Tutorial 9
1. Suppose
Y
has 5 covariates
X
1
, X
2
, X
3
, X
4
, X
5
denote any model
Y
=
β
0
+
β
i
X
i
+
β
j
X
j
+
β
k
X
k
+
ε
by (
ijk
).
All the models can be listed as (0), (1), (2), (3), (4),
(5), (12), (13), (14), (15), (23), (24), (25), (34), (35), (45), (123), (124), (125), (134),
(135), (145), (234), (235), (245), (345), (1234), (1235), (1245), (1345), (2345), (12345).
With
n
= 40, their respectively SSE are 95.2261, 73.3037, 95.2139, 90.8737, 69.2889,
73.6588, 73.0233, 70.9692, 53.2715, 42.4473, 90.8726, 69.2660, 73.2872, 63.9012, 72.6848,
25.0160, 70.5948, 52.8136, 42.4440, 49.9665, 42.4467, 0.2618, 63.7998, 72.4197, 24.7802,
24.4749, 49.3597, 42.4432, 0.2512, 0.2496, 24.3022, 0.2406.
And their respectively
BIC are 0.9596, 0.7902, 1.0517, 1.0050, 0.7338, 0.7950, 0.8786, 0.8500, 0.5632, 0.3361,
1.0972, 0.8257, 0.8822, 0.7451, 0.8739, 0.1927, 0.9370, 0.6468, 0.4282, 0.5914, 0.4283,
4.6600, 0.8358, 0.9625, 0.1099, 0.1223, 0.6714, 0.5204, 4.6093, 4.6157, 0.0372,
4.5600.
(a) Based on BIC, which model should be preferred
(b) Based on BIC and using forward selection, which model is selected.
Please
indicate which models are calculated
(c) Based on BIC and using backward elimination, which model is selected. Please
indicate which models are calculated
(d) using backward elimination and Ftest, start from the full model, test whether
X
2
and
X
3
can be removed simultaneously.
2. Consider the Gaoline consumption and automotive variables.
Y
: Miles/Gallon;
X
1
:
Displacements;
X
2
: Horsepower;
X
3
: Torque;
X
4
: compression ratio;
X
5
: real axle
ratio;
X
6
: Carburetor;
X
7
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 Fall '09
 XIAYingcun
 Regression Analysis, AIC, linear regression model, stepwise selection method

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