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Cointegration

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1128:(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. 4082: 4068: 4106: 4094: 1182:
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
340:(i.e. integrated of at least order one). The usual procedure for testing hypotheses concerning the relationship between non-stationary variables was to run 3858: 3482: 2123: 3256: 1267: 3695: 1198:
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
134: 17: 1420:"Co-integration and error correction: Representation, estimation and testing" 3024: 2876: 2496: 2291: 2203: 2188: 2183: 2148: 1768:"A Drunk and her Dog: An Illustration of Cointegration and Error Correction" 348: 337: 207: 1652: 1186:, and tests for cointegration with two unknown breaks are also available. 977:{\displaystyle \Delta y_{t}=\Delta x_{t}b+\alpha u_{t-1}+\varepsilon _{t}} 2540: 2158: 2035: 2030: 2025: 1997: 1313:
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.
194:) but some (cointegrating) vector of coefficients exists to form a 1534:"Asymptotic Properties of Residual Based Tests for Cointegration" 1887: 3856: 3423: 3170: 2469: 2239: 1856: 1800: 410:
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: 1576:Gregory, Allan W.; Hansen, Bruce E. (1996). 1793:An intuitive introduction to cointegration. 1684:"Cointegration and Error-Correction Models" 900:. The second stage regression is given as: 3866: 3853: 3770: 3576: 3445: 3420: 3191: 3167: 2895: 2678: 2479: 2466: 2249: 2236: 1875: 1866: 1853: 1819: 1805: 1797: 1532:Phillips, P. C. B.; Ouliaris, S. (1990). 1516: 1328: 1143: 1085: 1069: 1053: 1044: 1018: 997: 991: 968: 949: 930: 914: 905: 878: 867: 866: 863: 842: 833: 812: 803: 780: 759: 753: 732: 726: 705: 699: 675: 646: 640: 620: 596: 590: 575:{\displaystyle y_{t}-\beta x_{t}=u_{t}\,} 571: 565: 552: 536: 530: 506: 500: 480: 452: 446: 425: 419: 366: 307: 273: 240: 168: 121:is integrated of order less than d, then 1657:. Palgrave Macmillan. pp. 871–898. 1268:Journal of the Royal Statistical Society 149:If two or more series are individually 1723:. Princeton University Press. pp.  1228: 3783:Kaplan–Meier estimator (product limit) 1475:"ARDL Models - Part II - Bounds Tests" 1039:and we estimate a differences model: 7: 4093: 3793:Accelerated failure time (AFT) model 1117:Phillips–Ouliaris cointegration test 153:(in the time series sense) but some 4105: 3388:Analysis of variance (ANOVA, anova) 1177:Variable shifts in long time series 3483:Cochran–Mantel–Haenszel statistics 2109:Pearson product-moment correlation 1062: 1046: 923: 907: 835: 805: 25: 4104: 4092: 4080: 4067: 4066: 1690:Applied Econometrics Time Series 893:{\displaystyle {\hat {u}}_{t-1}} 3742:Least-squares spectral analysis 1497:Journal of Applied Econometrics 95:, and there exist coefficients 91:) are each integrated of order 2723:Mean-unbiased minimum-variance 1789:10.1080/00031305.1994.10476017 1154: 1148: 872: 377: 371: 318: 312: 284: 278: 251: 245: 179: 173: 1: 4036:Geographic information system 3252:Simultaneous equations models 1473:Giles, David (19 June 2013). 1239:Journal of Monetary Economics 404:Engle–Granger two-step method 198:linear combination of them. 3219:Coefficient of determination 2830:Uniformly most powerful test 1597:10.1016/0304-4076(69)41685-7 1405:10.1016/0304-4076(81)90079-8 1339:10.1016/0304-4076(74)90034-7 1252:10.1016/0304-3932(82)90012-5 1218:Stationary subspace analysis 851:{\displaystyle \Delta x_{t}} 821:{\displaystyle \Delta y_{t}} 664:Augmented Dickey–Fuller test 3788:Proportional hazards models 3732:Spectral density estimation 3714:Vector autoregression (VAR) 3148:Maximum posterior estimator 2380:Randomized controlled trial 1766:Murray, Michael P. (1994). 