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ECON 3371 FINANCIAL ECONOMETRICS
创建日期：2024-04-10 14:54:35
所属分类：金融Finance
标签： ECON 3371
FINANCIAL ECONOMETRICS

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ECON 3371 FINANCIAL ECONOMETRICS

INDICATIVE SOLUTIONS: PRACTICAL 3

Part I: Testing for ARCH effects First, we need to prepare our data for analysis in EViews. The data contained in the file

“SPDDailyreturns.xls” are daily log returns from the S&P 500 index. It is good practice to understand the properties 

of your data through visualization and inspecting sample summary statistics, so this is what we will be doing.


'Change working directory - adjust this to the directory where your data file is stored
cd J:\teaching\p3
' Import data
import SPDailyreturns.xls range="data" @rename DLs_p500 sp500
sp500.displayname Daily S&P 500 Returns
'Inspect data
sp500.line
-.12
-.08
-.04
.00
.04
.08
.12
250 500 750 1000 1250 1500 1750 2000 2250
Daily S&P 500 Returns

We can see periods of high and low volatility. These “volatility clusters” 

are indicative of ARCH effects can be captured by ARCH/GARCH models.

sp500.hist

0
100
200
300
400
500
600
700
800
900
-0.10 -0.05 0.00 0.05 0.10
Series: SP500
Sample 1 2262
Observations 2262
Mean -0.000211
Median 0.000413
Maximum 0.109572
Minimum -0.094695
Std. Dev. 0.013607
Skewness -0.127892
Kurtosis 11.90835
Jarque-Bera 7485.717
Probability 0.000000

The Jarque-Bera statistic indicates that our return series does not follow a normal distribution, 

rejecting the null hypothesis of a normal distribution at 1% significance.

Question 1 i) We run the OLS regression as specified and save the estimation results:
' Run OLS regression
equation est_ols.ls sp500 c sp500(-1) sp500(-2)
show est_ols
Dependent Variable: SP500
Method: Least Squares
Sample (adjusted): 3 2262
Included observations: 2260 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C -0.000238 0.000283 -0.841010 0.4004
SP500(-1) -0.092581 0.020927 -4.424025 0.0000
SP500(-2) -0.110442 0.020904 -5.283339 0.0000
R-squared 0.019063 Mean dependent var -0.000195
Adjusted R-squared 0.018193 S.D. dependent var 0.013588
S.E. of regression 0.013464 Akaike info criterion -5.776256
Sum squared resid 0.409152 Schwarz criterion -5.768659
Log likelihood 6530.169 Hannan-Quinn criter. -5.773484
F-statistic 21.93011 Durbin-Watson stat 1.990220
Prob(F-statistic) 0.000000
ii) First, we save the residuals of the model, square them and inspect their properties visually
'Create residuals
est_ols.makeresid resid_ols
series resid_ols_sq = resid_ols^2
' Visually inspect residual series
est_ols.resids(s)
-8
-6
-4
-2
0
2
4
6
8
250 500 750 1000 1250 1500 1750 2000 2250

Standardized ResidualsThe residuals appear not like a white noise series – the volatility clusters are still visible. 

These clusters are also visible in the squared residual series:

resid_ols_sq.line
.000
.002
.004
.006
.008
.010
.012
250 500 750 1000 1250 1500 1750 2000 2250
RESID_OLS_SQ
Second, we regress the squared residual series on their own lags; q indicates the order of the lags to include in the regression.
' Regress residuals on two lags of itself and show results
equation est_resid.ls resid_ols_sq c resid_ols_sq(-1) resid_ols_sq(-2)
show est_resid
Dependent Variable: RESID_OLS_SQ
Method: Least Squares
Sample (adjusted): 5 2262
Included observations: 2258 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 8.41E-05 1.15E-05 7.293471 0.0000
RESID_OLS_SQ(-1) 0.122616 0.019177 6.393975 0.0000
RESID_OLS_SQ(-2) 0.412616 0.019179 21.51427 0.0000
R-squared 0.206431 Mean dependent var 0.000181
Adjusted R-squared 0.205727 S.D. dependent var 0.000569
S.E. of regression 0.000507 Akaike info criterion -12.33552
Sum squared resid 0.000579 Schwarz criterion -12.32791
Log likelihood 13929.80 Hannan-Quinn criter. -12.33274
F-statistic 293.2965 Durbin-Watson stat 2.061663
Prob(F-statistic) 0.000000

The auxiliary model is a linear model (because it is linear in parameters) but it is a nonlinear equation.

