Infer conditional variances of conditional variance models
Infer conditional variances from a GARCH(1,1) model with known coefficients. When you use, and then do not use presample data, compare the results from infer
.
Specify a GARCH(1,1) model with known parameters. Simulate 101 conditional variances and responses (innovations) from the model. Set aside the first observation from each series to use as presample data.
Mdl = garch('Constant',0.01,'GARCH',0.8,'ARCH',0.15); rng default; % For reproducibility [vS,yS] = simulate(Mdl,101); y0 = yS(1); v0 = vS(1); y = yS(2:end); v = vS(2:end); figure subplot(2,1,1) plot(v) title('Conditional Variances') subplot(2,1,2) plot(y) title('Innovations')
Infer the conditional variances of y
without using presample data. Compare them to the known (simulated) conditional variances.
vI = infer(Mdl,y); figure plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vI,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - No Presamples') hold off
Notice the transient response (discrepancy) in the early time periods due to the absence of presample data.
Infer conditional variances using the set-aside presample innovation, y0
. Compare them to the known (simulated) conditional variances.
vE = infer(Mdl,y,'E0',y0); figure plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vE,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - Presample E') hold off
There is a slightly reduced transient response in the early time periods.
Infer conditional variances using the set-aside presample conditional variance, v0
. Compare them to the known (simulated) conditional variances.
vO = infer(Mdl,y,'V0',v0); figure plot(v) plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vO,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - Presample V') hold off
There is a much smaller transient response in the early time periods.
Infer conditional variances using both the presample innovation and conditional variance. Compare them to the known (simulated) conditional variances.
vEO = infer(Mdl,y,'E0',y0,'V0',v0); figure plot(v) plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vEO,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - Presamples') hold off
When you use sufficient presample innovations and conditional variances, the inferred conditional variances are exact (there is no transient response).
Infer conditional variances from an EGARCH(1,1) model with known coefficients. When you use, and then do not use presample data, compare the results from infer
.
Specify an EGARCH(1,1) model with known parameters. Simulate 101 conditional variances and responses (innovations) from the model. Set aside the first observation from each series to use as presample data.
Mdl = egarch('Constant',0.001,'GARCH',0.8,... 'ARCH',0.15,'Leverage',-0.1); rng default; % For reproducibility [vS,yS] = simulate(Mdl,101); y0 = yS(1); v0 = vS(1); y = yS(2:end); v = vS(2:end); figure subplot(2,1,1) plot(v) title('Conditional Variances') subplot(2,1,2) plot(y) title('Innovations')
Infer the conditional variances of y
without using any presample data. Compare them to the known (simulated) conditional variances.
vI = infer(Mdl,y); figure plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vI,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - No Presamples') hold off
Notice the transient response (discrepancy) in the early time periods due to the absence of presample data.
Infer conditional variances using the set-aside presample innovation, y0
. Compare them to the known (simulated) conditional variances.
vE = infer(Mdl,y,'E0',y0); figure plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vE,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - Presample E') hold off
There is a slightly reduced transient response in the early time periods.
Infer conditional variances using the set-aside presample variance, v0
. Compare them to the known (simulated) conditional variances.
vO = infer(Mdl,y,'V0',v0); figure plot(v) plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vO,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - Presample V') hold off
The transient response is almost eliminated.
Infer conditional variances using both the presample innovation and conditional variance. Compare them to the known (simulated) conditional variances.
vEO = infer(Mdl,y,'E0',y0,'V0',v0); figure plot(v) plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vEO,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - Presamples') hold off
When you use sufficient presample innovations and conditional variances, the inferred conditional variances are exact (there is no transient response).
Infer conditional variances from a GJR(1,1) model with known coefficients. When you use, and then do not use presample data, compare the results from infer
.
Specify a GJR(1,1) model with known parameters. Simulate 101 conditional variances and responses (innovations) from the model. Set aside the first observation from each series to use as presample data.
Mdl = gjr('Constant',0.01,'GARCH',0.8,'ARCH',0.14,... 'Leverage',0.1); rng default; % For reproducibility [vS,yS] = simulate(Mdl,101); y0 = yS(1); v0 = vS(1); y = yS(2:end); v = vS(2:end); figure subplot(2,1,1) plot(v) title('Conditional Variances') subplot(2,1,2) plot(y) title('Innovations')
Infer the conditional variances of y
without using any presample data. Compare them to the known (simulated) conditional variances.
vI = infer(Mdl,y); figure plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vI,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - No Presamples') hold off
Notice the transient response (discrepancy) in the early time periods due to the absence of presample data.
Infer conditional variances using the set-aside presample innovation, y0
. Compare them to the known (simulated) conditional variances.
vE = infer(Mdl,y,'E0',y0); figure plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vE,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - Presample E') hold off
There is a slightly reduced transient response in the early time periods.
Infer conditional variances using the set-aside presample conditional variance, vO
. Compare them to the known (simulated) conditional variances.
vO = infer(Mdl,y,'V0',v0); figure plot(v) plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vO,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - Presample V') hold off
There is a much smaller transient response in the early time periods.
