6.3 Mortality and the Economy
6.3.3 A GLM-Based Test for a Relationship Between
All of the tests for a relationship between economic fluctuations and mortality de- scribed in the previous section are clearly similar in nature.
• They all test for a significant relationship between an economic variable (in most cases the unemployment rate) and a standardised measure of mortality,
• after allowing, either directly or indirectly, for other, non-economic variables that are likely to have an impact on the mortality rate (for example, age and sex), and
• this is generally done through the use of linear regression models.
The model developed in this section is based on these same principles, but now we use a GLM framework.
In Section 4.2.4 a Poisson GLM of the form given in Equation (4.10) was pro- posed. This model can be rewritten as:
ln E(Dx,t) Ec x,tmsx,t =η; (6.17) where Dx,t Ec
x,tmsx,t is the mortality ratio for lives aged x in year t and η is a linear combination of the covariates. This same model structure is used again in this section.
This model is similar in form to those given in Equations (6.10) and (6.12) except that the mortality ratio is used in place of the age-adjusted mortality rate and a Poisson error structure is assumed instead of a normal error distribution. The mor- tality ratio and the age-adjusted mortality rate are two different but similar methods of standardising mortality data so that the mortality among different populations
Dependency Relationships 85 can be compared. Throughout the previous sections of this thesis, the mortality ratio has been used as the standardised measure of mortality and it is used here, again, for consistency. Furthermore, it is more appropriate to assume the number of deaths follows a Poisson distribution than a normal distribution, as the observed number of deaths is a discrete, non-negative random variable, whereas the normal distribution would imply that it was a continuous random variable with range over the entire number line (nevertheless, in a large sample, we expect there to be little difference between the Poisson and normal models, by the Central Limit Theorem). Based on the literature review provided in Section 6.3.2, the possibility of a relationship between mortality and the unemployment rate should be investigated. Furthermore, in later parts of this thesis, a number of other economic variables are considered (the price inflation rate, the share price index growth rate, the share dividend yield rate, the long-term interest rate, the short-term interest rate, and the property yield), and it is desirable to see if a significant relationship exists between mortality and any of these variables. In theory, this could be done by including all of the economic and non-economic variates in the linear predictor of the GLM given in Equation (6.17) and then testing to determine whether the economic variates are significant, given the presence of the non-economic variates in the model. However, from Table 5.19 it is clear that some of the economic variates are highly correlated with each other, and including all of them together in the one model is likely to give rise to multicollinearity problems. Thus, it is necessary to reduce the set of economic variates to be included in the model.
Another point that should be considered is that the existence of a statistically sig- nificant relationship between two variables does not necessarily imply that a causal relationship exists between them. It is easy to see how a causal relationship could exist between unemployment rates, interest rates, or price inflation and mortality, since the main source of income for the majority of the adult population is wages and salaries (see ABS (2007) for further details); most adults are exposed to movements in interest rates through either debts or investments; and all adults are directly exposed to movements in prices. However, it is more difficult to see how a causal re- lationship could exist between commercial property yields, share price growth rates or dividend yields and mortality.
According to Parlett and Rossiter (2004), in 2002, only 10.3% of Australian households owned investment property, and it seems likely that the majority of these properties were residential, rather than commercial properties (statistics were not available to confirm this), since residential property is typically cheaper than commercial property. Furthermore, according to ASX (2007), in 2006, 46% of Aus- tralian adults owned shares (down from 55% in 2004), either directly or indirectly, with people from households with a greater combined income more likely to in-
86 Dependency Relationships vest than those from households with a lower combined income. However, in spite of this, shares make up a relatively small proportion of the total assets of most Australian households. According to ABS (2006), in 2003–04, among “high wealth households8”, shares only made up 5.1% of total household assets, on average, with the percentage falling to 0.9% for “medium wealth households9” and 0.8% for “low wealth households10” For all households, on average, shares made up 3.4% of total household assets. With such low exposures to these assets by the population in gen- eral, it is difficult to see how a causal link could exist between the mortality ratio and fluctuations in the property yield, share price index growth rate or dividend yield.
Based on the above argument, it was decided to test only for relationships be- tween the mortality ratio and the unemployment rate, price inflation rate or interest rates. As was mentioned previously, however, in order to avoid multicollinearity problems, these four variables should not all be included in a single model. To de- termine which economic variates should be included in the model, a principal com- ponents analysis of the economic variates was conducted11 (using the correlation matrix, rather than the covariance matrix and implemented using S-Plus). Figure 6.1 presents the scree plot12 for the economic data described in Section 5.4. From this graph, it can be seen that over 95% of the variance of the data is explained by the first two principal components. The variates with the greatest absolute weight- ings in each of these components were, respectively, the short-term interest rate and the unemployment rate. Consequently, it was decided to include only the short-term interest rate and the unemployment rate in the linear predictor when testing for a significant relationship between mortality and economic fluctuations.
To test for a relationship between the mortality rate and any of these variables, two different linear predictors were considered in fitting the GLM in Equation (6.17) to this data:
Model 1:
η =α+βZ; (6.18)
8
High wealth households are defined as those in the highest household net worth quintile.
9Median wealth households are defined as those in the third household net worth quintile. 10Low wealth households are defined as those in the lowest household net worth quintile. 11
For more information on principal components analysis, see Rencher (1998) or Lattin et al. (2003).
12
Ascree plot is a diagnostic tool used in principal components analysis. It plots “the variance accounted for by each principal component in order from largest to smallest” (Lattin et al. (2003, p.113)).
Dependency Relationships 87
Figure 6.1: A Scree Plot for the Economic Data.
Variances 0.0 0.5 1.0 1.5 2.0 2.5
Comp.1 Comp.2 Comp.3 Comp.4
0.673 0.961 0.996 1 Model 2: η=α+βZ +γ1V +γ2V ×T ype+γ3V ×Sex +γ4V ×Age+γ5V ×Duration; (6.19) whereZ is a vector of the covariates: age, sex, duration and policy type, including all two-way interactions, and V is a vector of the economic covariates: unemployment rate and the short-term interest rate. The sex and policy type variables were both treated as categorical covariates, and as usual,α,βand theγ’s are model parameters to be estimated. Duration was fitted as a continuous variate for durations of less than 10 years, and as an indicator variable for durations of 10 years or greater. The reasoning behind this is given in Section 6.3.4.
Non-economic variables were allowed for explicitly in the model, rather than by fitting the model to subsets of the data, as this makes better use of the available data. Only age, sex, duration and policy type were used as (non-economic) explanatory variables and not race, educational attainment, income per capita or year, as were used in Ruhm (2000, 2003), Gerdtham and Ruhm (2002) and Neumayer (2004). We omitted the latter variables because they are not commonly collected by insurers and so are unlikely to be included in a pricing or reserving model.
Each of these models was fitted to the data and then, because the two models are nested, the residual deviances of the models were used to compare them, as was
88 Dependency Relationships done in Section 6.2. The results are presented in Section 6.3.5.