4154: 3548:Multivariate distributions 1968:Average absolute deviation 828:on the lagged regressors, 407: 4062: 3865: 3852: 3536:Structural equation model 3444: 3419: 3190: 3166: 2898: 2872:Score/Lagrange multiplier 2478: 2465: 2287:Sample size determination 2248: 2235: 1865: 1852: 1834: 1776:The American Statistician 1630:10.1007/s00181-007-0175-9 1113:Distributed Lags (ARDL). 1032:{\displaystyle \alpha =0} 662:for stationarity with an 37:property of a collection 4031:Environmental statistics 3553:Elliptical distributions 3346:Generalized linear model 3275:Simple linear regression 3045:Hodges–Lehmann estimator 2502:Probability distribution 2411:Stochastic approximation 1973:Coefficient of variation 3691:Cross-correlation (XCF) 3299:Non-standard predictors 2733:Lehmann–ScheffĂ© theorem 2406:Adaptive clinical trial 1681:Enders, Walter (2004). 1585:Journal of Econometrics 1392:Journal of Econometrics 1316:Journal of Econometrics 4087:Mathematics portal 3908:Engineering statistics 3816:Nelson–Aalen estimator 3393:Analysis of covariance 3280:Ordinary least squares 3204:Pearson product-moment 2608:Statistical functional 2519:Empirical distribution 2352:Controlled experiments 2081:Frequency distribution 1859:Descriptive statistics 1745:; Kim, In-Moo (1998). 1208:Error correction model 1161: 1095: 1033: 1007: 978: 894: 852: 822: 789: 788:{\displaystyle \beta } 769: 742: 715: 692:ordinary least squares 684: 683:{\displaystyle \beta } 656: 635:is known, we can test 629: 628:{\displaystyle \beta } 606: 576: 516: 489: 488:{\displaystyle \beta } 462: 435: 384: 342:ordinary least squares 325: 291: 258: 186: 4003:Population statistics 3945:System identification 3679:Autocorrelation (ACF) 3607:Exponential smoothing 3521:Discriminant analysis 3516:Canonical correlation 3380:Partition of variance 3242:Regression validation 3086:(Jonckheere–Terpstra) 2985:Likelihood-ratio test 2674:Frequentist inference 2586:Location–scale family 2507:Sampling distribution 2472:Statistical inference 2439:Cross-sectional study 2426:Observational studies 2385:Randomized experiment 2214:Stem-and-leaf display 2016:Central limit theorem 1610:Hatemi-J, A. (2008). 1162: 1096: 1034: 1008: 1006:{\displaystyle u_{t}} 979: 895: 853: 823: 790: 770: 768:{\displaystyle u_{t}} 743: 741:{\displaystyle x_{t}} 716: 714:{\displaystyle y_{t}} 685: 657: 655:{\displaystyle u_{t}} 630: 607: 605:{\displaystyle u_{t}} 577: 517: 515:{\displaystyle u_{t}} 490: 463: 461:{\displaystyle y_{t}} 436: 434:{\displaystyle x_{t}} 385: 326: 292: 259: 187: 4133:Mathematical finance 3926:Probabilistic design 3511:Principal components 3354:Exponential families 3306:Nonlinear regression 3285:General linear model 3247:Mixed effects models 3237:Errors and residuals 3214:Confounding variable 3116:Bayesian probability 3094:Van der Waerden test 3084:Ordered alternative 2849:Multiple comparisons 2728:Rao–Blackwellization 2691:Estimating equations 2647:Statistical distance 2365:Factorial experiment 1898:Arithmetic-Geometric 1160:{\displaystyle I(1)} 1142: 1122:Peter C. B. Phillips 1043: 1017: 990: 904: 862: 832: 802: 779: 752: 725: 698: 674: 668:Phillips–Perron test 639: 619: 589: 529: 499: 479: 470:order of integration 445: 418: 383:{\displaystyle I(1)} 365: 324:{\displaystyle I(1)} 306: 290:{\displaystyle I(1)} 272: 257:{\displaystyle I(1)} 239: 224:spurious correlation 185:{\displaystyle I(1)} 167: 159:order of integration 157:of them has a lower 3998:Official statistics 3921:Methods engineering 3602:Seasonal adjustment 3370:Poisson regressions 3290:Bayesian regression 3229:Regression analysis 3209:Partial correlation 3181:Regression analysis 2780:Prediction interval 2775:Likelihood interval 2765:Confidence interval 2757:Interval estimation 2718:Unbiased