9.2 Generalized linear models for reserving

Generalized Linear Models (GLMs) are a powerful tool for actuarial reserving. They extend ordinary linear regression to handle non-normal distributions, making them ideal for modeling insurance data like claim counts and amounts. GLMs consist of three key components: response variable distribution, linear predictor function, and link function.

GLMs offer flexibility in modeling different types of insurance data and can incorporate external factors that influence claims development. They also provide a framework for quantifying uncertainty in reserve estimates. However, it's crucial to carefully consider distributional assumptions and be aware of potential limitations like sensitivity to outliers and computational complexity.

Components of GLMs

• Generalized Linear Models (GLMs) extend ordinary linear regression to accommodate response variables with non-normal distributions
• GLMs consist of three main components: the response variable distribution, the linear predictor function, and the link function
• Understanding these components is crucial for applying GLMs effectively in actuarial reserving and other domains

Response variable distribution

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• Specifies the probability distribution of the response variable (claim counts, claim amounts, etc.)
• Common distributions include Poisson (for count data), Gamma (for continuous positive data), and Binomial (for binary data)
• The choice of distribution depends on the nature of the response variable and the underlying assumptions about its variability
• Examples: Poisson distribution for modeling claim counts, Gamma distribution for modeling average claim amounts

Linear predictor function

• Represents the systematic component of the model, relating the explanatory variables to the expected value of the response variable
• Combines the effects of multiple predictors through a linear combination of their values and associated coefficients
• Allows for the inclusion of categorical and continuous predictors, as well as interactions between them
• Example: $\eta = \beta_0 + \beta_1 \times \text{Development Period} + \beta_2 \times \text{Accident Period}$

Link function

• Connects the linear predictor to the expected value of the response variable, ensuring that the model's predictions are compatible with the response variable's distribution
• Common link functions include log (for Poisson and Gamma), logit (for Binomial), and identity (for Normal)
• The choice of link function depends on the response variable distribution and the desired interpretation of the model coefficients
• Examples: log link for Poisson and Gamma models, logit link for Binomial models

Model structure for reserving

• GLMs provide a flexible framework for modeling the development of claims over time, allowing for the incorporation of multiple factors that influence the reserving process
• The model structure typically includes development periods and accident periods as key factors, along with their interaction

Development periods as factors

• Represent the time elapsed since the occurrence of a claim until its settlement or reporting
• Captured as categorical variables in the model, with each level corresponding to a specific development period (e.g., 0-12 months, 12-24 months, etc.)
• Allow for the modeling of changes in claim development patterns over time
• Example: Development periods 1, 2, 3, ..., n as factor levels in the model

Accident periods as factors

• Represent the time period in which a claim occurred, typically measured in years or quarters
• Captured as categorical variables in the model, with each level corresponding to a specific accident period
• Allow for the modeling of differences in claim frequency and severity across accident periods
• Example: Accident years 2010, 2011, 2012, ..., 2020 as factor levels in the model

Interaction between development and accident periods

• Captures the potential variation in claim development patterns across different accident periods
• Allows for the modeling of changes in claim settlement speeds or reporting lags over time
• Interaction terms in the model enable the estimation of development factors specific to each accident period
• Example: Interaction term $\text{Development Period} \times \text{Accident Period}$ in the linear predictor

Model fitting and estimation

• Once the model structure is specified, the next step is to estimate the model parameters using the available data
• Maximum likelihood estimation (MLE) is the most common approach for fitting GLMs

Maximum likelihood estimation

• Involves finding the parameter values that maximize the likelihood of observing the given data under the assumed model
• Requires the specification of the log-likelihood function, which depends on the chosen response variable distribution and link function
• MLE provides consistent and asymptotically efficient parameter estimates under certain regularity conditions
• Example: Maximizing the Poisson log-likelihood to estimate the coefficients of a Poisson GLM

Iterative weighted least squares

• An alternative estimation approach that is equivalent to MLE for GLMs
• Iteratively solves a weighted least squares problem, where the weights are updated based on the current parameter estimates and the variance function of the response distribution
• Provides a computationally efficient way to fit GLMs, especially for large datasets
• Example: Using iteratively reweighted least squares (IRLS) to fit a Gamma GLM

Deviance and goodness of fit

• Deviance measures the discrepancy between the fitted model and a saturated model that perfectly fits the data
• Calculated as twice the difference in log-likelihoods between the saturated model and the fitted model
• Provides an overall assessment of the model's goodness of fit, with lower deviance indicating better fit
• Can be used to compare nested models and assess the significance of individual predictors
• Example: Comparing the deviance of a full model to that of a reduced model to assess the significance of the removed predictors

Over-dispersed Poisson model

• In some cases, the variability in the response variable may exceed what is expected under the assumed distribution, leading to over-dispersion
• The over-dispersed Poisson model accounts for this extra variability by introducing a scale parameter

