An easy way to create duration variables in binary cross-sectional time-series data

Abstract

In cross-sectional time-series data with a dichotomous dependent variable, failing to account for duration dependence when it exists can lead to faulty inferences. A common solution is to include duration dummies, polynomials, or splines to proxy for duration dependence. Because creating these is not easy for the common practitioner, I introduce a new command, mkduration, that is a straightforward way to generate a duration variable for binary cross-sectional time-series data in Stata. mkduration can handle various forms of missing data and allows the duration variable to easily be turned into common parametric and nonparametric approximations.

Keywords

st0621 mkduration binary cross-sectional time series event history duration

1 Introduction

It is well known that when one models a dichotomous dependent variable in binary cross- sectional time-series data (B-CSTS), failing to account for duration dependence—the phenomenon by which the occurrence of an event at time t in unit i may make the reoccurrence of an event at a future time point more or less likely—can have severe consequences for estimation (Beck, Katz, and Tucker 1998). At best, failing to model such dependence may induce serial autocorrelation, leading to standard errors that are anticonservative. At worst, it can produce omitted variable bias even if the included regressors are unrelated to the omitted duration dependence.

A common approach recommended by Beck, Katz, and Tucker (1998) when dealing with B-CSTS data—when the occurrence of events is relatively rare—is to estimate a logistic regression (LR) with duration dummies to proxy for any duration dependence.¹ While alternative approaches exist (cf. Zorn [2000]; Box-Steffensmeier and Jones [2004]; or fitting random-effects parametric survival models using the xtstreg command in Stata), in general, the Beck, Katz, and Tucker (1998) approach is appealing because of its simplicity; as of April 2020, their article had nearly 2,900 citations.² Despite this popularity, creating the duration variable from a dichotomous dependent variable is not straightforward. For one, such time-since-last-event variables are not as simple as techniques such as including the lag of a series. Moreover, missing data can lead to additional complications because it is unknown whether an event has occurred during this period. And creating nonparametric approximations of duration dependence through tools such as splines is not straightforward (see Carter and Signorino [2010]).

In this article, I introduce mkduration, an easy way to generate duration variables for B-CSTS data in Stata using a single command. It can also handle missing data—in effect interpolating or extrapolating—depending on what the user specifies. Moreover, it can produce several functional forms of duration commonly used in the literature. In the sections that follow, I first discuss duration dependence in the context of B-CSTS data, and then I introduce the mkduration command. I illustrate the utility of this command through an example using data from Philips (2020).

2 Duration dependence with B-CSTS

Consider a simple B-CSTS dataset in long form, like the one shown in table 1. y_it is a dichotomous dependent variable for unit i observed at time t that does not occur relatively often.³ This is commonly modeled using a generalized linear model with a logit link to account for the dichotomous nature of the dependent variable (Beck, Katz, and Tucker 1998). The problem that arises is with duration dependence, which exists if the Pr(y_it ) = 1 changes based on how long it has been since the last event (or entry into the sample). This is shown by the duration variable in table 1, which records the time since the last event in the data.⁴

Table 1.

Durations in B-CSTS data

Unit	Time	Event (y_it )	Duration
1	1	0	1
1	2	0	2
1	3	1	3
1	4	0	1
1	5	1	2
2	1	0	1
2	2	1	2
2	3	0	1
2	4	0	2
2	5	0	3
3	1	0	1
3	2	1	2
3	3	1	1
3	4	0	1
3	5	0	2
…	…	…	…

Failing to model duration dependence implies a constant hazard rate, meaning that the probability of event reoccurrence does not change over time. In other words, events are independent from one another. In real-world data, however, such an assumption is probably almost always violated. For instance, duration dependence has been argued to exist in topics as varied as conflict onsets (Clare 2010; Bapat and Zeigler 2016), pursuit of nuclear weapons (Way and Weeks 2014), and firm-level bankruptcies (Hillegeist et al. 2004). Failing to model duration dependence when it exists can lead to many problems. At best, the estimator will be inefficient, and the standard errors will be incorrect; at worst, biased and inconsistent estimates may result because failing to include duration when it exists is a form of omitted variable bias (Beck, Katz, and Tucker 1998).

