CHAPTER 1
SOLUTIONS TO PROBLEMS
1.1 (i) Ideally, we could randomly assign students to classes of different sizes. That is, each student is assigned a different class size without regard to any student characteristics such as ability and family background. For reasons we will see in Chapter 2, we would like substantial variation in class sizes (subject, of course, to ethical considerations and resource constraints).
(ii) A negative correlation means that larger class size is associated with lower performance. We might find a negative correlation because larger class size actually hurts performance. However, with observational data, there are other reasons we might find a negative relationship. For example, children from more affluent families might be more likely to attend schools with smaller class sizes, and affluent children generally score better on standardized tests. Another possibility is that, within a school, a principal might assign the better students to smaller classes. Or, some parents might insist their children are in the smaller classes, and these same parents tend to be more involved in their children’s education.
(iii) Given the potential for confounding factors – some of which are listed in (ii) – finding a negative correlation would not be strong evidence that smaller class sizes actually lead to better performance. Some way of controlling for the confounding factors is needed, and this is the subject of multiple regression analysis.
1.2 (i) Here is one way to pose the question: If two firms, say A and B, are identical in all respects except that firm A supplies job training one hour per worker more than firm B, by how much would firm A’s output differ from firm B’s?
(ii) Firms are likely to choose job training depending on the characteristics of workers. Some observed characteristics are years of schooling, years in the workforce, and experience in a particular job. Firms might even discriminate based on age, gender, or race. Perhaps firms choose to offer training to more or les s able workers, where “ability” might be difficult to quantify but where a manager has some idea about the relative abilities of different employees. Moreover, different kinds of workers might be attracted to firms that offer more job training on average, and this might not be evident to employers.
(iii) The amount of capital and technology available to workers would also affect output. So, two firms with exactly the same kinds of employees would generally have different outputs if they use different amounts of capital or technology. The quality of managers would also have an effect.
(iv) No, unless the amount of training is randomly assigned. The many factors listed in parts (ii) and (iii) can contribute to finding a positive correlation between output and training even if job training does not improve worker productivity.
1.3 It does not make sense to pose the question in terms of causality. Economists would assume that students choose a mix of studying and working (and other activities, such as attending class,
leisure, and sleeping) based on rational behavior, such as maximizing utility subject to the constraint that there are only 168 hours in a week. We can then use statistical methods to measure the association between studying and working, including regression analysis that we cover starting in Chapter 2. But we would not be claiming that one variable “causes” the other. They are both choice variables of the student.
CHAPTER 2
SOLUTIONS TO PROBLEMS
2.1 (i) Income, age, and family background (such as number of siblings) are just a few
possibilities. It seems that each of these could be correlated with years of education. (Income and education are probably positively correlated; age and education may be negatively correlated because women in more recent cohorts have, on average, more education; and number of siblings and education are probably negatively correlated.)
(ii) Not if the factors we listed in part (i) are correlated with educ . Because we would like to hold these factors fixed, they are part of the error term. But if u is correlated with educ then E(u|educ ) ≠ 0, and so SLR.4 fails.
2.2 In the equation y = β0 + β1x + u , add and subtract α0 from the right hand side to get y = (α0 + β0) + β1x + (u - α0). Call the new error e = u - α0, so that E(e ) = 0. The new intercept is α0 + β0, but the slope is still β1.
2.3 (i) Let y i = GPA i , x i = ACT i , and n = 8. Then x = 25.875, y =
3.2125, 1n
i =∑(x i – x )(y i – y ) =
5.8125, and 1
n
i =∑(x i – x )2 = 56.875. From equation (2.9), we obtain the slope as 1
?β= 5.8125/56.875 ≈ .1022, rounded to four places after the decimal. From (2.17), 0
?β = y – 1
?β
x ≈ 3.2125 – (.1022)25.875 ≈ .5681. So we can write GPA = .5681 + .1022 ACT
n = 8.
The intercept does not have a useful interpretation because ACT is not close to zero for the population of interest. If ACT is 5 points higher, GPA increases by .1022(5) = .511.
(ii) The fitted values and residuals — rounded to four decimal places — are given along with the observation number i and GPA in the following table:
You can verify that the residuals, as reported in the table, sum to -.0002, which is pretty close to zero given the inherent rounding error.
(iii) When ACT = 20, GPA = .5681 + .1022(20) ≈ 2.61.
(iv) The sum of squared residuals, 21
?n
i i u
=∑, is about .4347 (rounded to four decimal places), and the total sum of squares, 1
n
i =∑(y i – y )2, is about 1.0288. So the R -squared from the
regression is
R 2 = 1 – SSR/SST ≈ 1 – (.4347/1.0288) ≈ .577.
Therefore, about 57.7% of the variation in GPA is explained by ACT in this small sample of students.
2.4 (i) When cigs = 0, predicted birth weight is 119.77 ounces. When cigs = 20, bwght = 109.49. This is about an 8.6% drop.
(ii) Not necessarily. There are many other factors that can affect birth weight, particularly overall health of the mother and quality of prenatal care. These could be correlated with
cigarette smoking during birth. Also, something such as caffeine consumption can affect birth weight, and might also be correlated with cigarette smoking.
(iii) If we want a predicted bwght of 125, then cigs = (125 – 119.77)/( –.524) ≈–10.18, or about –10 cigarettes! This is nonsense, of course, and it shows what happens when we are trying to predict something as complicated as birth weight with only a single explanatory variable. The largest predicted birth weight is necessarily 119.77. Yet almost 700 of the births in the sample had a birth weight higher than 119.77.
(iv) 1,176 out of 1,388 women did not smoke while pregnant, or about 84.7%. Because we are using only cigs to explain birth weight, we have only one predicted birth weight at cigs = 0. The predicted birth weight is necessarily roughly in the middle of the observed birth weights at cigs = 0, and so we will under predict high birth rates.
2.5 (i) The intercept implies that when inc = 0, cons is predicted to be negative $124.84. This, of course, cannot be true, and reflects that fact that this consumption function might be a poor predictor of consumption at very low-income levels. On the other hand, on an annual basis, $124.84 is not so far from zero.
(ii) Just plug 30,000 into the equation: cons = –124.84 + .853(30,000) = 25,465.16 dollars.
(iii) The MPC and the APC are shown in the following graph. Even though the intercept is negative, the smallest APC in the sample is positive. The graph starts at an annual income level
increases housing prices.
(ii) If the city chose to locate the incinerator in an area away from more expensive neighborhoods, then log(dist) is positively correlated with housing quality. This would violate SLR.4, and OLS estimation is biased.
(iii) Size of the house, number of bathrooms, size of the lot, age of the home, and quality of the neighborhood (including school quality), are just a handful of factors. As mentioned in part (ii), these could certainly be correlated with dist [and log(dist)].
2.7 (i) When we condition on inc
becomes a constant. So E(u |inc
?e |inc
) = ?E(e |inc
?0 because E(e |inc ) = E(e ) = 0.
(ii) Again, when we condition on inc
becomes a constant. So Var(u |inc
?e |inc
2Var(e |inc ) = 2e σinc because Var(e |inc ) = 2e σ.
(iii) Families with low incomes do not have much discretion about spending; typically, a low-income family must spend on food, clothing, housing, and other necessities. Higher income people have more discretion, and some might choose more consumption while others more saving. This discretion suggests wider variability in saving among higher income families.
2.8 (i) From equation (2.66),
1β = 1n i i i x y =?? ???∑ / 21n i i x =?? ???
∑.
Plugging in y i = β0 + β1x i + u i gives
1β = 011()n i i i i x x u ββ=??++ ???∑/ 21n i i x =?? ???
∑.
After standard algebra, the numerator can be written as
2
0111
1
i
n n n
i i i i i i x x x u ββ===++∑∑∑.
Putting this over the denominator shows we can write 1β as
1β = β01n i i x =?? ???∑/ 21n i i x =?? ???
∑ + β1 + 1n i i i x u =?? ???∑/ 21n i i x =??
???∑.
Conditional on the x i , we have
E(1β) = β01n i i x =?? ???∑/ 21n i i x =??
???
∑ + β1
because E(u i ) = 0 for all i . Therefore, the bias in 1β is given by the first term in this equation. This bias is obviously zero when β0 = 0. It is also zero when 1n
i i x =∑ = 0, which is the same as
x = 0. In the latter case, regression through the origin is identical to regression with an intercept.
(ii) From the last expression for 1βin part (i) we have, conditional on the x i ,
Var(1β)
= 221n i i x -=?? ???∑Var 1n i i i x u =?? ???∑ = 2
21n i i x -=?? ???∑21Var()n i i i x u =?? ???
∑
= 2
21n i i x -=?? ???∑221n i i x σ=?? ???∑ = 2σ/ 21n
i i x =?? ???
∑.
(iii) From (2.57), Var(1?β) = σ2/21()n i i x x =??- ???
∑. From the hint, 21n i i x =∑ ≥ 21()n i i x x =-∑, and so
Var(1β) ≤ Var(1
?β). A more direct way to see this is to write 21
()n
i i x x =-∑ = 221
()n
i i x n x =-∑, which is less than 21
n
i i x =∑ unless x = 0.
(iv) For a given sample size, the bias in 1β increases as x increases (holding the sum of the
2i x fixed). But as x increases, the variance of 1
?β
increases relative to Var(1
β). The bias in 1
β is also small when 0β is small. Therefore, whether we prefer 1β or 1?β on a mean squared error basis depends on the sizes of 0β, x , and n (in addition to the size of 21
n
i i x =∑).
2.9 (i) We follow the hint, noting that 1c y = 1c y (the sample average of 1i c y is c 1 times the sample average of y i ) and 2c x = 2c x . When we regress c 1y i on c 2x i (including an intercept) we use equation (2.19) to obtain the slope:
22
11
121
1
12
22
22
211
1
1112
2
2
1
()()()()
()
()()()?.()
n n
i i
i
i
i i n
n
i
i i i n
i
i
i n
i
i c x c x c y c y c c x x y y c x c x c
x x x x y y c c c c x x ββ======----=
=
----=?=
-∑∑∑∑∑∑
From (2.17), we obtain the intercept as 0β = (c 1y ) – 1β(c 2x ) = (c 1y ) – [(c 1/c 2)1
?β](c 2x ) = c 1(y – 1
?β
x ) = c 10
?β) because the intercept from regressing y i on x i is (y – 1
?βx ).
