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Evaluation of the EU proposed farm income stabilisation tool by skew normal linear mixed models

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Abstract

The European Commission has introduced new risk management tools in the rural development pillar 2 of the Common Agricultural Policy. One of them consists in providing co-financing support to mutual funds compensating farmers who experience a severe drop in their income. This paper analyses this income stabilisation tool for a region in Belgium by means of a skew normal linear mixed model. Relying on the farm accountancy data network, this analysis focuses on estimating the probability that such a fund would need to intervene and, in that case, the expected amount of each farm income compensation. The predictive distribution of future incomes given past revenues trajectory is derived and used for evaluation purposes. Particular attention is paid to additional requirements that could be imposed to the income stabilisation tool.

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Acknowledgments

The authors wish to thank the anonymous referees and the editor for numerous constructive comments which have improved earlier versions of the present paper. The financial support from the Direction Générale opérationnelle de l’Agriculture, des Ressources naturelles et de l’Environnement of the Belgian Ministry of Wallonia is gratefully acknowledged.

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Authors and Affiliations

  1. Institute of Statistics, Biostatistics and Actuarial Science (ISBA), Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Mathieu Pigeon & Michel Denuit

  2. Département de mathématiques, Université du Québec à Montréal (UQAM), Montréal, Canada

    Mathieu Pigeon

  3. Earth and Life Institute (ELI), Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Bruno Henry de Frahan

Authors
  1. Mathieu Pigeon
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  2. Bruno Henry de Frahan
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  3. Michel Denuit
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Corresponding author

Correspondence to Mathieu Pigeon.

Appendix

Appendix

1.1 Preliminary results

Lemma 1

Let \({\varvec{X}} \sim N_n(\varvec{\mu }, \varvec{\varSigma })\). Then, for \(\mathbf {a}\) \((k \times 1)\) and \(\mathbf B\) \((k \times n)\), we have

$$\begin{aligned} E\! \left[ \varPhi _k\left( \mathbf {a} + \mathbf {B}{\varvec{X}}; \varvec{\nu }, \varvec{\varOmega }\right) \right]&= \varPhi _k\left( \mathbf {a}; \varvec{\nu } - \mathbf {B}\varvec{\mu }, \varvec{\varOmega } + \mathbf {B}\varvec{\varSigma }\mathbf {B}'\right) . \end{aligned}$$

Proof

Let \({\varvec{Z}} \sim N_k(\varvec{\nu }, \varvec{\varOmega })\) and

$$\begin{aligned} E\! \left[ \varPhi _k\left( \mathbf {a} + \mathbf {B}{\varvec{X}}; \varvec{\nu }, \varvec{\varOmega }\right) \right]&= E\! \left[ P\left( {\varvec{Z}} \le \mathbf {a} + \mathbf {Bx}| {\varvec{X}} = \mathbf {x}\right) \right] \\&= E\! \left[ P\left( {\varvec{Z}} - \mathbf {Bx} \le \mathbf {a} | {\varvec{X}} = \mathbf {x}\right) \right] \\&= E\! \left[ P\left( \mathbf {U} \le \mathbf {a} | {\varvec{X}} = \mathbf {x}\right) \right] \\&= P\left( \mathbf {U} \le \mathbf {a}\right) , \end{aligned}$$

where \(\mathbf {U} \sim N_k(\varvec{\nu } - \mathbf {B}\varvec{\mu }, \varvec{\varOmega } + \mathbf {B}\varvec{\varSigma }\mathbf {B}')\).

Lemma 2

Let \({\varvec{X}} \sim N_q(\varvec{\nu }, \varvec{\varOmega })\) and \({\varvec{Y}} \sim N_p(\varvec{\mu }, \varvec{\varSigma })\). Then,

$$\begin{aligned} \phi _p\left( {\varvec{y}}; \varvec{\mu } + \mathbf {Ax}, \varvec{\varSigma }\right) \phi _q\left( \mathbf {x}; \varvec{\nu }, \varvec{\varOmega }\right)&= \phi _p\left( {\varvec{y}}; \varvec{\mu } + \mathbf {A}\varvec{\nu }, \varvec{\varSigma } + \mathbf {A}\varvec{\varOmega }\mathbf {A}'\right) \\&\quad \times \phi _q\left( \mathbf {x}; \varvec{\nu } + \varvec{\varLambda }\mathbf {A}'\varvec{\varSigma }^{-1}\left( {\varvec{y}} - \varvec{\mu } - \mathbf {A}\varvec{\nu }\right) , \varvec{\varLambda }\right) , \end{aligned}$$

where \(\varvec{\varLambda } = \left( \varvec{\varOmega }^{-1} + \mathbf {A}'\varvec{\varSigma }^{-1}\mathbf {A}\right) ^{-1}\).

