US Wine Sales vs Population 1934
Download
Report
Transcript US Wine Sales vs Population 1934
Regression with Autocorrelated Errors
U.S. Wine Consumption and Adult
Population – 1934-2002
Data Description
• Y=U.S. Annual Wine Consumption (Millions of
Gallons)
• X=U.S. Adult Population (Millions of People)
• Years – 1934-2002 (Post Prohibition)
• Model:
Yt 0 1 X t t
t t 1 t 1 ... q t q
t ~ iid 0,
2
Ordinary Least Squares Regression
Regression Statistics
Multiple R
0.965612383
R Square
0.932407274
Adjusted R Square
0.931398427
Standard Error
48.64438813
Observations
69
Intercept
apop_m
ANOVA
df
1
67
68
Regression
Residual
Total
Coefficients
-347.9736
4.3092
SS
2186985.91
158540.53
2345526.43
Standard Error
t Stat
21.9895
-15.8245
0.1417
30.4012
^
P-value
0.0000
0.0000
MS
2186985.91
2366.28
Lower 95%
-391.8649
4.0263
F
924.23
Significance F
0.0000
Upper 95%
-304.0824
4.5921
^
W t 347.97 4.31Pt e 2366.28 48.64
n
Durbin - WatsonT est : DW
e e
t 2
2
t 1
t
n
e
t 1
2
t
19008.8044
0.11989871
158540.53
d L p 1, n 70 1.58 dU p 1, n 70 1.64
DW dU Autocorrelationis present
Residuals versus Time
150
100
Residuals
50
0
-50
-100
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69
Covariance Structure (q=1)
Yt 0 1 X t t
t t t 1 t ~ iid 0, 2
1 1
2
Assuming: E 1 0 V 1
1 2
2 2 2 1 2 2 2
2
E 2 0 V 2 V 2 V 1
1 2
1 2
1 2
2
2
2
COV 1 , 2 COV 1 , 2 1 V 1
1 2
In General:
2
E t 0 V t
1 2
k 2
COV t , t k
1 2
1
1
n 1
1
n2
2
2
V ε V
2
1
n 1
n2
1
n
2 1 2 V t V t COV t , t 1
COV t , t 1
V t
Generalized Least Squares (q=1)
1
2
V V Y V ε
1 2
n 1
1
n2
1 2
0
1
Define: T
0
0
n 1
n2
1
0
0
1
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
Consider the Case where n 4 :
1 2
T 1 V
0
0
T 1 VT1'
0
1
0
0
1
0
1 2
2 0
1 2 0
0
0
2
0
2
0 1
1
1 2
1 2
0
0
1
2
3
2
1
2
1
2 1 2
1 2
1 2
0
3
2
2
1 2
1
1 2
0
0
0
3 1 2 1 2
2 1 2 0
1 2 0
2
1 0
1
0
0
1 2
1 2
0
2 1 2
1 2
1 2
0
0
0
1
0
3 1 2
2 1 2
1 2
1 2
0
0
1
1 2
0
0
0
1 2
0
0
2 0
2 I V T 1Y T 1VT1' 2 I T ransformY* T 1 Y T 1 Xβ T 1ε
2
2
1 0
0
1
0
0
0
1 2
0
Estimated GLS (q=1)
^
1 n 2
(0) et
n t 1
^
^ 1
T
^ 2
^
( 0)
^
0
0
0
1 n
(1) et et 1
n t 2
^
^
0
0
0
0
1
0
0
0
1
0
0
^
(1)
0
1
0
0
0
^
( 0)
^
^
^
(0) (1)
0
0
0
0
1
^
0
^ 2
^
1
^
β EGLS
^ 1 ^ 1
X'
T
'
T
X
1
^ 1
X' T ' T
^ 1 ^ 1
V β EGLS e X' T ' T X
^
^
^ 2
^ ^
^
(
0
)
(1)
s2
n p '1
^ 1
^ 2
e
Y
^
1
SE i
^
ti
EGLS ,i
^
SE EGLS ,i
^
^
s 2 / ( 0)
1
^
^
^
^
Y X β EGLS ' T ' T Y X β EGLS
n p '1
ti
i
^
SE i
Estimated GLS (q=1) – Wine/Population Data
^
^
^
(0) 2297.69 (1) 2147.89 0.9348
^ 1
T
0
0.3552
- 0.9348
1
0
- 0.9348
0
0
0
0
^ 2
289.82
0
0
0
0 - 0.9348
1
0
0
0
- 0.9348 1
0
0
1
0
0
0
0
0
0
^ 2
- 347.23
β EGLS
e 280.5479
4.2540
^ ^
4.2540 4.2540
5482.225 - 31.515
V β EGLS
t
9.358
1
0.2066 0.4546
- 31.515 0.2066
^
s 2 4.39
SE i
^
^
^
s 2 / (0) .04372
ti
i
^
SE i
0.9348
21.38
0.04372
Transformed residuals versus Year - EGLS (q=1)
60
40
Residuals
20
0
-20
-40
-60
