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Cointegration

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1117:(1990) show that residual-based unit root tests applied to the estimated cointegrating residuals do not have the usual Dickey–Fuller distributions under the null hypothesis of no-cointegration. Because of the spurious regression phenomenon under the null hypothesis, the distribution of these tests have asymptotic distributions that depend on (1) the number of deterministic trend terms and (2) the number of variables with which co-integration is being tested. These distributions are known as Phillips–Ouliaris distributions and critical values have been tabulated. In finite samples, a superior alternative to the use of these asymptotic critical value is to generate critical values from simulations. 4071: 4057: 4095: 4083: 1171:
economic crises, changes in the people's preferences and behaviour accordingly, policy or regime alteration, and organizational or institutional developments. This is especially likely to be the case if the sample period is long. To take this issue into account, tests have been introduced for cointegration with one unknown
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Tests for cointegration assume that the cointegrating vector is constant during the period of study. In reality, it is possible that the long-run relationship between the underlying variables change (shifts in the cointegrating vector can occur). The reason for this might be technological progress,
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is a test for cointegration that allows for more than one cointegrating relationship, unlike the Engle–Granger method, but this test is subject to asymptotic properties, i.e. large samples. If the sample size is too small then the results will not be reliable and one should use Auto Regressive
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trends can be co-integrated only if there is a genuine relationship between the two. Thus the standard current methodology for time series regressions is to check all-time series involved for integration. If there are
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processes, Granger and Newbold showed that de-trending does not work to eliminate the problem of spurious correlation, and that the superior alternative is to check for co-integration. Two series with
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series, but it is more generally applicable and can be used for variables integrated of higher order (to detect correlated accelerations or other second-difference effects).
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For example, regressing the consumption series for any country (e.g. Fiji) against the GNP for a randomly selected dissimilar country (e.g. Afghanistan) might give a high
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The possible presence of cointegration must be taken into account when choosing a technique to test hypotheses concerning the relationship between two variables having
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Koop, G.; Strachan, R.; van Dijk, H.K.; Villani, M. (January 1, 2006). "Chapter 17: Bayesian Approaches to Cointegration". In Mills, T.C.; Patterson, K. (eds.).
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are cointegrated. Cointegration has become an important property in contemporary time series analysis. Time series often have trends—either deterministic or
329:(i.e. integrated of at least order one). The usual procedure for testing hypotheses concerning the relationship between non-stationary variables was to run 3847: 3471: 2112: 3245: 1256: 3684: 1187:
have been proposed to compute the posterior distribution of the number of cointegrating relationships and the cointegrating linear combinations.
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Yule, U. (1926). "Why do we sometimes get nonsense-correlations between time series? - A study in sampling and the nature of time series".
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is estimated, the critical values of this ADF test are non-standard, and increase in absolute value as more regressors are included.
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extends the cointegration technique beyond two variables, and occasionally to variables integrated at different orders.
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of this collection is integrated of order less than d, then the collection is said to be co-integrated. Formally, if (
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series on both sides of the regression relationship, then it is possible for regressions to give misleading results.
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Pesaran, M.H.; Shin, Y.; Smith, R.J. (2001). "Bounds testing approaches to the analysis of level relationships".
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Granger, Clive (1981). "Some Properties of Time Series Data and Their Use in Econometric Model Specification".
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If the variables are found to be cointegrated, a second-stage regression is conducted. This is a regression of
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If the variables are not cointegrated (if we cannot reject the null of no cointegration when testing
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series which are not directly causally related may nonetheless show a significant correlation.
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Mahdavi Damghani, Babak; et al. (2012). "The Misleading Value of Measured Correlation".
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Nelson, C.R; Plosser, C.I (1982). "Trends and random walks in macroeconomic time series".
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relationship (suggesting high explanatory power on Fiji's consumption from Afghanistan's
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The first to introduce and analyse the concept of spurious—or nonsense—regression was
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https://www.econ.queensu.ca/sites/econ.queensu.ca/files/wpaper/qed_wp_1227.pdf
123: 1409:"Co-integration and error correction: Representation, estimation and testing" 3013: 2865: 2485: 2280: 2192: 2177: 2172: 2137: 1757:"A Drunk and her Dog: An Illustration of Cointegration and Error Correction" 337: 326: 196: 1641: 1175:, and tests for cointegration with two unknown breaks are also available. 966:{\displaystyle \Delta y_{t}=\Delta x_{t}b+\alpha u_{t-1}+\varepsilon _{t}} 2529: 2147: 2024: 2019: 2014: 1986: 1302:
Granger, C.; Newbold, P. (1974). "Spurious Regressions in Econometrics".
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is unknown, we must first estimate it. This is typically done by using
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formalized the cointegrating vector approach, and coined the term.
