Granger Causality

metrics timeseries

这是一个多变量的自回归系统，类似于 Autoregression。取名为 Vector Autoregression。它的核心含义是，如果说 $Y$ 的过去值（lag 值）无法对 $X$ 的现值做任何贡献的话，we could say that $Y$ does not Granger cause variable $X$ .

In the example,

$$X_t={\beta }_{xx}{X}_{t-1}+{\beta }_{xy}{Y}_{t-1}+{\beta }_{xz}{Z}_{t-1}+{\gamma }_{xx}{X}_{t-2}+{\gamma }_{xy}{Y}_{t-2}+{\gamma }_{xz}{Z}_{t-2}+u_t$$

that is to say, ${\beta }_{xy}={\gamma }_{xy}=0$

But this still leaves possibility that $Y$ may still have some influence to $X$ through some indirect direction, like maybe through $Z$ . To rule out all, we say that variable $Y$ is not Granger Causally-Prior to variable $X$ iff the lags of $Y$ are not significant in the autoregressive equation for variable $X$ also not significant in all other autoregressions in the VAR system.

Prediction

Under this system, we basically are using $X_{t-1},Y_{t-1},Z_{t-1},X_{t-2},Y_{t-2},Z_{t-2}$ to predict $X_t,Y_t,Z_t$ . As said in page 2 in lecture notes, it is like the Engineering “black-box” approach.

Temporal Correlation

Temporal Correlation is the basis of Time-Series Analysis, it basically means the same variable has correlation in different period, i.e. $\text{Cov}(X_t,X_{t-k})\ne 0$ .

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Try to Answer Question 2(b) in Pset 4

GnC means Granger non-Causal, GnCP means Granger non-Causal Priority

We could form three equations from the question:

$X_t={\beta }_{xx}{X}_{t-1}+{\beta }_{xy}{Y}_{t-1}+{\beta }_{xz}{Z}_{t-1}+{\gamma }_{xx}{X}_{t-2}+{\gamma }_{xy}{Y}_{t-2}+{\gamma }_{xz}{Z}_{t-2}+u_t$

$Y_t={\beta }_{yx}{X}_{t-1}+{\beta }_{yy}{Y}_{t-1}+{\beta }_{yz}{Z}_{t-1}+{\gamma }_{yx}{X}_{t-2}+{\gamma }_{yy}{Y}_{t-2}+{\gamma }_{yz}{Z}_{t-2}+v_t$

$Z_t={\beta }_{zx}{X}_{t-1}+{\beta }_{zy}{Y}_{t-1}+{\beta }_{zz}{Z}_{t-1}+{\gamma }_{zx}{X}_{t-2}+{\gamma }_{zy}{Y}_{t-2}+{\gamma }_{zz}{Z}_{t-2}+w_t$

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Granger non-Causality (GnC)

$Y$ does not Granger-cause $X$： ${\beta }_{xy}={\gamma }_{xy}=0$

$X$ does not Granger-cause $Y$：${\beta }_{yx}={\gamma }_{yx}=0$

$Z$ does not Granger-cause $X$：${\beta }_{xz}={\gamma }_{xz}=0$

$X$ does not Granger-cause $Z$： ${\beta }_{zx}={\gamma }_{zx}=0$

$Z$ does not Granger-cause $Y$：${\beta }_{yz}={\gamma }_{yz}=0$

$Y$ does not Granger-cause $Z$：${\beta }_{zy}={\gamma }_{zy}=0$

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Granger non-Causal Priority (GnCP)

$Y$ is not Granger Causally-Prior to $X$：$Y$ ‘s lags are insignificant in every equation: ${\beta }_{xy}={\gamma }_{xy}={\beta }_{yy}={\gamma }_{yy}={\beta }_{zy}={\gamma }_{zy}=0$

$X$ is not Granger Causally-Prior to $Y$：${\beta }_{xx}={\gamma }_{xx}={\beta }_{yx}={\gamma }_{yx}={\beta }_{zx}={\gamma }_{zx}=0$

$Z$ is not Granger Causally-Prior to $X$：${\beta }_{xz}={\gamma }_{xz}={\beta }_{yz}={\gamma }_{yz}={\beta }_{zz}={\gamma }_{zz}=0$

We could also form other equations using the same logic.