# Weather Conditions and Telematics Panel Data in Monthly Motor Insurance Claim Frequency Models

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Context and Motivation

#### 1.2. Background

#### 1.3. Objectives

#### 1.4. Data and Structure

- -
- The first category contains the variables related to the basic features of the insured, such as gender, age, vehicle power, and driver’s licence age. These variables do not vary over time, so they are implicitly included in the panel models as individual effects. In this category, we also consider the location where the insured driver has mostly driven each month, which is generally constant over time because it is usually their place of residence. Specifically, we have an indicator of the province where the driver has driven the majority of the monthly total distance.
- -
- The second category includes time-varying variables related to the drivers’ behaviour and weather information. Weather information contains data on sunshine hours, average monthly temperatures, and wind. The drivers’ information includes features such as maximum and average speed and distance travelled obtained from daily information. All variables are disaggregated by road type: highway, national road, regional road, and urban road.
- -
- Finally, the third category includes claim frequency information.

#### 1.5. Description of the Data

## 2. Methods

- -
- Behavioural variables allow us to identify the most dangerous forms of conduct.
- -
- Variables related to the weather conditions that change with the area and month where the vehicle has been driven allow us to see which environments are the most dangerous.

#### 2.1. Software and Package

#### 2.2. Models Specification and Estimation

_{t}= (λ

_{it}) is the N × 1-dimensional vector of the state variables in period t, ${\u03f5}_{t}=\left({\u03f5}_{it}\right)$ is the N × 1-dimensional vector of the error terms and ${X}_{t}$ is the N × K matrix where the i-th row ${x}_{it}=({x}_{ikt})$ consists of the K covariates observed by individual i in month t.

## 3. Materials and Results

#### 3.1. Poisson Panel Data Model with Random Effects for the Number of Claims

#### 3.2. Poisson Panel Data Model with Fixed Effects for the Number of Claims

#### 3.3. Poisson Panel Data Model with Random Effects for At-Fault Third-Party Liability Claims

#### 3.4. Poisson Panel Data Model with Fixed Effects for At-Fault Third-Party Liability Claims

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Descriptive statistics: mean and standard deviation (SD) for 20,784 policyholders in seven observed months in 2018 and 2019.

Mean | Month 6 | Month 7 | Month 8 | Month 9 | Month 10 | Month 11 | Month 12 | Total |
---|---|---|---|---|---|---|---|---|

Telematics variables | ||||||||

Tele_km_total_urban | 361.745 | 353.987 | 345.384 | 331.785 | 322.242 | 303.230 | 283.157 | 328.790 |

Tele_speed_mean_urban | 19.422 | 19.540 | 19.833 | 20.077 | 20.340 | 20.476 | 20.384 | 20.010 |

Tele_speed_max_highway | 131.781 | 131.704 | 131.712 | 131.793 | 131.557 | 131.121 | 130.341 | 131.430 |

Weather conditions | ||||||||

Wind | 9.560 | 10.021 | 10.259 | 10.297 | 10.251 | 10.246 | 10.343 | 10.140 |

Temperature | 14.404 | 14.407 | 16.202 | 18.138 | 18.959 | 18.386 | 16.986 | 16.783 |

Sun | 6.682 | 7.219 | 8.118 | 8.702 | 8.706 | 8.289 | 7.740 | 7.922 |

Responses | ||||||||

Claims | 0.076 | 0.073 | 0.068 | 0.068 | 0.064 | 0.061 | 0.068 | 0.068 |

At-fault third-party liability | 0.004 | 0.004 | 0.004 | 0.004 | 0.003 | 0.002 | 0.003 | 0.004 |

S.D. | Month 6 | Month 7 | Month 8 | Month 9 | Month 10 | Month 11 | Month 12 | Total |

Telematics variables | ||||||||

Tele_km_total_urban | 310.455 | 297.183 | 293.006 | 275.432 | 265.530 | 245.056 | 230.455 | 230.455 |

Tele_speed_mean_urban | 5.524 | 5.598 | 5.715 | 5.764 | 5.928 | 6.056 | 6.075 | 6.075 |

Tele_speed_max_highway | 15.583 | 15.527 | 15.618 | 16.002 | 16.131 | 16.737 | 16.934 | 16.934 |

