Build SARIMA model equation with exogenous variable or regressors

I have a SARIMA model with one regressor (X):

> fit_arima.reg
Series: sales.ts
Regression with ARIMA(0,0,1)(2,1,1)[7] errors

Coefficients:
ma1    sar1    sar2     sma1     xreg
0.3078  0.3401  0.1026  -0.8621  -0.0015
s.e.  0.0305  0.0839  0.0636   0.0731   0.0016

sigma^2 estimated as 0.02782:  log likelihood=348.89
AIC=-685.79   AICc=-685.7   BIC=-656.73


I would like to use these coefficients to obtain the actual equation. I took the example from this post, this post, this post and look at the ARIMAX Model Muddle post from Rob Hyndman's blog, but still the fitted and predicted values generated from the R forecast differ from those reproduced by hand from the built equation. I would appreciate your help with the correct equation. Here the built formula:

(EDITED based on the reply of Dr. Reilly and this post):

# (1 - sar1*B7 - sar2*B14)*(1 - B7)*(1 - xreg*X(t))*Y(t) = (1 + ma1*B)*(1-sma1*B7)E(t)

# thus:

# Y(t) = sar1*(Yt-7) + sar2*(Yt-14) + xreg*(Xt) - sar1*xreg*(X-7) - sar2*xreg*(Xt-14) + (Yt-7) - sar1*(Yt-14) -sar2*(Yt-21) - xreg*(Xt-7) + sar1*xreg(Xt-14) + sar2*xreg*(Xt-21) + ma1*(Et-1) - sma1*(Et-7) - ma1*sma1*(Et-8)


Here the original transformed data:

