**************************************** *** Output for Problem A of the exam. *************************************** (A): SCA output (edited) -- input dd,jp. file 'd-fxja00.txt' JP , A 1063 BY 1 VARIABLE, IS STORED IN THE WORKSPACE -- jp=ln(jp) <== Take log-transformation -- diff old jp. new xjp. comp. <== Compute log returns 1 DIFFERENCE ORDERS ARE (1-B ) SERIES JP IS DIFFERENCED, THE RESULT IS STORED IN VARIABLE XJP SERIES XJP HAS 1062 ENTRIES -- xjp=xjp*100 <== Use percentages. -- desc xjp VARIABLE NAME IS XJP NUMBER OF OBSERVATIONS 1062 NUMBER OF MISSING VALUES 0 STATISTIC STD. ERROR STATISTIC/S.E. MEAN 0.0038 0.0185 0.2072 VARIANCE 0.3641 STD DEVIATION 0.6034 C. V. 157.2516 SKEWNESS -0.2920 0.0751 KURTOSIS 1.4874 0.1500 QUARTILE MINIMUM -2.9314 1ST QUARTILE -0.3453 MEDIAN 0.0000 3RD QUARTILE 0.3819 MAXIMUM 2.1127 -- acf xjp. maxl 12. NAME OF THE SERIES . . . . . . . . . . XJP TIME PERIOD ANALYZED . . . . . . . . . 1 TO 1062 MEAN OF THE (DIFFERENCED) SERIES . . . 0.0038 STANDARD DEVIATION OF THE SERIES . . . 0.6031 T-VALUE OF MEAN (AGAINST ZERO) . . . . 0.2073 AUTOCORRELATIONS 1- 12 -.02 .00 .02 .02 .03 -.02 .03 -.05 -.02 .03 .00 .01 ST.E. .03 .03 .03 .03 .03 .03 .03 .03 .03 .03 .03 .03 Q .3 .3 .6 .9 1.7 2.1 2.9 5.4 6.0 6.9 6.9 7.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 +----+----+----+----+----+----+----+----+----+----+ I 1 -0.02 + I + 2 0.00 + I + 3 0.02 + I + 4 0.02 + I + 5 0.03 + IX+ 6 -0.02 +XI + 7 0.03 + IX+ 8 -0.05 +XI + 9 -0.02 +XI + 10 0.03 + IX+ 11 0.00 + I + 12 0.01 + I + -- p=1-cdfc(7.0,12) -- print p P IS A 1 BY 1 VARIABLE .858 -- print xjp. span 1060,1062. time 1060 1061 1062 -.376 -.160 -.123 -- (B) Splus output > x=matrix(scan(file='d-fxja00.txt'),2) > jp=x[2,] > jp=log(jp) > xjp=diff(jp) > xjp=xjp*100 > summaryStats(xjp) Sample Quantiles: min 1Q median 3Q max -2.931 -0.3438 0 0.3825 2.113 Sample Moments: mean std skewness kurtosis 0.003837 0.6034 -0.2924 4.496 Number of Observations: 1062 > sqrt(6/1062) [1] 0.0751646 > sqrt(24/1062) [1] 0.1503292 > autocorTest(xjp,lag.n=12) Test for Autocorrelation: Ljung-Box Null Hypothesis: no autocorrelation Test Statistics: Test Stat 6.9959 p.value 0.8579 Dist. under Null: chi-square with 12 degrees of freedom Total Observ.: 1062 > xjp[1060:1062] [1] -0.3758343 -0.1601583 -0.1226473 ********************************** *** SCA output for Problem B *** ********************************** (Output is edited to simplify the presentation) -- input da,temp. file 'd-tempord.txt' -- acf temp. maxl 800 NAME OF THE SERIES . . . . . . . . . . TEMP TIME PERIOD ANALYZED . . . . . . . . . 1 TO 8395 MEAN OF THE (DIFFERENCED) SERIES . . . 49.5275 STANDARD DEVIATION OF THE SERIES . . . 19.9237 T-VALUE OF MEAN (AGAINST ZERO) . . . . 227.7648 AUTOCORRELATIONS 1- 12 .94 .88 .85 .83 .82 .81 .80 .80 .79 .79 .79 .78 ST.E. .01 .02 .02 .03 .03 .03 .03 .04 .04 .04 .04 .04 709-720 .68 .69 .69 .69 .70 .70 .70 .70 .71 .71 .71 .71 ST.E. .