Impulse Response Function
impresp.RdCompute the (orthogonalized) impulse response function of a VARMA model or a state space model. The impulse response coefficients are also called Power series parameters of the system.
Usage
impresp(obj, lag.max, H)
# S3 method for armamod
impresp(obj, lag.max = 12, H = NULL)
# S3 method for rmfdmod
impresp(obj, lag.max = 12, H = NULL)
# S3 method for stspmod
impresp(obj, lag.max = 12, H = NULL)
# S3 method for impresp
impresp(obj, lag.max = NULL, H = NULL)Arguments
- obj
armamod(),stspmod()object orimpresp()object. The last case may be used to transform the impulse response function to a different orthogonalization scheme.- lag.max
Maximum lag of the impulse response coefficients. This parameter is ignored in the case that
objis an impresp object.- H
An (n x n) (non singular) matrix which specifies a transformation of the noise. The noise \(u_t\) is transformed to \(H^{-1}u_t\) and the impulse response coefficients (\(k_j \rightarrow k_j H\)) and the (left) square root of the noise covariance matrix (\(L \rightarrow H^{-1}L\)) are transformed correspondingly.
The default caseH=NULLcorresponds to the identity matrix (i.e. no transformation).
ForH='chol', the transformation matrixH = t(chol(Sigma))is determined from the Choleski decomposition of the noise covariance \(\Sigma\). ForH='eigen'the symmetric square root of \(\Sigma\) (obtained from the the eigenvalue decomposition of \(\Sigma\)) is used. ForH='sigma_L'the left square root of the noise covariance, which is stored in the objectobj, is used. In these cases one obtains an orthogonalized impulse response function. Other orthogonalization schemes may be obtained by setting \(H\) to a suitable square root of \(\Sigma\).
Value
impresp object, i.e. a list with components
- irf
pseriesobject.- sigma_L
(n,n)-dimensional matrix which contains left square of noise covariance matrix.
- names
(n)-dimensional character vector or NULL
- label
character string or NULL
Details
The impulse response coefficients \((k_j \,|\, j \geq 0)\) define the map between the noise and the output process. If the model is stable then the stationary solution of the ARMA system, respectively state space system, is given by $$ y_t = \sum_{j \geq 0} k_j u_{t-j}. $$ For a state space system the impulse response coefficients are $$k_0 = D \mbox{ and }$$ $$k_j = CA^{j-1}B \mbox{ for }j >0.$$ For an ARMA model the coefficients are (recursively) computed by a comparison of coefficients in the equation $$ (a_0 + a_1 z + \cdots + a_p z^p)(k_0 + k_1 z + k_2 z^2 + \cdots ) = b_0 + b_1 z + \cdots + b_q z^q $$
The S3 methods impresp.* compute the coefficients \(k_j\) for
\(j = 0,\cdots,N\) and store the result, together with the left square root
(sigma_L) of the noise covariance \(\Sigma\), in a impresp object.
impresp objects contain the complete information
about the underlying model, provided that the maximum lag \(N\) is large enough.
This means that one may reconstruct the underlying model from an impulse response object.
Examples
# IRF from state space model ################################################
model = stspmod(stsp(A = c(0,0.2,1,-0.5), B = c(1,1,1,-1),
C = c(1,0,0,1)),
sigma_L = matrix(c(4, 1, 1, 3), 2, 2),
names = c('y1','y2'), label = 'test model')
# IRF
irf = impresp(model, lag.max=10)
irf
#> test model: Impulse response [2,2] with 10 lags
#> lag=0 [,1] [,2] lag=1 [,1] [,2] lag=2 [,1] [,2] lag=3 [,1] [,2]
#> [1,] 1 0 1 1 1.0 -1.0 -0.30 0.70
#> [2,] 0 1 1 -1 -0.3 0.7 0.35 -0.55
#> lag=4 [,1] [,2] lag=5 [,1] [,2] lag=6 [,1] [,2] lag=7 [,1]
#> [1,] 0.350 -0.550 -0.2350 0.4150 0.18750 -0.31750 -0.140750
#> [2,] -0.235 0.415 0.1875 -0.3175 -0.14075 0.24175 0.107875
#> [,2] lag=8 [,1] [,2] lag=9 [,1] [,2] lag=10 [,1]
#> [1,] 0.241750 0.1078750 -0.1843750 -0.08208750 0.1405375 0.06261875
#> [2,] -0.184375 -0.0820875 0.1405375 0.06261875 -0.1071438 -0.04772688
#> [,2]
#> [1,] -0.10714375
#> [2,] 0.08167938
# Orthogonalized IRF: Cholesky
irf_chol = impresp(model, lag.max = 10, H = 'chol')
irf_chol
#> test model: Impulse response [2,2] with 10 lags
#> lag=0 [,1] [,2] lag=1 [,1] [,2] lag=2 [,1] [,2] lag=3 [,1]
#> [1,] 4.123106 0.000000 5.820855 2.667892 2.42535625 -2.667892 -0.04850713
#> [2,] 1.697749 2.667892 2.425356 -2.667892 -0.04850713 1.867524 0.50932481
#> [,2] lag=4 [,1] [,2] lag=5 [,1] [,2] lag=6 [,1] [,2]
#> [1,] 1.867524 0.5093248 -1.467341 -0.2643638 1.1071751 0.2340469 -0.8470557
#> [2,] -1.467341 -0.2643638 1.107175 0.2340469 -0.8470557 -0.1698962 0.6449629
#> lag=7 [,1] [,2] lag=8 [,1] [,2] lag=9 [,1] [,2]
#> [1,] -0.1698962 0.6449629 0.13175748 -0.4918926 -0.09985798 0.3749389
#> [2,] 0.1317575 -0.4918926 -0.09985798 0.3749389 0.07628049 -0.2858479
#> lag=10 [,1] [,2]
#> [1,] 0.07628049 -0.2858479
#> [2,] -0.05811184 0.2179117
print(irf_chol$sigma_L) # Sigma is (approximately equal to) the identity matrix
#> [,1] [,2]
#> [1,] 0.9701425 0.2425356
#> [2,] -0.2425356 0.9701425
# IRF from VARMA model ################################################
model = armamod(sys = test_lmfd(dim = c(2,2), degrees = c(2,1)))
irf = impresp(model)
print(irf, digits = 2, format = 'iz|j')
#> Orthogonalized impulse response [2,2] with 12 lags
#> [,1] [,2]
#> lag=0 [1,] 0.67 -0.75
#> [2,] 0.40 0.38
#> lag=1 [1,] 2.55 -3.11
#> [2,] -1.70 -0.67
#> lag=2 [1,] 6.17 -6.29
#> [2,] -2.33 1.79
#> lag=3 [1,] 10.40 -14.72
#> [2,] -5.66 3.90
#> lag=4 [1,] 18.05 -28.80
#> [2,] -9.37 11.23
#> lag=5 [1,] 28.64 -55.07
#> [2,] -17.01 23.12
#> lag=6 [1,] 44.39 -98.58
#> [2,] -27.71 46.89
#> lag=7 [1,] 64.29 -170.36
#> [2,] -44.75 87.07
#> lag=8 [1,] 87.08 -281.50
#> [2,] -67.76 156.07
#> lag=9 [1,] 104.38 -446.66
#> [2,] -97.81 266.57
#> lag=10 [1,] 98.13 -674.86
#> [2,] -129.69 438.25
#> lag=11 [1,] 27.47 -960.95
#> [2,] -151.42 689.11
#> lag=12 [1,] -186.99 -1256.88
#> [2,] -131.32 1032.02