Mơ hình chuỗi thời gian đơn biến hay cịn được gọi với cái tên mơ hình ARIMA được sử dụng khi hành vi của biến số cần giải thích được quyết định bởi những thơng tin về giá trị của chính nó trong q khứ và/hoặc giá trị hiện tại và quá khứ của hạng nhiễu.
Mơ hình tự hồi qui là giá trị ước tính tương lai của mơ hình phân tích chuỗi thời gian chỉ phụ thuộc vào giá trị trong quá khứ.: Yt-1, Yt-2..., Yt-n và sai số ut. Mơ hình tự hồi quy bậc 1có dạng:
AR(1): Yt = µ + ϕ Yt-1 + ut
Trong mơ hình tự hồi qui q trình phụ thuộc vào tổng có trọng số của các giá trị quá khứ và số hạng ngẫu nhiên:
Mơ hình trung bình trượt (MA) là loại mơ hình đơn giản nhất trong nhóm các mơ hình chuỗi thời gian mà giá trị hiện tại của biến số Yt chỉ phụ thuộc vào sai số ut. Mơ
hình trung bình trượt bậc 1 có dạng:
MA(1): Yt = µ + θ1ut-1+ ut
Trong mơ hình trung bình trượt q trình phụ thuộc vào tổng có trọng số của các ngẫu nhiên hiện hành có độ trễ:
MA(q): Yt = µ + θ1ut-1 + θ2ut-2 + θ3ut-3 + …+ θqut-q+ ut Trong đó: ut là nhiễu trắng
µ (const) là hằng số t=1,2,3...n
ARIMA (1,1,1)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 11:12
Sample (adjusted): 12/08/2016 5/31/2021 Included observations: 1115 after adjustments Convergence achieved after 65 iterations MA Backcast: 12/07/2016
Variable Coefficient Std. Error t-Statistic Prob.
C -0.035059 0.142524 -0.245983 0.8057
AR(1) 0.804896 0.064169 12.54344 0.0000
MA(1) -0.767291 0.071414 -10.74420 0.0000
R-squared 0.029750 Mean dependent var 0.023303
Adjusted R-squared 0.028005 S.D. dependent var 4.020613
S.E. of regression 3.963914 Akaike info criterion 5.595028
Sum squared resid 17472.43 Schwarz criterion 5.608525
Log likelihood -3116.228 Hannan-Quinn criter. 5.600131
F-statistic 17.04832 Durbin-Watson stat 1.890245
Prob(F-statistic) 0.000000
Inverted AR Roots .80 Inverted MA Roots .77
SPSABt = -0.0351 + 0.8049AR(1) − 0.7673MA(1) + ȗt SPSABt = -0.0351 + 0.8049SPSABt-1 − 0.7673ȗt-1 + ȗt
SABt − SABt-1= -0.0351 + 0.8049 (SABt-1 − SABt-2) – 0.7673ȗt-1 + ȗt SABt = -0.0351 + 1.8049SABt-1 − 0.8049SABt-2 – 0.7673ȗt-1 + ȗt
ARIMA (1,1,1) (20,1,11)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 12:45
Sample (adjusted): 1/05/2017 5/31/2021 Included observations: 1096 after adjustments Convergence achieved after 11 iterations MA Backcast: 12/20/2016 1/04/2017
Variable Coefficient Std. Error t-Statistic Prob.
C -0.023707 0.113483 -0.208903 0.8346
AR(1) -0.216060 0.243681 -0.886650 0.3755
AR(20) -0.064365 0.029020 -2.217957 0.0268
MA(1) 0.287898 0.239214 1.203518 0.2290
MA(11) -0.054049 0.029529 -1.830394 0.0675
R-squared 0.013438 Mean dependent var -0.025500
Adjusted R-squared 0.009821 S.D. dependent var 3.916682
S.E. of regression 3.897401 Akaike info criterion 5.563049
Sum squared resid 16572.00 Schwarz criterion 5.585856
Log likelihood -3043.551 Hannan-Quinn criter. 5.571679
F-statistic 3.715207 Durbin-Watson stat 1.980966
Prob(F-statistic) 0.005194
Inverted AR Roots .85-.14i .85+.14i .77+.40i .77-.40i .61+.62i .61-.62i .39+.78i .39-.78i .13-.86i .13+.86i -.15+.86i -.15-.86i -.41-.78i -.41+.78i -.63-.62i -.63+.62i -.79-.40i -.79+.40i -.87+.14i -.87-.14i Inverted MA Roots .74 .62+.41i .62-.41i .29+.69i
.29-.69i -.14+.75i -.14-.75i -.53+.58i -.53-.58i -.77+.21i -.77-.21i
SPSABt = -0.0237 - 0.2161AR(1) – 0.0644AR(20) + 0.2879MA(1) – 0.054MA(11) + ȗt
SPSABt = -0.0237 - 0.2161SPSABt-1 – 0.0644SPSABt-20 + 0.2879ȗt-1 – 0.054ȗt-11 + ȗt SABt − SABt-1 = - 0.0237 - 0.2161(SABt-1 − SABt-2) – 0.0644(SABt-20 − SABt-21) + 0.2879ȗt-1 – 0.054 ȗt-11 + ȗt
SABt = - 0.0237 + 0.7839SABt-1 + 0.261SABt-2 – 0.0644SABt-20 + 0.0644SABt-21 + 0.2879ȗt-1 – 0.054 ȗt-11 + ȗt
ARIMA (1,1,1) (20,1,20)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 11:18
Included observations: 1096 after adjustments Convergence achieved after 48 iterations MA Backcast: 12/07/2016 1/04/2017
Variable Coefficient Std. Error t-Statistic Prob.
