Assistance with ARIMA prediction using postestimation commands

[Maria Sundaram]

Hi Statalist, I'm new to using ARIMA modeling and although I've read the ARIMA postestimation documentation, I'm a bit stumped on this one. Hoping folks can help me out--prior threads on this topic didn't have replies from OPs explaining if they had figured it out.

I'm modeling a wastewater signal of SARS-CoV-2 against daily new cases of COVID identified in the community. I have variables describing natural-log-transformed new cases, a wastewater classification signal (increasing, decreasing, or plateau), wastewater flow rate, and PMMOV (a proxy measure for amount of feces in wastewater). Please forgive the large number of values in the -dataex- step, it's meant to assist with running the models and generating predictions.

clear
input double newcases_mm float ln_newcases_mm double(sample_collect_date flow_rate_mm) byte classification_mm double ln_pmmov_conc_mm float avg_50day_log_mm
53  3.970292 23012 36.82 0 19.729013442993164 5.589388
74  4.304065 23013 40.77 0 19.377422332763672 5.586774
98 4.5849676 23014 39.41 0 19.590559005737305 5.575511
86  4.454347 23015 38.11 0  19.57074546813965 5.575034
70  4.248495 23016     . .                  . 5.570022
48  3.871201 23017 36.38 0 19.829322814941406 5.572013
42   3.73767 23018 36.56 0  19.92564582824707 5.567854
78  4.356709 23019  37.2 0 20.021602630615234 5.567013
86  4.454347 23020 36.36 0 19.782886505126953 5.562326
53  3.970292 23021 36.37 1  19.91169548034668 5.554551
48  3.871201 23022 35.59 1 19.538572311401367 5.550797
43    3.7612 23023     . .                  . 5.551994
33  3.496508 23024 35.29 1 19.350637435913086 5.564191
33  3.496508 23025 34.73 2 19.556203842163086 5.565587
46 3.8286414 23026 37.28 2  19.82225799560547 5.567031
65 4.1743875 23027  36.7 1 19.574539184570313  5.56625
49   3.89182 23028 36.53 1  19.75336265563965 5.561026
46 3.8286414 23029  36.9 1 19.682092666625977  5.55269
47 3.8501475 23030     . .                  . 5.538929
52 3.9512436 23031 35.94 0 19.754417419433594 5.544563
28 3.3322046 23032  36.6 0 19.604124069213867 5.532524
55 4.0073333 23033 38.14 0 19.855194091796875 5.523293
77 4.3438053 23034 37.44 0 19.717487335205078 5.526967
56 4.0253515 23035 36.96 0 19.794902801513672 5.530503
40 3.6888795 23036 36.58 0 19.616056442260742 5.515887
47 3.8501475 23037     . .                  . 5.513857
34 3.5263605 23038 35.39 0 19.668861389160156 5.515269
33  3.496508 23039 36.55 0  20.01917839050293 5.517579
45 3.8066626 23040  36.8 1 20.099346160888672 5.508532
52 3.9512436 23041 36.37 1 19.729877471923828 5.500669
57 4.0430512 23042  37.3 2 19.718799591064453 5.499311
50  3.912023 23043 36.87 2 20.093955993652344 5.491871
60 4.0943446 23044     . .                  . 5.489391
30 3.4011974 23045 35.87 2 19.843114852905273 5.495866
26 3.2580965 23046 36.92 2 19.161474227905273   5.4988
65 4.1743875 23047 36.56 2  19.96534538269043 5.492465
62 4.1271343 23048 37.61 1 19.562353134155273 5.497091
71   4.26268 23049 35.15 1  19.84638786315918 5.499902