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Error correction model § Engle and Granger 2-step approach
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variables. First, all of the series must be integrated of
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on non-stationary time series data, which Nobel laureate
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The six main methods for testing for cointegration are:
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Statistical property of collections of time series data
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showed to be a dangerous approach that could produce
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Autoregressive conditional heteroskedasticity (ARCH)
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New York: Wiley. pp.  8: 1565:Gregory, Allan W.; Hansen, Bruce E. (1996). 1782:An intuitive introduction to cointegration. 1673:"Cointegration and Error-Correction Models" 889:. The second stage regression is given as: 3855: 3842: 3759: 3565: 3434: 3409: 3180: 3156: 2884: 2667: 2468: 2455: 2238: 2225: 1864: 1855: 1842: 1808: 1794: 1786: 1521:Phillips, P. C. B.; Ouliaris, S. (1990). 1505: 1317: 1132: 1074: 1058: 1042: 1033: 1007: 986: 980: 957: 938: 919: 903: 894: 867: 856: 855: 852: 831: 822: 801: 792: 769: 748: 742: 721: 715: 694: 688: 664: 635: 629: 609: 585: 579: 564:{\displaystyle y_{t}-\beta x_{t}=u_{t}\,} 560: 554: 541: 525: 519: 495: 489: 469: 441: 435: 414: 408: 355: 296: 262: 229: 157: 110:is integrated of order less than d, then 1646:. Palgrave Macmillan. pp. 871–898. 1257:Journal of the Royal Statistical Society 138:If two or more series are individually 1712:. Princeton University Press. pp.  1217: 3772:Kaplan–Meier estimator (product limit) 1464:"ARDL Models - Part II - Bounds Tests" 1028:and we estimate a differences model: 7: 4082: 3782:Accelerated failure time (AFT) model 1106:Phillips–Ouliaris cointegration test 142:(in the time series sense) but some 4094: 3377:Analysis of variance (ANOVA, anova) 1166:Variable shifts in long time series 3472:Cochran–Mantel–Haenszel statistics 2098:Pearson product-moment correlation 1051: 1035: 912: 896: 824: 794: 14: 4093: 4081: 4069: 4056: 4055: 1679:Applied Econometrics Time Series 882:{\displaystyle {\hat {u}}_{t-1}} 3731:Least-squares spectral analysis 1486:Journal of Applied Econometrics 84:, and there exist coefficients 80:) are each integrated of order 2712:Mean-unbiased minimum-variance 1778:10.1080/00031305.1994.10476017 1143: 1137: 861: 366: 360: 307: 301: 273: 267: 240: 234: 168: 162: 1: 4025:Geographic information system 3241:Simultaneous equations models 1462:Giles, David (19 June 2013). 1228:Journal of Monetary Economics 393:Engle–Granger two-step method 187:linear combination of them. 3208:Coefficient of determination 2819:Uniformly most powerful test 1586:10.1016/0304-4076(69)41685-7 1394:10.1016/0304-4076(81)90079-8 1328:10.1016/0304-4076(74)90034-7 1241:10.1016/0304-3932(82)90012-5 1207:Stationary subspace analysis 840:{\displaystyle \Delta x_{t}} 810:{\displaystyle \Delta y_{t}} 653:Augmented Dickey–Fuller test 3777:Proportional hazards models 3721:Spectral density estimation 3703:Vector autoregression (VAR) 3137:Maximum posterior estimator 2369:Randomized controlled trial 1755:Murray, Michael P. (1994). 4143: 3537:Multivariate distributions 1957:Average absolute deviation 817:on the lagged regressors, 396: 4051: 3854: 3841: 3525:Structural equation model 3433: 3408: 3179: 3155: 2887: 2861:Score/Lagrange multiplier 2467: 2454: 2276:Sample size determination 2237: 2224: 1854: 1841: 1823: 1765:The American Statistician 1619:10.1007/s00181-007-0175-9 1102:Distributed Lags (ARDL). 1021:{\displaystyle \alpha =0} 651:for stationarity with an 26:property of a collection 4020:Environmental statistics 3542:Elliptical distributions 3335:Generalized linear model 3264:Simple linear regression 3034:Hodges–Lehmann estimator 2491:Probability distribution 2400:Stochastic approximation 1962:Coefficient of variation 3680:Cross-correlation (XCF) 3288:Non-standard predictors 2722:Lehmann–ScheffĂ© theorem 2395:Adaptive clinical trial 1670:Enders, Walter (2004). 