Beck, Katz, and Tucker (1998) note that a straightforward way to continue to model B-CSTS data in the logit framework—but also account for duration dependence—is to simply create a time-since-last-event variable (that is, the duration variable shown in table 1), which is then turned into a vector of dummy variables. These are then included in the logit generalized linear model⁵

\Pr (y_{i t} = 1 | x_{i t}, k_{i t}) = \frac{1}{1 + \exp {- (x_{i t} β + k_{i t} γ)}}

Now, in addition to the standard covariates (x _it is a matrix of k regressors with dimensions (N · T ) × k with k coefficients β), κ_it is now included and is a matrix of duration dummies with coefficients γ.⁶ An illustration of these dummy variables is shown in table 2. For instance, κ ₁ = 1 if the duration variable is equal to 1, κ ₂ = 1 if the duration variable is equal to 2, and so on.

Table 2.

Duration dummy variables

Unit	Time	Event (y_it )	Duration	κ ₁	κ ₂	κ ₃
1	1	0	1	1	0	0
1	2	0	2	0	1	0
1	3	1	3	0	0	1
1	4	0	1	1	0	0
1	5	1	2	0	1	0
2	1	0	1	1	0	0
2	2	1	2	0	1	0
2	3	0	1	1	0	0
2	4	0	2	0	1	0
2	5	0	3	0	0	1
3	1	0	1	1	0	0
3	2	1	2	0	1	0
3	3	1	1	1	0	0
3	4	0	1	1	0	0
3	5	0	2	0	1	0
…	…	…	…	…	…	…

Because it is likely that some κ_it may be perfectly collinear with y_it , separation is likely to lead to estimation issues when using maximum likelihood; this will force Stata to drop any collinear dummy variables. To alleviate this, Carter and Signorino (2010) advocate for a simple approach of incorporating duration, duration squared, and duration cubed in the model instead of either splines (another approach that Beck, Katz, and Tucker [1998] recommend) or dummy variables. While some consider κ_it to be nuisance parameters (Beck 2010), others contend that it is important to discuss and interpret the estimated dependence function as a feature of theoretical interest (Carter and Signorino 2010; Williams 2016).⁷ Regardless, both lines of reasoning agree that it is necessary to include some functional form of duration in the model to account for duration dependence.

One difficulty with implementing the advice above is that incorporating some functional form of duration dependence requires the creation of a duration variable, which is far less straightforward than taking lags or including time dummies in standard cross- sectional time-series data with a continuous dependent variable. This difficulty is compounded if some data are missing or if units enter or leave the sample at different times.

Below, I show a straightforward way to create a duration variable, even in the presence of missing data, using the command mkduration. The resulting variable can easily be included in the model through the use of dummy variables, basis functions—most commonly polynomials—or nonparametric approximations such as splines.

3 Accounting for dependence with mkduration

3.1 Syntax

The command syntax is

This command requires the specification of a single variable, eventvar, which is a dichotomous dependent variable where “1” indicates the presence of some event occurring at time t for unit i and “0” indicates the absence of this event; note that eventvar will become the dependent variable in the logit model. The data must also first be set using xtset.

3.2 Options

dname( string ) names the duration variable generated by mkduration. By default, the duration variable is called duration.

spline( string ) creates a spline to model duration dependence. The resulting spline variables in the model will include a spl1, spl2, and so on suffix. string can be one of the following:

spline(linear) creates a linear spline. In effect, a piecewise linear model is run across the duration series, which can then be included in the LR model. By default, five knots are used, meaning that the duration will be split into six equal segments based on percentiles of the data; the first knot will be placed at about the 16.66 percentile, the next at the 33.33 percentile, and so on. The assumption with a linear spline is that duration has a linear effect on the probability of an event occurring between the knots, although this effect may differ across knots. Increasing the number of knots allows for a more flexible approximation of the relationship but has the potential to overfit the data.

spline(cubic) creates a restricted cubic spline that creates a linear function before the first knot, a cubic polynomial function from the second and subsequent knots, while data past the final knot are assumed linear. By default, five knots are used and are placed along percentiles of duration, as recommended by Harrell (2015), which is the default in Stata.

nknots( # ) defines the number of knots to include. The default is nknots(5) for both the linear and cubic splines. # can range from 3 to 7 knots.⁸ Fewer knots are often more efficient but offer less flexibility in modeling duration dependence. Greater numbers of knots increase flexibility, at the cost of (potentially) decreasing efficiency and overfitting duration. nknots() can only be specified with spline().

