In a highly cited paper by Zhao (2006) it is stated that (Section 2)
An estimate which is consistent in terms of parameter estimation does not necessarily consistently selects the correct model (or even attempt to do so) where the reverse is also true. The former requires $$\hat\beta^n - \beta^n \xrightarrow{p}0, \quad \text{as} \quad n\to \infty $$ while the latter requires $$\Pr\left( \{i\colon \hat\beta_i^n \neq 0 \} = \{i\colon \beta_i^n \neq 0 \} \right) \to 1 \quad \text{as} \quad n\to \infty. $$
I do not understand how it could be possible that an estimator is consistent in parameter estimation but selects a wrong model? If asymptotically estimates converge to the true values then they also should converge to zeros whenever the true value is zero, hence selecting the right model.
EDIT: One more question related to this one is "If an estimator $\hat\beta^n$ has oracle properties does this imply that $\hat\beta^n - \beta^n \to 0$? That is, do oracle properties imply consistency in parameter estimation?". For clarity I list oracle properties below.
Define $\mathcal{A} = \{ j\colon \beta_j \neq 0 \}$ and further assume that $\vert \mathcal{A}\vert < p$ where $p$ is a number of coefficients in vector $\beta$. We say that fitting procedure $\delta$ that produces estimate $\hat\beta(\delta)$ is an oracle procedure if $\hat\beta(\delta)$ satisfies following oracle properties:
- Identifies the right subset model, $\{ j\colon \hat\beta_j \neq 0 \} = \mathcal{A}$
- Has optimal estimation rate, $\sqrt{n}\left( \hat\beta(\delta)_{\mathcal{A}} - \beta_{\mathcal{A}} \right) \xrightarrow{d} N(0, \Sigma)$, where $\Sigma$ is the covariance matrix of the true subset model.