Pertemuan 3 - Regresi dengan Peubah Lag

Packages

## Loading required package: nardl

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

## Loading required package: dynlm

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

## 
## Attaching package: 'MLmetrics'

## The following object is masked from 'package:dLagM':
## 
##     MAPE

## The following object is masked from 'package:base':
## 
##     Recall

## Loading required package: carData

Impor Data

data <- read_xlsx("C:/Users/asus/Documents/Semester 5/MPDW/Pertemuan 3/TUGAS PRAKTIKUM/Reksa Dana Astra .xlsx", sheet="Data P3 MPDW")
str(data)

## tibble [60 × 5] (S3: tbl_df/tbl/data.frame)
##  $ Tanggal: POSIXct[1:60], format: "2024-07-01" "2024-07-02" ...
##  $ t      : num [1:60] 1 2 3 4 5 6 7 8 9 10 ...
##  $ Yt     : num [1:60] 1.19e+12 1.20e+12 1.20e+12 1.24e+12 1.24e+12 ...
##  $ Y(t-1) : num [1:60] NA 1.19e+12 1.20e+12 1.20e+12 1.24e+12 ...
##  $ Xt     : num [1:60] 7.99e+08 8.08e+08 8.11e+08 8.37e+08 8.36e+08 ...

View(data)
data

## # A tibble: 60 × 5
##    Tanggal                 t            Yt      `Y(t-1)`         Xt
##    <dttm>              <dbl>         <dbl>         <dbl>      <dbl>
##  1 2024-07-01 00:00:00     1 1185358484195            NA 799414537.
##  2 2024-07-02 00:00:00     2 1197451692797 1185358484195 807556925.
##  3 2024-07-03 00:00:00     3 1202604152834 1197451692797 810847212.
##  4 2024-07-04 00:00:00     4 1241653581623 1202604152834 837024592.
##  5 2024-07-05 00:00:00     5 1240091556976 1241653581623 835870088.
##  6 2024-07-06 00:00:00     6 1248735927481 1240091556976 841681783.
##  7 2024-07-07 00:00:00     7 1255293716629 1248735927481 845947959.
##  8 2024-07-08 00:00:00     8 1269255043183 1255293716629 855213146.
##  9 2024-07-09 00:00:00     9 1260167548745 1269255043183 849049108.
## 10 2024-07-10 00:00:00    10 1216926208674 1260167548745 819852877.
## # ℹ 50 more rows

summary(data)

##     Tanggal                          t               Yt           
##  Min.   :2024-07-01 00:00:00   Min.   : 1.00   Min.   :9.099e+11  
##  1st Qu.:2024-07-15 18:00:00   1st Qu.:15.75   1st Qu.:1.087e+12  
##  Median :2024-07-30 12:00:00   Median :30.50   Median :1.237e+12  
##  Mean   :2024-07-30 12:00:00   Mean   :30.50   Mean   :1.198e+12  
##  3rd Qu.:2024-08-14 06:00:00   3rd Qu.:45.25   3rd Qu.:1.294e+12  
##  Max.   :2024-08-29 00:00:00   Max.   :60.00   Max.   :1.443e+12  
##                                                                   
##      Y(t-1)                Xt           
##  Min.   :9.099e+11   Min.   :611405760  
##  1st Qu.:1.086e+12   1st Qu.:730323070  
##  Median :1.238e+12   Median :832069921  
##  Mean   :1.198e+12   Mean   :805086713  
##  3rd Qu.:1.295e+12   3rd Qu.:871548676  
##  Max.   :1.443e+12   Max.   :968190397  
##  NA's   :1

Pembagian Data

#SPLIT DATA
train<-data[1:48,]
test<-data[49:60,]

#data time series
train.ts<-ts(train)
test.ts<-ts(test)
data.ts<-ts(data)

Model Koyck

Model Koyck didasarkan pada asumsi bahwa semakin jauh jarak lag peubah independen dari periode sekarang maka semakin kecil pengaruh peubah lag terhadap peubah dependen.

