Modeling Time in the Discrete Time Analysis using Left-truncated Data

I am doing multiple-spell, discrete-time analysis to examine factors shaping time to exiting a renter spell, with time-varying covariates at monthly interval. My data has left-truncated spells, meaning some people have already started their renter spell before the start of two-year observation period.

I have followed Stephen Jenkins' advice (http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf) on setting up left-truncated data for discrete time analysis, which makes observations in the left-truncated spell start to contribute to person-observation from the time when they enter the study.

My problem starts from here. I have now left with observations with elapsed months since risk onset much longer (25 percentile = 9 months, 50 percentile = 30 months, 75 percentile = 80 months, 99 percentile = 657 months) than two-year study period.

I am not very comfortable modeling time that goes way over study time period, but at the same time, I would like to keep left-truncated spells in my analysis. I would like to have your thoughts or direction on how to model time when left-truncated data makes elapsed time since risk onset far stretched beyond the length of observation period.

Here's event occurrence for each month since risk onset:

Code:

 tab timesince monthlymove

           |      monthlymove
 timesince |         0          1 |     Total
-----------+----------------------+----------
         1 |     3,915         35 |     3,950 
         2 |     3,864         39 |     3,903 
         3 |     3,745         64 |     3,809 
         4 |     3,525         31 |     3,556 
         5 |     3,236         41 |     3,277 
         6 |     3,247         48 |     3,295 
         7 |     3,040         68 |     3,108 
         8 |     2,996         74 |     3,070 
         9 |     2,867         44 |     2,911 
        10 |     2,673         74 |     2,747 
        11 |     2,556         45 |     2,601 
        12 |     2,435         59 |     2,494 
        13 |     1,952         87 |     2,039 
        14 |     1,806         43 |     1,849 
        15 |     1,682         48 |     1,730 
        16 |     1,573         45 |     1,618 
        17 |     1,441         28 |     1,469 
        18 |     1,401         33 |     1,434 
        19 |     1,343         26 |     1,369 
        20 |     1,333         10 |     1,343 
        21 |     1,299         19 |     1,318 
        22 |     1,270         21 |     1,291 
        23 |     1,246         16 |     1,262 
        24 |     1,225         19 |     1,244 
        25 |     1,074         29 |     1,103 
        26 |       987         25 |     1,012 
        27 |       933         12 |       945 
        28 |       905          9 |       914 
        29 |       884         10 |       894 
        30 |       837          6 |       843 
        31 |       817          8 |       825 
        32 |       828          6 |       834 
        33 |       800         12 |       812 
        34 |       785          5 |       790 
        35 |       776         10 |       786 
        36 |       760          7 |       767 
        37 |       736         17 |       753 
        38 |       729          4 |       733 
        39 |       712          8 |       720 
        40 |       695          9 |       704 
        41 |       676          8 |       684 
        42 |       639          5 |       644 
        43 |       616          6 |       622 
        44 |       608          3 |       611 
        45 |       600          4 |       604 
        46 |       566          6 |       572 
        47 |       566          5 |       571 
        48 |       539          3 |       542 
        49 |       526         10 |       536 
        50 |       514          2 |       516 
        51 |       516          5 |       521 
        52 |       500         10 |       510 
        53 |       488          7 |       495 
        54 |       487         10 |       497 
        55 |       488          1 |       489 
        56 |       489          7 |       496 
        57 |       493          2 |       495 
        58 |       469         13 |       482 
        59 |       452          7 |       459 
        60 |       438         11 |       449 
        61 |       416          2 |       418 
        62 |       397          5 |       402 
        63 |       387          6 |       393 
        64 |       381          5 |       386 
        65 |       358          8 |       366 
        66 |       361          2 |       363 
        67 |       351         10 |       361 
        68 |       336          3 |       339 
        69 |       324          6 |       330 
        70 |       326          1 |       327 
        71 |       315          2 |       317 
        72 |       309          6 |       315 
        73 |       301          5 |       306 
        74 |       294          2 |       296 
        75 |       285          1 |       286 
        76 |       277          5 |       282 
        77 |       283          3 |       286 
        78 |       273          1 |       274 
        79 |       273          2 |       275 
        80 |       265          5 |       270 
        81 |       252          1 |       253 
        82 |       247          2 |       249 
        83 |       242          1 |       243 
        84 |       245          0 |       245 
        85 |       243          1 |       244 
        86 |       239          2 |       241 
        87 |       237          0 |       237 
        88 |       227          3 |       230 
        89 |       225          2 |       227 
        90 |       217          5 |       222 
        91 |       207          3 |       210 
        92 |       205          1 |       206 
        93 |       206          0 |       206 
        94 |       200          0 |       200 
        95 |       193          0 |       193 
