- 참고 : Statistics and Data Analysis for Financial Engineering
1. 함수 설명
- 관측된 시계열에 자기회귀누적이동평균모형(Autoregressive Integrated Moving Average Model)을 구축하는 데 유용한 함수를 요약하면 다음과 같다.
| 함수 | 설명 |
|---|---|
arima() |
ARIMA모형의 차수 \(p\)와 \(q\), 차분 수 \(d\)를 지정하여 모형 구축 |
auto.arima() |
모형의 차수 지정없이 자동적으로 최적화된 모형 구축 |
acf() |
상관도표그림 |
pacf() |
부분상관도표그림 |
Box.test() |
Ljung-Box Test |
checkresiduals() |
잔차가 백색잡음과정의 가정을 만족하는지 확인할 때 사용 |
forecast() |
예측 |
2. Random Walk and Momentum Plot
# AR(1)과정을 따르는 시계열 생성
set.seed(4631)
y1 <- arima.sim(n = 500, # 생성하고자하는 시계열 개수
list(ar = c(0.4))) # AR(1)과정 : Y_t = 0.4Y_{t-1}
y1Time Series:
Start = 1
End = 500
Frequency = 1
[1] -0.912343409 -1.701603284 1.309603454 0.691301009 0.832229499
[6] 0.555481039 -0.026684342 -0.362545410 0.693315964 -0.075195230
[11] 0.122254548 -0.370393783 -0.254577654 0.187517312 -1.708057556
[16] 0.986697733 -0.640294098 -0.885591883 -0.993104578 -0.135172972
[21] -0.194449428 -0.123467230 -1.216275158 -0.931739467 0.031253153
[26] -1.499635290 1.319634259 2.398330132 -0.229766437 1.332681837
[31] 0.157756638 -0.103097574 -0.102247983 2.536262232 3.218027097
[36] 0.445937187 -0.177490207 -1.602786627 -0.913088486 -1.406607759
[41] 0.099831368 -1.448052841 0.277834307 0.308540220 0.604106907
[46] 0.372668105 1.287859364 1.821393863 3.624531543 0.972845982
[51] -0.896416641 -1.777569846 -0.695491554 -1.375497500 1.007413097
[56] 1.142459616 0.721150425 -0.400368442 -0.961452076 -0.969140832
[61] -1.968814756 -0.798978245 -1.022224506 -1.673391629 -0.705015657
[66] 0.731069223 -0.427970435 1.303036486 1.862548515 0.017150757
[71] 1.279744874 1.015345454 2.535688480 1.043884770 1.103283049
[76] 2.050819678 0.166049018 1.325027055 0.982567526 -1.866691371
[81] -1.395834381 -2.988217256 -0.158685151 -0.491788055 -0.277597713
[86] 1.604608492 0.982089512 0.125541809 0.635742936 -0.235507928
[91] 0.132245190 -0.483377174 0.351436414 2.254957441 -0.203113635
[96] 1.751899755 1.201800038 2.079376478 1.641068416 0.012711375
[101] 0.455767972 -0.247430434 -0.419432042 -1.387682589 -0.806270199
[106] -1.412385197 -1.392636594 1.346065579 0.313476772 0.845857651
[111] 0.029078760 0.054460112 -0.885253382 -0.146332643 -0.618382418
[116] -0.920818943 -0.338097474 -0.752403660 -1.223742893 0.689190649
[121] 0.539929591 1.258735015 0.022750117 -0.869336438 -0.828436846
[126] -2.166858885 0.273965986 -1.037722970 -0.589593679 1.943031697
[131] 0.861840955 -0.520984536 0.770711156 1.101170054 0.846006091
[136] -2.173509959 -1.102412181 0.596310901 -0.281340324 0.879172771
[141] -0.409098649 1.747917946 0.444448222 -0.304067456 -0.970378141
[146] -0.237309951 -0.342986461 -0.497046979 -0.938379568 -0.230699307
[151] -0.696098707 -0.528535492 1.179030894 1.775753907 0.135886565
[156] 0.219726895 1.743976297 3.234301835 0.144757587 -1.026764175
[161] -1.290638719 -0.293860740 0.213341896 0.656327137 0.115984113
[166] 1.718136766 0.476141379 -0.987470475 -1.852852123 -2.641018085
[171] -0.024482023 0.491991907 0.975704783 -0.002667929 -0.136539414
[176] -0.081797455 -0.702349988 -0.149531603 0.467953220 0.075355046
[181] -0.558518159 0.281974471 -0.899797597 -1.328805544 -0.027818037
[186] 0.802736204 -0.968890576 -0.312796925 0.867650846 -0.257391157
[191] 0.172430824 -0.313593354 1.450511226 0.445832929 -0.349042275
[196] -2.148408416 -0.898881311 0.151759003 -2.266476419 0.117458198
[201] 1.766492454 -0.526284231 0.066299695 -0.566873492 0.965266585
[206] 1.037151790 -1.587680491 1.813497932 1.255998632 0.161051227
[211] 3.054247824 0.952734586 0.452260141 0.203109914 -0.612809154
[216] -1.421695279 -0.075559582 -0.150847637 -0.294898135 -0.191676540
[221] -0.984143939 -0.954974484 -1.952447346 -0.407105502 -1.010789331
[226] -0.376014894 -0.639076989 -1.140124930 0.607587073 0.094643343
[231] 1.277664446 1.289747945 0.765406149 0.965295215 0.768660214
[236] 2.119481591 1.975781515 1.218813681 0.168313822 0.299093643
[241] 1.074462951 0.832784873 2.086258930 0.916655229 -0.164630750
[246] -1.621972456 -1.146651373 -1.195654157 -0.949149364 0.278622989
[251] -0.349154803 -0.063271529 0.487016245 -0.256592989 -0.198698480
[256] 0.139972608 -1.401984577 -0.451456758 0.546480685 -0.118633637
[261] 0.834943293 1.232088158 -0.628640225 -1.307409713 -1.459409232
[266] -0.035619335 -0.178265267 0.126249921 -1.136784165 -2.529288363
[271] 0.386093595 0.625958961 -1.444598418 -0.757761283 -0.958362055
[276] -1.367402052 0.002037527 0.624906595 -1.513101287 0.318593880
[281] 0.341663053 0.750263380 -1.475926277 -0.157797771 -0.245202005
[286] -0.537848659 0.110833231 -1.759603329 -0.086328201 -0.127624951
[291] -0.587355815 -2.306925358 -0.187697381 -1.456423114 -0.186679580
[296] -0.190593254 0.446225486 -0.710882725 0.669572420 -1.652750327
[301] -2.254961336 -0.242182465 0.667217541 0.935181469 1.634818270
[306] 0.312140714 -1.067727411 0.416256545 0.597315306 -2.083217594
[311] -0.686914533 1.523376113 -0.589944868 0.809860833 -0.978598351
[316] -2.291625802 -0.468682911 -1.008714414 -1.689834548 -0.417147451
[321] -0.803271601 -0.832076120 0.065363742 -0.093893644 -1.049059664
[326] -1.444510947 -1.076551021 0.159225511 0.899011492 0.495151288
[331] -2.182210398 -0.704122441 -1.850799443 -0.911979956 -0.364270108
[336] 0.833332861 0.886993545 0.118737971 -0.608727843 -1.085932642
[341] 0.539806089 1.230683997 1.145261304 0.979793271 0.981000516
[346] 0.858859490 0.425712618 1.451944450 1.040442611 1.270852691
[351] 0.001724409 0.104234966 -0.341138452 -1.138528598 -1.618354797
[356] -1.699497595 -0.243076411 -0.651603385 1.343148562 -1.751853045
[361] -2.139267445 -1.229228556 0.255161099 1.813250189 1.862858137
[366] 0.208573786 0.501641144 -1.252334702 -1.172962682 -1.129589843
[371] -0.067739846 -0.062815505 -1.508314314 -0.335414905 -0.353518485
[376] 0.208172483 0.262092705 0.531750971 -0.824122936 0.017032118
[381] -1.015173764 0.365974862 -1.400376808 -1.443808435 -0.451407292
[386] 0.066326783 -0.895822894 0.900912062 -0.037442219 -0.456481687
[391] 2.377575658 2.430206896 -1.619430146 0.987079742 0.281160126
[396] -1.018931312 -0.794530108 1.494620796 1.388568007 -0.936099757
[401] -1.617907620 -1.593275135 -0.836974326 -0.798359722 -1.113207216
[406] 0.511917371 -1.735132035 0.177605755 -1.184768329 0.188680990
[411] -0.878350456 -0.586789699 -0.784074820 0.617449054 2.593695064
[416] 0.547644160 -1.303269875 0.256800524 -0.885652387 -0.367378463
[421] -0.458222229 0.632855057 -2.123028093 -0.601463044 -1.402681000
[426] -1.015006105 0.672074035 0.441082153 0.108044662 1.241146031
[431] 0.182850562 0.532864396 2.021748358 1.453454116 -0.351721426
[436] -0.760864939 -0.342109875 0.709747073 0.678984727 -0.345892776
[441] 1.049415015 1.099538689 0.931988534 -0.307293897 0.324674165
[446] 1.332195826 0.397157781 0.460074576 0.722534790 0.911401034
[451] 1.052024961 -0.770966223 -1.270573476 0.567950193 0.663503429
[456] 1.895802731 0.792548728 0.743382531 0.966190659 -1.038656980
[461] -0.638670844 -0.815864620 0.311205909 0.085439372 1.169179615
[466] 0.045130403 1.334452836 2.056890984 0.061869779 0.993814932
[471] -0.051180302 0.181854157 -0.867732369 -1.729196924 -1.068033410
[476] -1.336379582 -2.486448663 -0.743950909 -0.780466420 0.422205583
[481] -0.603969049 -1.086293175 -2.017387073 -2.463539252 -1.517367558
[486] -1.439320711 0.757130325 1.029261284 -1.059658946 -0.542882678
[491] 0.053945454 0.336157628 -0.645807708 0.372731417 2.163157158
[496] 1.349402356 0.636951690 1.163041710 2.250858491 -0.545616025y2 <- cumsum(y1) # cumsum : 누적합
y3 <- cumsum(y2)
par(mfrow=c(3, 1)) # 3개의 그래프를 한 화면에 출력
plot(y1, type = "l",
ylab = expression(y[1]),
lwd = 1, main = "(a)")
plot(y2, type = "l",
xlab = "Time", ylab = expression(y[2]),
lwd = 1, main = "(b)")
plot(y3, type = "l",
xlab = "Time", ylab = expression(y[3]),
lwd = 1, main = "(c)")
Result! (a) 그래프는 평균 0 근처에서 무작위로 변하며, 정상시계열로 보인다.
