STA 6207 – Exam 1 – Fall 2015
PRINT Name _____________________
Unless stated otherwise, Questions are based on the following 2 regression models, where SIMPLE
REGRESSION refers to the case where p=1, and X is of full column rank (no linear dependencies among
predictor variables).
Model 1 : Yi   0  1 X i1     p X ip   i

i  1,..., n  i ~ NID 0,  2

Model 2 : Y  Xβ  ε X  n  p' β  p'1 ε ~ N 0,  2 I


d a ' x
d  x'Ax 
a
 2Ax (A symmetric)
dx
dx
__________________________________________________________________________________________
Given:
Q.1. Consider model 2.
p.1.a. Derive the least squares estimator for . Show all work.
p.1.b. Derive the mean and variance of the least squares estimator. Show all work.
Q.2. Consider the “centered” (with respect to the independent variable) model in scalar form:


Yi    1 X i  X   i
i  1,..., n  i ~ NID  0,  2 
p.2.a. Obtain the normal equations and the least squares estimated for the parameters  and 1. Show all work.
^ ^ 
p.2.b. Derive COV   ,  1 


n
 n
 n n
Hint: COV   aiYi ,  b jY j    aib j COV Yi , Y j  Show all work.
j 1
 i 1
 i 1 j 1
^
Q.3. For model 2, show: Y 'e = 0.
Show all work. Hint:  X'X  is symmetric.
-1
Q.4. The following measurements were obtained at n=5 locations on the earth’s surface, where Y=measured
gravity, and X= latitude. Consider fitting Model 2, containing an intercept, with p’=2. Note this includes the
fitted values and residuals from results of p.4.a, and makes use of both models 1 and model 2.
Obs#
1
2
3
4
5
X(Lat)
20
30
40
50
60
Y(Grav)
1.10
1.60
1.90
2.40
3.00
Y-hat
1.08
1.54
2.00
2.46
2.92
e
0.02
0.06
-0.10
-0.06
0.08
p.4.a. Obtain the following matrices and vectors (model 2).
X'X
X'Y
INV(X'X)
Beta-hat
p.4.b. Compute the residual sum of squares, and give its degrees of freedom (based on model 1).
p.4.c. Compute the regression sum of squares, and give its degrees of freedom (based on model 1).
p.4.d. Conduct the F-test, testing H0: 1 = 0 vs HA: 1 ≠ 0 at  = 0.05 significance level.
p.4.d.i. Test Statistic:
p.4.d.ii. Rejection Region: ________________________________________________________
Q.5. A linear regression was run on a set of data, based on model 1 with p=1 independent variable. You are
given only the following partial information:
ANOVA
df
Regression
Residual
Total
Intercept
X
SS
MS
5
Coefficients Standard Error
293.89
5.62
-1.65
0.13
F
P-value
44.2
t Stat
P-value
52.29
0.0000
-13.13
0.0000
p.5.a. Compute a 95% Confidence Interval for 1:
p.5.b. Give the F-statistic and rejection for testing H0: 1 = 0 vs HA: 1 ≠ 0 at  significance level. (Hint:
think of connection between t- and F-tests)
p.5.c. Compute the coefficient of determination, R2.
F Distributions Indexed by Numerator Degrees of Freedom
df |
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200

