# The Specification of Dynamic Discrete-Time Two-State Panel Data Models

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## Abstract

**:**

## 1. Introduction

## 2. Modeling Discrete-Time Two-State Panel Data

#### 2.1. Context, Data, and Likelihood

#### 2.2. Prototype DBR Models

#### 2.3. Prototype MSD Models

#### 2.4. The Relationship between DBR and MSD Models

## 3. Case Study

#### 3.1. Data

#### 3.2. Estimation Results

#### 3.3. In Sample Prediction

#### 3.4. Out of Sample Prediction

## 4. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | Typical topics include employment (e.g., Heckman 1981a; Hyslop 1999), unemployment (e.g., Arulampalam et al. 2000), poverty (e.g., Stevens 1999; Cappellari and Jenkins 2004), welfare dependency (e.g., Bane and Ellwood 1983), health (e.g., Halliday 2008), and peace and conflict between national states (e.g., Beck and Katz 1997; Beck et al. 2001). |

2 | The data we use come from the PSID survey years 1970–89, with the income and poverty observations corresponding to calendar years 1969–88. Years mentioned in the text refer to survey years. We use a balanced panel in order to abstract from attrition issues that may differentially affect the estimation methods. |

3 | |

4 | |

5 | Models (F) and (G) are conceptually the same, except that assumption (12) is not fully exploited in model (G), and the initial spells continue to be modeled separately from the structural spells even after m periods. Because model (G) does not nest the other specifications, a Wald test is not possible; however, the improvement in the log quasi-likelihood value from model (F) to (G) is huge, suggesting assumption (12) is also problematic. |

6 | These statistics are intended for indicative purposes, as their distribution is unclear, and using critical values from a chi-square distribution with “$\mathrm{df}$” degrees of freedom is likely to result in a conservative test (under-rejection). For models estimated by maximizing the complete likelihood function, Chernoff and Lehmann (1954) and Moore (1977) among others show that the critical value is somewhere between chi-squares with q and $q-l$ degrees of freedom, where q is the number of free terms in the test statistic and l is the number of estimated parameters. Andrews (1988) extends this to non-dynamic models estimated by maximizing the conditional likelihood function given covariates. However, these results do not apply to dynamic models estimated by maximizing a quasi-likelihood function using clustered samples. For convenience, we report the “maximum degrees of freedom” (i.e., q). |

7 | We exclude the DBR2 model, and show just the DBR1 and MSD1 model predictions here, as the DBR model predictions are comparatively similar. |

8 | The latter is the average person-average, which gives equal weight to each person who has a fresh spell. |

9 | There are exceptions across the age subsamples, and the MSD1 predicted rate is close to the actual for three of the five age groups. Note that because of differences in the outcome history and covariate values at the exit times, the models are not designed to fit the sample averages. |

Mean (Standard Error) | |
---|---|

Person-years | |

Aged 0–5 | 0.025 (0.0005) |

Aged 6–17 | 0.225 (0.001) |

Aged 18–24 | 0.204 (0.001) |

Aged 25–54 | 0.420 (0.002) |

Aged 55+ | 0.126 (0.001) |

Female head | 0.336 (0.001) |

Black head | 0.582 (0.002) |

Poor (${y}_{it}$) | 0.353 (0.001) |

Transition (${c}_{it}$) | 0.177 (0.001) |

No. person-years | 104,960 |

Persons | |

Transitions | 3.35 (0.032) |

No. persons | 5248 |

Spells | |

Duration of all spells | 4.59 (0.033) |

Duration of initial spells | 7.11 (0.084) |

Duration of fresh spells | 3.84 (0.032) |

No. spells | 22,849 |

DBR1 | DBR2 | ||||
---|---|---|---|---|---|

IC | Strl | IC1 | IC2 | Strl | |

Variables | |||||

${y}_{it-1}$ | 2.191 (0.040) | 2.426 (0.147) | 1.859 (0.051) | ||

${y}_{it-2}$ | 0.942 (0.050) | ||||

${y}_{it-1}{y}_{it-2}$ | 0.039 (0.075) | ||||

Aged 0–5 | 0.253 (0.104) | 0.549 (0.093) | 0.218 (0.100) | 0.163 (0.136) | 0.328 (0.104) |