estimators 2536:Model specification 2416:Up-and-down designs 2104:Partial correlation 2060:Index of dispersion 1978:Interquartile range 1617:Empirical Economics 522:. In other words: 357:spurious regression 4018:Spatial statistics 3898:Medical statistics 3798:First hitting time 3752:Whittle likelihood 3403:Degrees of freedom 3398:Multivariate ANOVA 3331:Heteroscedasticity 3143:Bayesian estimator 3108:Bayesian inference 2957:Kolmogorov–Smirnov 2842:Randomization test 2812:Testing hypotheses 2785:Tolerance interval 2696:Maximum likelihood 2591:Exponential family 2524:Density estimation 2484:Statistical theory 2444:Natural experiment 2390:Scientific control 2307:Survey methodology 1993:Standard deviation 1368:10.1002/wilm.10167 1190:Bayesian inference 1171:Multicointegration 1157: 1132:Multicointegration 1091: 1029: 1003: 974: 890: 848: 818: 785: 765: 738: 711: 680: 652: 625: 602: 572: 512: 485: 458: 431: 380: 355:). This is called 321: 287: 254: 212:linear regressions 182: 155:linear combination 77:linear combination 4120: 4119: 4058: 4057: 4054: 4053: 3993:National accounts 3963:Actuarial science 3955:Social statistics 3848: 3847: 3844: 3843: 3840: 3839: 3775:Survival function 3760: 3759: 3622:Granger causality 3463:Contingency table 3438:Survival analysis 3415: 3414: 3411: 3410: 3267:Linear regression 3162: 3161: 3158: 3157: 3133:Credible interval 3102: 3101: 2885: 2884: 2701:Method of moments 2570:Parametric family 2531:Statistical model 2461: 2460: 2457: 2456: 2375:Random assignment 2297:Statistical power 2231: 2230: 2227: 2226: 2076:Contingency table 2046: 2045: 1913:Generalized/power 1758:978-0-521-58782-2 1734:978-0-691-01018-2 1704:978-0-471-23065-6 1664:978-1-4039-4155-8 1213:Granger causality 875: 359:: two integrated 53:, ...,  16:(Redirected from 4145: 4108: 4107: 4096: 4095: 4085: 4084: 4070: 4069: 3973:Crime statistics 3867: 3854: 3771: 3737:Fourier analysis 3724:Frequency domain 3704: 3651: 3617:Structural break 3577: 3526:Cluster analysis 3473:Log-linear model 3446: 3421: 3362: 3336:Homoscedasticity 3192: 3168: 3087: 3079: 3071: 3070:(Kruskal–Wallis) 3055: 3040: 2995:Cross validation 2980: 2962:Anderson–Darling 2909: 2896: 2867:Likelihood-ratio 2859:Parametric tests 2837:Permutation test 2820:1- & 2-tails 2711:Minimum distance 2683:Point estimation 2679: 2630:Optimal decision 2581: 2480: 2467: 2449:Quasi-experiment 2399:Adaptive designs 2250: 2237: 2114:Rank correlation 1876: 1867: 1854: 1821: 1814: 1807: 1798: 1792: 1772: 1762: 1738: 1722: 1708: 1686: 1669: 1668: 1648: 1642: 1641: 1607: 1601: 1600: 1582: 1573: 1567: 1566: 1538: 1529: 1523: 1522: 1520: 1492: 1486: 1485: 1483: 1481: 1470: 1464: 1459: 1453: 1452: 1424: 1415: 1409: 1408: 1386: 1380: 1379: 1349: 1343: 1342: 1332: 1310: 1301: 1300: 1262: 1256: 1255: 1233: 1196:Bayesian methods 1184:structural break 1168: 1166: 1164: 1163: 1158: 1100: 1098: 1097: 1092: 1090: 1089: 1074: 1073: 1058: 1057: 1038: 1036: 1035: 1030: 1012: 1010: 1009: 1004: 1002: 1001: 983: 981: 980: 975: 973: 972: 960: 959: 935: 934: 919: 918: 899: 897: 896: 891: 889: 888: 877: 876: 868: 857: 855: 854: 849: 847: 846: 827: 825: 824: 819: 817: 816: 794: 792: 791: 786: 775:. However, when 774: 772: 771: 766: 764: 763: 747: 745: 744: 739: 737: 736: 720: 718: 717: 712: 710: 709: 689: 687: 686: 681: 661: 659: 658: 653: 651: 650: 634: 632: 631: 626: 611: 609: 608: 603: 601: 600: 581: 579: 578: 573: 570: 569: 557: 556: 541: 540: 521: 519: 518: 513: 511: 510: 494: 492: 491: 486: 467: 465: 464: 459: 457: 456: 440: 438: 437: 432: 430: 429: 391: 389: 387: 386: 381: 332: 330: 328: 327: 322: 298: 296: 294: 293: 288: 265: 263: 261: 260: 255: 193: 191: 189: 188: 183: 120: 63: 21: 4153: 4152: 4148: 4147: 4146: 4144: 4143: 4142: 4123: 4122: 4121: 4116: 4079: 4050: 