Quiz 1 (12 minutes)

Suppose the joint distribution of $\left(X_1,v\right)$ is bivariate normal, i.e., $$\left(\begin{array}{c} X_{1}\\ v \end{array}\right)\sim N\left(\left(\begin{array}{c} 1\\ 0 \end{array}\right),\left(\begin{array}{cc} 4 & 3\\ 3 & 9 \end{array}\right)\right).$$ A random sample $\{\left(X_{1t},v_t\right)\}$ was obtained from this joint distribution and $Y_t$ was obtained by $Y_t=2+3X_{1t}+v_t$.

• Do you think $Y_t=2+3 X_{1t}+v_t$ is a linear regression model? Show why or why not.
• If you only get to observe a random sample from $\{\left(X_{1t}, Y_t\right)\}$ and you obtain a least squares fit of $Y$ on $X_1$, what will least squares be recovering in large samples? Show your work.

One possible solution to Quiz 1

• No, $Y_t=2+3 X_{1t}+v_t$ is not a linear regression model, because $\langle v,X_1 \rangle =\mathsf{Cov}\left(v, X_{1}\right) \overset{\mathsf{IID}}{=} \mathsf{Cov}\left(v_t, X_{1t}\right)= 3\neq 0$.

• Because we have a random sample $\{\left(X_{1t},Y_t\right)\}$ from a bivariate normal distribution $$\left(\begin{array}{c} X_{1}\\ Y \end{array}\right)\sim N\left(\left(\begin{array}{c} 1\\ 5 \end{array}\right),\left(\begin{array}{cc} 4 & 15\\ 15 & 45 \end{array}\right)\right),$$ the least squares fit of $Y$ on $X_1$ will recover the best linear predictor $\beta^*_0+\beta_1^* X_{1t}= \dfrac{5}{4}+\dfrac{15}{4} X_{1t}$.

Exam strategies

• Because English is your second language, it requires extra time to understand or figure out what the question is asking.

• Therefore, in an exam, you should give yourself an allowance for problems with English comprehension.

• Because you have to “design” a solution, it also requires extra time to write down the components of a complete solution.

• A detailed solution is preferable but you need to decide what details are most important for the presentation of the solution.

• Do NOT waste your time writing extra detail that is irrelevant. You waste time for the problem you are solving and others you have NOT solved!

• Therefore, in an exam, you should give yourself an allowance for designing your solution.

• Conclusion: Time pressure is real.

Remarks

• This quiz is based on Lectures 1 to 5. Some ideas from Chapter 2 and Chapter 4 are needed to obtain the correct answer.

• It is not necessary that you produce the joint bivariate normal distribution of $X_1$ and $Y$. What is important is to know show that you can find $\beta_0^*$ and $\beta_1^*$. In addition, you should know the asymptotic behavior of $\widehat{\beta}_0$ and $\widehat{\beta}_1$.

• To calculate $\beta_0^*$ and $\beta_1^*$: $$\begin{eqnarray} \beta_1^* &=&\dfrac{\mathsf{Cov}\left(X_1,Y\right)}{\mathsf{Var}\left(X_1\right)}=\dfrac{\mathsf{Cov}\left(X_1,2+3X_1+v\right)}{\mathsf{Var}\left(X_1\right)}\\ &=&3+\dfrac{\mathsf{Cov}\left(X_1,v\right)}{\mathsf{Var}\left(X_1\right)}=3+\dfrac{3}{4}=\dfrac{15}{4} \\ \beta_0^* &=& \mathbb{E}\left(Y\right)-\beta_1^* \mathbb{E}\left(X_1\right) \\ &=& 2+3\mathbb{E}\left(X_1\right)+\mathbb{E}\left(v\right)-\dfrac{15}{4}\mathbb{E}\left(X_1\right)\\ &=& 2+3*1+0-\dfrac{15}{4}*1=\dfrac{5}{4} \end{eqnarray}$$

• Note that we can drop the subscript $t$ because we are in IID world.

Follow-up questions

• Is $Y_t=\dfrac{5}{4}+\dfrac{15}{4} X_{1t}+u_t$ a linear regression model? Show why or why not.

• Is $v_t$ an error from the best linear prediction of $Y$ using $X_1$? Show why or why not.

• Is $Y_t=\dfrac{5}{4}+\dfrac{15}{4} X_{1t}+u_t$ a correctly specified linear regression model? Show why or why not.

• As practice and check of past AE1 knowledge, try to obtain the bivariate normal displayed in the previous slide. Why is the joint distribution of $X_1$ and $Y$ also bivariate normal? How do you get the entries of the covariance matrix?

• In the next slide, you will see another version of the quiz. This version is one possible illustration of what some call omitted variable bias (pages 43, 104, Exercise 3.23) or $X_1$ is endogenous (page 245 Example 7.5).

Another version of the quiz (12 minutes)

Suppose the joint distribution of $\left(X_1,v\right)$ is bivariate normal, i.e., $$\left(\begin{array}{c} X_{1}\\ v \end{array}\right)\sim N\left(\left(\begin{array}{c} \mu_1\\ 0 \end{array}\right),\left(\begin{array}{cc} \sigma^2_1 & \sigma_{1v}\\ \sigma_{1v} & \sigma^2_v \end{array}\right)\right).$$ A random sample $\{\left(X_{1t},v_t\right)\}$ was obtained from this joint distribution and $Y_t$ was obtained by $Y_t=\beta_0+\beta_1X_{1t}+v_t$.

• Do you think $Y_t=\beta_0+\beta_1 X_{1t}+v_t$ is a linear regression model? Show why or why not.
• If you only get to observe a random sample from $\{\left(X_{1t}, Y_t\right)\}$ and you obtain a least squares fit of $Y$ on $X_1$, what will least squares be recovering in large samples? Show your work.