There are four additional options to account for various types of missing data. By default, the duration variable is created for all nonmissing values of the event variable; any gaps in the middle of the series are handled by replacing the duration variable with missings until the next event occurs.

strict takes a more stringent approach than the default at the beginning of the series (they both account for gaps in the middle of the series in the same way). Duration data may be left-censored, in that events may have occurred before the start of the sample. As such, the true underlying duration at the start of the sample is unknown, although it is quite common to ignore this and instead start the duration at t = 1 (that is, the default setting). Adding strict will leave the duration missing until the first observed event occurs because only then is the underlying duration truly known.

force forces the creation of duration data when gaps in time are present for one or more units in the middle of the series. By default, force fills in gaps only in the middle of a series. Specifying force will fill in any missing gaps in the duration variable that are preceded and succeeded by nonmissing values.⁹ To do this, it must assume that no event occurred during the gap. This is described in greater detail in the example below.

lfill and rfill fill in any missing values at the start and end of the series, respectively. lfill, rfill, or both can be specified only in addition to force.

When one specifies lfill in addition to force, the duration variable will start when the first time variable is observed, regardless of whether the event variable is missing. As with force, it is assumed that no events have occurred during this period.

rfill is similar to lfill but will fill in duration in all available time points after the event variable is observed. For instance, if an event variable is not observed after t = 10, but the dataset includes time up to t = 15, including both rfill and force will tell mkduration to fill in the duration variable all the way to t = 15. As with force and lfill, it is assumed that no events have occurred during this period.

4 Example

For an applied example, I use data from Philips (2020), who examines whether state governments in India time land reforms to occur just before state elections to appeal to voters. Passage of legislative land reforms is a relatively rare event, occurring in just 48 of the 515 state-years under observation, meaning that these B-CSTS data may exhibit some form of duration dependence; one intuitive expectation is that passage of reform in one year makes additional land reform passage quite unlikely in the near term.¹⁰ To start, we will create the duration variable using the dependent variable, landref, and then summarize it using a histogram. We will be sure to first xtset the data.

The histogram is shown in figure 1. Duration is a monotonically decreasing function with a maximum duration of 32 years, meaning that no land reform occurred during 32 years “at risk” for one of the states.

Figure 1.

Histogram of duration

By default, the generated duration variable is called duration, although this can be changed using dname(). The duration variable can quickly be turned into dummy variables κ_it using Stata’s categorical variable capabilities when specifying the model. In addition to the duration dummies (i. duration), predictors in the random-effects logit model include the following dichotomous variables: the year before an election (f1elecdum); the election year (elecdum); whether the state’s government is single-party dominant (onep_dom); whether the state’s government is a multiparty system of left, center, and right parties (multp_leftcenright); whether the state’s government is a two-party system of left and center parties (twop_leftcenter); and whether the state’s government is a two-party system of center and right parties (twop_centerright). There is also a continuous variable of the percentage of citizens in a state that own no land (noland).

The results are shown in table 3, model 1. As is clear from the table, because of perfect collinearity (no land reform ever occurs for many of the duration-years), many duration dummies fall out of the model, reducing the number of observations. For the other covariates, it appears that land reform is more likely in the year before a state legislative election. Land reform is also more likely in multiparty competitive political systems than it is for two-party or single-party competition.

Table 3.