C -0.024624 0.119317 -0.206373 0.8365
AR(1) -0.077168 0.295093 -0.261503 0.7938
AR(20) -0.026943 0.112884 -0.238677 0.8114
MA(1) 0.154900 0.292505 0.529565 0.5965
MA(20) -0.039484 0.115826 -0.340894 0.7332
R-squared 0.010302 Mean dependent var -0.025500
Adjusted R-squared 0.006673 S.D. dependent var 3.916682
S.E. of regression 3.903591 Akaike info criterion 5.566223
Sum squared resid 16624.69 Schwarz criterion 5.589030
Log likelihood -3045.290 Hannan-Quinn criter. 5.574852
F-statistic 2.839120 Durbin-Watson stat 1.992527
Prob(F-statistic) 0.023311
SPSABt = -0.0246 – 0.077AR(1) – 0.0269AR(20) + 0.1549MA(1) – 0.0395MA(20) + ȗt
SPSABt = -0.0246 – 0.077SPSABt-1 – 0.0269SPSABt-20 + 0.1549ȗt-1 – 0.0395ȗt-20 + ȗt SABt − SABt-1 = -0.0246 – 0.077(SABt-1 − SABt-2) – 0.0269(SABt-20 − SABt-21) + 0.1549ȗt-1 – 0.0395ȗt-20 + ȗt
SABt = -0.0246 – 0.923SABt-1 + 0.077SABt-2 – 0.0269SABt-20 + 0.0269SABt-21 + 0.1549ȗt-1 – 0.0395ȗt-20 + ȗt
ARIMA (1,1,1) (20,1,26)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 11:19
Sample (adjusted): 1/05/2017 5/31/2021 Included observations: 1096 after adjustments Convergence achieved after 12 iterations MA Backcast: 11/29/2016 1/04/2017
Variable Coefficient Std. Error t-Statistic Prob.
C -0.022006 0.122969 -0.178951 0.8580
AR(1) -0.045971 0.288418 -0.159390 0.8734
AR(20) -0.061262 0.029386 -2.084726 0.0373
MA(1) 0.122349 0.286965 0.426357 0.6699
MA(26) 0.033691 0.030265 1.113198 0.2659
R-squared 0.011292 Mean dependent var -0.025500
Adjusted R-squared 0.007667 S.D. dependent var 3.916682
Sum squared resid 16608.06 Schwarz criterion 5.588029
Log likelihood -3044.742 Hannan-Quinn criter. 5.573852
F-statistic 3.115085 Durbin-Watson stat 1.990858
Prob(F-statistic) 0.014605
SPSABt = -0.22 – 0.046AR(1) – 0.0612AR(20) + 0.1223MA(1) + 0.0337MA(26) + ȗt SPSABt = -0.22 – 0.046SPSABt-1 – 0.0612SPSABt-20 + 0.1223ȗt-1 + 0.0337ȗt-26 + ȗt SABt − SABt-1 = -0.22 – 0.046(SABt-1 − SABt-2) – 0.0612(SABt-20 − SABt-21) + 0.1223ȗt-1 + 0.0337ȗt-26 + ȗt
SABt = -0.22 + 0.954SABt-1 + 0.046SABt-2 – 0.0612SABt-20 + 0.0612SABt-21 + 0.1223ȗt-1 + 0.0337ȗt-26 + ȗt
ARIMA (1,1,1) (21,1,11)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 11:20
Sample (adjusted): 1/06/2017 5/31/2021 Included observations: 1095 after adjustments Convergence achieved after 20 iterations MA Backcast: 12/21/2016 1/05/2017
Variable Coefficient Std. Error t-Statistic Prob.
