubmat(arf,1,ny,k*ny+1,(k+1)*ny)
   do i=1,ny
      do j=1,ny
         compute rfcoeffs((j-1)*nlagsrf+k,i)=phi(i,j)
      end do j
   end do i
end do k
*
compute %modelsetcoeffs(reducedform,rfcoeffs)
compute factor=ainv*%diag(.5*%exp(lsv))
*
* Compute the decomposition of variance
*
errors(factor=factor,model=reducedform,steps=20)

Output

FIND Optimization - Estimation by BFGS

Convergence in 220 Iterations. Final criterion was 0.0000071 <= 0.0000100

Function Value 27494.2965

Variable Coeff Std Error T-Stat Signif

*************************************************************************************

1. C(1)(1,1) 1.00000000 0.00000000 0.00000 0.00000000

2. C(1)(1,2) 0.88608860 0.15090912 5.87167 0.00000000

3. C(1)(1,3) 0.66188629 0.11301619 5.85656 0.00000000

4. C(1)(1,4) -0.55734807 0.08096947 -6.88344 0.00000000

5. C(1)(1,5) 1.44405328 0.25018389 5.77197 0.00000001

6. C(1)(1,6) -0.32507938 0.05532773 -5.87552 0.00000000

7. C(1)(1,7) -1.03525787 0.17625910 -5.87350 0.00000000

8. C(1)(1,8) 0.06225983 0.01947542 3.19684 0.00138941

9. C(1)(1,9) 0.06896533 0.02081602 3.31309 0.00092272

10. C(1)(1,10) -0.28814985 0.04441216 -6.48809 0.00000000

11. C(1)(1,11) 0.04714660 0.01764817 2.67147 0.00755192

12. C(1)(1,12) 0.00529011 0.00460742 1.14817 0.25089824

13. C(1)(1,13) -0.85659486 0.16878318 -5.07512 0.00000039

14. C(1)(1,14) 0.03991730 0.01073563 3.71821 0.00020064

15. C(1)(1,15) 0.02722392 0.00795802 3.42094 0.00062405

16. C(2)(1,1) 0.00000000 0.00000000 0.00000 0.00000000

17. C(2)(1,2) 0.29334073 0.10269801 2.85634 0.00428552

18. C(2)(1,3) 0.24513065 0.08081600 3.03319 0.00241980

19. C(2)(1,4) -0.02979032 0.05752614 -0.51786 0.60455805

20. C(2)(1,5) -0.16574739 0.14550101 -1.13915 0.25464085

21. C(2)(1,6) 0.02732859 0.03175402 0.86063 0.38943975

22. C(2)(1,7) 0.07631517 0.09993858 0.76362 0.44509328

23. C(2)(1,8) -0.02332124 0.01636980 -1.42465 0.15425832

24. C(2)(1,9) -0.00215343 0.01634713 -0.13173 0.89519696

25. C(2)(1,10) -0.04939158 0.03268108 -1.51132 0.13070686

26. C(2)(1,11) -0.01698739 0.01484462 -1.14435 0.25247987

27. C(2)(1,12) 0.00603788 0.00577236 1.04600 0.29556173

28. C(2)(1,13) -0.56558628 0.15836053 -3.57151 0.00035493

29. C(2)(1,14) -0.00516671 0.00849876 -0.60794 0.54322899

30. C(2)(1,15) 0.00252478 0.00612786 0.41202 0.68032676

31. C(3)(1,1) 0.00000000 0.00000000 0.00000 0.00000000

32. C(3)(1,2) 0.05462361 0.07952849 0.68684 0.49218150

33. C(3)(1,3) 0.05395523 0.06338659 0.85121 0.39465334

34. C(3)(1,4) 0.18354499 0.06091329 3.01322 0.00258494

35. C(3)(1,5) 0.01684784 0.08250396 0.20421 0.83819216

36. C(3)(1,6) -0.02124364 0.01911538 -1.11134 0.26642328

37. C(3)(1,7) -0.07676716 0.06065154 -1.26571 0.20561752

38. C(3)(1,8) -0.03420957 0.01503417 -2.27545 0.02287866

39. C(3)(1,9) -0.04545800 0.01776632 -2.55866 0.01050761

40. C(3)(1,10) 0.03317730 0.02999711 1.10602 0.26871939

41. C(3)(1,11) -0.01325689 0.01562749 -0.84831 0.39626754

42. C(3)(1,12) 0.00053693 0.00465736 0.11529 0.90821815

43. C(3)(1,13) -0.10921400 0.13207048 -0.82694 0.40827264

44. C(3)(1,14) -0.00756226 0.00713797 -1.05944 0.28939869

45. C(3)(1,15) 0.00026825 0.00591722 0.04533 0.96384160

46. A(1)(1,1) 0.76593612 0.11366893 6.73831 0.00000000

47. A(1)(2,1) 1.02797310 0.14557166 7.06163 0.00000000

48. A(1)(3,1) 0.36877320 0.05359120 6.88123 0.00000000

49. A(1)(4,1) 0.68097388 0.10476948 6.49974 0.00000000

50. A(1)(5,1) -0.04267733 0.10540701 -0.40488 0.68556473

51. A(1)(6,1) 2.37268002 0.37129781 6.39023 0.00000000

52. A(1)(7,1) 0.63559929 0.10371229 6.12849 0.00000000

53. A(1)(8,1) 0.67644777 0.10452535 6.47161 0.00000000

54. A(1)(9,1) 0.63449738 0.09731038 6.52035 0.00000000

55. A(1)(10,1) 1.01286154 0.16043217 6.31333 0.00000000

56. A(1)(11,1) 1.35242937 0.20071194 6.73816 0.00000000

57. A(1)(12,1) 0.58030736 0.09801867 5.92038 0.00000000

58. A(1)(13,1) 0.85301529 0.12687585 6.72323 0.00000000

59. A(1)(14,1) 0.37002837 0.07426314 4.98267 0.00000063

60. A(1)(15,1) 0.28706690 0.07729959 3.71369 0.00020426

61. A(2)(1,1) 0.32078893 0.07675534 4.17937 0.00002923

62. A(2)(2,1) 0.25940194 0.11119563 2.33284 0.01965639

63. A(2)(3,1) 0.07943936 0.04110606 1.93255 0.05329212

64. A(2)(4,1) 0.03889366 0.06748864 0.57630 0.56441287

65. A(2)(5,1) 0.40209286 0.08807406 4.56539 0.00000499

66. A(2)(6,1) 0.20553597 0.21405520 0.96020 0.33695418

67. A(2)(7,1) -0.04663913 0.06419046 -0.72657 0.46748691

68. A(2)(8,1) 0.19876613 0.07201140 2.76020 0.00577653

69. A(2)(9,1) 0.17741511 0.06445665 2.75247 0.00591472

70. A(2)(10,1) 0.27248462 0.10532549 2.58707 0.00967954

71. A(2)(11,1) 0.09235544 0.13341858 0.69222 0.48879715

72. A(2)(12,1) 0.25527965 0.08460321 3.01738 0.00254974

73. A(2)(13,1) 0.33605946 0.09356435 3.59175 0.00032847

74. A(2)(14,1) -0.06480824 0.05756537 -1.12582 0.26024180

75. A(2)(15,1) 0.13631593 0.06533132 2.08653 0.03693041

76. A(3)(1,1) 0.05974859 0.02724203 2.19325 0.02828936

77. A(3)(2,1) -0.00286551 0.06018518 -0.04761 0.96202589

78. A(3)(3,1) 0.05632265 0.02542842 2.21495 0.02676357

79. A(3)(4,1) 0.01447838 0.02745716 0.52731 0.59797972

80. A(3)(5,1) -0.04803694 0.05701749 -0.84249 0.39951108

81. A(3)(6,1) -0.43137485 0.10833300 -3.98193 0.00006836

82. A(3)(7,1) -0.01871991 0.02600088 -0.71997 0.47154206

83. A(3)(8,1) -0.04892376 0.03014433 -1.62298 0.10459290

84. A(3)(9,1) 0.09044531 0.03542662 2.55303 0.01067895

85. A(3)(10,1) 0.08253662 0.06010583 1.37319 0.16969388

86. A(3)(11,1) 0.07501448 0.06941631 1.08065 0.27985449

87. A(3)(12,1) 0.24266367 0.07439450 3.26185 0.00110688