4012: 3949: 3935:quality control 3902: 3884:Clinical trials 3861: 3836: 3820: 3808:Hazard function 3802: 3756: 3718: 3702: 3665: 3661:Breusch–Godfrey 3649: 3626: 3566: 3541:Factor analysis 3487: 3468:Graphical model 3440: 3407: 3374: 3360: 3340: 3294: 3261: 3223: 3186: 3185: 3154: 3098: 3085: 3077: 3069: 3053: 3038: 3017:Rank statistics 3011: 2990:Model selection 2978: 2936:Goodness of fit 2930: 2907: 2881: 2853: 2806: 2751: 2740:Median unbiased 2668: 2579: 2512:Order statistic 2474: 2453: 2420: 2394: 2346: 2301: 2244: 2242:Data collection 2223: 2135: 2090: 2064: 2042: 2002: 1954: 1871:Continuous data 1861: 1848: 1830: 1825: 1770: 1765: 1759: 1741: 1735: 1711: 1705: 1680: 1677: 1675:Further reading 1672: 1665: 1650: 1649: 1645: 1609: 1608: 1604: 1580: 1575: 1574: 1570: 1555:10.2307/2938339 1536: 1531: 1530: 1526: 1509:10.1002/jae.616 1494: 1493: 1489: 1479: 1477: 1472: 1471: 1467: 1460: 1456: 1441:10.2307/1913236 1422: 1417: 1416: 1412: 1388: 1387: 1383: 1351: 1350: 1346: 1330:10.1.1.353.2946 1312: 1311: 1304: 1281:10.2307/2341482 1264: 1263: 1259: 1235: 1234: 1230: 1226: 1204: 1192: 1179: 1140: 1139: 1137: 1134: 1119: 1106: 1081: 1065: 1049: 1041: 1040: 1015: 1014: 993: 988: 987: 964: 945: 926: 910: 902: 901: 865: 860: 859: 838: 830: 829: 808: 800: 799: 777: 776: 755: 750: 749: 728: 723: 722: 701: 696: 695: 694:(by regressing 672: 671: 642: 637: 636: 617: 616: 612:is stationary. 592: 587: 586: 561: 548: 532: 527: 526: 502: 497: 496: 477: 476: 448: 443: 442: 421: 416: 415: 412: 406: 398: 363: 362: 360: 304: 303: 301: 270: 269: 267: 237: 236: 234: 233:For integrated 204: 165: 164: 162: 147: 139:Charles Plosser 108: 61: 52: 45: 38: 28: 23: 22: 15: 12: 11: 5: 4151: 4149: 4141: 4140: 4135: 4125: 4124: 4118: 4117: 4115: 4114: 4102: 4090: 4076: 4063: 4060: 4059: 4056: 4055: 4052: 4051: 4049: 4048: 4043: 4038: 4033: 4028: 4022: 4020: 4014: 4013: 4011: 4010: 4005: 4000: 3995: 3990: 3985: 3980: 3975: 3970: 3965: 3959: 3957: 3951: 3950: 3948: 3947: 3942: 3937: 3928: 3923: 3918: 3912: 3910: 3904: 3903: 3901: 3900: 3895: 3890: 3881: 3879:Bioinformatics 3875: 3873: 3863: 3862: 3857: 3850: 3849: 3846: 3845: 3842: 3841: 3838: 3837: 3835: 3834: 3828: 3826: 3822: 3821: 3819: 3818: 3812: 3810: 3804: 3803: 3801: 3800: 3795: 3790: 3785: 3779: 3777: 3768: 3762: 3761: 3758: 3757: 3755: 3754: 3749: 3744: 3739: 3734: 3728: 3726: 3720: 3719: 3717: 3716: 3711: 3706: 3698: 3693: 3688: 3687: 3686: 3684:partial (PACF) 3675: 3673: 3667: 3666: 3664: 3663: 3658: 3653: 3645: 3640: 3634: 3632: 3631:Specific tests 3628: 3627: 3625: 3624: 3619: 3614: 3609: 3604: 3599: 3594: 3589: 3583: 3581: 3574: 3568: 3567: 3565: 3564: 3563: 3562: 3561: 3560: 3545: 3544: 3543: 3533: 3531:Classification 3528: 3523: 3518: 3513: 3508: 3503: 3497: 3495: 3489: 3488: 3486: 3485: 3480: 3478:McNemar's test 3475: 3470: 3465: 3460: 3454: 3452: 3442: 3441: 3424: 3417: 3416: 3413: 3412: 3409: 3408: 3406: 3405: 3400: 3395: 3390: 3384: 3382: 3376: 3375: 3373: 3372: 3356: 3350: 3348: 3342: 3341: 3339: 3338: 3333: 3328: 3323: 3318: 3316:Semiparametric 3313: 3308: 3302: 3300: 3296: 3295: 3293: 3292: 3287: 3282: 3277: 3271: 3269: 3263: 3262: 3260: 3259: 3254: 3249: 3244: 3239: 3233: 3231: 3225: 3224: 3222: 3221: 3216: 3211: 3206: 3200: 3198: 3188: 3187: 3184: 3183: 3178: 3172: 3171: 3164: 3163: 3160: 3159: 3156: 3155: 3153: 3152: 3151: 3150: 3140: 3135: 3130: 3129: 3128: 3123: 3112: 3110: 3104: 3103: 3100: 3099: 3097: 3096: 3091: 3090: 3089: 3081: 3073: 3057: 3054:(Mann–Whitney) 3049: 3048: 3047: 3034: 3033: 3032: 3021: 3019: 3013: 3012: 3010: 3009: 3008: 3007: 3002: 2997: 2987: 2982: 2979:(Shapiro–Wilk) 2974: 2969: 2964: 2959: 2954: 2946: 2940: 2938: 2932: 2931: 2929: 2928: 2920: 2911: 2899: 2893: 2891:Specific tests 2887: 2886: 2883: 2882: 2880: 2879: 