(b) 그래프는 Random Walk로 보인다. → 1차 차분 필요
(c) 그래프는 Momentum (위 또는 아래로 움직이기 시작하면 그 방향으로 계속 움직이는 경향)으로 보인다. → 2차 차분 필요
3. CPI 데이터셋
CSV 파일에 저장되어 있는 CPI (계절 조정된 미국의 소비자 물가 지수) 데이터셋은 1913년 1월 31일부터 2001년 11월 30일까지 월별 CPI가 기록되어져 있다.
# 데이터 불러오기
CPI.dat <- read.csv("C:/Users/User/Desktop/CPI.dat.csv")
CPI.dat X.Y..m..d CPI
1 1913-01-31 9.80
2 1913-02-28 9.80
3 1913-03-31 9.80
4 1913-04-30 9.80
5 1913-05-31 9.70
6 1913-06-30 9.80
7 1913-07-31 9.90
8 1913-08-31 9.90
9 1913-09-30 10.00
10 1913-10-31 10.00
11 1913-11-30 10.10
12 1913-12-31 10.00
13 1914-01-31 10.00
14 1914-02-28 9.90
15 1914-03-31 9.90
16 1914-04-30 9.80
17 1914-05-31 9.90
18 1914-06-30 9.90
19 1914-07-31 10.00
20 1914-08-31 10.20
21 1914-09-30 10.20
22 1914-10-31 10.10
23 1914-11-30 10.20
24 1914-12-31 10.10
25 1915-01-31 10.10
26 1915-02-28 10.00
27 1915-03-31 9.90
28 1915-04-30 10.00
29 1915-05-31 10.10
30 1915-06-30 10.10
31 1915-07-31 10.10
32 1915-08-31 10.10
33 1915-09-30 10.10
34 1915-10-31 10.20
35 1915-11-30 10.30
36 1915-12-31 10.30
37 1916-01-31 10.40
38 1916-02-29 10.40
39 1916-03-31 10.50
40 1916-04-30 10.60
41 1916-05-31 10.70
42 1916-06-30 10.80
43 1916-07-31 10.80
44 1916-08-31 10.90
45 1916-09-30 11.10
46 1916-10-31 11.30
47 1916-11-30 11.50
48 1916-12-31 11.60
49 1917-01-31 11.70
50 1917-02-28 12.00
51 1917-03-31 12.00
52 1917-04-30 12.60
53 1917-05-31 12.80
54 1917-06-30 13.00
55 1917-07-31 12.80
56 1917-08-31 13.00
57 1917-09-30 13.30
58 1917-10-31 13.50
59 1917-11-30 13.50
60 1917-12-31 13.70
61 1918-01-31 14.00
62 1918-02-28 14.10
63 1918-03-31 14.00
64 1918-04-30 14.20
65 1918-05-31 14.50
66 1918-06-30 14.70
67 1918-07-31 15.10
68 1918-08-31 15.40
69 1918-09-30 15.70
70 1918-10-31 16.00
71 1918-11-30 16.30
72 1918-12-31 16.50
73 1919-01-31 16.50
74 1919-02-28 16.20
75 1919-03-31 16.40
76 1919-04-30 16.70
77 1919-05-31 16.90
78 1919-06-30 16.90
79 1919-07-31 17.40
80 1919-08-31 17.70
81 1919-09-30 17.80
82 1919-10-31 18.10
83 1919-11-30 18.50
84 1919-12-31 18.90
85 1920-01-31 19.30
86 1920-02-29 19.50
87 1920-03-31 19.70
88 1920-04-30 20.30
89 1920-05-31 20.60
90 1920-06-30 20.90
91 1920-07-31 20.80
92 1920-08-31 20.30
93 1920-09-30 20.00
94 1920-10-31 19.90
95 1920-11-30 19.80
96 1920-12-31 19.40
97 1921-01-31 19.00
98 1921-02-28 18.40
99 1921-03-31 18.30
100 1921-04-30 18.10
101 1921-05-31 17.70
102 1921-06-30 17.60
103 1921-07-31 17.70
104 1921-08-31 17.70
105 1921-09-30 17.50
106 1921-10-31 17.50
107 1921-11-30 17.40
108 1921-12-31 17.30
109 1922-01-31 16.90
110 1922-02-28 16.90
111 1922-03-31 16.70
112 1922-04-30 16.70
113 1922-05-31 16.70
114 1922-06-30 16.70
115 1922-07-31 16.80
116 1922-08-31 16.60
117 1922-09-30 16.60
118 1922-10-31 16.70
119 1922-11-30 16.80
120 1922-12-31 16.90
121 1923-01-31 16.80
122 1923-02-28 16.80
123 1923-03-31 16.80
124 1923-04-30 16.90
125 1923-05-31 16.90
126 1923-06-30 17.00
127 1923-07-31 17.20
128 1923-08-31 17.10
129 1923-09-30 17.20
130 1923-10-31 17.30
131 1923-11-30 17.30
132 1923-12-31 17.30
133 1924-01-31 17.30
134 1924-02-29 17.20
135 1924-03-31 17.10
136 1924-04-30 17.00
137 1924-05-31 17.00
138 1924-06-30 17.00
139 1924-07-31 17.10
140 1924-08-31 17.00
141 1924-09-30 17.10
142 1924-10-31 17.20
143 1924-11-30 17.20
144 1924-12-31 17.30
145 1925-01-31 17.30
146 1925-02-28 17.20
147 1925-03-31 17.30
148 1925-04-30 17.20
149 1925-05-31 17.30
150 1925-06-30 17.50
151 1925-07-31 17.70
152 1925-08-31 17.70
153 1925-09-30 17.70
154 1925-10-31 17.70
155 1925-11-30 18.00
156 1925-12-31 17.90
157 1926-01-31 17.90
158 1926-02-28 17.90
159 1926-03-31 17.80
160 1926-04-30 17.90
161 1926-05-31 17.80
162 1926-06-30 17.70
163 1926-07-31 17.50
164 1926-08-31 17.40
165 1926-09-30 17.50
166 1926-10-31 17.60
167 1926-11-30 17.70
168 1926-12-31 17.70
169 1927-01-31 17.50
170 1927-02-28 17.40
171 1927-03-31 17.30
172 1927-04-30 17.30
173 1927-05-31 17.40
174 1927-06-30 17.60
175 1927-07-31 17.30
176 1927-08-31 17.20
177 1927-09-30 17.30
178 1927-10-31 17.40
179 1927-11-30 17.30
180 1927-12-31 17.30
181 1928-01-31 17.30
182 1928-02-29 17.10
183 1928-03-31 17.10
184 1928-04-30 17.10
185 1928-05-31 17.20
186 1928-06-30 17.10
187 1928-07-31 17.10
188 1928-08-31 17.10
189 1928-09-30 17.30
190 1928-10-31 17.20
191 1928-11-30 17.20
192 1928-12-31 17.10
193 1929-01-31 17.10
194 1929-02-28 17.10
195 1929-03-31 17.00
196 1929-04-30 16.90
197 1929-05-31 17.00
198 1929-06-30 17.10
199 1929-07-31 17.30
200 1929-08-31 17.30
201 1929-09-30 17.30
202 1929-10-31 17.30
203 1929-11-30 17.30
204 1929-12-31 17.20
205 1930-01-31 17.10
206 1930-02-28 17.00
207 1930-03-31 16.90
208 1930-04-30 17.00
209 1930-05-31 16.90
210 1930-06-30 16.80
211 1930-07-31 16.60
212 1930-08-31 16.50
213 1930-09-30 16.60
214 1930-10-31 16.50
215 1930-11-30 16.40
216 1930-12-31 16.10
217 1931-01-31 15.90
218 1931-02-28 15.70
219 1931-03-31 15.60
220 1931-04-30 15.50
221 1931-05-31 15.30
222 1931-06-30 15.10
223 1931-07-31 15.10
224 1931-08-31 15.10
225 1931-09-30 15.00
226 1931-10-31 14.90
227 1931-11-30 14.70
228 1931-12-31 14.60
229 1932-01-31 14.30
230 1932-02-29 14.10
231 1932-03-31 14.00
232 1932-04-30 13.90
233 1932-05-31 13.70
234 1932-06-30 13.60
235 1932-07-31 13.60
236 1932-08-31 13.50
237 1932-09-30 13.40
238 1932-10-31 13.30
239 1932-11-30 13.20
240 1932-12-31 13.10
241 1933-01-31 12.90
242 1933-02-28 12.70
243 1933-03-31 12.60
244 1933-04-30 12.60
245 1933-05-31 12.60
246 1933-06-30 12.70
247 1933-07-31 13.10
248 1933-08-31 13.20
249 1933-09-30 13.20
250 1933-10-31 13.20
251 1933-11-30 13.20
252 1933-12-31 13.20
253 1934-01-31 13.20
254 1934-02-28 13.30
255 1934-03-31 13.30
256 1934-04-30 13.30
257 1934-05-31 13.30
258 1934-06-30 13.40
259 1934-07-31 13.40
260 1934-08-31 13.40
261 1934-09-30 13.60
262 1934-10-31 13.50
263 1934-11-30 13.50
264 1934-12-31 13.40
265 1935-01-31 13.60
266 1935-02-28 13.70
267 1935-03-31 13.70
268 1935-04-30 13.80
269 1935-05-31 13.80
270 1935-06-30 13.70
271 1935-07-31 13.70
272 1935-08-31 13.70
273 1935-09-30 13.70
274 1935-10-31 13.70
275 1935-11-30 13.80
276 1935-12-31 13.80
277 1936-01-31 13.80
278 1936-02-29 13.80
279 1936-03-31 13.70
280 1936-04-30 13.70
281 1936-05-31 13.70
282 1936-06-30 13.80
283 1936-07-31 13.90
284 1936-08-31 14.00
285 1936-09-30 14.00
286 1936-10-31 14.00
287 1936-11-30 14.00
288 1936-12-31 14.00
289 1937-01-31 14.10
290 1937-02-28 14.10
291 1937-03-31 14.20
