elca.rh: Extended (Stratified) Lee-Carter model (with a single extra...
In ilc: Lee-Carter Mortality Models using Iterative Fitting Algorithms
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
A purpose-built regression routine to fit the extended Lee-Carter model with an extra additive effect of an observable factor (other than age and period) on the log mortality mortality rates.
Usage
1
2
3
4
Arguments
dat
rhdata class multidimensional mortality data object
year
vector of years to be included in the regression (all available years by default)
age
vector of ages to be included in the regression (all available ages by default)
dec.conv
number of decimal places used to achieve convergence. The lower the value the faster the convergence of the fitting algorithm. Note: very high values could over fit the parameters.
error
type of error structure of the model choice (Poisson distribution of the errors by default)
restype
types of residuals, which also controls the type of the fitted value.
Thus, in the cases of logrates and rates the function returns as fitted values the log and untransformed mortality rates, respectively. Likewise, the choices of deaths and deviance correspond to the fitted number of deaths
scale
logical, if TRUE, re-scale the interaction parameters so that the k_t has drift parameter equal to 1 (see also lca)
interpolate
logical, if TRUE, replace before regression all zero or missing values in the mortality rates of dat argument by interpolation across calendar years (see also smooth.demogdata)
verbose
logical, it controls the amount of process information
spar
numerical smoothing spline parameter in the interval (0,1] (with a recommended value of 0.6). If it is not NULL, the interaction effects (i.e. β_x^{(0,1)}) are smoothed out after the initial regression. Consequently, the period and/or cohort effects are adjusted (smoothed out) accordingly.
ax.fix
vector of constant age effect to be used in the model (e.g. the fitted values of a standard LC regression to the experience of a large population). If NULL the base ax values are estimated from dat
Details
This function models the number of deaths for a group within a generalised Lee-Carter framework with a Poisson or Gaussian error structure. The methodology quantifies the differences in the mortality experience of population subgroups differentiated by an additional measurable covariate (other than age and period). Additional covariate, for instance, could be related to geographical, socio-economic or race differences.
Value
An object of class elca with the following components:
lca
list of fitted lca model objects by the level of the extra factor
age
vector of fitted ages
year
vector of fitted years
ag
parameter estimates of the effects of the extra factor
ax
parameter estimates (or ax.fix) of (mean) age-specific mortality rates across the entire fitting period
bx
parameter estimates of age-specific interaction effect between age and period
kt
parameter estimates of year-specific period trend of mortality rates
adjust
type of error structure used in fitting (e.g. "poisson" or "gaussian")
label
data label
call
copy of the R call to the model
conv.iter
number of iterations used to reach convergence
mdev
mean deviance of total and base lack of fit (see also lca)
model
string expression of the fitted model
df
degree of freedom of the fitted GLM model
Author(s)
Z. Butt and S. Haberman and H. L. Shang
References
Li, N. and Lee, R. D. (2005), ‘Cohort mortality forecasts for a group of populations: an extension of the Lee-Carter method’, Demography, 42(3), 575-594.
Renshaw and Haberman (2006), ‘A cohort-based extension to the Lee-Carter model for mortality reduction factors.’, Insurance: Mathematics and Economics, 38, 556-570.
See Also
Examples
1
2
3
4
5
Example output
Loading required package: demography
Loading required package: forecast
Registered S3 method overwritten by 'quantmod':
method from
as.zoo.data.frame zoo
Registered S3 methods overwritten by 'demography':
method from
print.lca e1071
summary.lca e1071
This is demography 1.22
Loading required package: rainbow
Loading required package: MASS
Loading required package: pcaPP
Loading required package: date
Original sample: Multidimensional Mortality data for: CMI [male]
Across covariates:
years: 1983 - 2003
ages: 50 - 108
X: base, a, b, c, d
Applied sample: Multidimensional Mortality data for: CMI [male]
Across covariates:
years: 1983 - 2003
ages: 50 - 100
X: base, a, b, c, d
Fitting model: [ LC(g) = a(x)+a(g)+b(x)*k(t) ]
- with poisson error structure and with deaths as weights -
Note: 1161 cells have 0/NA deaths and 127 have 0/NA exposure
out of a total of 5355 data cells.