Weather conditions | ||||||||

Wind | 1.802 | 1.877 | 1.734 | 1.666 | 1.657 | 1.739 | 1.864 | 1.864 |

Temperature | 4.936 | 5.311 | 5.827 | 5.894 | 5.864 | 6.024 | 6.080 | 6.080 |

Sun | 2.078 | 2.348 | 2.432 | 2.337 | 2.319 | 2.372 | 2.364 | 2.364 |

Responses | ||||||||

Claims | 0.491 | 0.481 | 0.467 | 0.466 | 0.440 | 0.407 | 0.443 | 0.443 |

At-fault third-party liability | 0.064 | 0.065 | 0.063 | 0.061 | 0.058 | 0.049 | 0.059 | 0.059 |

**Table A2.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values, for the total number of claims (left) and at-fault third-party liability (TPL) claims (right) per month and per insured based on the seven observed months in 2018 and 2019. The models consider the average number of sunshine hours per day as explanatory variable describing weather conditions.

Variable | All Claims | Only At-Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −2.688 | <0.001 | - | - | −6.049 | <0.001 | - | - |

lag(claims *) | −0.445 | <0.001 | −0.569 | <0.001 | −1.177 | 0.261 | −4.357 | <0.001 |

Sun | 0.001 | 0.979 | 0.004 | 0.438 | −0.025 | 0.209 | −0.003 | 0.887 |

Tele_km_total_urban ** | 0.116 | 0.018 | −0.223 | <0.001 | 0.789 | <0.001 | 0.015 | 0.951 |

Tele_speed_mean_urban | −0.020 | <0.001 | −0.020 | <0.001 | −0.022 | 0.014 | −0.028 | 0.043 |

Tele_speed_max_highway | 0.003 | <0.001 | −0.001 | 0.328 | 0.006 | 0.048 | −0.011 | 0.045 |

σ_{τ} | 0.158 | <0.001 | - | - | 1.000 | 0.159 | - | - |

AIC | 58,920.18 | 26,836.56 | 5787.84 | 1426.87 |

**Table A3.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values, for the total number of claims (left) and at-fault third-party liability (TPL) claims (right) per month and per insured based on the seven observed months in 2018 and 2019. The models consider the average monthly temperature as explanatory variable describing weather conditions.

Variable | All Claims | Only At-Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −2.680 | <0.001 | - | - | −6.022 | <0.001 | - | - |

lag(claims *) | −0.444 | <0.001 | −0.570 | <0.001 | −1.177 | 0.261 | −4.357 | <0.001 |

Temperature | −0.001 | 0.788 | 0.002 | 0.331 | −0.013 | 0.116 | −0.002 | 0.857 |

Tele_km_total_urban ** | 0.116 | 0.017 | −0.224 | <0.001 | 0.797 | <0.001 | 0.016 | 0.946 |

Tele_speed_mean_urban | −0.020 | <0.001 | −0.020 | <0.001 | −0.021 | 0.016 | −0.028 | 0.043 |

Tele_speed_max_highway | 0.003 | <0.001 | −0.001 | 0.317 | 0.005 | 0.056 | −0.011 | 0.046 |

σ_{τ} | 0.158 | <0.001 | - | - | 1.004 | 0.160 | - | - |

AIC | 58,920.11 | 26,836.22 | 5786.94 | 1426.86 |

**Table A4.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values. for the total number of claims (left) and at-fault third party liability (TPL) claims (right) per month and per insured based on the seven observed months in 2018 and 2019. Monthly maximum daily average wind speed is introduced as the weather-related regressor.

Variable | ALL Claims | Only at Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −2.839 | <0.001 | - | - | −6.920 | <0.001 | - | - |

lag(claims *) | −0.444 | <0.001 | −0.569 | <0.001 | −1.187 | 0.257 | −4.358 | <0.001 |

Wind max | 0.011 | 0.038 | 0.006 | 0.279 | 0.053 | 0.010 | 0.036 | 0.165 |

Tele_km_total_urban ** | 0.120 | 0.014 | −0.218 | <0.001 | 0.788 | <0.001 | 0.039 | 0.870 |

Tele_speed_mean_urban | −0.020 | <0.001 | −0.020 | <0.001 | −0.023 | 0.009 | −0.028 | 0.045 |

Tele_speed_max_highway | 0.003 | <0.001 | −0.001 | 0.381 | 0.006 | 0.034 | −0.011 | 0.043 |

σ_{τ} | 0.158 | <0.001 | - | - | 1.008 | 0.161 | - | - |

AIC | 58,915.90 | 26,836.00 | 5782.95 | 1424.98 |

**Table A5.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values, for the total number of claims (left) and at-fault third party liability (TPL) claims (right) per month and per insured based on the seven observed months in 2018 and 2019. The models consider the monthly maximum daily average of sunshine hours as explanatory variable describing weather conditions.

