中级计量经济学 作业四
（Due Date ：December 19， 2011）
(共117分)
无论如何英语1. (2’×5=10’)
根据20世纪70年代早期101个国家以美元计人均收入（x ）和预期寿命（y ）的数据，Sen 和 Srivastava 得到如下回归结果
()752.0R (2.42) (0.859) (4.73) ]7x ln D [36.3x ln 39.940.2y
ˆ2i i i i ==−−+−=
其中，如果lnx i >7, D i = 1，否则D i = 0。 注：当lnx i = 7 时，x = 1097美元（近似）。
（a ）以对数形式引入收入变量的原因是什么？
（b ）你如何解释lnx i 的系数9.39？
（c ）引入回归元D i (lnx i -7)的理由是什么？你又如何解释这个回归元的系数-3.36？
（d ）假定穷国与富国之间的分界线为人均收入1097美元，你如何推导出收入低于1097美元
国家的回归线和收入高于1097美元国家的回归线？
（e ）你从回归结果中能得出什么一般结论？
2. (2’×2=4’)
设时间序列｛X t ｝是由X δ δ t ε 生成的，如果ε 是一个零均值，同方差，序列不相关的白噪声，请问：
（a ）X t 是平稳时间序列吗？为什么？
（b ）X t – E(X t ) 是平稳时间序列吗？为什么？
3. (2’×4=8’)
服装设计研究生From the houhold budget survey of 1980 of the Dutch Central Bureau of Statistics, J. S. Cramer obtained the following logit model bad on a sample of 2820 houholds. (The results given here are bad on the method of maximum likelihood and are after the third iteration.)
The purpo of the logit model was to determine car ownership as a function of (logarithm of) income. Car ownership was a binary variable: Y = 1 if a houhold owns a car, zero otherwi.
L
2.77231 0.347582ln Income t = (−
3.35) (
4.05)
χ2(1 df) = 16.681 ( p value = 0.0000)
where L
= estimated logit and where ln Income is the logarithm of income. The χ2 measures the goodness of fit of the model.
(a) Interpret the estimated logit model.
(b) From the estimated logit model, how would you obtain the expression for the probability of car ow
nership?
(c) What is the probability that a houhold with an income of 20,000 will own a car? And at an income level of 25,000? What is the rate of change of probability at the income level of 20,000? (d) Comment on the statistical significance of the estimated logit model.
建造师考试科目4. (5’)
Let grad be a dummy variable for whether a student-athlete at a large university graduates in five years. Let hsGPA and SAT be high school grade point average and SAT score, respectively. Let study be the number of hours spent per week in an organized study hall. Suppo that, using data on 420 student-athletes, the following logit model is obtained:
effectiveP grad 1|hsGPA,SAT,study Λ 1.17 .24hsGPA .00058SAT .073study where Λ z exp z / 1 exp z is the logit function. Holding hsGPA fixed at 3.0 and SAT fixed at 1,200, compute the estimated difference in the graduation probability for someone who spent 10 hours per week in study hall and someone who spent 5 hours per week.
5. (5’)
假设两个时间序列Yt和Zt都是I(1)序列，但有一不为零的β，使Y βZ 是I(0)。证明：对于任何δ β，组合Y δZ 一定是I(1)。
6. (5’)
Let gM t be the annual growth in the money supply and let unem t be the unemployment rate. Assuming that unemt follows a stable AR(1) process, explain in detail how you would test whether gM Granger caus unem.
7. (10’× 4 = 40’)
The goal of this exerci is to help you gain experience with simple limited dependent variable estimation techniques by computing and then comparing OLS, probit, and logit estimates of a typical labor force participation equation. The numerical comparison of the various estimators is bad in large part on the work of Takeshi Amemiya (1981), who has derived relationships among them.
(a) Econometrics textbooks typically point out that if the dependent variable in an equation is a dichotomous dummy variable, and if the equation is estimated by OLS in which this dependent variable is related linearly to an intercept term, a number of regressors, and a stochastic error term, t
hen the resulting equation (often called a linear probability model) suffers from at least two defects: (1) the fitted values are not confined to the 0-1 interval, and so their interpretation as probabilities is inappropriate; and (2) the residuals from such an equation are heteroskedastic. Note that in our context, LFP is such a dichotomous dependent variable.