> data$transf [1] 3.340642 3.665769 3.483445 3.192846 3.210586 3.463296 3.794070 [8] 3.369216 3.051538 2.998695 3.069298 2.957607 3.360215 3.770705 [15] 3.430720 3.050380 3.124830 3.305566 3.220892 3.747101 3.878349 [22] 3.655427 3.037426 3.143951 3.137354 3.378216 3.740126 3.446692 [29] 2.987219 3.171141 3.226600 3.193125 3.686726 3.486147 3.075547 [36] 3.233504 3.339253 3.258637 3.675962 3.787460 3.399501 3.509203 [43] 3.738622 3.580241 3.614053 3.802089 3.838471 3.427486 3.134177 [50] 3.286456 3.443576 3.341435 3.772908 3.864452 3.480869 2.893207 [57] 3.187803 3.277609 3.231470 3.725503 3.857995 3.505693 3.156246 [64] 3.212454 3.256237 3.281488 3.691524 3.812780 3.400538 3.076276 [71] 3.218536 3.138618 3.158061 3.670988 3.594171 3.069668 3.076276 [78] 3.281033 3.351796 3.713491 3.889806 3.198107 3.254548 3.292699 [85] 3.371806 3.747179 3.731508 3.454235 3.361728 3.381115 3.496376 [92] 3.537693 3.656769 3.695131 3.292034 3.388279 3.425208 3.445293 [99] 3.639785 3.721728 3.608847 3.115278 3.464340 3.420451 3.148294 [106] 3.562887 3.599774 3.846523 3.627263 3.262925 3.342620 3.224792 [113] 3.597476 3.710033 3.787319 3.605951 3.530072 3.026533 3.198382 [120] 3.293584 3.383277 3.784902 3.497483 3.085291 3.109579 3.365113 [127] 3.396025 3.620240 3.862191 3.437909 3.120574 3.274850 3.127105 [134] 3.353916 3.697665 3.735599 3.420781 3.180413 3.299725 3.266702 [141] 3.317227 3.516403 3.608098 3.452553 3.425045 3.452400 3.388101 [148] 3.494850 3.574726 3.698449 3.421439 3.093071 2.913284 3.119256 [155] 3.384891 3.478133 3.658965 3.388989 3.077004 3.160769 3.098644 [162] 3.453777 3.579212 3.708846 3.443106 3.160469 3.264346 3.019947 [169] 3.510277 3.585235 3.771146 3.398808 3.077731 3.253822 3.342620 [176] 3.193125 3.582972 3.674861 3.678791 3.128399 3.168497 3.291147 [183] 3.236033 3.581039 3.598134 3.430720 3.040207 3.231724 3.386499 [190] 3.372175 3.716337 3.750663 3.360593 3.142076 3.151676 3.258158 [197] 3.590396 3.489958 3.599119 3.445760 3.226084 3.301898 3.496376 [204] 3.486855 3.492341 3.515874 3.322426 3.315970 3.292034 3.396896 [211] 3.373096 3.611617 3.727541 3.391464 3.181844 3.405517 3.405346 [218] 3.510277 3.651666 3.650793 3.547898 3.180413 3.342423 3.458336 [225] 3.526598 3.494294 3.680698 3.256718 3.264818 3.454235 3.346744 [232] 3.560385 3.495128 3.599774 3.418798 3.461499 3.389698 3.349472 [239] 3.583879 3.522966 3.650599 3.462398 3.504743 3.188647 3.473049 [246] 3.758761 3.735759 3.325105 3.098990 3.225051 3.284656 3.302764 [253] 3.724440 3.627468 3.499962 3.125806 3.150142 3.354301 3.359456 [260] 3.752740 3.772762 3.391817 3.050380 3.187239 3.319730 3.308351 [267] 3.666892 3.752586 3.605521 3.071514 3.164055 3.259355 3.411788 [274] 3.650696 3.840169 3.370513 3.204391 3.211921 3.244277 3.217221 [281] 3.510277 3.813314 3.348694 2.962369 3.152594 3.501880 3.540705 [288] 3.426999 3.802910 3.394101 3.349472 3.573684 3.421275 3.696269 [295] 3.729893 3.722222 3.566673 3.147058 3.159266 3.296226 3.663512 [302] 3.567849 3.708336 3.359266 3.107549 3.201943 3.242790 3.325105 [309] 3.681693 3.833784 3.513750 3.107888 3.080266 3.502564 3.438542 [316] 3.715335 3.793581 3.391288 2.997823 3.372175 3.295127 3.555457 [323] 3.630123 3.845160 3.544192 3.068557 3.121231 3.271609 3.510277 [330] 3.610447 3.750354 3.425371 3.140508 3.211388 3.507856 3.465532 [337] 3.772688 3.690285 3.655715 3.187803 3.330008 3.348110 3.601191 [344] 3.794279 3.868938 3.683227 3.343802 3.591621 3.724194 3.732233 [351] 3.823474 3.790356 3.646011 3.075182 3.458033 3.769156 3.822430 [358] 3.796366 3.889358 3.280578 3.431846 3.276692 3.297979 3.366796 [365] 3.571126 3.244030 2.982723 3.044932 3.168792 3.423574 3.652150 [372] 3.736476 3.516006 3.006894 3.097951 3.195069 3.269746 3.637890 [379] 3.761552 3.205204 3.063333 3.269980 3.226858 3.345570 3.551938 [386] 3.832381 3.344981 3.026533 3.047664 3.311754 3.522575 3.674034 [393] 3.822691 3.499275 2.999131 3.082067 3.265525 3.410271 3.746712 [400] 3.670431 3.330211 3.358696 3.308137 3.670802 3.781109 3.806587 [407] 3.847881 3.380392 3.106531 3.203577 3.175802 3.497759 3.632862 [414] 3.805161 3.371806 2.946943 2.958564 3.063709 3.138618 3.816042 [421] 3.591955 