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## 9.5 Exercises

#### 1. Simulation Study: Misspecification of Functional Form

As stated in Chapter 9.2, misspecification of the regression function violates assumption 1 of Key Concept 6.3 so that the OLS estimator will be biased and inconsistent. We have illustrated the bias of $\hat{\beta}_0$ for the example of the quadratic population regression function $Y_i = X_i^2$ and the linear model $Y_i = \beta_0 + \beta_1 X_i + u_i, \, u_i \sim \mathcal{N}(0,1)$ using 100 randomly generated observations. Strictly speaking, this finding could be just a coincidence because we consider just one estimate obtained using a single data set.

In this exercise, you have to generate simulation evidence for the bias of $\hat{\beta}_0$ in the model $Y_i = \beta_0 + \beta_1 X_i + u_i$ if the population regression function is $Y_i = X_i^2.$

Instructions:

Make sure to use the definitions suggested in the skeleton code in script.R to complete the following tasks:

• Generate 1000 OLS estimates of $\beta_0$ in the model above using a for() loop where $X_i \sim \mathcal{U}[-5,5]$, $u_i \sim \mathcal{N}(0,1)$ using samples of size $100$. Save the estimates in beta_hats.

• Compare the sample mean of the estimates to the true parameter using the == operator.

Hint:

You can generate random numbers from a uniform distribution using runif().

#### 2. Simulation Study: Errors-in-Variables Bias

Consider again the application of the classical measurement error model introduced in Chapter 9.2:

The single regressor $X_i$ is measured with error so that $\overset{\sim}{X}_i$ is observed instead. Thus one estimates $\beta_1$ in \begin{align*} Y_i =& \, \beta_0 + \beta_1 \overset{\sim}{X}_i + \underbrace{\beta_1 (X_i -\overset{\sim}{X}_i) + u_i}_{=v_i} \\ Y_i =& \, \beta_0 + \beta_1 \overset{\sim}{X}_i + v_i \end{align*} instead of $Y_i = \beta_0 + \beta_1 X_i + u_i,$

with the zero mean error $w_i$ being uncorrelated with $X_i$ and $u_i$. Then $\beta_1$ is inconsistently estimated by OLS: $\begin{equation} \widehat{\beta}_1 \xrightarrow{p}{\frac{\sigma_{X}^2}{\sigma_{X}^2 + \sigma_{w}^2}} \beta_1 \end{equation}$

Let $(X, Y) \sim \mathcal{N}\left[\begin{pmatrix}50\\ 100\end{pmatrix},\begin{pmatrix}10 & 5 \\ 5 & 10 \end{pmatrix}\right].$ Recall from (9.2) that $E(Y_i\vert X_i) = 75 + 0.5 X_i$ in this case. Further Assume that $\overset{\sim}{X_i} = X_i + w_i$ with $w_i \overset{i.i.d}{\sim} \mathcal{N}(0,10)$.

As mentioned in Exercise 1, Chapter 9.2 discusses the consequences of the measurement error for the OLS estimator of $\beta_1$ in this setting based on a single sample and and thus just one estimate. Strictly speaking, the conclusion made could be wrong because the oberseved bias may be due to random variation. A Monto Carlo simulation is more appropriate here.

Instructions:

Show that $\beta_1$ is estimated with a bias using a simulation study. Make sure to use the definitions suggested in the skeleton code in script.R to complete the following tasks:

• Generate 1000 estimates of $\beta_1$ in the simple regression model $Y_i = \beta_0 + \beta_1 X_i + u_i.$ Use rmvnorm() to generate samples of 100 random observations from the bivariate normal distribution stated above.

• Save the estimates in beta_hats.

• Compute the sample mean of the estimates.