An example of a consistent and biased estimator? Really stumped on this one. I would really like an example or situation where an estimator B would be both consistent and biased. 
 A: Consider any unbiased and consistent estimator $T_n$ and a sequence $\alpha_n$ converging to 1 ($\alpha_n$ need not to be random) and form $\alpha_nT_n$. It is biased, but consistent since $\alpha_n$ converges to 1.
From wikipedia:
Loosely speaking, an estimator $T_n$ of parameter $\theta$ is said to be consistent, if it converges in probability to the true value of the parameter:
$$\underset{n\to\infty}{\operatorname{plim}}\;T_n = \theta.$$
Now recall that the bias of an estimator is defined as:
$$\operatorname{Bias}_\theta[\,\hat\theta\,] = \operatorname{E}_\theta[\,\hat{\theta}\,]-\theta $$
The bias is indeed non zero, and the convergence in probability remains true.
A: The simplest example I can think of is the sample variance that comes intuitively to most of us, namely the sum of squared deviations divided by $n$ instead of $n-1$:
$$S_n^2 = \frac{1}{n} \sum_{i=1}^n \left(X_i-\bar{X} \right)^2$$ 
It is easy to show that $E\left(S_n^2 \right)=\frac{n-1}{n} \sigma^2$ and so the estimator is biased. But assuming finite variance $\sigma^2$, observe that the bias goes to zero as $n \to \infty$ because
$$E\left(S_n^2 \right)-\sigma^2 = -\frac{1}{n}\sigma^2 $$
It can also be shown that the variance of the estimator tends to zero and so the estimator converges in mean-square. Hence, it is also convergent in probability.
A: In a time series setting with a lagged dependent variable included as a regressor, the OLS estimator will be consistent but biased. The reason for this is that in order to show unbiasedness of the OLS estimator we need strict exogeneity, $E\left[\varepsilon_{t}\left|x_{1},\, x_{2,},\,\ldots,\, x_{T}\right.\right]
 $, i.e. that the error term, $\varepsilon_{t}
 $, in period $t
 $ is uncorrelated with all the regressors in all time periods. However, in order to show consistency of the OLS estimator we only need contemporanous exogeneity, $E\left[\varepsilon_{t}\left|x_{t}\right.\right]
 $, i.e. that the error term, $\varepsilon_{t}
 $, in period $t
 $ is uncorrelated with the regressors, $x_{t}
 $ in period $t
 $. Consider the AR(1) model: $y_{t}=\rho y_{t-1}+\varepsilon_{t},\;\varepsilon_{t}\sim N\left(0,\:\sigma_{\varepsilon}^{2}\right)$
  with $x_{t}=y_{t-1}
 $ from now on. 
First I show that strict exogeneity does not hold in a model with a lagged dependent variable included as a regressor. Let's look at the correlation between $\varepsilon_{t}
 $ and $x_{t+1}=y_{t}
 $ $$E\left[\varepsilon_{t}x_{t+1}\right]=E\left[\varepsilon_{t}y_{t}\right]=E\left[\varepsilon_{t}\left(\rho y_{t-1}+\varepsilon_{t}\right)\right]
 $$
$$=\rho E\left(\varepsilon_{t}y_{t-1}\right)+E\left(\varepsilon_{t}^{2}\right)
 $$
$$=E\left(\varepsilon_{t}^{2}\right)=\sigma_{\varepsilon}^{2}>0
 \ (Eq. (1)).$$
If we assume sequential exogeneity, $E\left[\varepsilon_{t}\mid y_{1},\: y_{2},\:\ldots\ldots,y_{t-1}\right]=0
 $, i.e. that the error term, $\varepsilon_{t}
 $, in period $t
 $ is uncorrelated with all the regressors in previous time periods and the current then the first term above, $\rho E\left(\varepsilon_{t}y_{t-1}\right)
 $, will dissapear. What is clear from above is that unless we have strict exogeneity the expectation  $E\left[\varepsilon_{t}x_{t+1}\right]=E\left[\varepsilon_{t}y_{t}\right]\neq0
 $. However, it should be clear that contemporaneous exogeneity, $E\left[\varepsilon_{t}\left|x_{t}\right.\right]
 $, does hold.
