Hi everyone, Thank you to everyone who replies to my Mortality Projection Questions. Q1) For the extrapolation method m_x,t= a_x + (b_x)*(k_t) , i read that for a_x "parameters of the model are estimated by Maximum Likelihood Estimation by fitting the model to historical data. Then the parameters are projected forwards". I am confused as to how parameters can be fitted and then projected ? Are projections made to find m_x at ages not in the historical data (eg if historical data only covers ages 0 to 40 years old)? or are parameters found for every age x in a_x, eg ages 0 to 120 years, from the historical data and projected forward into future years whislt combining them with other effects (b_x)*(k_t)? Q2) I am unsure about the cohort term. What does this mean for calculations in exam questions/how do I use it? where are its in real life applications? why does the cohort matter? I understand the time period t over which mortality applies, eg year 1900 to 1980. However, im unsure about the cohort "x-t".
Hello 1 I am assuming you're referring to the Lee-Carter model where we have: \( ln(m_{x,t} ) = a_x + b_x * k_t + \epsilon_{x, t} \) When fitting and projecting using a Lee-Carter model, we estimate the parameters \( a_x \) and \( b_x \) which, although differ by age, are then kept constant for projection purposes. So, for example, once we estimate \( a_{50} \) from the past data, this does not change when forecasting future mortality for 50-year-olds. However, \( k_t \) reflects the time trend, which we probably do want to change as we project into the future. There are a number of ways to forecast the \( k_t \) and we look at a couple in the notes. For example, we can forecast using the random walk model by assuming that: \( k_t - k_{t-1} = \mu + \epsilon_t \) where \( \mu \) is the average change in this parameter per unit time. So, using this model, we can estimate \( k_{t+s} \) as \( k_t + s * \mu \). It is this that drives the projection, as, again, \( a_x \) and \( b_x \) are kept constant. So, in summary, if we have past data across a range of ages for a range of prior years: 1. Estimate the \( a_x \) and \( b_x \) for each age 2. Estimate the parameters corresponding to each time in the historic period, \( k_t \) 3. Project \( k_t \) for future periods to project future mortality. 2 Cohorts relate to people of a certain age at a certain point in time, which we can think about as people being born in the same year. There is evidence in the UK of cohort effects where, for example, some cohorts exhibit higher improvements than others. Another example is the 'smoking cohort' mentioned in the notes. People born between 1890 and 1910 had higher than expected mortality in middle age. Now, although it may seem like it, any model that includes age and time is not necessarily able to separate out cohort effects (even though once you know someone's age and the current year you know their year of birth). It depends on the complexity of the model. The Lee-Carter model is not complex enough to allow for cohort specific effects, even though it includes age and time. For example, imagine that there is a cohort with really really high mortality. Let's try to see if this would get picked up in our fitted Lee-Carter model: The parameter \( a_x \) is estimated as the average of the log of the mortality rates for a particular age x over all the time periods we have data for. let's assume we have data for our special cohort for when they were age x. There will be a spike in mortality for this cohort at age x, however this won't really be captured in \(a_x \) because this is 'averaged out' along with all the other cohorts. Our cohort has an impact of course, \( a_x \) is higher because of it. However, when we then use the model, this is not specific to our cohort and this spike gets 'lost'. The parameter \( k_t \) is estimated by considering the observed data at each age for a particular time t. Again, even though we have a spike for our special cohort corresponding to the appropriate age at time t, this gets 'lost' in the 'averaging out' across all the ages for which we have data at time t. The parameter \( b_x \) represents the impact of the time trend for a particular age x (e.g. how does the time trend affect 50-year-olds in general). Our special cohort will again be 'averaged out'. Now, you may say, well we do have \( b_x * k_t \), which includes both age and time, so can we make this pick out our cohort? So let's see what happens if we try to represent our really high mortality cohort. Let's say our cohort is those people born in 1980 and we'll consider the year 2030 (so our cohort is age 50). So we need \(k_{2030} * b_{50} \) to be 'large'. However, no other cohort has this spike, so we similarly need to ensure \(k_{2030} * b_{x} \) is sensible (other ages at this time) and \(k_{t} * b_{50} \) is sensible (other cohorts when they are 50). How do we do this? If we make \( k_{2030} \) large, this in turn impacts all the other ages in this year. Alternatively, we could make \( b_{50} \) large, which in turn impacts everyone whenever they are age 50. Perhaps we can make them both kind of large, which then impacts all lives in 2030 as well as all lives when they are aged 50. None of these options are what we want. You can then extend this problem to all years, how would we do it for 2031, 2032? We have the same issue. It just isn't possible to do this in the way that we would like with the given formulation. So, we can't really pick out the mortality for this cohort accurately in this case. One extension to the Lee-Carter model includes an additional cohort term to allow for better capturing of these effects. In terms of calculations to do with cohorts, it is difficult to say specifically. One possible example could be providing you with some information on the cohort parameter and other parameters and expecting you to calculate some projected mortality rates. I hope this helps. Andy
