OLS estimation for Nonlinear model 
Consider the following model which may be nonlinear:
$Y_{t} = f (X_{t}, \beta_{0}) + \mu_{t}, \hspace{0.2cm} t=1, ..., T$
If we assume that:

*

*$\mu_{t}$ i.i.d with mean = $0$ and variance $\sigma_{0}^{2} < \infty$


*The parameter space $B$ is nonempty and compact


*$f (X_{t}, \beta)$ is non random for all $\beta \in B$


*$\frac{1}{T}\sum_{t=1}^{T}[f (X_{t}, \beta_{0})-f (X_{t}, \beta)]^2$ converges uniformly to a continuous function $A(\beta)$, which attains its minimum only in $\beta = \beta_{0}$
Let $\hat{\beta_{T}}$ be the OLS estimation for $\beta_{0}$.
Does $\hat{\beta_{T}}$ converges in probability to $\beta_{0}$ when $T\rightarrow \infty$?

I'm trying to solve this problem because our professor stated it as a "recommended excercise". I've tried consulting Amemiya's "Advanced Econometrics" book and other references to get an idea but I'm still stuck on the problem. I would appreciate any help.
 A: OLS is generally consistent for the linear projection coefficients $\beta^\star$, see e.g. any somewhat advanced econometrics textbook like Hayashi's. In the case of a simple regression without constant (for simplicity),
$$
\beta^\star=\frac{E(X\cdot Y)}{E(X^2)}
$$
Let us consider the case where $P(X_i=-1)=P(X_i=1)=0.5$ and $Y_i=\exp(X_i\beta)+u_i$, $u_i$ independent of $X_i$. Without having checked carefully, this example should fit your set of assumptions. Then,
$$
E(X\cdot Y)=E(X\exp(X\beta))=-0.5\exp(-\beta)+0.5\exp(\beta)
$$
and $E(X^2)=1$. So for $\beta=1$, OLS is consistent for $0.5[\exp(\beta)-\exp(-\beta)]\approx1.18$, not 1.
Here is an illustration for a sample size of $n=500$ and 10,000 replications to simulate the resulting distribution, which we see centers around 1.18, not 1.
 
library(MASS)
reps <- 10000
olse <- rep(NA,reps)

n <- 500
beta <- 1
plimols <- (exp(beta) - exp(-beta))/2

for (i in 1:reps) {
  X <- 2*(runif(n)>.5)-1
  u <- rnorm(n,sd=.1)
  y <- exp(beta*X) + u
  olse[i] <- lm(y~X-1)$coefficients  
}
truehist(olse,col="salmon")
abline(v=plimols,col="gold",lwd=3)
abline(v=beta,col="purple",lwd=3)
mean(olse)