(ii) We use the same approach from part (i) along with the fact that 1()c y + = c 1 + y and
2()c x + = c 2 + x . Therefore, 11()()i c y c y +-+ = (c 1 + y i ) – (c 1 + y ) = y i – y and (c 2 + x i ) –
2()c x + = x i – x . So c 1 and c 2 entirely drop out of the slope formula for the regression of (c 1 + y i )
on (c 2 + x i ), and 1
β = 1
?β
. The intercept is 0
β = 1
()c y + – 1
β2
()c x + = (c 1 + y ) – 1
?β(c 2 + x ) = (1?y x β-) + c 1 – c 21?β = 0?β + c 1 – c 21
?β, which is what we wanted to show.
(iii) We can simply apply part (ii) because 11log()log()log()i i c y c y =+. In other words, replace c 1 with log(c 1), y i with log(y i ), and set c 2 = 0.
(iv) Again, we can apply part (ii) with c 1 = 0 and replacing c 2 with log(c 2) and x i with log(x i ). If 01?? and ββ are the original intercept and slope, then 11?ββ= and 0021
??log()c βββ=-.
2.10 (i) This derivation is essentially done in equation (2.52), once (1/SST )x is brought inside the summation (which is valid because SST x does not depend on i ). Then, just define /SST i i x w d =.
(ii) Because 111
??Cov(,)E[()] ,u u βββ=- we show that the latter is zero. But, from part (i), ()
1111?E[()] =E E().n
n i i i i i i u wu u w u u ββ==??-=???
?∑∑ Because the i u are pairwise uncorrelated (they are independent), 22E()E(/)/i i u u u n n σ== (because E()0, i h u u i h =≠). Therefore,
22111E()(/)(/)0.n n n
i i i i i i i w u u w n n w σσ======∑∑∑
(iii) The formula for the OLS intercept is 0
??y x ββ=- and, plugging in 01y x u ββ=++ gives 0
1
1
1
1
???()().x u x u x β
ββββββ=++-=+--
(iv) Because 1
? and u β are uncorrelated, 222222201??Var()Var()Var()/(/SST )//SST x x
u x n x n x ββσσσσ=+=+=+, which is what we wanted to show.
(v) Using the hint and substitution gives ()220?Var()[SST /]/SST x x
n x βσ=+ ()()
21222212
11/SST /SST .n
n
i x i x i i n x x x n x σσ--==??=-+=????∑∑
2.11 (i) We would want to randomly assign the number of hours in the preparation course so that hours is independent of other factors that affect performance on the SAT. Then, we would collect information on SAT score for each student in the experiment, yielding a data set
{(,):1,...,}i i sat hours i n =, where n is the number of students we can afford to have in the study. From equation (2.7), we should try to get as much variation in i hours as is feasible.
(ii) Here are three factors: innate ability, family income, and general health on the day of the exam. If we think students with higher native intelligence think they do not need to prepare for the SAT, then ability and hours will be negatively correlated. Family income would probably be positively correlated with hours , because higher income families can more easily afford
preparation courses. Ruling out chronic health problems, health on the day of the exam should be roughly uncorrelated with hours spent in a preparation course.
(iii) If preparation courses are effective,1β should be positive: other factors equal, an increase in hours should increase sat .
(iv) The intercept, 0β, has a useful interpretation in this example: because E(u ) = 0, 0β is the average SAT score for students in the population with hours = 0.
CHAPTER 3
SOLUTIONS TO PROBLEMS
3.1 (i) hsperc is defined so that the smaller it is, the lower the student’s standing in high school. Everything else equal, the worse the student’s standing in high school , the lower is his/her expected college GPA. (ii) Just plug these values into the equation:
colgpa = 1.392 - .0135(20) + .00148(1050) = 2.676.
(iii) The difference between A and B is simply 140 times the coefficient on sat , because hsperc is the same for both students. So A is predicted to have a score .00148(140) ≈ .207 higher. (iv) With hsperc fixed, colgpa ? = .00148?sat . Now, we want to find ?sat such that
colgpa ? = .5, so .5 = .00148(?sat ) or ?sat = .5/(.00148) ≈ 338. Perhaps not surprisingly, a large ceteris paribus difference in SAT score – almost two and one-half standard deviations – is needed to obtain a predicted difference in college GPA or a half a point.
3.2 (i) Yes. Because of budget constraints, it makes sense that, the more siblings there are in a family, the less education any one child in the family has. To find the increase in the number of siblings that reduces predicted education by one year, we solve 1 = .094(?sibs ), so ?sibs = 1/.094 ≈ 10.6. (ii) Holding sibs and feduc fixed, one more year of mother’s education implies .131 years more of predicted education. So if a mother has four more years of education, her son is predicted to have about a half a year (.524) more years of education.
(iii) Since the number of siblings is the same, but meduc and feduc are both different, the coefficients on meduc and feduc both need to be accounted for. The predicted difference in education between B and A is .131(4) + .210(4) = 1.364.
3.3 (i) If adults trade off sleep for work, more work implies less sleep (other things equal), so 1β < 0. (ii) The signs of 2β and 3β are not obvious, at least to me. One could argue that more educated people like to get more out of life, and so, other things equal, they sleep less (2β < 0). The relationship between sleeping and age is more complicated than this model suggests, and economists are not in the best position to judge such things. (iii) Since totwrk is in minutes, we must convert five hours into minutes: ?totwrk = 5(60) = 300. Then sleep is predicted to fall by .148(300) = 4
4.4 minutes. For a week, 45 minutes less sleep is not an overwhelming change. (iv) More education implies less predicted time sleeping, but the effect is quite small. If we assume the difference between college and high school is four years, the college graduate sleeps about 45 minutes less per week, other things equal. (v) Not surprisingly, the three explanatory variables explain only about 11.3% of the variation in sleep . One important factor in the error term is general health. Another is marital status, and whether the person has children. Health (however we measure that), marital status, and number and ages of children would generally be correlated with totwrk . (For example, less healthy people would tend to work less.)
3.4 (i) A larger rank for a law school means that the school has less prestige; this lowers starting salaries. For example, a rank of 100 means there are 99 schools thought to be better. (ii) 1β > 0, 2β > 0. Both LSAT and GPA are measures of the quality of the entering class. No matter where better students attend law school, we expect them to earn more, on average. 3β, 4β > 0. The number of volumes in the law library and the tuition cost are both measures of the school quality. (Cost is less obvious than library volumes, but should reflect quality of the faculty, physical plant, and so on.) (iii) This is just the coefficient on GPA , multiplied by 100: 2
4.8%. (iv) This is an elasticity: a one percent increase in library volumes implies a .095% increase in predicted median starting salary, other things equal. (v) It is definitely better to attend a law school with a lower rank. If law school A has a ranking 20 less than law school B, the predicted difference in starting salary is 100(.0033)(20) = 6.6% higher for law school A.
3.5 (i) No. By definition, study + sleep + work + leisure = 168. Therefore, if we change study , we must change at least one of the other categories so that the sum is still 168. (ii) From part (i), we can write, say, study as a perfect linear function of the other
independent variables: study = 168 - sleep - work - leisure . This holds for every observation, so MLR.3 violated. (iii) Simply drop one of the independent variables, say leisure :
GPA = 0β + 1βstudy + 2βsleep + 3βwork + u .
Now, for example, 1β is interpreted as the change in GPA when study increases by one hour, where sleep , work , and u are all held fixed. If we are holding sleep and work fixed but increasing study by one hour, then we must be reducing leisure by one hour. The other slope parameters have a similar interpretation.
3.6 Conditioning on the outcomes of the explanatory variables, we have 1E()θ = E(1
?β + 2?β) = E(1
?β
) + E(2
?β) = β1 + β2 = 1
θ.
3.7 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 variance formulas for the ?j
β.) 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.
3.8 We can use Table 3.2. By definition, 2β > 0, and by assumption, Corr(x 1,x 2) < 0. Therefore, there is a negative 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.
3.9 (i) 1β < 0 because more pollution can be expected to lower housing values; note that 1β is the elasticity of 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.) (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 3.2 to 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 the importance of pollution. [E(1β) is more negative than 1β.] (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, -.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, 1β is closer to -.718.
3.10 (i) Because 1x is highly correlated with 2x and 3x , and these latter variables have large partial effects on y , the simple and multiple regression coefficients on 1x can differ by large amounts. We have not done this case explicitly, but given equation (3.46) and the discussion with a single omitted variable, the intuition is pretty straightforward.
(ii) Here we would expect 1β and 1
?β to be similar (subject, of course, to what we mean by “almost uncorrelated”). The amount of correlation bet ween 2x and 3x does not directly effect the multiple regression estimate on 1x if 1x is essentially uncorrelated with 2x and 3x . (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
2
x and 3x like increases the standard error of the coefficient on 1x substantially, so se(1
?β
) is likely to be much larger than se(1β).
(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(1?β) smaller than se(1β). The amount of correlation between 2x and 3x does not directly affect
se(1
?β
).
3.11 From equation (3.22) we have
11121
1
?,?
n
i i
i n
i i r y
r β===
∑∑
where the 1?i r
are defined in the problem. As usual, we must plug in the true model for y i :
10
1122331
121
1
?(.?
n
i i i i i
i n
i i r x x x u r β
ββββ==++++=
∑∑
The numerator of this expression simplifies because 11
?n
i i r
=∑ = 0, 121
?n
i i i r x =∑ = 0, and 111
?n
i i i r x =∑ = 2
11
?
n
i i r =∑. These all follow from the fact that the 1?i r
are the residuals from the regression of 1i x on 2i x : the 1?i r
have zero sample average and are uncorrelated in sample with 2i x . So the numerator of 1β can be expressed as
21131311
1
1
???.n
n
n
i i i i i i i i r
r x r u ββ===++∑∑∑
Putting these back over the denominator gives
13
1111132
21
1
1
1
??.?