88. A(3)(13,1) 0.12289642 0.03323768 3.69750 0.00021773

89. A(3)(14,1) 0.02077981 0.04547654 0.45693 0.64771798

90. A(3)(15,1) 0.02284120 0.05716195 0.39959 0.68946036

91. D(1)(1) -0.22300389 0.05807474 -3.83995 0.00012306

92. D(1)(2) -0.02232446 0.04692918 -0.47571 0.63428441

93. D(2)(1) 0.00399755 0.04667312 0.08565 0.93174475

94. D(2)(2) 0.00295576 0.04022111 0.07349 0.94141786

95. D(3)(1) -0.01939499 0.04084237 -0.47487 0.63487659

96. D(3)(2) 0.07402472 0.04113283 1.79965 0.07191584

97. D(4)(1) 0.05897782 0.04169979 1.41434 0.15726113

98. D(4)(2) -0.02226095 0.04385933 -0.50755 0.61176658

99. D(5)(1) -0.27770691 0.07046365 -3.94114 0.00008110

100. D(5)(2) 0.11391754 0.06139940 1.85535 0.06354588

101. D(6)(1) 0.01709008 0.04355031 0.39242 0.69474678

102. D(6)(2) -0.01141678 0.04433499 -0.25751 0.79678370

103. D(7)(1) -0.18856305 0.04284421 -4.40113 0.00001077

104. D(7)(2) -0.12243198 0.04393466 -2.78668 0.00532505

105. D(8)(1) -0.00102467 0.03874513 -0.02645 0.97890137

106. D(8)(2) 0.24210955 0.04003934 6.04679 0.00000000

107. D(9)(1) -0.18617817 0.04130179 -4.50775 0.00000655

108. D(9)(2) -0.02112896 0.04125407 -0.51217 0.60853440

109. D(10)(1) -0.03533337 0.04008080 -0.88155 0.37801832

110. D(10)(2) -0.07069406 0.04117543 -1.71690 0.08599764

111. D(11)(1) 0.21263385 0.03916955 5.42855 0.00000006

112. D(11)(2) -0.09766225 0.04178702 -2.33714 0.01943174

113. D(12)(1) -0.19686835 0.04297523 -4.58097 0.00000463

114. D(12)(2) -0.04089869 0.04312891 -0.94829 0.34298220

115. D(13)(1) -0.20829838 0.05147334 -4.04672 0.00005194

116. D(13)(2) -0.07328170 0.04295190 -1.70613 0.08798312

117. D(14)(1) -0.02884491 0.04350712 -0.66299 0.50733517

118. D(14)(2) -0.10281274 0.04334824 -2.37179 0.01770236

119. D(15)(1) -0.22594516 0.04291775 -5.26461 0.00000014

120. D(15)(2) -0.19055415 0.04294394 -4.43728 0.00000911

121. LSV(1) -13.45291737 0.17601677 -76.42975 0.00000000

122. LSV(2) -10.84572264 0.07214179 -150.33898 0.00000000

123. LSV(3) -10.31477447 0.05900654 -174.80730 0.00000000

124. LSV(4) -11.26328681 0.05370997 -209.70569 0.00000000

125. LSV(5) -9.27565296 0.18898884 -49.08043 0.00000000

126. LSV(6) -7.72592325 0.06670875 -115.81573 0.00000000

127. LSV(7) -9.90277283 0.06943673 -142.61577 0.00000000

128. LSV(8) -9.47790452 0.06830019 -138.76836 0.00000000

129. LSV(9) -10.08241007 0.06292268 -160.23491 0.00000000

130. LSV(10) -10.30251566 0.06642661 -155.09620 0.00000000

131. LSV(11) -9.33079325 0.06135397 -152.08134 0.00000000

132. LSV(12) -8.36668171 0.06374042 -131.26180 0.00000000

133. LSV(13) -12.82063262 0.13590749 -94.33353 0.00000000

134. LSV(14) -8.49814344 0.06303401 -134.81839 0.00000000

135. LSV(15) -8.19473623 0.06179457 -132.61257 0.00000000