292 1937-04-30 14.30
293 1937-05-31 14.40
294 1937-06-30 14.40
295 1937-07-31 14.50
296 1937-08-31 14.50
297 1937-09-30 14.60
298 1937-10-31 14.60
299 1937-11-30 14.50
300 1937-12-31 14.40
301 1938-01-31 14.20
302 1938-02-28 14.10
303 1938-03-31 14.10
304 1938-04-30 14.20
305 1938-05-31 14.10
306 1938-06-30 14.10
307 1938-07-31 14.10
308 1938-08-31 14.10
309 1938-09-30 14.10
310 1938-10-31 14.00
311 1938-11-30 14.00
312 1938-12-31 14.00
313 1939-01-31 14.00
314 1939-02-28 13.90
315 1939-03-31 13.90
316 1939-04-30 13.80
317 1939-05-31 13.80
318 1939-06-30 13.80
319 1939-07-31 13.80
320 1939-08-31 13.80
321 1939-09-30 14.10
322 1939-10-31 14.00
323 1939-11-30 14.00
324 1939-12-31 14.00
325 1940-01-31 13.90
326 1940-02-29 14.00
327 1940-03-31 14.00
328 1940-04-30 14.00
329 1940-05-31 14.00
330 1940-06-30 14.10
331 1940-07-31 14.00
332 1940-08-31 14.00
333 1940-09-30 14.00
334 1940-10-31 14.00
335 1940-11-30 14.00
336 1940-12-31 14.10
337 1941-01-31 14.10
338 1941-02-28 14.10
339 1941-03-31 14.20
340 1941-04-30 14.30
341 1941-05-31 14.40
342 1941-06-30 14.70
343 1941-07-31 14.70
344 1941-08-31 14.90
345 1941-09-30 15.10
346 1941-10-31 15.30
347 1941-11-30 15.40
348 1941-12-31 15.50
349 1942-01-31 15.70
350 1942-02-28 15.80
351 1942-03-31 16.00
352 1942-04-30 16.10
353 1942-05-31 16.30
354 1942-06-30 16.30
355 1942-07-31 16.40
356 1942-08-31 16.50
357 1942-09-30 16.50
358 1942-10-31 16.70
359 1942-11-30 16.80
360 1942-12-31 16.90
361 1943-01-31 16.90
362 1943-02-28 16.90
363 1943-03-31 17.20
364 1943-04-30 17.40
365 1943-05-31 17.50
366 1943-06-30 17.50
367 1943-07-31 17.40
368 1943-08-31 17.30
369 1943-09-30 17.40
370 1943-10-31 17.40
371 1943-11-30 17.40
372 1943-12-31 17.40
373 1944-01-31 17.40
374 1944-02-29 17.40
375 1944-03-31 17.40
376 1944-04-30 17.50
377 1944-05-31 17.50
378 1944-06-30 17.60
379 1944-07-31 17.70
380 1944-08-31 17.70
381 1944-09-30 17.70
382 1944-10-31 17.70
383 1944-11-30 17.70
384 1944-12-31 17.80
385 1945-01-31 17.80
386 1945-02-28 17.80
387 1945-03-31 17.80
388 1945-04-30 17.80
389 1945-05-31 17.90
390 1945-06-30 18.10
391 1945-07-31 18.10
392 1945-08-31 18.10
393 1945-09-30 18.10
394 1945-10-31 18.10
395 1945-11-30 18.10
396 1945-12-31 18.20
397 1946-01-31 18.20
398 1946-02-28 18.10
399 1946-03-31 18.30
400 1946-04-30 18.40
401 1946-05-31 18.50
402 1946-06-30 18.70
403 1946-07-31 19.80
404 1946-08-31 20.20
405 1946-09-30 20.40
406 1946-10-31 20.80
407 1946-11-30 21.30
408 1946-12-31 21.50
409 1947-01-31 21.48
410 1947-02-28 21.62
411 1947-03-31 22.00
412 1947-04-30 22.00
413 1947-05-31 21.95
414 1947-06-30 22.08
415 1947-07-31 22.23
416 1947-08-31 22.40
417 1947-09-30 22.84
418 1947-10-31 22.91
419 1947-11-30 23.06
420 1947-12-31 23.41
421 1948-01-31 23.68
422 1948-02-29 23.67
423 1948-03-31 23.50
424 1948-04-30 23.82
425 1948-05-31 24.01
426 1948-06-30 24.15
427 1948-07-31 24.40
428 1948-08-31 24.43
429 1948-09-30 24.36
430 1948-10-31 24.31
431 1948-11-30 24.16
432 1948-12-31 24.05
433 1949-01-31 24.01
434 1949-02-28 23.91
435 1949-03-31 23.91
436 1949-04-30 23.92
437 1949-05-31 23.91
438 1949-06-30 23.92
439 1949-07-31 23.70
440 1949-08-31 23.70
441 1949-09-30 23.75
442 1949-10-31 23.67
443 1949-11-30 23.70
444 1949-12-31 23.61
445 1950-01-31 23.51
446 1950-02-28 23.61
447 1950-03-31 23.64
448 1950-04-30 23.65
449 1950-05-31 23.77
450 1950-06-30 23.88
451 1950-07-31 24.07
452 1950-08-31 24.20
453 1950-09-30 24.34
454 1950-10-31 24.50
455 1950-11-30 24.60
456 1950-12-31 24.98
457 1951-01-31 25.38
458 1951-02-28 25.83
459 1951-03-31 25.88
460 1951-04-30 25.92
461 1951-05-31 25.99
462 1951-06-30 25.93
463 1951-07-31 25.91
464 1951-08-31 25.86
465 1951-09-30 26.03
466 1951-10-31 26.16
467 1951-11-30 26.32
468 1951-12-31 26.47
469 1952-01-31 26.45
470 1952-02-29 26.41
471 1952-03-31 26.39
472 1952-04-30 26.46
473 1952-05-31 26.47
474 1952-06-30 26.53
475 1952-07-31 26.68
476 1952-08-31 26.69
477 1952-09-30 26.63
478 1952-10-31 26.69
479 1952-11-30 26.69
480 1952-12-31 26.71
481 1953-01-31 26.64
482 1953-02-28 26.59
483 1953-03-31 26.63
484 1953-04-30 26.69
485 1953-05-31 26.70
486 1953-06-30 26.77
487 1953-07-31 26.79
488 1953-08-31 26.85
489 1953-09-30 26.89
490 1953-10-31 26.95
491 1953-11-30 26.85
492 1953-12-31 26.87
493 1954-01-31 26.94
494 1954-02-28 26.99
495 1954-03-31 26.93
496 1954-04-30 26.86
497 1954-05-31 26.93
498 1954-06-30 26.94
499 1954-07-31 26.86
500 1954-08-31 26.85
501 1954-09-30 26.81
502 1954-10-31 26.72
503 1954-11-30 26.78
504 1954-12-31 26.77
505 1955-01-31 26.77
506 1955-02-28 26.82
507 1955-03-31 26.79
508 1955-04-30 26.79
509 1955-05-31 26.77
510 1955-06-30 26.71
511 1955-07-31 26.76
512 1955-08-31 26.72
513 1955-09-30 26.85
514 1955-10-31 26.82
515 1955-11-30 26.88
516 1955-12-31 26.87
517 1956-01-31 26.83
518 1956-02-29 26.86
519 1956-03-31 26.89
520 1956-04-30 26.93
521 1956-05-31 27.03
522 1956-06-30 27.15
523 1956-07-31 27.29
524 1956-08-31 27.31
525 1956-09-30 27.35
526 1956-10-31 27.51
527 1956-11-30 27.51
528 1956-12-31 27.63
529 1957-01-31 27.67
530 1957-02-28 27.80
531 1957-03-31 27.86
532 1957-04-30 27.93
533 1957-05-31 28.00
534 1957-06-30 28.11
535 1957-07-31 28.19
536 1957-08-31 28.28
537 1957-09-30 28.32
538 1957-10-31 28.32
539 1957-11-30 28.41
540 1957-12-31 28.47
541 1958-01-31 28.64
542 1958-02-28 28.70
543 1958-03-31 28.87
544 1958-04-30 28.94
545 1958-05-31 28.94
546 1958-06-30 28.91
547 1958-07-31 28.89
548 1958-08-31 28.94
549 1958-09-30 28.91
550 1958-10-31 28.91
551 1958-11-30 28.95
552 1958-12-31 28.97
553 1959-01-31 29.01
554 1959-02-28 29.00
555 1959-03-31 28.97
556 1959-04-30 28.98
557 1959-05-31 29.04
558 1959-06-30 29.11
559 1959-07-31 29.15
560 1959-08-31 29.18
561 1959-09-30 29.25
562 1959-10-31 29.35
563 1959-11-30 29.35
564 1959-12-31 29.41
565 1960-01-31 29.37
566 1960-02-29 29.41
567 1960-03-31 29.41
568 1960-04-30 29.54
569 1960-05-31 29.57
570 1960-06-30 29.61
571 1960-07-31 29.55
572 1960-08-31 29.61
573 1960-09-30 29.61
574 1960-10-31 29.75
575 1960-11-30 29.78
576 1960-12-31 29.81
577 1961-01-31 29.84
578 1961-02-28 29.84
579 1961-03-31 29.84
580 1961-04-30 29.81
581 1961-05-31 29.84
582 1961-06-30 29.84
583 1961-07-31 29.92
584 1961-08-31 29.94
585 1961-09-30 29.98
586 1961-10-31 29.98
587 1961-11-30 29.98
588 1961-12-31 30.01
589 1962-01-31 30.04
590 1962-02-28 30.11
591 1962-03-31 30.17
