filter
Filter disturbances through conditional variance model
Description
returns the table or timetable Tbl2
= filter(Mdl
,Tbl1
)Tbl2
containing the results
from filtering the paths of disturbances in the input table or timetable
Tbl1
through Mdl
. The disturbance
variable in Tbl1
is associated with the model innovations
process through Mdl
. (since R2023a)
filter
selects the response variable named in
Mdl.SeriesName
or the sole variable in
Tbl1
. To select a different disturbance variable in
Tbl1
to filter through the model, use the
DisturbanceVariable
name-value argument.
[___] = filter(___,
specifies options using one or more name-value arguments in
addition to any of the input argument combinations in previous syntaxes.
Name,Value
)filter
returns the output argument combination for the
corresponding input arguments. For example, filter(Mdl,Z,Z0=PS)
filters the
numeric vector of disturbances Z
through the conditional
variance model Mdl
and specifies the numeric vector of
presample disturbance data PS
to initialize the model.
Examples
Filter Numeric Vector Containing Disturbance Path
Demonstrate that simulate
and filter
can return equal quantities. Supply data in a numeric vector.
Specify a GARCH(1,1) model with Gaussian innovations.
Mdl = garch(Constant=0.005,GARCH=0.8,ARCH=0.1);
Simulate the model using Monte Carlo simulation. Then, standardize the simulated innovations and filter them.
rng(1) % For reproducibility
[vs,es] = simulate(Mdl,100,E0=0,V0=0.05);
Z = es./sqrt(vs);
[vf,ef] = filter(Mdl,Z,Z0=0,V0=0.05);
Confirm that the outputs of simulate
and filter
are identical.
norm(vs-vf)
ans = 0
A norm of 0 indicates that the two outputs are identical.
Filter Timetable of Disturbance Data Through GARCH Model
Since R2023a
Fit a GARCH(1,1) model to the average weekly closing NASDAQ returns, and then filter a randomly generated series of disturbances through the estimated model. Supply timetables of data throughout the process.
Load the U.S. equity indices data Data_EquityIdx.mat
.
load Data_EquityIdx
The timetable DataTimeTable
contains the daily NASDAQ closing prices, among other indices.
Compute the weekly average closing prices of all timetable variables.
DTTW = convert2weekly(DataTimeTable,Aggregation="mean");
Compute the weekly returns.
DTTRet = price2ret(DTTW); DTTRet.Interval = []; T = height(DTTRet)
T = 626
Plot the weekly NASDAQ returns.
figure
plot(DTTRet.Time,DTTRet.NASDAQ)
title("NASDAQ Weekly Returns")
The returns exhibit volatility clustering.
When you plan to supply a timetable, you must ensure it has all the following characteristics:
The selected response variable is numeric and does not contain any missing values.
The timestamps in the
Time
variable are regular, and they are ascending or descending.
Remove all missing values from the timetable, relative to the NASDAQ returns series.
DTTRet = rmmissing(DTTRet,DataVariables="NASDAQ");
numobs = height(DTTRet)
numobs = 626
Because all sample times have observed NASDAQ returns, rmmissing
does not remove any observations.
Determine whether the sampling timestamps have a regular frequency and are sorted.
areTimestampsRegular = isregular(DTTRet,"weeks")
areTimestampsRegular = logical
1
areTimestampsSorted = issorted(DTTRet.Time)
areTimestampsSorted = logical
1
areTimestampsRegular = 1
indicates that the timestamps of DTTRet
represent a regular weekly sample. areTimestampsSorted = 1
indicates that the timestamps are sorted.
Specify a GARCH(1,1) model, and fit it to the series. Name the response series of the model NASDAQ
by using dot notation.
Mdl = garch(1,1);
Mdl.SeriesName = "NASDAQ";
EstMdl = estimate(Mdl,DTTRet);
GARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant 1.7406e-06 8.9077e-07 1.9541 0.050694 GARCH{1} 0.65947 0.059314 11.118 1.0229e-28 ARCH{1} 0.33773 0.079595 4.2431 2.2044e-05
estimate
fits the model to the response data in the NASDAQ
variable of DTTRet
because the name matches the name of the response variable in Mdl.SeriesName
. Alternatively, you can specify the response variable by using the ResponseVariable
name-value argument.
