arma2ma
Convert ARMA model to MA model
Description
returns the coefficients of the truncated, infinite-order MA model approximation to
an ARMA model with input AR and MA coefficients.ma
= arma2ma(ar0
,ma0
)
arma2ma:
Accepts:
Vectors or cell vectors of matrices in difference-equation notation.
LagOp
lag operator polynomials corresponding to the AR and MA polynomials in lag operator notation.
Accommodates time series models that are univariate or multivariate (i.e.,
numVars
variables compose the model), stationary or integrated, structural or in reduced form, and invertible.Assumes that the model constant c is 0.
Examples
Convert an ARMA model to an MA Model
Find the lag coefficients of the truncated, MA approximation of this univariate, stationary, and invertible ARMA model
The ARMA model is in difference-equation notation because the left side contains only and its coefficient 1. Create a vector containing the AR lag term coefficients in order starting from t - 1.
ar0 = [0.2 -0.1];
Alternatively, you can create a cell vector of the scalar coefficients.
Create a vector containing the MA lag term coefficient.
ma0 = 0.5;
Convert the ARMA model to an MA model by obtaining the coefficients of the truncated approximation of the infinite-lag polynomial.
ma = arma2ma(ar0,ma0)
ma = 1×4
0.7000 0.0400 -0.0620 -0.0164
ma
is a numeric vector because ar0
and ma0
are numeric vectors.
The approximate MA model truncated at 4 lags is
Convert an AR(3) Model to an MA(5) Model
Find the first five lag coefficients of the MA approximation of this univariate and stationary AR(3) model
The AR model is in difference-equation notation because the left side contains only and its coefficient of 1. Create a cell vector containing the AR lag term coefficient in order starting from t - 1. Because the second lag term of the MA model is missing, specify a 0
for its coefficient.
ar0 = {-0.2 0 0.5};
Convert the AR model to an MA model with at most five lag coefficients of the truncated approximation of the infinite-lag polynomial. Because there is no MA contribution, specify an empty cell ({}
) for the MA coefficients.
numLags = 5; ma0 = {}; ma = arma2ma(ar0,ma0,numLags)
ma=1×5 cell array
{[-0.2000]} {[0.0400]} {[0.4920]} {[-0.1984]} {[0.0597]}
ma
is a cell vector of scalars because at least one of ar0
and ma0
is a cell vector.
The approximate MA(5) model is
Convert a Structural VARMA model to a Structural VMA model
Find the coefficients of the truncated, structural VMA equivalent of the structural, stationary, and invertible VARMA model
where and .
The VARMA model is in lag operator notation because the response and innovation vectors are on opposite sides of the equation.
Create a cell vector containing the VAR matrix coefficients. Because this model is a structural model, start with the coefficient of and enter the rest in order by lag. Construct a vector that indicates the degree of the lag term for the corresponding coefficients.
var0 = {[1 0.2 -0.1; 0.03 1 -0.15; 0.9 -0.25 1],... [0.5 -0.2 -0.1; -0.3 -0.1 0.1; 0.4 -0.2 -0.05],... [0.05 -0.02 -0.01; -0.1 -0.01 -0.001; 0.04 -0.02 -0.005]}; var0Lags = [0 4 8];
Create a cell vector containing the VMA matrix coefficients. Because this model is a structural model, start with the coefficient of and enter the rest in order by lag. Construct a vector that indicates the degree of the lag term for the corresponding coefficients.
vma0 = {eye(3),...
[-0.02 0.03 0.3; 0.003 0.001 0.01; 0.3 0.01 0.01]};
vma0Lags = [0 4];
arma2ma
requires LagOp
lag operator polynomials for input arguments that comprise structural VAR or VMA models. Construct separate LagOp
polynomials that describe the VAR and VMA components of the VARMA model.
VARLag = LagOp(var0,'Lags',var0Lags); VMALag = LagOp(vma0,'Lags',vma0Lags);
VARLags
and VMALags
are LagOp
lag operator polynomials that describe the VAR and VMA components of the VARMA model.
