chapter 6时间序列Word格式.docx

1、C. Suppose that yt and zt are CI（1, 1）. Use the conditions in （6.19）, （6.20）, and （6.21） to write the error-correcting model. Compare your answer to （6.22） and （6.23）. Show that the error-correction model contains an intercept term. Imposing the restrictions necessary to ensure that yt and zt are CI

2、（1, 1）, the equations for yt and zt can be written as: yt = -a12a21/（1-a22）yt-1 + a21zt-1 + yt + a10 zt = a21yt-1 - （1 - a22）zt-1 + zt + a20 Normalizing the cointegrating vector with respect to yt-1: yt = y（yt-1 - zt-1） + yt + a10 zt = z（yt-1 - zt-1） + zt + a20 where: y = -a12a21/（1-a22）; z = a21; a

3、nd = （1-a22）/a21. Thus, the error-correcting equations for yt and zt each contain a drift term. Another way to answer the question is to note that the solutions for yt and zt obtained in Parts A and B contain the deterministic expressions （1-a22）a10 + a12a20 and a21a10 + （1-a11）a20, respectively. Si

4、nce the denominator contains a characteristic root equal to unity, the solution for each contains a deterministic trend. D. Show that yt and zt have the same deterministic time trend （i.e., show that the slope coefficient of the time trends are identical）. The constant in the numerator of the soluti

5、on for yt is: （1-a22）a10 + a12a20. Since 1- a22 = a12a21/（1-a11）, this constant can be rewritten as: a12/（1-a11）a21a10 + （1-a11）a20. Up to the expression a12/（1-a11）, this deterministic numerator expression is the same as that in the solution for zt. Given that the denominators are identical, yt and

6、 zt can be said to have the same deterministic time trend. E. What is the condition such that the slope of the trend is zero? Show that this condition is such that the constant can be included in the cointegrating vector. The yt sequence does not have a slope if （1-a22）a10 + a12a20 = 0. Solving for

7、a10 yields a10 = -a12a20/（1-a22）. Using this relationship, the error-correction equation for yt is: yt = y（yt-1 - zt-1） + yt - a12a20/（1-a22） = y（yt-1 - zt-1 + a20/a21） + yt Since z = a21, the error-correction model for zt can be written as: zt = z（yt-1 - zt-1 + a20/a21） + zt. Thus, the normalized l

8、ong-run equilibrium relationship is yt-1 - zt-1 + a20/a21. The cointegrating vector has an intercept although the yt and zt sequences do not contain deterministic trends. 2. The data file COINT6.PRN contains the three simulated series used in sections 5 and 9. The following programs will reproduce t

9、he results. Sample Program for RATS Users all 100 open data a:coint6.prn ;* The data disk is in drive a: data（format=prn,org=obs） / y z w table ;* Produce summary statistics for y, z and w set dy = y y（t1） ;* Take first-differences set dz = z z（t1） set dw = w w（t1） linreg dy ;* Perform Dickey-Fuller

10、 test # constant y1 * Perform Augmented Dickey-Fuller test # constant y1 dy1 to 4 * Repeat the four lines above for z and w. Alternatively, you can use the procedure entitled * DFUNIT.SRC. To use DFUNIT.SRC, type the statements source c:ratsdfunit.src ;* The procedure is assumed the dfunit.src proce

11、dure is in the :*RATS directory on drive c: dfunit（lags=4） / y linreg y / residy ;* Estimate the long-run equilibrium relationship using y as # constant z w ;* the left-hand-side variable. Save the residuals as residy set dresidy = residy residy1 ;* Obtain first-difference of the residuals linreg dr

12、esidy ;* Perform the Dickey-Fuller test of the residuals # residy1* Perform the Augmented Dickey-Fuller test # residy1 dresidy1 to 4 * Repeat the 7 lines above for z and w system 1 to 3 ;* Set up the system for the error-correction model variables dy dz dw lags 1 to 2 ;* Use 2 lags of dy, dz, and dw

13、 det constant residy1 ;* Include a constant and the error-correction term. You can end（system） ;* use the residuals from the other two equilibrium relations estimate（outsigma=vsigma） * Estimate the model. Vsigma is the variance/covariance matrix errors（impulses） 3 24 vsigma ;* Perform innovation acc

14、ounting using the error-correction # 1 ; # 2 ; # 3 ;* model * To reproduce the results in Section 9, use the CATS procedure or the downloadable file * entitled johansen.src. Note that the Johansen procedure in RATS does not allow you to use * the variable name w. Redefine w using the following state