3.107888 3.225309 3.189209 3.324282 3.721728 3.765966 [428] 2.967548 3.149527 3.293584 3.441695 3.572058 3.660771 3.424882 [435] 3.056524 3.249443 3.187803 3.339451 3.729893 3.819939 3.544440 [442] 3.190892 3.129690 3.250420 3.532372 3.713910 3.703205 3.194237 [449] 3.574957 3.401917 3.400538 3.503927 3.828982 3.585799 3.414973 [456] 3.313234 3.424882 3.464936 3.532627 3.668759 3.723045 3.334655 [463] 3.108565 3.146438 3.190892 3.372912 3.496930 3.540204 3.265290 [470] 3.034227 3.284882 3.320562 3.426674 3.721398 3.815445 3.441066 [477] 2.887617 3.318481 3.247237 3.506640 3.708931 3.677059 3.692142 [484] 3.260310 3.137037 3.133219 3.299507 3.644439 3.700098 3.416474 [491] 3.140194 3.235528 3.428783 3.412461 3.537819 3.522705 3.331427 [498] 3.093071 3.086004 3.388279 3.552911 3.663324 3.570193 3.661718 [505] 3.291369 3.569842 3.366796 3.694342 3.662947 3.551206 3.320354 [512] 2.972203 2.933993 3.230193 3.305136 3.564429 3.714330 3.623559 [519] 3.087071 3.176670 3.389166 3.306639 3.644242 3.683857 3.487704 [526] 2.905256 3.318481 3.296226 3.400711 3.514282 3.723620 3.190051 [533] 3.214314 3.344981 3.162863 3.698622 3.783332 3.458184 2.940018 [540] 3.026533 3.303412 3.255996 3.478855 3.650113 3.266467 3.070038 [547] 3.218798 3.276232 3.375115 3.606919 3.795254 3.178977 3.071514 [554] 3.176381 3.371437 3.323871 3.611192 3.634981 3.478422 3.249198 [561] 3.378034 3.347135 3.349083 3.401056 3.725095 3.490520 3.351603 [568] 3.149835 3.310481 3.269980 3.356790 3.480294 3.332438 3.225568 [575] 3.250664 3.308991 3.368473 3.707059 3.578066 3.431846 3.168203 [582] 3.260548 3.398634 3.610128 3.495267 3.683047 3.202488 3.315130 [589] 3.310481 3.300378 3.527372 3.553883 3.695919 3.456214 3.450095 [596] 3.255514 3.279667 3.488410 3.741703 3.669503 3.200303 3.296446 [603] 3.243782 3.138934 3.192010 3.705607 3.571942 3.375481 3.161667 [610] 3.013259 3.141450 3.432809 3.552668 3.658393 3.342225 2.907411 [617] 3.123198 3.224792 3.434729 3.523096 3.744136 3.334856 3.083861 [624] 3.065206 3.246252 3.042576 3.662852 3.812312 3.340841 3.124504 [631] 3.063333 3.210586 3.258637 3.640283 3.626443 3.436799 2.943000 [638] 3.178689 3.184407 3.326131 3.567026 3.781684 3.337459 3.024486 [645] 3.243534 3.360404 3.431846 3.581950 3.616055 3.416474 3.297104 [652] 3.384533 3.410777 3.667640 3.775756 3.758230 3.581039 3.128076 [659] 3.207904 3.170848 3.268110 3.716254 3.728273 3.412629 2.978637 [666] 3.103804 3.127105 3.396896 3.629104 3.662758 3.352183 2.991669 [673] 3.255273 3.216166 3.143327 3.531607 3.707911 3.454082 3.104487 [680] 3.123525 3.240050 3.339849 3.564548 3.673021 3.541454 3.063709 [687] 3.220892 3.172019 3.495960 3.652440 3.782831 3.519434 3.185259 [694] 3.211121 3.046105 3.439333 3.670802 3.738067 3.594945 3.038223 [701] 3.260787 3.510813 3.631545 3.814714 3.804821 3.658965 3.315551 [708] 3.538197 3.508664 3.681784 3.778079 3.715335 3.641474 3.489114 [715] 2.932981 3.518119 3.707655 3.734240 3.665206 3.775610 3.158362 [722] 3.498724 3.215109 3.611511 3.657725 3.220631 2.893207 2.990339 [729] 3.264109 3.018284 3.556423 3.706974 3.280578 3.091667 3.150449 [736] 3.056524 3.303412 3.542203 3.561459 3.392521 3.194514 2.958086 [743] 3.179264 3.348500 3.855337 3.396374 3.077731 3.188647 3.114277 [750] 3.150756 3.510679 3.777862 3.434729 3.033021 3.135451 3.578983 [757] 3.643354 3.825296 3.347720 3.111934 3.177536 3.428621 3.680879 [764] 3.496515 3.817698 3.282396 3.411956 3.431525 3.447623 3.436322 [771] 3.783689 3.830653 3.264582 3.067443 3.262688 3.233504 3.463296 [778] 3.413635 3.743196 3.359646 2.980003 3.144263 3.217747 3.514813 [785] 3.624798 3.338656 3.143951 3.016616 3.096910 3.232234 3.434729 [792] 3.632862 3.570309 3.196729 3.226858 3.363048 3.641573 3.753200 [799] 3.351796 2.991226 3.112270 3.548021 3.468790 3.670060 3.757168 [806] 3.803389 3.101403 3.168792 3.173478 3.150756 3.443576 3.667546 [813] 3.315340 3.315130 3.221936 3.388634 3.443419 3.481156 3.758988 [820] 3.358886 3.265054 3.306425 3.301030 3.594724 3.648945 3.631951 [827] 3.041393 3.290035 3.143951 2.919601 3.500922 3.648653 3.823930 [834] 3.319106 3.318898 3.242293 3.418964 3.605628 3.761025 3.615634 [841] 3.640084 3.161368 