Lecture 8 (Duration models) Flashcards
Briefly summarize the main points regarding durations models.
Why do we use duration models? What type of models do we use? What are the problems?
We use duration models when we like to look at duration outcomes such as unemployment duration. We can then specify a full model using MLE, but generally, in applied work, we do the partial likelihood (?) around our cox Regression model. But then, we need to keep in mind all the problems with unobserved heterogeneity. To be able to handle the problem with unobserved heterogeneity, we need to set up a mixed proportional hazard model, which is very painful to do in practice.
What are duration models?
Duration models are used when we have duration data as our outcome (dependent) variable. That is, something that measures the time some subject spends in a certain state, e.g., time in unemployment, time to employment (for refugees) etc.
What are the three main reasons to use duration models?
Three main reasons to use duration models (rather than usually linear OLS):
-
Censuring
- The duration outcome $T$, is often right [censured], i.e., we do not observe it after some point in time, only that it exists.
- We are interested in exit probabilities
- E.g., how exit-probability from unemployment changes over time and across individuals
- The independent variable $x,$ might change over time.
- Unemployment insurance benefits typically change with time in unemployment (e.g. at benefit exhaustion) and people may marry/divorce.
This can not be meaningfully implemented in a regression mode
- Unemployment insurance benefits typically change with time in unemployment (e.g. at benefit exhaustion) and people may marry/divorce.
Define the hazard rate and explain what it is?
\lambda(t) = Pr(T=t|T\geq t)
The probability of an exit in time $t$, given that the subject/individual has survived until up until the time $t$. See lecture notes for definitions in continuous time.
This hazard rate is a function of time and does thus change with time.
What different kinds of behaviors can a hazard rate exhibit?
- Constant hazard rate
- The likelihood of getting expelled is constant over time
- Duration dependent
- Hazard rate changes with time
- Positive duration dependence. Probability of termination increases with time. E.g., a football tournament gets increasingly harder the more teams you beat.
- Negative dependence
- Hazard rate changes with time
Define and explain “Survival rate”
$$
S(t) = Pr(T\geq t)
$$
This refers to the probability that the duration exceeds a specific value. E.g., “what is the probability of being unemployed longer than 4 weeks?”.
What is censuring?
We have two types of censoring.
- Left censuring: We do not observe the beginning of a spell.
- Right censuring: The date of the spell termination is not observed.
- We might only observe individuals under two years and do not know what happens after.
In this course, we mainly mention right censoring.
T_i = \min(T^_i, C^_i)
What are the different types of hazard models discussed in this class?
There are many ways of estimating hazard models. We mainly look at parametric and semi parametric (cox regression).
Parametric models:
The key thing here is that we make an explicit assumption regarding how our independent variable $x$ affects our duration outcome variable $T$. That is, we assume that the hazard has a particular form/distribution, e.g., exponential, logistic, or Weibull. Different distributions will implicitly assume different behaviors for the hazard rate, thus we could easily miss-specify our model while it is simple to estimate. If we choose the wrong distribution, we have miss-specification and thus bias.
Cox.regression:
This is a semi-parametric alternative to specifying a particular functional form for the survival model. This provides some protection against misspecification.
The idea is that we specify this “proportional model”, where we separate the impact of the baseline hazard on exits, and we then only model the specific order of when the spell ends. With this, we derive our partial function, where we get rid of the baseline hazard. We don’t need to assume any particular distribution!
Discuss the problem of duration dependence and unobserved heterogeneity in hazard models
we most likely have some sort of duration dependence in our hazard model. One big problem in duration models is if we have duration dependence, then might have unobserved heterogeneity. E.g.,
Studying unemployment, the most employable workers will disappear quickly from the unemployment pool.
That is, as time goes on, employable people will be selected out of the unemployment pool whilst less employable individuals will be left. Looking at our data, one might conclude that unemployment has a negative duration dependence, while in reality there is unobserved heterogeneity that is causing dynamic selection. This will affect our estimate for $\beta$. Unobserved heterogeneity will create problems even though we have $cov(x,e)=0.$
What problem will we have in duration models with dynamic treatment effects?
Unobserved heterogeneity in duration analysis will creat problems in estimating a treatment effect even though we have good randomization. The randomization will only be valid at time t. But as time goes and individuals exit, treatment and control groups will likely start to differ. The treatment estimate thus gets confounded when $t>1$. We will thus have violations agains SUTVA.
Explain the intuition in cox regression
See notion
How can we overcome the problem with unobserved heterogeneity in hazard models?
One way to try to solve the issues with unobserved heterogeneity (dynamic selection) is to estimate a mixed proportional hazard model. Here we explicitly model the unobserved error $v$:
$$
\lambda(t|x;\beta) = \lambda_0(t)\phi(x,\beta)v
$$
These models are however somewhat complicated to estimate in reality.
Another thing we can do if we have a problem with unobserved heterogeneity is that we instead can study the survival rate. However, The survival rate doesn’t say anything about the treatment effect dynamics. Then, we need to look at the hazard rate instead.