Generate 2 random, independent series of length T
from the standard Gaussian distribution. Store the matrix of series as one variable in DTTRet
.
rng(1) % For reproducibility
DTTRet.Z = randn(T,2);
DTTRet
contains a new variable called Z
containing a T
-by-2 matrix of five disturbance paths.
Filter the paths of disturbances through the estimated GARCH model. Specify the table variable name containing the disturbance paths.
Tbl2 = filter(EstMdl,DTTRet,DisturbanceVariable="Z")
Tbl2=626×5 timetable
Time NYSE NASDAQ Z NASDAQ_Variance NASDAQ_Response
___________ ___________ ___________ _____________________ ________________________ ________________________
12-Jan-1990 -0.0031597 -0.0026701 -0.64901 -0.50964 0.00059062 0.00045179 -0.015773 -0.010833
19-Jan-1990 -0.0038123 -0.0039103 1.1812 0.088893 0.00047526 0.00033931 0.02575 0.0016374
26-Jan-1990 -0.0040706 -0.0039139 -0.75845 -0.019698 0.00053909 0.00022641 -0.01761 -0.0002964
02-Feb-1990 -0.00099691 -0.0033847 -1.1096 -0.73807 0.00046199 0.00015108 -0.02385 -0.009072
09-Feb-1990 0.0022796 0.0031891 -0.84555 -1.1522 0.00049852 0.00012917 -0.018879 -0.013095
16-Feb-1990 -0.00021948 0.00037747 -0.57266 -1.9476 0.00045087 0.00014483 -0.01216 -0.023439
23-Feb-1990 -0.0022725 -0.0018693 -0.55868 0.026296 0.00034901 0.0002828 -0.010437 0.00044221
02-Mar-1990 0.0019481 0.0012208 0.17838 -0.82589 0.00026869 0.0001883 0.002924 -0.011333
09-Mar-1990 0.0022677 0.0026984 -0.19686 -0.71799 0.00018182 0.0001693 -0.0026545 -0.0093421
16-Mar-1990 0.00029781 0.0012667 0.58644 -1.941 0.00012403 0.00014286 0.006531 -0.0232
23-Mar-1990 0.00027271 0.00042646 -0.85189 0.98755 9.7937e-05 0.00027773 -0.0084306 0.016458
30-Mar-1990 0.00022176 -0.00052576 0.80032 -1.6631 9.0331e-05 0.00027637 0.0076065 -0.027648
06-Apr-1990 0.00016495 -0.0010113 -1.5094 2.0633 8.0852e-05 0.00044216 -0.013572 0.043387
13-Apr-1990 0.00050551 -0.00037366 0.87587 -2.082 0.00011727 0.00092908 0.009485 -0.063462
20-Apr-1990 -0.00072855 -0.00042758 -0.24279 0.27316 0.00010946 0.0019746 -0.0025402 0.012138
27-Apr-1990 -0.0039166 -0.0039974 0.16681 -2.3767 7.6106e-05 0.0013537 0.0014553 -0.087447
⋮
Tbl2
is a 626-by-5 timetable containing all variables in DTTRet
, the two filtered conditional variance paths NASDAQ_Variance
, and the two filtered response paths NASDAQ_Response
.
Filter Multiple Disturbance Paths Through EGARCH Model
Specify an EGARCH(1,1) model with Gaussian innovations.
Mdl = egarch(Constant=-0.1,GARCH=0.8,ARCH=0.3, ...
Leverage=-0.05);
Simulate 25 series of standard Gaussian observations for 100 periods.
rng(1); % For reproducibility
Z = randn(100,25);
Z
represents 25 paths of synchronized disturbances for 100 periods.
Obtain 25 paths of conditional variances by filtering the disturbance paths through the EGARCH(1,1) model.
V = filter(Mdl,Z);
Plot the paths of conditional variances.
figure; plot(V); title("Conditional Variance Paths"); xlabel("Periods");
Filter Disturbances Through GJR Model Specifying Presample Observations
Specify a GJR(1,2) model with Gaussian innovations.
Mdl = gjr(Constant=0.005,GARCH=0.8,ARCH={0.1 0.01}, ...
Leverage={0.05 0.01});
Simulate 25 series of standard Gaussian observations for 102 periods.
rng(1); % For reproducibility
Z = randn(102,25);
Z
represents 25 paths of synchronized disturbances for 102 periods.
Obtain 25, 100 period paths of conditional variances by filtering the disturbance paths through the GJR(1,2) model. Specify the first two disturbances as presample observations.