Starting values are:
age age.c int.c per per.c X4 X4.c
1 1 -1.286 0.02 1983 0 base 0
2 2 -2.508 0.02 1984 0 a 0
3 3 -3.028 0.02 1985 0 b 0
4 4 -3.343 0.02 1986 0 c 0
5 5 -3.153 0.02 1987 0 d 0
6 6 -3.149 0.02 1988 0
7 7 -3.402 0.02 1989 0
8 8 -2.956 0.02 1990 0
9 9 -3.553 0.02 1991 0
10 10 -3.091 0.02 1992 0
11 11 -3.202 0.02 1993 0
12 12 -3.233 0.02 1994 0
13 13 -3.141 0.02 1995 0
14 14 -3.051 0.02 1996 0
15 15 -2.832 0.02 1997 0
16 16 -2.681 0.02 1998 0
17 17 -2.739 0.02 1999 0
18 18 -2.530 0.02 2000 0
19 19 -2.407 0.02 2001 0
20 20 -2.436 0.02 2002 0
21 21 -2.171 0.02 2003 0
22 22 -2.142 0.02
23 23 -1.940 0.02
24 24 -1.754 0.02
25 25 -1.737 0.02
26 26 -1.583 0.02
27 27 -1.549 0.02
28 28 -1.300 0.02
29 29 -1.328 0.02
30 30 -1.230 0.02
31 31 -1.193 0.02
32 32 -0.936 0.02
33 33 -0.852 0.02
34 34 -0.879 0.02
35 35 -0.768 0.02
36 36 -0.612 0.02
37 37 -0.513 0.02
38 38 -0.465 0.02
39 39 -0.394 0.02
40 40 -0.266 0.02
41 41 -0.284 0.02
42 42 -0.202 0.02
43 43 -0.177 0.02
44 44 -0.043 0.02
45 45 0.119 0.02
46 46 0.102 0.02
47 47 -0.037 0.02
48 48 0.149 0.02
49 49 0.099 0.02
50 50 0.061 0.02
51 51 0.148 0.02
Iterative fit:
#iter Dev non-conv
1 6538592 0
2 29095571 1
3 4587766 0
4 1030918 0
5 962609.5 0
6 949241.1 0
7 949140.8 0
8 949282.3 1
9 949340.6 2
10 949360.1 3
11 949366.6 0
12 949368.8 0
13 949369.6 0
14 949369.8 0
15 949369.9 0
16 949370 0
17 949370 0
18 949370 0
19 949370 0
20 949370 0
21 949370 0
Iterations finished in: 21 steps
Updated values are:
age age.c int.c per per.c X4 X4.c
1 1 -3.57182 0.10173 1983 10.40003 base 0
2 2 -4.00941 0.04732 1984 14.39776 a 0.49457
3 3 -4.66980 0.02346 1985 11.09019 b 1.25965
4 4 -4.88614 0.02980 1986 12.61244 c -0.55938
5 5 -4.79139 0.00564 1987 6.20855 d 2.50562
6 6 -4.54121 0.04728 1988 8.27912
7 7 -4.92833 0.00628 1989 8.75253
8 8 -4.48164 0.01974 1990 8.76337
9 9 -4.74229 0.02496 1991 5.54912
10 10 -4.47916 0.00110 1992 3.45402
11 11 -4.49779 0.00941 1993 -0.45605
12 12 -4.55218 0.00782 1994 -4.30681
13 13 -4.46182 0.01542 1995 1.60598
14 14 -4.42328 0.00930 1996 -4.17017
15 15 -4.11704 0.00677 1997 -12.22838
16 16 -4.09382 0.02913 1998 -7.68399
17 17 -4.10641 0.02993 1999 -15.04592
18 18 -3.87744 0.01948 2000 -15.69551
19 19 -3.81737 0.01915 2001 -10.01521
20 20 -3.85569 0.01645 2002 -11.49669
21 21 -3.57086 0.01870 2003 -10.01438
22 22 -3.56004 0.01942
23 23 -3.29706 0.02017
24 24 -3.19681 0.02288
25 25 -3.15090 0.02172
26 26 -2.94881 0.00943
27 27 -2.90764 0.01705
28 28 -2.75753 0.02283
29 29 -2.69655 0.01681
30 30 -2.60415 0.01743
31 31 -2.61773 0.02969
32 32 -2.33704 0.01849
33 33 -2.37038 0.02361
34 34 -2.23598 0.01498
35 35 -2.15189 0.01120
36 36 -1.97745 0.00322
37 37 -2.00360 0.01541
38 38 -1.84680 0.00943
39 39 -1.79219 0.01515
40 40 -1.63259 0.01501
41 41 -1.67780 0.00650
42 42 -1.60457 0.01811
43 43 -1.59460 0.00472
44 44 -1.45978 0.01706
45 45 -1.30167 0.02744
46 46 -1.24017 0.01719
47 47 -1.40882 0.01141
48 48 -1.22433 0.01783
49 49 -1.27113 0.01216
50 50 -1.31165 0.02586
51 51 -1.14732 0.02891
total sums are:
bx kt
1 0
Warning messages:
1: A total of 1160 0/NA central mortality rates are re-estimated by the "interpolate" method.