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 721-732 .71 .72 .72 .72 .72 .72 .72 .72 .72 .71 .72 .72 ST.E. .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 733-744 .72 .71 .71 .71 .71 .71 .71 .71 .70 .70 .69 .69 ST.E. .22 .22 .22 .22 .22 .22 .22 .22 .23 .23 .23 .23 745-756 .68 .67 .67 .67 .66 .66 .65 .65 .64 .64 .64 .63 ST.E. .23 .23 .23 .23 .23 .23 .23 .23 .23 .23 .23 .23 -- acf temp. dfor 365. maxl 400 365 DIFFERENCE ORDERS. . . . . . . . . . . (1-B ) NAME OF THE SERIES . . . . . . . . . . TEMP TIME PERIOD ANALYZED . . . . . . . . . 1 TO 8395 MEAN OF THE (DIFFERENCED) SERIES . . . 0.1204 STANDARD DEVIATION OF THE SERIES . . . 12.7701 T-VALUE OF MEAN (AGAINST ZERO) . . . . 0.8450 AUTOCORRELATIONS 1- 12 .73 .45 .31 .23 .17 .14 .11 .10 .09 .08 .08 .09 ST.E. .01 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 349-360 -.03 -.04 -.05 -.06 -.06 -.07 -.07 -.06 -.07 -.08 -.08 -.10 ST.E. .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 361-372 -.13 -.18 -.23 -.34 -.46 -.32 -.20 -.14 -.09 -.07 -.07 -.06 ST.E. .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 -- tsm m1. model (1,2)temp(365)=(365)noise. -- estim m1. hold resi(r1). THE FOLLOWING ANALYSIS IS BASED ON TIME SPAN 1 THRU 8395 SUMMARY FOR UNIVARIATE TIME SERIES MODEL -- M1 ----------------------------------------------------------------------- VARIABLE TYPE OF ORIGINAL DIFFERENCING VARIABLE OR CENTERED 365 TEMP RANDOM ORIGINAL (1-B ) ----------------------------------------------------------------------- PARAMETER VARIABLE NUM./ FACTOR ORDER CONS- VALUE STD T LABEL NAME DENOM. TRAINT ERROR VALUE 1 TEMP MA 1 365 NONE .7949 .0067 119.20 2 TEMP AR 1 1 NONE .8574 .0109 78.32 3 TEMP AR 1 2 NONE -.1858 .0109 -16.97 EFFECTIVE NUMBER OF OBSERVATIONS . . 8028 R-SQUARE . . . . . . . . . . . . . . 0.882 RESIDUAL STANDARD ERROR. . . . . . . 0.684230E+01 -- acf r1. maxl 400. NAME OF THE SERIES . . . . . . . . . . R1 TIME PERIOD ANALYZED . . . . . . . . .368 TO 8395 MEAN OF THE (DIFFERENCED) SERIES . . . 0.1337 STANDARD DEVIATION OF THE SERIES . . . 6.8410 T-VALUE OF MEAN (AGAINST ZERO) . . . . 1.7516 AUTOCORRELATIONS 1- 12 .02 -.08 .04 .04 .03 .02 .01 .02 -.00 .02 .01 .02 ST.E. .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 361-372 -.00 -.03 -.02 .01 .01 .01 -.00 -.02 .01 .00 -.03 -.00 ST.E. .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 -- tsm m1. model (1,2)temp(365)=(2)(365)noise. <== Refined final model!! -- estim m1. hold resi(r1) THE FOLLOWING ANALYSIS IS BASED ON TIME SPAN 1 THRU 8395 SUMMARY FOR UNIVARIATE TIME SERIES MODEL -- M1 ----------------------------------------------------------------------- VARIABLE TYPE OF ORIGINAL DIFFERENCING VARIABLE OR CENTERED 365 TEMP RANDOM ORIGINAL (1-B ) ----------------------------------------------------------------------- PARAMETER VARIABLE NUM./ FACTOR ORDER CONS- VALUE STD T LABEL NAME DENOM. TRAINT ERROR VALUE 1 TEMP MA 1 2 NONE .1528 .0151 10.11 2 TEMP MA 2 365 NONE .7943 .0066 119.86 3 TEMP AR 1 1 NONE .8776 .0111 78.89 4 TEMP AR 1 2 NONE -.1294 .0123 -10.51 EFFECTIVE NUMBER OF OBSERVATIONS . . 