Variable | All Claims | Only At Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −2.723 | <0.001 | - | - | −6.032 | <0.001 | - | - |

lag(claims *) | −0.445 | <0.001 | −0.570 | <0.001 | −1.177 | 0.261 | −4.360 | <0.001 |

Sun max | 0.005 | 0.391 | 0.009 | 0.088 | −0.022 | 0.284 | 0.009 | 0.724 |

Tele_km_total_urban ** | 0.115 | 0.019 | −0.223 | <0.001 | 0.788 | <0.001 | 0.017 | 0.943 |

Tele_speed_mean_urban | −0.020 | <0.001 | −0.020 | <0.001 | −0.022 | 0.014 | −0.029 | 0.037 |

Tele_speed_max_highway | 0.003 | 0.001 | −0.001 | 0.295 | 0.006 | 0.047 | −0.011 | 0.040 |

σ_{τ} | 0.158 | <0.001 | - | - | 1.001 | 0.160 | - | - |

AIC | 58,919.45 | 26,834.25 | 5788.27 | 1426.77 |

**Table A6.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values, for the total number of claims (left) and at-fault third party liability (TPL) claims (right) per month and per insured based on the seven observed months in 2018 and 2019. The models consider the monthly maximum daily average temperature as explanatory variable describing weather conditions.

Variable | All Claims | Only at Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −2.702 | <0.001 | - | - | −5.981 | <0.001 | - | - |

lag(claims *) | −0.445 | <0.001 | −0.570 | <0.001 | −1.176 | 0.261 | −4.357 | <0.001 |

Temperature max | 0.001 | 0.581 | 0.004 | 0.053 | −0.014 | 0.084 | −0.001 | 0.881 |

Tele_km_total_urban ** | 0.114 | 0.020 | −0.229 | <0.001 | 0.805 | <0.001 | 0.017 | 0.942 |

Tele_speed_mean_urban | −0.020 | <0.001 | −0.020 | <0.001 | −0.021 | 0.016 | −0.028 | 0.042 |

Tele_speed_max_highway | 0.003 | <0.001 | −0.001 | 0.273 | 0.006 | 0.053 | −0.011 | 0.046 |

σ_{τ} | 0.158 | <0.001 | - | - | 1.007 | 0.161 | - | - |

AIC | 58,919.88 | 26,833.42 | 5786.42 | 1426.87 |

**Table A7.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values. for the total number of claims (left) and at-fault third party liability (TPL) claims (right) per month and per insured based on the eight observed months in 2018 and 2019, i.e., month 5 is included.

Variable | All Claims | Only at Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −3.049 | <0.001 | - | - | −7.309 | <0.001 | - | - |

lag(claims *) | −0.421 | <0.001 | −0.535 | <0.001 | −1.373 | 0.184 | −4.389 | <0.001 |

Wind | 0.019 | 0.006 | 0.014 | 0.055 | 0.077 | 0.002 | 0.063 | 0.054 |

Tele_km_total_urban ** | 0.064 | 0.145 | −0.275 | <0.001 | 0.727 | <0.001 | −0.042 | 0.842 |

Tele_speed_mean_urban | 0.016 | <0.001 | −0.016 | <0.001 | −0.018 | 0.024 | −0.018 | 0.148 |

Tele_speed_max_highway | 0.004 | <0.001 | <0.001 | 0.911 | 0.008 | 0.004 | −0.007 | 0.145 |

σ_{τ} | 0.183 | <0.001 | - | - | 1.151 | 0.125 | - | - |

AIC | 70,710.83 | 34,887.78 | 6893.57 | 1886.79 |

**Table A8.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values, for the total number of claims (left) and at-fault third party liability (TPL) claims (right) per month and per insured based on the eight observed months in 2018 and 2019, i.e., month 5 is included. The models consider the average number of sunshine hours per day as explanatory variable describing weather conditions.