Aged 6–17 | 0.601 (0.075) | 0.560 (0.043) | 0.558 (0.073) | 0.165 (0.077) | 0.449 (0.037) |

Aged 18–24 | 0.397 (0.113) | 0.186 (0.030) | 0.349 (0.111) | −0.13 (0.126) | 0.109 (0.029) |

Aged 55+ | −0.131 (0.172) | 0.272 (0.048) | −0.105 (0.170) | −0.236 (0.180) | 0.288 (0.043) |

Female Head | 1.076 (0.139) | 0.935 (0.047) | 1.085(0.138) | 0.744 (0.156) | 0.874 (0.045) |

Black Head | 1.429 (0.145) | 0.620 (0.055) | 1.48 (0.144) | 0.855 (0.155) | 0.527 (0.048) |

Random effects (mass points and probabilities) | |||||

${\nu}_{1}$ | −2.178 (0.157) | −3.186 (0.057) | −2.121 (0.163) | −2.686 (0.178) | −3.206 (0.057) |

${\nu}_{2}$ | −0.654 (0.150) | −1.604 (0.073) | −0.767 (0.155) | −1.607 (0.171) | −1.926 (0.076) |

${\pi}_{1}$ | 0.638 (0.023) | 0.640 (0.032) | |||

Statistics | |||||

No. persons | 5248 | 5248 | |||

No. years | 20 | 20 | |||

Log QL | −45,060.2 | −44,110.7 |

MSD1 | MSD2 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Initial | Initial Spells | Fresh Spells | Initial | Fresh Spells | ||||||

State | Entry | Exit | Entry | Exit | State | Entry | Exit | |||

Variables | ||||||||||

$1({d}_{it}\ge 2)$ | −0.070 | −0.425 | −0.507 | −0.562 | −0.509 | −0.544 | ||||

(0.161) | (0.168) | (0.070) | (0.068) | (0.075) | (0.069) | |||||

$1({d}_{it}\ge 3)$ | −0.490 | 0.234 | −0.365 | −0.188 | −0.373 | −0.163 | ||||

(0.185) | (0.191) | (0.093) | (0.094) | (0.096) | (0.097) | |||||

$1({d}_{it}\ge 4)$ | 0.007 | −0.046 | −0.186 | −0.290 | −0.228 | −0.321 | ||||

(0.192) | (0.202) | (0.115) | (0.119) | (0.114) | (0.120) | |||||

$1({d}_{it}\ge 5)$ | 0.003 | −0.028 | −0.051 | −0.169 | −0.068 | −0.258 | ||||

(0.209) | (0.236) | (0.130) | (0.142) | (0.117) | (0.135) | |||||

$1({d}_{it}\ge 6)$ | 0.257 | −0.051 | −0.309 | −0.056 | 0.029 | −0.055 | ||||

(0.162) | (0.203) | (0.113) | (0.136) | (0.091) | (0.114) | |||||

Aged 0–5 | 0.156 | 0.332 | −0.068 | −0.340 | −0.049 | 0.036 | −0.206 | −1.124 | ||

(0.100) | (0.121) | (0.120) | (0.200) | (0.226) | (0.168) | (0.429) | (0.646) | |||

Aged 6–17 | 0.512 | −0.098 | −0.440 | 0.511 | −0.188 | 0.497 | 0.431 | −0.272 | ||

(.071) | (.063) | (.061) | (.059) | (.057) | (.069) | (.053) | (.055) | |||

Aged 18–24 | 0.314 | 0.513 | 0.401 | 0.111 | 0.169 | 0.073 | 0.310 | 0.197 | ||

(0.111) | (0.060) | (0.067) | (0.044) | (0.044) | (0.100) | (0.040) | (0.040) | |||