Proof

Define \(\mathbf {z} = {\varvec{y}} - \varvec{\mu } - \mathbf {A}\varvec{\nu }\) and \(\mathbf {w} = \mathbf {x} - \varvec{\nu }\), we have

$$\begin{aligned}&(\mathbf {z} - \mathbf {Aw})'\varvec{\varSigma }^{-1}(\mathbf {z} - \mathbf {Aw}) + \mathbf {w}'\varvec{\varOmega }^{-1}\mathbf {w}\\&= \mathbf {z}\left( \varvec{\varSigma } + \mathbf {A}\varvec{\varOmega }\mathbf {A}'\right) ^{-1}\mathbf {z} + \left( \mathbf {w} - \varvec{\varLambda }\mathbf {A}'\varvec{\varSigma }^{-1}\mathbf {z}\right) '\varvec{\varLambda }^{-1}\left( \mathbf {w} - \varvec{\varLambda }\mathbf {A}'\varvec{\varSigma }^{-1}\mathbf {z}\right) , \end{aligned}$$

and \(|\varvec{\varSigma } + \mathbf {A}\varvec{\varOmega }\mathbf {A}'|| \varvec{\varLambda }| = |\varvec{\varSigma }||\varvec{\varOmega }|\).

1.2 Proof of Proposition 1

  1. (i)

    We have, with \(\mathbf {x} = {\varvec{y}} - {\varvec{X}}\varvec{\beta } - \varvec{\varOmega }\mathbf {t}\),

    $$\begin{aligned}&M_{{\varvec{Y}}}(\mathbf {t}) = E\! \left[ e^{\mathbf {t}'{\varvec{Y}}} \right] \\&= \int e^{\mathbf {t}'{\varvec{y}}}2^T\phi _T\left( {\varvec{y}}; {\varvec{X}}\varvec{\beta }, \varvec{\varOmega }^*\right) \varPhi _T\left( \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}({\varvec{y}} - {\varvec{X}}\varvec{\beta }); \mathbf {0}, \mathbf {I}_T - \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}\varvec{\varDelta }\right) \,d{\varvec{y}}\\&= 2^Te^{\mathbf {t}'{\varvec{X}}\varvec{\beta } + 0.5\mathbf {t}'\varvec{\varOmega }^*\mathbf {t}}\int \phi _T\left( \mathbf {x}; \mathbf {0},\varvec{\varOmega }^* \right) \varPhi _T\left( \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}(\mathbf {x} + \varvec{\varOmega }^*\mathbf {t}); \mathbf {0}, \mathbf {I}_T - \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}\varvec{\varDelta }\right) \,d\mathbf {x}\\&= 2^Te^{\mathbf {t}'{\varvec{X}}\varvec{\beta } + 0.5\mathbf {t}'\varvec{\varOmega }^*\mathbf {t}}E\! \left[ \varPhi _T\left( \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}({\varvec{X}} + \varvec{\varOmega }^*\mathbf {t}); \mathbf {0}, \mathbf {I}_T - \varvec{\varDelta }'\left( \varvec{\varOmega }^*\right) ^{-1}\varvec{\varDelta }\right) \right] , \end{aligned}$$

    where \(X \sim N_T\left( \mathbf {0}, \varvec{\varOmega }^*\right)\). By using Lemma 1, we obtain

    $$\begin{aligned}&= 2^Te^{\mathbf {t}'{\varvec{X}}\varvec{\beta } + 0.5\mathbf {t}'\varvec{\varOmega }^*\mathbf {t}}\varPhi _T\left( \varvec{\varDelta }'\mathbf {t}\right) . \end{aligned}$$
  2. (ii)

    The proof is direct from result (i).