Variable | All Claims | Only at Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −2.842 | <0.001 | - | - | −6.356 | <0.001 | - | - |

lag(claims *) | −0.420 | <0.001 | −0.534 | <0.001 | −1.354 | 0.190 | −4.375 | <0.001 |

Sun | 0.001 | 0.834 | 0.004 | 0.425 | −0.013 | 0.462 | 0.001 | 0.944 |

Tele_km_total_urban ** | 0.059 | 0.177 | −0.283 | <0.001 | 0.731 | <0.001 | −0.081 | 0.701 |

Tele_speed_mean_urban | −0.016 | <0.001 | −0.016 | <0.001 | −0.016 | 0.050 | −0.018 | 0.149 |

Tele_speed_max_highway | 0.004 | <0.001 | >−0.001 | 0.965 | 0.007 | 0.011 | −0.007 | 0.120 |

σ_{τ} | 0.183 | <0.001 | - | - | 1.139 | 0.123 | - | - |

AIC | 70,718.47 | 34,890.80 | 6902.71 | 1890.49 |

**Table A9.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values, for the total number of claims (left) and at-fault third party liability (TPL) claims (right) per month and per insured based on the eight observed months in 2018 and 2019, i.e., month 5 is included. The models consider the average monthly temperature as explanatory variable describing weather conditions.

Variable | All Claims | Only at Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −2.817 | <0.001 | - | - | −6.283 | <0.001 | - | - |

lag(claims *) | −0.420 | <0.001 | −0.534 | <0.001 | −1.355 | 0.190 | −4.373 | <0.001 |

Temperature | −0.002 | 0.377 | <0.001 | 0.982 | −0.010 | 0.163 | −0.005 | 0.545 |

Tele_km_total_urban ** | 0.061 | 0.168 | −0.284 | <0.001 | 0.738 | <0.001 | −0.081 | 0.698 |

Tele_speed_mean_urban | −0.016 | <0.001 | −0.016 | <0.001 | −0.015 | 0.054 | −0.017 | 0.170 |

Tele_speed_max_highway | 0.004 | <0.001 | >−0.001 | 0.987 | 0.007 | 0.012 | −0.007 | 0.136 |

σ_{τ} | 0.183 | <0.001 | - | - | 1.141 | 0.123 | - | - |

AIC | 70,717.74 | 34,891.44 | 6901.30 | 1890.13 |

**Table A10.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values. for the total number of claims (left) and at-fault third party liability (TPL) claims (right) per month and per insured based on the seven observed months in 2018 and 2019. Interactions are included.

Variable | All Claims | Only At Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −3.342 | <0.001 | - | - | −6.197 | 0.009 | - | - |

lag(claims *) | −0.445 | <0.001 | −0.569 | <0.001 | −1.203 | 0.251 | −4.370 | <0.001 |

Wind | 0.061 | 0.295 | 0.015 | 0.809 | −0.011 | 0.961 | −0.042 | 0.872 |

Tele_km_total_urban ** | 0.083 | 0.752 | −0.335 | 0.244 | 0.847 | 0.253 | −0.983 | 0.375 |

Tele_speed_mean_urban | −0.007 | 0.609 | <0.001 | 0.977 | −0.067 | 0.183 | −0.061 | 0.295 |

Tele_speed_max_highway | 0.005 | 0.295 | −0.004 | 0.419 | 0.005 | 0.762 | −0.012 | 0.551 |

Wind*Tele_km_total_urban ** | 0.004 | 0.883 | 0.011 | 0.678 | −0.005 | 0.941 | 0.103 | 0.324 |

Wind*Tele_speed_mean_urban | −0.001 | 0.291 | −0.002 | 0.143 | 0.004 | 0.388 | 0.003 | 0.563 |

Wind*Tele_speed_max_highway | >−0.001 | 0.799 | <0.001 | 0.538 | <0.001 | 0.922 | <0.001 | 0.931 |

σ_{τ} | 0.158 | <0.001 | - | - | 1.006 | 0.161 | - | - |

AIC | 58,916.38 | 26,837.19 | 5784.357 | 1426.86 |

**Table A11.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values, for the total number of claims (left) and at-fault third party liability (TPL) claims (right) per month and per insured based on the seven observed months in 2018 and 2019. The models consider the average number of sunshine hours per day as explanatory variable describing weather conditions. Interactions are included.

