---
title: "SF_vignette"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{SF_vignette}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup}
library(sufficientForecasting)
```

# 1. Motivation
Forecasting a single time series using high dimensional predictors has received 
a lot of interests in macroeconomics, finance, business and many other research
fields. It is usually reasonable to assume that a few underlying common factors
drive the forecasting target and the high-dimensional predictors. The use of principal
components effectively reduces the dimensionality and more importantly provides
a useful characterization of predictors. 

By assuming the linear forecasting function, Stock and Watson (1989, 2002a, 2002b)
demonstrated the validity of the estimated principal components in forecasting.
Bai and Ng (2006) conducted inferences on factor-augmented regressions to enable 
the forecast. Bair et al. (2006) applied the correlation screening to obtain
relevant predictors, and Bai and Ng (2008)  established the thresholding criteria 
to rule out predictors not informative for the target.

However, all of the aforementioned works may not perform well when the target and
the latent factors have possibly nonlinear relationship. The possibly nonlinear
and nonseparable forecasting function poses a significant challenge when extracting
the information relevant to the target. The package provides sufficient forecasting
(SF) procedure to make predictions. SF procedure obtains sufficient predictive indices
with provable theoretical guarantees, allowing for an unknown nonlinear forecasting 
function.

# 2. Examples
## 2.1 Data
The package contains existing datasets: `dataExample$y`, `dataExample$X`, and `dataExample$newX`. y is a 100 by 1 matrix, X is a 100 by 100 matrix. X and y 
are our training sets. Our goal is to predict what the value of y is when we 
know next predictors are newX.

## 2.2 SF.SIR
We can use SF.SIR to forecast.
```{r}
SF.SIR(y=dataExample$y,X=dataExample$X,newX=dataExample$newX)
```

## 2.3 SF.DR
Also, We can use SF.DR.
```{r}
SF.DR(y=dataExample$y,X=dataExample$X,newX=dataExample$newX)
```

# References
Bai, J. , and Ng, S. (2006), Confidence intervals for diffusion index forecasts and inference for factor-augmented regressions, *Econometrica* 74(4), 1133–1150.

Bai, J. , and Ng, S. (2008), Forecasting economic time series using targeted predictors, *Journal of Econometrics* 146, 304–317.

Bair, E. , Hastie, T. , Paul, D. , and Tibshirani, R. (2006), Prediction by supervised principal components, *Journal of the American Statistical Association* 101, 119–137.

Stock, J. H. , and Watson, M. W. (1989), New indexes of coincident and leading economic indicators, *NBER Macroeconomics Annual* 4, 351–409.

Stock, J. H. , and Watson, M. W. (2002a), Forecasting using principal components from a large number of predictors, *Journal of the American Statistical Association* 97, 1167–1179. 

Stock, J. H. , and Watson, M. W. (2002b), Macroeconomic forecasting using diffusion indexes, *Journal of Business & Economic Statistics* 20, 147–162.