1.3 Proof of Proposition 2

The marginal probability density function is given by (with \(\varvec{\varOmega } = \varvec{\varSigma } + \varvec{\varDelta }\varvec{\varDelta }'\))

$$\begin{aligned} f_{{\varvec{Y}}}({\varvec{y}})&= \int \limits _{\mathbb {R}^{T}} MSN({\varvec{y}}|\varvec{\varTheta }; {\varvec{X}}\varvec{\beta } + {\varvec{Z}}\varvec{\varTheta }, \varvec{\varSigma }, \varvec{\varDelta })N_T(\varvec{\varTheta }; \mathbf {0}, \varvec{\varPsi })\,\mathbf {d}\varvec{\varTheta }\\&= \int \limits _{\mathbb {R}^T} 2^T\phi _T\left( {\varvec{y}}; {\varvec{X}}\varvec{\beta } + {\varvec{Z}}\varvec{\varTheta }, \varvec{\varOmega }\right) \phi _T\left( \varvec{\varTheta }; \mathbf {0}, \varvec{\varPsi }\right) \\&\quad \times \varPhi _T\left( \varvec{\varDelta }'\varvec{\varOmega }^{-1}({\varvec{y}} - {\varvec{X}}\varvec{\beta } - {\varvec{Z}}\varvec{\varTheta }); \mathbf {0}, \mathbf {I}_T - \varvec{\varDelta }'\varvec{\varOmega }^{-1}\varvec{\varDelta }\right) \,\mathbf {d}\varvec{\varTheta }. \end{aligned}$$

By using Lemma 2, we have

$$\begin{aligned} \phi _T\left( {\varvec{y}}; {\varvec{X}}\varvec{\beta } + {\varvec{Z}}\varvec{\varTheta }, \varvec{\varOmega }\right) \phi _T\left( \varvec{\varTheta }; \mathbf {0}, \varvec{\varPsi }\right)&= \phi _T\left( {\varvec{y}}; {\varvec{X}}\varvec{\beta }, \varvec{\varOmega } + {\varvec{Z}}\varvec{\varPsi }{\varvec{Z}}'\right) \\ \phi _T\left( \varvec{\varTheta }; \varvec{\varLambda }{\varvec{Z}}'\varvec{\varOmega }^{-1}({\varvec{y}} - {\varvec{X}}\varvec{\beta }), \varvec{\varLambda }\right) , \end{aligned}$$

where \(\varvec{\varLambda } = \left( \varvec{\varPsi }^{-1} + {\varvec{Z}}'\mathbf {D}^{-1}\varvec{\varOmega }\right) ^{-1}\). So, the marginal probability density function can be written as

$$\begin{aligned} f_{{\varvec{Y}}}({\varvec{y}}) = 2^T\phi _T\left( {\varvec{y}}; {\varvec{X}}\varvec{\beta }, \varvec{\varOmega } + {\varvec{Z}}\varvec{\varPsi }{\varvec{Z}}'\right) \\ E\! \left[ \varPhi _T\left( \varvec{\varDelta }'\varvec{\varOmega }^{-1}({\varvec{y}} - {\varvec{X}}\varvec{\beta } - {\varvec{Z}}\mathbf {W}); \mathbf {0}, \mathbf {I}_T - \varvec{\varDelta }'\varvec{\varOmega }^{-1}\varvec{\varDelta }\right) \right] , \end{aligned}$$

where \(\mathbf {W} \sim N_T\left( \varvec{\varLambda }{\varvec{Z}}'\varvec{\varOmega }^{-1}({\varvec{y}} - {\varvec{X}}\varvec{\beta }), \varvec{\varLambda }\right)\). By using Lemma 1, we obtain

$$\begin{aligned} f_{{\varvec{Y}}}({\varvec{y}})&= 2^T\phi _T\left( {\varvec{y}}; {\varvec{X}}\varvec{\beta }, \varvec{\varOmega } + {\varvec{Z}}\varvec{\varPsi }{\varvec{Z}}'\right) \\&\quad \times \varPhi _T\left( \varvec{\varDelta }'\left( \varvec{\varOmega } + {\varvec{Z}}\varvec{\varPsi }{\varvec{Z}}'\right) ^{-1}({\varvec{y}} - {\varvec{X}}\varvec{\beta }); \mathbf {0}, \mathbf {I}_T - \varvec{\varDelta }'\left( \varvec{\varOmega } + {\varvec{Z}}\varvec{\varPsi }{\varvec{Z}}'\right) ^{-1}\varvec{\varDelta }\right) . \end{aligned}$$

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Pigeon, M., Henry de Frahan, B. & Denuit, M. Evaluation of the EU proposed farm income stabilisation tool by skew normal linear mixed models. Eur. Actuar. J. 4, 383–409 (2014). https://doi.org/10.1007/s13385-014-0097-9

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