Variance proportional to mean

• In the standard Poisson model, the variance is equal to the mean, but in the over-dispersed Poisson model, the variance is proportional to the mean
• The proportionality constant is called the scale parameter (φ), which is estimated from the data
• The variance of the response variable is given by $\text{Var}(Y) = \phi \times \mathbb{E}(Y)$
• Example: If the scale parameter is estimated to be 2, the variance of the response variable is twice its mean

Scale parameter estimation

• The scale parameter can be estimated using various methods, such as the method of moments or maximum likelihood
• A common approach is to estimate φ using the Pearson chi-square statistic divided by the degrees of freedom
• The estimated scale parameter is then used to adjust the standard errors of the model coefficients and the model's goodness of fit measures
• Example: Estimating the scale parameter as $\hat{\phi} = \frac{\sum_{i=1}^{n} (y_i - \hat{\mu}_i)^2 / \hat{\mu}_i}{n - p}$, where $y_i$ is the observed response, $\hat{\mu}_i$ is the predicted mean, $n$ is the sample size, and $p$ is the number of parameters

Pearson and deviance residuals

• Residuals in GLMs are calculated differently than in ordinary linear regression due to the non-normal response distributions
• Pearson residuals are standardized residuals that measure the difference between the observed and predicted values, scaled by the square root of the variance function
• Deviance residuals are based on the contribution of each observation to the model's deviance and are more sensitive to outliers than Pearson residuals
• Both types of residuals can be used to assess the model's fit and identify potential outliers or influential observations
• Example: Pearson residual for observation $i$ is calculated as $r_i^P = \frac{y_i - \hat{\mu}_i}{\sqrt{\hat{V}(\hat{\mu}_i)}}$, where $\hat{V}(\hat{\mu}_i)$ is the estimated variance function evaluated at the predicted mean

Gamma model

• The Gamma distribution is often used for modeling continuous, positive response variables, such as average claim amounts
• It is particularly useful when the variance of the response variable is expected to increase with its mean

Variance proportional to square of mean

• In the Gamma model, the variance of the response variable is proportional to the square of its mean
• The proportionality constant is the reciprocal of the shape parameter (α), which determines the distribution's skewness and variability
• The variance of the response variable is given by $\text{Var}(Y) = \frac{\mathbb{E}(Y)^2}{\alpha}$
• Example: If the shape parameter is estimated to be 4, the variance of the response variable is one-fourth of its squared mean

Shape and scale parameterization

• The Gamma distribution can be parameterized in terms of its shape (α) and scale (θ) parameters
• The shape parameter determines the distribution's skewness and variability, with larger values indicating a more symmetric and less variable distribution
• The scale parameter determines the distribution's spread, with larger values indicating a more dispersed distribution
• The mean of the Gamma distribution is given by $\mathbb{E}(Y) = \alpha \times \theta$, and its variance is $\text{Var}(Y) = \alpha \times \theta^2$
• Example: If the shape parameter is 4 and the scale parameter is 0.5, the mean of the response variable is 2, and its variance is 1

Log link function

• In the Gamma GLM, the log link function is commonly used to connect the linear predictor to the mean of the response variable
• The log link ensures that the predicted values are always positive, which is consistent with the support of the Gamma distribution
• The relationship between the linear predictor (η) and the expected value of the response variable (μ) is given by $\log(\mu) = \eta$
• Example: If the linear predictor is estimated to be 1.5, the predicted mean of the response variable is $\exp(1.5) \approx 4.48$

Model diagnostics

• After fitting a GLM, it is essential to assess the model's adequacy and identify potential issues or outliers
• Various diagnostic tools can be used to evaluate the model's fit and assumptions

Residual plots

• Plotting the residuals (Pearson or deviance) against the fitted values or explanatory variables can reveal patterns that indicate model misspecification or violation of assumptions
• Ideally, the residuals should be randomly scattered around zero, with no systematic patterns or trends
• Residual plots can help detect non-linearity, heteroscedasticity, or the presence of outliers
• Example: Plotting Pearson residuals against fitted values to check for a non-random pattern or increasing variability

Q-Q plots

• Quantile-Quantile (Q-Q) plots compare the distribution of the residuals to a theoretical distribution (e.g., standard normal)
• If the model assumptions are met, the points in the Q-Q plot should fall approximately along a straight line
• Deviations from the straight line indicate departures from the assumed distribution, such as skewness or heavy tails
• Example: Creating a Q-Q plot of the deviance residuals against the theoretical quantiles of the standard normal distribution

