Forecasts combination using regression, robust regression, constrained regression or variance based

Description

Combine different forecasts. Use simple average, Ordinary Least Squares (OLS), robust regression, inverse mean squared error (IMSE), constrained least squares (CLS), or simply use the best forecast based on the MSE metric.

Usage

Forecast_comb(obs, fhat, fhat_new= NULL, Averaging_scheme= c("simple", "ols", "robust",
"cls", "variance based", "best") )

Arguments

Details

Performs simple forecast averaging where each forecast carries equal weight: frac{1}{p} with p the column dimension of fhat. OLS forecast combination is based on

obs_t = const + ∑_{i = 1}^p w_{i} widehat{obs}_{it} + e_t,

where obs is the observed values and widehat{obs} is the forecast, one out of the p forecasts available.

Robust regression performs the same, but minimize different loss function, which is less sensitive to outliers (see quantile regression and references therein).

Constrained least squares minimize the sum of squared errors under the restriction that the weights sum up to 1, and that the forecasts themselves are unbiased (no intercept in the regression).

The variance-based method computes the mean squared error and weigh the forecasts according to their accuracy. Accurate forecasts (based on MSE metric) receive relatively more weight.

The best restric all the weights to zero apart from the best forecast, again based on the MSE. Essentially selecting only one forecast to be used.

Value

Forecast_comb returns a list that contains the following objects:

Author(s)

Eran Raviv (eeraviv@gmail.com)

References

Bates, J. M., Granger, C.W. (1969), The combination of forecasts, Operations Research Quarterly, 20(4), 451-468

Clemen, R.T. (1989) Combining forecasts: A review and annotated bibliography. International Journal of Forecasting 5, 559-583.

Koenker, R. (2005) Quantile Regression. Cambridge University Press.

Timmermann, A.G. (2006) Forecast combinations. In: Elliott, G., Granger, C.W., Timmermann, A. (Eds.), Handbook of Economic Forecasting, Elsevier, 135-196.

Examples

 library(MASS)
 tt <- NROW(Boston)/2
 TT <- NROW(Boston)
 y <- Boston[1:tt, 14] # dependent variable is columns number 14

 # Create two sets of explanatory variables
 x1 <- Boston[1:tt, 1:6] # The first 6 explanatories
 x2 <- Boston[1:tt, 7:13]# The last 6 explanatories

#create two forecasts based on the two different x1 and x2
 coef1 <- lm(y~as.matrix(x1))$coef
 coef2 <- lm(y~as.matrix(x2))$coef
 f1 <- t(coef1 %*% t(cbind(rep(1,tt), Boston[(tt+1):TT, 1:6] )))
 f2 <- t(coef2 %*% t(cbind(rep(1,tt), Boston[(tt+1):TT, 7:13] )))
 ff <- cbind(f1, f2)
scheme=c("simple", "ols", "robust", "variance based", "cls", "best")

example0 <- list()

 for ( i in 	scheme) {
 	example0[[i]] <- Forecast_comb(obs = Boston[(tt+1):TT, 14] ,
 	fhat = ff, Averaging_scheme = i)
 	cat(i, ":",  sqrt(mean((example0[[i]]$fitted - Boston[(tt+1):TT, 14] )^2)), "\n" )
 }

# Compare with
apply(ff, 2, function(x) {  sqrt(mean((x - Boston[(tt+1):TT, 14])^2)) }  )

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(ForecastCombinations)
Loading required package: quantreg
Loading required package: SparseM

Attaching package: 'SparseM'

The following object is masked from 'package:base':

    backsolve

Loading required package: quadprog
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/ForecastCombinations/Forecast_comb.Rd_%03d_medium.png", width=480, height=480)
> ### Name: Forecast_comb
> ### Title: Forecasts combination using regression, robust regression,
> ###   constrained regression or variance based
> ### Aliases: Forecast_comb
> 
> ### ** Examples
> 
>  library(MASS)
>  tt <- NROW(Boston)/2
>  TT <- NROW(Boston)
>  y <- Boston[1:tt, 14] # dependent variable is columns number 14
> 
>  # Create two sets of explanatory variables
>  x1 <- Boston[1:tt, 1:6] # The first 6 explanatories
>  x2 <- Boston[1:tt, 7:13]# The last 6 explanatories
> 
> #create two forecasts based on the two different x1 and x2
>  coef1 <- lm(y~as.matrix(x1))$coef
>  coef2 <- lm(y~as.matrix(x2))$coef
>  f1 <- t(coef1 %*% t(cbind(rep(1,tt), Boston[(tt+1):TT, 1:6] )))
>  f2 <- t(coef2 %*% t(cbind(rep(1,tt), Boston[(tt+1):TT, 7:13] )))
>  ff <- cbind(f1, f2)
> scheme=c("simple", "ols", "robust", "variance based", "cls", "best")
> 
> example0 <- list()
> 
>  for ( i in 	scheme) {
+  	example0[[i]] <- Forecast_comb(obs = Boston[(tt+1):TT, 14] ,
+  	fhat = ff, Averaging_scheme = i)
+  	cat(i, ":",  sqrt(mean((example0[[i]]$fitted - Boston[(tt+1):TT, 14] )^2)), "\n" )
+  }
simple : 7.163777 
ols : 5.785442 
robust : 5.946072 
variance based : 5.862556 
cls : 5.811881 
best : 6.072841 
> 
> # Compare with
> apply(ff, 2, function(x) {  sqrt(mean((x - Boston[(tt+1):TT, 14])^2)) }  )
[1] 11.685468  6.072841
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>