Different approaches to account for duration

	(1) Duration Dummies	(2) Cubic Polynomial	(3) Cubic Spline	(4) Linear Spline	(5) No Duration
Year before election	0.98**(0.39)	0.89**(0.36)	1.04***(0.38)	0.98***(0.38)	0.86**(0.36)
Election year	0.15 (0.45)	-0.01 (0.42)	0.35 (0.43)	0.20 (0.44)	0.03 (0.41)
Single-party dominant	0.31 (0.42)	0.25 (0.40)	0.34 (0.41)	0.33 (0.42)	0.43 (0.39)
Multiparty: Left-Center-Right	2.19**(0.97)	1.29 (0.87)	1.55* (0.91)	1.89**(0.93)	1.45* (0.86)
Two-party: Left-Center	0.74 (0.45)	0.75* (0.43)	0.75* (0.44)	0.73 (0.45)	0.96**(0.42)
Two-party: Center-Right	-0.08 (0.69)	-0.21 (0.67)	-0.19 (0.67)	-0.20 (0.68)	-0.43 (0.66)
Percentage owning no land	0.03 (0.02)	0.03 (0.02)	0.03 (0.02)	0.03 (0.02)	0.04* (0.02)
κ ₂	-0.08 (0.65)
κ ₃	0.29 (0.62)
κ ₄	1.11* (0.58)
κ ₅	0.86 (0.64)
κ ₆	-0.34 (0.88)
κ ₈	-0.81 (1.13)
κ ₁₁	-0.79 (1.13)
κ ₁₂	0.13 (0.91)
κ ₁₃	-0.16 (1.14)
κ ₁₄	0.35 (0.92)
κ ₁₆	0.24 (1.19)
κ ₁₇	1.54 (0.95)
Duration		-0.08 (0.21)
Duration2		0.00 (0.02)
Duration3		-0.00 (0.00)
Spline			0.90***(0.34)	-0.20 (0.62)
Spline			-30.07***(10.05)	0.72**(0.29)
Spline			58.93***(19.88)	-0.83***(0.23)
Spline			-35.20***(12.42)	0.49**(0.20)
Spline				-0.18 (0.14)
Constant	-3.31***(0.69)	-2.81***(0.69)	-4.66***(0.94)	-3.08***(1.14)	-3.36***(0.43)
	389	515	515	515	515
States	15	15	15	15	15
LR-Test (versus model 5)		3.95	13.28**	20.91***
χ ²	25.21***	21.09***	28.88***	33.00***	20.35***

NOTE: Dependent variable is equal to 1 if state i enacted land reform in year t, 0 otherwise. LR-test results not available for model 1 because of sample-size difference. Random-effects LR with standard errors in parentheses. Two-tailed tests.

* p < 0.10, ** p < 0.05, *** p < 0.01.

Instead of including duration dummies, we can use the recommendation of Carter and Signorino (2010) and create a cubic polynomial term of duration using Stata’s interaction capabilities:

The results using a cubic polynomial are shown in model 2 in table 3. None of the duration coefficients are statistically significant, which suggests they may not be needed. The results remain similar to those in model 1, although multiparty government is no longer statistically significant, while two-party governments (specifically, one left party and one centrist party) are associated with an increased probability of land reform, although this effect is statistically significant only at the 10% level.

As an additional functional form choice, users can choose to model duration using splines:

Each command shows (respectively) a restricted cubic spline, using the default of five knots, and a piecewise linear spline with four knots (meaning that duration will be partitioned into quintiles). Note, too, that by specifying dname( string ), we can change the name of the resulting duration spline variables that are created. All generated spline variables have an spl suffix followed by the spline number, for example, dlinear_spl1, dlinear_spl2, and so on. The results for the cubic and linear splines are shown in models 3 and 4 in table 3. Most of the splines are statistically significant in both models. Last, in model 5, a model without any form of duration is shown. Likelihood-ratio tests at the bottom of table 3 indicate that both the cubic and linear splines are preferred to the model with no duration dependence. Compared with incorporating a linear spline of duration (model 4), the model omitting a duration function (model 5) finds evidence that the percentage owning no land and two-party left-center governments make land reform more likely.