3.144574 3.604442 3.729651 3.414806 2.888179 [848] 3.014100 3.236537 3.563837 3.471438 3.695657 3.491222 3.042969 [855] 3.215373 3.169968 3.276692 3.563362 3.662663 3.378761 3.370883 [862] 3.343999 3.414973 3.304921 3.630224 3.697752 3.268812 3.032216 [869] 3.060698 3.548389 3.073352 3.534280 3.702344 3.328787 3.167022 [876] 3.233250 3.211121 3.645521 3.676328 3.456062 3.030195 3.044540 [883] 3.149835 3.325926 3.505693 3.620552 3.435844 3.058426 3.093422 [890] 3.244030 3.457276 3.520615 3.608526 3.352375 3.194514 3.154424 [897] 3.481443 3.294907 3.617420 3.685294 3.376029 3.035029 3.199481 [904] 3.439017 3.431846 3.360215 3.697229 3.432649 3.112270 3.082067 [911] 3.487563 3.236537 3.572291 3.731750 3.196729 3.111934 3.271144 [918] 3.188647 3.421275 3.667640 3.564903 3.305136 3.101403 3.163161 [925] 3.337260 3.311754 3.283527 3.761778 3.300813 3.341632 3.321184 [932] 3.335257 3.328583 3.631038 3.575419 3.466868 3.181844 3.216694 [939] 3.293141 3.635986 3.454845 3.561578 3.268812 3.085647  Here the values of the regressor or external variable: > data$xreg
[1]  6  3  4  4  3  4 10  9  9  7  9  4  3  3  8  3  2  2  3  3  2  5
[23]  5  8  2 -2  5  7  6  7 10 10  9  5  7  8  3  1  1  1  3  6  9  9
[45] 10 11  9 11 15 12 11  9  7  8 10  7  6  7  9  9  9  8  8  7 12 13
[67] 10 14 11 12 12 15 13  9 13 10  9  7  8  8 11 11 14 13 18 13 13 14
[89] 13 13 12 15 15 16 21 12 13 14 12 13 12 13 10 10 10 13 14 12 13 13
[111]  8  8  9 11 13 15 11 14  9 11 11 14 14 10 11 14 19 15 15 16 17 21
[133] 17 14 13 14 18 21 19 23 22 23 19 21 20 18 20 22 22 19 17 16 14 17
[155] 16 17 20 20 17 21 23 21 21 26 28 27 26 30 22 21 19 20 21 19 15 16
[177] 20 19 21 21 21 25 27 26 23 24 23 17 18 19 19 20 23 23 23 20 19 20
[199] 16 17 16 20 18 18 18 18 18 18 19 20 17 18 20 18 19 17 14 17 19 20
[221] 20 21 21 19 21 18 18 18 17 20 20 19 21 23 24 24 24 13 17 22 18 16
[243] 18 17 17 16 16 15 16 16 16 15 14 12 14 14 13 15 16 15 16 19 16 17
[265] 17 17 17 15 16 15 14 14 15 13 15 15 15 14 15 14 17 17 16 19 13 15
[287] 16 13 13 12 15 17 16 15 12 13 11  9 11 13 11 13 12  8  9  9  7 11
[309] 10  8  6  6 10 11 11  7  8  6 12 13 14 10  6  5  6  9  5  4  2  5
[331]  5  8  8  7  9 10  3  3  1  1  3  7  5  4  4  7  5  8 10 11 10  9
[353]  9  6  2  3  7 13 10  7  9  9 10  7  4  4  4  6  8  6  6  5  5  9
[375]  5  6  8  5  3  6  9 11 10  8  5  9 11 10  8  5  6  5  3  3  2  2
[397]  3  6  4  7  5  5  3  5  7  7  8  8  9  7  5  3  2  3  2 -1 -4 -2
[419] -2  2  7  8  8  7  7  9 11  9  9 10  9 10 -2 -1  0  6  8  9  9  7
[441]  9 11 12  6  7  6  7  7  9 11 10  9 14 14 10  9 12 10  9 10 15 12
[463] 12 16 21 25 24 21 20 13 14 12 12 10  8  6  6 11 11 14 18 18 21 23
[485] 23 17 14 15 12 15 17 20 13 14 15 17 17 19 19 17 16 17 21 23 22 16
[507] 18 18 20 21 22 16 15 18 18 19 18 18 19 15 19 19 20 17 16 21 22 21
[529] 18 20 21 22 23 21 22 23 23 25 24 22 23 24 28 28 28 25 18 20 20 22
[551] 25 27 26 21 22 24 20 24 26 28 27 28 31 26 20 18 20 21 23 26 29 26
[573] 26 28 28 20 14 16 19 20 21 22 20 17 19 19 21 23 22 19 20 16 16 16
[595] 17 18 16 17 18 21 20 21 17 17 18 16 17 19 18 19 13 17 16 18 20 21
[617] 19 19 19 14 11 11 12 14 18 19 14 15 13 12 18 16 14 18 10 12 13 17
[639] 19 18 20 21 12 14 17 13 13 12 15 16 10 13 14 11  9  5  6  7  7 10
[661] 10  8 10 11 14 13 11 11 11 11 11 12 12 12 12  9  8  8  6  3  4  4
[683]  7  7  6  6  7 12 11  9 11 13 11  7 12 12  9 10  9  7  7  4  3  2
[705]  4  8  9  8  7  9 11  9 11  5  8  6  8 10  9  8  8  5  4  2  4  8
[727]  9  7  3  3  7  9 10  7  8  7  2  4  4  3  3  1  3  9  8  5  3  4
[749]  1  1  1  2  3  6  9  7  9  9  5  7  9  8  9 11  9 11  8  9 10 11
[771] 11 13 13 15 15 15 11  9 11 11 10 11  9  8 12  8  8  7  9 12 12 10
[793]  7 10 10 12  9 10 11  9 10 10 11 14 16  8 10  7  6  7 10  8 10 12
[815]  8  9  9  8  6  8 11 13 16 17 19 20 20 22 16 15 13 13 10 11 14 15
[837] 13 11  9  9 10 12 12 13 12 14 13 14 16 15 12 15 16 18 18 18 21 19
[859] 20 19 16 14 14 20 17 24 22 17 15 16 16 15 15 16 16 13 14 16 17 16
[881] 18 17 17 17 18 19 17 22 22 18 17 19 30 21 19 19 19 22 24 21 18 19
[903] 20 22 23 22 20 18 18 22 21 20 17 21 20 22 28 28 32 21 17 19 22 17
[925] 19 21 21 20 22 20 19 20 20 20 18 18 14 18 15 19 16 18 17 18