        96 |       183          2 |       185 
        97 |       181          5 |       186 
        98 |       185          0 |       185 
        99 |       183          0 |       183 
       100 |       180          2 |       182 
       101 |       175          1 |       176 
       102 |       167          2 |       169 
       103 |       160          2 |       162 
       104 |       181          1 |       182 
       105 |       185          2 |       187 
       106 |       184          2 |       186 
       107 |       182          0 |       182 
       108 |       177          4 |       181 
       109 |       169          2 |       171 
       110 |       165          1 |       166 
       111 |       159          2 |       161 
       112 |       148          2 |       150 
       113 |       146          1 |       147 
       114 |       148          2 |       150 
       115 |       144          2 |       146 
       116 |       143          1 |       144 
       117 |       144          2 |       146 
       118 |       149          1 |       150 
       119 |       152          1 |       153 
       120 |       153          2 |       155 
       121 |       155          1 |       156 
       122 |       152          1 |       153 
       123 |       150          1 |       151 
       124 |       148          1 |       149 
       125 |       144          0 |       144 
       126 |       137          1 |       138 
       127 |       135          1 |       136 
       128 |       130          0 |       130 
       129 |       128          0 |       128 
       130 |       124          1 |       125 
       131 |       116          0 |       116 
       132 |       111          0 |       111 
       133 |       114          1 |       115 
       134 |       106          1 |       107 
       135 |       107          2 |       109 
       136 |       103          0 |       103 
       137 |       103          1 |       104 
       138 |       103          1 |       104 
       139 |       103          1 |       104 
       140 |       102          1 |       103 
       141 |        91          1 |        92 
       142 |        94          1 |        95 
       143 |        97          0 |        97 
       144 |        96          0 |        96 
       145 |        91          0 |        91 
       146 |        98          1 |        99 
       147 |        99          0 |        99 
       148 |       104          1 |       105 
       149 |       107          0 |       107 
       150 |       106          1 |       107 
       151 |       110          0 |       110 
       152 |       126          0 |       126 
       153 |       126          2 |       128 
       154 |       126          2 |       128 
       155 |       127          2 |       129 
       156 |       127          1 |       128 
       157 |       131          0 |       131 
       158 |       130          1 |       131 
       159 |       129          1 |       130 
       160 |       123          0 |       123 
       161 |       120          2 |       122 
       162 |       122          1 |       123 
       163 |       116          0 |       116 
       164 |       123          0 |       123 
       165 |       126          0 |       126 
       166 |       116          1 |       117 
       167 |       116          2 |       118 
       168 |       115          0 |       115 
       169 |       110          4 |       114 
       170 |       102          1 |       103 
       171 |       106          0 |       106 
       172 |       103          1 |       104 
       173 |        97          0 |        97 
       174 |        89          1 |        90 
       175 |        83          0 |        83 
       176 |        70          2 |        72 
       177 |        69          0 |        69 
       178 |        67          1 |        68 
       179 |        64          0 |        64 
       180 |        64          0 |        64 
       181 |        64          0 |        64 
       182 |        62          0 |        62 
       183 |        57          0 |        57 
       184 |        56          0 |        56 
       185 |        52          1 |        53 
       186 |        53          0 |        53 
       187 |        54          0 |        54 
       188 |        46          1 |        47 
       189 |        45          0 |        45 
       190 |        47          1 |        48 
       191 |        44          0 |        44 
       192 |        47          0 |        47 
       193 |        46          0 |        46 
       194 |        45          0 |        45 
       195 |        44          0 |        44 
       196 |        41          0 |        41 
       197 |        43          0 |        43 
       198 |        48          0 |        48 
       199 |        47          0 |        47 
       200 |        53          1 |        54 
       201 |        53          1 |        54 
       202 |        52          2 |        54 
       203 |        53          0 |        53 
       204 |        52          1 |        53 
       205 |        52          1 |        53 
       206 |        53          2 |        55 
       207 |        55          0 |        55 
       208 |        50          4 |        54 
       209 |        49          0 |        49 
       210 |        50          0 |        50 
       211 |        46          1 |        47 
       212 |        60          1 |        61 
       213 |        58          1 |        59 
       214 |        55          0 |        55 
       215 |        56          0 |        56 
       216 |        54          0 |        54 
       217 |        55          1 |        56 
       218 |        51          1 |        52 
       219 |        48          2 |        50 
       220 |        50          1 |        51 
       221 |        52          0 |        52 
       222 |        50          1 |        51 