1574:Journal of Econometrics 1381:Journal of Econometrics 1305:Journal of Econometrics 4076:Mathematics portal 3897:Engineering statistics 3805:Nelson–Aalen estimator 3382:Analysis of covariance 3269:Ordinary least squares 3193:Pearson product-moment 2597:Statistical functional 2508:Empirical distribution 2341:Controlled experiments 2070:Frequency distribution 1848:Descriptive statistics 1734:; Kim, In-Moo (1998). 1197:Error correction model 1150: 1084: 1022: 996: 967: 883: 841: 811: 778: 777:{\displaystyle \beta } 758: 731: 704: 681:ordinary least squares 673: 672:{\displaystyle \beta } 645: 624:is known, we can test 618: 617:{\displaystyle \beta } 595: 565: 505: 478: 477:{\displaystyle \beta } 451: 424: 373: 331:ordinary least squares 314: 280: 247: 175: 3992:Population statistics 3934:System identification 3668:Autocorrelation (ACF) 3596:Exponential smoothing 3510:Discriminant analysis 3505:Canonical correlation 3369:Partition of variance 3231:Regression validation 3075:(Jonckheere–Terpstra) 2974:Likelihood-ratio test 2663:Frequentist inference 2575:Location–scale family 2496:Sampling distribution 2461:Statistical inference 2428:Cross-sectional study 2415:Observational studies 2374:Randomized experiment 2203:Stem-and-leaf display 2005:Central limit theorem 1599:Hatemi-J, A. (2008). 1151: 1085: 1023: 997: 995:{\displaystyle u_{t}} 968: 884: 842: 812: 779: 759: 757:{\displaystyle u_{t}} 732: 730:{\displaystyle x_{t}} 705: 703:{\displaystyle y_{t}} 674: 646: 644:{\displaystyle u_{t}} 619: 596: 594:{\displaystyle u_{t}} 566: 506: 504:{\displaystyle u_{t}} 479: 452: 450:{\displaystyle y_{t}} 425: 423:{\displaystyle x_{t}} 374: 315: 281: 248: 176: 4122:Mathematical finance 3915:Probabilistic design 3500:Principal components 3343:Exponential families 3295:Nonlinear regression 3274:General linear model 3236:Mixed effects models 3226:Errors and residuals 3203:Confounding variable 3105:Bayesian probability 3083:Van der Waerden test 3073:Ordered alternative 2838:Multiple comparisons 2717:Rao–Blackwellization 2680:Estimating equations 2636:Statistical distance 2354:Factorial experiment 1887:Arithmetic-Geometric 1149:{\displaystyle I(1)} 1131: 1111:Peter C. B. Phillips 1032: 1006: 979: 893: 851: 821: 791: 768: 741: 714: 687: 663: 657:Phillips–Perron test 628: 608: 578: 518: 488: 468: 459:order of integration 434: 407: 372:{\displaystyle I(1)} 354: 313:{\displaystyle I(1)} 295: 279:{\displaystyle I(1)} 261: 246:{\displaystyle I(1)} 228: 213:spurious correlation 174:{\displaystyle I(1)} 156: 148:order of integration 146:of them has a lower 3987:Official statistics 3910:Methods engineering 3591:Seasonal adjustment 3359:Poisson regressions 3279:Bayesian regression 3218:Regression analysis 3198:Partial correlation 3170:Regression analysis 2769:Prediction interval 2764:Likelihood interval 2754:Confidence interval 2746:Interval estimation 2707:Unbiased estimators 2525:Model specification 2405:Up-and-down designs 2093:Partial correlation 2049:Index of dispersion 1967:Interquartile range 1606:Empirical Economics 511:. In other words: 346:spurious regression 4007:Spatial statistics 3887:Medical statistics 3787:First hitting time 3741:Whittle likelihood 3392:Degrees of freedom 3387:Multivariate ANOVA 3320:Heteroscedasticity 3132:Bayesian estimator 3097:Bayesian inference 2946:Kolmogorov–Smirnov 2831:Randomization test 2801:Testing hypotheses 2774:Tolerance interval 2685:Maximum likelihood 2580:Exponential family 2513:Density estimation 2473:Statistical theory 2433:Natural experiment 2379:Scientific control 2296:Survey methodology 1982:Standard deviation 1357:10.1002/wilm.10167 1179:Bayesian inference 1160:Multicointegration 1146: 1121:Multicointegration 1080: 1018: 992: 963: 879: 837: 807: 774: 754: 727: 700: 669: 641: 614: 591: 561: 501: 474: 447: 420: 369: 344:). This is called 310: 276: 243: 201:linear regressions 171: 144:linear combination 66:linear combination 4109: 4108: 4047: 4046: 4043: 4042: 3982:National accounts 3952:Actuarial science 3944:Social statistics 3837: 3836: 3833: 3832: 3829: 3828: 3764:Survival function 3749: 3748: 3611:Granger causality 3452:Contingency table 3427:Survival analysis 3404: 3403: 3400: 3399: 3256:Linear regression 3151: 3150: 3147: 3146: 3122:Credible interval 3091: 3090: 2874: 2873: 2690:Method of moments 2559:Parametric family 2520:Statistical model 2450: 2449: 2446: 