comb_EIG1 {GeomComb} | R Documentation |
Standard Eigenvector Forecast Combination
Description
Computes forecast combination weights according to the standard eigenvector approach by Hsiao and Wan (2014) and produces forecasts for the test set, if provided.
Usage
comb_EIG1(x)
Arguments
x |
An object of class |
Details
The standard eigenvector approach retrieves combination weights from the sample estimated mean squared prediction error matrix
as follows:
Suppose y_t
is the variable of interest, there are N
not perfectly collinear predictors,
\mathbf{f}_t = (f_{1t}, \ldots, f_{Nt})'
, \Sigma
is the (positive definite)
mean squared prediction error matrix of \mathbf{f}_t
and \mathbf{e}
is an N \times 1
vector of (1, \ldots, 1)'
.
The N
positive eigenvalues are then arranged in increasing order (\Phi_1 = \Phi_{min}, \Phi_2, \ldots, \Phi_N)
, and \mathbf{w^j}
is defined as the eigenvector corresponding to \Phi_j
. The combination weights \mathbf{w}^{EIG1} = (w_1, \ldots, w_N)'
are then
chosen corresponding to the minimum of \left(\frac{\Phi_1}{d_1^2}, \frac{\Phi_2}{d_2^2},\ldots,\frac{\Phi_N}{d_N^2}\right)
, denoted as \mathbf{w}^l
, where d_j = \mathbf{e}'\mathbf{w}^j
, as:
\mathbf{w}^{EIG1} = \frac{1}{d_l} \mathbf{w}^l
The combined forecast is then obtained by:
\hat{y}_t = {\mathbf{f}_{t}}'\mathbf{w}^{EIG1}
The difference to extant methods that minimize the population mean squared prediction error (e.g., Newbold and Granger, 1974) is the normalization function. While
previous approaches optimize MSPE under the constraint of \mathbf{e}'\mathbf{w} = 1
, Hsiao and Wan (2014) show that this is dominated by
using \mathbf{w}'\mathbf{w} = 1
as constraint in the optimization problem.
Value
Returns an object of class foreccomb_res
with the following components:
Method |
Returns the used forecast combination method. |
Models |
Returns the individual input models that were used for the forecast combinations. |
Weights |
Returns the combination weights obtained by applying the combination method to the training set. |
Fitted |
Returns the fitted values of the combination method for the training set. |
Accuracy_Train |
Returns range of summary measures of the forecast accuracy for the training set. |
Forecasts_Test |
Returns forecasts produced by the combination method for the test set. Only returned if input included a forecast matrix for the test set. |
Accuracy_Test |
Returns range of summary measures of the forecast accuracy for the test set. Only returned if input included a forecast matrix and a vector of actual values for the test set. |
Input_Data |
Returns the data forwarded to the method. |
Author(s)
Christoph E. Weiss and Gernot R. Roetzer
References
Hsiao, C., and Wan, S. K. (2014). Is There An Optimal Forecast Combination? Journal of Econometrics, 178(2), 294–309.
Newbold, P., and Granger, C. W. J. (1974). Experience with Forecasting Univariate Time Series and the Combination of Forecasts. Journal of the Royal Statistical Society, Series A, 137(2), 131–165.
See Also
foreccomb
,
plot.foreccomb_res
,
summary.foreccomb_res
,
comb_NG
,
accuracy
Examples
obs <- rnorm(100)
preds <- matrix(rnorm(1000, 1), 100, 10)
train_o<-obs[1:80]
train_p<-preds[1:80,]
test_o<-obs[81:100]
test_p<-preds[81:100,]
data<-foreccomb(train_o, train_p, test_o, test_p)
comb_EIG1(data)