107年 - [無官方正解]107 國立臺北大學_碩士班一般入學考試試題_經濟學系：統計學#146655

科目：研究所、轉學考(插大)◆統計學 | 年份：107年 | 選擇題數：6 | 申論題數：11

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所屬科目：研究所、轉學考(插大)◆統計學

選擇題 (6)

1. Consider the simple linear population model without intercept:

$$y_i = \beta x_i + \varepsilon_i$$

$$\varepsilon_i = \rho \varepsilon_{i-1} + u_i,$$

where $u_i \sim i.i.d(0, \sigma^2)$. We use OLS method to estimate parameter $\hat{\beta}$. After we obtain the $\hat{\beta}$, we use OLS residuals to generate Durbin-Watson statistic. Which approximation value does Durbin-Watson statistic converge under $T \to \infty$?

(A) \(2 - \hat{\rho}\)
(B) \(1 - 2\hat{\rho}\)
(C) \(2(1 - \hat{p})\)
(D) \(1 - \hat{p}\)
(E) \(2 - 3\hat{p}\)
2. Eric estimates a regression model as below:

$$y_i = \alpha + \beta_1 x_i + \beta_2 z_i + u_i,$$

Where $u_i \sim N(0, \sigma_u^2)$. If Eric regress $x_i$ on $z_i$, he find the following equation statistically significant:

$$x_i = \omega + \delta_2 z_i + \varepsilon_i.$$

Which problem he will face when he use OLS method to estimate $\hat{\beta}_1$ and $\hat{\beta}_2$?

(A) Multicollinearity
(B) Serial correlation
(C) Heteroskedasticity
(D) Nonlinear in parameter
(E) Random sample
3. Mary want to estimates a regression model as below:

$$y_i = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \varepsilon_i,$$

If she doubts the regression residuals $\hat{\varepsilon}_i$ with heteroskedasticity, she wants to use Breusch-Pagan test to detect whether heteroskedasticity exists in $\hat{\varepsilon}_i^2$. Which independent variables set she will choose to execute Breusch-Pagan test?

(A) \(x_{1i}, x_{2i}\)
(B) \(x_{n}, x_{2n}, x_{1n}^{2}, x_{2n}^{2}\)
(C) \(x_{n}, x_{2n}, x_{1n}^{2}, x_{2n}^{2}, x_{1n} x_{2n}\)
(D) \(x_{n}, x_{2n}\)
(E) \(x_{n}, x_{2n}, x_{1n} x_{2n}\)
4. Consider the simple regression

$$y_i = \alpha + \beta x_i + \varepsilon_i$$

And suppose population parameter $\beta = 2$ and least square estimator $\hat{\beta} = 2.3$. Now consider the regression

$$(x_i - y_i) = \alpha^* + \beta^* x_i + u_i.$$

What is the value of $\hat{\beta}^*$?

(A) 1
(B)-1
(C)-1.3
(D)-0.3
(E) 0.3
5. Consider the linear model with intercept:

$$y_t = \alpha + \beta x_t + u_t$$

If the $$A = \sum_{i=1}^{n} (x_i - \bar{x})^2$$ is positive definite and $$E(u_i \mid x_i) = 0, \forall t = 1, 2, \dots, T$$, which statement is true under sample size $$T \to \infty$$.

(A) \(\hat{\beta}\) is consistent, but we cannot judge whether \(\hat{\beta}\) is efficiency
(B) \(\hat{\beta}\) is unbiased and efficiency
(C) \(\hat{\beta}\) is consistent and efficiency
(D) \(\hat{\beta}\) is, but we cannot judge whether \(\hat{\beta}\) is efficiency
(E) \(\hat{\beta}\) is unbiased and consistent.
6. Consider the linear population model with intercept:

$$y_t = \alpha + \beta x_t + \varepsilon_t$$

The variance structure is known as below:

$$E(\varepsilon_t^2) = \sigma^2 h_t^2,$$

Which regression may obtain efficiency $$\hat{\beta}$$ estimator?