2445: 2364:Random assignment 2286:Statistical power 2220: 2219: 2216: 2215: 2065:Contingency table 2035: 2034: 1902:Generalized/power 1747:978-0-521-58782-2 1723:978-0-691-01018-2 1693:978-0-471-23065-6 1653:978-1-4039-4155-8 1202:Granger causality 864: 348:: two integrated 42:, ...,  4134: 4097: 4096: 4085: 4084: 4074: 4073: 4059: 4058: 3962:Crime statistics 3856: 3843: 3760: 3726:Fourier analysis 3713:Frequency domain 3693: 3640: 3606:Structural break 3566: 3515:Cluster analysis 3462:Log-linear model 3435: 3410: 3351: 3325:Homoscedasticity 3181: 3157: 3076: 3068: 3060: 3059:(Kruskal–Wallis) 3044: 3029: 2984:Cross validation 2969: 2951:Anderson–Darling 2898: 2885: 2856:Likelihood-ratio 2848:Parametric tests 2826:Permutation test 2809:1- & 2-tails 2700:Minimum distance 2672:Point estimation 2668: 2619:Optimal decision 2570: 2469: 2456: 2438:Quasi-experiment 2388:Adaptive designs 2239: 2226: 2103:Rank correlation 1865: 1856: 1843: 1810: 1803: 1796: 1787: 1781: 1761: 1751: 1727: 1711: 1697: 1675: 1658: 1657: 1637: 1631: 1630: 1596: 1590: 1589: 1571: 1562: 1556: 1555: 1527: 1518: 1512: 1511: 1509: 1481: 1475: 1474: 1472: 1470: 1459: 1453: 1448: 1442: 1441: 1413: 1404: 1398: 1397: 1375: 1369: 1368: 1338: 1332: 1331: 1321: 1299: 1290: 1289: 1251: 1245: 1244: 1222: 1185:Bayesian methods 1173:structural break 1157: 1155: 1153: 1152: 1147: 1089: 1087: 1086: 1081: 1079: 1078: 1063: 1062: 1047: 1046: 1027: 1025: 1024: 1019: 1001: 999: 998: 993: 991: 990: 972: 970: 969: 964: 962: 961: 949: 948: 924: 923: 908: 907: 888: 886: 885: 880: 878: 877: 866: 865: 857: 846: 844: 843: 838: 836: 835: 816: 814: 813: 808: 806: 805: 783: 781: 780: 775: 764:. However, when 763: 761: 760: 755: 753: 752: 736: 734: 733: 728: 726: 725: 709: 707: 706: 701: 699: 698: 678: 676: 675: 670: 650: 648: 647: 642: 640: 639: 623: 621: 620: 615: 600: 598: 597: 592: 590: 589: 570: 568: 567: 562: 559: 558: 546: 545: 530: 529: 510: 508: 507: 502: 500: 499: 483: 481: 480: 475: 456: 454: 453: 448: 446: 445: 429: 427: 426: 421: 419: 418: 380: 378: 376: 375: 370: 321: 319: 317: 316: 311: 287: 285: 283: 282: 277: 254: 252: 250: 249: 244: 182: 180: 178: 177: 172: 109: 52: 4142: 4141: 4137: 4136: 4135: 4133: 4132: 4131: 4112: 4111: 4110: 4105: 4068: 4039: 4001: 3938: 3924:quality control 3891: 3873:Clinical trials 3850: 3825: 3809: 3797:Hazard function 3791: 3745: 3707: 3691: 3654: 3650:Breusch–Godfrey 3638: 3615: 3555: 3530:Factor analysis 3476: 3457:Graphical model 3429: 3396: 3363: 3349: 3329: 3283: 3250: 3212: 3175: 3174: 3143: 3087: 3074: 3066: 3058: 3042: 3027: 3006:Rank statistics 3000: 2979:Model selection 2967: 2925:Goodness of fit 2919: 2896: 2870: 2842: 2795: 2740: 2729:Median unbiased 2657: 2568: 2501:Order statistic 2463: 2442: 2409: 2383: 2335: 2290: 2233: 2231:Data collection 2212: 2124: 2079: 2053: 2031: 1991: 1943: 1860:Continuous data 1850: 1837: 1819: 1814: 1759: 1754: 1748: 1730: 1724: 1700: 1694: 1669: 1666: 1664:Further reading 1661: 1654: 1639: 1638: 1634: 1598: 1597: 1593: 1569: 1564: 1563: 1559: 1544:10.2307/2938339 1525: 1520: 1519: 1515: 1498:10.1002/jae.616 1483: 1482: 1478: 1468: 1466: 1461: 1460: 1456: 1449: 1445: 1430:10.2307/1913236 1411: 1406: 1405: 1401: 1377: 1376: 1372: 1340: 1339: 1335: 1319:10.1.1.353.2946 1301: 1300: 1293: 1270:10.2307/2341482 1253: 1252: 1248: 1224: 1223: 1219: 1215: 1193: 1181: 1168: 1129: 1128: 1126: 1123: 1108: 1095: 1070: 1054: 1038: 1030: 1029: 1004: 1003: 982: 977: 976: 953: 934: 915: 899: 891: 890: 854: 849: 848: 827: 819: 818: 797: 789: 788: 766: 765: 744: 739: 738: 717: 712: 711: 690: 685: 684: 683:(by regressing 661: 660: 631: 626: 625: 606: 605: 601:is stationary. 