C -0.036523 0.133184 -0.274230 0.7840
AR(1) 0.370450 0.245818 1.507014 0.1321
AR(21) 0.043991 0.027576 1.595293 0.1109
MA(1) -0.295123 0.253004 -1.166474 0.2437
MA(11) -0.044853 0.029163 -1.537994 0.1243
R-squared 0.011517 Mean dependent var -0.025608
Adjusted R-squared 0.007890 S.D. dependent var 3.918470
S.E. of regression 3.902981 Akaike info criterion 5.565914
Sum squared resid 16604.26 Schwarz criterion 5.588738
Log likelihood -3042.338 Hannan-Quinn criter. 5.574551
F-statistic 3.175074 Durbin-Watson stat 1.988176
Prob(F-statistic) 0.013184
Inverted AR Roots .88 .85+.25i .85-.25i .73+.48i .73-.48i .56-.67i .56+.67i .33-.80i .33+.80i .08-.86i .08+.86i -.18+.84i -.18-.84i -.42+.74i -.42-.74i -.62-.58i -.62+.58i -.76+.37i -.76-.37i -.84+.13i -.84-.13i
Inverted MA Roots .79 .67+.40i .67-.40i .34-.68i .34+.68i -.08-.74i -.08+.74i -.47-.57i -.47+.57i -.70+.21i -.70-.21i
SPSABt = -0.0365 + 0.3704AR(1) – 0.044AR(21) – 0.2951MA(1) – 0.0449MA(11) + ȗt
SPSABt = -0.0365 + 0.3704SPSABt-1 – 0.044SPSABt-21 – 0.2951ȗt-1 – 0.0449ȗt-11 + ȗt SABt − SABt-1 = -0.0365 + 0.3704(SABt-1 − SABt-2) – 0.044(SABt-21 − SABt-22) – 0.2951ȗt-1 – 0.0449ȗt-11 + ȗt
SABt = -0.0365 + 1.3704SABt-1 − 0.3704SABt-2 – 0.044SABt-21 + 0.044SABt-22 – 0.2951ȗt-1 – 0.0449ȗt-11 + ȗt
ARIMA (1,1,1) (21,1,20)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 11:20
Sample (adjusted): 1/06/2017 5/31/2021 Included observations: 1095 after adjustments Convergence achieved after 13 iterations MA Backcast: 12/08/2016 1/05/2017
Variable Coefficient Std. Error t-Statistic Prob.
C -0.053093 0.146142 -0.363296 0.7165
AR(1) 0.693157 0.147291 4.706039 0.0000
AR(21) 0.067591 0.025618 2.638368 0.0084
MA(1) -0.642041 0.157749 -4.070016 0.0001
MA(20) -0.064225 0.029639 -2.166925 0.0305
R-squared 0.014120 Mean dependent var -0.025608
Adjusted R-squared 0.010502 S.D. dependent var 3.918470
S.E. of regression 3.897841 Akaike info criterion 5.563278
Sum squared resid 16560.55 Schwarz criterion 5.586102
Log likelihood -3040.895 Hannan-Quinn criter. 5.571915
F-statistic 3.902695 Durbin-Watson stat 1.944884
Prob(F-statistic) 0.003745
Inverted AR Roots .94 .89+.25i .89-.25i .77+.48i .77-.48i .58-.67i .58+.67i .35-.81i .35+.81i .09+.87i .09-.87i -.17+.85i -.17-.85i -.41+.76i -.41-.76i -.62+.59i -.62-.59i -.77+.38i -.77-.38i -.85+.13i -.85-.13i
Inverted MA Roots .92 .88+.26i .88-.26i .75+.50i .75-.50i .55-.69i .55+.69i .30-.82i .30+.82i .03-.86i .03+.86i -.24-.82i -.24+.82i -.49+.70i -.49-.70i -.68+.51i -.68-.51i -.80-.27i -.80+.27i -.85
SPSABt = -0.0531 + 0.6932AR(1) + 0.0676AR(21) – 0.642MA(1) – 0.0642MA(20) + ȗt
SPSABt = -0.0531 + 0.6932SPSABt-1 + 0.0676SPSABt-21 – 0.642ȗt-20 – 0.0642ȗt-20 + ȗt SABt − SABt-1 = -0.0531 + 0.6932(SABt-1 − SABt-2) + 0.0676(SABt-21 − SABt-22) – 0.642ȗt-20 – 0.0642ȗt-20 + ȗt
SABt = -0.0531 + 1.6932SABt-1 – 0.6932SABt-2 + 0.0676SABt-21 − 0.0676SABt-22 – 0.642ȗt-20 – 0.0642ȗt-20 + ȗt
ARIMA (1,1,1) (21,1,26)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 11:21
Sample (adjusted): 1/06/2017 5/31/2021 Included observations: 1095 after adjustments Convergence achieved after 12 iterations MA Backcast: 11/30/2016 1/05/2017
Variable Coefficient Std. Error t-Statistic Prob.
