Linear Combination Weights#

Consistent with Stock and Waston (1998)’s experiment on linear combination weights, the combined forecasts are the weighted averages of the single forecasts. Let $MS{E}_{t+h,t,i}=(1/v)\sum _{\tau =t-v}^t{e}_{\tau ,\tau -h,i}^2$ be the $i$th forecasting model’s MSE at time $t$, computed over a window of the previous $v$ periods. Then

When $\kappa =0$, $MS{E}_{t+h,t,i}^{\kappa }$ would be $1$, and the weight is assigned equally to every single forecast, suggesting the arithmetic average of forecasts. As $\kappa$ increases, an increasing amount of weight is assigned to models that perform well, and the weight is equal to the inverse of MSE when $\kappa =1$. The study tests two specific scenarios where $\kappa =0$ and $\kappa =1$.

Least squares estimators of the weights#

Least square estimators of the weights are computed by the ordinary least squares, regressing realizations of the target variable $y_{\tau }$ the N-vector of forecasts, ${\hat{y}}_{\tau }$ using data over the period $\tau =h,...,t$:

Built on Granger and Ramanathan [1984]’s proposed methods (excluding the one with constant term) and shrinkage methods, three specific formats are specified below:

$\text{(1) }{y}_{t+h}={\omega }_h^{{}^{\prime }}{\hat{y}}_{t+h,t},+{ϵ}_{t+h}$

$\text{(2) }{y}_{t+h}={\omega }_h^{{}^{\prime }}{\hat{y}}_{t+h,t},+{ϵ}_{t+h},\text{ s.t }{\omega }_h^{{}^{\prime }}\iota =1$

$\text{(3) }{y}_{t+h}={\omega }_h^{{}^{\prime }}{\hat{y}}_{t+h,t},+{ϵ}_{t+h}+\lambda |{\omega }_h^{{}^{\prime }}|$

Equation 1 represents the ordinary least square estimation without intercept. Equation 2 is the constrained least square estimation, where estimates must be positive, and the sum of estimates will be 1. Equation 3 is the LASSO where $\lambda |{\omega }_h^{{}^{\prime }}|$ would prevent the overfitting, and the procedure is not to perform variable selection.