2: In elca.rh(rfp.cmi, age = 50:100, interp = TRUE, dec = 3, verb = TRUE) :
There are 127 cells with 0/NA exposures, which are ignored in the current analysis.
Try reducing the fitted age range.
Alternatively, fit ELC model with error= "gaussian" .
3: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
4: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
5: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
6: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
7: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
------------------------------------------------------------
Extended Lee-Carter Regression:
Fitted Model: LC(g) = a(x)+a(g)+b(x)*k(t)
------------------------------------------------------------
Call: elca.rh(dat = rfp.cmi, age = 50:100, dec.conv = 3, interpolate = TRUE,
verbose = TRUE)
Error Structure: poisson
Data Source: CMI : male over
calendar years: (1983 - 2003) , ages: (50 - 100)
and groups: base a b c d
Deviance convergence in: 21 iterations
dev dev.c df df.c
1 Mean deviance base 258.473 df base 3673
2 Mean deviance total 204.598 df tot 4845
$ax
[1] -3.571824 -4.009415 -4.669796 -4.886136 -4.791390 -4.541212 -4.928325
[8] -4.481641 -4.742290 -4.479161 -4.497794 -4.552177 -4.461817 -4.423281
[15] -4.117038 -4.093815 -4.106408 -3.877443 -3.817370 -3.855689 -3.570859
[22] -3.560043 -3.297057 -3.196805 -3.150899 -2.948812 -2.907638 -2.757527
[29] -2.696552 -2.604152 -2.617727 -2.337037 -2.370380 -2.235981 -2.151886
[36] -1.977447 -2.003599 -1.846796 -1.792191 -1.632592 -1.677804 -1.604574
[43] -1.594599 -1.459783 -1.301673 -1.240173 -1.408820 -1.224327 -1.271128
[50] -1.311654 -1.147319
$bx
50 51 52 53 54 55
0.101729278 0.047322733 0.023460727 0.029796446 0.005639591 0.047283580
56 57 58 59 60 61
0.006279627 0.019740155 0.024955307 0.001098350 0.009409952 0.007819733
62 63 64 65 66 67
0.015422050 0.009303486 0.006766201 0.029129242 0.029926564 0.019480814
68 69 70 71 72 73
0.019154898 0.016454427 0.018695005 0.019419435 0.020170930 0.022884747
74 75 76 77 78 79
0.021723467 0.009429028 0.017051438 0.022830755 0.016806310 0.017434296
80 81 82 83 84 85
0.029693569 0.018488004 0.023607445 0.014978951 0.011199544 0.003222573
86 87 88 89 90 91
0.015408666 0.009430216 0.015152550 0.015013695 0.006496414 0.018111967
92 93 94 95 96 97
0.004718990 0.017056066 0.027439920 0.017185271 0.011410005 0.017831747
98 99 100
0.012158291 0.025864231 0.028913312
$kt
Time Series:
Start = 1983
End = 2003
Frequency = 1
1983 1984 1985 1986 1987 1988
10.4000310 14.3977617 11.0901931 12.6124446 6.2085516 8.2791188
1989 1990 1991 1992 1993 1994
8.7525259 8.7633664 5.5491212 3.4540233 -0.4560549 -4.3068125
1995 1996 1997 1998 1999 2000
1.6059850 -4.1701653 -12.2283837 -7.6839893 -15.0459194 -15.6955142
2001 2002 2003
-10.0152114 -11.4966937 -10.0143783
$ag
base a b c d
0.0000000 0.4945722 1.2596543 -0.5593774 2.5056202
attr(,"class")
[1] "coef"
ilc documentation built on May 2, 2019, 5:07 a.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
A purpose-built regression routine to fit the extended Lee-Carter model with an extra additive effect of an observable factor (other than age and period) on the log mortality mortality rates.