Koyck mengusulkan suatu metode untuk menduga model dinamis distributed lag dengan mengasumsikan bahwa semua koefisien \(\beta\) mempunyai tanda sama.

Model kyock merupakan jenis paling umum dari model infinite distributed lag dan juga dikenal sebagai geometric lag

\[ y_t=a(1-\lambda)+\beta_0X_t+\beta_1Z_t+\lambda Y_{t-1}+V_t \]

dengan \[V_t=u_t-\lambda u_{t-1}\]

Pemodelan

Pemodelan model Koyck dengan R dapat menggunakan dLagM::koyckDlm() . Fungsi umum dari koyckDlm adalah sebagai berikut.

koyckDlm(x , y , intercept)

Fungsi koyckDlm() akan menerapkan model lag terdistribusi dengan transformasi Koyck satu prediktor. Nilai x dan y tidak perlu sebagai objek time series (ts). intercept dapat dibuat TRUE untuk memasukkan intersep ke dalam model.

#MODEL KOYCK
model.koyck <- koyckDlm(x = train$Xt, y = train$Yt)
summary(model.koyck)

## 
## Call:
## "Y ~ (Intercept) + Y.1 + X.t"
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -1.293e+11 -8.772e+09  6.552e+08  1.139e+10  1.462e+11 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.793e+10  1.988e+11  -0.241    0.811
## Y.1         -5.321e-01  1.859e+00  -0.286    0.776
## X.t          2.335e+03  2.996e+03   0.779    0.440
## 
## Residual standard error: 4.069e+10 on 44 degrees of freedom
## Multiple R-Squared: 0.9272,  Adjusted R-squared: 0.9239 
## Wald test: 234.5 on 2 and 44 DF,  p-value: < 2.2e-16 
## 
## Diagnostic tests:
## NULL
## 
##                                 alpha     beta        phi
## Geometric coefficients:  -31282909302 2335.274 -0.5320623

AIC(model.koyck)

## [1] 2434.636

BIC(model.koyck)

## [1] 2442.036

Dari hasil tersebut, didapat bahwa peubah \(x_t\) dan \(y_{t-1}\) memiliki nilai \(P-Value<0.05\). Hal ini menunjukkan bahwa peubah \(x_t\) dan \(y_{t-1}\) berpengaruh signifikan terhadap \(y\). Adapun model keseluruhannya adalah sebagai berikut

\[ \hat{Y_t}=(-47.93×10^9)+2335X_t+(-0.5321)Y_{t-1} \]

Peramalan dan Akurasi

Berikut adalah hasil peramalan y untuk 7 periode kedepan menggunakan model koyck

fore.koyck <- forecast(model = model.koyck, x=test$Xt, h=12)
fore.koyck

## $forecasts
##  [1] 1.419131e+12 1.431643e+12 1.353324e+12 1.410529e+12 1.336465e+12
##  [6] 1.349697e+12 1.022896e+12 1.537421e+12 1.263759e+12 1.343138e+12
## [11] 1.138221e+12 1.210759e+12
## 
## $call
## forecast.koyckDlm(model = model.koyck, x = test$Xt, h = 12)
## 
## attr(,"class")
## [1] "forecast.koyckDlm" "dLagM"

mape.koyck <- MAPE(fore.koyck$forecasts, test$Yt)
mape.koyck

## [1] 0.03904357

#akurasi data training
GoF(model.koyck)

##              n         MAE         MPE       MAPE      sMAPE      MASE
## model.koyck 47 22921788551 0.002111983 0.02055211 0.02077043 0.5855693
##                      MSE     MRAE     GMRAE
## model.koyck 1.550198e+21 2.238182 0.8224979

length(fore.koyck$forecasts)

## [1] 12

length(test$Yt)

## [1] 12

Regression with Distributed Lag

Pemodelan model Regression with Distributed Lag dengan R dapat menggunakan dLagM::dlm() . Fungsi umum dari dlm adalah sebagai berikut.

dlm(formula , data , x , y , q , remove )

Fungsi dlm() akan menerapkan model lag terdistribusi dengan satu atau lebih prediktor. Nilai x dan y tidak perlu sebagai objek time series (ts). \(q\) adalah integer yang mewakili panjang lag yang terbatas.