Convert the VARMA model to a VMA model by obtaining the coefficients of the truncated approximation of the infinite-lag polynomial. Specify to return at most 12 lagged terms.
numLags = 12; VMA = arma2ma(VARLag,VMALag,numLags)
VMA = 3-D Lag Operator Polynomial: ----------------------------- Coefficients: [Lag-Indexed Cell Array with 4 Non-Zero Coefficients] Lags: [0 4 8 12] Degree: 12 Dimension: 3
VMA
is a LagOP
lag operator polynomial. All coefficients except those corresponding to lags 0, 4, 8, and 12 are 3-by-3 matrices of zeros.
Display the nonzero coefficients of the resulting VMA model.
lag2Idx = VMA.Lags + 1; % Lags start at 0. Add 1 to convert to indices. vmaCoeff = toCellArray(VMA); for j = 1:numel(lag2Idx) fprintf('___________Lag %d__________\n',lag2Idx(j) - 1) fprintf('%8.3f %8.3f %8.3f \n',vmaCoeff{lag2Idx(j)}) fprintf ('__________________________\n') end
___________Lag 0__________
0.943 -0.162 -0.889 -0.172 1.068 0.421 0.069 0.144 0.974
__________________________
___________Lag 4__________
-0.650 0.460 0.546 0.370 0.000 -0.019 0.383 -0.111 -0.312
__________________________
___________Lag 8__________
0.431 -0.138 -0.089 -0.170 0.122 0.065 -0.260 0.165 0.089
__________________________
___________Lag 12__________
-0.216 0.078 0.047 0.099 -0.013 -0.011 0.153 -0.042 -0.026
__________________________
Unconditional Mean of ARMA Model
Find the lag coefficients and constant of the truncated MA approximation of this univariate, stationary, and invertible ARMA model
The ARMA model is in difference-equation notation because the left side contains only and its coefficient of 1. Create separate vectors for the AR and MA lag term coefficients in order starting from t - 1.
ar0 = [0.2 -0.1]; ma0 = 0.5;
Convert the ARMA model to an MA model by obtaining the first five coefficients of the truncated approximation of the infinite-lag polynomial.
numLags = 5; ar = arma2ma(ar0,ma0,numLags)
ar = 1×5
0.7000 0.0400 -0.0620 -0.0164 0.0029
To compute the constant of the MA model, consider the ARMA model in lag operator notation.
or
Part of the conversion involves premultiplying both sides of the equation by the inverse of the AR lag operator polynomial, as in this equation.
To compute the inverse of AR lag operator polynomial, use the lag operator left-division object function mldivide
.
Phi = LagOp([1 -0.2 0.1]);
PhiInv = mldivide(Phi,1,'RelTol',1e-5);
PhiInv
is a LagOp
lag operator polynomial.
The application of lag operator polynomials to constants results in the product of the constant with the sum of the coefficients. Apply PhiInv
to the ARMA model constant to obtain the MA model constant.
maConstant = 1.5*sum(cell2mat(toCellArray(PhiInv)))
maConstant = 1.6667
The approximate MA model is
Since the unconditional expected value of all innovations is 0, the unconditional expected value (or mean) of the response series is
Input Arguments
ar0
— Autoregressive coefficients
numeric vector | cell vector of square, numeric matrices | LagOp
lag operator polynomial object
Autoregressive coefficients of the ARMA(p,q)
model, specified as a numeric vector, cell vector of square, numeric
matrices, or a LagOp
lag
operator polynomial object. If ar0
is a vector
(numeric or cell), then the coefficient of yt is
the identity. To specify a structural AR polynomial (i.e., the coefficient
of yt is not the identity),
use LagOp
lag operator polynomials.
For univariate time series models,
ar0
is a numeric vector, cell vector of scalars, or a one-dimensionalLagOp
lag operator polynomial. For vectors,ar0
has length p and the elements correspond to lagged responses composing the AR polynomial in difference-equation notation. That is,ar0(j)
orar0{j}
is the coefficient of yt-j.For
numVars
-dimensional time series models,ar0
is a cell vector ofnumVars
-by-numVars
numeric matrices or anumVars
-dimensionalLagOp
lag operator polynomial. For cell vectors:ar0
has length p.ar0
andma0
must containnumVars
-by-numVars
matrices.The elements of
ar0
correspond to the lagged responses composing the AR polynomial in difference equation notation. That is,ar0{j}
is the coefficient matrix of yt-j.Row k of an AR coefficient matrix contains the AR coefficients in the equation of the variable yk. Subsequently, column k must correspond to variable yk, and the column and row order of all autoregressive and moving average coefficients must be consistent.