47 3.8501475 23050 36.96 1 19.601425170898438 5.500391
50  3.912023 23051     . .                  .   5.4966
40 3.6888795 23052 36.93 1 19.761768341064453  5.48804
21 3.0445225 23053 37.63 1 19.732898712158203 5.488622
51 3.9318256 23054 38.72 1  19.75694465637207 5.494551
81  4.394449 23055 39.03 1 19.529640197753906 5.493357
66  4.189655 23056 40.93 1 19.523550033569336 5.494569
40 3.6888795 23057  39.3 1 19.716611862182617 5.486994
37  3.610918 23058     . .                  . 5.495099
37  3.610918 23059 37.98 1 19.674585342407227 5.488668
36  3.583519 23060 39.13 1 19.573528289794922 5.490763
64  4.158883 23061 39.06 1 19.695148468017578 5.494078
58  4.060443 23062 38.61 0  19.73009490966797  5.48185
47 3.8501475 23063 37.15 0 19.712005615234375 5.475732
47 3.8501475 23064 37.56 0 19.589067459106445 5.472705
61 4.1108737 23065     . .                  . 5.455986
42   3.73767 23066 37.53 0 19.763023376464844 5.458945
28 3.3322046 23067 37.87 0 19.266376495361328 5.453701
60 4.0943446 23068 55.33 0 18.668790817260742  5.45036
63 4.1431346 23069 48.31 0 19.124570846557617 5.448144
66  4.189655 23070 45.74 0  19.29648208618164 5.448114
56 4.0253515 23071 43.77 0 19.759674072265625 5.459624
51 3.9318256 23072     . .                  . 5.459912
38  3.637586 23073 40.81 0 18.988718032836914 5.460083
29  3.367296 23074 40.71 0 19.842342376708984 5.447379
57 4.0430512 23075     . .                  . 5.447135
59 4.0775375 23076 41.88 0 18.965791702270508 5.441201
52 3.9512436 23077 41.48 0  19.20622444152832 5.435276
50  3.912023 23078 40.85 0 19.745311737060547 5.434058
41  3.713572 23079     . .                  . 5.434228
30 3.4011974 23080 39.33 0  19.38507652282715   5.4336
39 3.6635616 23081 40.51 0 19.143775939941406 5.423797
49   3.89182 23082 40.07 0 19.348424911499023 5.423503
60 4.0943446 23083 39.89 0 19.626890182495117 5.424713
54  3.988984 23084 40.02 1 19.753995895385742 5.414766
58  4.060443 23085 40.99 1  19.58233070373535 5.412777
56 4.0253515 23086     . .                  . 5.411372
40 3.6888795 23087 40.15 1 19.679595947265625 5.412155
38  3.637586 23088 40.57 1 19.852903366088867 5.407754
47 3.8501475 23089 40.58 2  19.70294761657715 5.404565
70  4.248495 23090 40.13 2  19.12930679321289 5.405726
56 4.0253515 23091 40.13 2 19.576051712036133 5.409409
47 3.8501475 23092 40.21 2 19.753995895385742 5.407861
44   3.78419 23093     . .                  . 5.412015
37  3.610918 23094 39.45 2 19.705385208129883 5.408628
28 3.3322046 23095 39.56 2 19.036083221435547 5.401965
47 3.8501475 23096 40.62 2  19.68729019165039 5.396789
67  4.204693 23097 41.32 2 20.298250198364258 5.399424
65 4.1743875 23098 40.81 1 19.508838653564453 5.384326
48  3.871201 23099 40.62 1 19.558513641357422 5.380361
54  3.988984 23100     . .                  . 5.379776
33  3.496508 23101 47.54 0  19.53042984008789 5.370909
31  3.433987 23102 44.31 0  19.54975700378418 5.365918
53  3.970292 23103 44.02 0  19.54014015197754 5.366637
60 4.0943446 23104 43.37 0  19.22393798828125 5.354074
55 4.0073333 23105 45.17 0  19.46335792541504 5.350321