Outlier detection

• Outliers are observations that are poorly fit by the model or have a disproportionate influence on the parameter estimates
• Diagnostic measures, such as standardized residuals or Cook's distance, can be used to identify potential outliers
• Standardized residuals greater than 2 or 3 in absolute value may indicate outliers, while Cook's distance measures the impact of each observation on the model coefficients
• Outliers should be carefully examined and may require special treatment or removal if they are found to be erroneous or unrepresentative
• Example: Identifying observations with standardized Pearson residuals greater than 3 as potential outliers

Model selection

• When multiple GLMs are considered for a given problem, model selection techniques can be used to choose the most appropriate model
• Model selection involves balancing the model's complexity (number of parameters) with its goodness of fit

Nested vs non-nested models

• Nested models are hierarchical, with one model being a special case of the other (e.g., a model with an interaction term vs. a model without the interaction)
• Non-nested models have different predictors or structures and cannot be obtained by imposing constraints on the parameters of the other model
• Model selection techniques for nested models include likelihood ratio tests and F-tests, while non-nested models can be compared using information criteria
• Example: Comparing a model with main effects only to a model with main effects and an interaction term (nested models)

Likelihood ratio test

• The likelihood ratio test (LRT) compares the fit of two nested models by assessing the significance of the difference in their log-likelihoods
• The test statistic is calculated as -2 times the difference in log-likelihoods and follows a chi-square distribution under the null hypothesis
• A significant LRT indicates that the more complex model provides a significantly better fit than the simpler model
• Example: Using an LRT to determine if adding an interaction term to a GLM significantly improves the model's fit

Akaike information criterion (AIC)

• The AIC is an information-theoretic criterion that balances the model's fit with its complexity
• It is calculated as -2 times the log-likelihood plus 2 times the number of parameters in the model
• Lower AIC values indicate better models, considering both the goodness of fit and the model's parsimony
• The AIC can be used to compare both nested and non-nested models, making it a versatile tool for model selection
• Example: Selecting the GLM with the lowest AIC value among several candidate models with different predictors or link functions

Advantages of GLMs for reserving

• GLMs offer several benefits when applied to actuarial reserving problems, making them a powerful tool for estimating future claims liabilities

Flexibility in modeling

• GLMs can accommodate a wide range of response variable distributions, allowing for the modeling of different types of insurance data (e.g., claim counts, claim amounts)
• The choice of link function provides additional flexibility in specifying the relationship between the predictors and the response variable
• GLMs can easily incorporate categorical and continuous predictors, as well as interactions between them
• Example: Using a Poisson GLM with a log link to model claim counts and a Gamma GLM with a log link to model average claim amounts

Incorporation of external factors

• GLMs allow for the inclusion of external factors that may influence the claims development process, such as changes in regulations, economic conditions, or company policies
• By incorporating these factors as predictors in the model, actuaries can better capture the underlying drivers of claims experience and improve the accuracy of reserve estimates
• Example: Including indicators for major regulatory changes or economic recessions as predictors in a reserving GLM

Ability to quantify uncertainty

• GLMs provide a framework for quantifying the uncertainty associated with reserve estimates through the estimation of standard errors and confidence intervals
• The model's standard errors can be used to construct confidence intervals around the predicted reserves, giving a range of plausible values
• Bootstrapping or simulation techniques can be applied to GLMs to generate a distribution of reserve estimates, allowing for the assessment of reserve variability
• Example: Calculating 95% confidence intervals for the predicted reserves using the model's standard errors and a normal approximation

Limitations and considerations

• While GLMs are a powerful tool for reserving, it is important to be aware of their limitations and potential issues that may arise in their application

Appropriateness of distributional assumptions

• The validity of the GLM results depends on the appropriateness of the chosen response variable distribution and link function
• Misspecification of the distribution or link function can lead to biased parameter estimates and inaccurate reserve predictions
• It is crucial to carefully consider the nature of the data and the underlying assumptions when selecting the response distribution and link function
• Example: Using a Poisson distribution for modeling claim amounts, which are continuous and non-negative, may lead to poor model fit and biased estimates

Sensitivity to outliers

• GLMs, like other regression models, can be sensitive to outliers or influential observations
• Outliers can distort the parameter estimates and lead to over- or under-estimation of reserves
• It is important to identify and carefully examine outliers using diagnostic tools and consider their potential impact on the model results
• In some cases, it may be necessary to remove or downweight outliers to improve the model's robustness
• Example: An unusually large claim that significantly influences the model coefficients and results in overestimated reserves

Computational complexity

• As the number of predictors and the size of the dataset increase, the computational complexity of fitting GLMs can become a challenge
• The iterative nature of the estimation algorithms (e.g., IRLS) can be time-consuming for large datasets or complex model structures
• Efficient computational techniques, such as sparse matrix representations or parallel processing, may be necessary to handle large-scale reserving problems
• Example: Fitting a GLM with multiple interactions and a large number of observations may require significant computational resources and time