Using OLS estimation procedures and the MROZ data file for all 753 obrvations, estimate parameters of a linear probability model in which LFP is related linearly to an intercept term, the LWW1 (e note 1), KL6, K618, WA, WE, UN, CIT, the wife’s property income PRIN (e note 2), and a stochastic error term.
Do signs of the OLS estimated parameters make n? Why or why not?
Comment on the appropriateness of using the OLS estimated standard errors to conduct tests
of statistical significance.
Next retrieve and print out the fitted values from this estimated linear probability model. For
how many obrvations are the fitted values negative? For how many are they greater than 1?
Why does this complicate the interpretation of this model?
2012新课标文科数学What is R2 in this model? Does it have any uful interpretation? Why or why not?
(b) One possible estimation procedure that is more appropriate when the dependent variable is dichotomous is bad on the assumption that the cumulative distribution of the stochastic disturbances is the logistic; the resulting maximum likelihood estimator is usually called logit.
With LEP as the dependent variable and with a constant term, LWW1, KL6, K618, WA, WE, UN, CIT, and PRIN as explanatory variables, u the entire sample of 753 obrvations in the MROZ data file and estimate parameters bad on a logit maximum likelihood procedure.
Do the signs of the estimated logit parameters make n? Why or why not?
Which of the estimated logit parameters is significantly different from zero? Interpret.
Does the nonlinear logit computational algorithm in your computer software program reach
convergence quickly, after only a few iterations (say, fewer than five)?
Some computer programs provide “goodness-of-fit” output for the estimated logit model, such
as a pudo-R2 measure or a measure indicating what percent of the predictions are “correct”.
Check your computer output and manual for the interpretation of any such goodness-of-fit measures.
Finally, compare your logit estimates to the OLS or linear probability model estimates from
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part (a).
(c) Another common estimation procedure that is ud in the estimation of dichotomous dependent variable models is bad on the assumption that the cumulative distribution of the disturbances is normal; this is usually called the probit model.scatology
With LEP as the dependent variable and with a constant term, LWW1, KL6, K618, WA, WE, UN, CIT, and PRIN as explanatory variables, u the entire sample of 753 obrvations in the MROZ data file and estimate parameters bad on a probit maximum likelihood procedure.
Do the signs of the estimated probit parameters make n? Why or why not?
Which of the estimated probit parameters is significantly different from zero? Interpret.
Does the nonlinear probit computational algorithm in your computer software program reach
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convergence quickly, after only a few iterations? Is convergence more or less rapid than with the logit model?
As in part (b), interpret any goodness-of-fit measures that are provided as output by your
computer software program.
(d) Becau the cumulative normal distribution and the logistic distribution are very clo to each other, in most ca the logit and probit estimated models will be quite similar.
Are the sample maximized log-likelihoods in your estimated logit and probit similar? Whichnowadays
is larger?
What about the signs and statistical significance of the estimated parameters – are they similar?
The estimated effect of a change in a regressor in the probability of participating in the labor
⁄, is equal to 1 in the logit model and in the probit force,
model, where P is the probability of LFP, and are the estimated logit and probit coefficients, respectively, on the i th explanatory variable, and f(P) is the cumulative normal function corresponding to P. Evaluate the estimated derivatives for the logit and probit models, using the sample LFPR of 425/753 = 0.568 as an estimate of P and noting that f(P) = f(0.568) = 0.393. At this sample mean, are the estimated effects similar for the logit and probit models? What happens if you evaluate the effects at the tail of the distribution, such as at P = 0.9 where f(P) = 0.175?
Note 1: One of the variables that is often employed in empirical analys of labor force participation is the wage rate. However, the wage rate is typically not obrved for women who are not working. Some analysts have attempted to deal with this problem (albeit in an unsatisfactory manner) by estimating a wage determination equation using data on workers only and using the resulting parameters estimates and characteristics of the nonworking sample to construct fitted or predicted wages for each of the nonworkers.