35  3.555348 23106 43.82 0 19.712663650512695 5.345755
28 3.3322046 23107     . .                  . 5.344503
33  3.496508 23108 40.52 0 19.436201095581055 5.331028
25  3.218876 23109 39.95 0 19.593534469604492 5.329842
45 3.8066626 23110 42.01 0 19.673900604248047 5.331709
51 3.9318256 23111 41.36 0    19.984619140625 5.326061
45 3.8066626 23112 41.74 0   19.6107177734375 5.327641
23  3.135494 23113  41.3 0 19.706270217895508 5.321802
29  3.367296 23114     . .                  . 5.319568
31  3.433987 23115  39.1 0 19.781042098999023 5.320871
26 3.2580965 23116 43.06 0  19.55876922607422 5.313355
30 3.4011974 23117 43.02 0 19.697607040405273 5.306843
44   3.78419 23118 41.75 0 19.534378051757813 5.298716
19  2.944439 23119 41.34 0 19.718143463134766 5.297304
 .         . 23120 40.61 0 19.673900604248047 5.294681
 .         . 23121     . .                  . 5.290625
 .         . 23122 39.07 0 19.623292922973633 5.275259
 .         . 23123 39.42 0 20.168975830078125  5.27065
 .         . 23124 39.89 0  19.83283805847168 5.260292
 .         . 23125  39.7 0 19.724897384643555  5.24592
 .         . 23126 39.21 0  19.77507209777832 5.242687
 .         . 23127 39.67 . 19.695819854736328 5.239677
 .         . 23129 38.32 0 19.797531127929688 5.233493
 .         . 23130 39.91 0 19.800756454467773 5.232412
 .         . 23131  39.5 0  20.20104217529297 5.228647
 .         . 23132 39.67 0 19.692909240722656 5.224076
 .         . 23133 38.88 1 19.416004180908203 5.216482
 .         . 23134 38.84 1 19.470102310180664 5.214991
 .         . 23136 36.46 1 19.818706512451172 5.196811
 .         . 23137 38.33 1  19.70737648010254 5.196811
 .         . 23138 40.12 1 19.633573532104492 5.196811
 .         . 23139 39.49 1 19.726415634155273 5.194838
 .         . 23140 39.57 1 19.899555206298828 5.185842
 .         . 23141 39.27 1 19.757858276367188   5.1806
 .         . 23142     . .                  . 5.172689
 .         . 23143 36.91 1 19.942378997802734 5.160373
 .         . 23144  36.6 1 19.805936813354492 5.155474
 .         . 23145 38.43 1 19.832406997680664 5.144033
 .         . 23146 38.24 0 20.069339752197266 5.142966
 .         . 23147 37.15 0  19.92653465270996 5.130842
 .         . 23148 37.66 0  19.80196189880371 5.131242
 .         . 23150 35.47 0 19.778575897216797 5.098516
 .         . 23151 36.09 0  19.77507209777832 5.095707
 .         . 23152 37.39 0 20.032516479492188 5.096595
 .         . 23153 36.43 0  19.95470428466797 5.081789
 .         . 23154 36.91 0 19.929723739624023 5.068685
 .         . 23155 36.58 0 19.649620056152344  5.05501
 .         . 23157 34.13 0  19.59551239013672 5.032358
 .         . 23158 33.23 0  19.86675262451172  5.03197
 .         . 23159 35.03 0  19.80817413330078 5.031382
 .         . 23160 37.37 0 19.947607040405273 5.019902
 .         . 23161 38.32 0 19.996273040771484 5.007681
end
format %td sample_collect_date
label values classification_mm class
label def class 0 "Decreasing", modify
label def class 1 "Plateau", modify
label def class 2 "Increasing", modify

