Variance of difference of $x_{i,t}$ and $x_{i,t+1}$

We have $n$ observations of human performance before and after a training program. We have a random sample of $n$ individuals and assume population distributions are normal.

We thus have: $x_{1,t},\ x_{2,t},\ ...,\ x_{n,t}$ and $x_{1,t+1},\ x_{2,t+1},\ ...,\ x_{n,t+1}$. It is clear from the example that ${\rm Cov}(x_t,x_{t-1}) \neq 0$.

What is the convention and reasonable assumptions on the variance of the difference $d = x_{t} - x_{t+1}$?

$$\widehat{\rm var}(d) = \widehat{\rm var}(x_t)+\widehat{\rm var}( x_{t+1}) - 2\widehat{\rm cov}(x_t,x_{t+1}) ?$$

Or assume constant variance (homoscedasticity) over time?

$$\widehat{\rm var}( d) = 2\widehat{\rm var}(x) - 2\widehat{\rm cov}(x_t,x_{t+1}) ?$$

Is it correct to simply take the variance of $d$? How do you recommend start thinking (rigorously) about a problem like this?

• Thank you. It looks like you are asking people to offer opinions about changes in weight within some undescribed experiment. By providing no information about that experiment (is it a force feeding of mice? Observation of growth of human infants? Charting growth of normal human adults?) you can expect a whole range of answers, any of which might be correct. – whuber Oct 22 '14 at 15:26
• The thing is I don't have much more information myself. It is observations of human performance before and after a training program. We have a random sample of n individuals and assume population distributions are normal. – snoram Oct 22 '14 at 15:33
• Please, then, include that information in your question. Consider opening it up a little bit and asking readers about how one might go about deciding whether such data are homo- or hetero-scedastic. That might produce more useful and objective answers compared to asking vaguely about "conventions" and "assumptions." – whuber Oct 22 '14 at 15:39
• Seems the post got lost, am I allowed to repost it? – snoram Oct 22 '14 at 23:14
• The system says you deleted it--that means you should be able to undelete it. Since I infer from your comment that you want it undeleted, I will do that for you. – whuber Oct 23 '14 at 1:21