581: 576: 575: 550: 537: 521: 516: 515: 491: 486: 485: 466: 465: 437: 432: 431: 410: 405: 404: 401: 395: 387: 352: 351: 349: 293: 292: 290: 259: 258: 256: 226: 225: 223: 222:For integrated 193: 154: 153: 151: 136: 128:Charles Plosser 97: 50: 41: 34: 27: 17: 12: 11: 5: 4140: 4138: 4130: 4129: 4124: 4114: 4113: 4107: 4106: 4104: 4103: 4091: 4079: 4065: 4052: 4049: 4048: 4045: 4044: 4041: 4040: 4038: 4037: 4032: 4027: 4022: 4017: 4011: 4009: 4003: 4002: 4000: 3999: 3994: 3989: 3984: 3979: 3974: 3969: 3964: 3959: 3954: 3948: 3946: 3940: 3939: 3937: 3936: 3931: 3926: 3917: 3912: 3907: 3901: 3899: 3893: 3892: 3890: 3889: 3884: 3879: 3870: 3868:Bioinformatics 3864: 3862: 3852: 3851: 3846: 3839: 3838: 3835: 3834: 3831: 3830: 3827: 3826: 3824: 3823: 3817: 3815: 3811: 3810: 3808: 3807: 3801: 3799: 3793: 3792: 3790: 3789: 3784: 3779: 3774: 3768: 3766: 3757: 3751: 3750: 3747: 3746: 3744: 3743: 3738: 3733: 3728: 3723: 3717: 3715: 3709: 3708: 3706: 3705: 3700: 3695: 3687: 3682: 3677: 3676: 3675: 3673:partial (PACF) 3664: 3662: 3656: 3655: 3653: 3652: 3647: 3642: 3634: 3629: 3623: 3621: 3620:Specific tests 3617: 3616: 3614: 3613: 3608: 3603: 3598: 3593: 3588: 3583: 3578: 3572: 3570: 3563: 3557: 3556: 3554: 3553: 3552: 3551: 3550: 3549: 3534: 3533: 3532: 3522: 3520:Classification 3517: 3512: 3507: 3502: 3497: 3492: 3486: 3484: 3478: 3477: 3475: 3474: 3469: 3467:McNemar's test 3464: 3459: 3454: 3449: 3443: 3441: 3431: 3430: 3413: 3406: 3405: 3402: 3401: 3398: 3397: 3395: 3394: 3389: 3384: 3379: 3373: 3371: 3365: 3364: 3362: 3361: 3345: 3339: 3337: 3331: 3330: 3328: 3327: 3322: 3317: 3312: 3307: 3305:Semiparametric 3302: 3297: 3291: 3289: 3285: 3284: 3282: 3281: 3276: 3271: 3266: 3260: 3258: 3252: 3251: 3249: 3248: 3243: 3238: 3233: 3228: 3222: 3220: 3214: 3213: 3211: 3210: 3205: 3200: 3195: 3189: 3187: 3177: 3176: 3173: 3172: 3167: 3161: 3160: 3153: 3152: 3149: 3148: 3145: 3144: 3142: 3141: 3140: 3139: 3129: 3124: 3119: 3118: 3117: 3112: 3101: 3099: 3093: 3092: 3089: 3088: 3086: 3085: 3080: 3079: 3078: 3070: 3062: 3046: 3043:(Mann–Whitney) 3038: 3037: 3036: 3023: 3022: 3021: 3010: 3008: 3002: 3001: 2999: 2998: 2997: 2996: 2991: 2986: 2976: 2971: 2968:(Shapiro–Wilk) 2963: 2958: 2953: 2948: 2943: 2935: 2929: 2927: 2921: 2920: 2918: 2917: 2909: 2900: 2888: 2882: 2880:Specific tests 2876: 2875: 2872: 2871: 2869: 2868: 2863: 2858: 2852: 2850: 2844: 2843: 2841: 2840: 2835: 2834: 2833: 2823: 2822: 2821: 2811: 2805: 2803: 2797: 2796: 2794: 2793: 2792: 2791: 2786: 2776: 2771: 2766: 2761: 2756: 2750: 2748: 2742: 2741: 2739: 2738: 2733: 2732: 2731: 2726: 2725: 2724: 2719: 2704: 2703: 2702: 2697: 2692: 2687: 2676: 2674: 2665: 2659: 2658: 2656: 2655: 2650: 2645: 2644: 2643: 2633: 2628: 2627: 2626: 2616: 2615: 2614: 2609: 2604: 2594: 2589: 2584: 2583: 2582: 2577: 2572: 2556: 2555: 2554: 2549: 2544: 2534: 2533: 2532: 2527: 2517: 2516: 2515: 2505: 2504: 2503: 2493: 2488: 2483: 2477: 2475: 2465: 2464: 2459: 2452: 2451: 2448: 2447: 2444: 2443: 2441: 2440: 2435: 2430: 2425: 2419: 2417: 2411: 2410: 2408: 2407: 2402: 2397: 2391: 2389: 2385: 2384: 2382: 2381: 2376: 2371: 2366: 2361: 2356: 2351: 2345: 2343: 2337: 2336: 2334: 2333: 2331:Standard error 2328: 2323: 2318: 2317: 2316: 2311: 2300: 2298: 2292: 2291: 2289: 2288: 2283: 2278: 2273: 2268: 2263: 2261:Optimal design 2258: 2253: 2247: 2245: 2235: 2234: 2229: 2222: 2221: 2218: 2217: 2214: 2213: 2211: 2210: 2205: 2200: 2195: 2190: 2185: 2180: 2175: 2170: 2165: 2160: 2155: 2150: 2145: 2140: 2134: 2132: 2126: 2125: 2123: 2122: 2117: 2116: 2115: 2110: 2100: 2095: 2089: 2087: 2081: 2080: 2078: 2077: 2072: 2067: 2061: 2059: 2058:Summary tables 2055: 2054: 2052: 2051: 2045: 2043: 2037: 2036: 2033: 2032: 2030: 2029: 2028: 2027: 2022: 2017: 2007: 2001: 1999: 1993: 1992: 1990: 1989: 1984: 1979: 1974: 1969: 1964: 1959: 1953: 1951: 1945: 1944: 1942: 1941: 1936: 1931: 1930: 1929: 1924: 1919: 1914: 1909: 1904: 1899: 1894: 1892:Contraharmonic 1889: 1884: 1873: 1871: 1862: 1852: 1851: 1846: 1839: 1838: 1836: 1835: 1830: 1824: 1821: 1820: 1815: 1813: 1812: 1805: 1798: 1790: 1784: 1783: 1752: 1746: 1732:Maddala, G. S. 1728: 1722: 1702:Hayashi, Fumio 1698: 1692: 1665: 1662: 1660: 1659: 1652: 1632: 1613:(3): 497–505. 1591: 1557: 1538:(1): 165–193. 1513: 1492:(3): 289–326. 1476: 1454: 1443: 1424:(2): 251–276. 1399: 1388:(1): 121–130. 1370: 1333: 1312:(2): 111–120. 1291: 1246: 1235:(2): 139–162. 