Aged 55+ | −0.078 | 0.047 | −0.289 | 0.366 | −0.287 | 0.150 | 0.327 | −0.289 | ||

(0.170) | (0.080) | (0.157) | (0.061) | (0.060) | (0.153) | (0.052) | (0.060) | |||

Female Head | 1.086 | 0.906 | −0.732 | 0.881 | −0.817 | 1.195 | 0.838 | −0.802 | ||

(0.140) | (0.087) | (0.121) | (0.067) | (0.065) | (0.135) | (0.058) | (0.066) | |||

Black Head | 1.529 | 0.246 | −0.871 | 0.713 | −0.498 | 1.379 | 0.407 | −0.493 | ||

(0.142) | (0.084) | (0.118) | (0.072) | (0.063) | (0.144) | (0.059) | (0.063) | |||

Random effects (mass points and probabilities) | ||||||||||

${\nu}_{1}$ | −2.148 | −2.403 | 0.332 | −2.610 | 1.161 | −2.592 | −2.154 | 1.003 | ||

(0.181) | (0.126) | (0.193) | (0.102) | (0.094) | (0.179) | (0.099) | (0.109) | |||

${\nu}_{2}$ | −0.901 | −1.678 | −0.615 | −1.143 | 0.121 | −1.180 | −0.870 | 0.029 | ||

(0.147) | (0.134) | (0.147) | (0.093) | (0.076) | (0.180) | (0.130) | (0.088) | |||

${\pi}_{1}$ | 0.590 | 0.677 | ||||||||

(0.042) | (0.049) | |||||||||

Statistics | ||||||||||

No. persons | 5248 | 5248 | ||||||||

No. years | 20 | 16 | ||||||||

Log QL | −43,444.5 | −34,621.8 |

Model | Duration p Dependence | Heterogeneity | Log QL | No. Parms | ${\mathbf{H}}_{0}/{\mathbf{H}}_{\mathbf{A}}$ | Wald Statistic | $\mathbf{df}$ |
---|---|---|---|---|---|---|---|