Here the future values of the external variable used to forecast the dependent one:

> data\$future_xreg
[1] 18 19 20 23 25 28 28 28 20 19 20 23 19 19 21 19 16 16 17 17 17 17
[23] 17 18 18 18

• why don't you post the actual Y and X and the residuals from your model with model/coefficient summary and I will try and help you – IrishStat Sep 7 at 10:40
• [(1-B** 7)]Y(T) = +[X1(T)][(- .0015)] P1115099003 +[(1- .340B** 7- .103B** 14)]**-1 [(1+ .308B** 1)(1- .862B** 7)] [A(T)] with the sign of ma coefficients presented in conventional form – IrishStat Sep 7 at 11:19
• Thank you so much for your reply, Dr. Reilly. It is an honor. Your answers have responded basically all questions I have searched. I have 944 observations (after removing the outliers). I posted the log(10) transformed data of these observations. I tried the equation you posted, which is the same as the following: Y(T) = 0.340*(Y-7) + 0.103*(Y-14) + (Y-7) - 0.340*(Y-14) -0.103*(Y-21) + 0.308*(E-7) - 0.862*(E-7) - 0.308*0.862*(E-8), being "E" the residual terms, but again the fitted and predicted values differ from those generated by R – ChrisGila Sep 8 at 9:20
• I am having some issues trying to read the data from the tect. To be safe please email me a csv file for the y values ( I guess they will be in logs ) and the x values. I will use those two series and the model you specified to get fitted values and errors . Also send a second csv file with your fitted values and errors so i can exactly compare them. – IrishStat Sep 8 at 13:18

