**************************************************************** ** Problem A **************************************************************** ** SCA output for log return of SP500 index ** -- input dd,wmt,sp. file 'd-wmtsp9604.txt' SP , A 2267 BY 1 VARIABLE, IS STORED IN THE WORKSPACE -- sp=ln(sp+1)*100 -- desc sp VARIABLE NAME IS SP NUMBER OF OBSERVATIONS 2267 NUMBER OF MISSING VALUES 0 STATISTIC STD. ERROR STATISTIC/S.E. MEAN 0.0299 0.0252 1.1870 VARIANCE 1.4342 STD DEVIATION 1.1976 C. V. 40.1122 SKEWNESS -0.0939 0.0514 KURTOSIS 2.6824 0.1028 -- print sp. span 2266,2267. SP IS A 2267 BY 1 VARIABLE .008 -.134 -- acf sp. maxl 12 NAME OF THE SERIES . . . . . . . . . . SP TIME PERIOD ANALYZED . . . . . . . . . 1 TO 2267 MEAN OF THE (DIFFERENCED) SERIES . . . 0.0299 STANDARD DEVIATION OF THE SERIES . . . 1.1973 T-VALUE OF MEAN (AGAINST ZERO) . . . . 1.1873 AUTOCORRELATIONS 1- 12 -.01 -.03 -.03 .01 -.04 -.01 -.04 .01 .01 .02 -.03 .05 ST.E. .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 Q .4 2.2 4.1 4.2 8.6 8.9 12.8 13.2 13.3 13.9 15.7 22.3 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 +----+----+----+----+----+----+----+----+----+----+ I 1 -0.01 +I+ 2 -0.03 XI+ 3 -0.03 XI+ 4 0.01 +I+ 5 -0.04 XI+ 6 -0.01 +I+ 7 -0.04 XI+ 8 0.01 +I+ 9 0.01 +I+ 10 0.02 +I+ 11 -0.03 XI+ 12 0.05 +IX -- p=1-cdfc(13.9,10) -- print p P IS A 1 BY 1 VARIABLE .178 -- spsq=sp*sp -- acf spsq. maxl 12. NAME OF THE SERIES . . . . . . . . . . SPSQ TIME PERIOD ANALYZED . . . . . . . . . 1 TO 2267 MEAN OF THE (DIFFERENCED) SERIES . . . 1.4345 STANDARD DEVIATION OF THE SERIES . . . 3.1015 T-VALUE OF MEAN (AGAINST ZERO) . . . . 22.0216 AUTOCORRELATIONS 1- 12 .18 .17 .16 .11 .17 .11 .13 .12 .08 .11 .11 .10 ST.E. .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 Q 72.4 137 199 227 297 324 363 395 409 439 465 490 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 +----+----+----+----+----+----+----+----+----+----+ I 1 0.18 +I+XXX 2 0.17 +I+XXX 3 0.16 +I+XXX 4 0.11 +I+XX 5 0.17 +I+XXX 6 0.11 +I+XX 7 0.13 +I+XX 8 0.12 +I+XX 9 0.08 +I+X 10 0.11 +I+XX 11 0.11 +I+XX 12 0.10 +I+XX ** Splus output *** > sp=log(da[,3]+1)*100 > summaryStats(sp) Sample Moments: mean std skewness kurtosis 0.02986 1.198 -0.09398 5.687 Number of Observations: 2267 > autocorTest(sp,lag=10) Test for Autocorrelation: Ljung-Box Null Hypothesis: no autocorrelation Test Statistics: Test Stat 13.8858 p.value 0.1783 Dist. under Null: chi-square with 10 degrees of freedom > archTest(sp,lag=12) Test for ARCH Effects: LM Test Null Hypothesis: no ARCH effects Test Statistics: Test Stat 207.586 p.value 0.000 Dist. under Null: chi-square with 12 degrees of freedom ************************************************************ ** Problem B ** ************************************************************ -- input yy,mm,dd,pg. file 'eps-pg.txt' YY , A 45 BY 1 VARIABLE, IS STORED IN THE WORKSPACE .... PG , A 45 BY 1 VARIABLE, IS STORED IN THE WORKSPACE -- pg=ln(pg) % Log-transformation -- acf pg. maxl 12. NAME OF THE SERIES . . . . . . . . . . PG TIME PERIOD ANALYZED . . . . . . . . . 