Given that interpreting the various approaches to duration in table 3 is not straight- forward, we can instead plot the dummy variables, splines, and cubic polynomials to better understand the underlying nature of duration dependence in the data (Carter and Signorino 2010; Williams 2016). Here we fit each model and use margins to generate the predicted probability of conflict across duration, setting all other covariates to their modes or means:

The resulting plot of these durations is shown in figure 2. The estimated duration for land reform appears to be nonmonotonic for all specifications except the cubic polynomial; the predicted probability of land reform increases through the first four or five years after a previous land reform and then tends to decline. For the dummy and spline durations, there appears to be another period about a dozen years after a previous land reform in which reform once again becomes more likely. After about 20 years after land reform passage, there is only a small probability of an additional land reform. Figure 2 also shows how the inclusion of the duration dummies—especially in the context of separation—can result in “bumpy” durations; moreover, in this example, we are unable to obtain predicted probabilities beyond κ ₁₇ because of separation issues.

Figure 2.

Different durations generated using mkduration NOTE: Figure shows the predicted probability of land reform across duration, holding other covariates at their mean or modal value, using models 1–4 from table 3, with 95% confidence intervals shown.

Missing data

One issue with duration dependence has to do with missing data. I discuss three types specific to B-CSTS data, using a stylized example shown in table 4. First, the event variable may be missing at the beginning of the series; for instance, in table 4, the event series is not observed for t = 1, 2. Second, data may be missing at the end of the series. In table 4, data are not observed for time points t = 17 to t = 20. Third, data could be missing during the interval in which the series is observed; the event in table 4 is not observed for the interval t = 7, 8, although prior and future values are observed. mkduration has several options for handling “left” (missing at the beginning of the series), “interval” (missing in the middle of the series), and “right” (missing at the end of the series) forms of missing data.

Table 4.

mkduration and approaches to missing data

Unit	Time	Event	Default	strict	force	force & lfill	force & rfill	force, lfill, & rfill
1	1	.	.	.	.	1	.	1
1	2	.	.	.	.	2	.	2
1	3	0	1	.	1	3	1	3
1	4	0	2	.	2	4	2	4
1	5	1	3	.	3	5	3	5
1	6	0	1	1	1	1	1	1
1	7	.	2	2	2	2	2	2
1	8	.	.	.	3	3	3	3
1	9	1	.	.	4	4	4	4
1	10	0	1	1	1	1	1	1
1	11	0	2	2	2	2	2	2
1	12	0	3	3	3	3	3	3
1	13	1	4	4	4	4	4	4
1	14	0	1	1	1	1	1	1
1	15	1	2	2	2	2	2	2
1	16	0	1	1	1	1	1	1
1	17	.	2	2	2	2	2	2
1	18	.	.	.	.	.	3	3
1	19	.	.	.	.	.	4	4
1	20	.	.	.	.	.	5	5

By default, mkduration will start the duration at the first nonmissing event variable and create a missing duration value for any instances between the missing values until after the next observed event. This is shown by the “Default” column. Note, too, that by default, the duration variable will revert to missing one period after the last observed value.¹¹

A stricter interpretation might lead us to replace the duration with missings until the first event is actually observed because events may have occurred before the start of the sample (that is, left-censoring). Using the strict option will not start the duration variable until after the first event has been observed. For instance, because it is unknown how long it has been since the last event for nonmissing values at t = 3 through t = 5 in table 4, these are coded as missing in the strict column.

If the user is comfortable assuming that no events have occurred during the unobserved middle time period t = 7, 8, he or she can use the force option to fill in observed periods that contain missing values. As shown in table 4, including this option will fill in the duration variable for time points t = 8, 9 (that is, up until the next event occurs)..

In addition to using the force option, one may use two other options. Adding lfill will start the duration at the first time period, not the first observed value of the event variable. As shown in table 4, this will start the duration at t = 1, even though the event variable is not observed until t = 3. As with force, it is assumed that no event has occurred during this time. As with lfill, one can use the option rfill to continue the duration series after the last observed event variable. In table 4, this means that values t = 18 to t = 20 are filled in, even though the last observed event variable is at t = 16. As with force and lfill, it is assumed that no events are occurring during this time. Last, one can use the force, lfill, and rfill options together to fill in left, interval, and right forms of missingness.

