2.10(iii) From (2.57), Var(1ˆβ) = σ2/21()n i i x x =⎛⎫- ⎪⎝⎭∑. 由提示：： 21n i i x=∑ ≥21()n i i x x =-∑, and so Var(1β ) ≤ Var(1ˆβ). A more direct way to see this is to write(一个更直接的方式看到这是编写) 21()n ii x x =-∑ = 221()n i i x n x =-∑, which is less than21n i i x=∑unless x = 0.(iv)给定的c 2i x 但随着x 的增加， 1ˆβ的方差与Var(1β )的相关性也增加.0β小时1β 的偏差也小.因此, 在均方误差的基础上不管我们选择0β还是1β 要取决于0β,x ,和n 的大小 (除了 21n i i x=∑的大小).3.7We can use Table 3.2. By definition, 2β > 0, and by assumption, Corr(x 1,x 2) < 0. Therefore, there is anegative bias in 1β : E(1β ) < 1β. This means that, on average across different random samples, the simple regression estimator underestimates the effect of the training program. It is even possible that E(1β ) is negative even though 1β > 0. 我们可以使用表3.2。

根据定义，> 0，由假设，科尔（X1，X2）<0。

因此，有一个负偏压为：E （）<。

这意味着，平均在不同的随机抽样，简单的回归估计低估的培训计划的效果。

E （下），它甚至可能是负的，即使>0。

我们可以使用表格3.2。

根据定义,> 0,通过假设,柯尔(x1,x2)< 0。

因此,有一种负面的偏见:E()<。

这意味着,平均跨不同的随机样本,简单的回归估计低估了培训项目的效果。

甚至可能让E()是负的,尽管> 0。

3.8 Only (ii), omitting an important variable, can cause bias, and this is true only when the omitted variable is correlated with the included explanatory variables. The homoskedasticity assumption, MLR.5, played no role in showing that the OLS estimators are unbiased. (Homoskedasticity was used to obtain the usual varianceformulas for the ˆjβ.) Further, the degree of collinearity between the explanatory variables in the sample, even if it is reflected in a correlation as high as .95, does not affect the Gauss-Markov assumptions. Only if there is a perfect linear relationship among two or more explanatory variables is MLR.3 violated. 只有3.8(ii),遗漏重要变量,会造成偏见确实是这样,只有当省略变量就与包括解释变量。

homoskedasticity 的假设,多元线性回归。

5,没有发挥作用在显示OLS 估计量是公正的。

(Homoskedasticity 是用来获取通常的方差公式。

)进一步,共线的程度解释变量之间的样品中,即使它是反映在尽可能高的相关性。

95年,不影响的高斯-马尔可夫假定。

只要有一个完美的线性关系在两个或更多的解释变量是多元线性回归。

三违反了。

3.9 (i) Because 1x is highly correlated with 2x and 3x , and these latter variables have large partial effectson y , the simple and multiple regression coefficients on 1x can differ by large amounts. We have not done thiscase explicitly, but given equation (3.46) and the discussion with a single omitted variable, the intuition is pretty straightforward. 因为 是高度相关,和这些后面的变量有很大部分影响y,简单和多元回归系数的差异可大量。

我们还没有做到,这种情况下显式,但鉴于方程(3.46)和以讨论单个变量遗漏,直觉是相当简单的。

(ii) Here we wouldexpect 1β and 1ˆβ to be similar (subject, of course, to what we mean by “almost uncorrelated”). The amount of correlation between 2x and 3x does not directly effect the multiple regression estimate on 1x if 1x is essentially uncorrelated with 2x and 3x .这里我们将期待和相似(主题,当然对我们所说的“几乎不相关的”)。

相关性的数量,但不会直接影响了多元回归估计如果本质上是不相关的和。

(iii) (iii) In this case we are (unnecessarily) introducing multicollinearity into the regression: 2x and 3x have small partial effects on y and yet 2x and 3x are highly correlated with 1x . Adding 2x and 3x likeincreases the standard error of the coefficient on 1x substantially, so se(1ˆβ) is likely to be much larger than se(1β ).在这种情况下我们(不必要的)引入重合放入回归:,有微小的部分影响,但y,是高度相关的。

添加和像增加标准错误的系数显著,所以se()可能会远远大于se()。

(iv) In this case, adding 2x and 3x will decrease the residual variance without causing much collinearity(because 1x is almost uncorrelated with 2x and 3x ), so we should see se(1ˆβ) smaller than se(1β ). The amount of correlation between 2x and 3x does not directly affect se(1ˆβ).在这种情况下,添加和将减少剩余方差,也没有引起共线(因为几乎是不相关的,),所以我们应该看到se()小于se()。

相关性的数量,但不会直接影响se()。

3.11 (i)1β < 0 because more pollution can be expected to lower housing values; note that 1β is the elasticityof price with respect to nox .2β is probably positive because rooms roughly measures the size of a house.(However, it does not allow us to distinguish homes where each room is large from homes where each room is small.) < 0,因为更多的污染可以预期较低的房屋价值;注意,价格弹性对氮氧化物。

可能是积极的因为房间粗略地度量大小的房子。

(然而,不允许我们自己去辨别的家中,每个房间都是大从家中,每个房间小。

)(ii) If we assume that rooms increases with quality of the home, then log(nox ) and rooms are negatively correlated when poorer neighborhoods have more pollution, something that is often true. We can use Table 3.2to determine the direction of the bias. If 2β > 0 and Corr(x 1,x 2) < 0, the simple regression estimator 1β has a downward bias. But because 1β < 0, this means that the simple regression, on average, overstates theimportance of pollution. [E(1β ) is more negative than 1β.]如果我们假设房间随质量的家里,然后日志(nox)和房间反比当没那么富裕的社区有更多的污染,这往往是正确的。

我们可以使用表3.2来确定方向的偏见。

如果> 0和柯尔(x1,x2)< 0,那么简单的(iii) This is what we expect from the typical sample based on our analysis in part (ii). The simple regression estimate, -1.043, is more negative (larger in magnitude) than the multiple regression estimate, -0.718. As those estimates are only for one sample, we can never know which is closer to 1β. But if this is a“typical” sample,1βis closer to -0.718. 这是我们期待的东西从典型的示例基于我们的分析部分(ii)。