I then define the data as time-series on my sample collection date variable:

tsset(sample_collect_date)

And run an ARIMA model with these values. The AR and MA values were determined by me using the full dataset which I don't present here.

arima D.ln_newcases_mm classification_mm flow_rate_mm ln_pmmov_conc_mm, ar(3) ma(1 7)
//now predict the remaining cases
predict d_newcase_est
arima D.ln_newcases_mm classification_mm flow_rate_mm ln_pmmov_conc_mm, ar(3) ma(1 7)
predict newcase_est_2, y

However, when I graph the differenced, and non-differenced, predicted values, I see a difference in how far out the model predicts. The non-differenced values are only provided to the end of the actual data I have for new cases, whereas I wanted to have them moving forwards for the entire length of time that I have wastewater data for.

Here's the -tsline- graph:


I'm having a really hard time understanding why the non-differenced model predicted values (produced by predict newcase_est_2, y and shown in goldenrod) are not going past approximately mid-April 2023, but the differenced model predicted values (produced by predict d_newcase_est and shown in seafoam green) are. This is likely some embarrassingly simple mathematical thing, but I would be so grateful if someone could explain. I've also tried using the dynamic() option of predict, to no avail.

[Andrew Musau]

Here is the distinction from the documentation of arima:

xb, the default, calculates the predictions from the model. If D.depvar is the dependent variable, these predictions are of D.depvar and not of depvar itself.

y specifies that predictions of depvar be made, even if the model was specified in terms of, say, D.depvar.

So how Stata calculates the predictions of the dependent variable in levels is to take the predicted change in the dependent variable, \(\Delta \ln y_t\), and add it to the previous observed log value, \(\ln y_{t-1}\), effectively reconstructing the level from the differenced model.

\[
\ln y_t = \ln y_{t-1} + \Delta \ln y_t
\]

For missing periods after mid-April 2023, the previous observed value \(\ln y_{t-1}\) is not available. One could "cheat" by recursively using the predicted value from the previous period in place of the observed \(\ln y_{t-1}\):

\[
\ln \hat y_t = \ln \hat y_{t-1} + \Delta \ln y_t,
\]

starting from the last observed \(\ln y_t\). This is effectively an out-of-sample forecast. While this approach is feasible, it is only recommended if one understands that the predictions will accumulate any errors from previous periods, especially for models with MA terms, and therefore may become less accurate the further one forecasts beyond the observed sample.