V = filter(Mdl,Z(3:end,:),Z0=Z(1:2,:));
Plot the paths of conditional variances.
figure plot(V) title("Conditional Variance Paths"); xlabel("Periods");
Input Arguments
Z
— Disturbance paths zt
numeric column vector | numeric matrix
Disturbance paths zt used to
drive the output innovation process
εt, specified as a
numobs
-by-1 numeric vector or
numobs
-by-numpaths
numeric matrix.
Given the variance process
σt2,
the innovation process is
As a column vector, Z
represents a single path of the
underlying disturbance series.
As a matrix, the rows of Z
correspond to periods. The
columns correspond to separate paths. The observations across any row occur
simultaneously.
The last element or row of Z
contains the latest
observation.
Tbl1
— Time series data
table | timetable
Since R2023a
Time series data containing observed disturbance variable
zt, associated with the
model innovations process εt,
specified as a table or timetable with numvars
variables
and numobs
rows. You can optionally select a disturbance
variable by using the DisturbanceVariable
name-value
argument.
Given the variance process σt2, the innovation process is
The selected variable is a single path (numobs
-by-1
vector) or multiple paths
(numobs
-by-numpaths
matrix) of
numobs
observations of disturbance data. Each row is
an observation, and measurements in each row occur simultaneously.
Each path (column) of the selected variable is independent of the other paths.
If Tbl1
is a timetable, it must represent a sample
with a regular datetime time step (see isregular
), and the datetime
vector Tbl1.Time
must be strictly ascending or
descending.
If Tbl1
is a table, the last row contains the latest
observation.
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: filter(Mdl,Z,Z0=[1 1;0.5 0.5],V0=[1 0.5;1 0.5])
specifies two equivalent presample paths of disturbances and two different presample
paths of conditional variances.
DisturbanceVariable
— Variable to select from Tbl1
to treat as disturbance variable zt
string scalar | character vector | integer | logical vector
Since R2023a
Variable to select from Tbl1
to treat as the
disturbance variable zt to
filter through Mdl
, specified as one of the
following data types:
String scalar or character vector containing a variable name in
Tbl1.Properties.VariableNames
Variable index (positive integer) to select from
Tbl1.Properties.VariableNames
A logical vector, where
DisturbanceVariable(
selects variablej
) = true
fromj
Tbl1.Properties.VariableNames
The selected variable must be a numeric vector and cannot contain
missing values (NaN
).
If Tbl1
has one variable, the default specifies
that variable. Otherwise, the default matches the variable to names in
Mdl.SeriesName
.
Example: DisturbanceVariable="StockRateDist"
Example: DisturbanceVariable=[false false true
false]
or DisturbanceVariable=3
selects
the third table variable as the disturbance variable.
Data Types: double
| logical
| char
| cell
| string
Z0
— Presample disturbance paths zt
numeric column vector | numeric matrix
Presample disturbance paths
zt, specified as a
numpreobs
-by-1 numeric vector or
numpreobs
-by-numprepaths
matrix. Z0
provides initial values for the input
disturbance paths Z
. Use Z0
only
when you supply the numeric array of disturbances
Z
.
numpreobs
is the number of presample observations.
numprepaths
is the number of presample response
paths.
Each row is a presample observation, and measurements in each row
occur simultaneously. The last row contains the latest presample
observation. numpreobs
must be at least
Mdl.Q
. If numpreobs
>
Mdl.Q
, filter
uses the
latest required number of observations only.
If
Z0
is a column vector, it represents a single path of the underlying disturbance series.filter
applies it to each output path.If
Z0
is a matrix, each column represents a presample path of the underlying disturbance series.numprepaths
must be at leastnumpaths
. Ifnumprepaths
>numpaths
,filter
uses the firstsize(Z,2)
columns only.
By default, filter
sets any necessary
presample disturbances to an independent sequence of standardized
disturbances drawn from Mdl.Distribution
.
Data Types: double
V0
— Positive presample conditional variance paths σt2
positive column vector | positive matrix
Positive presample conditional variance paths
σt2,
specified as a numpreobs
-by-1 positive column vector
or numpreobs
-by-numprepaths
positive matrix. V0
provides initial values for the
conditional variances in the model. Use V0
only when
you supply the numeric array of disturbances
Z
.
To initialize the conditional variance model,
numpreobs
must be at least max([Mdl.P
Mdl.Q])
. If numpreobs
>
max([Mdl.P Mdl.Q])
,
filter
uses the latest required number of
observations only. The last element or row contains the latest
observation.