Usage
1 2 3 4 |
Arguments
dat |
|
year |
vector of years to be included in the regression (all available years by default) |
age |
vector of ages to be included in the regression (all available ages by default) |
dec.conv |
number of decimal places used to achieve convergence. The lower the value the faster the convergence of the fitting algorithm. Note: very high values could over fit the parameters. |
error |
type of error structure of the model choice (Poisson distribution of the errors by default) |
restype |
types of residuals, which also controls the type of the fitted value.
Thus, in the cases of |
scale |
logical, if TRUE, re-scale the interaction parameters so that the k_t has drift parameter equal to 1 (see also |
interpolate |
logical, if TRUE, replace before regression all zero or missing values in the mortality rates of |
verbose |
logical, it controls the amount of process information |
spar |
numerical smoothing spline parameter in the interval (0,1] (with a recommended value of 0.6). If it is not NULL, the interaction effects (i.e. β_x^{(0,1)}) are smoothed out after the initial regression. Consequently, the period and/or cohort effects are adjusted (smoothed out) accordingly. |
ax.fix |
vector of constant age effect to be used in the model (e.g. the fitted values of a standard LC regression to the experience of a large population). If NULL the base ax values are estimated from |
Details
This function models the number of deaths for a group within a generalised Lee-Carter framework with a Poisson or Gaussian error structure. The methodology quantifies the differences in the mortality experience of population subgroups differentiated by an additional measurable covariate (other than age and period). Additional covariate, for instance, could be related to geographical, socio-economic or race differences.
Value
An object of class elca with the following components:
lca |
list of fitted |
age |
vector of fitted ages |
year |
vector of fitted years |
ag |
parameter estimates of the effects of the extra factor |
ax |
parameter estimates (or ax.fix) of (mean) age-specific mortality rates across the entire fitting period |
bx |
parameter estimates of age-specific interaction effect between age and period |
kt |
parameter estimates of year-specific period trend of mortality rates |
adjust |
type of error structure used in fitting (e.g. "poisson" or "gaussian") |
label |
data label |
call |
copy of the R call to the model |
conv.iter |
number of iterations used to reach convergence |
mdev |
mean deviance of total and base lack of fit (see also |
model |
string expression of the fitted model |
df |
degree of freedom of the fitted GLM model |
Author(s)
Z. Butt and S. Haberman and H. L. Shang
References
Li, N. and Lee, R. D. (2005), ‘Cohort mortality forecasts for a group of populations: an extension of the Lee-Carter method’, Demography, 42(3), 575-594. Renshaw and Haberman (2006), ‘A cohort-based extension to the Lee-Carter model for mortality reduction factors.’, Insurance: Mathematics and Economics, 38, 556-570.
See Also
Examples
1 2 3 4 5 |
Example output
Loading required package: demography
Loading required package: forecast
Registered S3 method overwritten by 'quantmod':
method from
as.zoo.data.frame zoo
Registered S3 methods overwritten by 'demography':
method from
print.lca e1071
summary.lca e1071
This is demography 1.22
Loading required package: rainbow
Loading required package: MASS
Loading required package: pcaPP
Loading required package: date
Original sample: Multidimensional Mortality data for: CMI [male]
Across covariates:
years: 1983 - 2003
ages: 50 - 108
X: base, a, b, c, d
Applied sample: Multidimensional Mortality data for: CMI [male]
Across covariates:
years: 1983 - 2003
ages: 50 - 100
X: base, a, b, c, d
Fitting model: [ LC(g) = a(x)+a(g)+b(x)*k(t) ]
- with poisson error structure and with deaths as weights -
Note: 1161 cells have 0/NA deaths and 127 have 0/NA exposure
out of a total of 5355 data cells.