15、ment set x = wratsjohansen.src ;* It is assumed the johansen.src procedure is in the RATS directory on drive c: johansen.src（lags=2） / # y z x Note that johansen.src may inappropriately add seasonal dummy variables to your model. Moreover, there is no simple way to choose the form of the intercept t

16、erm. If you use RATS, your answers will be slightly different from those reported in the text. For example, the max and trace statistics will be reported as: lambda, lambdamax and trace test 0.32496 0.13401 0.02536 38.51272 14.10061 2.51767 2.51767 16.61829 55.131013. The file COINT_PPP.XLS contains

17、 quarterly values of German, Japanese, and Canadianwholesale prices and bilateral exchange rates with the United States. The file also contains the U.S. wholesale price level. The names on the individual series should be self-evident. For example, p_us is the U.S. price level and ex_g is the German

18、exchange rate with the United States. All variables except the mark/dollar exchange rates run from 1973:Q4 to 2001:Q4 and all have been normalized to equal 100 in 1973:Q4.A. Form the log of each variable. Estimate the long-run relationship between Canadaand the United States as log（ex_ca） = 4.12 + 0

19、.937 log（p_ca） 0.830log（p_us）Do the point estimates of the slope coefficients seem to be consistent with long-run PPP?Answer: Although the point estimates seem to be consistent with long-run PPP, you need to be a bit careful. There is a natural tendency to think that 0.937 is approximately equal to

20、unity and 0.830 is approximately equal to minus one. However, inference on the cointegrating is unwarranted since the residuals from the regression are serially correlated and prices are not necessarily weakly exogeneous. B. Since the residuals from the equilibrium regression contain a unit root, sh

21、ocks to the real exchange rate never decay. Hence, long-run PPP fails. C. A RATS program that can perform the indicated tests iscal 1973 4 4 ;* The data set begins in 1973Q4 and ends in 2004Q4all 2001:4open data a:coint_ppp.xlsdata（org=obs,format=xls）* Next, take the log of each variablelog ex_g / l

22、ex_g ; log ex_ca / lex_ca ; log ex_j / lex_jlog p_g / lp_g ; log p_j / lp_j ; log p_ca / lp_ca ; log p_us / lp_us* You should now test each for a unit root* The long-run relationship for the Canadian-U.S. rate can be obtained usinglin lex_ca / resids ; # constant lp_ca lp_us* Now, test the residuals

23、 for a unit rootdif resids / drlin dr ; # resids1 dr1 to 3* Similarly, PPP for the German-U.S. rate can be tested usinglin lex_g / resids ; # constant lp_g lp_us # resids1 dr1 to 44. The second, fourth, and fifth columns of the file labeled INT_RATES.XLS contain the interest rates paid on U.S. 3-mon

24、th, 3-year, and 10-year U.S. government securities. The data run from 1954:7 to 2002:12. These columns are labeled TBILL, r3, and r10, respectively.RATS PROGRAM cal 1954 7 12 ;* The data set runs from July 1954 to December 2002all 2002:12int_rates.xls*To test each series for a unit root using dfunit

25、.srcsource（noecho） c:winratsdfunit.srcdfunit（ttest,lags=12） tbilldfunit（ttest,lags=12） r3dfunit（ttest,lags=12） r10* The long-run relationship can be estimated using the T-bill rate as the dependent variable. linreg tbill / resids ; * Save the residuals as resids# constant r3 r10* Perform the Engle-G

26、ranger test on residsdiff resids / dresids ;linreg dresids ;# resids11 dresids11 to 9* Repeat using the 10-year rate as the dependent variablelinreg r10 / resids10# constant r3 tbilldif resids10 / dresids10 lin dresids10 ; # resids101 dresids101 to 4 ; * Note the lag length of 4* Note that some woul

27、d use 12 lags # resids101 dresids101 to 12* To estimate the error-correction model you need to difference the variablesdiff tbill / dtbill ; diff r3 / dr3 ; diff r10 / dr10* Beginning with RATS 5.0, you can estimate the error-correction model as a system of equations. Note that the resisuals from pa

28、rt B （i.e., resids） are used as the error-correction termssystem 1 to 3variables dtbill dr3 dr10lags 1 to 12det resids1end（system）estimate（noftests,outsigma=v） / 1* The multivariate AIC and SBC are calculated usingcompute aic = %nobs * %logdet + 2*（38*3）compute sbc = %nobs * %logdet + 38*3*log（%nobs）display aic = aic sbc = sbc* No