Infer conditional variances using both the presample innovation and conditional variance. Compare them to the known (simulated) conditional variances.
vEO = infer(Mdl,y,'E0',y0,'V0',v0); figure plot(v) plot(1:100,v,'r','LineWidth',2) hold on plot(1:100,vEO,'k:','LineWidth',1.5) legend('Simulated','Inferred','Location','NorthEast') title('Inferred Conditional Variances - Presamples') hold off
When you use sufficient presample innovations and conditional variances, the inferred conditional variances are exact (there is no transient response).
Infer the loglikelihood objective function values for an EGARCH(1,1) and EGARCH(2,1) model fit to NASDAQ Composite Index returns. To identify which model is the more parsimonious, adequate fit, conduct a likelihood ratio test.
Load the NASDAQ data included with the toolbox, and convert the index to returns. Set aside the first two observations to use as presample data.
load Data_EquityIdx
nasdaq = DataTable.NASDAQ;
r = price2ret(nasdaq);
r0 = r(1:2);
rn = r(3:end);
Fit an EGARCH(1,1) model to the returns, and infer the loglikelihood objective function value.
Mdl1 = egarch(1,1);
EstMdl1 = estimate(Mdl1,rn,'E0',r0);
EGARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant -0.13518 0.022134 -6.1073 1.0132e-09 GARCH{1} 0.98386 0.0024268 405.42 0 ARCH{1} 0.19997 0.013993 14.29 2.5181e-46 Leverage{1} -0.060243 0.0056558 -10.652 1.7133e-26
[~,logL1] = infer(EstMdl1,rn,'E0',r0);
Fit an EGARCH(2,1) model to the returns, and infer the loglikelihood objective function value.
Mdl2 = egarch(2,1);
EstMdl2 = estimate(Mdl2,rn,'E0',r0);
EGARCH(2,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant -0.14559 0.028436 -5.1202 3.0528e-07 GARCH{1} 0.85307 0.14018 6.0854 1.1619e-09 GARCH{2} 0.12952 0.13838 0.93596 0.34929 ARCH{1} 0.21969 0.029465 7.456 8.9218e-14 Leverage{1} -0.067936 0.01088 -6.2444 4.2554e-10
[~,logL2] = infer(EstMdl2,rn,'E0',r0);
Conduct a likelihood ratio test, with the more parsimonious EGARCH(1,1) model as the null model, and the EGARCH(2,1) model as the alternative. The degree of freedom for the test is 1, because the EGARCH(2,1) model has one more parameter than the EGARCH(1,1) model (an additional GARCH term).
[h,p] = lratiotest(logL2,logL1,1)
h = logical
0
p = 0.2256
The null hypothesis is not rejected (h = 0
). At the 0.05 significance level, the EGARCH(1,1) model is not rejected in favor of the EGARCH(2,1) model.
A GARCH(P, Q) model is nested within a GJR(P, Q) model. Therefore, you can perform a likelihood ratio test to compare GARCH(P, Q) and GJR(P, Q) model fits.
Infer the loglikelihood objective function values for a GARCH(1,1) and GJR(1,1) model fit to NASDAQ Composite Index returns. Conduct a likelihood ratio test to identify which model is the more parsimonious, adequate fit.
Load the NASDAQ data included with the toolbox, and convert the index to returns. Set aside the first two observations to use as presample data.
load Data_EquityIdx
nasdaq = DataTable.NASDAQ;
r = price2ret(nasdaq);
r0 = r(1:2);
rn = r(3:end);
Fit a GARCH(1,1) model to the returns, and infer the loglikelihood objective function value.
Mdl1 = garch(1,1);
EstMdl1 = estimate(Mdl1,rn,'E0',r0);
GARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant 2.005e-06 5.4298e-07 3.6926 0.00022196 GARCH{1} 0.88333 0.0084537 104.49 0 ARCH{1} 0.10924 0.0076666 14.249 4.5739e-46
[~,logL1] = infer(EstMdl1,rn,'E0',r0);
Fit a GJR(1,1) model to the returns, and infer the loglikelihood objective function value.
Mdl2 = gjr(1,1);
EstMdl2 = estimate(Mdl2,rn,'E0',r0);
GJR(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant 2.4761e-06 5.6998e-07 4.3441 1.3983e-05 GARCH{1} 0.881 0.0095132 92.608 0 ARCH{1} 0.064038 0.0091887 6.9692 3.1882e-12 Leverage{1} 0.089305 0.0099246 8.9984 2.2901e-19
[~,logL2] = infer(EstMdl2,rn,'E0',r0);
Conduct a likelihood ratio test, with the more parsimonious GARCH(1,1) model as the null model, and the GJR(1,1) model as the alternative. The degree of freedom for the test is 1, because the GJR(1,1) model has one more parameter than the GARCH(1,1) model (a leverage term).
[h,p] = lratiotest(logL2,logL1,1)
h = logical
1
p = 4.5808e-10
The null hypothesis is rejected (h = 1
). At the 0.05 significance level, the GARCH(1,1) model is rejected in favor of the GJR(1,1) model.