       223 |        54          0 |        54 
       224 |        59          0 |        59 
       225 |        58          0 |        58 
       226 |        60          1 |        61 
       227 |        64          1 |        65 
       228 |        65          0 |        65 
       229 |        70          0 |        70 
       230 |        68          0 |        68 
       231 |        69          0 |        69 
       232 |        68          1 |        69 
       233 |        67          1 |        68 
       234 |        62          2 |        64 
       235 |        63          0 |        63 
       236 |        55          0 |        55 
       237 |        52          0 |        52 
       238 |        54          0 |        54 
       239 |        50          0 |        50 
       240 |        48          0 |        48 
       241 |        53          0 |        53 
       242 |        53          1 |        54 
       243 |        51          2 |        53 
       244 |        49          0 |        49 
       245 |        45          3 |        48 
       246 |        44          0 |        44 
       247 |        40          0 |        40 
       248 |        46          0 |        46 
       249 |        46          0 |        46 
       250 |        45          1 |        46 
       251 |        41          1 |        42 
       252 |        41          0 |        41 
       253 |        37          0 |        37 
       254 |        37          0 |        37 
       255 |        35          1 |        36 
       256 |        34          1 |        35 
       257 |        31          2 |        33 
       258 |        34          0 |        34 
       259 |        34          0 |        34 
       260 |        32          0 |        32 
       261 |        32          0 |        32 
       262 |        28          0 |        28 
       263 |        27          1 |        28 
       264 |        26          0 |        26 
       265 |        21          0 |        21 
       266 |        20          0 |        20 
       267 |        23          0 |        23 
       268 |        23          0 |        23 
       269 |        24          0 |        24 
       270 |        24          0 |        24 
       271 |        27          0 |        27 
       272 |        28          0 |        28 
       273 |        28          0 |        28 
       274 |        28          0 |        28 
       275 |        30          1 |        31 
       276 |        28          1 |        29 
       277 |        29          0 |        29 
       278 |        27          0 |        27 
       279 |        28          0 |        28 
       280 |        28          0 |        28 
       281 |        28          1 |        29 
       282 |        28          0 |        28 
       283 |        24          0 |        24 
       284 |        24          0 |        24 
       285 |        24          1 |        25 
       286 |        25          0 |        25 
       287 |        25          0 |        25 
       288 |        26          0 |        26 
       289 |        27          1 |        28 
       290 |        27          0 |        27 
       291 |        25          0 |        25 
       292 |        27          0 |        27 
       293 |        26          0 |        26 
       294 |        27          0 |        27 
       295 |        28          0 |        28 
       296 |        29          0 |        29 
       297 |        30          0 |        30 
       298 |        30          0 |        30 
       299 |        28          0 |        28 
       300 |        26          0 |        26 
       301 |        26          0 |        26 
       302 |        25          0 |        25 
       303 |        25          0 |        25 
       304 |        27          0 |        27 
       305 |        25          1 |        26 
       306 |        24          1 |        25 
       307 |        25          0 |        25 
       308 |        23          0 |        23 
       309 |        23          0 |        23 
       310 |        23          0 |        23 
       311 |        23          0 |        23 
       312 |        22          0 |        22 
       313 |        18          2 |        20 
       314 |        19          0 |        19 
       315 |        19          0 |        19 
       316 |        17          0 |        17 
       317 |        19          0 |        19 
       318 |        15          0 |        15 
       319 |        13          0 |        13 
       320 |        10          0 |        10 
       321 |         9          0 |         9 
       322 |        11          1 |        12 
       323 |        12          0 |        12 
       324 |        12          0 |        12 
       325 |        11          1 |        12 
       326 |        12          0 |        12 
       327 |        15          0 |        15 
       328 |        16          1 |        17 
       329 |        18          0 |        18 
       330 |        19          0 |        19 
       331 |        20          0 |        20 
       332 |        26          1 |        27 
       333 |        26          1 |        27 
       334 |        28          0 |        28 
       335 |        32          0 |        32 
       336 |        32          0 |        32 
       337 |        35          0 |        35 
       338 |        33          1 |        34 
       339 |        33          0 |        33 
       340 |        35          0 |        35 
       341 |        35          0 |        35 
       342 |        35          0 |        35 
       343 |        35          0 |        35 
       344 |        33          2 |        35 
       345 |        35          0 |        35 
       346 |        31          2 |        33 
       347 |        27          2 |        29 
       348 |        29          0 |        29 
