Forecast Combination

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 \(MSE_{t+h,t,i} = (1/v)\sum_{\tau=t-v}^{t} e^{2}_{\tau,\tau−h,i}\) be the \(i\)th forecasting model’s MSE at time \(t\), computed over a window of the previous \(v\) periods. Then

\[ \hat{y}_{t+h,t} = \sum_{i=1}^{N} \hat{\omega}_{t+h,t,i} \hat{y}_{t+h,t,i}, \text{ where } \hat{\omega}_{t+h,t,i} = \frac{(1/MSE_{t+h,t,i}^{\kappa})}{\sum_{j=1}^{N} (1/MSE_{t+h,t,j}^{\kappa})} \]

When \(\kappa = 0\), \(MSE_{t+h,t,i}^{\kappa}\) 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 \(\tau = h, . . . , t\):

\[ \hat{\omega}_{t+h,t} = (\sum_{\tau=1}^{t-h} \hat{y}_{\tau+h,\tau} \hat{y}_{\tau+h,\tau}')^{-1} \sum_{\tau=1}^{t-h} \hat{y}_{\tau+h,\tau} y_{\tau+h} \]

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

\(\text{(1) } y_{t+h}= \omega_{h}^{'} \hat{y}_{t+h, t}, + \epsilon_{t+h}\)

\(\text{(2) } y_{t+h}= \omega_{h}^{'} \hat{y}_{t+h, t}, + \epsilon_{t+h}, \text{ s.t } \omega_{h}^{'} \iota = 1\)

\(\text{(3) } y_{t+h}= \omega_{h}^{'} \hat{y}_{t+h, t}, + \epsilon_{t+h} + \lambda |\omega_{h}^{'}| \)

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 \(\lambda |\omega_{h}^{'}|\) would prevent the overfitting, and the procedure is not to perform variable selection.