Statistical models A.Y. 2013/14
Written exam of March 24, 2014.
1. Consider the general linear model Y = Xβ + E, where Y = (y1, . . . , yn) is a vector in Rn and X is a matrix n × p.
(a) Write down the usual assumptions on the error term E.
(b) Using the assumption that the errors are normally distributed, write down the likelihood (or better its logarithm) of the observed data (y1, . . . , yn).
(c) Write down the formulae for the usual estimates of β and of the variance σ2of the error terms; are these maximum likelihood estimates?
(d) Write down the statement (including its exact assumptions) of Gauss-Markov theorem concerning the estimate of β.
2. Consider the linear model Y = Xβ + E where
Y =
y1
... y12
∈ R12, X =
1 z1 0 0 1 z2 0 0 1 z3 0 0 1 z4 0 0 1 z5 1 0 1 z6 1 0 1 z7 1 0 1 z8 1 0 1 z9 0 1 1 z10 0 1 1 z11 0 1 1 z12 0 1
where z1, . . . , znare observations of a variable Z; and a qualitative variable W is equal to A on the first 4 observations, to B in the following 4 and to C in the last 4.
(a) How would the regression model be represented in terms of the variables Y , Z and W ? (b) How can we test if there is a (linear) effect of the variable Z on Y ? and how about the
effect of the variable W ? State the procedure to be used.
3. Write down the following models as linear models, possibly after some transformation? In all cases εi represent independent and equidistributed error terms, E(εi) = 0; a, b, c . . . parameters to be estimated.
(a) yi =
(a + b(xi− 30) + εi if xi < 30 a + c(30 − xi) + εi if xi > 30 (b) yi = axbi(1 + εi)
(c) yi = a + bxi xi+ 1 + εi
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