592 1962-04-30 30.21
593 1962-05-31 30.24
594 1962-06-30 30.21
595 1962-07-31 30.22
596 1962-08-31 30.28
597 1962-09-30 30.42
598 1962-10-31 30.38
599 1962-11-30 30.38
600 1962-12-31 30.38
601 1963-01-31 30.44
602 1963-02-28 30.48
603 1963-03-31 30.51
604 1963-04-30 30.48
605 1963-05-31 30.51
606 1963-06-30 30.61
607 1963-07-31 30.69
608 1963-08-31 30.75
609 1963-09-30 30.72
610 1963-10-31 30.75
611 1963-11-30 30.78
612 1963-12-31 30.88
613 1964-01-31 30.94
614 1964-02-29 30.91
615 1964-03-31 30.94
616 1964-04-30 30.95
617 1964-05-31 30.98
618 1964-06-30 31.01
619 1964-07-31 31.02
620 1964-08-31 31.05
621 1964-09-30 31.08
622 1964-10-31 31.12
623 1964-11-30 31.21
624 1964-12-31 31.25
625 1965-01-31 31.28
626 1965-02-28 31.28
627 1965-03-31 31.31
628 1965-04-30 31.38
629 1965-05-31 31.48
630 1965-06-30 31.61
631 1965-07-31 31.58
632 1965-08-31 31.55
633 1965-09-30 31.62
634 1965-10-31 31.65
635 1965-11-30 31.75
636 1965-12-31 31.85
637 1966-01-31 31.88
638 1966-02-28 32.08
639 1966-03-31 32.18
640 1966-04-30 32.28
641 1966-05-31 32.35
642 1966-06-30 32.38
643 1966-07-31 32.45
644 1966-08-31 32.65
645 1966-09-30 32.75
646 1966-10-31 32.85
647 1966-11-30 32.88
648 1966-12-31 32.92
649 1967-01-31 32.90
650 1967-02-28 33.00
651 1967-03-31 33.00
652 1967-04-30 33.10
653 1967-05-31 33.10
654 1967-06-30 33.30
655 1967-07-31 33.40
656 1967-08-31 33.50
657 1967-09-30 33.60
658 1967-10-31 33.70
659 1967-11-30 33.90
660 1967-12-31 34.00
661 1968-01-31 34.10
662 1968-02-29 34.20
663 1968-03-31 34.30
664 1968-04-30 34.40
665 1968-05-31 34.50
666 1968-06-30 34.70
667 1968-07-31 34.90
668 1968-08-31 35.00
669 1968-09-30 35.10
670 1968-10-31 35.30
671 1968-11-30 35.40
672 1968-12-31 35.60
673 1969-01-31 35.70
674 1969-02-28 35.80
675 1969-03-31 36.10
676 1969-04-30 36.30
677 1969-05-31 36.40
678 1969-06-30 36.60
679 1969-07-31 36.80
680 1969-08-31 36.90
681 1969-09-30 37.10
682 1969-10-31 37.30
683 1969-11-30 37.50
684 1969-12-31 37.70
685 1970-01-31 37.90
686 1970-02-28 38.10
687 1970-03-31 38.30
688 1970-04-30 38.50
689 1970-05-31 38.60
690 1970-06-30 38.80
691 1970-07-31 38.90
692 1970-08-31 39.00
693 1970-09-30 39.20
694 1970-10-31 39.40
695 1970-11-30 39.60
696 1970-12-31 39.80
697 1971-01-31 39.90
698 1971-02-28 39.90
699 1971-03-31 40.00
700 1971-04-30 40.10
701 1971-05-31 40.30
702 1971-06-30 40.50
703 1971-07-31 40.60
704 1971-08-31 40.70
705 1971-09-30 40.80
706 1971-10-31 40.90
707 1971-11-30 41.00
708 1971-12-31 41.10
709 1972-01-31 41.20
710 1972-02-29 41.40
711 1972-03-31 41.40
712 1972-04-30 41.50
713 1972-05-31 41.60
714 1972-06-30 41.70
715 1972-07-31 41.80
716 1972-08-31 41.90
717 1972-09-30 42.10
718 1972-10-31 42.20
719 1972-11-30 42.40
720 1972-12-31 42.50
721 1973-01-31 42.70
722 1973-02-28 43.00
723 1973-03-31 43.40
724 1973-04-30 43.70
725 1973-05-31 43.90
726 1973-06-30 44.20
727 1973-07-31 44.20
728 1973-08-31 45.00
729 1973-09-30 45.20
730 1973-10-31 45.60
731 1973-11-30 45.90
732 1973-12-31 46.30
733 1974-01-31 46.80
734 1974-02-28 47.30
735 1974-03-31 47.80
736 1974-04-30 48.10
737 1974-05-31 48.60
738 1974-06-30 49.00
739 1974-07-31 49.30
740 1974-08-31 49.90
741 1974-09-30 50.60
742 1974-10-31 51.00
743 1974-11-30 51.50
744 1974-12-31 51.90
745 1975-01-31 52.30
746 1975-02-28 52.60
747 1975-03-31 52.80
748 1975-04-30 53.00
749 1975-05-31 53.10
750 1975-06-30 53.50
751 1975-07-31 54.00
752 1975-08-31 54.20
753 1975-09-30 54.60
754 1975-10-31 54.90
755 1975-11-30 55.30
756 1975-12-31 55.60
757 1976-01-31 55.80
758 1976-02-29 55.90
759 1976-03-31 56.00
760 1976-04-30 56.10
761 1976-05-31 56.40
762 1976-06-30 56.70
763 1976-07-31 57.00
764 1976-08-31 57.30
765 1976-09-30 57.60
766 1976-10-31 57.90
767 1976-11-30 58.10
768 1976-12-31 58.40
769 1977-01-31 58.70
770 1977-02-28 59.30
771 1977-03-31 59.60
772 1977-04-30 60.00
773 1977-05-31 60.20
774 1977-06-30 60.50
775 1977-07-31 60.80
776 1977-08-31 61.10
777 1977-09-30 61.30
778 1977-10-31 61.60
779 1977-11-30 62.00
780 1977-12-31 62.30
781 1978-01-31 62.70
782 1978-02-28 63.00
783 1978-03-31 63.40
784 1978-04-30 63.90
785 1978-05-31 64.50
786 1978-06-30 65.00
787 1978-07-31 65.50
788 1978-08-31 65.90
789 1978-09-30 66.50
790 1978-10-31 67.10
791 1978-11-30 67.50
792 1978-12-31 67.90
793 1979-01-31 68.50
794 1979-02-28 69.20
795 1979-03-31 69.90
796 1979-04-30 70.60
797 1979-05-31 71.40
798 1979-06-30 72.20
799 1979-07-31 73.00
800 1979-08-31 73.70
801 1979-09-30 74.40
802 1979-10-31 75.20
803 1979-11-30 76.00
804 1979-12-31 76.90
805 1980-01-31 78.00
806 1980-02-29 79.00
807 1980-03-31 80.10
808 1980-04-30 80.90
809 1980-05-31 81.70
810 1980-06-30 82.50
811 1980-07-31 82.60
812 1980-08-31 83.20
813 1980-09-30 83.90
814 1980-10-31 84.70
815 1980-11-30 85.60
816 1980-12-31 86.40
817 1981-01-31 87.20
818 1981-02-28 88.00
819 1981-03-31 88.60
820 1981-04-30 89.10
821 1981-05-31 89.70
822 1981-06-30 90.50
823 1981-07-31 91.50
824 1981-08-31 92.20
825 1981-09-30 93.10
826 1981-10-31 93.40
827 1981-11-30 93.80
828 1981-12-31 94.10
829 1982-01-31 94.40
830 1982-02-28 94.70
831 1982-03-31 94.70
832 1982-04-30 95.00
833 1982-05-31 95.90
834 1982-06-30 97.00
835 1982-07-31 97.50
836 1982-08-31 97.70
837 1982-09-30 97.70
838 1982-10-31 98.10
839 1982-11-30 98.00
840 1982-12-31 97.70
841 1983-01-31 97.90
842 1983-02-28 98.00
843 1983-03-31 98.10
844 1983-04-30 98.80
845 1983-05-31 99.20
846 1983-06-30 99.40
847 1983-07-31 99.80
848 1983-08-31 100.10
849 1983-09-30 100.40
850 1983-10-31 100.80
851 1983-11-30 101.10
852 1983-12-31 101.40
853 1984-01-31 102.10
854 1984-02-29 102.60
855 1984-03-31 102.90
856 1984-04-30 103.30
857 1984-05-31 103.50
858 1984-06-30 103.70
859 1984-07-31 104.10
860 1984-08-31 104.40
861 1984-09-30 104.70
862 1984-10-31 105.10
863 1984-11-30 105.30
864 1984-12-31 105.50
865 1985-01-31 105.70
866 1985-02-28 106.30
867 1985-03-31 106.80
868 1985-04-30 107.00
869 1985-05-31 107.20
870 1985-06-30 107.50
871 1985-07-31 107.70
872 1985-08-31 107.90
873 1985-09-30 108.10
874 1985-10-31 108.50
875 1985-11-30 109.00
876 1985-12-31 109.50
877 1986-01-31 109.90
878 1986-02-28 109.70
879 1986-03-31 109.10
880 1986-04-30 108.70
881 1986-05-31 109.00
882 1986-06-30 109.40
883 1986-07-31 109.50
884 1986-08-31 109.60
885 1986-09-30 110.00
886 1986-10-31 110.20
887 1986-11-30 110.40
888 1986-12-31 110.80
889 1987-01-31 111.50
