Principal Components

386—Chapter 12. Groups

ponent series edit field, each separated by a space. Note that you cannot store more principal components than there are series in the group. You may also store the eigenvalues and eigenvectors in a named vector and matrix.

The principal component view displays output that looks as follows:

Date: 10/31/00 Time: 16:05

Sample: 1 74

Included observations: 74

Correlation of X1 X2 X3 X4

The column headed by “Comp1” and “Vector1” corresponds to the first principal component, “Comp2” and “Vector2” denote the second principal component and so on. The row labeled “Eigenvalue” reports the eigenvalues of the sample second moment matrix in descending order from left to right. The Variance Prop. row displays the variance proportion explained by each principal component. This value is simply the ratio of each eigenvalue to the sum of all eigenvalues. The Cumulative Prop. row displays the cumulative sum of the Variance Prop. row from left to right and is the variance proportion explained by principal components up to that order.

The second part of the output table displays the eigenvectors corresponding to each eigenvalue. The first principal component is computed as a linear combination of the series in the group with weights given by the first eigenvector. The second principal component is the linear combination with weights given by the second eigenvector and so on.

Correlations, Covariances, and Correlograms

Correlations and Covariances display the correlation and covariance matrices of the series in the group. The Common Sample view drops observations for which any one of the series has missing data in the current sample. The Pairwise Samples view computes each of the second moments using all non-missing observations for the relevant series. Note that Pairwise Samples returns entries that correspond to @cov(x,y) and @cor(x,y)

388—Chapter 12. Groups

Granger Causality

Correlation does not necessarily imply causation in any meaningful sense of that word. The econometric graveyard is full of magnificent correlations, which are simply spurious or meaningless. Interesting examples include a positive correlation between teachers’ salaries and the consumption of alcohol and a superb positive correlation between the death rate in the UK and the proportion of marriages solemnized in the Church of England. Economists debate correlations which are less obviously meaningless.

The Granger (1969) approach to the question of whether x causes y is to see how much of the current y can be explained by past values of y and then to see whether adding lagged values of x can improve the explanation. y is said to be Granger-caused by x if x helps in the prediction of y , or equivalently if the coefficients on the lagged x ’s are statistically significant. Note that two-way causation is frequently the case; x Granger causes y and y Granger causes x .

It is important to note that the statement “ x Granger causes y ” does not imply that y is the effect or the result of x . Granger causality measures precedence and information content but does not by itself indicate causality in the more common use of the term.

When you select the Granger Causality view, you will first see a dialog box asking for the number of lags to use in the test regressions. In general, it is better to use more rather than fewer lags, since the theory is couched in terms of the relevance of all past information.

You should pick a lag length, l , that corresponds to reasonable beliefs about the longest time over which one of the variables could help predict the other.

EViews runs bivariate regressions of the form:

yt = α0 + α1yt − 1 + … + αlyt − l + β1xt − 1 + … + βlx−l + t

(12.11)

xt = α0 + α1xt − 1 + … + αlxt − l + β1yt − 1 + … + βly−l + ut

for all possible pairs of (x, y ) series in the group. The reported F-statistics are the Wald statistics for the joint hypothesis:

for each equation. The null hypothesis is that x does not Granger-cause y in the first regression and that y does not Granger-cause x in the second regression. The test results are given by: