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RegARIMA model, pre-adjustment in X13 and TRAMO-SEATS

Source: R/jregarima.R, R/regarima.R
regarima.Rd

The regarima/regarima_x13/regarima_tramoseats functions remove deterministic effects from the input series (e.g.calendar effects, outliers) using a multivariate regression model with arima errors. The jregarima/jregarima_x13/jregarima_tramoseats functions do the same computation but return the Java objects instead of a formatted output.

Usage

jregarima(series, spec = NA)

jregarima_tramoseats(
  series,
  spec = c("TRfull", "TR0", "TR1", "TR2", "TR3", "TR4", "TR5")
)

jregarima_x13(series, spec = c("RG5c", "RG0", "RG1", "RG2c", "RG3", "RG4c"))

regarima(series, spec = NA)

regarima_tramoseats(
  series,
  spec = c("TRfull", "TR0", "TR1", "TR2", "TR3", "TR4", "TR5")
)

regarima_x13(series, spec = c("RG5c", "RG0", "RG1", "RG2c", "RG3", "RG4c"))

Arguments

series

an univariate time series

spec

the model specification. For the function:

  • regarima: an object of class c("regarima_spec","X13") or c("regarima_spec","TRAMO_SEATS"). See the functions regarima_spec_x13 and regarima_spec_tramoseats.

  • regarima_x13: the name of a predefined X13 'JDemetra+' model specification (see Details). The default value is "RG5c".

  • regarima_tramoseats:the name of a predefined TRAMO-SEATS 'JDemetra+' model specification (see Details). The default value is "TRfull".

Value

The jregarima/jregarima_x13/jregarima_tramoseats functions return a jSA object that contains the result of the pre-adjustment method without any formatting. Therefore, the computation is faster than with the regarima/regarima_x13/regarima_tramoseats functions. The results of the seasonal adjustment can be extracted with the function get_indicators.

The regarima/regarima_x13/regarima_tramoseats functions return an object of class "regarima"

and sub-class "X13" or "TRAMO_SEATS". regarima_x13 returns an object of class c("regarima","X13") and regarima_tramoseats, an object of class c("regarima","TRAMO_SEATS"). For the function regarima, the sub-class of the object depends on the used method that is defined by the spec object class.

An object of class "regarima" is a list containing the following components:

specification

a list with the model specification as defined by the spec argument. See also the Value of the regarima_spec_x13 and regarima_spec_tramoseats functions.

arma

a vector containing the orders of the autoregressive (AR), moving average (MA), seasonal AR and seasonal MA processes, as well as the regular and seasonal differencing orders (P,D,Q) (BP,BD,BQ).

arima.coefficients

a matrix containing the estimated regular and seasonal AR and MA coefficients, as well as the associated standard errors and t-statistics values. The estimated coefficients can be also extracted with the function coef (whose output also includes the regression coefficients).

regression.coefficients

a matrix containing the estimated regression variables (i.e.: mean, calendar effect, outliers and user-defined regressors) coefficients, as well as the associated standard errors and t-statistics values. The estimated coefficients can be also extracted with the function coef (whose output also includes the arima coefficients).

loglik

a matrix containing the log-likelihood of the RegARIMA model as well as the associated model selection criteria statistics (AIC, AICC, BIC and BICC) and parameters (np = number of parameters in the likelihood, neffectiveobs = number of effective observations in the likelihood). These statistics can also be extracted with the function logLik.

model

a list containing information on the model specification after its estimation (spec_rslt), as well as the decomposed elements of the input series (ts matrix, effects). The model specification includes information on the estimation method (Model) and time span (T.span), whether the original series was log transformed (Log transformation) and details on the regression part of the RegARIMA model i.e. if it includes a Mean, Trading days effects (if so, it provides the number of regressors), Leap year effect, Easter effect and whether outliers were detected (Outliers (if so, it provides the number of outliers). The decomposed elements of the input series contain the linearised series (y_lin) and the deterministic components i.e.: trading days effect (tde), Easter effect (ee), other moving holidays effect (omhe) and outliers effect (total - out, related to irregular - out_i, related to trend - out_t, related to seasonal - out_s).

residuals

the residuals (time series). They can be also extracted with the function residuals.

residuals.stat

a list containing statistics on the RegARIMA residuals. It provides the residuals standard error (st.error) and the results of normality, independence and linearity of the residuals (tests) - object of class c("regarima_rtests","data.frame").

forecast

a ts matrix containing the forecast of the original series (fcst) and its standard error (fcsterr).

Details

When seasonally adjusting with X13 and TRAMO-SEATS, the first step consists in pre-adjusting the original series with a RegARIMA model, where the original series is corrected for any deterministic effects and missing observations. This step is also referred to as the linearization of the original series.

The RegARIMA model (model with ARIMA errors) is specified as such:

$$z_t = y_t\beta + x_t$$

where:

  • \(z_t\) is the original series;

  • \(\beta = (\beta_1,...,\beta_n)\) is a vector of regression coefficients;

  • \(y_t = (y_{1t},...,y_{nt})\) are \(n\) regression variables (outliers, calendar effects, user-defined variables);

  • \(x_t\) is a disturbance that follows the general ARIMA process: \(\phi(B)\delta(B)x_t = \theta(B)a_t\); where \(\phi(B), \delta(B)\) and \(\theta(B)\) are finite polynomials in \(B\) and \(a_t\) is a white noise variable with zero mean and a constant variance.

The polynomial \(\phi(B)\) is a stationary autoregressive (AR) polynomial in \(B\), which is a product of the stationary regular AR polynomial in \(B\) and the stationary seasonal polynomial in \(B^s\):

$$\phi(B)=\phi_p(B)\Phi_{bp}(B^s)=(1+\phi_1B+...+\phi_pB^p)(1+\Phi_1B^s+...+\Phi_{bp}B^{bps})$$

where:

  • \(p\) is the number of regular AR terms (here and in 'JDemetra+', \(p \le 3\));

  • \(bp\) is the number of seasonal AR terms (here and in 'JDemetra+', \(bp \le 1\));

  • \(s\) is the number of observations per year (ie. The time series frequency).

The polynomial \(\theta(B)\) is an invertible moving average (MA) polynomial in \(B\), which is a product of the invertible regular MA polynomial in \(B\) and the invertible seasonal MA polynomial in \(B^s\):

$$\theta(B)=\theta_q(B)\Theta_{bq}(B^s)=(1+\theta_1B+...+\theta_qB^q)(1+\Theta_1B^s+...+\Theta_{bq}B^{bqs})$$

where:

  • \(q\) is the number of regular MA terms (here and in 'JDemetra+', \(q \le 3\));

  • \(bq\) is the number of seasonal MA terms (here and in 'JDemetra+', \(bq \le 1\)).

The polynomial \(\delta(B)\) is the non-stationary AR polynomial in \(B\) (unit roots):

$$\delta(B) = (1-B)^d(1-B^s)^{d_s}$$

where:

  • \(d\) is the regular differencing order (here and in 'JDemetra+', \(d \le 1\));

  • \(d_s\) is the seasonal differencing order (here and in 'JDemetra+', \(d_s \le 1\)).

NB. The notations used for AR and MA processes, as well as the model denoted as ARIMA \((P,D,Q)(BP,BD,BQ)\), are consistent with those in 'JDemetra+'.

The available predefined 'JDemetra+' X13 and TRAMO-SEATS model specifications are described in the tables below:

X13:

Identifier |Log/level detection |Outliers detection |Calendar effects |ARIMARG0 |NA |
NA |NA |Airline(+mean)RG1 |automatic |AO/LS/TC |NA |
Airline(+mean)RG2c |automatic |AO/LS/TC |2 td vars + Easter |Airline(+mean)RG3 |
automatic |AO/LS/TC |NA |automaticRG4c |automatic |AO/LS/TC |
2 td vars + Easter |automaticRG5c |automatic |AO/LS/TC |7 td vars + Easter |automatic

TRAMO-SEATS:

Identifier |Log/level detection |Outliers detection |Calendar effects |ARIMATR0 |NA |NA |
NA |Airline(+mean)TR1 |automatic |AO/LS/TC |NA |Airline(+mean)TR2 |
automatic |AO/LS/TC |2 td vars + Easter |Airline(+mean)TR3 |automatic |AO/LS/TC |NA |
automaticTR4 |automatic |AO/LS/TC |2 td vars + Easter |automaticTR5 |automatic |
AO/LS/TC |7 td vars + Easter |automaticTRfull |automatic |AO/LS/TC |automatic |automatic

References

More information and examples related to 'JDemetra+' features in the online documentation: https://jdemetra-new-documentation.netlify.app/

BOX G.E.P. and JENKINS G.M. (1970), "Time Series Analysis: Forecasting and Control", Holden-Day, San Francisco.

BOX G.E.P., JENKINS G.M., REINSEL G.C. and LJUNG G.M. (2015), "Time Series Analysis: Forecasting and Control", John Wiley & Sons, Hoboken, N. J., 5th edition.

Examples

# \donttest{
 # X13 method
myseries <- ipi_c_eu[, "FR"]
myreg <- regarima_x13(myseries, spec ="RG5c")
summary(myreg)
#> y = regression model + arima (2, 1, 1, 0, 1, 1)
#> 
#> Model: RegARIMA - X13
#> Estimation span: from 1-1990 to 12-2020
#> Log-transformation: no
#> Regression model: no mean, trading days effect(7), leap year effect, Easter effect, outliers(4)
#> 
#> Coefficients:
#> ARIMA: 
#>             Estimate Std. Error  T-stat Pr(>|t|)    
#> Phi(1)     0.0003269  0.1077296   0.003   0.9976    
#> Phi(2)     0.1688192  0.0740996   2.278   0.0233 *  
#> Theta(1)  -0.5485606  0.1016550  -5.396 1.24e-07 ***
#> BTheta(1) -0.6660849  0.0422242 -15.775  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Regression model: 
#>               Estimate Std. Error  T-stat Pr(>|t|)    
#> Monday         0.55932    0.22801   2.453 0.014638 *  
#> Tuesday        0.88221    0.22832   3.864 0.000132 ***
#> Wednesday      1.03996    0.22930   4.535 7.85e-06 ***
#> Thursday       0.04943    0.22944   0.215 0.829549    
#> Friday         0.91132    0.22988   3.964 8.88e-05 ***
#> Saturday      -1.57769    0.22775  -6.927 1.99e-11 ***
#> Leap year      2.15403    0.70527   3.054 0.002425 ** 
#> Easter [1]    -2.37950    0.45391  -5.242 2.71e-07 ***
#> TC (4-2020)  -35.59245    2.17330 -16.377  < 2e-16 ***
#> AO (3-2020)  -20.89026    2.18013  -9.582  < 2e-16 ***
#> AO (5-2011)   13.49850    1.85694   7.269 2.28e-12 ***
#> LS (11-2008) -12.54901    1.63554  -7.673 1.60e-13 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> 
#> Residual standard error: 2.218 on 342 degrees of freedom
#> Log likelihood = -799.1, aic =  1632, aicc =  1634, bic(corrected for length) = 1.855
#> 
plot(myreg)







myspec1 <- regarima_spec_x13(myreg, tradingdays.option = "WorkingDays")
myreg1 <- regarima(myseries, myspec1)

myspec2 <- regarima_spec_x13(myreg, usrdef.outliersEnabled = TRUE,
             usrdef.outliersType = c("LS", "AO"),
             usrdef.outliersDate = c("2008-10-01", "2002-01-01"),
             usrdef.outliersCoef = c(36, 14),
             transform.function = "None")
myreg2 <- regarima(myseries, myspec2)
myreg2
#> y = regression model + arima (2, 1, 1, 0, 1, 1)
#> Log-transformation: no
#> Coefficients:
#>           Estimate Std. Error
#> Phi(1)     0.07859      0.114
#> Phi(2)     0.19792      0.076
#> Theta(1)  -0.48272      0.111
#> BTheta(1) -0.65916      0.043
#> 
#>               Estimate Std. Error
#> Monday         0.64094      0.228
#> Tuesday        0.81794      0.229
#> Wednesday      1.05374      0.229
#> Thursday       0.06981      0.228
#> Friday         0.93434      0.228
#> Saturday      -1.63686      0.226
#> Leap year      2.11550      0.697
#> Easter [1]    -2.38135      0.451
#> AO (9-2008)   31.95554      2.924
#> LS (9-2008)  -57.04093      2.657
#> TC (4-2020)  -35.62104      2.120
#> AO (3-2020)  -21.00931      2.145
#> AO (5-2011)   13.21877      1.832
#> TC (9-2008)   23.44654      4.001
#> TC (12-2001) -20.47521      2.922
#> AO (12-2001)  17.13461      2.962
#> TC (2-2002)   10.61731      1.937
#> 
#> Fixed outliers: 
#>              Coefficients
#> LS (10-2008)           36
#> AO (1-2002)            14
#> 
#> 
#> Residual standard error: 2.178 on 337 degrees of freedom
#> Log likelihood = -792.6, aic =  1629 aicc =  1632, bic(corrected for length) = 1.901
#> 

myspec3 <- regarima_spec_x13(myreg, automdl.enabled = FALSE,
             arima.p = 1, arima.q = 1,
             arima.bp = 0, arima.bq = 1,
             arima.coefEnabled = TRUE,
             arima.coef = c(-0.8, -0.6, 0),
             arima.coefType = c(rep("Fixed", 2), "Undefined"))
s_arimaCoef(myspec3)
#>                Type Value
#> Phi(1)        Fixed  -0.8
#> Theta(1)      Fixed  -0.6
#> BTheta(1) Undefined   0.0
myreg3 <- regarima(myseries, myspec3)
summary(myreg3)
#> y = regression model + arima (1, 1, 1, 0, 1, 1)
#> 
#> Model: RegARIMA - X13
#> Estimation span: from 1-1990 to 12-2020
#> Log-transformation: yes
#> Regression model: no mean, trading days effect(6), no leap year effect, Easter effect, outliers(3)
#> 
#> Coefficients:
#> ARIMA: 
#>           Estimate Std. Error T-stat Pr(>|t|)    
#> Phi(1)     -0.8000     0.0000     NA       NA    
#> Theta(1)   -0.6000     0.0000     NA       NA    
#> BTheta(1)  -0.6977     0.0399 -17.49   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Regression model: 
#>              Estimate Std. Error  T-stat Pr(>|t|)    
#> Monday       0.006317   0.001791   3.526 0.000476 ***
#> Tuesday      0.007824   0.001793   4.363 1.68e-05 ***
#> Wednesday    0.010528   0.001802   5.841 1.16e-08 ***
#> Thursday     0.001857   0.001811   1.025 0.306022    
#> Friday       0.010099   0.001812   5.574 4.90e-08 ***
#> Saturday    -0.018439   0.001781 -10.354  < 2e-16 ***
#> Easter [1]  -0.020593   0.003515  -5.859 1.06e-08 ***
#> TC (4-2020) -0.475720   0.031229 -15.233  < 2e-16 ***
#> AO (3-2020) -0.213355   0.023246  -9.178  < 2e-16 ***
#> AO (5-2011)  0.143705   0.015529   9.254  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> 
#> Residual standard error: 0.0256 on 347 degrees of freedom
#> Log likelihood = 802.3, aic =  1733, aicc =  1734, bic(corrected for length) = -7.15
#> 
plot(myreg3)







 # TRAMO-SEATS method
myspec <- regarima_spec_tramoseats("TRfull")
myreg <- regarima(myseries, myspec)
myreg
#> y = regression model + arima (2, 1, 0, 0, 1, 1)
#> Log-transformation: no
#> Coefficients:
#>           Estimate Std. Error
#> Phi(1)      0.4032      0.051
#> Phi(2)      0.2883      0.051
#> BTheta(1)  -0.6641      0.042
#> 
#>             Estimate Std. Error
#> Week days     0.6994      0.032
#> Leap year     2.3231      0.690
#> Easter [6]   -2.5154      0.436
#> AO (5-2011)  13.4679      1.787
#> TC (4-2020) -22.2128      2.205
#> TC (3-2020) -21.0391      2.217
#> AO (5-2000)   6.7386      1.794
#> 
#> 
#> Residual standard error: 2.326 on 348 degrees of freedom
#> Log likelihood = -816.1, aic =  1654 aicc =  1655, bic(corrected for length) = 1.852
#> 

myspec2 <- regarima_spec_tramoseats(myspec, tradingdays.mauto = "Unused",
             tradingdays.option = "WorkingDays",
             easter.type = "Standard",
             automdl.enabled = FALSE, arima.mu = TRUE)
myreg2 <- regarima(myseries, myspec2)

var1 <- ts(rnorm(length(myseries))*10, start = start(myseries), frequency = 12)
var2 <- ts(rnorm(length(myseries))*100, start = start(myseries), frequency = 12)
var <- ts.union(var1, var2)
myspec3 <- regarima_spec_tramoseats(myspec,
             usrdef.varEnabled = TRUE, usrdef.var = var)
s_preVar(myspec3)
#> $series
#>                  var1         var2
#> Jan 1990 -14.00043517  -96.0449159
#> Feb 1990   2.55317055  -97.5423042
#> Mar 1990 -24.37263611  -33.8576503
#> Apr 1990  -0.05571287  115.2347074
#> May 1990   6.21552721   40.5101183
#> Jun 1990  11.48411606  -47.0922497
#> Jul 1990 -18.21817661  -13.3251019
#> Aug 1990  -2.47325302  122.6682356
#> Sep 1990  -2.44199607   33.2943995
#> Oct 1990  -2.82705449  -34.7088466
#> Nov 1990  -5.53699384   -9.8550690
#> Dec 1990   6.28982042    3.4766060
#> Jan 1991  20.65024895   38.6127022
#> Feb 1991 -16.30989402    2.0831228
#> Mar 1991   5.12426950    0.7586777
#> Apr 1991 -18.63011492   93.0844030
#> May 1991  -5.22012515  -68.4749941
#> Jun 1991  -0.52601910   33.7401513
#> Jul 1991   5.42996343  -41.2137704
#> Aug 1991  -9.14074827   93.4261130
#> Sep 1991   4.68154420  184.0316741
#> Oct 1991   3.62951256  -70.4819663
#> Nov 1991 -13.04543545    0.8510312
#> Dec 1991   7.37776321  203.4189886
#> Jan 1992  18.88504929 -134.1686068
#> Feb 1992  -0.97445104  115.8979182
#> Mar 1992  -9.35847354  -20.3208958
#> Apr 1992  -0.15950311  -37.8028555
#> May 1992  -8.26788954  173.6111043
#> Jun 1992 -15.12399651  -84.5247816
#> Jul 1992   9.35363190  -96.1571493
#> Aug 1992   1.76488611  101.7491053
#> Sep 1992   2.43685465 -149.6053742
#> Oct 1992  16.23548883 -118.4818730
#> Nov 1992   1.12038083   63.0234373
#> Dec 1992  -1.33997013  210.1252514
#> Jan 1993 -19.10087468  -61.3736810
#> Feb 1993  -2.79237242 -163.4638272
#> Mar 1993  -3.13445978   -1.0441117
#> Apr 1993  10.67307879  -65.6506139
#> May 1993   0.70034850  -66.9533441
#> Jun 1993  -6.39123324  -47.8589028
#> Jul 1993  -0.49964899  131.9456316
#> Aug 1993  -2.51483443   63.6562761
#> Sep 1993   4.44797116   51.4327782
#> Oct 1993  27.55417575 -175.1375113
#> Nov 1993   0.46531380   89.3597518
#> Dec 1993   5.77709069   22.3038372
#> Jan 1994   1.18194874   58.0816593
#> Feb 1994 -19.11720491  -17.7821421
#> Mar 1994   8.62086482   74.0966708
#> Apr 1994  -2.43236740  -99.7443079
#> May 1994  -2.06087195 -293.8977561
#> Jun 1994   0.19177592   71.9015661
#> Jul 1994   0.29560754  -69.8005041
#> Aug 1994   5.49827542 -189.4125843
#> Sep 1994 -22.74114857    7.6299249
#> Oct 1994  26.82557184   87.5308501
#> Nov 1994  -3.61221255   45.3827393
#> Dec 1994   2.13355750  -85.0716906
#> Jan 1995  10.74345882   56.6201613
#> Feb 1995  -6.65088249  115.2211954
#> Mar 1995  11.13952419  -75.6197377
#> Apr 1995  -2.45896412  -48.9258334
#> May 1995 -11.77563309 -116.6052337
#> Jun 1995  -9.75850616  -47.9668950
#> Jul 1995  10.65057320   11.5348218
#> Aug 1995   1.31670635 -176.8048407
#> Sep 1995   4.88628809 -140.7638919
#> Oct 1995 -16.99450568   70.9178461
#> Nov 1995 -14.70736306 -124.0842940
#> Dec 1995   2.84150344  -36.8327348
#> Jan 1996  13.37320413   46.2080093
#> Feb 1996   2.36696283  -32.2833101
#> Mar 1996  13.18293384 -128.7214810
#> Apr 1996   5.23909788 -103.0040247
#> May 1996   6.06748047  151.4089316
#> Jun 1996  -1.09935672   34.6903586
#> Jul 1996   1.72181715  177.9441542
#> Aug 1996  -0.90327287   38.6630924
#> Sep 1996  19.24343341  -91.8695239
#> Oct 1996  12.98392759 -158.4336488
#> Nov 1996   7.48791268   -8.4058892
#> Dec 1996   5.56224329 -208.5070889
#> Jan 1997  -5.48257264    0.3567992
#> Feb 1997  11.10534893  -35.5770822
#> Mar 1997 -26.12334333  114.6359751
#> Apr 1997  -1.55693776  -22.1188446
#> May 1997   4.33889790  101.8179021
#> Jun 1997  -3.81951112  -26.3719295
#> Jul 1997   4.24187575  165.8542305
#> Aug 1997  10.63101996  -77.4086771
#> Sep 1997  10.48712620  -92.3937880
#> Oct 1997  -0.38102895  -27.5533378
#> Nov 1997   4.86148920  -59.3399688
#> Dec 1997  16.72882611  -12.2285891
#> Jan 1998  -3.54361164  117.9784246
#> Feb 1998   9.46347886   64.1037374
#> Mar 1998  13.16826356  -62.9588508
#> Apr 1998  -2.96640025  -80.7734971
#> May 1998  -3.87213575  -86.0489929
#> Jun 1998  -7.85432656 -216.9238693
#> Jul 1998 -10.56736867 -137.5836518
#> Aug 1998  -7.95541430  -49.3132472
#> Sep 1998 -17.56275428  -58.1652027
#> Oct 1998  -6.90537897  -16.7229304
#> Nov 1998  -5.58541994   48.5993129
#> Dec 1998  -5.36663326 -133.3395796
#> Jan 1999   2.27127133  -26.1965625
#> Feb 1999   9.78454920   65.2386303
#> Mar 1999  -2.08882651   74.8854971
#> Apr 1999 -13.99410460   89.6560285
#> May 1999   2.58537288  148.9300424
#> Jun 1999  -4.41799453  -65.9403481
#> Jul 1999   5.68599861   53.7283179
#> Aug 1999  21.26850459   74.6803067
#> Sep 1999   4.24858441  189.6317084
#> Oct 1999 -16.84281532 -206.0070725
#> Nov 1999   2.49401784    6.4543870
#> Dec 1999  10.72838252  -26.5147403
#> Jan 2000  20.39369263  -44.7344531
#> Feb 2000   4.49453778 -141.0700927
#> Mar 2000  13.91814046  -50.6418882
#> Apr 2000   4.26566547  -26.9761838
#> May 2000   1.07583992 -108.5154918
#> Jun 2000   0.22294733   36.2159127
#> Jul 2000   6.03611011  -33.5672143
#> Aug 2000  -2.62650573  136.3804498
#> Sep 2000  -5.28264082  -71.1524136
#> Oct 2000   1.92149422   66.2178797
#> Nov 2000 -11.46199669   29.1130223
#> Dec 2000   8.46184665   19.7958000
#> Jan 2001   0.81719629 -120.3566106
#> Feb 2001 -13.05117010   -3.9817044
#> Mar 2001  -9.44912060   68.6982465
#> Apr 2001   4.54341594   70.5267007
#> May 2001  -8.55202501   99.1441680
#> Jun 2001  -2.86895219  114.4248971
#> Jul 2001   8.94961626 -123.8910243
#> Aug 2001   0.67304440  265.4898333
#> Sep 2001  -1.62676337  -15.6917189
#> Oct 2001  -8.27310169  -42.3490117
#> Nov 2001  18.76505621  -19.8387058
#> Dec 2001   7.66440199  -89.4802407
#> Jan 2002   9.79956696   90.4269119
#> Feb 2002  13.21780992    7.9649210
#> Mar 2002 -11.19710829 -125.8827223
#> Apr 2002   5.14599819  102.5685106
#> May 2002 -15.09099836  -73.0778603
#> Jun 2002  15.32741480  -19.0145507
#> Jul 2002   4.29147371   52.8864693
#> Aug 2002   1.22103414   55.0210535
#> Sep 2002 -11.38012401   54.9684337
#> Oct 2002  -5.58015129  -65.9542372
#> Nov 2002  10.52538537    5.7421706
#> Dec 2002   6.77683644 -280.8010508
#> Jan 2003   0.38499547  -91.2259753
#> Feb 2003  -3.56381187  -78.2379163
#> Mar 2003   7.82844102  -66.4104924
#> Apr 2003   8.04411616   62.6309770
#> May 2003 -19.00060823  -50.7248206
#> Jun 2003   9.35784286   27.0361335
#> Jul 2003  -3.09051503   46.7476865
#> Aug 2003   2.63066677   72.3994958
#> Sep 2003 -17.90591856   61.3836939
#> Oct 2003  -7.88258845  -61.7869202
#> Nov 2003 -11.33021669   22.0724902
#> Dec 2003   3.63652568  112.7926598
#> Jan 2004  -2.85887914  181.3454336
#> Feb 2004   5.17669134   -8.3825685
#> Mar 2004  -1.02908670  136.7706666
#> Apr 2004  -9.74069593  -62.7434620
#> May 2004  12.70672301  -21.6629150
#> Jun 2004   9.60864787  -68.3713824
#> Jul 2004   7.68721370  -44.4702734
#> Aug 2004  10.35930771   60.6489806
#> Sep 2004  -4.73887074   62.4183075
#> Oct 2004 -12.75334875  -69.5431074
#> Nov 2004  -3.05620674  -78.3639078
#> Dec 2004  22.11769487  -95.3123859
#> Jan 2005 -10.41668381  179.2756071
#> Feb 2005 -11.46523850   34.8976696
#> Mar 2005 -16.75327303   25.9103768
#> Apr 2005  15.25938655  -80.5951897
#> May 2005   5.54185515   10.5664701
#> Jun 2005  19.93110265  -33.3599682
#> Jul 2005  -1.54120740  164.1847970
#> Aug 2005  25.64408338  -64.3905859
#> Sep 2005  10.61999145   58.7020562
#> Oct 2005  11.42694878  -15.0403088
#> Nov 2005  11.23838843 -171.0821848
#> Dec 2005  -3.97001493  143.1032558
#> Jan 2006  -8.23261151 -264.5212268
#> Feb 2006  -5.78884625 -103.2457405
#> Mar 2006  17.63789378  -70.7466431
#> Apr 2006   1.32992146  -70.0560014
#> May 2006   3.76499328   53.7885439
#> Jun 2006  11.38707653  -31.6332175
#> Jul 2006  12.41263075  -83.9622754
#> Aug 2006   6.12090945 -135.4928062
#> Sep 2006  -4.29380087  -81.7568272
#> Oct 2006  13.60461327  -63.4400003
#> Nov 2006  -0.70857431   81.5949433
#> Dec 2006  -2.72153684   30.2795706
#> Jan 2007 -24.46680029  180.7086625
#> Feb 2007   0.65486641  -89.4026756
#> Mar 2007 -10.98508902   -4.6428211
#> Apr 2007  -6.33178176  -47.1179138
#> May 2007 -20.63654451  -52.6692630
#> Jun 2007  26.48932029   -9.5134908
#> Jul 2007 -11.53398386 -249.5364809
#> Aug 2007  -3.40637876   16.6889217
#> Sep 2007   7.86362576   35.0492384
#> Oct 2007 -12.70513110  143.3701009
#> Nov 2007   5.42141549   76.5906803
#> Dec 2007   0.75105900  116.7520670
#> Jan 2008   5.58514422  -13.6943429
#> Feb 2008   4.15406399  -51.4902044
#> Mar 2008 -14.52299769  151.9744468
#> Apr 2008   9.41206122  -32.8491678
#> May 2008  -3.38935872   -5.3671506
#> Jun 2008  -0.75574247  -56.3524635
#> Jul 2008   0.40204392  -74.3908963
#> Aug 2008   1.24301066  -10.9041651
#> Sep 2008  -9.98432551  -56.0829227
#> Oct 2008  12.33390065   18.8001549
#> Nov 2008   3.40424488   74.8850942
#> Dec 2008  -4.72702482 -191.6538316
#> Jan 2009   7.08753061   23.6095847
#> Feb 2009 -15.28958715   62.8953415
#> Mar 2009   2.37425345   41.7925676
#> Apr 2009 -13.12814246  197.6758477
#> May 2009   7.47028587  -50.6286298
#> Jun 2009 -15.62518435 -110.9968853
#> Jul 2009   0.71053360  -94.8705723
#> Aug 2009  -6.39534770   47.6843757
#> Sep 2009  -8.45195739  -79.5201560
#> Oct 2009   6.75244698   23.4326923
#> Nov 2009  11.53375794 -122.2451097
#> Dec 2009 -16.86504742 -245.3647354
#> Jan 2010  -9.02814949 -148.9260814
#> Feb 2010  13.17633698  -43.2147734
#> Mar 2010  11.00189745  -94.2554006
#> Apr 2010  12.03767839  -12.1450799
#> May 2010 -14.31270777  133.6446798
#> Jun 2010  13.82910861  -86.0356182
#> Jul 2010   0.03125940   66.6537820
#> Aug 2010  -0.77886824 -142.1534746
#> Sep 2010   4.41428226  117.0056168
#> Oct 2010   1.28922896 -140.4714543
#> Nov 2010  -8.30214260  110.1708096
#> Dec 2010  -5.03592910   69.7986263
#> Jan 2011 -11.93641182  -86.4349803
#> Feb 2011  -7.51723323 -109.1470351
#> Mar 2011  14.55841403   -3.7051465
#> Apr 2011  -8.28603533   81.0053792
#> May 2011   2.89774460  -49.9355412
#> Jun 2011  -4.80053484   94.8031588
#> Jul 2011  -6.04829354  -17.4245957
#> Aug 2011  14.60110180 -110.6235952
#> Sep 2011   1.49679354  -94.5985005
#> Oct 2011 -14.33321100   28.9089591
#> Nov 2011  -0.10303319   87.6913145
#> Dec 2011  -2.12236035 -114.8903940
#> Jan 2012  -9.06340179 -113.7612756
#> Feb 2012 -21.02152479 -143.7246735
#> Mar 2012  18.93360464  -49.4143476
#> Apr 2012  -9.68125837   84.0801808
#> May 2012  -1.02603036   79.1534124
#> Jun 2012   2.39959572  -16.8848948
#> Jul 2012   0.60898893   61.2722104
#> Aug 2012 -21.77576028  -77.1158924
#> Sep 2012  -1.17860143   88.8628993
#> Oct 2012   1.12294787    1.3214477
#> Nov 2012   0.07886198   22.5339515
#> Dec 2012  18.77743872  -72.9915210
#> Jan 2013  21.58756554 -122.2487070
#> Feb 2013   7.09714522   40.6805171
#> Mar 2013   7.66983379  -75.1012223
#> Apr 2013  -3.08211421  -16.2116540
#> May 2013  10.12001849   35.2010126
#> Jun 2013  -9.19051597  -28.9058300
#> Jul 2013   5.63380077   10.4662227
#> Aug 2013   3.22482749   72.0186531
#> Sep 2013   3.66674363  -61.1046082
#> Oct 2013  11.29835153 -110.6914072
#> Nov 2013  -9.41498076   53.4803326
#> Dec 2013   2.17837643   73.6067968
#> Jan 2014  14.15412293 -122.2501574
#> Feb 2014  -3.83733048  102.1415310
#> Mar 2014  -1.74086374   46.5165158
#> Apr 2014  -2.21744517   79.0472705
#> May 2014 -10.09528722  -13.0264801
#> Jun 2014   4.80725266  -93.0285334
#> Jul 2014  16.04407328  -36.4851004
#> Aug 2014 -15.15024529   15.3872493