clear
input double newcases_mm float ln_newcases_mm double(sample_collect_date flow_rate_mm) byte classification_mm double ln_pmmov_conc_mm float avg_50day_log_mm
53  3.970292 23012 36.82 0 19.729013442993164 5.589388
74  4.304065 23013 40.77 0 19.377422332763672 5.586774
98 4.5849676 23014 39.41 0 19.590559005737305 5.575511
86  4.454347 23015 38.11 0  19.57074546813965 5.575034
70  4.248495 23016     . .                  . 5.570022
48  3.871201 23017 36.38 0 19.829322814941406 5.572013
42   3.73767 23018 36.56 0  19.92564582824707 5.567854
78  4.356709 23019  37.2 0 20.021602630615234 5.567013
86  4.454347 23020 36.36 0 19.782886505126953 5.562326
53  3.970292 23021 36.37 1  19.91169548034668 5.554551
48  3.871201 23022 35.59 1 19.538572311401367 5.550797
43    3.7612 23023     . .                  . 5.551994
33  3.496508 23024 35.29 1 19.350637435913086 5.564191
33  3.496508 23025 34.73 2 19.556203842163086 5.565587
46 3.8286414 23026 37.28 2  19.82225799560547 5.567031
65 4.1743875 23027  36.7 1 19.574539184570313  5.56625
49   3.89182 23028 36.53 1  19.75336265563965 5.561026
46 3.8286414 23029  36.9 1 19.682092666625977  5.55269
47 3.8501475 23030     . .                  . 5.538929
52 3.9512436 23031 35.94 0 19.754417419433594 5.544563
28 3.3322046 23032  36.6 0 19.604124069213867 5.532524
55 4.0073333 23033 38.14 0 19.855194091796875 5.523293
77 4.3438053 23034 37.44 0 19.717487335205078 5.526967
56 4.0253515 23035 36.96 0 19.794902801513672 5.530503
40 3.6888795 23036 36.58 0 19.616056442260742 5.515887
47 3.8501475 23037     . .                  . 5.513857
34 3.5263605 23038 35.39 0 19.668861389160156 5.515269
33  3.496508 23039 36.55 0  20.01917839050293 5.517579
45 3.8066626 23040  36.8 1 20.099346160888672 5.508532
52 3.9512436 23041 36.37 1 19.729877471923828 5.500669
57 4.0430512 23042  37.3 2 19.718799591064453 5.499311
50  3.912023 23043 36.87 2 20.093955993652344 5.491871
60 4.0943446 23044     . .                  . 5.489391
30 3.4011974 23045 35.87 2 19.843114852905273 5.495866
26 3.2580965 23046 36.92 2 19.161474227905273   5.4988
65 4.1743875 23047 36.56 2  19.96534538269043 5.492465
62 4.1271343 23048 37.61 1 19.562353134155273 5.497091
71   4.26268 23049 35.15 1  19.84638786315918 5.499902
47 3.8501475 23050 36.96 1 19.601425170898438 5.500391
50  3.912023 23051     . .                  .   5.4966
40 3.6888795 23052 36.93 1 19.761768341064453  5.48804
21 3.0445225 23053 37.63 1 19.732898712158203 5.488622
51 3.9318256 23054 38.72 1  19.75694465637207 5.494551
81  4.394449 23055 39.03 1 19.529640197753906 5.493357
66  4.189655 23056 40.93 1 19.523550033569336 5.494569
40 3.6888795 23057  39.3 1 19.716611862182617 5.486994
37  3.610918 23058     . .                  . 5.495099
37  3.610918 23059 37.98 1 19.674585342407227 5.488668
36  3.583519 23060 39.13 1 19.573528289794922 5.490763
64  4.158883 23061 39.06 1 19.695148468017578 5.494078
58  4.060443 23062 38.61 0  19.73009490966797  5.48185
47 3.8501475 23063 37.15 0 19.712005615234375 5.475732
47 3.8501475 23064 37.56 0 19.589067459106445 5.472705
61 4.1108737 23065     . .                  . 5.455986
42   3.73767 23066 37.53 0 19.763023376464844 5.458945
28 3.3322046 23067 37.87 0 19.266376495361328 5.453701
60 4.0943446 23068 55.33 0 18.668790817260742  5.45036
63 4.1431346 23069 48.31 0 19.124570846557617 5.448144
66  4.189655 23070 45.74 0  19.29648208618164 5.448114
56 4.0253515 23071 43.77 0 19.759674072265625 5.459624
51 3.9318256 23072     . .                  . 5.459912
38  3.637586 23073 40.81 0 18.988718032836914 5.460083
29  3.367296 23074 40.71 0 19.842342376708984 5.447379
57 4.0430512 23075     . .                  . 5.447135
59 4.0775375 23076 41.88 0 18.965791702270508 5.441201
52 3.9512436 23077 41.48 0  19.20622444152832 5.435276
50  3.912023 23078 40.85 0 19.745311737060547 5.434058
41  3.713572 23079     . .                  . 5.434228
30 3.4011974 23080 39.33 0  19.38507652282715   5.4336
39 3.6635616 23081 40.51 0 19.143775939941406 5.423797
49   3.89182 23082 40.07 0 19.348424911499023 5.423503
60 4.0943446 23083 39.89 0 19.626890182495117 5.424713
54  3.988984 23084 40.02 1 19.753995895385742 5.414766
