When we estimate a nonlinear equation how much must be the amount of statistics(t , D.W, R^2 and F) ?

Question

When we estimate a nonlinear equation how much must be the amount of statistics(t , D.W, R^2 and F) to accept

that estimation as a good estimation and prove that there is relation between dependent and independet variables? I will be very thankfull if you send me an exampel.

Answer 1

While an R-squared of 0.95 suggests that about 95% of the variation of your dependent variable y can be "explained" through variations in your independent variables (x1, x2, x3, etc.), the t statistics indicates the marginal explanatory power of each of the independent variables (x1, x2, x3, ...) is low. This is most likely due to multicollinearity. Multicollinearity does not reduce the predictive power or reliability of the model as a whole; it only affects calculations regarding individual predictors.

Answer 2

The resemblance to the 'within' moment matrix from the analysis of variance context is notable and convenient. Inserting the parts and collecting terms produces

Δγ = ()(){}111iNTititiitiit−==Δ−−ΣΣxxxx × (){}()11NTiitititiiitq==λ−ΣΣxx

and

Δαi = ()1/ + iTititiiitqγ=′−λΔΔΣx𐀄

Denote the matrix in the preceding as

V = -[Hγγ]-1 = Asy.Var[bMLE].

Then,

Asy.Cov[ai,aj] = ()() + = + ijiiijijs−=−=′ΔΔ11xVx ij

and

Asy.Cov[bMLE,ai] = -Vix.

Each of these involves a moderate amount of computation, but can easily be obtained with existing software and, most important for our purposes, involves computations that are linear in N and K. We note as well that the preceding extends directly to any other simple index function model, such as the binomial logit model [change derivatives λit to (1- Λit) and Δit to -Λit(1 - Λit) where Λit is the logit CDF] and the Poisson regression model [replace λit with (yit - mit) and Δit with -mit where mit = exp(β′xit + αi)]. Extension to models that involve ancillary parameters, such as the tobit model, are a bit more complicated, but not excessively so.

The preceding provides the estimator and asymptotic variances for all estimated parameters in the model. For inference purposes, note that the unconditional log likelihood function is computed. Thus, a test for homogeneity is straightforward using the likelihood ratio test. Finally, one would normally want to compute marginal effects for the estimated probit model. The conditional mean in the model is

E[zit | xit] = Φ(β′xit + αi)

so the slopes in the model are