new question(s) on static vs dynamic models

[kleintob]

• idea: compare U1-U0 to u1-u0; if they don't differ much then estimating a static model should be fine too, and the inner loop tolerance should not matter much
• for the current parameter values the difference is small
• AP: try to find parameter values such that the difference is bigger (call this parameters 2)
• ask students to assess dependence of importance of inner loop tolerance on parameter values
• ask students to estimate a static model and check when this gives good estimates

[jabbring]

They differ in level, but not much in slopes. If you increase the inner tolerance a lot, our procedures calculates U1-U0=u1-u0, so that estimates of beta_0 (intercept of u1-u0) are off, but beta_1 and delta_1 are not.

[jabbring]

• idea: compare U1-U0 to u1-u0; if they don't differ much then estimating a static model should be fine too, and the inner loop tolerance should not matter much

Qualification: This requires that the procedure for calculating U1-U0 gives something closer to u1-u0 (in a way that matters to our purpose) if the inner tolerance is larger. This is definitely true for our fixedPoint.m if the tolerance is so high it iterates only once (but not so ridiculously high it never iterates), in which cases it gives U1-U0=u1-u0. Not sure what theory says in cases it iterates more than once, but not often enough. We have monotone convergence to the fixed point in the space of [U0,U1] which, by the triangle inequality, implies monotone convergence of U1-U0 from u1-u0 (in our implementation) to the fixed point. This takes us some way but, as we emphasise in class, this convergence is monotone in the sense that the (sup, max) distance to the fixed point decreases along the way; there is no guarantee that the level (as in the previous comment) converges monotonically or that the slopes stay close along the way.