Y
— Response dataResponse data, specified as a numeric column vector or matrix.
As a column vector, Y
represents a single path of the
underlying series.
As a matrix, the rows of Y
correspond to periods and
the columns correspond to separate paths. The observations across any row
occur simultaneously.
infer
infers the conditional variances of
Y
. Y
usually represents an
innovation series with mean 0 and variances characterized by
Mdl
. It is the continuation of the presample
innovation series E0
. Y
can also
represent a time series of innovations with mean 0 plus an offset. If
Mdl
has a nonzero offset, then the software stores
its value in the Offset
property
(Mdl.Offset
).
infer
assumes that observations across any row occur
simultaneously.
The last observation of any series is the latest observation.
NaN
s indicate missing values. infer
removes
missing values. infer
uses list-wise deletion to remove any
NaN
s. Removing NaN
s in the data reduces
the sample size. Removing missing values, can also create irregular time
series.
Specify optional
comma-separated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
'E0',[1 1;0.5 0.5],'V0',[1 0.5;1 0.5]
specifies two
equivalent presample paths of innovations and two, different presample paths of
conditional variances.'E0'
— Presample innovationsPresample innovations, specified as the comma-separated pair
consisting of 'E0'
and a numeric column vector or
matrix. The presample innovations provide initial values for the
innovations process of the conditional variance model
Mdl
, and derive from a distribution with mean
0.
E0
must contain at least Mdl.Q
elements or rows. If E0
contains extra rows, then
infer
uses the latest
Mdl.Q
only.
The last element or row contains the latest presample innovation.
If E0
is a column vector, it represents a
single path of the underlying innovation series.
infer
applies
E0
to each inferred path.
If E0
is a matrix, then each column
represents a presample path of the underlying innovation series.
E0
must have at least as many columns as
Y
. If E0
has more
columns than necessary, infer
uses
the first size(Y,2)
columns only.
The defaults are:
For GARCH(P,Q) and
GJR(P,Q) models,
infer
sets any necessary
presample innovations to the square root of the average squared
value of the offset-adjusted response series
Y
.
For EGARCH(P,Q) models,
infer
sets any necessary presample
innovations to zero.
Example: 'E0',[1 1;0.5 0.5]
Data Types: double
'V0'
— Presample conditional variancesPresample conditional variances, specified as the comma-separated pair
consisting of 'V0'
and a numeric column vector or
matrix with positive entries. V0
provides initial
values for the conditional variances in the model.
If V0
is a column vector, then
infer
applies it to each output
path.
If V0
is a matrix, then each column
represents a presample path of conditional variances.
V0
must have at least as many columns as
Y
. If V0
has more
columns than required, infer
uses the
first size(Y,2)
columns only.
For GARCH(P,Q) and
GJR(P,Q) models,
V0
must have at least
Mdl.P
rows (or elements) to initialize
the variance equation.
For EGARCH(P,Q) models,
V0
must have at least
max(Mdl.P,Mdl.Q)
rows to initialize the
variance equation.
If the number of rows in V0
exceeds the necessary
number, then infer
uses the latest, required
number of observations only.
The last element row contains the latest observation.
By default, infer
sets any necessary
observations to the average squared value of the offset-adjusted
response series Y
.
Example: 'V0',[1 0.5;1 0.5]
Data Types: double
NaN
s indicate missing values.
infer
removes missing values. The
software merges the presample data (E0
and
V0
) separately from the input response data
(Y
), and then uses list-wise deletion to remove
any rows containing at least one NaN
. Removing
NaN
s in the data reduces the sample size.
Removing missing values can also create irregular time series.
infer
assumes that you synchronize
presample data such that the latest observation of each presample series
occurs simultaneously.
If you do not specify E0
and V0
,
then infer
derives the necessary presample
observations from the unconditional, or long-run, variance of the
offset-adjusted response process.
For all conditional variance models, V0
is the sample average of the squared disturbances of the
offset-adjusted response data Y
.
For GARCH(P,Q) and
GJR(P,Q) models,
E0
is the square root of the average
squared value of the offset-adjusted response series
Y
.
For EGARCH(P,Q)
models, E0
is
0
.
These specifications minimize initial transient effects.
V
— Conditional variancesConditional variances inferred from the response data
Y
, returned as a numeric column vector or
matrix.
The dimensions of V
and Y
are
equivalent. If Y
is a matrix, then the columns of
V
are the inferred conditional variance paths
corresponding to the columns of Y
.
Rows of V
are periods corresponding to the periodicity
of Y
.
logL
— Loglikelihood objective function values[1] Bollerslev, T. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics. Vol. 31, 1986, pp. 307–327.
[2] Bollerslev, T. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” The Review of Economics and Statistics. Vol. 69, 1987, pp. 542–547.
[3] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
[4] Enders, W. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, 1995.
[5] Engle, R. F. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica. Vol. 50, 1982, pp. 987–1007.
[6] Glosten, L. R., R. Jagannathan, and D. E. Runkle. “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.” The Journal of Finance. Vol. 48, No. 5, 1993, pp. 1779–1801.
[7] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
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