C -0.049224 0.174179 -0.282604 0.7775
AR(1) 0.734381 0.143553 5.115743 0.0000
AR(21) 0.031136 0.020992 1.483224 0.1383
MA(1) -0.690854 0.153162 -4.510620 0.0000
MA(26) 0.036864 0.025655 1.436889 0.1510
R-squared 0.012945 Mean dependent var -0.025608
Adjusted R-squared 0.009323 S.D. dependent var 3.918470
S.E. of regression 3.900161 Akaike info criterion 5.564469
Sum squared resid 16580.27 Schwarz criterion 5.587293
Log likelihood -3041.547 Hannan-Quinn criter. 5.573105
F-statistic 3.573825 Durbin-Watson stat 1.933770
Prob(F-statistic) 0.006638
Inverted AR Roots .92 .87-.23i .87+.23i .74+.46i .74-.46i .57+.65i .57-.65i .34-.77i .34+.77i .09+.83i .09-.83i -.16+.82i -.16-.82i -.40+.73i -.40-.73i -.60-.57i -.60+.57i -.74-.36i -.74+.36i -.81+.13i -.81-.13i
Inverted MA Roots .92-.10i .92+.10i .86-.30i .86+.30i .76-.49i .76+.49i .61+.65i .61-.65i .44+.77i .44-.77i .24+.85i .24-.85i .02+.87i .02-.87i -.19-.85i -.19+.85i -.39+.77i -.39-.77i -.56+.66i -.56-.66i -.70+.50i -.70-.50i -.80-.31i -.80+.31i -.85-.11i -.85+.11i
SPSABt = -0.4922 + 0.7344AR(1) + 0.0311AR(21) – 0.6909MA(1) – 0.0367MA(26) + ȗt
SPSABt = -0.4922 + 0.7344SPSABt-1 + 0.0311SPSABt-21 – 0.6909ȗt-1 – 0.0367ȗt-26 + ȗt SABt − SABt-1 = -0.4922 + 0.7344(SABt-1 − SABt-2) + 0.0311(SABt-21 − SABt-22) – 0.6909ȗt-1 – 0.0367ȗt-26 + ȗt
SABt = -0.4922 + 1.7344SABt-1 – 0.7344SABt-2 + 0.0311SABt-21 − 0.0311SABt-22 – 0.6909ȗt-1 – 0.0367ȗt-26 + ȗt
ARIMA (1,1,1) (35,1,11)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 11:21
Sample (adjusted): 2/02/2017 5/31/2021 Included observations: 1081 after adjustments Convergence achieved after 8 iterations MA Backcast: 1/11/2017 1/25/2017
Variable Coefficient Std. Error t-Statistic Prob.
C -0.044353 0.114135 -0.388602 0.6976
AR(1) -0.328075 0.219991 -1.491314 0.1362
AR(35) -0.064246 0.028565 -2.249141 0.0247
MA(1) 0.395051 0.214133 1.844885 0.0653
MA(11) -0.055986 0.028599 -1.957620 0.0505
R-squared 0.013347 Mean dependent var -0.048940
Adjusted R-squared 0.009679 S.D. dependent var 3.919430
S.E. of regression 3.900416 Akaike info criterion 5.564658
Sum squared resid 16369.45 Schwarz criterion 5.587718
Log likelihood -3002.698 Hannan-Quinn criter. 5.573389
F-statistic 3.638847 Durbin-Watson stat 1.967761
Prob(F-statistic) 0.005933
Inverted AR Roots .91-.08i .91+.08i .88+.25i .88-.25i .82-.40i .82+.40i .74-.54i .74+.54i .63+.67i .63-.67i .50-.77i .50+.77i .35+.85i .35-.85i .20+.90i .20-.90i .03-.92i .03+.92i -.13-.91i -.13+.91i -.30-.88i -.30+.88i -.45+.81i -.45-.81i -.59-.72i -.59+.72i -.71+.61i -.71-.61i -.80-.47i -.80+.47i -.88-.32i -.88+.32i -.92-.16i -.92+.16i -.94
Inverted MA Roots .74 .62+.41i .62-.41i .29-.69i .29+.69i -.14+.75i -.14-.75i -.54-.57i -.54+.57i -.78-.21i -.78+.21i
SPSABt = -0.4444 – 0.3281AR(1) – 0.0642AR(35) + 0.3951MA(1) – 0.056MA(11) + ȗt
SPSABt = -0.4444 – 0.3281SPSABt-1 – 0.0642SPSABt-35 + 0.3951ȗt-1 – 0.056ȗt-11 + ȗt SABt − SABt-1 = -0.4444 – 0.3281(SABt-1 − SABt-2) – 0.0642(SABt-35 − SABt-36) + 0.3951ȗt-1 – 0.056ȗt-11 + ȗt
SABt = -0.4444 + 0.6719SABt-1 + 0.3281SABt-2 – 0.0642SABt-35 + 0.0642SABt-36 + 0.3951ȗt-1 – 0.056ȗt-11 + ȗt
ARIMA (1,1,1) (35,1,20)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 11:23
Sample (adjusted): 2/02/2017 5/31/2021 Included observations: 1081 after adjustments Convergence achieved after 11 iterations MA Backcast: 12/28/2016 1/25/2017
Variable Coefficient Std. Error t-Statistic Prob.