5 Conclusion

In this article, I have introduced a new command, mkduration, for a simple, less error-prone way to create a duration variable in B-CSTS data when duration dependence is suspected. Replicating an example application that uses B-CSTS data, I have shown that this command allows users to easily account for duration dependence any way they choose, such as dummies, splines, or polynomials. Moreover, depending on the additional assumptions users are willing to make, mkduration can easily account for dependence in the context of missing data in different ways.

Supplemental Material

Supplemental Material, st0621 - An easy way to create duration variables in binary cross-sectional time-series data

Supplemental Material, st0621 for An easy way to create duration variables in binary cross-sectional time-series data by Andrew Q. Philips in The Stata Journal

Footnotes

6 Acknowledgments

I thank the editor and an anonymous reviewer for their thoughtful comments and suggestions. Inspiration to write this program came from students in panel-data courses at CU Boulder and the IPSA-USP Summer School held in S˜ao Paulo, Brazil. Despite this, all errors and omissions are my own.

7 Programs and supplemental materials

To install a snapshot of the corresponding software files as they existed at the time of publication of this article, type

Notes

References

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. 1998. Taking time seriously: Time-series-cross-section analysis with a binary dependent variable. American Journal of Political Science 42: 1260–1288.

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. 2004. Event History Modeling: A Guide for Social Scientists. Cambridge: Cambridge University Press.

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2010. Ideological fractionalization and the international conflict behavior of parliamentary democracies. International Studies Quarterly 54: 965–987. https://doi.org/10.1111/j.1468-2478.2010.00622.x.

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Jr . 2015. Regression Modeling Strategies: With Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis. 2nd ed. Cham, Switzerland: Springer.

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. 2004. Assessing the probability of bankruptcy. Review of Accounting Studies 9: 5–34. https://doi.org/10.1023/B:RAST.0000013627.90884.b7.

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2020. Just in time: Political policy cycles of land reform. Politics 40: 207–226. https://doi.org/10.1177/0263395719859459.

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. 2014. Making it personal: Regime type and nuclear proliferation. American Journal of Political Science 58: 705–719. https://doi.org/10.1111/ajps.12080.

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Supplementary Material

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0.00 MB

Unit	Time	Event (y_it )	Duration
1	1	0	1
1	2	0	2
1	3	1	3
1	4	0	1
1	5	1	2
2	1	0	1
2	2	1	2
2	3	0	1
2	4	0	2
2	5	0	3
3	1	0	1
3	2	1	2
3	3	1	1
3	4	0	1
3	5	0	2
…	…	…	…

Unit	Time	Event (y_it )	Duration	κ ₁	κ ₂	κ ₃
1	1	0	1	1	0	0
1	2	0	2	0	1	0
1	3	1	3	0	0	1
1	4	0	1	1	0	0
1	5	1	2	0	1	0
2	1	0	1	1	0	0
2	2	1	2	0	1	0
2	3	0	1	1	0	0
2	4	0	2	0	1	0
2	5	0	3	0	0	1
3	1	0	1	1	0	0
3	2	1	2	0	1	0
3	3	1	1	1	0	0
3	4	0	1	1	0	0
3	5	0	2	0	1	0
…	…	…	…	…	…	…

Unit	Time	Event	Default	strict	force	force & lfill	force & rfill	force, lfill, & rfill
1	1	.	.	.	.	1	.	1
1	2	.	.	.	.	2	.	2
1	3	0	1	.	1	3	1	3
1	4	0	2	.	2	4	2	4
1	5	1	3	.	3	5	3	5
1	6	0	1	1	1	1	1	1
1	7	.	2	2	2	2	2	2
1	8	.	.	.	3	3	3	3
1	9	1	.	.	4	4	4	4
1	10	0	1	1	1	1	1	1
1	11	0	2	2	2	2	2	2
1	12	0	3	3	3	3	3	3
1	13	1	4	4	4	4	4	4
1	14	0	1	1	1	1	1	1
1	15	1	2	2	2	2	2	2
1	16	0	1	1	1	1	1	1
1	17	.	2	2	2	2	2	2
1	18	.	.	.	.	.	3	3
1	19	.	.	.	.	.	4	4
1	20	.	.	.	.	.	5	5