 Given that it is linear in model, we can use OLS to estimate it.

iii) We can compute the test statistic and the associated p-value manually:
' Obtain R-squared and number of observation from regression results
scalar r2 = est_resid.@r2
scalar T = est_resid.@regobs
' Compute test statistic and associated p-value
scalar Tr2 = T*r2 ' test statistic (= 466.1213)

scalar Tr2_p = @chisq(Tr2, 2) ' p-value (= 0.0000) In actual practice, we use pre-written 

commands to do this for us. In our case, this is EViews test for ARCH effects:

est_ols.hettest(type=ARCH, lags=2)
Heteroskedasticity Test: ARCH
F-statistic 293.2965 Prob. F(2,2255) 0.0000
Obs*R-squared 466.1213 Prob. Chi-Square(2) 0.0000

You can see that it yields the same result as our manual computation. Alternatively, 

we could have used the routine est_ols.archtest(2) to get the equivalent result.

iv) The null of no arch effects is rejected at the 1% significance level.

v) Determine the number of lags of the squared residual that should appear in the model. 

This could bedone using a likelihood ratio test that would compare an unrestricted version 

of the squared residual model with, say, four or five variables, and a restricted version with fewer variables.

Specify the nature of the AR model to be used for the estimation process of the model 

for the returns. It may be that more lags are required than was the case in the OLS model 

estimated above. In both cases for deriving the variance and mean equations,

 a general-to-specific modelling approach for the models can be adopted in which the initial model is 

estimated with a large number of variables that are then tested for restrictions.

PART II: Modelling conditional mean and conditional variance
1) We estimate the ARMA(5, 0) model with ARCH(1) variance process:
smpl @all ' not required but useful for contrasting with next step
equation est_ar5arch1.arch(1, 0) sp500 c AR(1 to 5)
show est_ar5arch1
Dependent Variable: SP500
Method: ML ARCH - Normal distribution (BFGS / Marquardt steps)
Sample (adjusted): 6 2262
Included observations: 2257 after adjustments
Convergence achieved after 36 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(7) + C(8)*RESID(-1)^2
Variable Coefficient Std. Error z-Statistic Prob.
C -0.000141 0.000178 -0.795535 0.4263
AR(1) -0.226978 0.011772 -19.28175 0.0000
AR(2) -0.109372 0.012877 -8.493355 0.0000
AR(3) 0.023995 0.008865 2.706577 0.0068
AR(4) -0.022661 0.008414 -2.693187 0.0071
AR(5) 0.057594 0.010497 5.486941 0.0000
Variance Equation
C 0.000108 2.62E-06 41.19644 0.0000
RESID(-1)^2 0.439309 0.030829 14.24981 0.0000
R-squared -0.004349 Mean dependent var -0.000212
Adjusted R-squared -0.006580 S.D. dependent var 0.013583
S.E. of regression 0.013628 Akaike info criterion -5.936011
Sum squared resid 0.418058 Schwarz criterion -5.915730
Log likelihood 6706.789 Hannan-Quinn criter. -5.928610
Durbin-Watson stat 1.730620
Inverted AR Roots .49 .12+.54i .12-.54i -.48-.40i
-.48+.40i

The mean equation is appears well-specified with significant coefficients on the AR term. 

The variance equation is also appears well-specified with values on the coefficients 

that meet the restrictions placed on them by the ARCH model.

2) Next, we estimate the ARMA(2, 0) model with ARCH(1) variance process, which

 can be viewed as arestricted version of the previous model.

smpl @first+5 @last ' adjust sample period to make models comparable
equation est_ar2arch1.arch(1, 0) sp500 c AR(1 to 2)
smpl @all
show est_ar2arch1
Dependent Variable: SP500
Method: ML ARCH - Normal distribution (BFGS / Marquardt steps)
Sample: 6 2262
Included observations: 2257
Convergence achieved after 20 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(4) + C(5)*RESID(-1)^2
Variable Coefficient Std. Error z-Statistic Prob.
C -0.000138 0.000163 -0.844887 0.3982
AR(1) -0.241872 0.010140 -23.85300 0.0000
AR(2) -0.110732 0.011905 -9.301113 0.0000
Variance Equation
C 0.000108 2.71E-06 39.91425 0.0000
RESID(-1)^2 0.451482 0.029376 15.36930 0.0000
R-squared -0.002721 Mean dependent var -0.000212
Adjusted R-squared -0.003610 S.D. dependent var 0.013583
S.E. of regression 0.013608 Akaike info criterion -5.933075
Sum squared resid 0.417381 Schwarz criterion -5.920399
Log likelihood 6700.475 Hannan-Quinn criter. -5.928449
Durbin-Watson stat 1.702477
Inverted AR Roots -.12+.31i -.12-.31i
We manually compute the Likelihood Ratio test and contrast it with the EViews routines.
' Compute lrtest and associated p-value
scalar lrtest = -2 * (est_ar2arch1.@logl - est_ar5arch1.@logl) ' test statistic
scalar lrtest_p = @chisq(lrtest, 3) ' p-value
There are two ways of computing the same Likelihood Ratio test, depending on whether you test
adding or dropping terms from the respective model:
' Again, in practice, we would use a pre-programmed routine
est_ar5arch1.testdrop AR(3 to 5)
est_ar2arch1.testadd AR(3 to 5)
All of these computations give us the same answer: there is significant evidence against the
restricted ARMA(2, 0) model. Therefore, our preferred model is the ARMA(5, 0) one:


3) First, we compute our model residuals and visualize the standardized residuals. From the graph,
it is clear that the volatility clusters are still present in the residual series; however, they also appear
less pronounced.
' generate standardized residuals
est_ar5arch1.makeresids(s) resids_ar5arch1
' Autocorrelation test
resids_ar5arch1.correl(12)
There appears to be serial dependence in the residuals, indicative of unmodelled influences (like
omitted variables). The Q-statistic indicates, at all lags, rejection of the null that the residuals
follow a white noise process.


' normality test
resids_ar5arch1.hist

The Jarque-Bera statistic indicates that the null of the residuals following a normal distribution is
rejected at the 1% significance level.

' ARCH test
est_ar5arch1.archtest(5)
The test for ARCH effects rejects the null hypothesis of no ARCH effects in the residual series:

4) First, we estimate the ARMA(5, 0) mean model with the ARCH(3) variance process:
equation est_ar5arch3.arch(3, 0) sp500 c AR(1 to 5)
show est_ar5arch3


The coefficients in both the mean and variance equation appear appropriate in magnitude and
Heteroskedasticity Test: ARCH
F-statistic 38.50401 Prob. F(5,2246) 0.0000
Obs*R-squared 177.7944 Prob. Chi-Square(5) 0.0000
Test Equation:
Dependent Variable: WGT_RESID^2
Method: Least Squares
Sample (adjusted): 11 2262
Included observations: 2252 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.574736 0.063846 9.001945 0.0000
WGT_RESID^2(-1) -0.089438 0.020950 -4.269167 0.0000
WGT_RESID^2(-2) 0.143249 0.020917 6.848346 0.0000
WGT_RESID^2(-3) 0.140747 0.020923 6.726793 0.0000
WGT_RESID^2(-4) 0.113029 0.020916 5.403938 0.0000
WGT_RESID^2(-5) 0.119448 0.020969 5.696555 0.0000
R-squared 0.078950 Mean dependent var 1.001150
Adjusted R-squared 0.076899 S.D. dependent var 2.467253
S.E. of regression 2.370491 Akaike info criterion 4.566732
Sum squared resid 12620.79 Schwarz criterion 4.581971
Log likelihood -5136.140 Hannan-Quinn criter. 4.572294
F-statistic 38.50401 Durbin-Watson stat 2.017424
Prob(F-statistic) 0.000000
Dependent Variable: SP500
Method: ML ARCH - Normal distribution (BFGS / Marquardt steps)
Sample (adjusted): 6 2262
Included observations: 2257 after adjustments
Convergence achieved after 52 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(7) + C(8)*RESID(-1)^2 + C(9)*RESID(-2)^2 + C(10)*RESID(
-3)^2
Variable Coefficient Std. Error z-Statistic Prob.
C 0.000188 0.000158 1.190416 0.2339
AR(1) -0.084313 0.020163 -4.181618 0.0000
AR(2) -0.086865 0.018701 -4.644999 0.0000
AR(3) -0.028070 0.015027 -1.868011 0.0618
AR(4) 0.035926 0.014127 2.543090 0.0110
AR(5) -0.075986 0.014675 -5.178003 0.0000
Variance Equation
C 5.48E-05 2.67E-06 20.56119 0.0000
RESID(-1)^2 0.117557 0.017352 6.774857 0.0000
RESID(-2)^2 0.349902 0.027200 12.86393 0.0000
RESID(-3)^2 0.239821 0.027562 8.701097 0.0000
R-squared 0.008994 Mean dependent var -0.000212
Adjusted R-squared 0.006793 S.D. dependent var 0.013583
S.E. of regression 0.013537 Akaike info criterion -6.135706
Sum squared resid 0.412504 Schwarz criterion -6.110355
Log likelihood 6934.144 Hannan-Quinn criter. -6.126454
Durbin-Watson stat 2.023685
Inverted AR Roots .43-.34i .43+.34i -.15+.61i -.15-.61i
-.63
highly significant. Next, we generate the standardized residuals and visually inspect them
generate standardized residuals
est_ar5arch3.makeresids(s) resids_ar5arch3
' Visually inspect residuals
est_ar5arch3.resids(s)
' autocorrelation and omitted variables
resids_ar5arch3.correl(12)
' normality test
resids_ar5arch3.hist
' ARCH test
est_ar5arch3.archtest(5)

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CIVL2010 ECMT2150 CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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