协整时间序列的多期领先与滞后缺失值插补技术问询

Great question! The issue with na.interpolation() and na.interp() is that they're univariate interpolation methods—they only look at FB's own past/future values, completely ignoring the cointegrating relationships with MSFT and SPY that you want to leverage. Let's fix that by building a cointegration-aware imputation workflow:

Step 1: Validate Cointegration First #

Before we jump into imputation, we need to confirm that FB, MSFT, and SPY actually share a long-term cointegrating relationship. We'll use the Johansen test (via the urca package) for this:

# Load required packages
library(xts)
library(urca)
library(tsDyn)
library(ggplot2)
library(reshape2)
library(data.table)
library(quantmod)

# Recreate your initial data prep (from your code)
tickers <- c("MSFT", "SPY","FB")
dataEnv <- new.env()
getSymbols(tickers, from="2010-06-30", to="2015-06-30", env=dataEnv)
plist <- eapply(dataEnv, Ad)
pframe <- as.data.frame(do.call(merge, plist))
pframe <- as.data.table(pframe,keep.rownames = T)
pframe$FB.Adjusted[ (nrow(pframe) - 150):nrow(pframe) ] <- NA

# Convert to xts for easier time series handling
pxts <- xts(pframe[,-1], order.by = as.Date(pframe$rn))

# Run Johansen cointegration test
joh_test <- ca.jo(pxts[, c("FB.Adjusted", "MSFT.Adjusted", "SPY.Adjusted")], 
                  type = "trace", K = 2, ecdet = "none")
summary(joh_test)

Check the test results: if the trace statistic exceeds the critical value at your chosen significance level (e.g., 5%), you've confirmed a cointegrating relationship. Note the number of cointegrating relations (r) for the next step.

Step 2: Impute with a Vector Error Correction Model (VECM) #

Cointegrated time series are best modeled with a VECM, which captures both their long-term equilibrium and short-term adjustments. We'll use this model to predict FB's missing values using the complete MSFT and SPY data:

# Split data into training (non-missing FB) and missing segments
train_data <- pxts[1:(nrow(pxts)-151), ]
missing_dates <- index(pxts[(nrow(pxts)-150):nrow(pxts), ])

# Fit VECM on training data (use `r` from Johansen test, e.g., r=1)
vecm_model <- VECM(train_data[, c("FB.Adjusted", "MSFT.Adjusted", "SPY.Adjusted")], 
                   lag = 2, r = 1)

# Extract actual MSFT/SPY values for the missing period (we need these to predict FB)
actual_msft_spy <- pxts[missing_dates, c("MSFT.Adjusted", "SPY.Adjusted")]

# Predict FB's missing values using the VECM
fb_imputations <- predict(vecm_model, newdata = actual_msft_spy, n.ahead = 151)

# Replace missing FB values in the original dataset
pxts_imputed <- pxts
pxts_imputed[missing_dates, "FB.Adjusted"] <- fb_imputations[, "FB.Adjusted"]

# Convert back to data.table for plotting
pframe_imputed <- data.table(rn = index(pxts_imputed), coredata(pxts_imputed))

# Visualize the result
meltdf <- melt(pframe_imputed, id="rn")
ggplot(meltdf, aes(x=rn, y=value, colour=variable, group=variable)) + 
  geom_line() +
  ggtitle("FB Imputation Using Cointegration with MSFT & SPY") +
  theme(axis.text.x = element_text(angle=45, hjust=1))

Step 3: Optional: Multiple Imputation for Uncertainty #

If you want to account for imputation uncertainty (e.g., for downstream statistical analysis), you can use mice with a custom VECM-based imputation function:

library(mice)

# Define a custom VECM imputation function
vecm_impute <- function(y, x, ...) {
  # y = missing variable (FB), x = complete variables (MSFT, SPY)
  combined <- cbind(y, x)
  complete_cases <- na.omit(combined)
  
  # Fallback if too few complete observations
  if(nrow(complete_cases) < 20) return(y)
  
  # Fit VECM and predict missing values
  vecm_fit <- VECM(complete_cases, lag=2, r=1)
  missing_idx <- which(is.na(y))
  if(length(missing_idx) == 0) return(y)
  
  pred <- predict(vecm_fit, newdata = x[missing_idx, ], n.ahead=length(missing_idx))
  y[missing_idx] <- pred[, colnames(y)]
  return(y)
}

# Run multiple imputation (5 datasets by default)
imp <- mice(pframe[, -1], 
            method = list(FB.Adjusted = "vecm_impute", 
                          MSFT.Adjusted = "pmm", 
                          SPY.Adjusted = "pmm"),
            m = 5)

# Extract one imputed dataset (or pool results with `pool()` for analysis)
pframe_mice_imputed <- complete(imp, 1)
pframe_mice_imputed$rn <- pframe$rn

# Plot
meltdf <- melt(pframe_mice_imputed, id="rn")
ggplot(meltdf, aes(x=rn, y=value, colour=variable, group=variable)) + 
  geom_line() +
  ggtitle("Multiple Imputation of FB (VECM-Based)") +
  theme(axis.text.x = element_text(angle=45, hjust=1))

Key Notes #

• Always validate cointegration first—this approach only makes sense if the series share a long-term equilibrium.
• Adjust the lag order (lag=2) based on AIC/BIC (use VARselect() from the vars package to find the optimal lag).
• The VECM imputation uses the actual observed values of MSFT and SPY during the missing period, ensuring FB's imputed values stay aligned with the group's cointegrating relationship.

内容的提问来源于stack exchange，提问作者BlockChained