Can I use Wilcoxon Signed-Rank paired Test to compare binary data? I have two series of binary outcomes:
      Series1 Series2
Test1 0       1
Test2 1       1
Test3 1       1
Test4 0       1
Test5 0       0
...

I want to know whether or not the outcomes in the two series were significantly different. 
Can I use the Wilcoxon signed rank test for this purpose?
I.e., in R:

wilcox.test(data1, data2, paired=TRUE)

 A: For binary and independent variables you should use a chi-square test if the Central Limit Theorem's assumptions are not violated or Fisher's exact test if they are.
Here is a piece of code that tests the CLT assumptions and runs the relevant test depending on the outcome. 
I've used a rule of thumb value of 5. Feel free to change my number "5" with something a higher value for more conservative or less if you want to be less conservative:
# Generate some random binary outcomes for Series 1 and 2
set.seed(1241535)
Series1 <- rbinom(150, 1, 0.5)
Series2 <- rbinom(200, 1, 0.6)

# All tests need to be confirmed
pooled_p <- (sum(Series1) + sum(Series2))/(length(Series1) + length(Series2))

test1 <- (length(Series1) * pooled_p) >= 5
test2 <- (length(Series2) * pooled_p) >= 5
test3 <- (length(Series1) * (1 - pooled_p)) >= 5
test4 <- (length(Series2) * (1 - pooled_p)) >= 5

final_test <- all(test1, test2, test3, test4)

# Chi-square or Fisher's exact test
x    <- c(sum(Series1), sum(Series2))
n    <- c(length(Series1), length(Series2))
mash <- rbind(c(sum(Series1), length(Series1) - sum(Series1)),
              c(sum(Series2), length(Series2) - sum(Series2)))

if(final_test == T){

  ## With Yate's continuity correction

  prop.test(x,n)
  #Exactly the same as:
  chisq.test(mash)

}else{

  # Fisher's exact test
  fisher.test(mash)

}

If your variables are not independent e.g: if series 1 and 2 are measurements of the same individual before and after an intervention, then a McNemar's test is more appropriate:
set.seed(1241535)
Series1 <- rbinom(200, 1, 0.5)
Series2 <- rbinom(200, 1, 0.6)


tab <-
  matrix(c(sum(Series1 == 1 & Series2 == 1), 
           sum(Series1 == 0 & Series2 == 1), 
           sum(Series1 == 1 & Series2 == 0), 
           sum(Series1 == 0 & Series2 == 0)
           ),
         nrow = 2,
         dimnames = list("Series1" = c("1", "0"),
                         "Series2" = c("1", "0")))

tab
mcnemar.test(tab)

The latter might be the relevant one for your case as I've noticed that you specify paired = TRUE in your code.