?
n
n
i i i
i i n
n
i i i i r x ru
r r βββ=====++
∑∑∑∑
Conditional on all sample values on x 1, x 2, and x 3, only the last term is random due to its dependence on u i . But E(u i ) = 0, and so
13
11132
1
1
?E()=+,?
n
i i i n
i i r x r βββ
==∑∑
which is what we wanted to show. Notice that the term multiplying 3β is the regression
coefficient from the simple regression of x i 3 on 1?i r
.
3.12 (i) The shares, by definition, add to one. If we do not omit one of the shares then the equation would suffer from perfect multicollinearity. The parameters would not have a ceteris paribus interpretation, as it is impossible to change one share while holding all of the other shares fixed. (ii) Because each share is a proportion (and can be at most one, when all other shares are zero), it makes little sense to increase share p by one unit. If share p increases by .01 – which is equivalent to a one percentage point increase in the share of property taxes in total revenue –
holding share I , share S , and the other factors fixed, then growth increases by 1β(.01). With the other shares fixed, the excluded share, share F , must fall by .01 when share p increases by .01.
3.13 (i) For notational simplicity, define s zx = 1();n
i i i z z x =-∑ this is not quite the sample
covariance between z and x because we do not divide by n – 1, but we are only using it to simplify notation. Then we can write 1β as
1
1().n
i
i
i zx
z z y
s β=-=
∑
This is clearly a linear function of the y i : take the weights to be w i = (z i -z )/s zx . To show unbiasedness, as usual we plug y i = 0β + 1βx i + u i into this equation, and simplify:
11
1011
1
1
1()()
()()()n
i
i i i zx
n
n
i zx i i
i i zx
n
i
i
i zx
z z x u s z z s z z u s z z u
s β
βββββ====-++=
-++-=
-=+
∑∑∑∑
where we use the fact that 1
()n
i i z z =-∑ = 0 always. Now s zx is a function of the z i and x i and the
expected value of each u i is zero conditional on all z i and x i in the sample. Therefore, conditional on these values,
1
111()E()
E()n
i
i
i zx
z z u s βββ=-=+
=∑
because E(u i ) = 0 for all i . (ii) From the fourth equation in part (i) we have (again conditional on the z i and x i in the sample),
21
1
1222
2
1
2()()()
()()
n n
i
i
i
i i i zx
zx
n
i
i zx
z z u z
z Var u Var Var
s
s z z s βσ===--==
-=∑∑∑
because of the homoskedasticity assumption [Var(u i ) = σ2 for all i ]. Given the definition of s zx , this is what we wanted to show.
(iii) We know that Var(1
?β) = σ2/21
[()].n
i i x x =-∑ Now we can rearrange the inequality in the hint, drop x from the sample covariance, and cancel n -1
everywhere, to get 22
1
[()]/n
i zx i z z s =-∑ ≥
21
1/[()].n
i i x x =-∑ When we multiply through by σ2 we get Var(1β) ≥ Var(1
?β), which is what we wanted to show.
CHAPTER 4
4.1 (i) and (iii) generally cause the t statistics not to have a t distribution under H 0.
Homoskedasticity is one of the CLM assumptions. An important omitted variable violates Assumption MLR.3. The CLM assumptions contain no mention of the sample correlations among independent variables, except to rule out the case where the correlation is one.
4.2 (i) H 0:3β = 0. H 1:3β > 0. (ii) The proportionate effect on salary is .00024(50) = .012. To obtain the percentage effect, we multiply this by 100: 1.2%. Therefore, a 50 point ceteris paribus increase in ros is predicted to increase salary by only 1.2%. Practically speaking, this is a very small effect for such a large change in ros . (iii) The 10% critical value for a one-tailed test, using df = ∞, is obtained from Table G.2 as 1.282. The t statistic on ros is .00024/.00054 ≈ .44, which is well below the critical value. Therefore, we fail to reject H 0 at the 10% significance level. (iv) Based on this sample, the estimated ros coefficient appears to be different from zero only because of sampling variation. On the other hand, including ros may not be causing any harm; it depends on how correlated it is with the other independent variables (although these are very significant even with ros in the equation).
4.3 (i) Holding profmarg fixed, rdintens ? = .321 ?log(sales ) =
(.321/100)[100log()sales ??] ≈ .00321(%?sales ). Therefore, if %?sales = 10,
rdintens ? ≈ .032, or only about 3/100 of a percentage point. For such a large percentage increase in sales, this seems like a practically small effect. (ii) H 0:1β = 0 versus H 1:1β > 0, where 1β is the population slope on log(sales ). The t statistic is .321/.216 ≈ 1.486. The 5% critical value for a one-tailed test, with df = 32 – 3 = 29, is obtained from Table G.2 as 1.699; so we cannot reject H 0 at the 5% level. But the 10% critical value is 1.311; since the t statistic is above this value, we reject H 0 in favor of H 1 at the 10% level. (iii) Not really. Its t statistic is only 1.087, which is well below even the 10% critical value for a one-tailed test.
4.4 (i) H 0:3β = 0. H 1:3β ≠ 0. (ii) Other things equal, a larger population increases the demand for rental housing, which should increase rents. The demand for overall housing is higher when average income is higher, pushing up the cost of housing, including rental rates. (iii) The coefficient on log(pop ) is an elasticity. A correct statement is that “a 10% increase in population increases rent by .066(10) = .66%.” (iv) With df = 64 – 4 = 60, the 1% critical value for a two-tailed test is 2.660. The t
statistic is about 3.29, which is well above the critical value. So 3β is statistically different from zero at the 1% level.
4.5 (i) .412 ± 1.96(.094), or about .228 to .596. (ii) No, because the value .4 is well inside the 95% CI. (iii) Yes, because 1 is well outside the 95% CI.
4.6 (i) With df = n – 2 = 86, we obtain the 5% critical value from Table G.2 with df = 90. Because each test is two-tailed, the critical value is 1.987. The t statistic for H 0:0β = 0 is about -
.89, which is much less than 1.987 in absolute value. Therefore, we fail to reject 0β = 0. The t statistic for H 0: 1β = 1 is (.976 – 1)/.049 ≈ -.49, which is even less significant. (Remember, we reject H 0 in favor of H 1 in this case only if |t | > 1.987.) (ii) We use the SSR form of the F statistic. We are testing q = 2 restrictions and the df in the unrestricted model is 86. We are given SSR r = 209,448.99 and SSR ur = 165,644.51. Therefore,
(209,448.99165,644.51)8611.37,165,644.512F -??
=
?≈ ???
which is a strong rejection of H 0: from Table G.3c, the 1% critical value with 2 and 90 df is 4.85. (iii) We use the R -squared form of the F statistic. We are testing q = 3 restrictions and there are 88 – 5 = 83 df in the unrestricted model. The F statistic is [(.829 – .820)/(1 –
.829)](83/3) ≈ 1.46. The 10% critical value (again using 90 denominator df in Table G.3a) is 2.15, so we fail to reject H 0 at even the 10% level. In fact, the p -value is about .23. (iv) If heteroskedasticity were present, Assumption MLR.5 would be violated, and the F statistic would not have an F distribution under the null hypothesis. Therefore, comparing the F statistic against the usual critical values, or obtaining the p -value from the F distribution, would not be especially meaningful.
4.7 (i) While the standard error on hrsemp has not changed, the magnitude of the coefficient has increased by half. The t statistic on hrsemp has gone from about –1.47 to –2.21, so now the coefficient is statistically less than zero at the 5% level. (From Table G.2 the 5% critical value with 40 df is –1.684. The 1% critical value is –2.423, so the p -value is between .01 and .0
5.) (ii) If we add and subtract 2βlog(employ ) from the right-hand-side and collect terms, we have
log(scrap ) = 0β + 1βhrsemp + [2βlog(sales) – 2βlog(employ )] + [2βlog(employ ) + 3βlog(employ )] + u
= 0β + 1βhrsemp + 2βlog(sales /employ )
+ (2β + 3β)log(employ ) + u ,
where the second equality follows from the fact that log(sales /employ ) = log(sales ) – log(employ ). Defining 3θ ≡ 2β + 3β gives the result. (iii) No. We are interested in the coefficient on log(employ ), which has a t statistic of .2, which is very small. Therefore, we conclude that the size of the firm, as measured by employees, does not matter, once we control for training and sales per employee (in a logarithmic functional form). (iv) The null hypothesis in the model from part (ii) is H 0:2β = –1. The t statistic is [–.951 – (–1)]/.37 = (1 – .951)/.37 ≈ .132; this is very small, and we fail to reject whether we specify a one- or two-sided alternative.
4.8 (i) We use Property VAR.3 from Appendix B: Var(1?β - 32?β) = Var (1
?β) + 9 Var (2
?β
) – 6 Cov (1
?β,2
?β).
(ii) t = (1?β- 32?β - 1)/se(1?β- 32?β), so we need the standard error of 1
?β - 32?β. (iii) Because 1θ = 1β – 3β2, we can write 1β = 1θ + 3β2. Plugging this into the population model gives y = 0β + (1θ + 3β2)x 1 + 2βx 2 + 3βx 3 + u = 0β + 1θx 1 + 2β(3x 1 + x 2) + 3βx 3 + u .
This last equation is what we would estimate by regressing y on x 1, 3x 1 + x 2, and x 3. The coefficient and standard error on x 1 are what we want.
