Dear list users,
I want to use apply a MA(2) process (x=beta1*epsilon_(t-1) +
beta2*epsilon_(t-1) + epsilon_(t)) to a given time series (x), and I want to
estimate the two parameters beta1, beta2 and the variance of the random
variable epsilon_(t).
If I use
MA2_1 <- Arima(x, order=c(0,0,2))
I get the following result
[1] "MA2_1"
Series: x
ARIMA(0,0,2) with non-zero mean
Coefficients:
ma1 ma2 intercept
-0.0279 0.0783 5.3737
s.e. 0.0667 0.0622 0.0245
sigma^2 estimated as 0.1284: log likelihood=-92.63
AIC=193.25 AICc=193.43 BIC=207.11
[1] 0 2 0 0 1 0 0
From this straightforward analysis V[epsilon]=0.1284, beta1=-0.0279 and
beta2=0.0783.
I also tried to use a DLM representation of ARIMA models and estimate the
unknown parameters by maximum likelihood through the dlm package (in particular
applying the example at section 3.2.6, page 115, of "Dynamic Linear Models with
R" by Petris, Petrone and Campagnoli:
arma_parameters <- function(x)
{
buildGap <- function(u)
{
gap <- dlmModARMA(ma = u[2 : 3], sigma2 = u[1])
return(gap)
}
init <- c(0.005, 0.004, 0.003)
outMLE <- dlmMLE(x, init, buildGap)
dlmGap <- buildGap(outMLE$par)
}
and this gives:
[1] "outMLE"
$par
[1] 1.00816794 0.02349296 0.02364788
$value
[1] 3089.196
$counts
function gradient
10 10
$convergence
[1] 0
$message
[1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
[1] "dlmGap"
$FF
[,1] [,2] [,3]
[1,] 1 0 0
$V
[,1]
[1,] 0
$GG
[,1] [,2] [,3]
[1,] 0 1 0
[2,] 0 0 1
[3,] 0 0 0
$W
[,1] [,2] [,3]
[1,] 1.00816794 0.0236848488 0.0238410337
[2,] 0.02368485 0.0005564272 0.0005600964
[3,] 0.02384103 0.0005600964 0.0005637899
$m0
[1] 0 0 0
$C0
[,1] [,2] [,3]
[1,] 1e+07 0e+00 0e+00
[2,] 0e+00 1e+07 0e+00
[3,] 0e+00 0e+00 1e+07
In this case
V[epsilon]=W[1,1]=1.00816794
beta1=W[2,1]/W[1,1]=0.02349296
beta2=W[3,1]/W[1,1]=0.02364788
I presume that these two approaches should give comparable results, but this
does not happen.
Is the model that I used correct? And does it make sense to perform this kind
of comparison?
This is the log of a rainfall time series (which has already been
deseasonalised):
[1] 6.014937 4.978801 5.654592 5.616771 5.612398 5.837147 5.121580 5.832176
[9] 5.205654 5.355642 5.405376 6.257859 5.516247 5.500850 4.708629 5.482304
[17] 5.689684 5.727824 4.779123 5.289277 5.217107 5.976351 4.630838 5.683240
[25] 5.345678 5.906179 5.605434 5.497578 5.898801 5.660875 5.111988 5.571013
[33] 5.949340 5.374352 4.841033 5.995706 5.661223 5.458734 4.454347 5.795754
[41] 5.995706 5.596939 5.399971 5.908898 5.282696 5.438514 5.528635 6.022721
[49] 5.524257 5.519459 4.957235 5.547518 5.080783 5.411200 5.056883 5.798183
[57] 5.086361 5.536547 5.220356 5.141664 5.847017 5.052417 5.734635 5.340419
[65] 5.724238 5.634432 5.685958 5.307773 5.817706 5.134032 4.987708 5.110179
[73] 5.423628 5.347108 4.859037 5.556828 5.487283 5.661223 5.732370 5.469325
[81] 5.726848 5.419207 5.172187 5.608006 5.130490 5.586874 5.171052 5.683240
[89] 4.674696 5.286245 5.342813 5.370638 5.432411 5.748118 6.355239 5.557986
[97] 5.399067 5.222516 5.279644 5.425390 5.540871 5.917818 5.132853 5.689007
[105] 5.900993 5.007296 5.102911 5.778271 5.318120 5.927726 5.066385 5.716699
[113] 5.511815 4.714921 5.383577 5.319100 5.269403 5.354698 5.145749 5.204556
[121] 5.878296 5.070161 5.441552 5.213304 5.450180 5.695750 4.893352 5.425390
[129] 5.682559 5.487283 4.213608 5.751620 5.432411 5.379436 5.700444 5.580484
[137] 5.357529 5.319100 4.532599 5.603225 5.208393 5.254888 5.017280 5.349961
[145] 4.374498 5.187944 5.585374 5.716370 3.561046 5.119789 5.163070 5.422745
[153] 5.863915 5.651436 4.762174 5.655642 4.797442 5.735927 4.911183 5.240688
[161] 5.148076 5.477300 4.572647 5.493473 5.437644 4.854371 4.908233 4.755313
[169] 5.582744 5.527841 5.613128 5.211124 5.275049 5.462984 5.016617 5.981919
[177] 5.566817 5.094364 5.314191 5.712742 5.299317 5.452325 4.691348 5.851628
[185] 5.410753 5.488938 5.660179 5.900993 5.380819 5.256453 4.781641 5.531807
[193] 5.497578 5.274537 4.325456 5.271973 5.077047 5.258536 5.280662 5.247024
[201] 5.995208 4.700480 4.991113 5.457029 5.194622 5.487283 5.197391 5.747161
[209] 5.842094 5.372497 5.306781 5.641907 5.565286 5.259057 5.241218 4.759607
[217] 4.550714 5.230574 4.470495 5.664348 4.846547 5.771130 4.823502 5.598422
[225] 5.627621 5.547518 5.596939 5.468482 5.536940 5.606170 5.281680 5.656691
[233] 5.283204 5.752255 5.192401 4.550714
Thank you for your attention and your help
Stefano
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