1216: 1214: 1211: 1210: 1209: 1204: 1199: 1192: 1189: 1180: 1177: 1167: 1164: 1145: 1142: 1139: 1136: 1122: 1119: 1107: 1104: 1094: 1091: 1077: 1073: 1069: 1066: 1061: 1057: 1053: 1050: 1045: 1041: 1037: 1017: 1014: 1011: 989: 985: 960: 956: 952: 947: 944: 941: 937: 933: 930: 927: 922: 918: 914: 911: 906: 902: 898: 876: 873: 870: 863: 860: 834: 830: 826: 804: 800: 796: 773: 751: 747: 724: 720: 697: 693: 668: 638: 634: 613: 588: 584: 572: 571: 557: 553: 549: 544: 540: 536: 533: 528: 524: 498: 494: 473: 444: 440: 417: 413: 394: 391: 386: 383: 368: 365: 362: 359: 309: 306: 303: 300: 275: 272: 269: 266: 242: 239: 236: 233: 192: 189: 170: 167: 164: 161: 135: 132: 46: 39: 32: 15: 13: 10: 9: 6: 4: 3: 2: 4139: 4128: 4125: 4123: 4120: 4119: 4117: 4102: 4101: 4092: 4090: 4089: 4080: 4078: 4077: 4072: 4066: 4064: 4063: 4054: 4053: 4050: 4036: 4033: 4031: 4030:Geostatistics 4028: 4026: 4023: 4021: 4018: 4016: 4013: 4012: 4010: 4008: 4004: 3998: 3997:Psychometrics 3995: 3993: 3990: 3988: 3985: 3983: 3980: 3978: 3975: 3973: 3970: 3968: 3965: 3963: 3960: 3958: 3955: 3953: 3950: 3949: 3947: 3945: 3941: 3935: 3932: 3930: 3927: 3925: 3921: 3918: 3916: 3913: 3911: 3908: 3906: 3903: 3902: 3900: 3898: 3894: 3888: 3885: 3883: 3880: 3878: 3874: 3871: 3869: 3866: 3865: 3863: 3861: 3860:Biostatistics 3857: 3853: 3849: 3844: 3840: 3822: 3821:Log-rank test 3819: 3818: 3816: 3812: 3806: 3803: 3802: 3800: 3798: 3794: 3788: 3785: 3783: 3780: 3778: 3775: 3773: 3770: 3769: 3767: 3765: 3761: 3758: 3756: 3752: 3742: 3739: 3737: 3734: 3732: 3729: 3727: 3724: 3722: 3719: 3718: 3716: 3714: 3710: 3704: 3701: 3699: 3696: 3694: 3692:(Box–Jenkins) 3688: 3686: 3683: 3681: 3678: 3674: 3671: 3670: 3669: 3666: 3665: 3663: 3661: 3657: 3651: 3648: 3646: 3645:Durbin–Watson 3643: 3641: 3635: 3633: 3630: 3628: 3627:Dickey–Fuller 3625: 3624: 3622: 3618: 3612: 3609: 3607: 3604: 3602: 3601:Cointegration 3599: 3597: 3594: 3592: 3589: 3587: 3584: 3582: 3579: 3577: 3576:Decomposition 3574: 3573: 3571: 3567: 3564: 3562: 3558: 3548: 3545: 3544: 3543: 3540: 3539: 3538: 3535: 3531: 3528: 3527: 3526: 3523: 3521: 3518: 3516: 3513: 3511: 3508: 3506: 3503: 3501: 3498: 3496: 3493: 3491: 3488: 3487: 3485: 3483: 3479: 3473: 3470: 3468: 3465: 3463: 3460: 3458: 3455: 3453: 3450: 3448: 3447:Cohen's kappa 3445: 3444: 3442: 3440: 3436: 3432: 3428: 3424: 3420: 3416: 3411: 3407: 3393: 3390: 3388: 3385: 3383: 3380: 3378: 3375: 3374: 3372: 3370: 3366: 3360: 3356: 3352: 3346: 3344: 3341: 3340: 3338: 3336: 3332: 3326: 3323: 3321: 3318: 3316: 3313: 3311: 3308: 3306: 3303: 3301: 3300:Nonparametric 3298: 3296: 3293: 3292: 3290: 3286: 3280: 3277: 3275: 3272: 3270: 3267: 3265: 3262: 3261: 3259: 3257: 3253: 3247: 3244: 3242: 3239: 3237: 3234: 3232: 3229: 3227: 3224: 3223: 3221: 3219: 3215: 3209: 3206: 3204: 3201: 3199: 3196: 3194: 3191: 3190: 3188: 3186: 3182: 3178: 3171: 3168: 3166: 3163: 3162: 3158: 3154: 3138: 3135: 3134: 3133: 3130: 3128: 3125: 3123: 3120: 3116: 3113: 3111: 3108: 3107: 3106: 3103: 3102: 3100: 3098: 3094: 3084: 3081: 3077: 3071: 3069: 3063: 3061: 3055: 3054: 3053: 3050: 3049:Nonparametric 3047: 3045: 3039: 3035: 3032: 3031: 3030: 3024: 3020: 3019:Sample median 3017: 3016: 3015: 3012: 3011: 3009: 3007: 3003: 2995: 2992: 2990: 2987: 2985: 2982: 2981: 2980: 2977: 2975: 2972: 2970: 2964: 2962: 2959: 2957: 2954: 2952: 2949: 2947: 2944: 2942: 2940: 2936: 2934: 2931: 2930: 2928: 2926: 2922: 2916: 2914: 2910: 2908: 2906: 2901: 2899: 2894: 2890: 2889: 2886: 2883: 2881: 2877: 2867: 2864: 2862: 2859: 2857: 2854: 2853: 2851: 2849: 2845: 2839: 2836: 2832: 2829: 2828: 2827: 2824: 2820: 2817: 2816: 2815: 2812: 2810: 2807: 2806: 2804: 2802: 2798: 2790: 2787: 2785: 2782: 2781: 2780: 2777: 2775: 2772: 2770: 2767: 2765: 2762: 2760: 2757: 2755: 2752: 2751: 2749: 2747: 2743: 2737: 2734: 2730: 2727: 2723: 2720: 2718: 2715: 2714: 2713: 2710: 2709: 2708: 2705: 2701: 2698: 2696: 2693: 2691: 2688: 2686: 2683: 2682: 2681: 2678: 2677: 2675: 2673: 2669: 2666: 2664: 2660: 2654: 2651: 2649: 2646: 2642: 2639: 2638: 2637: 2634: 2632: 2629: 2625: 2624:loss function 2622: 2621: 2620: 2617: 2613: 2610: 2608: 2605: 2603: 2600: 2599: 2598: 2595: 2593: 2590: 2588: 2585: 2581: 2578: 2576: 2573: 2571: 2565: 2562: 2561: 2560: 2557: 2553: 2550: 2548: 2545: 2543: 2540: 2539: 2538: 2535: 2531: 2528: 2526: 2523: 2522: 2521: 2518: 2514: 2511: 2510: 2509: 2506: 2502: 2499: 2498: 2497: 2494: 2492: 2489: 2487: 2484: 2482: 2479: 2478: 2476: 2474: 2470: 2466: 2462: 2457: 2453: 2439: 2436: 2434: 2431: 2429: 2426: 2424: 2421: 2420: 2418: 2416: 2412: 2406: 2403: 2401: 2398: 2396: 2393: 2392: 2390: 2386: 2380: 2377: 2375: 2372: 2370: 2367: 2365: 2362: 2360: 2357: 2355: 2352: 2350: 2347: 2346: 2344: 2342: 2338: 2332: 2329: 2327: 2326:Questionnaire 2324: 2322: 2319: 2315: 2312: 2310: 2307: 2306: 2305: 2302: 2301: 2299: 2297: 2293: 2287: 2284: 2282: 2279: 