4.9 (i) With df = 706 – 4 = 702, we use the standard normal critical value (df = ∞ in Table G.2), which is 1.96 for a two-tailed test at the 5% level. Now t educ = -11.13/
5.88 ≈ -1.89, so |t educ | = 1.89 < 1.96, and we fail to reject H 0: educ β = 0 at the 5% level. Also, t age ≈ 1.52, so age is also statistically insignificant at the 5% level. (ii) We need to compute the R -squared form of the F statistic for joint significance. But F = [(.113 - .103)/(1 - .113)](702/2) ≈ 3.9
6. The 5% critical value in the F 2,702 distribution can be obtained from Table G.3b with denominator df = ∞: cv = 3.00. Therefore, educ and age are jointly significant at the 5% level (3.96 > 3.00). In fact, the p -value is about .019, and so educ and age are jointly significant at the 2% level. (iii) Not really. These variables are jointly significant, but including them only changes the coefficient on totwrk from –.151 to –.148. (iv) The standard t and F statistics that we used assume homoskedasticity, in addition to the other CLM assumptions. If there is heteroskedasticity in the equation, the tests are no longer valid.
4.10 (i) We need to compute the F statistic for the overall significance of the regression with n = 142 and k = 4: F = [.0395/(1 – .0395)](137/4) ≈ 1.41. The 5% critical value with 4
numerator df and using 120 for the numerator df , is 2.45, which is well above the value of F . Therefore, we fail to reject H 0: 1β = 2β = 3β = 4β = 0 at the 10% level. No explanatory
variable is individually significant at the 5% level. The largest absolute t statistic is on dkr , t dkr ≈ 1.60, which is not significant at the 5% level against a two-sided alternative. (ii) The F statistic (with the same df ) is now [.0330/(1 – .0330)](137/4) ≈ 1.17, which is even lower than in part (i). None of the t statistics is significant at a reasonable level.
(iii) It seems very weak. There are no significant t statistics at the 5% level (against a two-sided alternative), and the F statistics are insignificant in both cases. Plus, less than 4% of the variation in return is explained by the independent variables.
4.11 (i) In columns (2) and (3), the coefficient on profmarg is actually negative, although its t statistic is only about –1. It appears that, once firm sales and market value have been controlled for, profit margin has no effect on CEO salary. (ii) We use column (3), which controls for the most factors affecting salary. The t
statistic on log(mktval ) is about 2.05, which is just significant at the 5% level against a two-sided alternative. (We can use the standard normal critical value, 1.96.) So log(mktval ) is statistically significant. Because the coefficient is an elasticity, a ceteris paribus 10% increase in market value is predicted to increase salary by 1%. This is not a huge effect, but it is not negligible, either. (iii) These variables are individually significant at low significance levels, with t ceoten ≈ 3.11 and t comten ≈ –2.79. Other factors fixed, another year as CEO with the company increases salary by about 1.71%. On the other hand, another year with the company, but not as CEO, lowers salary by about .92%. This second finding at first seems surprising, but could be related to the “superstar” effect: firms that hire CEOs from outside the company often go after a small pool of highly regarded candidates, and salaries of these people are bid up. More non-CEO years with a company makes it less likely the person was hired as an outside superstar.
CHAPTER 5
5.1 Write y = 0β + 1βx 1 + u , and take the expected value: E(y ) = 0β + 1βE(x 1) + E(u ), or μy = 0β + 1βμx since E(u ) = 0, where μy = E(y ) and μx = E(x 1). We can rewrite this as 0β = μy -
1
βμx . Now, 0?β
= y - 1?β1x . Taking the plim of this we have plim(0?β) = plim(y - 1?β1
x ) = plim(y ) – plim(1
?β)?plim(1x ) = μy - 1βμx , where we use the fact that plim(y ) = μy and plim(1
x ) = μx by the law of large numbers, and plim(1?β
) = 1
β. We have also used the parts of Property PLIM.2 from Appendix C.
5.2 A higher tolerance of risk means more willingness to invest in the stock market, so 2β > 0. By assumption, funds and risktol are positively correlated. Now we use equation (5.5), where δ1 > 0: plim(1β) = 1β + 2βδ1 > 1β, so 1β has a positive inconsistency (asymptotic bias). This makes sense: if we omit risktol from the regression and it is positively correlated with funds , some of the estimated effect of funds is actually due to the effect of risktol .
5.3 The variable cigs has nothing close to a normal distribution in the population. Most people do not smoke, so cigs = 0 for over half of the population. A normally distributed random
variable takes on no particular value with positive probability. Further, the distribution of cigs is skewed, whereas a normal random variable must be symmetric about its mean.
5.4 Write y = 0β + 1βx + u , and take the expected value: E(y ) = 0β + 1βE(x ) + E(u ), or μy =
0β + 1βμx , since E(u ) = 0, where μy = E(y ) and μx = E(x ). We can rewrite this as 0β = μy -
1βμx . Now, 0β = y - 1βx . Taking the plim of this we have plim(0β) = plim(y - 1βx ) = plim(y ) – plim(1β)?plim(x ) = μy - 1βμx , where we use the fact that plim(y ) = μy and plim(x ) = μx by the law of large numbers, and plim(1β) = 1β. We have also used the parts of the Property PLIM.2 from Appendix C.
CHAPTER 6
6.1 The generality is not necessary. The t statistic on roe 2 is only about -.30, which shows that roe 2 is very statistically insignificant. Plus, having the squared term has only a minor effect on the slope even for large values of roe . (The approximate slope is .0215 - .00016 roe , and even when roe = 25 – about one standard deviation above the average roe in the sample – the slope is .211, as compared with .215 at roe = 0.)
6.2 By definition of the OLS regression of c 0y i on c 1x i 1, , c k x ik , i = 2, , n , the j β solve
00
111111
001111001111
[()()()]0
()[()()()]0
()[()()...()]0.
n
i
i k k ik i n
i i i k k ik i n
k ik
i i k k ik i c y c x c x c x
c y c x c x c x
c y c x c x β
ββββββββ===----=---
-=----=∑∑
∑
[We obtain these from equations (3.13), where we plug in the scaled dependent and independent
variables.] We now show that if 0β = 00
?c β and j β = 0(/)j j c c β, j = 1,…,k , then these k + 1 first order conditions are satisfied, which proves the result because we know that the OLS estimates are the unique solutions to the FOCs (once we rule out perfect collinearity in the independent variables). Plugging in these guesses for the j β gives the expressions
00
011110100
0111101
?
??[()(/)()...(/)()]?
??()[()(/)()...(/)()],n
i
i k k k ik
i n
j ij
i
i k k k ik
i c y c c c c x c c c x c x c y c c c c x c c c x ββββββ==--------∑∑
for j = 1,2,…,k . Simple cancellation shows we can write these equations as
01101
?
??[()]n
i
i k ik
i c y c c x c x βββ=----∑ and
00
01101
?
??()[()...],1,2,...,n
j ij
i
i k ik i c x c y c c x c x j k βββ=----=∑
or, factoring out constants,
0011
1
???()n
i i k ik i c y x x βββ=??---- ?
??
∑ and
0011
1???()n
j i i k ik i c c y x x βββ=??---- ?
??
∑, j = 1, 2,
But the terms multiplying c 0 and c 0c j are identically zero by the first order conditions for the ?j
β since, by definition, they are obtained from the regression y i on x i 1, , x ik , i = 1,2,..,n . So we
have shown that 0β = c 00
?β and j β = (c 0/c j ) ?j β, j = 1, , k solve the requisite first order conditions.
6.3 (i) The turnaround point is given by 1
?β/(2|2?β|), or .0003/(.000000014) ≈ 21,428.57; remember, this is sales in millions of dollars. (ii) Probably. Its t statistic is about –1.89, which is significant against the one-sided alternative H 0: 1β < 0 at the 5% level (cv ≈ –1.70 with df = 29). In fact, the p -value is about .036. (iii) Because sales gets divided by 1,000 to obtain salesbil , the corresponding coefficient gets multiplied by 1,000: (1,000)(.00030) = .30. The standard error gets multiplied by the same factor. As stated in the hint, salesbil 2 = sales /1,000,000, and so the coefficient on the quadratic gets multiplied by one million: (1,000,000)(.0000000070) = .0070; its standard error also gets multiplied by one million. Nothing happens to the intercept (because rdintens has not been rescaled) or to the R 2:
rdintens = 2.613 + .30 salesbil – .0070 salesbil 2 (0.429) (.14) (.0037) n = 32, R 2 = .1484. (iv) The equation in part (iii) is easier to read because it contains fewer zeros to the right of the decimal. Of course the interpretation of the two equations is identical once the different scales are accounted for.
