113 年 國立臺北大學經濟學系碩士班《統計學（含計量經濟學）》

What is the true value of $s^2$ (estimated error variance) in equation (1)?

(A) $4$
(B) $4.76$
(C) $4.55$
(D) $5$

Continue Question 1, Robert estimates the other equation by ordinary least squares (OLS) method:

y=6.0+3.2x_1 \tag{2}

where $SSE=140$. If Robert tests the following hypothesis:

$H_0:\beta_2=\beta_3=0$

what is the value of $F$-statistic?

(A) $3$
(B) $4.4$
(C) $7$
(D) $4.2$

Suppose that the true model of $y$ is

$y=\beta_1x_1+\beta_2x_2+u,$

but you estimated an incorrect model by OLS:

y=\beta_1x_1+v \tag{3}

The true parameter $\beta_2<0$, and $E(x_2\mid x_1)$ has a linear relationship:

E(x_2\mid x_1)=\delta_0+\delta_1x_1 \tag{4}

If we have the prior information that $\delta_1$ of Equation (4) has negative bias, what is the following condition satisfied for OLS estimator $\hat{\beta}_1$ in equation (3)?

(A) $|\hat{\beta}_1|=|\beta_1|$
(B) $|\hat{\beta}_1|>|\beta_1|$
(C) $|\hat{\beta}_1|<|\beta_1|$
(D) cannot decide

Consider the model:

$y_t=\alpha+\rho x_t+\varepsilon_t,$

where $t=1,\ldots,T$. If we want to obtain consistent estimator $\hat{\rho}$ for $T\to\infty$, which necessary condition must it have?

(A) $\operatorname{cov}(\varepsilon_t,x_t)=0$
(B) $\operatorname{Var}(\varepsilon_t\mid 1,x_t)=\sigma^2$
(C) $E(\varepsilon_t\mid 1,x_t)=0$
(D) $E(\varepsilon_t\varepsilon_s\mid x_t,x_s)=0$

Sample selection bias occurs when

(A) data are collected from a population by simple random sampling.
(B) The choice between two samples is made by the researcher.
(C) Samples are chosen to be small rather than large.
(D) The availability of the data is influenced by a selection process that is related to the value of the dependent variable.

The major flaw of the linear probability model is that

(A) the actuals can only be $0$ and $1$, but the predicted are almost always different from that.
(B) People do not always make clear-cut decisions.
(C) The regression $R^2$ cannot be used as a measure of fit.
(D) The predicted values can lie above $1$ and below $0$.

If $E(z^*\mid x_1,x_2)=0$, $\hat{\beta}_0$, $\hat{\beta}_1$ and $\hat{\beta}_2$ are the OLS estimators from the regression $y$ on $(1,x_1,x_2)$.

Whether $\hat{\beta}_0$, $\hat{\beta}_1$ and $\hat{\beta}_2$ are consistent estimators?

Now suppose that there is an observable variable $z$ that is redundant in (5), so that

$E(u\mid x_1,x_2,z^*,z)=0,$

and

E(z^*\mid x_1,x_2,z)=\delta_0+\delta_1z \tag{7}

How to obtain consistent estimators of $\beta_1$ and $\beta_2$?

A test for a certain disease is $90\%$ accurate. If a person has the disease, there's a $90\%$ chance the test will be positive. If a person doesn't have the disease, there's a $10\%$ chance the test will be positive.

If $1\%$ of the population has the disease, what is the probability that a person has the disease given they tested positive?

Given a joint density function

$f(x,y)=\frac{1}{2}$

for $0\leq x\leq 2$ and $0\leq y\leq 2-x$, find the marginal density function of $f_Y(y)$ for $0\leq y\leq 2$ to calculate the expected value (mean) of $y$.

A car manufacturer claims that their new electric car model has an average range of at least $300$ miles on a single charge. A random sample of $40$ cars was tested, and it was found that they had an average range of $295$ miles with a standard deviation of $15$ miles.

At a $5\%$ significance level, calculate the "$t$ value" to test the manufacturer's claim.

We have two six-sided dice, one blue and one green. Each die can show a number from $1$ to $6$ with equal probability.

We're interested in two variables: the total (sum) of the numbers that show up, and the difference (blue minus green). Let's call the total '$X=\text{blue}+\text{green}$' and the difference '$Y=\text{blue}-\text{green}$'.

We want to find out the covariance between $X$ and $Y$.

Let $X$ and $Y$ be two distinct random variables. The mean of $X$ is given as $10$, and its variance is $9$. The mean of $Y$ is $0$, and its variance is $25$. The correlation coefficient between $X$ and $Y$ is established at $0.5$.

Please compute the variance of the linear combination $3X+2Y$.

A population has a mean of $500$ and a standard deviation of $100$. Using Chebyshev's inequality, determine the minimum proportion of the population that lies within the range $300$ to $700$.

Suppose a variable $Z$ follows normal distribution. The mean for $Z$ is $2$ and variance is $0.25$. We define

$Y=e^Z.$

Suppose the lifetime of a machine component is $Y$. Find the probability that a randomly chosen component lasts less than $e^3$ hours.