Variable | All Claims | Only at Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −3.655 | <0.001 | - | - | −7.481 | <0.001 | - | - |

lag(claims *) | −0.445 | <0.001 | −0.569 | <0.001 | −1.207 | 0.250 | −4.366 | <0.001 |

Sun | 0.120 | 0.004 | 0.141 | 0.001 | 0.156 | 0.350 | 0.377 | 0.047 |

Tele_km_total_urban ** | −0.366 | 0.012 | −0.717 | <0.001 | 0.336 | 0.468 | −0.759 | 0.239 |

Tele_speed_mean_urban | −0.002 | 0.807 | 0.002 | 0.833 | 0.005 | 0.872 | 0.030 | 0.397 |

Tele_speed_max_highway | 0.009 | <0.001 | 0.005 | 0.059 | 0.014 | 0.157 | 0.005 | 0.644 |

Sun*Tele_km_total_urban ** | 0.059 | <0.001 | 0.061 | 0.001 | 0.055 | 0.306 | 0.093 | 0.205 |

Sun*Tele_speed_mean_urban | −0.002 | 0.012 | −0.003 | 0.005 | −0.003 | 0.350 | −0.007 | 0.083 |

Sun*Tele_speed_max_highway | −0.001 | 0.013 | −0.001 | 0.009 | −0.001 | 0.389 | −0.002 | 0.123 |

σ_{τ} | 0.158 | <0.001 | - | - | 0.970 | 0.151 | - | - |

AIC | 58,904.37 | 26,818.82 | 5791.49 | 1426.45 |

**Table A12.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values, for the total number of claims (left) and at-fault third party liability (TPL) claims (right) per month and per insured based on the seven observed months in 2018 and 2019. The models consider the average monthly temperature as explanatory variable describing weather conditions. Interactions are included.

Variable | All Claims | Only at Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −2.952 | <0.001 | - | - | −6.480 | <0.001 | - | - |

lag(claims *) | −0.445 | <0.001 | −0.570 | <0.001 | −1.190 | 0.256 | −4.367 | <0.001 |

Temperature | 0.015 | 0.358 | 0.022 | 0.201 | 0.016 | 0.812 | 0.090 | 0.236 |

Tele_km_total_urban ** | −0.071 | 0.573 | −0.460 | 0.001 | 0.650 | 0.102 | −0.494 | 0.362 |

Tele_speed_mean_urban | −0.003 | 0.640 | −0.002 | 0.764 | 0.022 | 0.372 | 0.039 | 0.194 |

Tele_speed_max_highway | 0.003 | 0.123 | −0.001 | 0.758 | 0.003 | 0.720 | −0.008 | 0.400 |

Temperature*Tele_km_total_urban ** | 0.011 | 0.111 | 0.014 | 0.054 | 0.008 | 0.711 | 0.029 | 0.301 |

Temperature*Tele_speed_mean_urban | −0.001 | 0.005 | −0.001 | 0.005 | −0.003 | 0.061 | −0.004 | 0.014 |

Temperature*Tele_speed_max_highway | <0.001 | 0.992 | >−0.001 | 0.795 | <0.001 | 0.740 | >−0.001 | 0.743 |

σ_{τ} | 0.158 | <0.001 | - | - | 0.988 | 0.157 | - | - |

AIC | 58,916.62 | 26,831.48 | 5789.44 | 1425.98 |

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**Figure 1.**Plots of the average maximum speed per individual from month 6 to month 12 in terms of whether or not individuals reported claims at some point over the entire period. From left to right there is a graph for the maximum speed on highways, national roads, and urban roads, respectively.

**Figure 2.**Plots of the average mean speed per individual from month 6 to month 12 in terms of whether or not individuals reported claims at some point over the entire period. From left to right there is a graph for the average mean speed on highways, national roads and urban roads, respectively.

**Figure 3.**Percentage monthly frequency of claims (light grey) and percentage of at-fault third-party liability frequency of monthly claims from month 6 to month 12. The monthly average temperature in degrees Celsius (scale on the left axis) and monthly average wind speed in km/h (scale also on the left axis) are plotted, together with the average monthly sunshine daily hours (scale on the right axis).