#> Sep 2014 -14.16023914   41.3154818
#> Oct 2014   8.76777327  248.0823360
#> Nov 2014   6.24132413 -217.9956742
#> Dec 2014  21.12277288   42.0874578
#> Jan 2015  -3.56124416  -35.7528325
#> Feb 2015 -10.64464209  -64.6861514
#> Mar 2015  10.77116538   -5.0141801
#> Apr 2015  11.81575567   41.6942847
#> May 2015   1.98392095  -63.2587542
#> Jun 2015  -4.00405249  115.0146673
#> Jul 2015   6.16154281  -23.5475907
#> Aug 2015  19.74156748 -164.3107386
#> Sep 2015  18.84662324 -150.3382146
#> Oct 2015 -15.88620547 -205.0584847
#> Nov 2015  -5.39923164  -75.3198229
#> Dec 2015 -11.69461464  -13.4141958
#> Jan 2016   5.59105989  100.5782847
#> Feb 2016 -18.19347247  216.7186798
#> Mar 2016   3.93343972  232.2556540
#> Apr 2016   0.42134106 -102.0423391
#> May 2016  11.79664177    4.8814436
#> Jun 2016  -2.56921176  -77.1888628
#> Jul 2016 -10.56336098  -78.5235068
#> Aug 2016   1.98777205  -72.6603031
#> Sep 2016   6.50533552   68.1878032
#> Oct 2016   3.43913337  -22.9843287
#> Nov 2016  14.77532312 -151.0601724
#> Dec 2016   0.72025698  -58.3727687
#> Jan 2017  21.26444534 -202.2918454
#> Feb 2017 -14.76196906   40.3504676
#> Mar 2017   4.07888500   55.0015549
#> Apr 2017  13.93977798    2.8357122
#> May 2017   3.60278296   89.3165020
#> Jun 2017   6.54550251  -37.6555496
#> Jul 2017  10.52155422   60.5884808
#> Aug 2017 -19.79555125   -0.4874726
#> Sep 2017  12.08385605  -52.0796373
#> Oct 2017  -1.69280084  -63.9018598
#> Nov 2017   2.95029753  -63.5894137
#> Dec 2017  12.66340587   10.6586975
#> Jan 2018 -11.35343257  117.6914248
#> Feb 2018 -11.31053798   44.7391153
#> Mar 2018   1.09993384  227.2954766
#> Apr 2018   8.52905410   13.6058206
#> May 2018  -2.34337862 -199.9039133
#> Jun 2018  20.86688567  -42.0500870
#> Jul 2018  -1.10919371  -37.8407395
#> Aug 2018 -13.92847056  122.0774789
#> Sep 2018 -11.42290768 -154.1030292
#> Oct 2018  17.04608737  -31.0310122
#> Nov 2018  -0.80073634   -2.0108184
#> Dec 2018  -4.37281240 -239.0200336
#> Jan 2019  -1.19215094   88.9865359
#> Feb 2019   7.86462865 -148.2813325
#> Mar 2019  -5.78945246   44.5750348
#> Apr 2019  -1.45426885  136.9775856
#> May 2019   5.26457991   -2.0110027
#> Jun 2019  17.33578110  -10.9217587
#> Jul 2019  14.48657220   26.4661745
#> Aug 2019  15.18193149   30.3848264
#> Sep 2019  -3.84007254  -18.3388483
#> Oct 2019  18.27125177   55.9649672
#> Nov 2019  -5.51491750  -18.6553842
#> Dec 2019  -8.65753541  -81.2275372
#> Jan 2020  -3.43831481 -164.0581672
#> Feb 2020  10.62876458   50.7922478
#> Mar 2020   8.13058204  175.4336961
#> Apr 2020  18.03483361   59.2400202
#> May 2020  -1.05068694  101.6713288
#> Jun 2020   9.82453362   12.1620586
#> Jul 2020 -17.13302622 -107.8067265
#> Aug 2020  -8.32019527 -114.3565720
#> Sep 2020  11.00491882  -52.9643677
#> Oct 2020  -1.73820106  -68.1273156
#> Nov 2020   1.78812018  -20.2447559
#> Dec 2020  -6.98429449  168.4495721
#> 
#> $description
#>           type coeff
#> var1 Undefined    NA
#> var2 Undefined    NA
#> 
myreg3 <- regarima(myseries, myspec3)
myreg3
#> y = regression model + arima (2, 1, 1, 0, 1, 1)
#> Log-transformation: no
#> Coefficients:
#>            Estimate Std. Error
#> Phi(1)    -0.002136      0.111
#> Phi(2)     0.173889      0.074
#> Theta(1)  -0.531219      0.106
#> BTheta(1) -0.669839      0.042
#> 
#>                Estimate Std. Error
#> r.var1        8.032e-03      0.009
#> r.var2        7.784e-05      0.001
#> Week days     6.885e-01      0.031
#> Leap year     2.303e+00      0.697
#> Easter [6]   -2.540e+00      0.453
#> TC (4-2020)  -2.122e+01      2.148
#> TC (3-2020)  -2.093e+01      2.149
#> AO (5-2011)   1.338e+01      1.816
#> LS (11-2008) -1.228e+01      1.646
#> AO (5-2000)   6.213e+00      1.820
#> 
#> 
#> Residual standard error: 2.206 on 344 degrees of freedom
#> Log likelihood = -797.2, aic =  1624 aicc =  1626, bic(corrected for length) = 1.812
#> 
# }