Why did fixing the covariance value cause AMOS to mis-estimate variance?

Question

I have 100 data points on two variables, a and b. The correlation between the two is .3 and the SD is 1.

When I run the following in AMOS I get sensible estimates (.995 and 1.001) for the variances of both a and b, even though I've fixed the covariance to an inaccurate value.

The estimates of variances stay about the same if I fix the covariance to be .3.

However, if I fix the covariance to other values then the variance estimates can become wildly different. For instance, if I fix the covariance to .95 the variance estimates become 1.57 and 1.58 respectively. If I fix the covariance to .15 the variance estimates become .95 and .96 respectively.

Why does this happen? In case it helps, here is the data in a Google spreadsheet.

Answer 1

The covariance is (about) 0.3, which is why that works.

So the covariance = 0.3/variance. (The variances are the same, which makes it easier).

When you fix the covariance to some other value, it's trying to get the covariance back to 0.3 / variance. It can't change the covariance, because you've fixed that. But by changing the variances, it can get closer. However, when it changes the variances, they become wrong, and this adds error to the model, so it doesn't make the variances as high as 3.