Pemodelan (Lag=2)

model.dlm <- dlm(x = train$Xt,y = train$Yt , q = 2)
summary(model.dlm)

## 
## Call:
## lm(formula = model.formula, data = design)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -2.782e+09 -7.190e+08 -5.682e+07  7.527e+08  5.285e+09 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  8.146e+09  2.045e+09   3.984 0.000264 ***
## x.t          1.490e+03  4.939e+00 301.669  < 2e-16 ***
## x.1         -2.599e-01  6.777e+00  -0.038 0.969597    
## x.2         -1.350e+01  5.165e+00  -2.614 0.012361 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.576e+09 on 42 degrees of freedom
## Multiple R-squared:  0.9999, Adjusted R-squared:  0.9999 
## F-statistic: 1.342e+05 on 3 and 42 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 2084.746 2093.889

AIC(model.dlm)

## [1] 2084.746

BIC(model.dlm)

## [1] 2093.889

Dari hasil diatas, didapat bahwa \(P-value\) dari intercept dan \(x_{t-1}<0.05\). Hal ini menunjukkan bahwa intercept dan \(x_{t-1}\) berpengaruh signifikan terhadap \(y\). Adapun model keseluruhan yang terbentuk adalah sebagai berikut

\[ \hat{Y_t}=-9.6779+0.3179X_t+1.5276X_{t-1}+0.2651X_{t-2} \]

Peramalan dan Akurasi

Berikut merupakan hasil peramalan \(y\) untuk 12 hari ke depan

fore.dlm <- forecast(model = model.dlm, x=test$Xt, h=12)
fore.dlm

## $forecasts
##  [1] 1.420804e+12 1.420709e+12 1.375135e+12 1.385056e+12 1.357630e+12
##  [6] 1.340844e+12 1.137065e+12 1.354613e+12 1.356486e+12 1.312258e+12
## [11] 1.208460e+12 1.185577e+12
## 
## $call
## forecast.dlm(model = model.dlm, x = test$Xt, h = 12)
## 
## attr(,"class")
## [1] "forecast.dlm" "dLagM"

mape.dlm <- MAPE(fore.dlm$forecasts, test$Yt)
#akurasi data training
GoF(model.dlm)

##            n        MAE          MPE         MAPE        sMAPE       MASE
## model.dlm 46 1108624931 6.597528e-06 0.0009241822 0.0009242661 0.02778521
##                    MSE      MRAE     GMRAE
## model.dlm 2.267687e+18 0.3191474 0.0479808

Lag Optimum

#penentuan lag optimum 
finiteDLMauto(formula = Yt ~ Xt,
              data = data.frame(train), q.min = 1, q.max = 8,
              model.type = "dlm", error.type = "AIC", trace = FALSE)

##   q - k    MASE      AIC      BIC   GMRAE   MBRAE R.Adj.Sq    Ljung-Box
## 8     8 0.01645 1786.194 1804.772 0.03295 0.05191  0.99995 8.558516e-08

Berdasarkan output tersebut, lag optimum didapatkan ketika lag=8. Selanjutnya dilakukan pemodelan untuk lag=8

#model dlm dengan lag optimum
model.dlm2 <- dlm(x = train$Xt,y = train$Yt , q = 8)
summary(model.dlm2)

## 
## Call:
## lm(formula = model.formula, data = design)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -1.882e+09 -4.730e+08  4.384e+07  5.391e+08  1.970e+09 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.209e+10  1.752e+09   6.898 1.17e-07 ***
## x.t          1.489e+03  3.498e+00 425.499  < 2e-16 ***
## x.1         -2.329e-03  4.693e+00   0.000    1.000    
## x.2         -6.407e-01  4.702e+00  -0.136    0.893    
## x.3         -5.488e+00  5.154e+00  -1.065    0.295    
## x.4         -2.105e+00  5.297e+00  -0.398    0.694    
## x.5         -1.168e+00  5.297e+00  -0.220    0.827    
## x.6          7.197e-01  5.227e+00   0.138    0.891    
## x.7         -4.083e+00  5.459e+00  -0.748    0.460    
## x.8         -4.111e+00  4.382e+00  -0.938    0.356    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.056e+09 on 30 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 9.525e+04 on 9 and 30 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##        AIC      BIC
## 1 1786.194 1804.772

