;;; 20 point time series from page 287 (first row is X_1, X_2, X_3, X_4;
;;; second row is X_5, X_6, X_7, X_8; etc)
 71.0  63.0  70.0  88.0 
 99.0  90.0 110.0 135.0
128.0 154.0 156.0 141.0
131.0 132.0 141.0 104.0
136.0 146.0 124.0 129.0

sample mean = 117.4

p = 1
\phi_{1,1} = 0.8000556894184591
innovations variance = 281.68066194644916
forward prediction errors:
-17.2774160109835    -3.876970495635831    8.522639678434956   5.121637268902695
-12.678975314700354  14.521525890065778   23.520412101696596  -3.480980133764881
 28.119409692164332   9.317961767284398   -7.282149611552519  -5.281314270275637
  3.719242623908956  11.919186934490497  -32.28131427027564   29.32074623820735 
 13.71896417681666  -16.281592717367932    6.319632449838169
backward prediction errors:
 -2.876970495635831 -16.477360321565044  -23.878362731097305 -14.678975314700354
  3.521525890065778 -21.479587898303404  -21.48098013376488    9.119409692164332
-18.682038232715602   5.717850388447481   19.718685729724363  12.719242623908956
  1.9191869344904973 -4.281314270275637   34.32074623820735  -28.281035823183338
 -4.281592717367932  23.319632449838167   -2.6806459972541266

p = 2
\phi_{1,2} = 0.7155199935944124
\phi_{2,2} = 0.10566226444248342
innovations variance = 278.5358342168134 
forward prediction errors:
 -3.5729832783327353 10.263674882046246    7.644679146249439 -11.127961543253798
 14.149433490228596  25.789993998322696   -1.211251130387283  27.15583221371152
 11.29194823135418   -7.886310631339214   -7.364835256307996   2.375298646273381
 11.716401297103804 -31.828940909688395   25.694338473322517  16.707202462673205
-15.82918993543039    3.855627279221851
backward prediction errors:
 -2.467321013890251 -17.37788172901584   -24.41952652258258  -13.339286072138766
  1.987148581361278 -23.964807901589058  -21.113171890351985   6.148249189304334
-19.666595173035358   6.4872988064130706  20.276721354754088  12.326259026255732
  0.6597786526789691 -0.8703975052988588  31.222649795534934 -29.730612643911098
 -2.5612427621205893 22.651885774744066

p = 3
\phi_{1,3} =  0.699055326284677
\phi_{2,3} = -0.0058325886174639685
\phi_{3,3} =  0.15582353261696202
innovations variance = 271.7727130583027 
forward prediction errors:
 10.648141558530689  10.352562066664447   -7.322824655671383  16.228008168577496
 25.4803494865402     2.523029895325209   30.44576124241531   10.333906323267406
 -4.821792296929151  -8.375709073465082   -0.7842917051141924  9.795680071680922
-31.93174995009409   25.829966887379175   11.841978873870683 -11.196460846389645
  4.2547291743051066
backward prediction errors:
 -4.066643091642678 -18.569102639307648  -22.685528244087074 -15.544100783114935
 -2.031539389627613 -23.776066471565823  -25.34468959704601    4.388697925766869
-18.43772239114538    7.634913453192931   19.90659392867148   10.50056798658306
  5.619476664683155  -4.874180094467892   28.619274487654387 -27.264052349707473
 -3.1620402252232642

p = 4
\phi_{1,4} =  0.7680043283336528
\phi_{2,4} = -0.00841340011481915
\phi_{3,4} =  0.4651424545722902
\phi_{4,4} = -0.4424813177510415
innovations variance = 218.5624106002292 
forward prediction errors:
  8.553148472651225 -15.539305660966573    6.190085737255377  18.602375288772507
  1.624111669139636  19.925296019140507   -0.8806453276246256 -2.879875455524564
-16.534056773426972   2.5940148605697706  18.60397598517436  -27.285444790256417
 28.316480327039404   9.685245242714634    1.4670334419864322 -7.809104636626834
backward prediction errors:
  0.514172213714451 -21.809315742608938  -15.504937805180237  -4.269522165553555
 -0.915145796818842 -10.304385917088311  -20.772129109610827   2.255144916299837
-22.14381717907156    7.287879026012792   24.24099935505647   -3.628634809431325
 17.04875445047646    0.36567432242240194 23.665049738195954 -25.38141417798715


;;; for the fitted AR(4) model, here are the corresponding estimated
;;; acvs for lags 0 to 20

lag  0 Burg acvs estimate = 782.64
lag  1 Burg acvs estimate = 626.1555847664627
lag  2 Burg acvs estimate = 530.7223546444701
lag  3 Burg acvs estimate = 489.3059104234309
lag  4 Burg acvs estimate = 316.27184468267575
lag  5 Burg acvs estimate = 208.5807697145015
lag  6 Burg acvs estimate = 150.29223774159786
lag  7 Burg acvs estimate = 44.27295374890565
lag  8 Burg acvs estimate = -10.18726002415869
lag  9 Burg acvs estimate = -30.58213934107883
lag 10 Burg acvs estimate = -69.30978291471806
lag 11 Burg acvs estimate = -77.3013955482757
lag 12 Burg acvs estimate = -68.5020545511937
lag 13 Burg acvs estimate = -70.66640406042121
lag 14 Burg acvs estimate = -58.98364578259735
lag 15 Burg acvs estimate = -42.36394096088959
lag 16 Burg acvs estimate = -34.598502285030435
lag 17 Burg acvs estimate = -22.38260891257773
lag 18 Burg acvs estimate = -10.504955653421078
lag 19 Burg acvs estimate = -5.227517422267727
lag 20 Burg acvs estimate = 0.9717156223241297