1 TO 45 MEAN OF THE (DIFFERENCED) SERIES . . . -0.1324 STANDARD DEVIATION OF THE SERIES . . . 0.6300 T-VALUE OF MEAN (AGAINST ZERO) . . . . -1.4092 AUTOCORRELATIONS -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 +----+----+----+----+----+----+----+----+----+----+ I %% Show strong 1 -0.03 + XI + seasonality. 2 0.01 + I + 3 -0.03 + XI + 4 0.86 + IXXXXXX+XXXXXXXXXXXXXXX 5 -0.07 + XXI + 6 -0.04 + XI + 7 -0.07 + XXI + 8 0.73 + IXXXXXXXXXXX+XXXXXX 9 -0.11 + XXXI + 10 -0.09 + XXI + 11 -0.11 + XXXI + 12 0.60 + IXXXXXXXXXXXXX+X -- iden pg. dfor 4. maxl 16. 4 DIFFERENCE ORDERS. . . . . . . . . . . (1-B ) NAME OF THE SERIES . . . . . . . . . . PG TIME PERIOD ANALYZED . . . . . . . . . 1 TO 45 MEAN OF THE (DIFFERENCED) SERIES . . . 0.0972 STANDARD DEVIATION OF THE SERIES . . . 0.0559 T-VALUE OF MEAN (AGAINST ZERO) . . . . 11.1461 AUTOCORRELATIONS -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 +----+----+----+----+----+----+----+----+----+----+ I 1 0.37 + IXXXXXXX+X 2 0.28 + IXXXXXXX + 3 0.17 + IXXXX + 4 -0.13 + XXXI + 5 0.00 + I + 6 0.14 + IXXX + 7 -0.01 + I + 8 0.03 + IX + 9 0.15 + IXXXX + 10 0.00 + I + 11 0.14 + IXXXX + 12 0.00 + I + 13 -0.29 + XXXXXXXI + 14 -0.19 + XXXXXI + 15 -0.29 + XXXXXXXI + 16 -0.29 + XXXXXXXI + PARTIAL AUTOCORRELATIONS -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 +----+----+----+----+----+----+----+----+----+----+ I 1 0.37 + IXXXXXXX+X 2 0.16 + IXXXX + 3 0.03 + IX + 4 -0.28 +XXXXXXXI + 5 0.10 + IXXX + 6 0.25 + IXXXXXX + 7 -0.11 + XXXI + 8 -0.14 + XXXI + 9 0.22 + IXXXXXX + 10 0.04 + IX + 11 0.04 + IX + 12 -0.24 + XXXXXXI + 13 -0.31 XXXXXXXXI + 14 0.08 + IXX + 15 -0.03 + XI + 16 -0.20 + XXXXXI + -- tsm m1. model (1)pg(4)=c+(4)noise. % This model is chosen among 3 -- % possible candidate models. estim m1. hold resi(r1). method exact. THE FOLLOWING ANALYSIS IS BASED ON TIME SPAN 1 THRU 45 NONLINEAR ESTIMATION TERMINATED DUE TO: RELATIVE CHANGE IN THE STANDARD ERROR LESS THAN 0.1000D-02 SUMMARY FOR UNIVARIATE TIME SERIES MODEL -- M1 ----------------------------------------------------------------------- VARIABLE TYPE OF ORIGINAL DIFFERENCING VARIABLE OR CENTERED 4 PG RANDOM ORIGINAL (1-B ) ----------------------------------------------------------------------- PARAMETER VARIABLE NUM./ FACTOR ORDER CONS- VALUE STD T LABEL NAME DENOM. TRAINT ERROR VALUE 1 C CNST 1 0 NONE .0508 .0152 3.35 2 PG MA 1 4 NONE .3073 .1484 2.07 3 PG AR 1 1 NONE .4712 .1446 3.26 EFFECTIVE NUMBER OF OBSERVATIONS . . 40 R-SQUARE . . . . . . . . . . . . . . 0.994 RESIDUAL STANDARD ERROR. . . . . . . 0.501563E-01 -- acf r1. NAME OF THE SERIES . . . . . . . . . . R1 TIME PERIOD ANALYZED . . . . . . . . . 6 TO 45 MEAN OF THE (DIFFERENCED) SERIES . . . 0.0001 STANDARD DEVIATION OF THE SERIES . . . 0.0501 T-VALUE OF MEAN (AGAINST ZERO) . . . . 