2874: 2869: 2863: 2861: 2855: 2854: 2852: 2851: 2846: 2845: 2844: 2834: 2833: 2832: 2822: 2816: 2814: 2808: 2807: 2805: 2804: 2803: 2802: 2797: 2787: 2782: 2777: 2772: 2767: 2761: 2759: 2753: 2752: 2750: 2749: 2744: 2743: 2742: 2737: 2736: 2735: 2730: 2715: 2714: 2713: 2708: 2703: 2698: 2687: 2685: 2676: 2670: 2669: 2667: 2666: 2661: 2656: 2655: 2654: 2644: 2639: 2638: 2637: 2627: 2626: 2625: 2620: 2615: 2605: 2600: 2595: 2594: 2593: 2588: 2583: 2567: 2566: 2565: 2560: 2555: 2545: 2544: 2543: 2538: 2528: 2527: 2526: 2516: 2515: 2514: 2504: 2499: 2494: 2488: 2486: 2476: 2475: 2470: 2463: 2462: 2459: 2458: 2455: 2454: 2452: 2451: 2446: 2441: 2436: 2430: 2428: 2422: 2421: 2419: 2418: 2413: 2408: 2402: 2400: 2396: 2395: 2393: 2392: 2387: 2382: 2377: 2372: 2367: 2362: 2356: 2354: 2348: 2347: 2345: 2344: 2342:Standard error 2339: 2334: 2329: 2328: 2327: 2322: 2311: 2309: 2303: 2302: 2300: 2299: 2294: 2289: 2284: 2279: 2274: 2272:Optimal design 2269: 2264: 2258: 2256: 2246: 2245: 2240: 2233: 2232: 2229: 2228: 2225: 2224: 2222: 2221: 2216: 2211: 2206: 2201: 2196: 2191: 2186: 2181: 2176: 2171: 2166: 2161: 2156: 2151: 2145: 2143: 2137: 2136: 2134: 2133: 2128: 2127: 2126: 2121: 2111: 2106: 2100: 2098: 2092: 2091: 2089: 2088: 2083: 2078: 2072: 2070: 2069:Summary tables 2066: 2065: 2063: 2062: 2056: 2054: 2048: 2047: 2044: 2043: 2041: 2040: 2039: 2038: 2033: 2028: 2018: 2012: 2010: 2004: 2003: 2001: 2000: 1995: 1990: 1985: 1980: 1975: 1970: 1964: 1962: 1956: 1955: 1953: 1952: 1947: 1942: 1941: 1940: 1935: 1930: 1925: 1920: 1915: 1910: 1905: 1903:Contraharmonic 1900: 1895: 1884: 1882: 1873: 1863: 1862: 1857: 1850: 1849: 1847: 1846: 1841: 1835: 1832: 1831: 1826: 1824: 1823: 1816: 1809: 1801: 1795: 1794: 1763: 1757: 1743:Maddala, G. S. 1739: 1733: 1713:Hayashi, Fumio 1709: 1703: 1676: 1673: 1671: 1670: 1663: 1643: 1624:(3): 497–505. 1602: 1568: 1549:(1): 165–193. 1524: 1503:(3): 289–326. 1487: 1465: 1454: 1435:(2): 251–276. 1410: 1399:(1): 121–130. 1381: 1344: 1323:(2): 111–120. 1302: 1257: 1246:(2): 139–162. 1227: 1225: 1222: 1221: 1220: 1215: 1210: 1203: 1200: 1191: 1188: 1178: 1175: 1156: 1153: 1150: 1147: 1133: 1130: 1118: 1115: 1105: 1102: 1088: 1084: 1080: 1077: 1072: 1068: 1064: 1061: 1056: 1052: 1048: 1028: 1025: 1022: 1000: 996: 971: 967: 963: 958: 955: 952: 948: 944: 941: 938: 933: 929: 925: 922: 917: 913: 909: 887: 884: 881: 874: 871: 845: 841: 837: 815: 811: 807: 784: 762: 758: 735: 731: 708: 704: 679: 649: 645: 624: 599: 595: 583: 582: 568: 564: 560: 555: 551: 547: 544: 539: 535: 509: 505: 484: 455: 451: 428: 424: 405: 402: 397: 394: 379: 376: 373: 370: 320: 317: 314: 311: 286: 283: 280: 277: 253: 250: 247: 244: 203: 200: 181: 178: 175: 172: 146: 143: 57: 50: 43: 26: 24: 18:Co-integration 14: 13: 10: 9: 6: 4: 3: 2: 4150: 4139: 4136: 4134: 4131: 4130: 4128: 4113: 4112: 4103: 4101: 4100: 4091: 4089: 4088: 4083: 4077: 4075: 4074: 4065: 4064: 4061: 4047: 4044: 4042: 4041:Geostatistics 4039: 4037: 4034: 4032: 4029: 4027: 4024: 4023: 4021: 4019: 4015: 4009: 4008:Psychometrics 4006: 4004: 4001: 3999: 3996: 3994: 3991: 3989: 3986: 3984: 3981: 3979: 3976: 3974: 3971: 3969: 3966: 3964: 3961: 3960: 3958: 3956: 3952: 3946: 3943: 3941: 3938: 3936: 3932: 3929: 3927: 3924: 3922: 3919: 3917: 3914: 3913: 3911: 3909: 3905: 3899: 3896: 3894: 3891: 3889: 3885: 3882: 3880: 3877: 3876: 3874: 3872: 3871:Biostatistics 3868: 3864: 3860: 3855: 3851: 3833: 3832:Log-rank test 3830: 3829: 3827: 3823: 3817: 3814: 3813: 3811: 3809: 3805: 3799: 3796: 3794: 3791: 3789: 3786: 3784: 3781: 3780: 3778: 3776: 3772: 3769: 3767: 3763: 3753: 3750: 3748: 3745: 3743: 3740: 3738: 3735: 3733: 3730: 3729: 3727: 3725: 3721: 3715: 3712: 3710: 3707: 3705: 3703:(Box–Jenkins) 3699: 3697: 3694: 3692: 3689: 3685: 3682: 3681: 3680: 3677: 