2277: 2274: 2272: 2269: 2267: 2264: 2262: 2259: 2257: 2254: 2252: 2249: 2248: 2246: 2244: 2240: 2236: 2232: 2227: 2223: 2209: 2206: 2204: 2201: 2199: 2196: 2194: 2191: 2189: 2186: 2184: 2181: 2179: 2176: 2174: 2171: 2169: 2166: 2164: 2161: 2159: 2156: 2154: 2153:Control chart 2151: 2149: 2146: 2144: 2141: 2139: 2136: 2135: 2133: 2131: 2127: 2121: 2118: 2114: 2111: 2109: 2106: 2105: 2104: 2101: 2099: 2096: 2094: 2091: 2090: 2088: 2086: 2082: 2076: 2073: 2071: 2068: 2066: 2063: 2062: 2060: 2056: 2050: 2047: 2046: 2044: 2042: 2038: 2026: 2023: 2021: 2018: 2016: 2013: 2012: 2011: 2008: 2006: 2003: 2002: 2000: 1998: 1994: 1988: 1985: 1983: 1980: 1978: 1975: 1973: 1970: 1968: 1965: 1963: 1960: 1958: 1955: 1954: 1952: 1950: 1946: 1940: 1937: 1935: 1932: 1928: 1925: 1923: 1920: 1918: 1915: 1913: 1910: 1908: 1905: 1903: 1900: 1898: 1895: 1893: 1890: 1888: 1885: 1883: 1880: 1879: 1878: 1875: 1874: 1872: 1870: 1866: 1863: 1861: 1857: 1853: 1849: 1844: 1840: 1834: 1831: 1829: 1826: 1825: 1822: 1818: 1811: 1806: 1804: 1799: 1797: 1792: 1791: 1788: 1779: 1775: 1771: 1767: 1766: 1758: 1753: 1749: 1743: 1739: 1738: 1733: 1729: 1725: 1719: 1715: 1710: 1709: 1703: 1699: 1695: 1689: 1685: 1681: 1680: 1674: 1668: 1667: 1663: 1655: 1649: 1645: 1644: 1636: 1633: 1628: 1624: 1620: 1616: 1612: 1608: 1607: 1602: 1595: 1592: 1587: 1583: 1580:(1): 99–126. 1579: 1575: 1568: 1561: 1558: 1553: 1549: 1545: 1541: 1537: 1533: 1532: 1524: 1517: 1514: 1508: 1503: 1499: 1495: 1491: 1487: 1480: 1477: 1465: 1458: 1455: 1452: 1447: 1444: 1439: 1435: 1431: 1427: 1423: 1419: 1418: 1410: 1403: 1400: 1395: 1391: 1387: 1383: 1382: 1374: 1371: 1366: 1362: 1358: 1354: 1350: 1346: 1345: 1337: 1334: 1329: 1325: 1320: 1315: 1311: 1307: 1306: 1298: 1296: 1292: 1287: 1283: 1279: 1275: 1271: 1267: 1263: 1259: 1258: 1250: 1247: 1242: 1238: 1234: 1230: 1229: 1221: 1218: 1212: 1208: 1205: 1203: 1200: 1198: 1195: 1194: 1190: 1188: 1186: 1178: 1176: 1174: 1165: 1163: 1161: 1140: 1134: 1120: 1118: 1116: 1112: 1105: 1103: 1100: 1099:Johansen test 1093:Johansen test 1092: 1090: 1075: 1071: 1067: 1064: 1059: 1055: 1048: 1043: 1039: 1015: 1012: 1009: 987: 983: 973: 958: 954: 950: 945: 942: 939: 935: 931: 928: 925: 920: 916: 909: 904: 900: 874: 871: 868: 858: 832: 828: 802: 798: 785: 771: 749: 745: 722: 718: 695: 691: 682: 666: 658: 654: 636: 632: 611: 602: 586: 582: 555: 551: 547: 542: 538: 534: 531: 526: 522: 514: 513: 512: 496: 492: 471: 463: 460: 442: 438: 415: 411: 400: 392: 390: 384: 382: 363: 357: 347: 343: 339: 334: 332: 328: 323: 304: 298: 270: 264: 237: 231: 220: 218: 214: 210: 206: 205:Clive Granger 202: 198: 190: 188: 186: 165: 159: 149: 145: 141: 133: 131: 129: 125: 121: 117: 113: 108: 105: +  104: 101: +  100: 95: 91: 87: 83: 79: 75: 71: 67: 64:. Next, if a 63: 60: 56: 49: 45: 38: 31: 25: 21: 20:Cointegration 4098: 4086: 4067: 4060: 3972:Econometrics 3922: / 3905:Chemometrics 3882:Epidemiology 3875: / 3848:Applications 3690:ARIMA model 3637:Q-statistic 3600: 3586:Stationarity 3482:Multivariate 3425: / 3421: / 3419:Multivariate 3417: / 3357: / 3353: / 3127:Bayes factor 3026:Signed rank 2938: 2912: 2904: 2892: 2587:Completeness 2423:Cohort study 2321:Opinion poll 2256:Missing data 2243:Study design 2198:Scatter plot 2120:Scatter plot 2113:Spearman's ρ 2075:Grouped data 1772:(1): 37–39. 1769: 1763: 1736: 1708:Econometrics 1707: 1678: 1642: 1635: 1610: 1604: 1594: 1577: 1573: 1560: 1535: 1531:Econometrica 1529: 1516: 1489: 1485: 1479: 1467:. Retrieved 1457: 1446: 1421: 1417:Econometrica 1415: 1402: 1385: 1379: 1373: 1351:(1): 64–73. 1348: 1342: 1336: 1309: 1303: 1264:(1): 11–63. 1261: 1255: 1249: 1232: 1226: 1220: 1182: 1169: 1159: 1124: 1115:Sam Ouliaris 1109: 1096: 974: 786: 603: 573: 461: 402: 388: 335: 324: 221: 217:Robert Engle 209:Paul Newbold 194: 137: 134:Introduction 119: 115: 111: 106: 102: 98: 93: 89: 85: 81: 77: 73: 69: 61: 47: 43: 36: 29: 19: 18: 4127:Time series 4100:WikiProject 4015:Cartography 3977:Jurimetrics 3929:Reliability 3660:Time domain 3639:(Ljung–Box) 3561:Time-series 3439:Categorical 3423:Time-series 3415:Categorical 3350:(Bernoulli) 3185:Correlation 3165:Correlation 2961:Jarque–Bera 2933:Chi-squared 2695:M-estimator 2648:Asymptotics 2592:Sufficiency 2359:Interaction 2271:Replication 2251:Effect size 2208:Violin plot 2188:Radar