8028 R-SQUARE . . . . . . . . . . . . . . 0.883 RESIDUAL STANDARD ERROR. . . . . . . 0.680042E+01 -- acf r1. maxl 400. NAME OF THE SERIES . . . . . . . . . . R1 TIME PERIOD ANALYZED . . . . . . . . .368 TO 8395 MEAN OF THE (DIFFERENCED) SERIES . . . 0.1146 STANDARD DEVIATION OF THE SERIES . . . 6.7995 T-VALUE OF MEAN (AGAINST ZERO) . . . . 1.5099 AUTOCORRELATIONS 1- 12 -.00 -.00 -.02 .01 .00 .00 -.01 .01 -.01 .02 .01 .02 ST.E. .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 Q .0 .0 2.7 3.5 3.6 3.8 4.2 5.4 6.0 9.9 10.2 12.6 361-372 -.00 -.03 -.02 .01 .01 .01 .00 -.01 .01 .00 -.02 -.00 ST.E. .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 Q 444 451 455 455 457 457 457 459 460 460 465 465 -- p=1-cdfc(12.6,8) -- print p 0.126 -- r1s=r1*r1 -- acf r1s. maxl 70 NAME OF THE SERIES . . . . . . . . . . R1S TIME PERIOD ANALYZED . . . . . . . . .368 TO 8395 MEAN OF THE (DIFFERENCED) SERIES . . . 46.2458 STANDARD DEVIATION OF THE SERIES . . . 75.5217 T-VALUE OF MEAN (AGAINST ZERO) . . . . 54.8661 AUTOCORRELATIONS 1- 12 .10 .08 .09 .10 .08 .08 .09 .07 .07 .05 .08 .07 ST.E. .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 Q 77.3 131 201 277 331 384 445 480 514 531 578 616 ************************************ *** Splus output for Problem C ** ************************************ > x=matrix(scan(file='m-decile1510.txt'),4) > dec1=x[2,] > jan=rep(c(1,0,0,0,0,0,0,0,0,0,0,0),44) <== create indicator for January > fit=OLS(dec1~jan) > summary(fit) Call: OLS(formula = dec1 ~ jan) Residuals: Min 1Q Median 3Q Max -0.2916 -0.0409 -0.0016 0.0383 0.4011 Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 0.0066 0.0032 2.0774 0.0382 jan 0.1365 0.0110 12.4191 0.0000 Regression Diagnostics: R-Squared 0.2267 Adjusted R-Squared 0.2253 Durbin-Watson Stat 1.4600 Residual Diagnostics: Stat P-Value Jarque-Bera 287.0815 0.0000 Ljung-Box 67.4913 0.0000 Residual standard error: 0.06981 on 526 degrees of freedom F-statistic: 154.2 on 1 and 526 degrees of freedom, the p-value is 0 > acf(fit$residuals,lag.max=24) Call: acf(x = fit$residuals, lag.max = 24) Autocorrelation matrix: lag fit 1 0 1.0000 2 1 0.2665 3 2 0.0191 4 3 0.0301 5 4 0.0409 6 5 0.0610 7 6 0.0131 8 7 -0.0054 9 8 -0.0867 10 9 -0.0171 11 10 0.0261 12 11 0.0993 13 12 0.0412 14 13 -0.0503 15 14 -0.0474 16 15 -0.0036 17 16 -0.0038 18 17 0.0204 19 18 -0.0467 20 19 -0.0978 21 20 -0.0263 22 21 0.0033 23 22 -0.0305 24 23 -0.0080 25 24 -0.0667 > fit1=OLS(dec1~jan+ar(1)) > summary(fit1) Call: OLS(formula = dec1 ~ jan + ar(1)) Residuals: Min 1Q Median 3Q Max -0.2800 -0.0367 -0.0017 0.0338 0.4176 Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 0.0019 0.0031 0.6077 0.5437 jan 0.1427 0.0106 13.3995 0.0000 lag1 0.2476 0.0368 6.7354 0.0000 Regression Diagnostics: R-Squared 0.2934 Adjusted R-Squared 0.2907 Durbin-Watson Stat 1.9395 Residual Diagnostics: Stat P-Value