Restricting your sample to workers (the first 428 obrvations), take the natural logarithm of the wife’s wage rate variable WW and call this log-transformed variable LWW. Then, for the entire sample of 753 obrvations, construct the square of the wife’s experience variable and call it AX2, that is, generate AX2 = AX*AX. Next, following the human capital literature on wage determination and using only 418 obrvation from the working sample, estimate by OLS a typical wage determination equation in which LWW is regresd on a constant term, WA, WE, CIT, AX, and AX2. Then u the parameter estimates from this equation and values of the WA, WE, CIT, AX, and AX2 variables for the 325 women in the nonworking sample to generate the predicted or fitted log-wage for the nonworkers. Call this fitted log-wage variable for the nonworkers FLWW. Finally, for the entire sample of 753 obrvations, generate a variable called LWW1 for which the first 428 obrvations (the working sample) LWW1 = LWW and for which the last 325 obrvations (the nonworking sample) LWW1 = FLWW from above. To ensure that you have computed the data ries LWW1 correctly, compute and print out its mean and standard deviation; they should equal 1.10432 and 0.58268, respectively.
Note 2:In the model of labor supply estimated by Mroz, it is assumed that in making her labor supply decisions the wife takes as given the houhold’s entire nonlabor income plus her husband’s labor in
come. Morz calls this sum the wife’s property income and computes it as total family income minus the labor income earned by the wife. For the entire sample of 753 obrvations, compute this property income variable (named, say, PRIN) as PRIN = FAMINC – (WHRS*WW). Also calculate and print its mean and standard deviation; the should equal 20129 and 11635, repectively.
8. (10’× 4 = 40’)
The purpo of this exerci is to give you hands-on experience in modeling the investment time ries using a procedure called the Box-Jenkins time ries approach.
There is a data file called KOPCKE that contains quarterly data ries 1952:1 – 1986:4 on real investment in equipment and structures (IE and IS). Note: Since the investment ries are asonally adjusted, the ries should not contain a asonal component.
(a) The first task is to generate a transformation of the investment time ries that is stationary. Using 1956:1-1979:4 data, plot the raw data ries for IE and IS. Do the data appear to follow a trend? What does this imply concerning stationary? Next construct the sample autocorrelations (and their standard errors) for IE and IS where the maximal lag m is 16. Determine whether each ries is stationary. If a ries is nonstationary, then difference the data one, two, or more times until the sa
mple autocorrelation function indicates that the differenced data are stationary, that is, until the sample autocorrelation functions go to zero as the length of the lag increa. Denote this degree of differencing by d.
(b) Having generated a stationary time ries for IE and IS, now identify the order of the moving average (MA) process, denote q, and the order of the autoregressive (AR) process, denoted p. Recall that spikes in the sample autocorrelation function often indicate MA components, while obrvations that are highly correlated with surrounding one, resulting in discernible up-and-down patterns, suggest that an AR process may be generating the data. Plot the sample autocorrelation function from your stationary ries, and examine its properties. Is there any evidence of an MA component? An AR component? Why or why not? The partial autocorrelation function can be also ud for guidance in determining the order of the AR portion of the ARIMA process. Using the stationary data ries in IE and IS, calculate partial autocorrelation functions for time displacements up to 12 quarters, and choo reasonable values for p and q; in particular, choo two ts of p, q for IE and two for IS. Defend your choice.
(c) Having identified veral possible ARIMA(p, d, q) process underlying the IE and IS data ries, now estimate their parameters. Specifically, for each of your ARIMA(p, d, q) specifications chon in
part (b) for IE and IS, u the Box-Jenkins estimation technique and estimate the ARIMA parameters employing quarterly investment data, 1956:1-1979:4. From your four alternative ARIMA(p, d, q) specifications from part (b), choo one preferred model among the for IE and one for IS.
(d) Now employ eight more recent obrvations, redo the identification and estimation of ARIMA models, and then perform an ex post dynamic forecast.
First, using the quarterly IE and IS data, 1956:1-1981:4, redo part (a) and generate a stationary
data ries.
Next, as in part (b), identify veral plausible ARIMA(p, d, q) specifications. Estimate
parameters of the specification using Box-Jenkins procedures, and choo a preferred specification for IE and IS. Compare the final specifications to tho of part (c). Are the specification robust? How do you interpret the results?
Finally, on the basis of your preferred specifications for IE and IS, u the Box-Jenkins
forecasting procedure and forecast for 20 quarters following the estimation period, that is, construct a dynamic forecast over 1982:1-1986:4, using forecasted rather than actual lagged values of investment as appropriate. Compare the forecasted with the actual investment data ries over this time period, compute the percent root mean squared error (%RMSE) using following equation, examine whether any residuals are particularly large, and comment on properties of the forecasts. How well do your ARIMA models perform?
%
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where is the fitted value of investment in time t bad on the regression estimates and T is the sample size.