If
V0
is a column vector, it represents a single path of the conditional variance series.filter
applies it to each output path.If
V0
is a matrix,numprepaths
must be at leastnumpaths
. Ifnumprepaths
>numpaths
,filter
uses the firstsize(Z,2)
columns only.
By default, filter
sets any necessary
presample conditional variances to the unconditional variance of the
process.
Data Types: double
Presample
— Presample data
table | timetable
Since R2023a
Presample data containing paths of innovation
εt or conditional
variance
σt2
series to initialize the model, specified as a table or timetable, the
same type as Tbl1
, with
numprevars
variables and
numpreobs
rows. Use
Presample
only when you supply a table or
timetable of data Tbl1
.
Each selected variable is a single path
(numpreobs
-by-1 vector) or multiple paths
(numpreobs
-by-numprepaths
matrix) of numpreobs
observations representing the
presample of the disturbance or conditional variance series for
DisturbanceVariable
, the selected disturbance
variable in Tbl1
.
Each row is a presample observation, and measurements in each row
occur simultaneously. numpreobs
must be one of the
following values:
Mdl.Q
whenPresample
provides only presample disturbancesmax([Mdl.P Mdl.Q])
whenPresample
provides presample conditional variances
If you supply more rows than necessary,
filter
uses the latest required number of
observations only.
If Presample
is a timetable, all the following
conditions must be true:
Presample
must represent a sample with a regular datetime time step (seeisregular
).The inputs
Tbl1
andPresample
must be consistent in time such thatPresample
immediately precedesTbl1
with respect to the sampling frequency and order.The datetime vector of sample timestamps
Presample.Time
must be ascending or descending.
If Presample
is a table, the last row contains
the latest presample observation.
By default, filter
sets any necessary
presample disturbances to an independent sequence of standardized
disturbances drawn from Mdl.Distribution
, and it
sets any necessary presample conditional variances to the unconditional
variance of the process characterized by
Mdl
.
If you specify the Presample
, you must specify
the presample disturbance or conditional variance names by using the
PresampleDisturbanceVariable
or
PresampleVarianceVariable
name-value
argument.
PresampleDisturbanceVariable
— Variable of Presample
containing presample disturbance paths zt
string scalar | character vector | integer | logical vector
Since R2023a
Variable of Presample
containing presample
disturbance paths zt,
specified as one of the following data types:
String scalar or character vector containing a variable name in
Presample.Properties.VariableNames
Variable index (positive integer) to select from
Presample.Properties.VariableNames
A logical vector, where
PresampleDisturbanceVariable(
selects variablej
) = true
fromj
Presample.Properties.VariableNames
The selected variable must be a numeric matrix and cannot contain
missing values (NaN
s).
If you specify presample disturbance data by using the
Presample
name-value argument, you must specify
PresampleDisturbanceVariable
.
Example: PresampleDisturbanceVariable="StockRateDist0"
Example: PresampleDisturbanceVariable=[false false true
false]
or
PresampleDisturbanceVariable=3
selects the third
table variable as the presample disturbance variable.
Data Types: double
| logical
| char
| cell
| string
PresampleVarianceVariable
— Variable of Presample
containing data for the presample conditional variances σt2
string scalar | character vector | integer | logical vector
Since R2023a
Variable of Presample
containing data for the presample conditional
variances
σt2,
specified as one of the following data types:
String scalar or character vector containing a variable name in
Presample.Properties.VariableNames
Variable index (positive integer) to select from
Presample.Properties.VariableNames
A logical vector, where
PresampleVarianceVariable(
selects variablej
) = true
fromj
Presample.Properties.VariableNames
The selected variable must be a numeric vector and cannot contain missing values
(NaN
s).
If you specify presample conditional variance data by using the Presample
name-value argument, you must specify PresampleVarianceVariable
.
Example: PresampleVarianceVariable="StockRateVar0"
Example: PresampleVarianceVariable=[false false true false]
or PresampleVarianceVariable=3
selects the third table variable as the presample conditional variance variable.
Data Types: double
| logical
| char
| cell
| string
Note
NaN
values inZ
,Z0
, andV0
indicate missing values.filter
removes missing values from specified data by list-wise deletion.For the presample,
filter
horizontally concatenatesZ0
andV0
, and then it removes any row of the concatenated matrix containing at least oneNaN
.For in-sample data
Z
,filter
removes any row containing at least oneNaN
.