AIC(model.dlm2)

## [1] 1786.194

BIC(model.dlm2)

## [1] 1804.772

Dari hasil tersebut terdapat beberapa peubah yang berpengaruh signifikan terhadap taraf nyata 5% yaitu \(x_t\) , \(x_{t-2}\) , \(x_{t-4}\) , \(x_{t-6}\). Adapun keseluruhan model yang terbentuk adalah

\[ \hat{Y_t}=11.36×10^9+1489X_t+...+(-4.545)X_{t-6} \]

Adapun hasil peramalan 12 periode kedepan menggunakan model tersebut adalah sebagai berikut

#peramalan dan akurasi
fore.dlm2 <- forecast(model = model.dlm2, x=test$Xt, h=12)
mape.dlm2<- MAPE(fore.dlm2$forecasts, test$Yt)
#akurasi data training
GoF(model.dlm2)

##             n       MAE          MPE         MAPE        sMAPE       MASE
## model.dlm2 40 724213641 3.741338e-06 0.0006139175 0.0006139164 0.01645336
##                     MSE     MRAE   GMRAE
## model.dlm2 8.356356e+17 0.168003 0.03295

Model tersebut merupakan model yang sangat baik dengan nilai MAPE yang kurang dari 10%.

Model Autoregressive

Peubah dependen dipengaruhi oleh peubah independen pada waktu sekarang, serta dipengaruhi juga oleh peubah dependen itu sendiri pada satu waktu yang lalu maka model tersebut disebut autoregressive (Gujarati 2004).

Pemodelan

Pemodelan Autoregressive dilakukan menggunakan fungsi dLagM::ardlDlm() . Fungsi tersebut akan menerapkan autoregressive berordo \((p,q)\) dengan satu prediktor. Fungsi umum dari ardlDlm() adalah sebagai berikut.

ardlDlm(formula = NULL , data = NULL , x = NULL , y = NULL , p = 1 , q = 1 , remove = NULL )

Dengan \(p\) adalah integer yang mewakili panjang lag yang terbatas dan \(q\) adalah integer yang merepresentasikan ordo dari proses autoregressive.

model.ardl <- ardlDlm(x = train$Xt, y = train$Yt, p = 1 , q = 1)
summary(model.ardl)

## 
## Time series regression with "ts" data:
## Start = 2, End = 48
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data, start = 1)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -427216010  -53657039     944397   64276418  334261035 
## 
## Coefficients:
##               Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) -8.520e+07  1.659e+08   -0.513     0.61    
## X.t          1.488e+03  3.842e-01 3873.073   <2e-16 ***
## X.1         -1.500e+03  1.596e+01  -93.954   <2e-16 ***
## Y.1          1.008e+00  1.081e-02   93.244   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 122400000 on 43 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 2.229e+07 on 3 and 43 DF,  p-value: < 2.2e-16

AIC(model.ardl)

## [1] 1889.728

BIC(model.ardl)

## [1] 1898.979

model.ardl <- ardlDlm(formula = Yt ~ Xt, 
                         data = train,p = 1 , q = 1)
summary(model.ardl)

## 
## Time series regression with "ts" data:
## Start = 2, End = 48
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -427216010  -53657039     944397   64276418  334261035 
## 
## Coefficients:
##               Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) -8.520e+07  1.659e+08   -0.513     0.61    
## Xt.t         1.488e+03  3.842e-01 3873.073   <2e-16 ***
## Xt.1        -1.500e+03  1.596e+01  -93.954   <2e-16 ***
## Yt.1         1.008e+00  1.081e-02   93.244   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 122400000 on 43 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 2.229e+07 on 3 and 43 DF,  p-value: < 2.2e-16

