# model fitting from MW(2010)
# Minor change to highlight OLS PC4 instead of OLS PC5

my_pc_reg <- function(y, gpc, ppc, gpc_out, ppc_out, num_grid_pc, num_proxy_pc){

  k = num_grid_pc
  p = num_proxy_pc
  tmpg <- data.frame(gpc[,1:k])
  if(k==1){colnames(tmpg) <- "PC1"}
  yreg <- lm(y~., data=tmpg)

  tmpghat <- tmpg[1:dim(ppc_out)[1],]
  if(k==1){tmpghat <- data.frame(PC1=tmpghat)}
  greg <- list(NULL)
  tmpx <- data.frame(ppc[,1:p])
  if(p==1){colnames(tmpx) <- "PC1"}
  for(i in 1:k){
    greg[[i]] <- lm(tmpg[,i]~.,  data=tmpx)   
    tmpghat[,i] <- predict(greg[[i]], newdata=ppc_out)
  }
  yhat <- predict(yreg, newdata=tmpghat)
}

# Run proxy pca:
proxy_pc <- prcomp(proxy, center=TRUE, scale=TRUE)
grid_pc <- prcomp(grid, center=TRUE, scale=TRUE)

# Build insample dataframe:
my_years <- intersect(rownames(temp), intersect(rownames(grid), rownames(proxy)))
yy_in <- temp[my_years,1]
names(yy_in) <- my_years
proxy_in <- data.frame( proxy[my_years,] )
gpc_in <- data.frame( grid_pc$x[my_years,] )
ppc_in <- data.frame( proxy_pc$x[my_years,] )

# Build out of sample dataframe:
proxy_out <- data.frame( proxy )
ppc_out <- data.frame( proxy_pc$x )

# FIT VARIOUS MODELS:
library(MASS)
library(lars)
myfrac <- seq(0, 1, length=201)
yhat <- list(NULL)
  # Intercept:
  mylm <- lm(yy_in~1)
  yhat[[1]] <- predict(mylm, data.frame(ppc_out))

  # OLS_PC1:
  mylm <- lm(yy_in~., data=data.frame(x=ppc_in[,1]))
  yhat[[2]] <- predict(mylm, data.frame(x=ppc_out[,1]))

  # OLS_PC4: 
  mylm <- lm(yy_in~., data=data.frame(ppc_in[,1:4]))
  yhat[[3]] <- predict(mylm, data.frame(ppc_out[,1:4]))

  # OLS_PC10:
  mylm <- lm(yy_in~., data=data.frame(ppc_in[,1:10]))
  yhat[[4]] <- predict(mylm, data.frame(ppc_out[,1:10]))

  # OLS_PC20:
  mylm <- lm(yy_in~., data=data.frame(ppc_in[,1:20]))
  yhat[[5]] <- predict(mylm, data.frame(ppc_out[,1:20]))

  # LASSO_PROXY95
  # Cross-validate:
  cv.mean <- matrix(NA,length(myfrac),numcv_lars)
  cv.se <- matrix(NA,length(myfrac),numcv_lars)
  for(j in 1:numcv_lars){
    tmp <- cv.lars(as.matrix(proxy_in), yy_in, K=K_lars, fraction=myfrac, type="lasso", use.Gram=FALSE, plot.it=FALSE)
    cv.mean[,j] <- tmp$cv
    cv.se[,j] <- tmp$cv.error
 }
 # Create Lasso Model
  tmp <- rowMeans(cv.mean)
  tmp2 <- (1:length(myfrac))[tmp==min(tmp)]
  cv.frac <- myfrac[tmp2]
  full_lasso <- lars(as.matrix(proxy_in), yy_in, type="lasso", use.Gram=FALSE)
  lasso_yhat <- predict(full_lasso, as.matrix(proxy_out), type="fit", s=cv.frac, mode="fraction")
  yhat[[6]] <- lasso_yhat$fit

  # LASSO_PC95
  # Cross-validate:
  cv.mean <- matrix(NA,length(myfrac),numcv_lars)
  cv.se <- matrix(NA,length(myfrac),numcv_lars)
  for(j in 1:numcv_lars){
    tmp <- cv.lars(as.matrix(ppc_in), yy_in, K=K_lars, fraction=myfrac, type="lasso", use.Gram=FALSE, plot.it=FALSE)
    cv.mean[,j] <- tmp$cv
    cv.se[,j] <- tmp$cv.error
 }
 # Create Lasso Model
  tmp <- rowMeans(cv.mean)
  tmp2 <- (1:length(myfrac))[tmp==min(tmp)]
  cv.frac <- myfrac[tmp2]
  full_lasso <- lars(as.matrix(ppc_in), yy_in, type="lasso", use.Gram=FALSE)
  lasso_yhat <- predict(full_lasso, as.matrix(ppc_out), type="fit", s=cv.frac, mode="fraction")
  yhat[[7]] <- lasso_yhat$fit

  # stepAIC_Proxy95:
  f0 <- lm(yy_in~1, data=proxy_in)
  f1 <- lm(yy_in~., data=proxy_in)
  f <- stepAIC(f0, scope=list(lower=f0, upper=f1), trace=FALSE)
  yhat[[8]] <- predict(f, data.frame(proxy_out))

  # stepAIC_PC95:
  f0 <- lm(yy_in~1, data=ppc_in)
  f1 <- lm(yy_in~., data=ppc_in)
  f <- stepAIC(f0, scope=list(lower=f0, upper=f1), trace=FALSE)
  yhat[[9]] <- predict(f, data.frame(ppc_out))

  # stepBIC_Proxy95:
  f0 <- lm(yy_in~1, data=proxy_in)
  f1 <- lm(yy_in~., data=proxy_in)
  f <- stepAIC(f0, scope=list(lower=f0, upper=f1), trace=FALSE, k=log(dim(proxy_in)[2]))
  yhat[[10]] <- predict(f, data.frame(proxy_out))

  # stepBIC_PC95:
  f0 <- lm(yy_in~1, data=ppc_in)
  f1 <- lm(yy_in~., data=ppc_in)
  f <- stepAIC(f0, scope=list(lower=f0, upper=f1), trace=FALSE, k=log(dim(ppc_in)[2]))
  yhat[[11]] <- predict(f, data.frame(ppc_out))

  cnt <- 12
  for(aa in numgridPC){
    for(bb in numproxyPC){
     yhat[[cnt]] <- my_pc_reg(yy_in, gpc_in, ppc_in, gpc_out, ppc_out, aa, bb)
      cnt <- cnt+1
    }
  }

# smooth results (loess, span =1/20 ~ 50 years)
x_in=as.numeric(rownames(ppc_out))
yhat_sm = yhat
for(i in 2:length(yhat)){
    tmplo <- loess(yhat[[i]]~x_in, span=.05)
    tmpy <- predict(tmplo, x_in, se=TRUE) 
    yhat_sm[[i]]= tmpy$fit
}