13. Régression linéaire simple
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(a) non linearité
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(b) non homoscedasticité
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(c) non normalité
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(d) modèle de régression linéaire
Figure 13.6 – Hypothèses du modèle de régression linéaire simple.
On va voir à présent que l’on peut reformuler les hypothèses d’un modèle
de régression simple en fonction des résidus (ce qui facilitera la généralisation
de ces hypothèses au cas de la régression multiple dans le chapitre suivant). On
a vu que l’hypothèse de linéarité peut être définie de façon équivalente par :
mean(Y |X = x) = β 0 + β 1 x ou par mean(ε|X = x) = 0.
Par ailleurs, la variable résiduelle dans le groupe X = x, que nous notons
par ε|X = x, peut s’obtenir à partir de la variable Y dans le groupe X =
x, que nous notons par Y |X = x, par simple soustraction de sa moyenne
mean(Y |X = x), de sorte que la variabilité et la forme de ces deux distributions
sont identiques. Il s’ensuit que l’hypothèse d’homoscédasticité peut être définie
de façon équivalente par :
variance(Y |X = x) = σ
2
ε
ou par variance(ε|X = x) = σ
2
ε .
De même, l’hypothèse de normalité revient de manière équivalente à supposer
que la variable Y |X = x est normale ou à supposer que la variable ε|X = x
est normale. Ainsi, les trois hypothèses ci-dessus reviennent à supposer que les
résidus ε i = y i −β 0 −β 1 x i de notre échantillon (i = 1, · · · , n) proviennent d’une
même distribution, qui est normale avec moyenne nulle et variance σ
2
ε .