(A) Regress \(y_{t}\) on 1, \(x_{t}\)
(B) Regress \( y_{t} / h_{t} \) on \( y_{t} / h_{t}, x_{t} / h_{t} \)
(C) Regress \( y_{t} / h_{t}^{2} \) on \( y_{t} / h_{t}^{2}, x_{t} / h_{t}^{2} \)
(D) Regress \( y_{t} \) on \( y_{t} / h_{t}, x_{t} / h_{t} \)
(E) Regress \( y_{t} / h_{t} \) on 1, \( x_{t} \)

申論題 (11)

II、計算問答題(每大題 10 分，共 20 分。)

1. Mary has $$T = 40$$ observations on variables $$y$$, $$x_1$$, and $$x_2$$. She estimates the following equations by least squares:

$$y = -2.0 + 3.5x_1 - 1.5x_2, \tag{1}$$

(0.7) (0.3)

Residual sum of squares (SSR)=140, $$R^2 = 0.8$$ where the numbers in parentheses are standard errors. What is the true value of SST in equation (1)? What is the value of $$s^2$$ (estimated error variance.)?
2. Consider the simple regression model, assume to satisfy ideal conditions:

$$y_t = \beta x_t + \varepsilon_t \tag{2}$$

We now want to consider another estimator $$\tilde{\beta}$$, defined as follow. You pick two observations at random; say, $$t = 3$$ and $$t = 6$$.

$$\tilde{\beta} = (y_6 - y_3)/(x_6 - x_3) \tag{3}$$

(a). Consider the estimator \(\tilde{\beta}\). Whether is \(\tilde{\beta}\) unbiased?
(b). Compared with OLS estimator \(\hat{\beta}\), which estimators is efficiency? Please explain.
Ⅲ、填空題(每格 5 分，共 50 分。)

1. Let random variable $X$ follow a geometric distribution with parameter $p$, that is,

$$P(X=x) = pq^{n-1}, \quad p+q=1, \quad x=1, 2, 3, \dots$$

Calculate the probability of $P(X > m)$, where $m$ is a positive integer. (1)
2. Suppose $S_n = X_1 + X_2 + \dots + X_n$, where $E(X_i) = \frac{1}{n}$, $Var(X_i) = \frac{1}{n} - \frac{1}{n^2}$ and $Cov(X_i, X_j) = \frac{1}{n(n-1)} - \frac{1}{n^2}$ for $i \neq j$. Then, $Var(S_n) =$ (2).
3. Suppose that the probability density function of $X$ is $f(x) = 2x \exp(-x^2)$, $0 \le x < \infty$. Find the probability density function of $Y = X^2$, $f(y) =$ (3).
4. Suppose that the continuous random variables $(X, Y, Z)$ have a valid joint probability density function

$$f(x, y, z) = Kxyz^2, \quad 0 \le x, y \le 1, \quad 0 \le z \le 3.$$

Then, $E(Y) =$ (4) and $E(Y|X = \frac{1}{2}, Z = 1) =$ (5). Of course, you should find the value of $K$ first.
5. If random variables $X_1, X_2, \dots, X_n$ are i.i.d. $N(\mu, \sigma^2)$, and $S^2$ is their sample variance. Then, the distribution of $\frac{(n-1)S^2}{\sigma^2}$ is (6).
6. Suppose you play a game of chance where you win $w$ dollars with probability $p$ but lose $v$ dollars with probability $1-p$. Denote $X_i$ the random variable. Then, how large should $p$ be so that you, a reasonable person, are willing to play the game? (7)
7. Let $X_1, \dots, X_n, X_{n+1}$ be a random sample from a normal population having an unknown mean $\mu$ and variance 1, and $\bar{X}_n$ be the average of the first $n$ variables. Then, the distribution of $X_{n+1} - \bar{X}_n$ is (8). If $\bar{X}_{200} = 5$, then the 95% confidence interval for $X_{201}$ is (9).
8. In a multiple linear regression, a dependent variable is explained by a constant and 4 independent variables using 35 observations. The $R^2$ of this regression is 0.6. To test the significance of this regression, we can obtain the $F$ statistic equal to (10).