**Table 1.**Description of the variables finally considered in the analysis. They are all measured from the policyholder’s monthly information. Spanish data set, seven complete months observed in 2018 and 2019.

Variable | Description |
---|---|

Telematics variables | |

Tele_km_total_urban | Kilometres travelled on urban roads |

Tele_speed_mean_urban | Mean speed on urban roads |

Tele_speed_mean_national | Mean speed on national roads |

Tele_speed_mean_highway | Mean speed on the highways |

Tele_speed_max_urban | Maximum speed on urban roads |

Tele_speed_max_national | Maximum speed on national roads |

Tele_speed_max_highway | Maximum speed on the highways |

Weather conditions | |

Temperature | Average monthly temperature (degrees Celsius) |

Sun | Average number of hours of sunshine per day during the month |

Wind | Average monthly wind speed (km/h) |

Response | |

Claims | Number of claims (of any type) |

At-fault claims | Number of at-fault third-party liability claims |

**Table 2.**Cross-table with the total number of claims (rows) and at-fault third-party liability (TPL) claims (columns) per insured during the seven observed months in 2018 and 2019.

All Claims | At-Fault TPL Claims | Total | |||
---|---|---|---|---|---|

0 | 1 | 2 | 3 or More | ||

0 | 16,235 | 0 | 0 | 0 | 16,235 |

1 | 2694 | 0 | 0 | 0 | 2694 |

2 | 475 | 0 | 0 | 0 | 475 |

3 or more | 868 | 502 | 9 | 1 | 1380 |

Total | 20,272 | 502 | 9 | 1 | 20,784 |

**Table 3.**Cross-table with the parameter estimates of random effects and fixed-effects Poisson panel data models, with p-values, for the total number of claims (left) and at-fault third-party liability (TPL) claims (right) per month and per insured based on the seven observed months in 2018 and 2019.

Variable | All Claims | Only At-Fault TPL Claims | ||||||
---|---|---|---|---|---|---|---|---|

Random Effects | Fixed Effects | Random Effects | Fixed Effects | |||||

Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | Parameter | p-Value | |

Intercept | −2.942 | <0.001 | - | - | −7.226 | <0.001 | - | - |

lag(claims *) | −0.445 | <0.001 | −0.569 | <0.001 | −1.200 | 0.252 | −4.370 | <0.001 |

Wind | 0.022 | 0.003 | 0.016 | 0.066 | 0.087 | 0.001 | 0.078 | 0.033 |

Tele_km_total_urban ** | 0.123 | 0.012 | −0.212 | <0.001 | 0.789 | <0.001 | 0.071 | 0.764 |

Tele_speed_mean_urban | −0.020 | <0.001 | −0.020 | <0.001 | −0.024 | 0.007 | −0.028 | 0.041 |

Tele_speed_max_highway | 0.004 | <0.001 | −0.001 | 0.407 | 0.007 | 0.021 | −0.011 | 0.047 |

σ_{τ} | 0.158 | <0.001 | - | - | 1.008 | 0.161 | - | - |

AIC | 58,911.58 | 26,833.79 | 5779.10 | 1420.46 |

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## Share and Cite

**MDPI and ACS Style**

Reig Torra, J.; Guillen, M.; Pérez-Marín, A.M.; Rey Gámez, L.; Aguer, G.
Weather Conditions and Telematics Panel Data in Monthly Motor Insurance Claim Frequency Models. *Risks* **2023**, *11*, 57.
https://doi.org/10.3390/risks11030057

**AMA Style**

Reig Torra J, Guillen M, Pérez-Marín AM, Rey Gámez L, Aguer G.
Weather Conditions and Telematics Panel Data in Monthly Motor Insurance Claim Frequency Models. *Risks*. 2023; 11(3):57.
https://doi.org/10.3390/risks11030057

**Chicago/Turabian Style**

Reig Torra, Jan, Montserrat Guillen, Ana M. Pérez-Marín, Lorena Rey Gámez, and Giselle Aguer.
2023. "Weather Conditions and Telematics Panel Data in Monthly Motor Insurance Claim Frequency Models" *Risks* 11, no. 3: 57.
https://doi.org/10.3390/risks11030057