890 1987-02-28 111.90
891 1987-03-31 112.30
892 1987-04-30 112.80
893 1987-05-31 113.10
894 1987-06-30 113.60
895 1987-07-31 113.90
896 1987-08-31 114.40
897 1987-09-30 114.80
898 1987-10-31 115.10
899 1987-11-30 115.50
900 1987-12-31 115.70
901 1988-01-31 116.10
902 1988-02-29 116.20
903 1988-03-31 116.60
904 1988-04-30 117.20
905 1988-05-31 117.60
906 1988-06-30 118.10
907 1988-07-31 118.60
908 1988-08-31 119.00
909 1988-09-30 119.60
910 1988-10-31 120.00
911 1988-11-30 120.40
912 1988-12-31 120.80
913 1989-01-31 121.30
914 1989-02-28 121.70
915 1989-03-31 122.30
916 1989-04-30 123.20
917 1989-05-31 123.80
918 1989-06-30 124.10
919 1989-07-31 124.60
920 1989-08-31 124.60
921 1989-09-30 124.90
922 1989-10-31 125.50
923 1989-11-30 125.90
924 1989-12-31 126.40
925 1990-01-31 127.60
926 1990-02-28 128.10
927 1990-03-31 128.60
928 1990-04-30 129.00
929 1990-05-31 129.20
930 1990-06-30 130.00
931 1990-07-31 130.60
932 1990-08-31 131.70
933 1990-09-30 132.60
934 1990-10-31 133.50
935 1990-11-30 133.80
936 1990-12-31 134.30
937 1991-01-31 134.80
938 1991-02-28 134.90
939 1991-03-31 134.90
940 1991-04-30 135.20
941 1991-05-31 135.70
942 1991-06-30 136.10
943 1991-07-31 136.30
944 1991-08-31 136.70
945 1991-09-30 137.10
946 1991-10-31 137.30
947 1991-11-30 137.90
948 1991-12-31 138.30
949 1992-01-31 138.40
950 1992-02-29 138.70
951 1992-03-31 139.20
952 1992-04-30 139.50
953 1992-05-31 139.80
954 1992-06-30 140.20
955 1992-07-31 140.60
956 1992-08-31 140.90
957 1992-09-30 141.20
958 1992-10-31 141.80
959 1992-11-30 142.20
960 1992-12-31 142.40
961 1993-01-31 142.80
962 1993-02-28 143.20
963 1993-03-31 143.40
964 1993-04-30 143.90
965 1993-05-31 144.30
966 1993-06-30 144.40
967 1993-07-31 144.60
968 1993-08-31 144.90
969 1993-09-30 145.10
970 1993-10-31 145.70
971 1993-11-30 146.00
972 1993-12-31 146.40
973 1994-01-31 146.40
974 1994-02-28 146.80
975 1994-03-31 147.20
976 1994-04-30 147.30
977 1994-05-31 147.60
978 1994-06-30 148.00
979 1994-07-31 148.50
980 1994-08-31 149.10
981 1994-09-30 149.40
982 1994-10-31 149.50
983 1994-11-30 149.90
984 1994-12-31 150.20
985 1995-01-31 150.60
986 1995-02-28 151.00
987 1995-03-31 151.30
988 1995-04-30 151.90
989 1995-05-31 152.20
990 1995-06-30 152.50
991 1995-07-31 152.70
992 1995-08-31 153.00
993 1995-09-30 153.20
994 1995-10-31 153.70
995 1995-11-30 153.80
996 1995-12-31 154.10
997 1996-01-31 154.80
998 1996-02-29 155.10
999 1996-03-31 155.60
1000 1996-04-30 156.20
1001 1996-05-31 156.50
1002 1996-06-30 156.80
1003 1996-07-31 157.10
1004 1996-08-31 157.30
1005 1996-09-30 157.80
1006 1996-10-31 158.30
1007 1996-11-30 158.80
1008 1996-12-31 159.20
1009 1997-01-31 159.50
1010 1997-02-28 159.90
1011 1997-03-31 159.90
1012 1997-04-30 160.00
1013 1997-05-31 160.10
1014 1997-06-30 160.30
1015 1997-07-31 160.50
1016 1997-08-31 160.80
1017 1997-09-30 161.30
1018 1997-10-31 161.60
1019 1997-11-30 161.80
1020 1997-12-31 161.90
1021 1998-01-31 162.10
1022 1998-02-28 162.20
1023 1998-03-31 162.20
1024 1998-04-30 162.40
1025 1998-05-31 162.70
1026 1998-06-30 162.90
1027 1998-07-31 163.20
1028 1998-08-31 163.50
1029 1998-09-30 163.50
1030 1998-10-31 163.90
1031 1998-11-30 164.20
1032 1998-12-31 164.50
1033 1999-01-31 164.80
1034 1999-02-28 164.80
1035 1999-03-31 165.00
1036 1999-04-30 166.00
1037 1999-05-31 166.00
1038 1999-06-30 166.10
1039 1999-07-31 166.70
1040 1999-08-31 167.10
1041 1999-09-30 167.80
1042 1999-10-31 168.20
1043 1999-11-30 168.50
1044 1999-12-31 168.90
1045 2000-01-31 169.40
1046 2000-02-29 170.20
1047 2000-03-31 171.20
1048 2000-04-30 171.10
1049 2000-05-31 171.30
1050 2000-06-30 172.20
1051 2000-07-31 172.70
1052 2000-08-31 172.80
1053 2000-09-30 173.60
1054 2000-10-31 173.90
1055 2000-11-30 174.30
1056 2000-12-31 174.60
1057 2001-01-31 175.70
1058 2001-02-28 176.20
1059 2001-03-31 176.30
1060 2001-04-30 176.80
1061 2001-05-31 177.50
1062 2001-06-30 177.90
1063 2001-07-31 177.40
1064 2001-08-31 177.50
1065 2001-09-30 178.20
1066 2001-10-31 177.60
1067 2001-11-30 177.60# 1977-01-31 ~ 1987-12-31 CPI만 추출
CPI <- as.matrix(CPI.dat$CPI)[769:900,]
CPI [1] 58.7 59.3 59.6 60.0 60.2 60.5 60.8 61.1 61.3 61.6
[11] 62.0 62.3 62.7 63.0 63.4 63.9 64.5 65.0 65.5 65.9
[21] 66.5 67.1 67.5 67.9 68.5 69.2 69.9 70.6 71.4 72.2
[31] 73.0 73.7 74.4 75.2 76.0 76.9 78.0 79.0 80.1 80.9
[41] 81.7 82.5 82.6 83.2 83.9 84.7 85.6 86.4 87.2 88.0
[51] 88.6 89.1 89.7 90.5 91.5 92.2 93.1 93.4 93.8 94.1
[61] 94.4 94.7 94.7 95.0 95.9 97.0 97.5 97.7 97.7 98.1
[71] 98.0 97.7 97.9 98.0 98.1 98.8 99.2 99.4 99.8 100.1
[81] 100.4 100.8 101.1 101.4 102.1 102.6 102.9 103.3 103.5 103.7
[91] 104.1 104.4 104.7 105.1 105.3 105.5 105.7 106.3 106.8 107.0
[101] 107.2 107.5 107.7 107.9 108.1 108.5 109.0 109.5 109.9 109.7
[111] 109.1 108.7 109.0 109.4 109.5 109.6 110.0 110.2 110.4 110.8
[121] 111.5 111.9 112.3 112.8 113.1 113.6 113.9 114.4 114.8 115.1
[131] 115.5 115.7 [,1]
[1,] 0.0101695792
[2,] 0.0050462681
[3,] 0.0066889882
[4,] 0.0033277901
[5,] 0.0049710127
[6,] 0.0049464239
[7,] 0.0049220772
[8,] 0.0032679768
[9,] 0.0048820276
[10,] 0.0064725145
[11,] 0.0048270407
[12,] 0.0064000218
[13,] 0.0047732788
[14,] 0.0063291351
[15,] 0.0078554999
[16,] 0.0093458624
[17,] 0.0077220461
[18,] 0.0076628727
[19,] 0.0060882989
[20,] 0.0090635062
[21,] 0.0089820963
[22,] 0.0059435539
[23,] 0.0059084367
[24,] 0.0087977107
[25,] 0.0101671174
[26,] 0.0100647866
[27,] 0.0099644953
[28,] 0.0112677248
[29,] 0.0111421766
[30,] 0.0110193952
[31,] 0.0095433580
[32,] 0.0094531426
[33,] 0.0106952891
[34,] 0.0105821093
[35,] 0.0117725362
[36,] 0.0142029502
[37,] 0.0127390258
[38,] 0.0138280016
[39,] 0.0099379700
[40,] 0.0098401778
[41,] 0.0097442915
[42,] 0.0012113872
[43,] 0.0072376673
[44,] 0.0083782656
[45,] 0.0094899882
[46,] 0.0105696815
[47,] 0.0093023927
[48,] 0.0092166551
[49,] 0.0091324836
[50,] 0.0067950431
[51,] 0.0056274769
[52,] 0.0067114346
[53,] 0.0088790816
[54,] 0.0109891216
[55,] 0.0076211583
[56,] 0.0097140537
[57,] 0.0032171610
[58,] 0.0042735108
[59,] 0.0031931906
[60,] 0.0031830266
[61,] 0.0031729270
[62,] 0.0000000000
[63,] 0.0031628914
[64,] 0.0094290903
[65,] 0.0114049966
[66,] 0.0051413995
[67,] 0.0020491810
[68,] 0.0000000000
[69,] 0.0040858075
[70,] -0.0010198879
[71,] -0.0030659196
[72,] 0.0020449905
[73,] 0.0010209291
[74,] 0.0010198879
[75,] 0.0071102382
[76,] 0.0040404095
[77,] 0.0020140994