A (DBR1) | 1 | Opposite | −45,060.2 | 18 | |||

B | 1 | Flexible | −44,968.7 | 25 | A / B | 118.6 | 7 |

C (DBR2) | 2 | Opposite | −44,110.7 | 29 | A / C | 817.5 | 11 |

D | 2 | Flexible | −44,003.7 | 43 | C / D | 167.2 | 14 |

E | 6 | Opposite | −43,832.9 | 45 | C / E | 184.9 | 16 |

F | 6 | Flexible | −43,709.2 | 59 | E / F | 199.8 | 14 |

G (MSD1) | 6 | Flexible | −43,444.5 | 61 |

No. | No. Transitions | ||||||||
---|---|---|---|---|---|---|---|---|---|

Years Poor | 0 | 1 | 2 | 3 | 4+ Even | 5+ Odd | Total | ||

Actual data | |||||||||

0 | 255 | 0 | 0 | 0 | 0 | 0 | 255 | ||

1 | 0 | 201 | 735 | 0 | 0 | 0 | 936 | ||

2–5 | 0 | 232 | 315 | 291 | 526 | 168 | 1532 | ||

6–10 | 0 | 88 | 68 | 209 | 325 | 322 | 1012 | ||

11–15 | 0 | 55 | 38 | 95 | 299 | 290 | 777 | ||

16–19 | 0 | 88 | 155 | 92 | 190 | 62 | 587 | ||

20 | 149 | 0 | 0 | 0 | 0 | 0 | 149 | ||

Total | 404 | 664 | 1311 | 687 | 1340 | 842 | 5248 | ||

DBR1 predictions | |||||||||

0 | 473.2 | 0 | 0 | 0 | 0 | 0 | 473.2 | ||

1 | 0 | 123.3 | 388.6 | 0 | 0 | 0 | 511.8 | ||

2–5 | 0 | 131.4 | 320.1 | 369.0 | 580.0 | 205.1 | 1605.5 | ||

6–10 | 0 | 31.0 | 62.4 | 194.6 | 475.5 | 458.6 | 1221.9 | ||

11–15 | 0 | 23.0 | 53.7 | 131.8 | 365.8 | 274.0 | 848.2 | ||

16–19 | 0 | 49.9 | 179.2 | 79.1 | 172.2 | 37.4 | 517.7 | ||

20 | 69.9 | 0 | 0 | 0 | 0 | 0 | 69.9 | ||

Total | 543.0 | 358.5 | 1003.8 | 774.4 | 1593.4 | 975.0 | 5248.0 | ||

GOF = 973.5 ($23\mathrm{df}$) | |||||||||

DBR2 predictions | |||||||||

0 | 527.5 | 0 | 0 | 0 | 0 | 0 | 527.5 | ||

1 | 0 | 113.8 | 440.7 | 0 | 0 | 0 | 554.5 | ||

2–5 | 0 | 166.8 | 228.2 | 334.0 | 588.0 | 212.9 | 1529.9 | ||

6–10 | 0 | 68.0 | 84.0 | 208.0 | 406.1 | 402.6 | 1168.6 | ||

11–15 | 0 | 44.4 | 72.8 | 136.4 | 315.5 | 265.0 | 834.1 | ||

16–19 | 0 | 58.6 | 179.1 | 76.7 | 179.6 | 45.1 | 539.0 | ||

20 | 94.6 | 0 | 0 | 0 | 0 | 0 | 94.6 | ||

Total | 622.1 | 451.5 | 1004.7 | 755.0 | 1489.1 | 925.6 | 5248.0 | ||

GOF = 613.6 ($23\mathrm{df}$) | |||||||||

MSD1 predictions | |||||||||

0 | 336.4 | 0 | 0 | 0 | 0 | 0 | 336.4 | ||

1 | 0 | 156.4 | 625.0 | 0 | 0 | 0 | 781.4 | ||

2–5 | 0 | 217.0 | 318.9 | 306.5 | 582.9 | 167.3 | 1592.5 | ||

6–10 | 0 | 105.2 | 73.4 | 187.8 | 341.6 | 350.5 | 1058.4 | ||

11–15 | 0 | 68.5 | 54.4 | 121.5 | 288.2 | 262.0 | 794.5 | ||

16–19 | 0 | 67.6 | 175.0 | 75.2 | 188.1 | 43.1 | 548.9 | ||

20 | 136.1 | 0 | 0 | 0 | 0 | 0 | 136.1 | ||

Total | 472.5 | 614.5 | 1246.6 | 690.9 | 1400.8 | 822.8 | 5248.0 | ||

GOF = 98.9 ($23\mathrm{df}$) |

No. Spells | Actual Data | DBR1 Predictions | MSD1 Predictions | ||||||
---|---|---|---|---|---|---|---|---|---|