58  4.060443 23085 40.99 1  19.58233070373535 5.412777
56 4.0253515 23086     . .                  . 5.411372
40 3.6888795 23087 40.15 1 19.679595947265625 5.412155
38  3.637586 23088 40.57 1 19.852903366088867 5.407754
47 3.8501475 23089 40.58 2  19.70294761657715 5.404565
70  4.248495 23090 40.13 2  19.12930679321289 5.405726
56 4.0253515 23091 40.13 2 19.576051712036133 5.409409
47 3.8501475 23092 40.21 2 19.753995895385742 5.407861
44   3.78419 23093     . .                  . 5.412015
37  3.610918 23094 39.45 2 19.705385208129883 5.408628
28 3.3322046 23095 39.56 2 19.036083221435547 5.401965
47 3.8501475 23096 40.62 2  19.68729019165039 5.396789
67  4.204693 23097 41.32 2 20.298250198364258 5.399424
65 4.1743875 23098 40.81 1 19.508838653564453 5.384326
48  3.871201 23099 40.62 1 19.558513641357422 5.380361
54  3.988984 23100     . .                  . 5.379776
33  3.496508 23101 47.54 0  19.53042984008789 5.370909
31  3.433987 23102 44.31 0  19.54975700378418 5.365918
53  3.970292 23103 44.02 0  19.54014015197754 5.366637
60 4.0943446 23104 43.37 0  19.22393798828125 5.354074
55 4.0073333 23105 45.17 0  19.46335792541504 5.350321
35  3.555348 23106 43.82 0 19.712663650512695 5.345755
28 3.3322046 23107     . .                  . 5.344503
33  3.496508 23108 40.52 0 19.436201095581055 5.331028
25  3.218876 23109 39.95 0 19.593534469604492 5.329842
45 3.8066626 23110 42.01 0 19.673900604248047 5.331709
51 3.9318256 23111 41.36 0    19.984619140625 5.326061
45 3.8066626 23112 41.74 0   19.6107177734375 5.327641
23  3.135494 23113  41.3 0 19.706270217895508 5.321802
29  3.367296 23114     . .                  . 5.319568
31  3.433987 23115  39.1 0 19.781042098999023 5.320871
26 3.2580965 23116 43.06 0  19.55876922607422 5.313355
30 3.4011974 23117 43.02 0 19.697607040405273 5.306843
44   3.78419 23118 41.75 0 19.534378051757813 5.298716
19  2.944439 23119 41.34 0 19.718143463134766 5.297304
 .         . 23120 40.61 0 19.673900604248047 5.294681
 .         . 23121     . .                  . 5.290625
 .         . 23122 39.07 0 19.623292922973633 5.275259
 .         . 23123 39.42 0 20.168975830078125  5.27065
 .         . 23124 39.89 0  19.83283805847168 5.260292
 .         . 23125  39.7 0 19.724897384643555  5.24592
 .         . 23126 39.21 0  19.77507209777832 5.242687
 .         . 23127 39.67 . 19.695819854736328 5.239677
 .         . 23129 38.32 0 19.797531127929688 5.233493
 .         . 23130 39.91 0 19.800756454467773 5.232412
 .         . 23131  39.5 0  20.20104217529297 5.228647
 .         . 23132 39.67 0 19.692909240722656 5.224076
 .         . 23133 38.88 1 19.416004180908203 5.216482
 .         . 23134 38.84 1 19.470102310180664 5.214991
 .         . 23136 36.46 1 19.818706512451172 5.196811
 .         . 23137 38.33 1  19.70737648010254 5.196811
 .         . 23138 40.12 1 19.633573532104492 5.196811
 .         . 23139 39.49 1 19.726415634155273 5.194838
 .         . 23140 39.57 1 19.899555206298828 5.185842
 .         . 23141 39.27 1 19.757858276367188   5.1806
 .         . 23142     . .                  . 5.172689
 .         . 23143 36.91 1 19.942378997802734 5.160373
 .         . 23144  36.6 1 19.805936813354492 5.155474
 .         . 23145 38.43 1 19.832406997680664 5.144033
 .         . 23146 38.24 0 20.069339752197266 5.142966
 .         . 23147 37.15 0  19.92653465270996 5.130842
 .         . 23148 37.66 0  19.80196189880371 5.131242
 .         . 23150 35.47 0 19.778575897216797 5.098516
 .         . 23151 36.09 0  19.77507209777832 5.095707
 .         . 23152 37.39 0 20.032516479492188 5.096595
 .         . 23153 36.43 0  19.95470428466797 5.081789
 .         . 23154 36.91 0 19.929723739624023 5.068685
 .         . 23155 36.58 0 19.649620056152344  5.05501
 .         . 23157 34.13 0  19.59551239013672 5.032358
 .         . 23158 33.23 0  19.86675262451172  5.03197
 .         . 23159 35.03 0  19.80817413330078 5.031382
 .         . 23160 37.37 0 19.947607040405273 5.019902
 .         . 23161 38.32 0 19.996273040771484 5.007681
end
format %td sample_collect_date
label values classification_mm class
label def class 0 "Decreasing", modify
label def class 1 "Plateau", modify
label def class 2 "Increasing", modify