0.0151 AUTOCORRELATIONS 1- 12 -.13 .22 .12 -.02 -.01 .23 -.04 -.01 .11 -.09 .14 .05 ST.E. .16 .16 .17 .17 .17 .17 .18 .18 .18 .18 .18 .18 Q .8 2.9 3.6 3.6 3.6 6.3 6.4 6.4 7.0 7.5 8.6 8.8 13- 24 -.29 .03 -.16 -.23 .05 -.09 -.12 -.06 .03 -.11 .03 -.18 ST.E. .18 .20 .20 .20 .21 .21 .21 .21 .21 .21 .21 .21 Q 13.9 13.9 15.7 19.3 19.5 20.2 21.3 21.6 21.7 22.8 22.9 26.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 +----+----+----+----+----+----+----+----+----+----+ I 1 -0.13 + XXXI + 2 0.22 + IXXXXX + 3 0.12 + IXXX + 4 -0.02 + I + 5 -0.01 + I + 6 0.23 + IXXXXXX + 7 -0.04 + XI + 8 -0.01 + I + 9 0.11 + IXXX + 10 -0.09 + XXI + 11 0.14 + IXXX + 12 0.05 + IX + 13 -0.29 + XXXXXXXI + 14 0.03 + IX + 15 -0.16 + XXXXI + 16 -0.23 + XXXXXXI + 17 0.05 + IX + 18 -0.09 + XXI + 19 -0.12 + XXXI + 20 -0.06 + XI + 21 0.03 + IX + 22 -0.11 + XXXI + 23 0.03 + IX + 24 -0.18 + XXXXI + -- fore m1. nofs 4. hold forecast(fst) ---------------------------------- 4 FORECASTS, BEGINNING AT 45 ---------------------------------- TIME FORECAST STD. ERROR ACTUAL IF KNOWN 46 -0.0922 0.0502 47 1.3738 0.0554 48 0.1978 0.0566 49 0.2197 0.0568 -- eps=exp(fst) % Take anti-log transformation. -- print eps EPS IS A 4 BY 1 VARIABLE .912 3.950 1.219 1.246 ** Splus output *** ** Note that the analysis is on (1-B^4)x_t ** > da=read.table('eps-pg.txt') > pg=log(da[,4]) > wt=diff(pg,lag=4) % seasonally differenced series. > > m3 = list(list(order=c(1,0,0)),list(order=c(0,0,1),period=4)) > m3fit=arima.mle(wt,model=m3,xreg=1) > m3fit Call: arima.mle(x = wt, model = m3, xreg = 1) Method: Maximum Likelihood Model : Coefficients: Variance-Covariance Matrix: ar(1) ma(4) ar(1) 0.019623555 0.002196155 % This part provides estimates of variances. ma(4) 0.002196155 0.022848735 Coeffficients for regressor(s): intercept [1] 0.09608 Optimizer has converged > m3fit$model [[1]]: [[1]]$order: [1] 1 0 0 [[1]]$ar: [1] 0.4727593 [[1]]$ndiff: [1] 0 [[2]]: [[2]]$order: [1] 0 0 1 [[2]]$period: [1] 4 [[2]]$ndiff: [1] 0 [[2]]$ma: [1] 0.3096479 %% For AR(1) part, the const = (1-phi)*mu, where mu is reg.coef. > const=m3fit$reg.coef*(1-m3fit$model[[1]]$ar[1]) > const [1] 0.05065575 > arima.forecast(wt,m3fit$model,4) $mean: [1] 0.008713 -0.012479 -0.040042 -0.034786 % forecasts of wt. $std.err: [1] 0.086118 0.095257 0.097182 0.097607 > pg[40:45] %% pg(t) = pg(t-1)+wt(t) [1] -0.040822 0.029559 -0.174353 1.278152 0.113329 0.122218 ** Output of R is similar to that of Splus. ** ********************************************************************* ** Problem C ** ******************************************************************** > da=read.table("d-wmtsp9904.txt") > wmt=log(da[,2]+1)*100 > sp5=log(da[,3]+1)*100 > > m1=OLS(wmt~sp5) > summary(m1) Call: OLS(formula = wmt ~ sp5) Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 0.0205 0.0463 0.4432 0.6577 sp5 0.9606 0.0370 25.9670 0.0000 Regression Diagnostics: R-Squared 0.3093 Adjusted R-Squared 0.3088 Durbin-Watson Stat 2.0558 Residual Diagnostics: Stat P-Value Jarque-Bera 696.5068 0.0000 Ljung-Box 46.6781 