Unit	Time	Event (y_it )	Duration
1	1	0	1
1	2	0	2
1	3	1	3
1	4	0	1
1	5	1	2
2	1	0	1
2	2	1	2
2	3	0	1
2	4	0	2
2	5	0	3
3	1	0	1
3	2	1	2
3	3	1	1
3	4	0	1
3	5	0	2
…	…	…	…

Unit	Time	Event (y_it )	Duration	κ ₁	κ ₂	κ ₃
1	1	0	1	1	0	0
1	2	0	2	0	1	0
1	3	1	3	0	0	1
1	4	0	1	1	0	0
1	5	1	2	0	1	0
2	1	0	1	1	0	0
2	2	1	2	0	1	0
2	3	0	1	1	0	0
2	4	0	2	0	1	0
2	5	0	3	0	0	1
3	1	0	1	1	0	0
3	2	1	2	0	1	0
3	3	1	1	1	0	0
3	4	0	1	1	0	0
3	5	0	2	0	1	0
…	…	…	…	…	…	…

Unit	Time	Event	Default	strict	force	force & lfill	force & rfill	force, lfill, & rfill
1	1	.	.	.	.	1	.	1
1	2	.	.	.	.	2	.	2
1	3	0	1	.	1	3	1	3
1	4	0	2	.	2	4	2	4
1	5	1	3	.	3	5	3	5
1	6	0	1	1	1	1	1	1
1	7	.	2	2	2	2	2	2
1	8	.	.	.	3	3	3	3
1	9	1	.	.	4	4	4	4
1	10	0	1	1	1	1	1	1
1	11	0	2	2	2	2	2	2
1	12	0	3	3	3	3	3	3
1	13	1	4	4	4	4	4	4
1	14	0	1	1	1	1	1	1
1	15	1	2	2	2	2	2	2
1	16	0	1	1	1	1	1	1
1	17	.	2	2	2	2	2	2
1	18	.	.	.	.	.	3	3
1	19	.	.	.	.	.	4	4
1	20	.	.	.	.	.	5	5

Unit	Time	Event (y_it )	Duration
1	1	0	1
1	2	0	2
1	3	1	3
1	4	0	1
1	5	1	2
2	1	0	1
2	2	1	2
2	3	0	1
2	4	0	2
2	5	0	3
3	1	0	1
3	2	1	2
3	3	1	1
3	4	0	1
3	5	0	2
…	…	…	…

Unit	Time	Event (y_it )	Duration	κ ₁	κ ₂	κ ₃
1	1	0	1	1	0	0
1	2	0	2	0	1	0
1	3	1	3	0	0	1
1	4	0	1	1	0	0
1	5	1	2	0	1	0
2	1	0	1	1	0	0
2	2	1	2	0	1	0
2	3	0	1	1	0	0
2	4	0	2	0	1	0
2	5	0	3	0	0	1
3	1	0	1	1	0	0
3	2	1	2	0	1	0
3	3	1	1	1	0	0
3	4	0	1	1	0	0
3	5	0	2	0	1	0
…	…	…	…	…	…	…

Unit	Time	Event	Default	strict	force	force & lfill	force & rfill	force, lfill, & rfill
1	1	.	.	.	.	1	.	1
1	2	.	.	.	.	2	.	2
1	3	0	1	.	1	3	1	3
1	4	0	2	.	2	4	2	4
1	5	1	3	.	3	5	3	5
1	6	0	1	1	1	1	1	1
1	7	.	2	2	2	2	2	2
1	8	.	.	.	3	3	3	3
1	9	1	.	.	4	4	4	4
1	10	0	1	1	1	1	1	1
1	11	0	2	2	2	2	2	2
1	12	0	3	3	3	3	3	3
1	13	1	4	4	4	4	4	4
1	14	0	1	1	1	1	1	1
1	15	1	2	2	2	2	2	2
1	16	0	1	1	1	1	1	1
1	17	.	2	2	2	2	2	2
1	18	.	.	.	.	.	3	3
1	19	.	.	.	.	.	4	4
1	20	.	.	.	.	.	5	5