计量经济学题库 一、单项选择题（每小题1分） 1．计量经济学是下列哪门学科的分支学科（C）。 A．统计学 B．数学 C．经济学 D．数理统计学 2．计量经济学成为一门独立学科的标志是（B）。 A．1930年世界计量经济学会成立B．1933年《计量经济学》会刊出版 C．1969年诺贝尔经济学奖设立 D．1926年计量经济学（Economics）一词构造出来 3．外生变量和滞后变量统称为（D）。 A．控制变量 B．解释变量 C．被解释变量 D．前定变量4．横截面数据是指（A）。 A．同一时点上不同统计单位相同统计指标组成的数据B．同一时点上相同统计单位相同统计指标组成的数据 C．同一时点上相同统计单位不同统计指标组成的数据D．同一时点上不同统计单位不同统计指标组成的数据 5．同一统计指标，同一统计单位按时间顺序记录形成的数据列是（C）。 A．时期数据 B．混合数据 C．时间序列数据 D．横截面数据6．在计量经济模型中，由模型系统内部因素决定，表现为具有一定的概率分布的随机变量，其数值受模型中其他变量影响的变量是（ A ）。 A．内生变量 B．外生变量 C．滞后变量 D．前定变量7．描述微观主体经济活动中的变量关系的计量经济模型是（ A ）。 A．微观计量经济模型 B．宏观计量经济模型 C．理论计量经济模型 D．应用计量经济模型 8．经济计量模型的被解释变量一定是（ C ）。 A．控制变量 B．政策变量 C．内生变量 D．外生变量9．下面属于横截面数据的是（ D ）。 A．1991－2003年各年某地区20个乡镇企业的平均工业产值 B．1991－2003年各年某地区20个乡镇企业各镇的工业产值 C．某年某地区20个乡镇工业产值的合计数 D．某年某地区20个乡镇各镇的工业产值 10．经济计量分析工作的基本步骤是（ A ）。 A．设定理论模型→收集样本资料→估计模型参数→检验模型B．设定模型→估计参数→检验模型→应用
第二章简单线性回归模型 2.1 （1）①首先分析人均寿命与人均GDP的数量关系，用Eviews分析: Dependent Variable: Y Method: Least Squares Date: 12/23/15 Time: 14:37 Sam pie: 1 22 In eluded observatio ns: 22 Variable Coeffieie nt Std. Error t-Statistie P rob. C 56.64794 1.960820 28.88992 0.0000 X1 0.128360 0.027242 4.711834 0.0001 R-squared 0.526082 Mean dependent var 62.50000 Adjusted R-squared 0.502386 S.D.dependent var 10.08889 S.E. of regressi on 7.116881 Akaike info eriteri on 6.849324 Sum squared resid 1013.000 Sehwarz eriteri on 6.948510 Log likelihood -73.34257 Hannan-Quinn eriter. 6.872689 F-statistie 22.20138 Durbin-Wats on stat 0.629074 P rob(F-statistie) 0.000134 有上可知，关系式为y=56.64794+0.128360x i ②关于人均寿命与成人识字率的关系，用 Eviews分析如下: Dependent Variable: Y Method: Least Squares Date: 12/23/15 Time: 15:01 Sam pie: 1 22 In eluded observatio ns: 22 Variable Coeffieie nt Std. Error t-Statistie P rob. C 38.79424 3.532079 10.98340 0.0000 X2 0.331971 0.046656 7.115308 0.0000 R-squared 0.716825 Mean dependent var 62.50000 Adjusted R-squared 0.702666 S.D.dependentvar 10.08889 S.E. of regressi on 5.501306 Akaike info eriterio n 6.334356 Sum squared resid 605.2873 Sehwarz eriterio n 6.433542 Log likelihood -67.67792 Hannan-Quinn eriter. 6.357721 F-statistie 50.62761 Durbin-Wats on stat 1.846406 P rob(F-statistie) 0.000001 由上可知，关系式为y=38.79424+0.331971X2 ③关于人均寿命与一岁儿童疫苗接种率的关系，用Eviews分析如下:
计量经济学题库 计算与分析题（每小题10分） 1 X:年均汇率（日元/美元） Y:汽车出口数量（万辆） 问题：（1）画出X 与Y 关系的散点图。 （2）计算X 与Y 的相关系数。其中X 129.3＝，Y 554.2＝，2X X 4432.1∑（－）＝，2Y Y 68113.6∑ （－）＝，()()X X Y Y ∑－－＝16195.4 （3）采用直线回归方程拟和出的模型为 ?81.72 3.65Y X =+ t 值 1.2427 7.2797 R 2=0.8688 F=52.99 解释参数的经济意义。 2．已知一模型的最小二乘的回归结果如下： i i ?Y =101.4-4.78X 标准差 （45.2） （1.53） n=30 R 2=0.31 其中，Y ：政府债券价格（百美元），X ：利率（%）。 回答以下问题：（1）系数的符号是否正确，并说明理由；（2）为什么左边是i ?Y 而不是i Y ； （3）在此模型中是否漏了误差项i u ；（4）该模型参数的经济意义是什么。 3．估计消费函数模型i i i C =Y u αβ++得 i i ?C =150.81Y + t 值 （13.1）（18.7） n=19 R 2=0.81 其中，C ：消费（元） Y ：收入（元） 已知0.025(19) 2.0930t =，0.05(19) 1.729t =，0.025(17) 2.1098t =，0.05(17) 1.7396t =。 问：（1）利用t 值检验参数β的显著性（α＝0.05）；（2）确定参数β的标准差；（3）判断一下该模型的拟合情况。 4．已知估计回归模型得 i i ?Y =81.7230 3.6541X + 且2X X 4432.1∑ （－）＝，2Y Y 68113.6∑（－）＝， 求判定系数和相关系数。 5．有如下表数据
思考题答案 第一章 绪论 思考题 1.1怎样理解产生于西方国家的计量经济学能够在中国的经济理论研究和现代化建设中发挥重要作用？ 答：计量经济学的产生源于对经济问题的定量研究，这是社会经济发展到一定阶段的客观需要。计量经济学的发展是与现代科学技术成就结合在一起的，它反映了社会化大生产对各种经济因素和经济活动进行数量分析的客观要求。经济学从定性研究向定量分析的发展，是经济学逐步向更加精密、更加科学发展的表现。我们只要坚持以科学的经济理论为指导，紧密结合中国经济的实际，就能够使计量经济学的理论与方法在中国的经济理论研究和现代化建设中发挥重要作用。 1.2理论计量经济学和应用计量经济学的区别和联系是什么？ 答：计量经济学不仅要寻求经济计量分析的方法，而且要对实际经济问题加以研究，分为理论计量经济学和应用计量经济学两个方面。 理论计量经济学是以计量经济学理论与方法技术为研究内容，目的在于为应用计量经济学提供方法论。所谓计量经济学理论与方法技术的研究，实质上是指研究如何运用、改造和发展数理统计方法，使之成为适合测定随机经济关系的特殊方法。 应用计量经济学是在一定的经济理论的指导下，以反映经济事实的统计数据为依据，用计量经济方法技术研究计量经济模型的实用化或探索实证经济规律、分析经济现象和预测经济行为以及对经济政策作定量评价。 1.3怎样理解计量经济学与理论经济学、经济统计学的关系？ 答：1、计量经济学与经济学的关系。联系：计量经济学研究的主体—经济现象和经济关系的数量规律；计量经济学必须以经济学提供的理论原则和经济运行规律为依据；经济计量分析的结果：对经济理论确定的原则加以验证、充实、完善。区别：经济理论重在定性分析,并不对经济关系提供数量上的具体度量；计量经济学对经济关系要作出定量的估计，对经济理论提出经验的内容。 2、计量经济学与经济统计学的关系。联系：经济统计侧重于对社会经济现象的描述性计量；经济统计提供的数据是计量经济学据以估计参数、验证经济理论的基本依据；经济现象不能作实验，只能被动地观测客观经济现象变动的既成事实，只能依赖于经济统计数据。区别：经济统计学主要用统计指标和统计分析方法对经济现象进行描述和计量；计量经济学主要利用数理统计方法对经济变量间的关系进行计量。 1.4在计量经济模型中被解释变量和解释变量的作用有什么不同？ 答：在计量经济模型中，解释变量是变动的原因，被解释变量是变动的结果。被解释变量是模型要分析研究的对象。解释变量是说明被解释变量变动主要原因的变量。 1.5一个完整的计量经济模型应包括哪些基本要素？你能举一个例子吗？ 答：一个完整的计量经济模型应包括三个基本要素：经济变量、参数和随机误差项。 例如研究消费函数的计量经济模型：u βX αY ++= 其中，Y 为居民消费支出，X 为居民家庭收入，二者是经济变量；α和β为参数；u 是随机误差项。 1.6假如你是中央银行货币政策的研究者，需要你对增加货币供应量促进经济增长提出建议，
APPENDIX A SOLUTIONS TO PROBLEMS A.1 (i) $566. (ii) The two middle numbers are 480 and 530; when these are averaged, we obtain 505, or $505. (iii) 5.66 and 5.05, respectively. (iv) The average increases to $586 while the median is unchanged ($505). A.3 If price = 15 and income = 200, quantity = 120 – 9.8(15) + .03(200) = –21, which is nonsense. This shows that linear demand functions generally cannot describe demand over a wide range of prices and income. A.5 The majority shareholder is referring to the percentage point increase in the stock return, while the CEO is referring to the change relative to the initial return of 15%. To be precise, the shareholder should specifically refer to a 3 percentage point increase. $45,935.80.≈ $40,134.84. When exper = 5, salary = exp[10.6 + .027(5)] ≈A.7 (i) When exper = 0, log(salary) = 10.6; therefore, salary = exp(10.6) (ii) The approximate proportionate increase is .027(5) = .135, so the approximate percentage change is 13.5%. 14.5%, so the exact percentage increase is about one percentage point higher.≈(iii) 100[(45,935.80 – 40,134.84)/40,134.84) A.9 (i) The relationship between yield and fertilizer is graphed below. (ii) Compared with a linear function, the function yield has a diminishing effect, and the slope approaches zero as fertilizer gets large. The initial pound of fertilizer has the largest effect, and each additional pound has an effect smaller than the previous pound.