3676: 3674: 3672: 3668: 3662: 3659: 3657: 3656:Durbin–Watson 3654: 3652: 3646: 3644: 3641: 3639: 3638:Dickey–Fuller 3636: 3635: 3633: 3629: 3623: 3620: 3618: 3615: 3613: 3612:Cointegration 3610: 3608: 3605: 3603: 3600: 3598: 3595: 3593: 3590: 3588: 3587:Decomposition 3585: 3584: 3582: 3578: 3575: 3573: 3569: 3559: 3556: 3555: 3554: 3551: 3550: 3549: 3546: 3542: 3539: 3538: 3537: 3534: 3532: 3529: 3527: 3524: 3522: 3519: 3517: 3514: 3512: 3509: 3507: 3504: 3502: 3499: 3498: 3496: 3494: 3490: 3484: 3481: 3479: 3476: 3474: 3471: 3469: 3466: 3464: 3461: 3459: 3458:Cohen's kappa 3456: 3455: 3453: 3451: 3447: 3443: 3439: 3435: 3431: 3427: 3422: 3418: 3404: 3401: 3399: 3396: 3394: 3391: 3389: 3386: 3385: 3383: 3381: 3377: 3371: 3367: 3363: 3357: 3355: 3352: 3351: 3349: 3347: 3343: 3337: 3334: 3332: 3329: 3327: 3324: 3322: 3319: 3317: 3314: 3312: 3311:Nonparametric 3309: 3307: 3304: 3303: 3301: 3297: 3291: 3288: 3286: 3283: 3281: 3278: 3276: 3273: 3272: 3270: 3268: 3264: 3258: 3255: 3253: 3250: 3248: 3245: 3243: 3240: 3238: 3235: 3234: 3232: 3230: 3226: 3220: 3217: 3215: 3212: 3210: 3207: 3205: 3202: 3201: 3199: 3197: 3193: 3189: 3182: 3179: 3177: 3174: 3173: 3169: 3165: 3149: 3146: 3145: 3144: 3141: 3139: 3136: 3134: 3131: 3127: 3124: 3122: 3119: 3118: 3117: 3114: 3113: 3111: 3109: 3105: 3095: 3092: 3088: 3082: 3080: 3074: 3072: 3066: 3065: 3064: 3061: 3060:Nonparametric 3058: 3056: 3050: 3046: 3043: 3042: 3041: 3035: 3031: 3030:Sample median 3028: 3027: 3026: 3023: 3022: 3020: 3018: 3014: 3006: 3003: 3001: 2998: 2996: 2993: 2992: 2991: 2988: 2986: 2983: 2981: 2975: 2973: 2970: 2968: 2965: 2963: 2960: 2958: 2955: 2953: 2951: 2947: 2945: 2942: 2941: 2939: 2937: 2933: 2927: 2925: 2921: 2919: 2917: 2912: 2910: 2905: 2901: 2900: 2897: 2894: 2892: 2888: 2878: 2875: 2873: 2870: 2868: 2865: 2864: 2862: 2860: 2856: 2850: 2847: 2843: 2840: 2839: 2838: 2835: 2831: 2828: 2827: 2826: 2823: 2821: 2818: 2817: 2815: 2813: 2809: 2801: 2798: 2796: 2793: 2792: 2791: 2788: 2786: 2783: 2781: 2778: 2776: 2773: 2771: 2768: 2766: 2763: 2762: 2760: 2758: 2754: 2748: 2745: 2741: 2738: 2734: 2731: 2729: 2726: 2725: 2724: 2721: 2720: 2719: 2716: 2712: 2709: 2707: 2704: 2702: 2699: 2697: 2694: 2693: 2692: 2689: 2688: 2686: 2684: 2680: 2677: 2675: 2671: 2665: 2662: 2660: 2657: 2653: 2650: 2649: 2648: 2645: 2643: 2640: 2636: 2635:loss function 2633: 2632: 2631: 2628: 2624: 2621: 2619: 2616: 2614: 2611: 2610: 2609: 2606: 2604: 2601: 2599: 2596: 2592: 2589: 2587: 2584: 2582: 2576: 2573: 2572: 2571: 2568: 2564: 2561: 2559: 2556: 2554: 2551: 2550: 2549: 2546: 2542: 2539: 2537: 2534: 2533: 2532: 2529: 2525: 2522: 2521: 2520: 2517: 2513: 2510: 2509: 2508: 2505: 2503: 2500: 2498: 2495: 2493: 2490: 2489: 2487: 2485: 2481: 2477: 2473: 2468: 2464: 2450: 2447: 2445: 2442: 2440: 2437: 2435: 2432: 2431: 2429: 2427: 2423: 2417: 2414: 2412: 2409: 2407: 2404: 2403: 2401: 2397: 2391: 2388: 2386: 2383: 2381: 2378: 2376: 2373: 2371: 2368: 2366: 2363: 2361: 2358: 2357: 2355: 2353: 2349: 2343: 2340: 2338: 2337:Questionnaire 2335: 2333: 2330: 2326: 2323: 2321: 2318: 2317: 2316: 2313: 2312: 2310: 2308: 2304: 2298: 2295: 2293: 2290: 2288: 2285: 2283: 2280: 2278: 2275: 2273: 2270: 2268: 2265: 2263: 2260: 2259: 2257: 2255: 2251: 2247: 2243: 2238: 2234: 2220: 2217: 2215: 2212: 2210: 2207: 2205: 2202: 2200: 2197: 2195: 2192: 2190: 2187: 2185: 2182: 2180: 2177: 2175: 2172: 2170: 2167: 2165: 2164:Control chart 2162: 2160: 2157: 2155: 2152: 2150: 2147: 2146: 2144: 2142: 2138: 2132: 2129: 2125: 2122: 2120: 2117: 2116: 2115: 2112: 2110: 2107: 2105: 2102: 2101: 2099: 2097: 2093: 2087: 2084: 2082: 2079: 2077: 2074: 2073: 2071: 2067: 2061: 2058: 2057: 2055: 2053: 2049: 2037: 2034: 2032: 2029: 2027: 2024: 2023: 2022: 2019: 2017: 2014: 2013: 2011: 2009: 2005: 