tsset(sample_collect_date)
arima D.ln_newcases_mm classification_mm flow_rate_mm ln_pmmov_conc_mm, ar(3) ma(1 7)
predict xb, xb
predict xby, y 
gen xby_manual = L.ln_newcases_mm + xb
sum xby xby_manual

Res.:

. arima D.ln_newcases_mm classification_mm flow_rate_mm ln_pmmov_conc_mm, ar(3) ma(1 7)
(note:  insufficient memory or observations to estimate usual
starting values [2])

Number of gaps in sample = 16
(note: filtering over missing observations)

(setting optimization to BHHH)
Iteration 0:  Log likelihood = -32.625229  
Iteration 1:  Log likelihood = -29.451087  
Iteration 2:  Log likelihood = -22.423784  
Iteration 3:  Log likelihood = -19.470923  
Iteration 4:  Log likelihood = -18.683271  
(switching optimization to BFGS)
Iteration 5:  Log likelihood = -18.524373  
Iteration 6:  Log likelihood = -18.074577  
Iteration 7:  Log likelihood = -18.015648  
Iteration 8:  Log likelihood = -18.002153  
Iteration 9:  Log likelihood = -18.001914  
Iteration 10: Log likelihood =   -18.0019  
Iteration 11: Log likelihood = -18.001897  
Iteration 12: Log likelihood = -18.001897  

ARIMA regression

Sample: 03jan2023 thru 19apr2023, but with gaps
                                                Number of obs     =         91
                                                Wald chi2(6)      =      33.50
Log likelihood = -18.0019                       Prob > chi2       =     0.0000

-----------------------------------------------------------------------------------
                  |                 OPG
 D.ln_newcases_mm | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
ln_newcases_mm    |
classification_mm |     .03241   .0291286     1.11   0.266    -.0246811     .089501
     flow_rate_mm |   .0078983   .0097961     0.81   0.420    -.0113017    .0270982
 ln_pmmov_conc_mm |   .0046808   .1226074     0.04   0.970    -.2356254    .2449869
            _cons |  -.4750386   2.590892    -0.18   0.855    -5.553094    4.603017
------------------+----------------------------------------------------------------
ARMA              |
               ar |
              L3. |   -.368622   .1313778    -2.81   0.005    -.6261179   -.1111262
                  |
               ma |
              L1. |    -.27124   .1577723    -1.72   0.086     -.580468    .0379881
              L7. |    .489072   .1243148     3.93   0.000     .2454195    .7327246
------------------+----------------------------------------------------------------
           /sigma |   .2846762   .0250163    11.38   0.000     .2356451    .3337073
-----------------------------------------------------------------------------------
Note: The test of the variance against zero is one sided, and the two-sided
      confidence interval is truncated at zero.

. predict xb, xb
(19 missing values generated)

. predict xby, y
(54 missing values generated)

. gen xby_manual = L.ln_newcases_mm + xb
(54 missing values generated)

. sum xby xby_manual

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
         xby |         92    3.792196    .3005984   2.904406   4.535826
  xby_manual |         92    3.792196    .3005984   2.904406   4.535826

[Maria Sundaram]

Dear Andrew, I apologize for the delay in replying but just wanted to thank you so much both for this explanation and for the code. It helped immensely for this ARIMA newbie.