四、简答题（每小题5分） 令狐采学 1．简述计量经济学与经济学、统计学、数理统计学学科间的关系。 2．计量经济模型有哪些应用？ 3．简述建立与应用计量经济模型的主要步调。4．对计量经济模型的检验应从几个方面入手？ 5．计量经济学应用的数据是怎样进行分类的？6．在计量经济模型中，为什么会存在随机误差项？ 7．古典线性回归模型的基本假定是什么？8．总体回归模型与样本回归模型的区别与联系。 9．试述回归阐发与相关阐发的联系和区别。 10．在满足古典假定条件下，一元线性回归模型的普通最小二乘估计量有哪些统计性质？11．简述BLUE 的含义。 12．对多元线性回归模型，为什么在进行了总体显著性F 检验之后，还要对每个回归系数进行是否为0的t 检验？ 13.给定二元回归模型：01122t t t t y b b x b x u =+++，请叙述模型的古典假定。 14.在多元线性回归阐发中，为什么用修正的决定系数衡量估计模型对样本观测值的拟合优度？ 15.修正的决定系数2R 及其作用。16.罕见的非线性回归模型有几种情况？ 17.观察下列方程并判断其变量是否呈线性，系数是否呈线性，或
都是或都不是。 ①t t t u x b b y ++=310②t t t u x b b y ++=log 10 ③t t t u x b b y ++=log log 10④t t t u x b b y +=)/(10 18. 观察下列方程并判断其变量是否呈线性，系数是否呈线性，或都是或都不是。 ①t t t u x b b y ++=log 10②t t t u x b b b y ++=)(210 ③t t t u x b b y +=)/(10④t b t t u x b y +-+=)1(11 0 19.什么是异方差性？试举例说明经济现象中的异方差性。 20.产生异方差性的原因及异方差性对模型的OLS 估计有何影响。21.检验异方差性的办法有哪些？ 22.异方差性的解决办法有哪些？23.什么是加权最小二乘法？它的基本思想是什么？ 24.样天职段法（即戈德菲尔特——匡特检验）检验异方差性的基来源根基理及其使用条件。 25．简述DW 检验的局限性。26．序列相关性的后果。27．简述序列相关性的几种检验办法。 28．广义最小二乘法（GLS ）的基本思想是什么？29．解决序列相关性的问题主要有哪几种办法？ 30．差分法的基本思想是什么？31．差分法和广义差分法主要区别是什么？ 32．请简述什么是虚假序列相关。33．序列相关和自相关的概念和规模是否是一个意思？ 34．DW 值与一阶自相关系数的关系是什么？35．什么是多重共线
计量经济学练习题 第一章导论 一、单项选择题 ⒈计量经济研究中常用的数据主要有两类：一类是时间序列数据，另一类是【 B 】 A 总量数据 B 横截面数据 C平均数据 D 相对数据 ⒉横截面数据是指【A 】 A 同一时点上不同统计单位相同统计指标组成的数据 B 同一时点上相同统计单位相同统计指标组成的数据 C 同一时点上相同统计单位不同统计指标组成的数据 D 同一时点上不同统计单位不同统计指标组成的数据 ⒊下面属于截面数据的是【D 】 A 1991-2003年各年某地区20个乡镇的平均工业产值 B 1991-2003年各年某地区20个乡镇的各镇工业产值 C 某年某地区20个乡镇工业产值的合计数 D 某年某地区20个乡镇各镇工业产值 ⒋同一统计指标按时间顺序记录的数据列称为【B 】 A 横截面数据 B 时间序列数据 C 修匀数据D原始数据 ⒌回归分析中定义【 B 】 A 解释变量和被解释变量都是随机变量 B 解释变量为非随机变量，被解释变量为随机变量 C 解释变量和被解释变量都是非随机变量 D 解释变量为随机变量，被解释变量为非随机变量 二、填空题 ⒈计量经济学是经济学的一个分支学科，是对经济问题进行定量实证研究的技术、方法和相关理论，可以理解为数学、统计学和_经济学_三者的结合。 ⒉现代计量经济学已经形成了包括单方程回归分析，联立方程组模型，时间序列分 析三大支柱。
⒊经典计量经济学的最基本方法是回归分析。 计量经济分析的基本步骤是：理论（或假说）陈述、建立计量经济模型、收集数据、计量经济模型参数的估计、检验和模型修正、预测和政策分析。 ⒋常用的三类样本数据是截面数据、时间序列数据和面板数据。 ⒌经济变量间的关系有不相关关系、相关关系、因果关系、相互影响关系和恒 等关系。 三、简答题 ⒈什么是计量经济学？它与统计学的关系是怎样的？ 计量经济学就是对经济规律进行数量实证研究，包括预测、检验等多方面的工作。计量经济学是一种定量分析，是以解释经济活动中客观存在的数量关系为内容的一门经济学学科。 计量经济学与统计学密切联系，如数据收集和处理、参数估计、计量分析方法设计，以及参数估计值、模型和预测结果可靠性和可信程度分析判断等。可以说，统计学的知识和方法不仅贯穿计量经济分析过程，而且现代统计学本身也与计量经济学有不少相似之处。例如，统计学也通过对经济数据的处理分析，得出经济问题的数字化特征和结论，也有对经济参数的估计和分析，也进行经济趋势的预测，并利用各种统计量对分析预测的结论进行判断和检验等，统计学的这些内容与计量经济学的内容都很相似。反过来，计量经济学也经常使用各种统计分析方法，筛选数据、选择变量和检验相关结论，统计分析是计量经济分析的重要内容和主要基础之一。 计量经济学与统计学的根本区别在于，计量经济学是问题导向和以经济模型为核心的，而统计学则是以经济数据为核心，且常常是数据导向的。典型的计量经济学分析从具体经济问题出发，先建立经济模型，参数估计、判断、调整和预测分析等都是以模型为基础和出发点；典型的统计学研究则并不一定需要从具体明确的问题出发，虽然也有一些目标，但可以是模糊不明确的。虽然统计学并不排斥经济理论和模型，有时也会利用它们，但统计学通常不一定需要特定的经济理论或模型作为基础和出发点，常常是通过对经济数据的统计处理直接得出结论，统计学侧重的工作是经济数据的采集、筛选和处理。 此外，计量经济学不仅是通过数据处理和分析获得经济问题的一些数字特征，而且是借助于经济思想和数学工具对经济问题作深刻剖析。经过计量经济分析实证检验的经济理论和模型，能够对分析、研究和预测更广泛的经济问题起重要作用。计量经济学从经济理论和经济模型出发进行计量经济分析的过程，也是对经济理论证实或证伪的过程。这些是以处理数
计量经济学题库一、单项选择题（每小题1分） 1．计量经济学是下列哪门学科的分支学科（C）。 A．统计学B．数学C．经济学D．数理统计学 2．计量经济学成为一门独立学科的标志是（B）。 A．1930年世界计量经济学会成立B．1933年《计量经济学》会刊出版 C．1969年诺贝尔经济学奖设立D．1926年计量经济学（Economics）一词构造出来 3．外生变量和滞后变量统称为（D）。 A．控制变量B．解释变量C．被解释变量D．前定变量4．横截面数据是指（A）。 A．同一时点上不同统计单位相同统计指标组成的数据B．同一时点上相同统计单位相同统计指标组成的数据 C．同一时点上相同统计单位不同统计指标组成的数据D．同一时点上不同统计单位不同统计指标组成的数据 5．同一统计指标，同一统计单位按时间顺序记录形成的数据列是（C）。 A．时期数据B．混合数据C．时间序列数据D．横截面数据6．在计量经济模型中，由模型系统内部因素决定，表现为具有一定的概率分布的随机变量，其数值受模型中其他变量影响的变量是（）。 A．内生变量B．外生变量C．滞后变量D．前定变量7．描述微观主体经济活动中的变量关系的计量经济模型是（）。 A．微观计量经济模型B．宏观计量经济模型C．理论计量经济模型D．应用计量经济模型 8．经济计量模型的被解释变量一定是（）。 A．控制变量B．政策变量C．内生变量D．外生变量9．下面属于横截面数据的是（）。
A．1991－2003年各年某地区20个乡镇企业的平均工业产值 B．1991－2003年各年某地区20个乡镇企业各镇的工业产值 C．某年某地区20个乡镇工业产值的合计数D．某年某地区20个乡镇各镇的工业产值 10．经济计量分析工作的基本步骤是（）。 A．设定理论模型→收集样本资料→估计模型参数→检验模型B．设定模型→估计参数→检验模型→应用模型 C．个体设计→总体估计→估计模型→应用模型D．确定模型导向→确定变量及方程式→估计模型→应用模型 11．将内生变量的前期值作解释变量，这样的变量称为（）。 A．虚拟变量B．控制变量C．政策变量D．滞后变量 12．（）是具有一定概率分布的随机变量，它的数值由模型本身决定。 A．外生变量B．内生变量C．前定变量D．滞后变量 13．同一统计指标按时间顺序记录的数据列称为（）。 A．横截面数据B．时间序列数据C．修匀数据D．原始数据 14．计量经济模型的基本应用领域有（）。 A．结构分析、经济预测、政策评价B．弹性分析、乘数分析、政策模拟 C．消费需求分析、生产技术分析、D．季度分析、年度分析、中长期分析 15．变量之间的关系可以分为两大类，它们是（）。 A．函数关系与相关关系B．线性相关关系和非线性相关关系 C．正相关关系和负相关关系D．简单相关关系和复杂相关关系 16．相关关系是指（）。 A．变量间的非独立关系B．变量间的因果关系C．变量间的函数关系D．变量间不确定性
计量经济学课后习题答 案汇总 标准化工作室编码[XX968T-XX89628-XJ668-XT689N]
计量经济学练习题 第一章导论 一、单项选择题 ⒈计量经济研究中常用的数据主要有两类：一类是时间序列数据，另一类是【 B 】 A 总量数据 B 横截面数据 C平均数据 D 相对数据 ⒉横截面数据是指【 A 】 A 同一时点上不同统计单位相同统计指标组成的数据 B 同一时点上相同统计单位相同统计指标组成的数据 C 同一时点上相同统计单位不同统计指标组成的数据 D 同一时点上不同统计单位不同统计指标组成的数据 ⒊下面属于截面数据的是【 D 】 A 1991-2003年各年某地区20个乡镇的平均工业产值 B 1991-2003年各年某地区20个乡镇的各镇工业产值 C 某年某地区20个乡镇工业产值的合计数 D 某年某地区20个乡镇各镇工业产值 ⒋同一统计指标按时间顺序记录的数据列称为【 B 】 A 横截面数据 B 时间序列数据 C 修匀数据 D原始数据 ⒌回归分析中定义【 B 】 A 解释变量和被解释变量都是随机变量 B 解释变量为非随机变量，被解释变量为随机变量 C 解释变量和被解释变量都是非随机变量 D 解释变量为随机变量，被解释变量为非随机变量 二、填空题 ⒈计量经济学是经济学的一个分支学科，是对经济问题进行定量实证研究的技术、方法和相关理论，可以理解为数学、统计学和_经济学_三者的结合。 ⒉现代计量经济学已经形成了包括单方程回归分析，联立方程组模型，时间序列 分析三大支柱。
