Chapter 5 – Quiz 2 – E600 Mathematics

1. Which of the following properties of the OLS estimator $\hat{\beta}$ is wrong?

If $\mathbb{E}(e_i|X_i) = 0$ , $\hat{\beta}$ is consistent.

The OLS estimator $\hat{\beta}=(X'X)^{-1}X'y$ is feasible if the determinant of the matrix X is zero.

The OLS estimator is justified as a linear approximation to the conditional expectation function $\mathbb{E}(Y|X)$

Assuming a linear conditional expectation function $\mathbb{E}(Y|X) = X \beta$ justifies estimation by OLS.

2. Which of the following statements are false?

The CLT implies that for a sequence of iid random variables $X_i$ with finite fourth moments $\frac{1}{\sqrt{n}}\sum_i X_i \xrightarrow{d} N(0, Var(X_i))$

$X_n\xrightarrow{p} X$ is a different notation to say $\lim_{n\rightarrow \infty}P(|X_n-X|<\epsilon)=0$

The WLLN for an iid sequence of random variables $X_i$ with finite second moments states that $\frac{1}{n}\sum_i X_i \xrightarrow{d}\mathbb{E}(X_i)$

$X_n \xrightarrow{d} X$ is equivalent to saying $\lim_{n \rightarrow \infty} F_{X_n}(x)= F_{X}(x)$

3. Which of the following transformations of conditional expectations is wrong?

$\mathbb{E}(\mathbb{E}(XY|Y) )= Y\mathbb{E}(X|Y)$

$\mathbb{E}(\mathbb{E}(\alpha f(Y)X|Y))= \alpha\mathbb{E}(Xf(Y))$

If X and Y are independent: $\mathbb{E}(X|Y)= \mathbb{E}(X)$

$\mathbb{E}(XY|Z )= \mathbb{E}(X|Z) \mathbb{E}(Y|Z)$

4. The conditional expectation $\mathbb{E}(Y|X)$ is …

a function $g(Y)$

necessarily either discrete or continuos

itself a random variable

the same as $\mathbb{E}(Y|X = x)$

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