provides the structure .. this was based upon 200 observations AND the form of the model you posted AND the parameters you posted where p115099003 is the name of the causal series. The equation predicts the 201 value .. the next value and then then be used recursively. Compare the coefficients and lag structure with your equation .

Following is your representation ( which omits the lag effect of X implied by the AR structure in the model

Your approach only has 7 contributions while the output from AUTOBOX ( a piece of software that I have helped to develop is here which correctly uses 9 conytributions ... your 7 plus the two lags of X for lag periods 7 and 14.

The noise model acts like a common multiplier on all variables in the model .

It is very important to be able to express/write the model in layman's terms i.e. a regression format in order to be able to "sell/explain" the solution.

Please let me know if this correction clears up your problem. I not I will actually use your data to perform a clinical analysis.

If my answer has been of help upvote it if you can and accept it to bring it the attention of others. Let me know how this works for you.

• yes with the two coefficients .000510 for lag 7 of X and .000154 for lag 14 of X as MULTIPLIERS of the appropriate X value. – IrishStat Sep 8 at 17:31
• Thank you! The average error of the fitted values (equation) vs the fitted values generated by R is of 129. It totally improved, although still the values are unmatched. I would like to know how to get those two coefficients of X in R. – ChrisGila Sep 8 at 17:52
• .0015 * .34 = .000510 & .0015 * .103 = .000154 reflecting the lagged effects of X when the algebra is done to clear fractions – IrishStat Sep 8 at 19:08
• Dr. Reilly, in fact, I considered the lagged effects of the regressor. It was reffered in my first (unedited) post and took the example from the following link, which is refered in my post stats.stackexchange.com/questions/116842/… – ChrisGila Sep 8 at 21:26
• i am not referring to lag effects of the regressor but for explicit inclusion of daily indicators i.e. 6 dummiy incators) and/or monthly indicators in addition to possible time trends or level shifts – IrishStat Sep 8 at 21:33