Correlation between two time series - volume and proportion My company makes widgets. We take a random sample of these widgets every month and count how many are good and bad. Some people think that we are worse at making widgets in the months where we make more - in other words, the months where we make more widgets have a higher proportion of bad widgets. 
Is calculating the Pearson correlation coefficient between the number of widgets made in a month and the proportion of bad widgets (in the monthly sample) an appropriate way to check this theory?
I am unsure whether it is OK to use the Pearson correlation coefficient because the dataset is a time series.
I have data on the number of widgets made every month, the number of widgets sampled every month and the proportion of bad widgets found in the sample every month.

 A: The issue with using correlation measures for time series, is that one doesn't know whether those correlations are statistically significant or simply a function of time.
For instance, if one analyses the movement of oil and the S&P 500 index over a period of 50 years, one will generally find a strong correlation between the two series. However, this does not indicate that the movement is necessarily statistically significant - it simply means that the two series have trended upwards over time - as we would expect over the longer-term.
So, how can one determine whether a correlation is significant or due to chance? The way we can test for this is by determining if two time series are cointegrated.
A cointegrated pair is one where the movements between two time series are statistically significant, i.e. not simply due to chance. Imagine a drunk woman walking with her dog on a leash. The two might walk in opposite directions at times, but because the dog is on a leash, the woman and the dog will ultimately end up going in the same directions.
If you are using R, the coint.test available from the aTSA package is a good way of testing for cointegration using the Engle-Granger method.
Consider the following variables X and y:
X <- matrix(rnorm(200),100,2)
y <- 0.3*X[,1] + 1.2*X[,2] + rnorm(100)

A cointegration test is run for X and y:
coint.test(y,X)

Here are the generated results:
Response: y 
Input: X 
Number of inputs: 2 
Model: y ~ X + 1 
------------------------------- 
Engle-Granger Cointegration Test 
alternative: cointegrated 

Type 1: no trend 
    lag      EG p.value 
   4.00   -4.76    0.01 
----- 
 Type 2: linear trend 
    lag      EG p.value 
  4.000   0.555   0.100 
----- 
 Type 3: quadratic trend 
    lag      EG p.value 
    4.0    -1.9     0.1 
----------- 
Note: p.value = 0.01 means p.value <= 0.01 
    : p.value = 0.10 means p.value >= 0.10

We can see that we have a p-value < 0.05 for type 1: no trend. However, the linear trend and quadratic trend are both statistically insignificant with a p-value > 0.05. In this regard, this indicates that no trend exists between the two time series, and they are not cointegrated.
If you have access to R, you might find the above of use in checking for the existence of cointegration between volume and proportion.