chart 2168:Forest plot 2158:Correlogram 2108:Kendall's τ 1507:10983/25617 55:time series 24:statistical 4116:Categories 3967:Demography 3685:ARMA model 3490:Regression 3067:(Friedman) 3028:(Wilcoxon) 2966:Normality 2956:Lilliefors 2903:Student's 2779:Resampling 2653:Robustness 2641:divergence 2631:Efficiency 2569:(monotone) 2564:Likelihood 2481:Population 2314:Stratified 2266:Population 2085:Dependence 2041:Count data 1972:Percentile 1949:Dispersion 1882:Arithmetic 1817:Statistics 1213:References 457:both have 397:See also: 327:unit roots 185:stationary 140:integrated 124:stochastic 96:such that 3348:Logistic 3115:posterior 3041:Rank sum 2789:Jackknife 2784:Bootstrap 2602:Bootstrap 2537:Parameter 2486:Statistic 2281:Statistic 2193:Run chart 2178:Pie chart 2173:Histogram 2163:Fan chart 2138:Bar chart 2020:L-moments 1907:Geometric 1627:153437469 1365:154550363 1314:CiteSeerX 1286:126346450 1072:ε 1052:Δ 1036:Δ 1010:α 955:ε 943:− 932:α 913:Δ 897:Δ 872:− 862:^ 825:Δ 795:Δ 772:β 667:β 612:β 535:β 532:− 472:β 338:R-squared 197:Udny Yule 4062:Category 3755:Survival 3632:Johansen 3355:Binomial 3310:Isotonic 2897:(normal) 2542:location 2349:Blocking 2304:Sampling 2183:Q–Q plot 2148:Box plot 2130:Graphics 2025:Skewness 2015:Kurtosis 1987:Variance 1917:Heronian 1912:Harmonic 1704:(2000). 1469:4 August 1191:See also 1183:Several 1002:), then 4088:Commons 4035:Kriging 3920:Process 3877:studies 3736:Wavelet 3569:General 2736:Plug-in 2530:L space 2309:Cluster 2010:Moments 1828:Outline 1684:319–386 1552:2938339 1438:1913236 1344:Wilmott 1278:2341482 1156:⁠ 1127:⁠ 379:⁠ 350:⁠ 320:⁠ 291:⁠ 286:⁠ 257:⁠ 253:⁠ 224:⁠ 191:History 181:⁠ 152:⁠ 35:,  3957:Census 3547:Normal 3495:Manova 3315:Robust 3065:2-way 3057:1-way 2895:-test 2566:  2143:Biplot 1934:Median 1927:Lehmer 1869:Center 1744:  1720:  1716:–669. 1690:  1650:  1625:  1550:  1436:  1363:  1316:  1284:  1276:  659:. If 574:where 118:, and 3581:Trend 3110:prior 3052:anova 2941:-test 2915:-test 2907:-test 2814:Power 2759:Pivot 2552:shape 2547:scale 1997:Shape 1977:Range 1922:Heinz 1897:Cubic 1833:Index 1760:(PDF) 1623:S2CID 1570:(PDF) 1548:JSTOR 1526:(PDF) 1434:JSTOR 1412:(PDF) 1361:S2CID 1282:S2CID 1274:JSTOR 385:Tests 59:order 22:is a 3814:Test 3014:Sign 2866:Wald 1939:Mode 1877:Mean 1742:ISBN 1718:ISBN 1688:ISBN 1648:ISBN 1471:2014 1349:2012 1113:and 1097:The 484:and 430:and 207:and 2994:BIC 2989:AIC 1774:doi 1714:623 1615:doi 1582:doi 1540:doi 1502:hdl 1494:doi 1426:doi 1390:doi 1353:doi 1324:doi 1266:doi 1237:doi 710:on 655:or 604:If 403:If 342:GNP 53:of 4118:: 1770:48 1768:. 1762:. 1686:. 1676:. 1621:. 1611:35 1609:. 1603:. 1578:70 1576:. 1572:. 1546:. 1536:58 1534:. 1528:. 1500:. 1490:16 1488:. 1432:. 1422:55 1420:. 1414:. 1386:16 1384:. 1359:. 1347:. 1322:. 1308:. 1294:^ 1280:. 1272:. 1262:89 1260:. 1233:10 1231:. 114:, 107:cZ 103:bY 99:aX 2939:G 2913:F 2905:t 2893:Z 2612:V 2607:U 1809:e 1802:t 1795:v 1780:. 1776:: 1750:. 1726:. 1696:. 1656:. 1629:. 1617:: 1588:. 1584:: 1554:. 1542:: 1510:. 1504:: 1496:: 1473:. 1440:. 1428:: 1396:. 1392:: 1367:. 1355:: 1330:. 1326:: 1310:2 1288:. 1268:: 1243:. 1239:: 1144:) 1141:1 1138:( 1135:I 1076:t 1068:+ 1065:b 1060:t 1056:x 1049:= 1044:t 1040:y 1016:0 1013:= 988:t 984:u 959:t 951:+ 946:1 940:t 936:u 929:+ 926:b 921:t 917:x 910:= 905:t 901:y 875:1 869:t 859:u 833:t 829:x 803:t 799:y 750:t 746:u 723:t 719:x 696:t 692:y 637:t 633:u 587:t 583:u 556:t 552:u 548:= 543:t 539:x 527:t 523:y 497:t 493:u 462:d 443:t 439:y 416:t 412:x 367:) 364:1 361:( 358:I 308:) 305:1 302:( 299:I 274:) 271:1 268:( 265:I 241:) 238:1 235:( 232:I 169:) 166:1 163:( 160:I 120:Z 116:Y 112:X 94:c 92:, 90:b 88:, 86:a 82:d 78:Z 76:, 74:Y 72:, 70:X 62:d 51:) 48:k 44:X 40:2 37:X 33:1 30:X 28:(

Index

statistical
time series
order
linear combination
stochastic
Charles Plosser
integrated
linear combination
order of integration
stationary
Udny Yule
linear regressions
Clive Granger
Paul Newbold
spurious correlation
Robert Engle
unit roots
ordinary least squares
R-squared
GNP
spurious regression
Error correction model § Engle and Granger 2-step approach
order of integration
Augmented Dickey–Fuller test
Phillips–Perron test
ordinary least squares
Johansen test
Peter C. B. Phillips
Sam Ouliaris
structural break

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