⒊经典计量经济学的最基本方法是回归分析。 计量经济分析的基本步骤是：理论（或假说）陈述、建立计量经济模型、收集数据、计量经济模型参数的估计、检验和模型修正、预测和政策分析。 ⒋常用的三类样本数据是截面数据、时间序列数据和面板数据。 ⒌经济变量间的关系有不相关关系、相关关系、因果关系、相互影响关系 和恒等关系。 三、简答题 ⒈什么是计量经济学它与统计学的关系是怎样的 计量经济学就是对经济规律进行数量实证研究，包括预测、检验等多方面的工作。计量经济学是一种定量分析，是以解释经济活动中客观存在的数量关系为内容的一门经济学学科。 计量经济学与统计学密切联系，如数据收集和处理、参数估计、计量分析方法设计，以及参数估计值、模型和预测结果可靠性和可信程度分析判断等。可以说，统计学的知识和方法不仅贯穿计量经济分析过程，而且现代统计学本身也与计量经济学有不少相似之处。例如，统计学也通过对经济数据的处理分析，得出经济问题的数字化特征和结论，也有对经济参数的估计和分析，也进行经济趋势的预测，并利用各种统计量对分析预测的结论进行判断和检验等，统计学的这些内容与计量经济学的内容都很相似。反过来，计量经济学也经常使用各种统计分析方法，筛选数据、选择变量和检验相关结论，统计分析是计量经济分析的重要内容和主要基础之一。 计量经济学与统计学的根本区别在于，计量经济学是问题导向和以经济模型为核心的，而统计学则是以经济数据为核心，且常常是数据导向的。典型的计量经济学分析从具体经济问题出发，先建立经济模型，参数估计、判断、调整和预测分析等都是以模型为基础和出发点；典型的统计学研究则并不一定需要从具体明确的问题出发，虽然也有一些目标，但可以是模糊不明确的。虽然统计学并不排斥经济理论和模型，有时也会利用它们，但统计学通常不一定需要特定的经济理论或模型作为基础和出发点，常常是通过对经济数据的统计处理直接得出结论，统计学侧重的工作是经济数据的采集、筛选和处理。 此外，计量经济学不仅是通过数据处理和分析获得经济问题的一些数字特征，而且是借助于经济思想和数学工具对经济问题作深刻剖析。经过计量经济分析实证检验的经济理论和模型，能够对分析、研究和预测更广泛的经济问题起重要作用。计量经济学从
2．已知一模型的最小二乘的回归结果如下： i i ?Y =101.4-4.78X 标准差 （） （） n=30 R 2 = 其中，Y ：政府债券价格（百美元），X ：利率（%）。 回答以下问题：（1）系数的符号是否正确，并说明理由；（2）为什么左边是i ?Y 而不是i Y ； （3）在此模型中是否漏了误差项i u ；（4）该模型参数的经济意义是什么。 13．假设某国的货币供给量Y 与国民收入X 的历史如系下表。 某国的货币供给量X 与国民收入Y 的历史数据 根据以上数据估计货币供给量Y 对国民收入X 的回归方程，利用Eivews 软件输出结果为： Dependent Variable: Y Variable Coefficient Std. Error t-Statistic Prob. X C R-squared Mean dependent var Adjusted R-squared . dependent var . of regression F-statistic Sum squared resid Prob(F-statistic) 问：（1）写出回归模型的方程形式，并说明回归系数的显著性（） 。 （2）解释回归系数的含义。 （2）如果希望1997年国民收入达到15，那么应该把货币供给量定在什么水平 14．假定有如下的回归结果 t t X Y 4795.06911.2?-= 其中，Y 表示美国的咖啡消费量（每天每人消费的杯数），X 表示咖啡的零售价格（单位：美元/杯），t 表示时间。问： （1）这是一个时间序列回归还是横截面回归做出回归线。 （2）如何解释截距的意义它有经济含义吗如何解释斜率（3）能否救出真实的总体回归函数 （4）根据需求的价格弹性定义： Y X ?弹性＝斜率，依据上述回归结果，你能救出对咖啡需求的价格弹性吗如果不能，计算此弹性还需要其他什么信息 15．下面数据是依据10组X 和Y 的观察值得到的： 1110=∑i Y ，1680 =∑i X ，204200=∑i i Y X ，315400 2=∑ i X ，133300 2 =∑i Y 假定满足所有经典线性回归模型的假设，求0β，1β的估计值； 16.根据某地1961—1999年共39年的总产出Y 、劳动投入L 和资本投入K 的年度数据，运用普通最小二乘法估计得出了下列回归方程： ，DW= 式下括号中的数字为相应估计量的标准误。 (1)解释回归系数的经济含义； (2)系数的符号符合你的预期吗为什么 17.某计量经济学家曾用1921~1941年与1945~1950年（1942~1944年战争期间略去）美国国内消费Ｃ和工资收入Ｗ、非工资－非农业收入
第一章绪论 参考重点： 计量经济学的一般建模过程 第一章课后题（1.4.5） 1.什么是计量经济学？计量经济学方法与一般经济数学方法有什么区别？ 答：计量经济学是经济学的一个分支学科，是以揭示经济活动中客观存在的数量关系为内容的分支学科，是由经济学、统计学和数学三者结合而成的交叉学科。 计量经济学方法揭示经济活动中各个因素之间的定量关系，用随机性的数学方程加以描述；一般经济数学方法揭示经济活动中各个因素之间的理论关系，用确定性的数学方程加以描述。 4.建立与应用计量经济学模型的主要步骤有哪些？ 答：建立与应用计量经济学模型的主要步骤如下：(1)设定理论模型，包括选择模型所包含的变量，确定变量之间的数学关系和拟定模型中待估参数的数值范围；(2)收集样本数据，要考虑样本数据的完整性、准确性、可比性和—致性；(3)估计模型参数；(4)检验模型，包括经济意义检验、统计检验、计量经济学检验和模型预测检验。 5.模型的检验包括几个方面？其具体含义是什么？ 答：模型的检验主要包括：经济意义检验、统计检验、计量经济学检验、模型的预测检验。在经济意义检验中，需要检验模型是否符合经济意义，检验求得的参数估计值的符号与大小是否与根据人们的经验和经济理论所拟订的期望值相符合；在统计检验中，需要检验模型参数估计值的可靠性，即检验模型的统计学性质；在计量经济学检验中，需要检验模型的计量经济学性质，包括随机扰动项的序列相关检验、异方差性检验、解释变量的多重共线性检验等；模型的预测检验主要检验模型参数估计量的稳定性以及对样本容量变化时的灵敏度，以确定所建立的模型是否可以用于样本观测值以外的范围。 第二章经典单方程计量经济学模型：一元线性回归模型参考重点： 1.相关分析与回归分析的概念、联系以及区别？ 2.总体随机项与样本随机项的区别与联系？
计量经济学题库(超完整版)及答案 一、单项选择题（每小题1分） 1．计量经济学是下列哪门学科的分支学科（C ）。 A ．统计学 B ．数学 C ．经济学 D ．数理统计学 2．计量经济学成为一门独立学科的标志是（B ）。 A ．1930年世界计量经济学会成立 B ．1933年《计量经济学》会刊出版 C ．1969年诺贝尔经济学奖设立 D ．1926年计量经济学（Economics ）一词构造出来3．外生变量和滞后变量统称为（D ）。 A ．控制变量 B ．解释变量 C ．被解释变量 D ．前定变量 4．横截面数据是指（A ）。 A ．同一时点上不同统计单位相同统计指标组成的数据 B ．同一时点上相同统计单位相同统计指标组成的数据 C ．同一时点上相同统计单位不同统计指标组成的数据 D ．同一时点上不同统计单位不同统计指标组成的数据 5．同一统计指标，同一统计单位按时间顺序记录形成的数据列是（C ）。 A ．时期数据 B ．混合数据 C ．时间序列数据 D ．横截面数据 6．在计量经济模型中，由模型系统内部因素决定，表现为具有一定的概率分布的随机变量，其数值受模型中其他变量影响的变量是（）。 A ．内生变量 B ．外生变量 C ．滞后变量 D ．前定变量 7．描述微观主体经济活动中的变量关系的计量经济模型是（）。 A ．微观计量经济模型 B ．宏观计量经济模型 C ．理论计量经济模型 D ．应用计量经济模型 8．经济计量模型的被解释变量一定是（）。 A ．控制变量 B ．政策变量 C ．内生变量 D ．外生变量 9．下面属于横截面数据的是（）。 A ．1991－2003年各年某地区20个乡镇企业的平均工业产值 B ．1991－2003年各年某地区20个乡镇企业各镇的工业产值 C ．某年某地区20个乡镇工业产值的合计数 D ．某年某地区20个乡镇各镇的工业产值10．经济计量分析工作的基本步骤是（）。 A ．设定理论模型→收集样本资料→估计模型参数→检验模型 B ．设定模型→估计参数→检验模型→应用模型 C ．个体设计→总体估计→估计模型→应用模型 D ．确定模型导向→确定变量及方程式→估计模型→应用模型 11．将内生变量的前期值作解释变量，这样的变量称为（）。 A ．虚拟变量 B ．控制变量 C ．政策变量 D ．滞后变量 12．（）是具有一定概率分布的随机变量，它的数值由模型本身决定。 A ．外生变量 B ．内生变量 C ．前定变量 D ．滞后变量 13．同一统计指标按时间顺序记录的数据列称为（）。 A ．横截面数据 B ．时间序列数据 C ．修匀数据 D ．原始数据 14．计量经济模型的基本应用领域有（）。 A ．结构分析、经济预测、政策评价 B ．弹性分析、乘数分析、政策模拟 C ．消费需求分析、生产技术分析、 D ．季度分析、年度分析、中长期分析 15．变量之间的关系可以分为两大类，它们是（）。 A ．函数关系与相关关系 B ．线性相关关系和非线性相关关系
1.3 某市居民家庭人均年收入服从4000X =元， 1200σ=元的正态分布， 求该市居民家庭人均年收入：（1）在5000—7000元之间的概率；（2）超过8000元的概率；（3）低于3000元的概率。 （1） ()() ()()()2,0,15000700050007000( ) 2.50.835( 2.5)62 X N X X X N X X X X P X P F F X X P σσ σ σ σ σ-∴---∴<<=< < --=<<= Q :: 根据附表1可知 ()0.830.5935F =，()2.50.9876F = ()0.98760.5935 500070000.1971 2 P X -∴<<= = PS ： ()()5000700050007000( ) 55( 2.5) 2.5660.99380.79760.1961 X X X X P X P X X P σ σ σ σ---<<=< < -??=<<=Φ-Φ ? ??=-=