Initial State | Initial State | Initial State | |||||||

Not Poor | Poor | Not Poor | Poor | Not Poor | Poor | ||||

1 | 255 | 149 | 473.2 | 69.9 | 336.4 | 136.1 | |||

2 | 159 | 505 | 99.2 | 259.3 | 132.7 | 481.8 | |||

3 | 1089 | 222 | 739.0 | 264.8 | 989.7 | 257.0 | |||

4 | 214 | 473 | 230.4 | 544.0 | 203.0 | 487.9 | |||

5 | 455 | 271 | 600.0 | 370.8 | 518.5 | 265.3 | |||

6 | 129 | 398 | 211.2 | 458.3 | 158.7 | 323.3 | |||

7 | 219 | 155 | 261.8 | 225.5 | 227.4 | 175.4 | |||

8 | 81 | 139 | 84.4 | 174.2 | 79.3 | 155.6 | |||

9 | 119 | 73 | 60.9 | 61.1 | 81.8 | 79.5 | |||

10 | 31 | 47 | 14.9 | 28.5 | 31.9 | 51.0 | |||

11 | 21 | 16 | 6.2 | 6.2 | 21.2 | 22.2 | |||

12 | 6 | 10 | 1.2 | 2.5 | 8.2 | 11.7 | |||

13 | 7 | 3 | 0.5 | 0.5 | 3.5 | 4.8 | |||

14 | 0 | 1 | 0.0 | 0.0 | 1.1 | 1.9 | |||

15 | 0 | 1 | 0.0 | 0.0 | 0.4 | 0.8 | |||

16 | 0 | 0 | 0.0 | 0.0 | 0.1 | 0.2 | |||

17 | 0 | 0 | 0.0 | 0.0 | 0.0 | 0.1 | |||

Total | 2785 | 2463 | 2782.7 | 2465.4 | 2793.7 | 2454.3 | |||

GOF | 564.6 | 481.9 | 66.1 | 32.6 | |||||

($\mathrm{df}$) | (11) | (11) | (11) | (11) |

Actual Data | DBR1 prd | MSD1 prd | |||||||
---|---|---|---|---|---|---|---|---|---|

Initial | Fresh | Initial | Fresh | Initial | Fresh | ||||

Spells | Spells | Spells | Spells | Spells | Spells | ||||

Status: not poor | |||||||||

Avg spell duration | 8.23 | 4.85 | 8.43 | 4.70 | 8.10 | 4.92 | |||

No. spells | 2785 | 9277 | 2783 | 9510 | 2794 | 9266 | |||

Status: Poor | |||||||||

Avg spell duration | 5.86 | 2.72 | 4.48 | 2.96 | 5.88 | 2.67 | |||

No. spells | 2463 | 8324 | 2465 | 8684 | 2454 | 8368 |

Sample | First-Year Exit Rate | No. Years Poor | No. Transitions | |||||||
---|---|---|---|---|---|---|---|---|---|---|

from Non-Poor | Next Decade | Next Decade | ||||||||

Data: ^{a} Spells | Data: ^{b} Persons | DBR1 | MSD1 | DBR1 | MSD1 | DBR1 | MSD1 | |||

All | 0.34 | 0.25 | 0.21 | 0.30 | 3.19 | 2.38 | 1.82 | 2.86 | ||

Female head | 0.40 | 0.28 | 0.30 | 0.41 | 4.58 | 3.00 | 2.16 | 4.01 | ||

Black head | 0.38 | 0.30 | 0.25 | 0.36 | 3.91 | 2.79 | 2.01 | 3.52 | ||

Aged 0–5 | 0.24 | 0.19 | 0.24 | 0.20 | 3.81 | 2.32 | 1.99 | 2.83 | ||

Aged 6–17 | 0.37 | 0.24 | 0.26 | 0.37 | 3.86 | 2.79 | 2.00 | 3.35 | ||

Aged 18–24 | 0.29 | 0.24 | 0.20 | 0.29 | 3.07 | 2.31 | 1.80 | 2.71 | ||

Aged 25–54 | 0.34 | 0.25 | 0.16 | 0.25 | 2.72 | 2.12 | 1.69 | 2.52 | ||

Aged 55+ | 0.39 | 0.31 | 0.20 | 0.32 | 3.37 | 2.50 | 1.86 | 3.19 |

^{a}Average across all (9277) new non-poor spells;

^{b}average (person-average first-year exit rate) across all (4993) persons who have a new non-poor spell.

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## Share and Cite

**MDPI and ACS Style**

Gørgens, T.; Hyslop, D.R.
The Specification of Dynamic Discrete-Time Two-State Panel Data Models. *Econometrics* **2019**, *7*, 1.
https://doi.org/10.3390/econometrics7010001

**AMA Style**

Gørgens T, Hyslop DR.
The Specification of Dynamic Discrete-Time Two-State Panel Data Models. *Econometrics*. 2019; 7(1):1.
https://doi.org/10.3390/econometrics7010001

**Chicago/Turabian Style**

Gørgens, Tue, and Dean Robert Hyslop.
2019. "The Specification of Dynamic Discrete-Time Two-State Panel Data Models" *Econometrics* 7, no. 1: 1.
https://doi.org/10.3390/econometrics7010001