0.0351 > archTest(m1$residuals,lag=10) Test for ARCH Effects: LM Test Null Hypothesis: no ARCH effects Test Statistics: Test Stat 147.1794 p.value 0.0000 Dist. under Null: chi-square with 10 degrees of freedom Total Observ.: 1508 > > m2=garch(wmt~-1+sp5,~garch(1,1),cond.dist='t') Warning messages: The estimated asymptotic variance is not well-defined. in: garch(wmt ~ -1 + sp5, ~ garch(1, 1), cond.dist = "t") > summary(m2) Call: garch(formula.mean = wmt ~ -1 + sp5, formula.var = ~ garch(1, 1), cond.dist = "t") Mean Equation: wmt ~ -1 + sp5 Conditional Variance Equation: ~ garch(1, 1) Conditional Distribution: t with estimated parameter 6.412057 and standard error 0.9888125 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) sp5 9.386e-01 0.027984 33.5397 0.000e+00 A -4.286e-05 0.003129 -0.0137 4.945e-01 ARCH(1) 3.770e-02 0.007969 4.7304 1.226e-06 GARCH(1) 9.629e-01 0.007388 130.3438 0.000e+00 -------------------------------------------------------------- AIC(5) = 5487.012 BIC(5) = 5513.605 Normality Test: -------------------------------------------------------------- Jarque-Bera P-value Shapiro-Wilk P-value 388.5 0 0.9846 0.03289 Ljung-Box test for standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 14.44 0.2734 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 17.38 0.1358 12 > m3=garch(wmt~-1+sp5,~egarch(2,1),leverage=T,cond.dist='t') > summary(m3) Call: garch(formula.mean = wmt ~ -1 + sp5, formula.var = ~ egarch(2, 1), leverage = T, cond.dist = "t") Mean Equation: wmt ~ -1 + sp5 Conditional Variance Equation: ~ egarch(2, 1) Conditional Distribution: t with estimated parameter 6.792428 and standard error 1.110971 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) sp5 0.93964 0.027459 34.219 0.000e+00 A -0.07062 0.014090 -5.012 3.007e-07 ARCH(1) 0.28157 0.064359 4.375 6.492e-06 ARCH(2) -0.18974 0.063337 -2.996 1.391e-03 GARCH(1) 0.99901 0.001978 505.048 0.000e+00 LEV(1) -0.39957 0.187722 -2.129 1.673e-02 LEV(2) -0.61677 0.333972 -1.847 3.249e-02 -------------------------------------------------------------- AIC(8) = 5478.147 BIC(8) = 5520.695 Normality Test: -------------------------------------------------------------- Jarque-Bera P-value Shapiro-Wilk P-value 223.4 0 0.982 0.0001688 Ljung-Box test for standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 14.97 0.2433 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 12.64 0.396 12 > predict(m3,3) $series.pred: [1] 0 0 0 $sigma.pred: [1] 0.7081759 0.7267398 0.7409259 $asymp.sd: [1] 3.792484 **************************************************************** ** Problem D ** **************************************************************** ** Daily log returns, in percentage, of Home Depot Stock. ** > da=read.table('d-hd9504.txt') % Load data > hd=log(da[,2]+1) % Convert into log return > hd=hd*100 % Use percentage log returns > autocorTest(hd,lag=10) Test for Autocorrelation: Ljung-Box Null Hypothesis: no autocorrelation Test Statistics: Test Stat 19.4319 p.value 0.0351 > acf(hd,lag.max=10) Autocorrelation matrix: lag hd 1 0 1.0000 2 1 0.0265 % The Q(10) is marginally significant, but 3 2 -0.0507 % all ACFs are small, so the mean equation 4 3 -0.0018 % is set to a constant. 