Jarque-Bera 594.6510 0.0000 Ljung-Box 30.3557 0.2984 Residual standard error: 0.06686 on 524 degrees of freedom F-statistic: 108.8 on 2 and 524 degrees of freedom, the p-value is 0 > acf((fit1$residuals)^2,lag.max=12) Call: acf(x = (fit1$residuals)^2, lag.max = 12) Autocorrelation matrix: lag fit1 1 0 1.0000 2 1 0.1652 3 2 0.0368 4 3 0.0254 5 4 0.0165 6 5 0.0103 7 6 -0.0157 8 7 -0.0121 9 8 0.0151 10 9 0.0622 11 10 0.0472 12 11 0.0704 13 12 0.0350 > archTest(fit1$residuals,lag.n=5) Test for ARCH Effects: LM Test Null Hypothesis: no ARCH effects Test Statistics: Test Stat 14.3949 p.value 0.0133 Dist. under Null: chi-square with 5 degrees of freedom Total Observ.: 527 > fit2=garch(dec1~jan+ar(1),~garch(1,1),cond.dist='ged') Convergence reached. > summary(fit2) Call: garch(formula.mean = dec1 ~ jan + ar(1), formula.var = ~ garch(1, 1), cond.dist = "ged") Mean Equation: dec1 ~ jan + ar(1) Conditional Variance Equation: ~ garch(1, 1) Conditional Distribution: ged with estimated parameter 1.140752 and standard error 0.0734265 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) C 0.0012242 0.0023391 0.5234 0.30046 AR(1) 0.3360363 0.0331234 10.1450 0.00000 jan 0.1124912 0.0076735 14.6597 0.00000 A 0.0004554 0.0002535 1.7961 0.03653 ARCH(1) 0.1053558 0.0509585 2.0675 0.01959 GARCH(1) 0.7904901 0.0918632 8.6051 0.00000 -------------------------------------------------------------- AIC(7) = -1436.628 BIC(7) = -1406.744 Normality Test: -------------------------------------------------------------- Jarque-Bera P-value Shapiro-Wilk P-value 766.1 0 0.9553 0 Ljung-Box test for standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 8.539 0.7417 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 3.747 0.9876 12 Lagrange multiplier test: -------------------------------------------------------------- Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 Lag 7 Lag 8 Lag 9 -0.316 -0.1671 -0.6171 -0.3743 -0.6711 -0.8427 -0.7048 -0.05478 -0.02364 Lag 10 Lag 11 Lag 12 C 1.034 0.2581 0.1121 -0.1026 TR^2 P-value F-stat P-value 3.598 0.9897 0.3293 0.9994 ****************************************** *** Splus output for Problem D *** ****************************************** > module(finmetrics) S+FinMetrics Version 1.0 for Linux 2.2.12 : 2002 > x=matrix(scan(file='d-yhoo9703.txt'),2) > yhoo=log(1+x[2,]) > > autocorTest(yhoo,lag.n=12) Test for Autocorrelation: Ljung-Box Null Hypothesis: no autocorrelation Test Statistics: Test Stat 16.2726 p.value 0.1791 Dist. under Null: chi-square with 12 degrees of freedom Total Observ.: 1761 > archTest(yhoo,lag.n=12) Test for ARCH Effects: LM Test Null Hypothesis: no ARCH effects Test Statistics: Test Stat 84.4676 p.value 0.0000 Dist. under Null: chi-square with 12 degrees of freedom Total Observ.: 1761 > fit=garch(yhoo~1,~garch(1,1)) Convergence R-Square = 2.476282e-05 is less than tolerance = 0.0001 Convergence reached. > summary(fit) Call: garch(formula.mean = yhoo ~ 1, formula.var = ~ garch(1, 1)) Mean Equation: yhoo ~ 1 Conditional Variance Equation: ~ garch(1, 1) Conditional Distribution: gaussian -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) C 3.241e-03 1.082e-03 2.994 0.001395 A 3.185e-05 9.238e-06 3.448 0.000289 ARCH(1) 6.681e-02 7.205e-03 9.273 0.000000 GARCH(1) 9.228e-01 8.313e-03 111.008 0.000000 -------------------------------------------------------------- AIC(4) = -5698.64 BIC(4) = -5676.745 Normality Test: -------------------------------------------------------------- Jarque-Bera P-value Shapiro-Wilk P-value 318.9 0 0.9789 3.362e-12 Ljung-Box test for standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 15.38 0.2213 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 10.41 0.5799 12 Lagrange multiplier test: -------------------------------------------------------------- Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 Lag 7 Lag 8 Lag 9 Lag 10 -0.2033 -0.2484 0.2694 0.06366 0.196 -0.9115 -1.455 -0.772 1.441 -0.6872 Lag 11 Lag 12 C -0.9363 -0.1382 1.845 TR^2 P-value F-stat P-value 10.85 0.5419 0.9924 0.5632 > predit(fit,5) Problem: Couldn't find a function definition for "predit" > predict(fit,5) $series.pred: [1] 0.003240637 0.003240637 0.003240637 0.003240637 0.003240637 $sigma.pred: [1] 0.02765700 0.02808530 0.02850279 0.02890999 0.02930738 $asymp.sd: [1] 0.05526032 > fit1=garch(yhoo~1,~egarch(1,1),leverage=T,cond.dist='ged') Convergence R-Square = 3.823533e-05 is less than tolerance = 0.0001 Convergence reached. > summary(fit1) Call: garch(formula.mean = yhoo ~ 1, formula.var = ~ egarch(1, 1), leverage = T, cond.dist = "ged") Mean Equation: yhoo ~ 1 Conditional Variance Equation: ~ egarch(1, 1) Conditional Distribution: ged with estimated parameter 1.445849 and standard error 0.05991048 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) C 0.0012 0.001004 1.195 1.161e-01 A -0.4108 0.089946 -4.567 2.650e-06 ARCH(1) 0.2037 0.030807 6.613 2.496e-11 GARCH(1) 0.9583 0.012394 77.320 0.000e+00 LEV(1) -0.2430 0.078605 -3.091 1.014e-03 -------------------------------------------------------------- AIC(6) = -5779.968 BIC(6) = -5747.126 Normality Test: -------------------------------------------------------------- Jarque-Bera P-value Shapiro-Wilk P-value 217.1 0 0.9817 2.928e-06 Ljung-Box test for standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 15.37 0.2217 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 7.736 0.8054 12 Lagrange multiplier test: -------------------------------------------------------------- Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 Lag 7 Lag 8 Lag 9 Lag 10 -0.7664 -0.2433 0.2679 -0.1959 0.5049 -1.004 -1.468 -0.599 1.29 -0.5052 Lag 11 Lag 12 C -1.032 -0.1693 0.7717 TR^2 P-value F-stat P-value 8.42 0.7515 0.7691 0.7821 > predict(fit1,5) $series.pred: [1] 0.001199869 0.001199869 0.001199869 0.001199869 0.001199869 $sigma.pred: [1] 0.02643190 0.02729338 0.02814527 0.02898657 0.02981637 $asymp.sd: [1] 0.05101367