       349 |        26          1 |        27 
       350 |        26          0 |        26 
       351 |        26          0 |        26 
       352 |        23          1 |        24 
       353 |        23          0 |        23 
       354 |        22          1 |        23 
       355 |        21          1 |        22 
       356 |        18          0 |        18 
       357 |        16          0 |        16 
       358 |        14          0 |        14 
       359 |        14          0 |        14 
       360 |        13          0 |        13 
       361 |        14          0 |        14 
       362 |        16          0 |        16 
       363 |        15          0 |        15 
       364 |        13          0 |        13 
       365 |        13          0 |        13 
       366 |        14          0 |        14 
       367 |        16          0 |        16 
       368 |        15          0 |        15 
       369 |        14          1 |        15 
       370 |        14          0 |        14 
       371 |        15          0 |        15 
       372 |        15          0 |        15 
       373 |        15          0 |        15 
       374 |        15          0 |        15 
       375 |        15          0 |        15 
       376 |        16          0 |        16 
       377 |        15          0 |        15 
       378 |        14          0 |        14 
       379 |        16          0 |        16 
       380 |        16          0 |        16 
       381 |        17          0 |        17 
       382 |        18          0 |        18 
       383 |        18          0 |        18 
       384 |        17          1 |        18 
       385 |        16          1 |        17 
       386 |        17          0 |        17 
       387 |        17          0 |        17 
       388 |        16          0 |        16 
       389 |        20          0 |        20 
       390 |        19          0 |        19 
       391 |        16          0 |        16 
       392 |        17          0 |        17 
       393 |        17          0 |        17 
       394 |        18          1 |        19 
       395 |        18          0 |        18 
       396 |        19          0 |        19 
       397 |        19          1 |        20 
       398 |        20          0 |        20 
       399 |        21          0 |        21 
       400 |        23          0 |        23 
...........................................................
       500 |         8          0 |         8 
       501 |         8          0 |         8 
       502 |         8          0 |         8 
       503 |         8          0 |         8 
       504 |         8          0 |         8 
       505 |         8          1 |         9 
       506 |         8          0 |         8 
       507 |        10          0 |        10 
       508 |         9          0 |         9 
       509 |         9          0 |         9 
       510 |         9          0 |         9 
       511 |        11          0 |        11 
       512 |        15          0 |        15 
       513 |        17          0 |        17 
       514 |        16          0 |        16 
       515 |        16          0 |        16 
       516 |        15          0 |        15 
       517 |        15          0 |        15 
       518 |        15          0 |        15 
       519 |        14          0 |        14 
       520 |        14          0 |        14 
       521 |        14          0 |        14 
       522 |        15          0 |        15 
       523 |        14          0 |        14 
       524 |        12          0 |        12 
       525 |        12          0 |        12 
       526 |        14          0 |        14 
       527 |        14          1 |        15 
       528 |        14          0 |        14 
       529 |        15          0 |        15 
       530 |        16          0 |        16 
       531 |        15          0 |        15 
       532 |        13          2 |        15 
       533 |        13          0 |        13 
       534 |        12          0 |        12 
       535 |        11          0 |        11 
       536 |        10          0 |        10 
       537 |        10          0 |        10 
       538 |        10          0 |        10 
       539 |        12          0 |        12 
       540 |        12          0 |        12 
       541 |        12          0 |        12 
       542 |        13          0 |        13 
       543 |        13          0 |        13 
       544 |        12          0 |        12 
       545 |        13          0 |        13 
       546 |        14          0 |        14 
       547 |        14          0 |        14 
       548 |        14          0 |        14 
       549 |        14          0 |        14 
       550 |        12          0 |        12 
       551 |        11          0 |        11 
       552 |        10          1 |        11 
       553 |         9          0 |         9 
       554 |         9          0 |         9 
       555 |         9          0 |         9 
       556 |        10          0 |        10 
       557 |         9          0 |         9 
       558 |         9          0 |         9 
       559 |         9          0 |         9 
       560 |         9          0 |         9 
       561 |         8          0 |         8 
       562 |         8          0 |         8 
       563 |         8          0 |         8 
       564 |         8          1 |         9 
       565 |         8          0 |         8 
       566 |         7          0 |         7 
       567 |         6          0 |         6 
       568 |         7          0 |         7 
       569 |         8          0 |         8 
       570 |         6          0 |         6 
       571 |         7          0 |         7 
       572 |         7          0 |         7 
       573 |         8          0 |         8 