Now let's look at the bias of the OLS estimator when estimating the AR(1) model specified above. The OLS estimator of $\rho
 $, $\hat{\rho}
 $ is given as:
$$\hat{\rho}=\frac{\frac{1}{T}\sum_{t=1}^{T}y_{t}y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}=\frac{\frac{1}{T}\sum_{t=1}^{T}\left(\rho y_{t-1}+\varepsilon_{t}\right)y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}=\rho+\frac{\frac{1}{T}\sum_{t=1}^{T}\varepsilon_{t}y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}
\ (Eq. (2))$$
Then take conditional expectation on all previous, contemporaneous and future values, $E\left[\varepsilon_{t}\left|y_{1},\, y_{2,},\,\ldots,\, y_{T-1}\right.\right]
 $, of $Eq. (2)$:
$$E\left[\hat{\rho}\left|y_{1},\, y_{2,},\,\ldots,\, y_{T-1}\right.\right]=\rho+\frac{\frac{1}{T}\sum_{t=1}^{T}\left[\varepsilon_{t}\left|y_{1},\, y_{2,},\,\ldots,\, y_{T-1}\right.\right]y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}
 $$
However, we know from $Eq. (1)$ that $E\left[\varepsilon_{t}y_{t}\right]=E\left(\varepsilon_{t}^{2}\right)
 $ such that $\left[\varepsilon_{t}\left|y_{1},\, y_{2,},\,\ldots,\, y_{T-1}\right.\right]\neq0
 $ meaning that $\frac{\frac{1}{T}\sum_{t=1}^{T}\left[\varepsilon_{t}\left|y_{1},\, y_{2,},\,\ldots,\, y_{T-1}\right.\right]y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}\neq0
 $ and hence $E\left[\hat{\rho}\left|y_{1},\, y_{2,},\,\ldots,\, y_{T-1}\right.\right]\neq\rho
 $ but is biased: $E\left[\hat{\rho}\left|y_{1},\, y_{2,},\,\ldots,\, y_{T-1}\right.\right]=\rho+\frac{\frac{1}{T}\sum_{t=1}^{T}\left[\varepsilon_{t}\left|y_{1},\, y_{2,},\,\ldots,\, y_{T-1}\right.\right]y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}=\rho+\frac{\frac{1}{T}\sum_{t=1}^{T}E\left(\varepsilon_{t}^{2}\right)y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}=$$\rho+\frac{\frac{1}{T}\sum_{t=1}^{T}\sigma_{\varepsilon}^{2}y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}
 $.
All I assume to show consistency of the OLS estimator in the AR(1) model is contemporanous exogeneity, $E\left[\varepsilon_{t}\left|x_{t}\right.\right]=E\left[\varepsilon_{t}\left|y_{t-1}\right.\right]=0
 $ which leads to the moment condition, $E\left[\varepsilon_{t}x_{t}\right]=0
 $ with $x_{t}=y_{t-1}
 $. As before, we have that the OLS estimator of $\rho
 $, $\hat{\rho}
 $ is given as: $$\hat{\rho}=\frac{\frac{1}{T}\sum_{t=1}^{T}y_{t}y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}=\frac{\frac{1}{T}\sum_{t=1}^{T}\left(\rho y_{t-1}+\varepsilon_{t}\right)y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}=\rho+\frac{\frac{1}{T}\sum_{t=1}^{T}\varepsilon_{t}y_{t-1}}{\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}
 $$
Now assume that $plim\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}=\sigma_{y}^{2}
 $ and $\sigma_{y}^{2}
 $ is positive and finite, $0<\sigma_{y}^{2}<\infty
 $.
Then, as $T\rightarrow\infty
 $ and as long as a law of large numbers (LLN) applies we have that $p\lim\frac{1}{T}\sum_{t=1}^{T}\varepsilon_{t}y_{t-1}=E\left[\varepsilon_{t}y_{t-1}\right]=0
 $. Using this result we have: $$\underset{T\rightarrow\infty}{p\lim\hat{\rho}}=\rho+\frac{p\lim\frac{1}{T}\sum_{t=1}^{T}\varepsilon_{t}y_{t-1}}{p\lim\frac{1}{T}\sum_{t=1}^{T}y_{t}^{2}}=\rho+\frac{0}{\sigma_{y}^{2}}=\rho
 $$
Thereby it has been shown that the OLS estimator of $p
 $, $\hat{\rho}
 $ in the AR(1) model is biased but consistent. Note that this result holds for all regressions where the lagged dependent variable is included as a regressor.
A: A simple example would be estimating the parameter $\theta > 0$ given $n$ i.i.d. observations $y_i \sim \text{Uniform}\left[0, \,\theta\right]$.
Let $\hat{\theta}_n = \max\left\{y_1, \ldots, y_n\right\}$.  For any finite $n$ we have $\mathbb{E}\left[\theta_n\right] < \theta$ (so the estimator is biased), but in the limit it will equal $\theta$ with probability one (so it is consistent).