1999: 1996: 1994: 1991: 1989: 1986: 1984: 1981: 1979: 1976: 1974: 1971: 1969: 1966: 1965: 1963: 1961: 1957: 1951: 1948: 1946: 1943: 1939: 1936: 1934: 1931: 1929: 1926: 1924: 1921: 1919: 1916: 1914: 1911: 1909: 1906: 1904: 1901: 1899: 1896: 1894: 1891: 1890: 1889: 1886: 1885: 1883: 1881: 1877: 1874: 1872: 1868: 1864: 1860: 1855: 1851: 1845: 1842: 1840: 1837: 1836: 1833: 1829: 1822: 1817: 1815: 1810: 1808: 1803: 1802: 1799: 1790: 1786: 1782: 1778: 1777: 1769: 1764: 1760: 1754: 1750: 1749: 1744: 1740: 1736: 1730: 1726: 1721: 1720: 1714: 1710: 1706: 1700: 1696: 1692: 1691: 1685: 1679: 1678: 1674: 1666: 1660: 1656: 1655: 1647: 1644: 1639: 1635: 1631: 1627: 1623: 1619: 1618: 1613: 1606: 1603: 1598: 1594: 1591:(1): 99–126. 1590: 1586: 1579: 1572: 1569: 1564: 1560: 1556: 1552: 1548: 1544: 1543: 1535: 1528: 1525: 1519: 1514: 1510: 1506: 1502: 1498: 1491: 1488: 1476: 1469: 1466: 1463: 1458: 1455: 1450: 1446: 1442: 1438: 1434: 1430: 1429: 1421: 1414: 1411: 1406: 1402: 1398: 1394: 1393: 1385: 1382: 1377: 1373: 1369: 1365: 1361: 1357: 1356: 1348: 1345: 1340: 1336: 1331: 1326: 1322: 1318: 1317: 1309: 1307: 1303: 1298: 1294: 1290: 1286: 1282: 1278: 1274: 1270: 1269: 1261: 1258: 1253: 1249: 1245: 1241: 1240: 1232: 1229: 1223: 1219: 1216: 1214: 1211: 1209: 1206: 1205: 1201: 1199: 1197: 1189: 1187: 1185: 1176: 1174: 1172: 1151: 1145: 1131: 1129: 1127: 1123: 1116: 1114: 1111: 1110:Johansen test 1104:Johansen test 1103: 1101: 1086: 1082: 1078: 1075: 1070: 1066: 1059: 1054: 1050: 1026: 1023: 1020: 998: 994: 984: 969: 965: 961: 956: 953: 950: 946: 942: 939: 936: 931: 927: 920: 915: 911: 885: 882: 879: 869: 843: 839: 813: 809: 796: 782: 760: 756: 733: 729: 706: 702: 693: 677: 669: 665: 647: 643: 622: 613: 597: 593: 566: 562: 558: 553: 549: 545: 542: 537: 533: 525: 524: 523: 507: 503: 482: 474: 471: 453: 449: 426: 422: 411: 403: 401: 395: 393: 374: 368: 358: 354: 350: 345: 343: 339: 334: 315: 309: 281: 275: 248: 242: 231: 229: 225: 221: 217: 216:Clive Granger 213: 209: 201: 199: 197: 176: 170: 160: 156: 152: 144: 142: 140: 136: 132: 128: 124: 119: 116: +  115: 112: +  111: 106: 102: 98: 94: 90: 86: 82: 78: 75:. Next, if a 74: 71: 67: 60: 56: 49: 42: 36: 32: 31:Cointegration 19: 4109: 4097: 4078: 4071: 3983:Econometrics 3933: / 3916:Chemometrics 3893:Epidemiology 3886: / 3859:Applications 3701:ARIMA model 3648:Q-statistic 3611: 3597:Stationarity 3493:Multivariate 3436: / 3432: / 3430:Multivariate 3428: / 3368: / 3364: / 3138:Bayes factor 3037:Signed rank 2949: 2923: 2915: 2903: 2598:Completeness 2434:Cohort study 2332:Opinion poll 2267:Missing data 2254:Study design 2209:Scatter plot 2131:Scatter plot 2124:Spearman's ρ 2086:Grouped data 1783:(1): 37–39. 1780: 1774: 1747: 1719:Econometrics 1718: 1689: 1653: 1646: 1621: 1615: 1605: 1588: 1584: 1571: 1546: 1542:Econometrica 1540: 1527: 1500: 1496: 1490: 1478:. Retrieved 1468: 1457: 1432: 1428:Econometrica 1426: 1413: 1396: 1390: 1384: 1362:(1): 64–73. 1359: 1353: 1347: 1320: 1314: 1275:(1): 11–63. 1272: 1266: 1260: 1243: 1237: 1231: 1193: 1180: 1170: 1135: 1126:Sam Ouliaris 1120: 1107: 985: 797: 614: 584: 472: 413: 399: 346: 335: 232: 228:Robert Engle 220:Paul Newbold 205: 148: 145:Introduction 130: 126: 122: 117: 113: 109: 104: 100: 96: 92: 88: 84: 80: 72: 58: 54: 47: 40: 30: 29: 4138:Time series 4111:WikiProject 4026:Cartography 3988:Jurimetrics 3940:Reliability 3671:Time domain 3650:(Ljung–Box) 3572:Time-series 3450:Categorical 3434:Time-series 3426:Categorical 3361:(Bernoulli) 3196:Correlation 3176:Correlation 2972:Jarque–Bera 2944:Chi-squared 2706:M-estimator 2659:Asymptotics 2603:Sufficiency 2370:Interaction 2282:Replication 2262:Effect size 2219:Violin plot 2199:Radar chart 2179:Forest plot 2169:Correlogram 2119:Kendall's τ 1518:10983/25617 66:time series 