       574 |         8          0 |         8 
       575 |         8          0 |         8 
       576 |         8          0 |         8 
       577 |         9          0 |         9 
       578 |         9          0 |         9 
       579 |         9          0 |         9 
       580 |         8          0 |         8 
       581 |         8          0 |         8 
       582 |         9          0 |         9 
       583 |        10          0 |        10 
       584 |        11          0 |        11 
       585 |         9          0 |         9 
       586 |         9          0 |         9 
       587 |        10          0 |        10 
       588 |        10          0 |        10 
       589 |         9          1 |        10 
       590 |         9          0 |         9 
       591 |        10          0 |        10 
       592 |        10          0 |        10 
       593 |         7          2 |         9 
       594 |         6          0 |         6 
       595 |         5          0 |         5 
       596 |         5          0 |         5 
       597 |         5          0 |         5 
       598 |         5          0 |         5 
       599 |         4          0 |         4 
       600 |         3          0 |         3 
       601 |         3          0 |         3 
       602 |         3          0 |         3 
       603 |         3          0 |         3 
       604 |         3          0 |         3 
       605 |         4          0 |         4 
       606 |         3          0 |         3 
       607 |         3          0 |         3 
       608 |         2          1 |         3 
       609 |         1          1 |         2 
       610 |         2          0 |         2 
       611 |         2          0 |         2 
       612 |         2          0 |         2 
       613 |         2          0 |         2 
       614 |         2          0 |         2 
       615 |         2          0 |         2 
       616 |         2          0 |         2 
       617 |         2          0 |         2 
       618 |         2          0 |         2 
       619 |         2          0 |         2 
       620 |         1          0 |         1 
       621 |         1          0 |         1 
       622 |         2          0 |         2 
       623 |         2          0 |         2 
       624 |         2          0 |         2 
       625 |         2          0 |         2 
       626 |         2          0 |         2 
       627 |         2          0 |         2 
       628 |         2          0 |         2 
       629 |         2          0 |         2 
       630 |         3          0 |         3 
       631 |         5          0 |         5 
       632 |         6          0 |         6 
       633 |         7          0 |         7 
       634 |         6          0 |         6 
       635 |         6          0 |         6 
       636 |         6          0 |         6 
       637 |         6          0 |         6 
       638 |         6          0 |         6 
       639 |         6          0 |         6 
       640 |         6          0 |         6 
       641 |         5          1 |         6 
       642 |         5          0 |         5 
       643 |         5          0 |         5 
       644 |         6          1 |         7 
       645 |         6          0 |         6 
       646 |         5          0 |         5 
       647 |         4          0 |         4 
       648 |         4          0 |         4 
       649 |         4          0 |         4 
       650 |         5          0 |         5 
       651 |         5          0 |         5 
       652 |         6          0 |         6 
       653 |         6          0 |         6 
       654 |         6          0 |         6 
       655 |         5          0 |         5 
       656 |         3          0 |         3 
       657 |         4          0 |         4 
       658 |         5          0 |         5 
       659 |         5          0 |         5 
       660 |         5          0 |         5 
       661 |         5          0 |         5 
       662 |         8          0 |         8 
       663 |         9          0 |         9 
       664 |        13          0 |        13 
       665 |        14          1 |        15 
       666 |        16          0 |        16 
       667 |        19          0 |        19 
       668 |        31          0 |        31 
       669 |        36          0 |        36 
       670 |        37          0 |        37 
       671 |        39          1 |        40 
       672 |        44          0 |        44 
       673 |        47          0 |        47 
       674 |        45          0 |        45 
       675 |        45          0 |        45 
       676 |        44          0 |        44 
       677 |        43          0 |        43 
       678 |        42          0 |        42 
       679 |        42          0 |        42 
       680 |        40          0 |        40 
       681 |        37          0 |        37 
       682 |        36          0 |        36 
       683 |        36          0 |        36 
       684 |        36          0 |        36 
       685 |        35          0 |        35 
       686 |        33          0 |        33 
       687 |        32          0 |        32 
       688 |        28          0 |        28 
       689 |        27          0 |        27 
       690 |        23          2 |        25 
       691 |        20          0 |        20 
       692 |        11          0 |        11 
       693 |         9          0 |         9 
       694 |         8          0 |         8 
       695 |         5          0 |         5 
       696 |         2          0 |         2 
-----------+----------------------+----------
     Total |   110,911      1,608 |   112,519 

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