ISSUE

What is variance decomposition and what is it used for?

TOTAL VARIANCE

It is necessary to start with the law of total variance in order to understand and use variance decomposition.

If the conditional expected value of (Y,X) is linear, the expected value of Y in relations to X may be given by:

(1) E(Y|X) = a+ bX

It follows that the covariance of Y and X is:

(2) b = Cov(Y,X) / Var (X)

The intercept: b is provided by:

(3) a = E(Y) – [Cov(Y,X) / Vr(X)] (E(X))

The explained component is:

(4) Corr(Y,Y)^2 = [Var(Y|X)] / Var (Y)

VARIANCE DECOMPOSITION

Recall the linear model in a form of Y = a + bX + c where “a” is the intercept, “b” is the slope and “c” is the forecast error; in the multivariable forecast, the formula may be rewritten in a formal statement by:

(5) Y = b(o) + b(1)X(t-1) + b(p)X(t-p) + u(t)

…where b(o) = intercept, and u(t) forecast error which may be decomposed in kp matrix as:

b(1) =

[b1 b2 … b(p-1) b(p)

I(k) 0 … 0 0

0 I(k) … 0 0

… …. I(k) 0 0

0 …. …. I(k) 0]

Y = [Y1

…

Yp]

b(o) = [b

0

…

0]

u(t) = [u(t)

0

…

0]

THE USE OF VARIANCE DECOMPOSITION

In econometrics and other time series analysis, variance decomposition is used to help in the interpretation of vector autoregression (VAR) analysis. The decomposition helps the researcher to see how much does the variance of each variable contribute to the other variable in the autoregression. Variance decomposition helps to verify the magnitude of forecast error variance of each variable that had been resulted from exogeneous shock.

REFERENCES

Neil A. Weiss, A Course in Probability, Addison–Wesley, 2005, pages, 380-383; 385–386.

Bowsher, C.G. and P.S. Swain, Proc Natl Acad Sci USA, 2012: 109, E1320–29.

Lütkepohl, H. (2007) New Introduction to Multiple Time Series Analysis, Springer. p. 63.

Thanks

It explains the relative importance of each exogenous shock to variables in total variance error of the endogenous variable. It also shows the strength of the endogenous variable in accounting for its own variance. 

You may estimate impulse response functions after estimating the VAR model and identifying the monetary or fiscal policy shocks. You can identify the monetary or fiscal policy shocks either by using the 'Choleskey Decomposition' method or by using the AB model used by Amisano-Giannini in their 'Structural VAR' modeling. If you can precisely identify the policy shock in consideration, the impulse response functions will reflect the response of the non-policy variable in question due to shocks to policy variable in concern. Forecast Error Variance Decomposition will indicate how much of the variations in one of the endogenous variables included in the estimated VAR model are accounted for by other endogenous variables. FEVD analysis may support your findings that you find from your 'impulse response' analysis. Thanks.