Starting values are:
age age.c int.c per per.c X4 X4.c
1 1 -1.286 0.02 1983 0 base 0
2 2 -2.508 0.02 1984 0 a 0
3 3 -3.028 0.02 1985 0 b 0
4 4 -3.343 0.02 1986 0 c 0
5 5 -3.153 0.02 1987 0 d 0
6 6 -3.149 0.02 1988 0
7 7 -3.402 0.02 1989 0
8 8 -2.956 0.02 1990 0
9 9 -3.553 0.02 1991 0
10 10 -3.091 0.02 1992 0
11 11 -3.202 0.02 1993 0
12 12 -3.233 0.02 1994 0
13 13 -3.141 0.02 1995 0
14 14 -3.051 0.02 1996 0
15 15 -2.832 0.02 1997 0
16 16 -2.681 0.02 1998 0
17 17 -2.739 0.02 1999 0
18 18 -2.530 0.02 2000 0
19 19 -2.407 0.02 2001 0
20 20 -2.436 0.02 2002 0
21 21 -2.171 0.02 2003 0
22 22 -2.142 0.02
23 23 -1.940 0.02
24 24 -1.754 0.02
25 25 -1.737 0.02
26 26 -1.583 0.02
27 27 -1.549 0.02
28 28 -1.300 0.02
29 29 -1.328 0.02
30 30 -1.230 0.02
31 31 -1.193 0.02
32 32 -0.936 0.02
33 33 -0.852 0.02
34 34 -0.879 0.02
35 35 -0.768 0.02
36 36 -0.612 0.02
37 37 -0.513 0.02
38 38 -0.465 0.02
39 39 -0.394 0.02
40 40 -0.266 0.02
41 41 -0.284 0.02
42 42 -0.202 0.02
43 43 -0.177 0.02
44 44 -0.043 0.02
45 45 0.119 0.02
46 46 0.102 0.02
47 47 -0.037 0.02
48 48 0.149 0.02
49 49 0.099 0.02
50 50 0.061 0.02
51 51 0.148 0.02
Iterative fit:
#iter Dev non-conv
1 6538592 0
2 29095571 1
3 4587766 0
4 1030918 0
5 962609.5 0
6 949241.1 0
7 949140.8 0
8 949282.3 1
9 949340.6 2
10 949360.1 3
11 949366.6 0
12 949368.8 0
13 949369.6 0
14 949369.8 0
15 949369.9 0
16 949370 0
17 949370 0
18 949370 0
19 949370 0
20 949370 0
21 949370 0
Iterations finished in: 21 steps
Updated values are:
age age.c int.c per per.c X4 X4.c
1 1 -3.57182 0.10173 1983 10.40003 base 0
2 2 -4.00941 0.04732 1984 14.39776 a 0.49457
3 3 -4.66980 0.02346 1985 11.09019 b 1.25965
4 4 -4.88614 0.02980 1986 12.61244 c -0.55938
5 5 -4.79139 0.00564 1987 6.20855 d 2.50562
6 6 -4.54121 0.04728 1988 8.27912
7 7 -4.92833 0.00628 1989 8.75253
8 8 -4.48164 0.01974 1990 8.76337
9 9 -4.74229 0.02496 1991 5.54912
10 10 -4.47916 0.00110 1992 3.45402
11 11 -4.49779 0.00941 1993 -0.45605
12 12 -4.55218 0.00782 1994 -4.30681
13 13 -4.46182 0.01542 1995 1.60598
14 14 -4.42328 0.00930 1996 -4.17017
15 15 -4.11704 0.00677 1997 -12.22838
16 16 -4.09382 0.02913 1998 -7.68399
17 17 -4.10641 0.02993 1999 -15.04592
18 18 -3.87744 0.01948 2000 -15.69551
19 19 -3.81737 0.01915 2001 -10.01521
20 20 -3.85569 0.01645 2002 -11.49669
21 21 -3.57086 0.01870 2003 -10.01438
22 22 -3.56004 0.01942
23 23 -3.29706 0.02017
24 24 -3.19681 0.02288
25 25 -3.15090 0.02172
26 26 -2.94881 0.00943
27 27 -2.90764 0.01705
28 28 -2.75753 0.02283
29 29 -2.69655 0.01681
30 30 -2.60415 0.01743
31 31 -2.61773 0.02969
32 32 -2.33704 0.01849
33 33 -2.37038 0.02361
34 34 -2.23598 0.01498
35 35 -2.15189 0.01120
36 36 -1.97745 0.00322
37 37 -2.00360 0.01541
38 38 -1.84680 0.00943
39 39 -1.79219 0.01515
40 40 -1.63259 0.01501
41 41 -1.67780 0.00650
42 42 -1.60457 0.01811
43 43 -1.59460 0.00472
44 44 -1.45978 0.01706
45 45 -1.30167 0.02744
46 46 -1.24017 0.01719
47 47 -1.40882 0.01141
48 48 -1.22433 0.01783
49 49 -1.27113 0.01216
50 50 -1.31165 0.02586
51 51 -1.14732 0.02891
total sums are:
bx kt
1 0
Warning messages:
1: A total of 1160 0/NA central mortality rates are re-estimated by the "interpolate" method.