35:statistical 4127:Categories 3978:Demography 3696:ARMA model 3501:Regression 3078:(Friedman) 3039:(Wilcoxon) 2977:Normality 2967:Lilliefors 2914:Student's 2790:Resampling 2664:Robustness 2652:divergence 2642:Efficiency 2580:(monotone) 2575:Likelihood 2492:Population 2325:Stratified 2277:Population 2096:Dependence 2052:Count data 1983:Percentile 1960:Dispersion 1893:Arithmetic 1828:Statistics 1224:References 468:both have 408:See also: 338:unit roots 196:stationary 151:integrated 135:stochastic 107:such that 3359:Logistic 3126:posterior 3052:Rank sum 2800:Jackknife 2795:Bootstrap 2613:Bootstrap 2548:Parameter 2497:Statistic 2292:Statistic 2204:Run chart 2189:Pie chart 2184:Histogram 2174:Fan chart 2149:Bar chart 2031:L-moments 1918:Geometric 1638:153437469 1376:154550363 1325:CiteSeerX 1297:126346450 1083:ε 1063:Δ 1047:Δ 1021:α 966:ε 954:− 943:α 924:Δ 908:Δ 883:− 873:^ 836:Δ 806:Δ 783:β 678:β 623:β 546:β 543:− 483:β 349:R-squared 208:Udny Yule 4073:Category 3766:Survival 3643:Johansen 3366:Binomial 3321:Isotonic 2908:(normal) 2553:location 2360:Blocking 2315:Sampling 2194:Q–Q plot 2159:Box plot 2141:Graphics 2036:Skewness 2026:Kurtosis 1998:Variance 1928:Heronian 1923:Harmonic 1715:(2000). 1480:4 August 1202:See also 1194:Several 1013:), then 4099:Commons 4046:Kriging 3931:Process 3888:studies 3747:Wavelet 3580:General 2747:Plug-in 2541:L space 2320:Cluster 2021:Moments 1839:Outline 1695:319–386 1563:2938339 1449:1913236 1355:Wilmott 1289:2341482 1167:⁠ 1138:⁠ 390:⁠ 361:⁠ 331:⁠ 302:⁠ 297:⁠ 268:⁠ 264:⁠ 235:⁠ 202:History 192:⁠ 163:⁠ 46:,  3968:Census 3558:Normal 3506:Manova 3326:Robust 3076:2-way 3068:1-way 2906:-test 2577:  2154:Biplot 1945:Median 1938:Lehmer 1880:Center 1755:  1731:  1727:–669. 1701:  1661:  1636:  1561:  1447:  1374:  1327:  1295:  1287:  670:. If 585:where 129:, and 3592:Trend 3121:prior 3063:anova 2952:-test 2926:-test 2918:-test 2825:Power 2770:Pivot 2563:shape 2558:scale 2008:Shape 1988:Range 1933:Heinz 1908:Cubic 1844:Index 1771:(PDF) 1634:S2CID 1581:(PDF) 1559:JSTOR 1537:(PDF) 1445:JSTOR 1423:(PDF) 1372:S2CID 1293:S2CID 1285:JSTOR 396:Tests 70:order 33:is a 3825:Test 3025:Sign 2877:Wald 1950:Mode 1888:Mean 1753:ISBN 1729:ISBN 1699:ISBN 1659:ISBN 1482:2014 1360:2012 1124:and 1108:The 495:and 441:and 218:and 3005:BIC 3000:AIC 1785:doi 1725:623 1626:doi 1593:doi 1551:doi 1513:hdl 1505:doi 1437:doi 1401:doi 1364:doi 1335:doi 1277:doi 1248:doi 721:on 666:or 615:If 414:If 353:GNP 64:of 4129:: 1781:48 1779:. 1773:. 1697:. 1687:. 1632:. 1622:35 1620:. 1614:. 1589:70 1587:. 1583:. 1557:. 1547:58 1545:. 1539:. 1511:. 1501:16 1499:. 1443:. 1433:55 1431:. 1425:. 1397:16 1395:. 1370:. 1358:. 1333:. 1319:. 1305:^ 1291:. 1283:. 1273:89 1271:. 1244:10 1242:. 125:, 118:cZ 114:bY 110:aX 2950:G 2924:F 2916:t 2904:Z 2623:V 2618:U 1820:e 1813:t 1806:v 1791:. 1787:: 1761:. 1737:. 1707:. 1667:. 1640:. 1628:: 1599:. 1595:: 1565:. 1553:: 1521:. 1515:: 1507:: 1484:. 1451:. 1439:: 1407:. 1403:: 1378:. 1366:: 1341:. 1337:: 1321:2 1299:. 1279:: 1254:. 1250:: 1155:) 1152:1 1149:( 1146:I 1087:t 1079:+ 1076:b 1071:t 1067:x 1060:= 1055:t 1051:y 1027:0 1024:= 999:t 995:u 970:t 962:+ 957:1 951:t 947:u 940:+ 937:b 932:t 928:x 921:= 916:t 912:y 886:1 880:t 870:u 844:t 840:x 814:t 810:y 761:t 757:u 734:t 730:x 707:t 703:y 648:t 644:u 598:t 594:u 567:t 563:u 559:= 554:t 550:x 538:t 534:y 508:t 504:u 473:d 454:t 450:y 427:t 423:x 378:) 375:1 372:( 369:I 319:) 316:1 313:( 310:I 285:) 282:1 279:( 276:I 252:) 249:1 246:( 243:I 180:) 177:1 174:( 171:I 131:Z 127:Y 123:X 105:c 103:, 101:b 99:, 97:a 93:d 89:Z 87:, 85:Y 83:, 81:X 73:d 62:) 59:k 55:X 51:2 48:X 44:1 41:X 39:( 20:)

Index

Co-integration
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

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