A vector autoregression treats every series as a function of all series' own lags — no "independent/dependent" asymmetry. This tool fits each equation by OLS, checks stability through the companion-matrix eigenvalues (all inside the unit circle or forecasts explode), and computes Granger-causality F tests and Johansen reduced-rank statistics for cointegration.
Information criteria follow Lütkepohl's convention. Granger causality is prediction, not philosophy: "x Granger-causes y" means x's lags improve y's forecasts. The Johansen trace statistics are reported for comparison against the published Osterwald-Lumenau/MacKinnon tables for your number of series and deterministic specification — this site deliberately does not hardcode a table it cannot verify.
Frequently asked questions
How many series can I paste?
Two to four. The lag-count explosion (K² p parameters per equation) makes larger systems need structural priors rather than unconstrained VAR.
What is cointegration in one sentence?
Several non-stationary series whose linear combination IS stationary — they drift together. The Johansen rank r counts how many such stationary combinations exist; r ≥ 1 means a VECM, not a VAR in differences.
中文说明
向量自回归把每个序列都写成所有序列自身滞后的函数——没有“自变量/因变量”的不对称。本工具对每个方程做 OLS，用伴随矩阵特征值检查稳定性（全部落在单位圆内，否则预测发散），并计算 Granger 因果 F 检验与 Johansen 降秩协整统计量。
信息准则采用 Lütkepohl 口径。Granger 因果是预测意义上的、不是哲学意义上的：“x Granger 引致 y”指 x 的滞后改进 y 的预测。Johansen 迹统计量供对照已发表的 Osterwald-Lumenau/MacKinnon 临界值表（按你的序列数与确定性设定）——本站刻意不硬编码无法亲自验证的表。
常见问题
最多能贴几个序列？
二到四个。参数量随 K² p 爆炸，更大的系统需要结构性先验，而不是无约束 VAR。
一句话解释协整？
几个非平稳序列的某个线性组合本身平稳——它们同向漂移。Johansen 的秩 r 是这种平稳组合的个数；r ≥ 1 就该用 VECM 而非对差分建 VAR。