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69
DW 1.67 d L p 1, n 70 1.58 dU p 1, n 70 1.64
DW dU Do not concludeAutocorrelation is present
Untransformed data and Fitted Equation from EGLS
700
600
500
Wine Sales
400
Y
Yhat_w
300
200
100
0
50
70
90
110
130
150
Population
170
190
210
230
250
Estimated GLS (General q) - I
1) Fit OrdinaryLeast Squares Regression and Obtain OLS Residuals :
β OLS X' X X' Y eOLS Y X β OLS
^
^
1
2) Est imatetheautocovariances to lag q and thefollowing qxq mat rixand qx1 vector:
^
^
^
^
(
0
)
(
1
)
(
q
1
)
^
^ (1)
e
e
^
^
t
t
h
^
^
^
(0) (q 2) γ (2)
(h) t h 1
h 0,1,...,q Γ q (1)
q
n
^
^
^ (q 1) ^ (q 2)
(0)
(q )
3) Est imatetheCoefficients of thelagged errors- qx1 vectorandV t 2 :
n
^
^ 1 ^
^ 2
^
^ ^
ρ Γ q γ q (0) ρ' γ q
^ 1
^ 1
^
^
4) Obtain theCholeskyDecomposition of Γ q : Γ q P q ' P q
^ 1/2
^
5) Obtain thetransformation mat rix: T V
^ 2 ^
T11 P q
^
q
0
T2
0
0
(q q)
^
1
q (n q)
T12 0
(assuming n 2(q 1)) :
T1 T11
0
0
0
0
0
q 1 1
0
0
0
0
1
^
1
^
0
0
0
0 q
0
0
0
0
q 1
^
^
0
^
1
^
0
0
0
1
T12
q n
(n q) n
^
T
T 1 ( n n)
T2
Estimated GLS (General q) - II
1
^ ^
^ ^
6) Obt ain t heest imat edgeneralized least squares est imat e: β EGLS X' T' T X X' T' T Y
7) Obt ain t heEst imat ederror variance from t he t ransformed model:
^
^
^
^ ^
Y X β EGLS ' T' T Y X β EGLS
^ 2
e
n p ' q
^
^ ^ ^
8) Obt ain t heest imat edvariance- covariancemat rixof β EGLS : V β EGLS e X' T' T X
^
2
^
1
^
9) Obt ain est imat edt - st at ist icsfor regression coefficient s : ti
EGLS ,i
^
SE EGLS ,i
10) Obt ain residual MS for est imat esof aut oregressive paramet ers and t hest andarderrors:
^ ^
^
(
0
)
ρ
'
γ
q
^
^
s2
SE i s 2 ii
n p ' q
11) Obt ain t- st at ist icsfor t he i :
^
^
ti
i
^
SE i
^ 1
ii i diagonal elementof Γ q
th
and comparewit h t het - dist ribut ion wit h df n p 'q
SAS Proc Autoreg Output
The AUTOREG Procedure
Dependent Variable
wine
Ordinary Least Squares Estimates
SSE
158540.525
DFE
MSE
2366
Root MSE
SBC
738.318203
AIC
Regress R-Square
0.9324
Total R-Square
Durbin-Watson
0.1199
Variable
Intercept
adpop
DF
1
1
Estimate
-347.9736
4.3092
Standard
Error
21.9895
0.1417
t Value
-15.82
30.40
67
48.64439
733.84999
0.9324
Approx
Pr > |t|
<.0001
<.0001
Estimates of Autocorrelations
Lag
Covariance
Correlation
0
1
2297.7
2147.9
1.000000
0.934807
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
|
|
|********************|
|******************* |
SAS Proc Autoreg Output
Preliminary MSE
289.8
Estimates of Autoregressive Parameters
Standard
Lag
Coefficient
Error
t Value
1
-0.934807
0.043717
-21.38
SSE
MSE
SBC
Regress R-Square
Durbin-Watson
Yule-Walker Estimates
18516.1612
DFE
280.54790
Root MSE
596.454422
AIC
0.5702
Total R-Square
1.6728
Variable
DF
Estimate
Standard
Error
Intercept
adpop
1
1
-347.2297
4.2540
74.0420
0.4546
66
16.74956
589.752103
0.9921
t Value
Approx
Pr > |t|
-4.69
9.36
<.0001
<.0001