This type of data reduction reduces the effective sample size and can create an irregular time series.
For numeric data inputs,
filter
assumes that you synchronize the presample data such that the latest observations occur simultaneously.filter
issues an error when any table or timetable input contains missing values.
Output Arguments
V
— Filtered conditional variance paths σt2
numeric column vector | numeric matrix
Filtered conditional variance paths
σt2,
returned as a numobs
-by-1 numeric column vector or
numobs
-by-numpaths
numeric matrix.
V
represents the conditional variances of the
mean-zero, heteroscedastic innovations associated with
Y
. filter
returns
V
only when you supply the input
Z
.
The dimensions of V
and Z
are
equivalent. If Z
is a matrix, then the columns of
V
are the conditional variance paths corresponding to
the columns of Z
.
Rows of V
are periods corresponding to the periodicity
of Z
.
Y
— Filtered response paths yt
numeric column vector | numeric matrix
Filtered response paths yt,
returned as a numobs
-by-1 numeric column vector or
numobs
-by-numpaths
.
Y
usually represents a mean-zero, heteroscedastic
time series of innovations with conditional variances given in
V
. filter
returns
Y
only when you supply the input
Z
.
Y
can also represent a time series of mean-zero,
heteroscedastic innovations plus an offset. If Mdl
includes an offset, then filter
adds the offset to
the underlying mean-zero, heteroscedastic innovations. Therefore,
Y
represents a time series of offset-adjusted
innovations.
If Z
is a matrix, then the columns of
Y
are the response paths corresponding to the columns
of Z
.
Rows of Y
are periods corresponding to the periodicity
of Z
.
Tbl2
— Filtered conditional variance σt2 and
response yt paths
table | timetable
Since R2023a
Filtered conditional variance
σt2
and response yt paths, returned as
a table or timetable, the same data type as Tbl1
.
filter
returns Tbl2
only
when you supply the input Tbl1
.
Tbl2
contains the following variables:
The filtered conditional variances paths, which are in a
numobs
-by-numpaths
numeric matrix, with rows representing observations and columns representing independent paths, each corresponding to the input observations and paths of the disturbance variable inTbl1
.filter
names the filtered conditional variance variable inTbl2
, whereresponseName
_Variance
isresponseName
Mdl.SeriesName
. For example, ifMdl.SeriesName
isStockReturns
,Tbl2
contains a variable for the corresponding filtered response paths with the nameStockReturns_Variance
.The filtered response paths, which are in a
numobs
-by-numpaths
numeric matrix, with rows representing observations and columns representing independent paths, each corresponding to the input observations and paths of the disturbance variable inTbl1
.filter
names the filtered response variable inTbl2
, whereresponseName
_Response
isresponseName
Mdl.SeriesName
. For example, ifMdl.SeriesName
isStockReturns
,Tbl2
contains a variable for the corresponding filtered conditional variance paths with the nameStockReturns_Response
.All variables
Tbl1
.
If Tbl1
is a timetable, row times of
Tbl1
and Tbl2
are
equal.
Alternatives
filter
generalizes simulate
. Both function filter a series of disturbances to produce
output responses and conditional variances. However, simulate
autogenerates a series of mean-zero, unit-variance, independent and identically
distributed (iid) disturbances according to the distribution in the conditional variance
model object, Mdl
. In contrast, filter
lets you directly specify your own disturbances.
References
[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] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
Version History
Introduced in R2012aR2023a: filter
accepts input data in tables and timetables, and returns results in tables and timetables
In addition to accepting input data (in-sample and presample) in numeric arrays,
filter
accepts input data in tables or regular
timetables. When you supply data in a table or timetable, the following conditions
apply:
filter
chooses the default in-sample disturbance series on which to operate, but you can use the specified optional name-value argument to select a different series.If you specify optional presample disturbance or conditional variance data to initialize the model, you must also specify the presample disturbance or conditional variance series name.
filter
returns results in a table or timetable.
Name-value arguments to support tabular workflows include:
DisturbanceVariable
specifies the variable name of the disturbance paths in the input dataTbl1
to filter through the model.Presample
specifies the input table or timetable of presample disturbance and conditional variance data.PresampleDisturbanceVariable
specifies the variable name of the disturbance paths to select fromPresample
.PresampleVarianceVariable
specifies the variable name of the conditional variance paths to select fromPresample
.
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