For
LagOp
lag operator polynomials:The first element of the
Coefficients
property corresponds to the coefficient of yt (to accommodate structural models). All other elements correspond to the coefficients of the subsequent lags in theLags
property.To construct a univariate model in reduced form, specify
1
for the first coefficient. FornumVars
-dimensional multivariate models, specifyeye(numVars)
for the first coefficient.When you work from a model in difference-equation notation, negate the AR coefficient of the lagged terms to construct the lag-operator polynomial equivalent. For example, consider . The model is in difference-equation notation. To convert to an MA model, enter the following into the command window.
ma = arma2ma([0.5 -0.8], [-0.6 0.08]);
The ARMA model in lag operator notation is The AR coefficients of the lagged responses are negated compared to the corresponding coefficients in difference-equation format. In this form, to obtain the same result, enter the following into the command window.
ar0 = LagOp({1 -0.5 0.8}); ma0 = LagOp({1 -0.6 0.08}); ma = arma2ma(ar0, ma0);
It is a best practice for ar0
to constitute
a stationary or unit-root stationary (integrated) time series model.
ma0
— Moving average coefficients
numeric vector | cell vector of square, numeric matrices | LagOp
lag operator polynomial object
Moving average coefficients of the ARMA(p,q)
model, specified as a numeric vector, cell vector of square, numeric
matrices, or a LagOp
lag
operator polynomial object. If ma0
is a vector
(numeric or cell), then the coefficient of εt is
the identity. To specify a structural MA polynomial (i.e., the coefficient
of εt is not the identity),
use LagOp
lag operator polynomials.
For univariate time series models,
ma0
is a numeric vector, cell vector of scalars, or a one-dimensionalLagOp
lag operator polynomial. For vectors,ma0
has length q and the elements correspond to lagged innovations composing the AR polynomial in difference-equation notation. That is,ma0(j)
orma0{j}
is the coefficient of εt-j.For
numVars
-dimensional time series models,ma0
is a cell vector of numericnumVars
-by-numVars
numeric matrices or anumVars
-dimensionalLagOp
lag operator polynomial. For cell vectors:ma0
has length q.ar0
andma0
must both containnumVars
-by-numVars
matrices.The elements of
ma0
correspond to the lagged responses composing the AR polynomial in difference equation notation. That is,ma0{j}
is the coefficient matrix of yt-j.
For
LagOp
lag operator polynomials:The first element of the
Coefficients
property corresponds to the coefficient of εt (to accommodate structural models). All other elements correspond to the coefficients of the subsequent lags in theLags
property.To construct a univariate model in reduced form, specify
1
for the first coefficient. FornumVars
-dimensional multivariate models, specifyeye(numVars)
for the first coefficient.
If the ARMA model is strictly an AR model, then specify []
or {}
.
It is a best practice for ma0
to constitute
an invertible time series model.
numLags
— Maximum number of lag-term coefficients to return
positive integer
Maximum number of lag-term coefficients to return, specified as a positive integer.
If you specify 'numLags'
, then arma2ma
truncates
the output polynomial at a maximum of numLags
lag
terms, and then returns the remaining coefficients. As a result, the
output vector has numLags
elements or is at most
a degree numLags
LagOp
lag
operator polynomial.
By default, arma2ma
determines the
number of lag coefficients to return by the stopping criteria of mldivide
.
Data Types: double
Output Arguments
ma
— Lag-term coefficients of the truncated MA model
numeric vector | cell vector of square, numeric matrices | LagOp
lag operator polynomial object
Lag-term coefficients of the truncated MA model approximation
of the ARMA model, returned as a numeric vector, cell vector of square,
numeric matrices, or a LagOp
lag
operator polynomial object. ma
has numLags
elements,
or is at most a degree numLags
LagOp
lag
operator polynomial.
