Asymptotics Primer - University of Washington · Asymptotics Primer Econ 583 Econometric Theory I...
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1 Asymptotics Primer Econ 583 Econometric Theory I Fall 2008 Consistency of Sample Mean from iid N(0,1) Data Sample Size, T Sample Mean 0 200 400 600 800 1000 0.0 0.2 0.4 0.6
Transcript of Asymptotics Primer - University of Washington · Asymptotics Primer Econ 583 Econometric Theory I...
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Asymptotics Primer
Econ 583Econometric Theory I
Fall 2008
Consistency of Sample Mean from iid N(0,1) Data
Sample Size, T
Sam
ple
Mea
n
0 200 400 600 800 1000
0.0
0.2
0.4
0.6
2
Consistency of Sample Mean from iid U(-1,1) Data
Sample Size, T
Sam
ple
Mea
n
0 200 400 600 800 1000
0.0
0.2
0.4
0.6
Consistency of Sample Mean from iid Chi-Square 1 Data
Sample Size, T
Sam
ple
Mea
n
0 200 400 600 800 1000
0.0
0.2
0.4
0.6
0.8
1.0
1.2
2[ (1)] 1E χ =
3
Sample Mean from iid Student's t with 1 df
Sample Size, T
Sam
ple
Mea
n
0 200 400 600 800 1000
-20
24
68 Student’s t with 1 df = Cauchy
Expected value of Cauchy rv does not exist because tails are too thick
N(0,1) Data
Sample Size, T
Sam
ple
Mea
n
0 200 400 600 800 1000
-3-1
01
2
U(-1,1) Data
Sample Size, T
Sam
ple
Mea
n
0 200 400 600 800 1000
-1.0
0.0
0.5
1.0
Chi-Square 1 Data
Sample Size, T
Sam
ple
Mea
n
0 200 400 600 800 1000
01
23
45
6
Cauchy Data
Sample Size, T
Sam
ple
Mea
n
0 200 400 600 800 1000
-600
-200
0
100 Simulated Sequences of Yn
4
T = 25
mu.hat
dens
ity
-1.0 -0.5 0.0 0.5 1.0
0.0
0.5
1.0
1.5
2.0
T = 50
mu.hat
dens
ity
-1.0 -0.5 0.0 0.5 1.0
0.0
0.5
1.0
1.5
2.0
2.5
T = 100
mu.hat
dens
ity
-1.0 -0.5 0.0 0.5 1.0
01
23
4
T = 1000
mu.hat
dens
ity
-1.0 -0.5 0.0 0.5 1.0
02
46
810
12
Distribution of Sample Mean for iid N(0,1) data
( )SD XT
σ=
-2 0 2
0.0
0.1
0.2
0.3
0.4
T=10
z.10
-4 -2 0 2
0.0
0.1
0.2
0.3
T=25
z.25
-3 -2 -1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
T=50
z.50
-3 -2 -1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
T=100
z.100
for ~ (0,1)iXT X iid Nμσ−⎛ ⎞
⎜ ⎟⎝ ⎠
5
2for ~ (1)iXT X iidμ χσ−⎛ ⎞
⎜ ⎟⎝ ⎠
-2 0 2 4 6
0.0
0.1
0.2
0.3
0.4
T=10
z.10
-2 -1 0 1 2 3
0.0
0.1
0.2
0.3
0.4
T=25
z.25
-2 0 2 4
0.0
0.1
0.2
0.3
0.4
T=50
z.50
-2 0 2
0.0
0.1
0.2
0.3
0.4
T=100
z.100