C -0.044442 0.113917 -0.390125 0.6965
AR(1) -0.194310 0.238776 -0.813775 0.4160
AR(35) -0.059035 0.029229 -2.019710 0.0437
MA(1) 0.268201 0.234460 1.143912 0.2529
MA(20) -0.065391 0.029929 -2.184864 0.0291
R-squared 0.014214 Mean dependent var -0.048940
Adjusted R-squared 0.010549 S.D. dependent var 3.919430
S.E. of regression 3.898701 Akaike info criterion 5.563779
Sum squared resid 16355.06 Schwarz criterion 5.586839
Log likelihood -3002.222 Hannan-Quinn criter. 5.572510
F-statistic 3.878698 Durbin-Watson stat 1.979968
Prob(F-statistic) 0.003908
SPSABt = -0.4444 – 0.1943AR(1) – 0.0590AR(35) + 0.2682MA(1) – 0.0654MA(20) + ȗt
SPSABt = -0.4444 – 0.1943SPSABt-1 – 0.0590SPSABt-35 + 0.2682ȗt-1 – 0.0654ȗt-20 + ȗt SABt − SABt-1 = -0.4444 – 0.1943(SABt-1 − SABt-2) – 0.0590(SABt-35 – SABt-36) + 0.2682ȗt-1 – 0.0654ȗt-20 + ȗt
SABt = -0.4444 + 0.8057SABt-1 + 0.1943SABt-2 – 0.0590SABt-35 + 0.0590SABt-36 + 0.2682ȗt-1 – 0.0654ȗt-20 + ȗt
ARIMA (1,1,1) (35,1,26)
Dependent Variable: SPSAB Method: Least Squares Date: 06/10/21 Time: 11:23
Sample (adjusted): 2/02/2017 5/31/2021 Included observations: 1081 after adjustments Convergence achieved after 20 iterations MA Backcast: 12/20/2016 1/25/2017
Variable Coefficient Std. Error t-Statistic Prob.
C -0.044944 0.124411 -0.361253 0.7180
AR(1) -0.071462 0.291883 -0.244833 0.8066
AR(35) -0.055811 0.029484 -1.892931 0.0586
MA(1) 0.147489 0.289771 0.508986 0.6109
MA(26) 0.033658 0.030320 1.110081 0.2672
R-squared 0.010509 Mean dependent var -0.048940
Adjusted R-squared 0.006831 S.D. dependent var 3.919430
S.E. of regression 3.906021 Akaike info criterion 5.567530
Sum squared resid 16416.53 Schwarz criterion 5.590590
Log likelihood -3004.250 Hannan-Quinn criter. 5.576261
F-statistic 2.857009 Durbin-Watson stat 1.988348
Prob(F-statistic) 0.022626
SPSABt = -0.0449 – 0.0715AR(1) – 0.0558AR(35) + 0.1475MA(1) + 0.0337MA(26) + ȗt
SPSABt = -0.0449 – 0.0715SPSABt-1 – 0.0558SPSABt-35 + 0.1475ȗt-1 + 0.0337ȗt-26 + ȗt SABt − SABt-1 = -0.0449 – 0.0715(SABt-1 − SABt-2) – 0.0558(SABt-35 − SABt-36) + 0.1475ȗt-1 + 0.0337ȗt-26 + ȗt
SABt = -0.0449 + 0.9285SABt-1 + 0.0715SABt-2 – 0.0558SABt-35 + 0.0558SABt-36 + 0.1475ȗt-1 + 0.0337ȗt-26 + ȗt