# comb_EIG1: Standard Eigenvector Forecast Combination In ceweiss/GeomComb: Forecast Combination Methods

## 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

 1 comb_EIG1(x) 

## Arguments

 x An object of class foreccomb. Contains training set (actual values + matrix of model forecasts) and optionally a test set.

## 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, f_t = (f_{1t}, …, f_{Nt})', Σ is the (positive definite) mean squared prediction error matrix of f_t and e is an N * 1 vector of (1, …, 1)'. The N positive eigenvalues are then arranged in increasing order (Φ_1 = Φ_{min}, Φ_2, …, Φ_N), and w^j is defined as the eigenvector corresponding to Φ_j. The combination weights w = (w_1, …, w_N)' are then chosen corresponding to the minimum of (Φ_1/d_1^2, Φ_2/d_2^2, …, Φ_N/d_N^2), denoted as w^l, where d_j = e'w^j, as:

w = 1/d_l w^l

The combined forecast is then obtained by:

\hat{y}_t = (f_t)'w

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 e'w = 1, Hsiao and Wan (2014) show that this is dominated by using w'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.

foreccomb, plot.foreccomb_res, summary.foreccomb_res, comb_NG, accuracy
 1 2 3 4 5 6 7 8 9 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)