[78,] 0.0040160697
[79,] 0.0030015030
[80,] 0.0029925209
[81,] 0.0039761484
[82,] 0.0029717704
[83,] 0.0029629651
[84,] 0.0068796340
[85,] 0.0048852076
[86,] 0.0029197101
[87,] 0.0038797333
[88,] 0.0019342366
[89,] 0.0019305025
[90,] 0.0038498604
[91,] 0.0028776998
[92,] 0.0028694424
[93,] 0.0038131600
[94,] 0.0019011413
[95,] 0.0018975338
[96,] 0.0018939400
[97,] 0.0056603925
[98,] 0.0046926412
[99,] 0.0018709079
[100,] 0.0018674142
[101,] 0.0027945989
[102,] 0.0018587366
[103,] 0.0018552881
[104,] 0.0018518524
[105,] 0.0036934483
[106,] 0.0045977092
[107,] 0.0045766670
[108,] 0.0036463122
[109,] -0.0018214941
[110,] -0.0054844744
[111,] -0.0036730987
[112,] 0.0027560881
[113,] 0.0036630078
[114,] 0.0009136593
[115,] 0.0009128253
[116,] 0.0036429913
[117,] 0.0018165309
[118,] 0.0018132371
[119,] 0.0036166405
[120,] 0.0062978166
[121,] 0.0035810244
[122,] 0.0035682464
[123,] 0.0044424773
[124,] 0.0026560441
[125,] 0.0044111232
[126,] 0.0026373642
[127,] 0.0043802085
[128,] 0.0034904049
[129,] 0.0026098318
[130,] 0.0034692142
[131,] 0.0017301042 [,1]
[1,] -5.123311e-03
[2,] 1.642720e-03
[3,] -3.361198e-03
[4,] 1.643223e-03
[5,] -2.458879e-05
[6,] -2.434673e-05
[7,] -1.654100e-03
[8,] 1.614051e-03
[9,] 1.590487e-03
[10,] -1.645474e-03
[11,] 1.572981e-03
[12,] -1.626743e-03
[13,] 1.555856e-03
[14,] 1.526365e-03
[15,] 1.490362e-03
[16,] -1.623816e-03
[17,] -5.917335e-05
[18,] -1.574574e-03
[19,] 2.975207e-03
[20,] -8.140984e-05
[21,] -3.038542e-03
[22,] -3.511721e-05
[23,] 2.889274e-03
[24,] 1.369407e-03
[25,] -1.023307e-04
[26,] -1.002914e-04
[27,] 1.303230e-03
[28,] -1.255483e-04
[29,] -1.227813e-04
[30,] -1.476037e-03
[31,] -9.021540e-05
[32,] 1.242146e-03
[33,] -1.131798e-04
[34,] 1.190427e-03
[35,] 2.430414e-03
[36,] -1.463924e-03
[37,] 1.088976e-03
[38,] -3.890032e-03
[39,] -9.779219e-05
[40,] -9.588633e-05
[41,] -8.532904e-03
[42,] 6.026280e-03
[43,] 1.140598e-03
[44,] 1.111723e-03
[45,] 1.079693e-03
[46,] -1.267289e-03
[47,] -8.573756e-05
[48,] -8.417154e-05
[49,] -2.337440e-03
[50,] -1.167566e-03
[51,] 1.083958e-03
[52,] 2.167647e-03
[53,] 2.110040e-03
[54,] -3.367963e-03
[55,] 2.092895e-03
[56,] -6.496893e-03
[57,] 1.056350e-03
[58,] -1.080320e-03
[59,] -1.016402e-05
[60,] -1.009952e-05
[61,] -3.172927e-03
[62,] 3.162891e-03
[63,] 6.266199e-03
[64,] 1.975906e-03
[65,] -6.263597e-03
[66,] -3.092218e-03
[67,] -2.049181e-03
[68,] 4.085808e-03
[69,] -5.105695e-03
[70,] -2.046032e-03
[71,] 5.110910e-03
[72,] -1.024061e-03
[73,] -1.041233e-06
[74,] 6.090350e-03
[75,] -3.069829e-03
[76,] -2.026310e-03
[77,] 2.001970e-03
[78,] -1.014567e-03
[79,] -8.982067e-06
[80,] 9.836274e-04
[81,] -1.004378e-03
[82,] -8.805259e-06
[83,] 3.916669e-03
[84,] -1.994426e-03
[85,] -1.965497e-03
[86,] 9.600232e-04
[87,] -1.945497e-03
[88,] -3.734050e-06
[89,] 1.919358e-03
[90,] -9.721606e-04
[91,] -8.257400e-06
[92,] 9.437176e-04
[93,] -1.912019e-03
[94,] -3.607481e-06
[95,] -3.593816e-06
[96,] 3.766453e-03
[97,] -9.677513e-04
[98,] -2.821733e-03
[99,] -3.493761e-06
[100,] 9.271848e-04
[101,] -9.358623e-04
[102,] -3.448493e-06
[103,] -3.435721e-06
[104,] 1.841596e-03
[105,] 9.042609e-04
[106,] -2.104222e-05
[107,] -9.303549e-04
[108,] -5.467806e-03
[109,] -3.662980e-03
[110,] 1.811376e-03
[111,] 6.429187e-03
[112,] 9.069197e-04
[113,] -2.749348e-03
[114,] -8.340113e-07
[115,] 2.730166e-03
[116,] -1.826460e-03
[117,] -3.293802e-06
[118,] 1.803403e-03
[119,] 2.681176e-03
[120,] -2.716792e-03
[121,] -1.277799e-05
[122,] 8.742309e-04
[123,] -1.786433e-03
[124,] 1.755079e-03
[125,] -1.773759e-03
[126,] 1.742844e-03
[127,] -8.898036e-04
[128,] -8.805731e-04
[129,] 8.593824e-04
[130,] -1.739110e-03Caution! 함수 diff()를 이용하여 시계열을 차분할 수 있으며, 옵션 diff에 차분 횟수를 입력하면 된다.
par(mfrow=c(3, 1)) # 3개의 그래프를 한 화면에 출력
plot(ts(log(CPI), # log(CPI)를 ts로 변환
start = c(1977, 1), # 시계열의 시작 날짜 / c(1977, 1) : 1977년 1월
frequency = 12), # 주기 / 12 : 월별 시계열로 1년에 12번 관측
xlab = "year", ylab = "log(CPI)", # 축 이름
type = "b", # 점과 선을 함께 표시
main = "(a)") # 제목
plot(ts(as.vector(CPI_diff1), # log(CPI)를 1번 차분한 시계열을 ts로 변환
start = c(1977, 2), # 시계열의 시작 날짜 / c(1977, 2) : 1977년 2월
frequency = 12),
xlab = "year", ylab = expression(paste(Delta," log(CPI)")),
type = "b",
main = "(b)")
plot(ts(as.vector(CPI_diff2), # log(CPI)를 2번 차분한 시계열을 ts로 변환
start = c(1977, 3), # 시계열의 시작 날짜 / c(1977, 3) : 1977년 3월
frequency = 12),
xlab ="year", ylab = expression(paste(Delta^2," log(CPI)")),
type = "b",
main = "(c)")
Caution! 함수 ts()를 이용하여 시계열 객체로 변환할 수 있으며, 옵션 start에는 시계열의 시작 날짜, 옵션 frequency에는 주기를 입력한다.
Result! (a) 그래프를 통해 원 시계열 log(CPI)는 Momentum 현상을 보인다는 것을 알 수 있다.
(b) 그래프를 통해 1번 차분한 log(CPI)는 Momentum 현상은 보이지 않으나, 시간이 흐름에 따라 평균이 변한다는 것을 알 수 있다.
(c) 그래프를 통해 2번 차분한 log(CPI)는 시간이 흐름에 따라 평균이 0에서 변하지 않는다는 것을 알 수 있다.
3-1. ARIMA 모형
Call:
arima(x = CPI_diff2, order = c(0, 0, 2))
Coefficients:
ma1 ma2 intercept
-0.3433 -0.3694 0e+00
s.e. 0.0831 0.0837 1e-04
sigma^2 estimated as 5.062e-06: log likelihood = 607.81, aic = -1207.62Result! 2번 차분한 log(CPI)에 대해 구축된 MA(2) 모형, 즉, 원 시계열 log(CPI)에 대해 구축된 ARIMA(0,2,2) 모형은 \((1-B)^2Y_t=\epsilon_t-0.3433\epsilon_{t-1}-0.3694\epsilon_{t-2}\)이다.
# 잔차를 이용한 모형 진단
Box.test(fit_ma$resid, # 잔차
lag = 20,
type = "Ljung-Box",
fitdf = 2) # 추정된 theta 개수
Box-Ljung test
data: fit_ma$resid
X-squared = 26.956, df = 18, p-value = 0.07983Result! 귀무가설 \(H_0 : \rho(1)=\rho(2)=\cdots=\rho(20)=0\)에 대한 검정 결과에 따르면, \(p\)값이 0.07983이므로 유의수준 0.05에서 \(p\)값이 0.05보다 크기 때문에 귀무가설을 기각하지 못한다. 즉, 잔차에 대해 시차 20까지의 자기상관계수 \(\rho(1), \rho(2), \cdots, \rho(20)\) 중 유의한 자기상관계수가 적어도 1개 존재한다는 증거가 부족하며, 2번 차분한 log(CPI)에 대해 MA(2) 모형을 가정하는 것이 적절하다.
par(mfrow=c(2,2)) # 1행에 2개의 그래프를 출력 -> 총 2개의 행으로 4개의 그래프가 출력됨
acf(log(CPI),main = "(a) log(CPI)")
acf(CPI_diff1, main = expression(paste("(b) ",Delta," log(CPI)")))
acf(CPI_diff2, main = expression(paste("(c) ",Delta^2," log(CPI)")))
acf(fit_ma$resid, main = "(d) residuals, ARIMA(0,2,2)")
Result! (a) 그래프를 통해 원 시계열 log(CPI)의 자기상관계수 ACF는 천천히 감소하고 있다는 것을 알 수 있으며, 이는 원 시계열이 비정상시계열임을 의미한다.
(c) 그래프를 통해 2번 차분한 log(CPI)의 자기상관계수 ACF는 처음 2개의 시차에서 큰 자기상관을 가지고, 그 이후는 작은 자기상관을 가진다는 것을 알 수 있다.
(d) 그래프를 통해 2번 차분한 log(CPI)에 구축된 MA(2) 모형의 잔차는 자기상관이 존재하지 않다는 것을 알 수 있다. 이는 2번 차분한 log(CPI)에 MA(2) 모형, 즉, 원 시계열 log(CPI)에 ARIMA(0,2,2) 모형을 가정하는 것이 적절하다는 것을 의미한다.
4. Mishkin 데이터셋
Package "Ecdat"에서 제공하는 Mishkin 데이터셋은 1950년 2월부터 1990년 12월 사이에 인플레이션율에 대한 시계열 데이터셋이다.
# 패키지 설치
pacman::p_load("Ecdat")
# 데이터 불러오기
data(Mishkin, package = "Ecdat")
y <- as.vector(Mishkin[,1]) # 월별 인플레이션율 추출
y [1] -3.552289 5.247540 1.692860 5.064298 6.719322 11.668920
[7] 9.912501 8.346786 6.517766 4.865085 16.076321 19.154240
[13] 14.061910 4.650814 1.546310 4.627010 -1.540355 1.540355
[19] 0.000000 7.672257 6.102576 6.209472 4.533152 0.000000
[25] -7.564866 0.000000 19.570490 -1.494151 1.494151 0.000000
[31] -1.494151 0.000000 1.494151 0.000000 -1.494151 -2.993894
[37] -6.010291 3.008908 1.501630 2.997634 4.482457 2.979029
[43] 2.971549 1.483071 2.960655 -4.443726 -1.484906 2.967978
[49] -1.483071 -1.484906 -2.839906 2.839906 1.484906 0.000000
[55] -1.484906 -2.839906 -2.982291 2.982291 -2.982291 0.000000
[61] 0.000000 1.492020 0.000000 0.000000 0.000000 2.978693
[67] -2.978693 4.330177 0.000000 1.484906 -2.836390 0.000000
[73] 0.000000 2.836390 1.483071 5.914023 5.751562 8.774637
[79] -1.457907 1.457907 5.682521 1.449363 2.893485 0.000000
[85] 2.886525 2.748830 2.873018 2.866156 7.006195 5.678584
[91] 1.415482 1.413814 0.000000 4.103441 0.000000 7.020216
[97] 1.399128 6.844860 2.777502 0.000000 1.386248 1.384744
[103] 0.000000 0.000000 1.383148 1.381556 -1.381556 1.381556
[109] -1.381556 0.000000 1.381556 1.254555 5.504627 2.742874
[115] -1.370653 2.739743 3.973921 1.363010 -2.727571 1.364560
[121] 0.000000 6.799529 0.000000 2.586158 0.000000 0.000000
[127] 1.352397 5.394306 1.344819 0.000000 0.000000 1.343314
[133] -1.343314 1.343314 0.000000 1.341812 5.230890 -1.335972
[139] 2.670459 0.000000 0.000000 0.000000 0.000000 2.664529
[145] 2.658626 2.532271 0.000000 0.000000 2.647071 0.000000
[151] 6.592472 -1.315600 0.000000 -1.317044 1.317044 1.315600
[157] 1.194725 0.000000 0.000000 5.242814 3.917133 0.000000
[163] 1.302875 1.183181 0.000000 2.598952 0.000000 0.000000
[169] -1.298772 1.298772 2.593336 0.000000 2.587653 1.291782
[175] -1.291782 2.582175 1.171857 2.574118 1.284992 -1.284992
[181] 0.000000 1.284992 3.846739 2.557659 6.254851 1.270154
[187] -2.541654 2.541654 0.000000 2.536194 3.679570 0.000000
[193] 7.553715 2.393494 4.999561 0.000000 2.491991 2.486741
[199] 7.317456 1.233940 4.811423 0.000000 0.000000 0.000000
[205] 1.227743 2.451723 2.446724 3.549967 3.649961 4.849325
[211] 2.307577 3.617287 3.606416 2.289339 3.588767 4.768305
[217] 3.456034 4.735883 3.539687 4.596686 3.515803 3.399432
[223] 3.495549 3.485477 5.681739 3.458990 2.300467 3.338259
[229] 4.574945 6.726781 5.662605 3.282331 7.861482 3.252032
[235] 4.457231 4.339972 4.424734 5.408054 7.570396 2.182552
[241] 5.439077 3.153226 7.553640 4.197689 3.211568 5.236804
[247] 2.126986 4.242622 6.240278 4.110415 5.237093 2.088382
[253] 5.110830 5.183215 6.096802 5.041712 6.133401 2.962434
[259] 1.016681 1.015820 2.029064 1.013247 3.952596 1.009064
[265] 5.945445 3.007194 2.999539 2.901685 2.984980 2.977574
[271] 2.880218 4.934430 2.950948 3.834231 2.934329 5.758887
[277] 8.630559 11.440700 9.436906 6.536329 7.438794 3.659775
[283] 19.304211 1.838323 7.242186 8.105994 8.051480 9.785962
[289] 15.814170 12.122030 6.883693 12.010690 10.111450 7.566704
[295] 13.394280 12.342670 9.006326 9.680984 7.246837 4.855838
[301] 6.371189 4.738003 4.719480 5.566699 9.340268 12.388900
[307] 4.665991 5.350513 6.864336 6.061113 5.269925 3.040442
[313] 3.720887 3.778037 5.200997 6.673568 7.379198 5.857414
[319] 5.093520 5.072093 5.117023 3.575525 5.014266 6.435108
[325] 12.054450 6.336976 9.123064 6.255971 7.616233 4.105433
[331] 4.782352 2.760079 4.065748 5.420399 4.033883 6.056223
[337] 7.374409 8.668254 9.873637 10.451070 11.012750 5.793180
[343] 5.119508 7.025764 7.623123 6.306279 6.273224 9.946648
[349] 11.106510 10.389710 11.519510 13.766670 11.877010 9.987637
[355] 9.318593 10.410780 9.167216 6.281625 9.620259 14.005740
[361] 14.956960 14.772750 8.707769 10.170030 8.572340 7.396264
[367] 9.999892 11.967070 7.265405 7.174647 7.693233 10.614180
[373] 15.060890 9.936591 6.880011 7.334265 6.843737 9.197388
[379] 8.644333 9.498251 5.220479 4.722268 4.229963 6.531282
[385] 3.765132 1.835451 1.875277 8.318006 11.909480 8.179443
[391] 2.243804 5.880264 5.358331 1.808168 0.903063 2.664138
[397] 0.409372 0.818200 8.558391 6.479959 4.032262 4.820824
[403] 4.002672 5.979086 3.176779 1.981179 1.582568 6.702875
[409] 5.491976 2.736532 5.843219 3.492417 3.868475 3.856044
[415] 4.994358 5.737063 3.048542 0.000000 0.760985 2.279948
[421] 4.924980 5.281344 4.883469 4.490115 3.729027 1.860177
[427] 2.599458 3.703707 3.692311 4.048485 2.935736 3.659655
[433] -3.293165 -5.508809 -2.579503 3.683244 5.869799 0.365932
[439] 2.193139 5.828813 1.089709 1.088720 1.087844 7.226761
[445] 4.308729 5.364409 6.405601 4.251591 4.236581 3.167622
[451] 6.310269 6.277260 3.126359 1.040348 0.000000 3.115431
[457] 3.107574 5.161263 6.164397 4.092111 5.095515 5.073969
[463] 5.052708 8.040197 3.999936 0.998057 1.993322 5.960278
[469] 4.944349 6.888122 7.823956 6.804426 2.904398 2.897386
[475] 1.927679 3.846190 5.746220 2.862824 1.904730 12.307830
[481] 5.638225 6.544558 1.863323 2.789618 6.484031 4.610040
[487] 10.992540 9.988676 7.212614 2.693695 0.000000# 시계열 그림
plot(y, type = "l")
Result! 시간의 흐름에 따라 평균이 변하므로 원 시계열은 비정상시계열임이 의심된다.
4-1. 단위근 검정
# 패키지 설치
pacman::p_load("tseries")
# Dickey-Fuller test
adf.test(y)
Augmented Dickey-Fuller Test
data: y
Dickey-Fuller = -3.8651, Lag order = 7, p-value = 0.01576
alternative hypothesis: stationary# Phillips-Perron test
pp.test(y)
Phillips-Perron Unit Root Test
data: y
Dickey-Fuller Z(alpha) = -248.75, Truncation lag parameter =
5, p-value = 0.01
alternative hypothesis: stationary# KPSS test
kpss.test(y)
KPSS Test for Level Stationarity
data: y
KPSS Level = 2.51, Truncation lag parameter = 5, p-value =
0.01Caution! 단위근 검정을 수행하기 위해 Package "tseries"에서 제공하는 함수 adf.test(), pp.test(), kpss.test()를 사용한다.
Result! 1. Dickey-Fuller test를 수행했을 때, \(p\)값이 0.01576이므로 유의수준 0.05에서 \(p\)값이 0.05보다 작기 때문에 귀무가설을 기각한다. 즉, 관측된 시계열은 정상성을 만족한다.
2. Phillips-Perron test를 수행했을 때, \(p\)값이 0.01이므로 유의수준 0.05에서 \(p\)값이 0.05보다 작기 때문에 귀무가설을 기각한다. 즉, 관측된 시계열은 정상성을 만족한다.
3. KPSS test를 수행했을 때, \(p\)값이 0.01이므로 유의수준 0.05에서 \(p\)값이 0.05보다 작기 때문에 귀무가설을 기각한다. 즉, 단위근이 존재하므로 관측된 시계열은 비정상성을 가진다.
4-2. ARIMA 모형 구축
# 패키지 설치
pacman::p_load("forecast")
auto.arima(y,
max.p = 5, max.q = 5,
ic = "bic") # BIC 기준으로 BIC가 가장 작은 모형을 최적 모형으로 선택Series: y
ARIMA(1,1,1)
Coefficients:
ar1 ma1
0.2383 -0.8772
s.e. 0.0550 0.0269
sigma^2 = 8.587: log likelihood = -1221.62
AIC=2449.25 AICc=2449.29 BIC=2461.83Result! 함수 auto.arima()를 이용하여 BIC 기준으로 최적의 모형을 판단했을 때, ARIMA(1,1,1) 모형이 선택되었다. 추정된 모수 결과를 이용하면 구축된 ARIMA(1,1,1) 모형은 \((1-B)Y_t = 0.2383Y_{t-1}+\epsilon_t -0.8772\epsilon_{t-1}\)이다.
Box.test(fitARIMA111$resid,
lag = 15,
fitdf = 2) # 추정한 phi와 theta 개수
Box-Pierce test
data: fitARIMA111$resid
X-squared = 14.442, df = 13, p-value = 0.3435Result! 잔차의 자기상관계수 ACF 그래프를 보면 시차 0을 제외하고 막대의 끝이 파란색 선을 넘어가지 않으므로 다른 시차에서 자기상관계수가 통계적으로 유의하다는 증거가 부족하다.
“Ljung-Box” 검정 결과에 따르면, 귀무가설 \(H_0 : \rho(1)=\rho(2)=\cdots=\rho(15)=0\)에 대해 \(p\)값이 0.3435이므로 유의수준 0.05에서 \(p\)값이 0.05보다 크기 때문에 귀무가설을 기각하지 못한다. 즉, 잔차에 대해 시차 15까지의 자기상관계수 \(\rho(1), \rho(2), \cdots, \rho(15)\) 중 유의한 자기상관계수가 적어도 1개 존재한다는 증거가 부족하며, 해당 시계열에 대해 ARIMA(1,1,1) 모형을 가정하는 것이 적절하다.