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t.05
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.676
1.671
1.667
1.664
1.662
1.660
1.659
1.658
1.657
1.656
1.655
1.654
1.654
1.653
1.653
1.653
1.645
t.025
2
 .05
F.05,1
F.05,2
F.05,3
F.05,4
F.05,5
F.05,6
F.05,7
F.05,8
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.021
2.009
2.000
1.994
1.990
1.987
1.984
1.982
1.980
1.978
1.977
1.976
1.975
1.974
1.973
1.973
1.972
1.960
3.841
5.991
7.815
9.488
11.070
12.592
14.067
15.507
16.919
18.307
19.675
21.026
22.362
23.685
24.996
26.296
27.587
28.869
30.144
31.410
32.671
33.924
35.172
36.415
37.652
38.885
40.113
41.337
42.557
43.773
55.758
67.505
79.082
90.531
101.879
113.145
124.342
135.480
146.567
157.610
168.613
179.581
190.516
201.423
212.304
223.160
233.994
---
161.448
18.513
10.128
7.709
6.608
5.987
5.591
5.318
5.117
4.965
4.844
4.747
4.667
4.600
4.543
4.494
4.451
4.414
4.381
4.351
4.325
4.301
4.279
4.260
4.242
4.225
4.210
4.196
4.183
4.171
4.085
4.034
4.001
3.978
3.960
3.947
3.936
3.927
3.920
3.914
3.909
3.904
3.900
3.897
3.894
3.891
3.888
3.841
199.500
19.000
9.552
6.944
5.786
5.143
4.737
4.459
4.256
4.103
3.982
3.885
3.806
3.739
3.682
3.634
3.592
3.555
3.522
3.493
3.467
3.443
3.422
3.403
3.385
3.369
3.354
3.340
3.328
3.316
3.232
3.183
3.150
3.128
3.111
3.098
3.087
3.079
3.072
3.066
3.061
3.056
3.053
3.049
3.046
3.043
3.041
2.995
215.707
19.164
9.277
6.591
5.409
4.757
4.347
4.066
3.863
3.708
3.587
3.490
3.411
3.344
3.287
3.239
3.197
3.160
3.127
3.098
3.072
3.049
3.028
3.009
2.991
2.975
2.960
2.947
2.934
2.922
2.839
2.790
2.758
2.736
2.719
2.706
2.696
2.687
2.680
2.674
2.669
2.665
2.661
2.658
2.655
2.652
2.650
2.605
224.583
19.247
9.117
6.388
5.192
4.534
4.120
3.838
3.633
3.478
3.357
3.259
3.179
3.112
3.056
3.007
2.965
2.928
2.895
2.866
2.840
2.817
2.796
2.776
2.759
2.743
2.728
2.714
2.701
2.690
2.606
2.557
2.525
2.503
2.486
2.473
2.463
2.454
2.447
2.441
2.436
2.432
2.428
2.425
2.422
2.419
2.417
2.372
230.162
19.296
9.013
6.256
5.050
4.387
3.972
3.687
3.482
3.326
3.204
3.106
3.025
2.958
2.901
2.852
2.810
2.773
2.740
2.711
2.685
2.661
2.640
2.621
2.603
2.587
2.572
2.558
2.545
2.534
2.449
2.400
2.368
2.346
2.329
2.316
2.305
2.297
2.290
2.284
2.279
2.274
2.271
2.267
2.264
2.262
2.259
2.214
233.986
19.330
8.941
6.163
4.950
4.284
3.866
3.581
3.374
3.217
3.095
2.996
2.915
2.848
2.790
2.741
2.699
2.661
2.628
2.599
2.573
2.549
2.528
2.508
2.490
2.474
2.459
2.445
2.432
2.421
2.336
2.286
2.254
2.231
2.214
2.201
2.191
2.182
2.175
2.169
2.164
2.160
2.156
2.152
2.149
2.147
2.144
2.099
236.768
19.353
8.887
6.094
4.876
4.207
3.787
3.500
3.293
3.135
3.012
2.913
2.832
2.764
2.707
2.657
2.614
2.577
2.544
2.514
2.488
2.464
2.442
2.423
2.405
2.388
2.373
2.359
2.346
2.334
2.249
2.199
2.167
2.143
2.126
2.113
2.103
2.094
2.087
2.081
2.076
2.071
2.067
2.064
2.061
2.058
2.056
2.010
238.883
19.371
8.845
6.041
4.818
4.147
3.726
3.438
3.230
3.072
2.948
2.849
2.767
2.699
2.641
2.591
2.548
2.510
2.477
2.447
2.420
2.397
2.375
2.355
2.337
2.321
2.305
2.291
2.278
2.266
2.180
2.130
2.097
2.074
2.056
2.043
2.032
2.024
2.016
2.010
2.005
2.001
1.997
1.993
1.990
1.987
1.985
1.938
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