The data types and orientations of ar0
and ma0
determine
the data type and orientation of ma
. If ar0
or ma0
are
of the same data type or have the same orientation, then ma
shares
the common data type or orientation. If at least one of ar0
or ma0
is
a LagOp
lag operator polynomial, then ma
is
a LagOp
lag operator polynomial. Otherwise, if
at least one of ar0
or ma0
is
a cell vector, then ma
is a cell vector. If ar0
and ma0
are
cell or numeric vectors and at least one is a row vector, then ma
is
a row vector.
If ma
is a cell or numeric vector, then the
order of the elements of ma
corresponds to the
order of the coefficients of the lagged innovations in difference-equation notation starting with
the coefficient of εt-1.
The resulting MA model is in reduced form.
If ma
is a LagOp
lag operator
polynomial, then the order of the coefficients of ma
corresponds
to the order of the coefficients of the lagged innovations in lag operator notation starting
with the coefficient of εt.
If Θ0 ≠ InumVars,
then the resulting MA model is structural.
More About
Difference-Equation Notation
A linear time series model written in difference-equation notation positions the present value of the response and its structural coefficient on the left side of the equation. The right side of the equation contains the sum of the lagged responses, present innovation, and lagged innovations with corresponding coefficients.
In other words, a linear time series written in difference-equation notation is
where
yt is a
numVars
-dimensional vector representing the responses ofnumVars
variables at time t, for all t and fornumVars
≥ 1.εt is a
numVars
-dimensional vector representing the innovations at time t.Φj is the
numVars
-by-numVars
matrix of AR coefficients of the response yt-j, for j = 0,...,p.Θk is the
numVars
-by-numVars
matrix of MA coefficients of the innovation εt-k., k = 0,...,q.c is the n-dimensional model constant.
Φ0 = Θ0 = InumVars, which is the
numVars
-dimensional identity matrix, for models in reduced form.
Lag Operator Notation
A time series model written in lag operator notation positions a p-degree lag operator polynomial on the present response on the left side of the equation. The right side of the equation contains the model constant and a q-degree lag operator polynomial on the present innovation.
In other words, a linear time series model written in lag operator notation is
where
yt is a
numVars
-dimensional vector representing the responses ofnumVars
variables at time t, for all t and fornumVars
≥ 1., which is the autoregressive, lag operator polynomial.
L is the back-shift operator, in other words, .
Φj is the
numVars
-by-numVars
matrix of AR coefficients of the response yt-j, for j = 0,...,p.εt is a
numVars
-dimensional vector representing the innovations at time t., which is the moving average, lag operator polynomial.
Θk is the
numVars
-by-numVars
matrix of MA coefficients of the innovation εt-k., k = 0,...,q.c is the
numVars
-dimensional model constant.Φ0 = Θ0 = I
numVars
, which is thenumVars
-dimensional identity matrix, for models in reduced form.
When comparing lag operator notation to difference-equation notation, the signs of the lagged AR coefficients appear negated relative to the corresponding terms in difference-equation notation. The signs of the moving average coefficients are the same and appear on the same side.
For more details on lag operator notation, see Lag Operator Notation.
Tips
Algorithms
The software computes the infinite-lag polynomial of the resulting MA model according to this equation in lag operator notation:
where and
arma2ma
approximates the MA model coefficients whetherar0
andma0
compose a stable polynomial (a polynomial that is stationary or invertible). To check for stability, useisStable
.isStable
requires aLagOp
lag operator polynomial as input. For example, ifar0
is a vector, enter the following code to checkar0
for stationarity.ar0LagOp = LagOp([1 -ar0]); isStable(ar0LagOp)
A
0
indicates that the polynomial is not stable.You can similarly check whether the MA approximation to the ARMA model (
ma
) is invertible.
References
[1] 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.
[2] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[3] Lutkepohl, H. New Introduction to Multiple Time Series Analysis. Springer-Verlag, 2007.
Version History
Introduced in R2015a
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