5 4 -0.0253 6 5 -0.0227 7 6 -0.0358 8 7 -0.0118 9 8 0.0278 10 9 0.0041 11 10 -0.0324 > m1 = garch(hd~1,~garch(1,1),cond.dist='t') > summary(m1) Call: garch(formula.mean = hd ~ 1, formula.var = ~ garch(1, 1), cond.dist = "t") Mean Equation: hd ~ 1 Conditional Variance Equation: ~ garch(1, 1) Conditional Distribution: t with estimated parameter 6.209725 and standard error 0.5740383 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) C 0.08183 0.035867 2.282 1.130e-02 A 0.04163 0.015912 2.616 4.471e-03 ARCH(1) 0.04856 0.008476 5.730 5.627e-09 GARCH(1) 0.94394 0.009203 102.568 0.000e+00 -------------------------------------------------------------- AIC(5) = 10801.89 BIC(5) = 10831.05 Ljung-Box test for standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 16.74 0.1595 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 2.051 0.9993 12 > m3=garch(hd~1,~egarch(1,1),leverage=T,cond.dist='t') > summary(m3) Call: garch(formula.mean = hd ~ 1, formula.var = ~ egarch(1, 1), leverage = T, cond.dist = "t") Mean Equation: hd ~ 1 Conditional Variance Equation: ~ egarch(1, 1) Conditional Distribution: t with estimated parameter 6.684178 and standard error 0.6430938 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) C 0.04918 0.035687 1.378 8.414e-02 A -0.06316 0.012746 -4.955 3.849e-07 ARCH(1) 0.10325 0.017275 5.977 1.301e-09 GARCH(1) 0.98935 0.003414 289.800 0.000e+00 LEV(1) -0.54636 0.143523 -3.807 7.207e-05 -------------------------------------------------------------- AIC(6) = 10759.36 BIC(6) = 10794.35 Ljung-Box test for standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 15.92 0.1949 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 2.982 0.9957 12 > predict(m3,5) $series.pred: [1] 0.04918154 0.04918154 0.04918154 0.04918154 0.04918154 $sigma.pred: [1] 1.183769 1.195266 1.206750 1.218221 1.229677 $asymp.sd: [1] 2.465928 > m4=garch(hd~1+var.in.mean,~garch(1,1),cond.dist='t') > summary(m4) Call: garch(formula.mean = hd ~ 1 + var.in.mean, formula.var = ~ garch(1, 1), cond.dist = "t") Mean Equation: hd ~ 1 + var.in.mean Conditional Variance Equation: ~ garch(1, 1) Conditional Distribution: t with estimated parameter 6.213237 and standard error 0.574506 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) C 0.058003 0.064232 0.9030 1.833e-01 ARCH-IN-MEAN 0.006294 0.013896 0.4529 3.253e-01 A 0.041880 0.015986 2.6198 4.425e-03 ARCH(1) 0.048967 0.008517 5.7492 5.027e-09 GARCH(1) 0.943522 0.009240 102.1157 0.000e+00 -------------------------------------------------------------- AIC(6) = 10803.68 BIC(6) = 10838.67 Ljung-Box test for standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 16.45 0.1716 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 2.019 0.9994 12