Solved – Treating missing values in panel data set

Question

You can often see in the empirical papers that do some kind of the regression analysis on economic data something like this "we drop the companies from our sample which do not have the observation in each year of the reporting period". Doing so the author reduces his/her sample. Another way to deal with missing values is to impute them.

My questions therefore are: what is more correct to drop or replace missing values while doing regression analysis?

Besides, is it better to use some sophisticated methods that allow us to account for missing values in dependent/independent variables or impute the values?

For example, I've panel data for 100 companies in the period of 6 last years. For them I've collected 5 variables such as profit, sales etc. For some companies I have missing values in 1-2 periods and for other in 4-5 periods. I am going to conduct the time series/cross-sectional analysis. Is it better 1). to leave all as it is, 2). to drop companies which have missing values in 4-5 periods (than the sample reduces by 50%) and rest of missing values replace with the mean values?

Besides what to do if the missing values are dependent variables? Does the imputation methods differ for dependent or independent variables?

Answer 1

Imputation is very useful for improving the accuracy of your parameter estimates in situations where a significant amount of data would otherwise be deleted. Consider that in a study with, for example, 100 observations and four regressors, each with a 10% missing observation rate, you'll only be missing 10% of the data but on average you'll be deleting about 34% of the observations if you drop each observation with one or more missing values - which is what happens if you just run the data through a standard regression package. You'll be deleting much more data (2.4x in fact) than is actually missing. In addition, unless your data is missing completely at random, case deletion can introduce bias into your parameter estimates.

It is typically better to use an imputation algorithm that captures at least the covariance structure of the data and generates random numbers (rather than replacing with mean or median values.) This holds true especially if you're going to be doing some estimation using the imputed data, because you'll get more accurate estimates of the covariance matrix of the parameters. Replacing by the mean value will give you overly optimistic standard errors, sometimes by quite a bit.

I've included an example using the default imputation method from the mice package in R. The example has a regression with 100 observations and four regressors, each with a 10% chance of a missing value at every observation. We compare the std. errors of the estimates for the complete-data regression (no missing values), the case deletion regression (delete any observation with a missing value), mean imputation (replace the missing value by the mean of the variable), and a good quality imputation routine that estimates the covariance matrix of the data and generates random values. I've constructed nonlinear relationships between the regressors such that mice isn't going to model them using their true relationships, just to add a layer of inaccuracy to the whole thing. I've run the entire process 100 times and averaged the standard errors of the four methods for each of the parameters for comparative purposes.

Here's the code, with a comparison of the standard errors at the bottom:

results <- data.frame(se_x1 = rep(0,400),
                      se_x2 = rep(0,400),
                      se_x3 = rep(0,400),
                      se_x4 = rep(0,400),
                      method = c(rep("Complete data",100),
                                 rep("Case deletion",100),
                                 rep("Mean value imputation", 100),
                                 rep("Randomized imputation", 100)))
N <- 100
pct_missing <- 0.1
for (i in 1:100) {
  x1 <- 4 + rnorm(N)
  x2 <- 0.025*x1^2 + rnorm(N)
  x3 <- 0.2*x1^1.3 + 0.04*x2^0.7 + rnorm(N)
  x4 <- 0.4*x1^0.3 - 0.3*x2^1.1 + rnorm(N)
  e <- rnorm(N, 0, 1.5)
  y <- x1 + x2 + x3 + e   # The coefficient of x4 = 0

  # Complete data regression 
  mc <- summary(lm(y~x1+x2+x3+x4))
  results[i,1:4] <- mc$coefficients[2:5,2]

  # Cause data to be missing
  x1[rbinom(N,1,pct_missing)==1] <- NA
  x2[rbinom(N,1,pct_missing)==1] <- NA
  x3[rbinom(N,1,pct_missing)==1] <- NA
  x4[rbinom(N,1,pct_missing)==1] <- NA

  # Case deletion 
  mm <- summary(lm(y~x1+x2+x3+x4))
  results[i+100,1:4] <- mm$coefficients[2:5,2]

  # Mean value imputation
  x1m <- x1; x1m[is.na(x1m)] <- mean(x1, na.rm=TRUE)
  x2m <- x2; x2m[is.na(x2m)] <- mean(x2, na.rm=TRUE)
  x3m <- x3; x3m[is.na(x3m)] <- mean(x3, na.rm=TRUE)
  x4m <- x4; x4m[is.na(x4m)] <- mean(x4, na.rm=TRUE)

  mmv <- summary(lm(y~x1m+x2m+x3m+x4m))
  results[i+200,1:4] <- mmv$coefficients[2:5,2]

  # Imputation; I'm only using 1 of the 5 multiple imputations
  # It would be better to use all the multiple imputations, though.
  imp <- mice(cbind(y,x1,x2,x3,x4), printFlag=FALSE)
  x1[is.na(x1)] <- as.numeric(imp$imp$x1[,1])
  x2[is.na(x2)] <- as.numeric(imp$imp$x2[,1])
  x3[is.na(x3)] <- as.numeric(imp$imp$x3[,1])
  x4[is.na(x4)] <- as.numeric(imp$imp$x4[,1])

  mi <- summary(lm(y~x1+x2+x3+x4))
  results[i+300,1:4] <- mi$coefficients[2:5,2]
}

options(digits = 3)
results <- data.table(results)
results[, .(se_x1 = mean(se_x1),
            se_x2 = mean(se_x2),
            se_x3 = mean(se_x3),
            se_x4 = mean(se_x4)), by = method]

And the output:

                  method se_x1 se_x2 se_x3 se_x4
1:         Complete data 0.208 0.278 0.192 0.193
2:         Case deletion 0.267 0.359 0.244 0.250
3: Mean value imputation 0.231 0.301 0.212 0.217
4: Randomized imputation 0.213 0.271 0.195 0.198

Note that the complete data method is as good as you can get with this data. Case deletion results in considerably less accurate parameter estimates, but the randomized imputation of mice gets you almost all the way back to the accuracy you would get with complete data. (These numbers are a little optimistic, as I'm not using the full multiple imputation approach, but this is just a simple example.) The mean value imputation in this case appears to have improved things considerably relative to case deletion, but is actually overly optimistic.

So the tl;dr version is: impute, unless you'd only be missing a very small fraction of your cases using case deletion (like 1%). The big caveat is: understand the assumptions that are required for imputation first! If data is not missing at random, and I'm using that phrase non-technically so look up what imputation requires in this respect, imputation won't help you, and may make things worse. But that's a topic for another question. Here are a couple of links which might be helpful: overview of imputation, missing data rates and imputation, different imputation algorithms.