4-3. 예측
pred <- forecast(fitARIMA111,
h = 100) # 미래 100시점까지 예측
pred Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
492 3.706101 -0.04167452 7.453876 -2.025627 9.437828
493 4.589298 0.60462429 8.573972 -1.504735 10.683331
494 4.799773 0.73892466 8.860620 -1.410758 11.010303
495 4.849930 0.73788314 8.961978 -1.438903 11.138764
496 4.861884 0.70417897 9.019588 -1.496777 11.220544
497 4.864732 0.66298827 9.066476 -1.561281 11.290745
498 4.865411 0.62034731 9.110475 -1.626854 11.357676
499 4.865573 0.57768877 9.153457 -1.692180 11.423325
500 4.865611 0.53534501 9.195878 -1.756959 11.488182
501 4.865620 0.49338614 9.237855 -1.821135 11.552376
502 4.865623 0.45182011 9.279425 -1.884706 11.615951
503 4.865623 0.41064046 9.320606 -1.947685 11.678931
504 4.865623 0.36983763 9.361409 -2.010088 11.741334
505 4.865623 0.32940172 9.401845 -2.071929 11.803176
506 4.865623 0.28932308 9.441924 -2.133224 11.864471
507 4.865623 0.24959239 9.481654 -2.193987 11.925233
508 4.865623 0.21020076 9.521046 -2.254231 11.985478
509 4.865623 0.17113966 9.560107 -2.313970 12.045217
510 4.865623 0.13240091 9.598846 -2.373216 12.104462
511 4.865623 0.09397664 9.637270 -2.431981 12.163227
512 4.865623 0.05585933 9.675387 -2.490276 12.221523
513 4.865623 0.01804173 9.713205 -2.548113 12.279360
514 4.865623 -0.01948312 9.750730 -2.605502 12.336749
515 4.865623 -0.05672191 9.787969 -2.662454 12.393701
516 4.865623 -0.09368109 9.824928 -2.718978 12.450225
517 4.865623 -0.13036685 9.861614 -2.775084 12.506331
518 4.865623 -0.16678519 9.898032 -2.830781 12.562028
519 4.865623 -0.20294187 9.934189 -2.886078 12.617325
520 4.865623 -0.23884244 9.970089 -2.940983 12.672230
521 4.865623 -0.27449228 10.005739 -2.995505 12.726752
522 4.865623 -0.30989655 10.041143 -3.049651 12.780898
523 4.865623 -0.34506028 10.076307 -3.103430 12.834676
524 4.865623 -0.37998830 10.111235 -3.156847 12.888094
525 4.865623 -0.41468527 10.145932 -3.209912 12.941158
526 4.865623 -0.44915574 10.180402 -3.262630 12.993876
527 4.865623 -0.48340408 10.214651 -3.315008 13.046255
528 4.865623 -0.51743452 10.248681 -3.367053 13.098300
529 4.865623 -0.55125118 10.282498 -3.418771 13.150018
530 4.865623 -0.58485803 10.316105 -3.470169 13.201415
531 4.865623 -0.61825894 10.349506 -3.521251 13.252498
532 4.865623 -0.65145763 10.382704 -3.572024 13.303271
533 4.865623 -0.68445775 10.415704 -3.622493 13.353740
534 4.865623 -0.71726281 10.448509 -3.672664 13.403911
535 4.865623 -0.74987622 10.481123 -3.722542 13.453789
536 4.865623 -0.78230132 10.513548 -3.772132 13.503379
537 4.865623 -0.81454132 10.545788 -3.821439 13.552686
538 4.865623 -0.84659937 10.577846 -3.870467 13.601714
539 4.865623 -0.87847849 10.609725 -3.919222 13.650469
540 4.865623 -0.91018167 10.641428 -3.967708 13.698955
541 4.865623 -0.94171177 10.672958 -4.015929 13.747176
542 4.865623 -0.97307161 10.704318 -4.063890 13.795137
543 4.865623 -1.00426392 10.735511 -4.111595 13.842841
544 4.865623 -1.03529134 10.766538 -4.159047 13.890294
545 4.865623 -1.06615646 10.797403 -4.206251 13.937498
546 4.865623 -1.09686182 10.828108 -4.253211 13.984457
547 4.865623 -1.12740986 10.858657 -4.299930 14.031177
548 4.865623 -1.15780297 10.889050 -4.346412 14.077659
549 4.865623 -1.18804350 10.919290 -4.392661 14.123908
550 4.865623 -1.21813371 10.949380 -4.438680 14.169927
551 4.865623 -1.24807582 10.979322 -4.484473 14.215719
552 4.865623 -1.27787201 11.009119 -4.530042 14.261289
553 4.865623 -1.30752437 11.038771 -4.575391 14.306638
554 4.865623 -1.33703499 11.068282 -4.620524 14.351771
555 4.865623 -1.36640586 11.097653 -4.665443 14.396689
556 4.865623 -1.39563896 11.126886 -4.710151 14.441398
557 4.865623 -1.42473620 11.155983 -4.754651 14.485898
558 4.865623 -1.45369947 11.184946 -4.798947 14.530194
559 4.865623 -1.48253060 11.213777 -4.843040 14.574287
560 4.865623 -1.51123138 11.242478 -4.886934 14.618181
561 4.865623 -1.53980356 11.271050 -4.930632 14.661878
562 4.865623 -1.56824885 11.299496 -4.974135 14.705382
563 4.865623 -1.59656893 11.327816 -5.017447 14.748694
564 4.865623 -1.62476545 11.356012 -5.060570 14.791816
565 4.865623 -1.65283999 11.384087 -5.103506 14.834753
566 4.865623 -1.68079414 11.412041 -5.146258 14.877505
567 4.865623 -1.70862943 11.439876 -5.188829 14.920075
568 4.865623 -1.73634736 11.467594 -5.231220 14.962466
569 4.865623 -1.76394940 11.495196 -5.273433 15.004680
570 4.865623 -1.79143700 11.522684 -5.315472 15.046719
571 4.865623 -1.81881157 11.550058 -5.357338 15.088584
572 4.865623 -1.84607448 11.577321 -5.399033 15.130279
573 4.865623 -1.87322710 11.604474 -5.440559 15.171806
574 4.865623 -1.90027075 11.631517 -5.481919 15.213165
575 4.865623 -1.92720674 11.658453 -5.523114 15.254360
576 4.865623 -1.95403634 11.685283 -5.564146 15.295393
577 4.865623 -1.98076080 11.712007 -5.605018 15.336264
578 4.865623 -2.00738135 11.738628 -5.645730 15.376977
579 4.865623 -2.03389918 11.765146 -5.686286 15.417532
580 4.865623 -2.06031549 11.791562 -5.726686 15.457933
581 4.865623 -2.08663142 11.817878 -5.766933 15.498179
582 4.865623 -2.11284812 11.844095 -5.807028 15.538274
583 4.865623 -2.13896670 11.870213 -5.846973 15.578219
584 4.865623 -2.16498824 11.896235 -5.886769 15.618016
585 4.865623 -2.19091383 11.922160 -5.926419 15.657666
586 4.865623 -2.21674452 11.947991 -5.965924 15.697170
587 4.865623 -2.24248134 11.973728 -6.005285 15.736531
588 4.865623 -2.26812530 11.999372 -6.044504 15.775750
589 4.865623 -2.29367742 12.024924 -6.083582 15.814829
590 4.865623 -2.31913866 12.050385 -6.122522 15.853769
591 4.865623 -2.34450999 12.075757 -6.161324 15.892571plot(pred)
Caution! 예측은 Package "forecast"에서 제공하는 함수 forecast()를 이용하여 수행할 수 있다.
Result! 원 시계열은 비정상성을 가지기 때문에 예측 구간은 발산한다는 것을 알 수 있다.
Result! 1번 차분한 시계열은 시간의 흐름에 따라 평균이 변하지 않고 분산도 일정해 보이므로 정상시계열로 보인다.
# 1번 차분한 시계열에 MA(3) 모형 구축
fit_diff <- arima(diff(y),
order = c(0, 0, 3))
# 잔차를 이용한 모형 진단
acf(fit_diff$resid) 
Box.test(fit_diff$resid,
lag = 15,
fitdf = 3)
Box-Pierce test
data: fit_diff$resid
X-squared = 13.346, df = 12, p-value = 0.3444Result! 잔차의 자기상관계수 ACF 그래프를 보면 시차 0을 제외하고 막대의 끝이 파란색 선을 넘어가지 않으므로 다른 시차에서 자기상관계수가 통계적으로 유의하다는 증거가 부족하다.
“Ljung-Box” 검정 결과에 따르면, 귀무가설 \(H_0 : \rho(1)=\rho(2)=\cdots=\rho(15)=0\)에 대해 \(p\)값이 0.3444이므로 유의수준 0.05에서 \(p\)값이 0.05보다 크기 때문에 귀무가설을 기각하지 못한다. 즉, 잔차에 대해 시차 15까지의 자기상관계수 \(\rho(1), \rho(2), \cdots, \rho(15)\) 중 유의한 자기상관계수가 적어도 1개 존재한다는 증거가 부족하며, 해당 시계열에 대해 MA(3) 모형을 가정하는 것이 적절하다. 이는 1번 차분한 시계열은 정상시계열임을 의미한다.
# 예측
pred.diff <- forecast(fit_diff,
h = 100) # 미래 100시점까지 예측
pred.diff Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
491 3.4279770957 -0.3093655 7.165320 -2.287795 9.143749
492 0.8929253001 -3.5301468 5.315997 -5.871580 7.657430
493 0.5146289502 -3.9250766 4.954335 -6.275315 7.304573
494 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
495 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
496 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
497 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
498 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
499 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
500 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
501 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
502 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
503 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
504 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
505 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
506 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
507 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
508 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
509 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
510 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
511 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
512 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
513 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
514 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
515 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
516 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
517 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
518 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
519 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
520 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
521 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
522 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
523 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
524 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
525 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
526 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
527 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
528 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
529 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
530 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
531 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
532 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
533 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
534 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
535 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
536 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
537 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
538 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
539 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
540 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
541 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
542 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
543 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
544 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
545 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
546 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
547 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
548 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
549 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
550 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
551 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
552 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
553 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
554 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
555 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
556 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
557 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
558 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
559 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
560 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
561 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
562 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
563 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
564 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
565 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
566 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
567 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
568 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
569 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
570 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
571 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
572 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
573 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
574 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
575 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
576 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
577 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
578 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
579 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
580 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
581 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
582 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
583 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
584 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
585 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
586 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
587 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
588 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
589 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880
590 -0.0001564659 -4.4582307 4.457918 -6.818193 6.817880plot(pred.diff)
Result! 원 시계열을 1번 차분한 시계열은 정상성을 가지기 때문에 예측 구간은 수렴한다는 것을 알 수 있다.