在附表1中，()() F Z P x x z σ=-< （2）()80001080003X X X X X P X P P σσσ?? ??--->=>=> ? ?? ? ? ? =0.0004 （3）()3000530006 X X X X X P X P P σσσ???? ---<=<=<- ? ?? ? ? ? =0.2023 ()030001050300036X X X X X X P X P P σ σσσ???? ----<<=<< =-<<- ? ? ???? =0.2023-0.0004=0.20191.4 据统计70岁的老 人在5年内正常死亡概率为0.98，因事故死亡的概率为0.02。保险公司开办老人事故死亡保险，参加者需缴纳保险费100元。若5年内因事故死亡，公司要赔偿a 元。应如何测算出a ，才能使公司可期望获益；若有1000人投保，公司可期望总获益多少？ 设公司从一个投保者得到的收益为X ，则
四、简答题（每小题5分） 1．简述计量经济学与经济学、统计学、数理统计学学科间的关系。2．计量经济模型有哪些应用？ 3．简述建立与应用计量经济模型的主要步骤。 4．对计量经济模型的检验应从几个方面入手？ 5．计量经济学应用的数据是怎样进行分类的？ 6．在计量经济模型中，为什么会存在随机误差项？ 7．古典线性回归模型的基本假定是什么？ 8．总体回归模型与样本回归模型的区别与联系。 9．试述回归分析与相关分析的联系和区别。 10．在满足古典假定条件下，一元线性回归模型的普通最小二乘估计量有哪些统计性质？ 11．简述BLUE 的含义。 12．对于多元线性回归模型，为什么在进行了总体显著性F 检验之后，还要对每个回归系数进行是否为0的t 检验？ 13.给定二元回归模型： 01122t t t t y b b x b x u =+++，请叙述模型的古典假定。 14.在多元线性回归分析中，为什么用修正的决定系数衡量估计模型对样本观测值的拟合优度？ 15.修正的决定系数2R 及其作用。 16.常见的非线性回归模型有几种情况？ 17.观察下列方程并判断其变量是否呈线性，系数是否呈线性，或都是或都不是。 ①t t t u x b b y ++=310 ②t t t u x b b y ++=log 10 ③ t t t u x b b y ++=log log 10 ④t t t u x b b y +=)/(10 18. 观察下列方程并判断其变量是否呈线性，系数是否呈线性，或都是或都不是。 ①t t t u x b b y ++=log 10 ②t t t u x b b b y ++=)(210 ③ t t t u x b b y +=)/(10 ④t b t t u x b y +-+=)1(110 19.什么是异方差性？试举例说明经济现象中的异方差性。
1、已知一模型的最小二乘的回归结果如下： i i ?Y =101.4-4.78X （）（） n=30 R 2= 其中，Y ：政府债券价格（百美元），X ：利率（%）。 回答以下问题： （1）系数的符号是否正确，并说明理由；（2）为什么左边是i ?Y 而不是i Y ； （3）在此模型中是否漏了误差项i u ；（4）该模型参数的经济意义是什么。 答：（1）系数的符号是正确的，政府债券的价格与利率是负相关关系，利率的上升会引起政府债券价格的下降。 （2）i Y 代表的是样本值，而i ?Y 代表的是给定i X 的条件下i Y 的期望值，即?(/)i i i Y E Y X =。此模型是根据样本数据得出的回归结果，左边应当是i Y 的期望值，因此是i ?Y 而不是i Y 。 （3）没有遗漏，因为这是根据样本做出的回归结果，并不是理论模型。 （4）截距项表示在X 取0时Y 的水平，本例中它没有实际意义；斜率项表明利率X 每上升一个百分点，引起政府债券价格Y 降低478美元。 2、有10户家庭的收入（X ，元）和消费（Y ，百元）数据如下表： 10户家庭的收入（X ）与消费（Y ）的资料 X 20 30 33 40 15 13 26 38 35 43 Y 7 9 8 11 5 4 8 10 9 10 若建立的消费Y 对收入X 的回归直线的Eviews 输出结果如下： Dependent Variable: Y Adjusted R-squared F-statistic （1）说明回归直线的代表性及解释能力。 （2）在95%的置信度下检验参数的显着性。（0.025(10) 2.2281t =，0.05(10) 1.8125t =，0.025(8) 2.3060t =，0.05(8) 1.8595t =） （3）在95%的置信度下，预测当X ＝45（百元）时，消费（Y ）的置信区间。（其
第1章计量经济学的性质与经济数据 1.1 复习笔记 一、计量经济学 由于计量经济学主要考虑在搜集和分析非实验经济数据时的固有问题，计量经济学已从数理统计分离出来并演化成一门独立学科。 1．非实验数据是指并非从对个人、企业或经济系统中的某些部分的控制实验而得来的数据。非实验数据有时被称为观测数据或回顾数据，以强调研究者只是被动的数据搜集者这一事实。 2．实验数据通常是在实验环境中获得的，但在社会科学中要得到这些实验数据则困难得多。 二、经验经济分析的步骤 经验分析就是利用数据来检验某个理论或估计某种关系。 1．对所关心问题的详细阐述 在某些情形下，特别是涉及到对经济理论的检验时，就要构造一个规范的经济模型。经济模型总是由描述各种关系的数理方程构成。 2．经济模型变成计量模型 先了解一下计量模型和经济模型有何关系。与经济分析不同，在进行计量经济分析之前，必须明确函数的形式。
通过设定一个特定的计量经济模型，就解决了经济模型中内在的不确定性。 在多数情况下，计量经济分析是从对一个计量经济模型的设定开始的，而没有考虑模型构造的细节。一旦设定了一个计量模型，所关心的各种假设便可用未知参数来表述。 3．搜集相关变量的数据 4．用计量方法来估计计量模型中的参数，并规范地检验所关心的假设 在某些情况下，计量模型还用于对理论的检验或对政策影响的研究。 三、经济数据的结构 1．横截面数据 （1）横截面数据集，就是在给定时点对个人、家庭、企业、城市、州、国家或一系列其他单位采集的样本所构成的数据集。有时，所有单位的数据并非完全对应于同一时间段。在一个纯粹的横截面分析中，应该忽略数据搜集中细小的时间差别。 （2）横截面数据的重要特征 ①假定它们是从样本背后的总体中通过随机抽样而得到的。 当抽取的样本（特别是地理上的样本）相对总体而言太大时，可能会导致另一种偏离随机抽样的情况。这种情形中潜在的问题是，总体不够大，所以不能合理地假定观测值是独立抽取的。 ②数据排序不影响计量分析这一事实，是由随机抽样而得到横截面数据集的一个重要特征。 2．时间序列数据 （1）时间序列数据集，是由对一个或几个变量不同时间的观测值所构成。与横截面数据的排序不同，时间序列对观测值按时间先后排序，这也传递了潜在的重要信息。
计量经济学第一次作业 第二章P85 8.用SPSS软件对10名同学的成绩数据进行录入，分析得r=0.875,这说明学生的课堂练习和期终考试有密切的关系，一般平时练习成绩较高者，期终成绩也高。 9.(1)一元线性回归模型如下：Y i=?0+?1X i+u i 其中，Y i表示财政收入，X i表示国民生产总值，u i为随机扰动项， ?0 ?1为待估参数。 由Eviews软件得散点图如下图： （2）Y i =-1354.856+0.179672X i Sê:(655.7254) (0.007082) t:(-2.066194) (25.37152) R2=0.958316 F=643.7141 df=28 斜率? 1 =0.179672表示国民生产总值每增加1亿元，财政收入增加0.179672亿元。（3）可决系数R2=0.958316表示在财政收入Y的总变差中由模型作出的解释部分占95.8316%，即有95.8316%由国民生产总值来解释，同时说明样本回归模型对样本数据的拟合程度较高。 R2=ESS/(ESS+RSS) ESS=RSS*R2/(1-R2)=(1.91E+08)*0.958316/(1-0.958316)=44.02E+08 F=(n-2)ESS/RSS,ESS=F*RSS/(n-2)=4.39*E09 (4)Sê(? 0)=655.7245 Sê(? 1 )=0.007082
?1的95%的置信区间是： [?1-t 0.025(28)Sê(?1),?1+t 0.025(28)Sê(?1)] 代入数值得： [0.179672-2.048*0.007082,0.179672+2.048*0.007082] 即：[0.165,0.194] 同理可得，?0的95%置信区间为[-2697.78,-11.93] （5）①原假设H 0:?0=0 备择假设：H 1:?0≠0 则?0的t 值为：t 0=-2.066194 当ɑ=0.05时 t ɑ/2(28)=2.048 |t 0|=2.066194＞t ɑ/2(28)=2.048 故拒绝原假设H 0，表明模型应保留截距项。 ②原假设H 0:?1=0 备择假设：H 1:?1≠0 当ɑ=0.05时 t ɑ/2(28)=2.048 因为|t 1|=25.37152＞t ɑ/2(28)=2.048 故拒绝原假设H 0 表明国民生产总值的变动对国家财政收入有显著影响. 计量经济学第二次作业 第二章9.(10) 、建立X 与t 的趋势模型，其回归分析结果如下： Dependent Variable: X Method: Least Squares Date: 04/19/10 Time: 22:03 Sample: 1978 2008 Dependent Variable: Y Method: Least Squares Date: 04/10/10 Time: 17:31 Sample: 1978 2007 C -1354.856 655.7254 -2.066194 0.0482 R-squared 0.958316 Mean dependent var 10049.04 Adjusted R-squared 0.956827 S.D. dependent var 12585.51 S.E. of regression 2615.036 Akaike info criterion 18.64028 Sum squared resid 1.91E+08 Schwarz criterion 18.73370 Log likelihood -277.6043 F-statistic 643.7141