2: In elca.rh(rfp.cmi, age = 50:100, interp = TRUE, dec = 3, verb = TRUE) :
There are 127 cells with 0/NA exposures, which are ignored in the current analysis.
Try reducing the fitted age range.
Alternatively, fit ELC model with error= "gaussian" .
3: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
4: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
5: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
6: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
7: In FUN(array(newX[, i], d.call, dn.call), ...) :
Please assign column name for the data matrix.
------------------------------------------------------------
Extended Lee-Carter Regression:
Fitted Model: LC(g) = a(x)+a(g)+b(x)*k(t)
------------------------------------------------------------
Call: elca.rh(dat = rfp.cmi, age = 50:100, dec.conv = 3, interpolate = TRUE,
verbose = TRUE)
Error Structure: poisson
Data Source: CMI : male over
calendar years: (1983 - 2003) , ages: (50 - 100)
and groups: base a b c d
Deviance convergence in: 21 iterations
dev dev.c df df.c
1 Mean deviance base 258.473 df base 3673
2 Mean deviance total 204.598 df tot 4845
$ax
[1] -3.571824 -4.009415 -4.669796 -4.886136 -4.791390 -4.541212 -4.928325
[8] -4.481641 -4.742290 -4.479161 -4.497794 -4.552177 -4.461817 -4.423281
[15] -4.117038 -4.093815 -4.106408 -3.877443 -3.817370 -3.855689 -3.570859
[22] -3.560043 -3.297057 -3.196805 -3.150899 -2.948812 -2.907638 -2.757527
[29] -2.696552 -2.604152 -2.617727 -2.337037 -2.370380 -2.235981 -2.151886
[36] -1.977447 -2.003599 -1.846796 -1.792191 -1.632592 -1.677804 -1.604574
[43] -1.594599 -1.459783 -1.301673 -1.240173 -1.408820 -1.224327 -1.271128
[50] -1.311654 -1.147319
$bx
50 51 52 53 54 55
0.101729278 0.047322733 0.023460727 0.029796446 0.005639591 0.047283580
56 57 58 59 60 61
0.006279627 0.019740155 0.024955307 0.001098350 0.009409952 0.007819733
62 63 64 65 66 67
0.015422050 0.009303486 0.006766201 0.029129242 0.029926564 0.019480814
68 69 70 71 72 73
0.019154898 0.016454427 0.018695005 0.019419435 0.020170930 0.022884747
74 75 76 77 78 79
0.021723467 0.009429028 0.017051438 0.022830755 0.016806310 0.017434296
80 81 82 83 84 85
0.029693569 0.018488004 0.023607445 0.014978951 0.011199544 0.003222573
86 87 88 89 90 91
0.015408666 0.009430216 0.015152550 0.015013695 0.006496414 0.018111967
92 93 94 95 96 97
0.004718990 0.017056066 0.027439920 0.017185271 0.011410005 0.017831747
98 99 100
0.012158291 0.025864231 0.028913312
$kt
Time Series:
Start = 1983
End = 2003
Frequency = 1
1983 1984 1985 1986 1987 1988
10.4000310 14.3977617 11.0901931 12.6124446 6.2085516 8.2791188
1989 1990 1991 1992 1993 1994
8.7525259 8.7633664 5.5491212 3.4540233 -0.4560549 -4.3068125
1995 1996 1997 1998 1999 2000
1.6059850 -4.1701653 -12.2283837 -7.6839893 -15.0459194 -15.6955142
2001 2002 2003
-10.0152114 -11.4966937 -10.0143783
$ag
base a b c d
0.0000000 0.4945722 1.2596543 -0.5593774 2.5056202
attr(,"class")
[1] "coef"
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