AIC(model.ardl)

## [1] 1889.728

BIC(model.ardl)

## [1] 1898.979

Hasil di atas menunjukkan bahwa selain peubah \(x_{t-1}\), hasil uji t menunjukkan nilai-p pada peubah \(\ge0.05\) Hal ini menunjukkan bahwa peubah \(x_{t-1}\) berpengaruh signifikan terhadap \(y_t\), sementara \(x_t\) dan \(y_{t-1}\) berpengaruh signifikan terhadap \(y_t\). Model keseluruhannya adalah sebagai berikut:

\[ \hat{Y}=-8,352×10^7+1488X_t+(-1500)X_{t-1}+1.008Y_{t-1} \]

Peramalan dan Akurasi

fore.ardl <- forecast(model = model.ardl, x=test$Xt, h=12)
fore.ardl

## $forecasts
##  [1] 1.426432e+12 1.426559e+12 1.381119e+12 1.391229e+12 1.363646e+12
##  [6] 1.347181e+12 1.143654e+12 1.360871e+12 1.361149e+12 1.319170e+12
## [11] 1.215727e+12 1.192669e+12
## 
## $call
## forecast.ardlDlm(model = model.ardl, x = test$Xt, h = 12)
## 
## attr(,"class")
## [1] "forecast.ardlDlm" "dLagM"

Data di atas merupakan hasil peramalan untuk 5 periode ke depan menggunakan Model Autoregressive dengan \(p=1\) dan \(q=1\).

mape.ardl <- MAPE(fore.ardl$forecasts, test$Yt)
mape.ardl

## [1] 0.000472012

#akurasi data training
GoF(model.ardl)

##             n      MAE           MPE         MAPE       sMAPE       MASE
## model.ardl 47 83846637 -2.706864e-07 7.015278e-05 7.01524e-05 0.00214198
##                    MSE       MRAE       GMRAE
## model.ardl 1.37023e+16 0.01260164 0.003963212

Berdasarkan akurasi di atas, terlihat bahwa nilai MAPE keduanya tidak jauh berbeda. Artinya, model regresi dengan distribusi lag ini tidak overfitted atau underfitted

Lag Optimum

#penentuan lag optimum
model.ardl.opt <- ardlBoundOrders(data = data.frame(data), ic = "AIC", 
                                  formula = Yt ~ Xt )
min_p=c()
for(i in 1:6){
  min_p[i]=min(model.ardl.opt$Stat.table[[i]])
}
q_opt=which(min_p==min(min_p, na.rm = TRUE))
p_opt=which(model.ardl.opt$Stat.table[[q_opt]] == 
              min(model.ardl.opt$Stat.table[[q_opt]], na.rm = TRUE))
data.frame("q_optimum" = q_opt, "p_optimum" = p_opt, 
           "AIC"=model.ardl.opt$min.Stat)

##   q_optimum p_optimum      AIC
## 1         1        15 1772.889

Dari tabel di atas, dapat terlihat bahwa nilai AIC terendah didapat ketika \(p=6\) dan \(q=1\), yaitu sebesar -20,56587. Artinya, model autoregressive optimum didapat ketika \(p=6\) dan \(q=1\).

Selanjutnya dapat dilakukan pemodelan dengan nilai \(p\) dan \(q\) optimum seperti inisialisasi di langkah sebelumnya.

Pemodelan DLM & ARDL dengan Library `dynlm`

Pemodelan regresi dengan peubah lag tidak hanya dapat dilakukan dengan fungsi pada packages dLagM , tetapi terdapat packages dynlm yang dapat digunakan. Fungsi dynlm secara umum adalah sebagai berikut.

dynlm(formula, data, subset, weights, na.action, method = "qr",
  model = TRUE, x = FALSE, y = FALSE, qr = TRUE, singular.ok = TRUE,
  contrasts = NULL, offset, start = NULL, end = NULL, ...)

Untuk menentukan formula model yang akan digunakan, tersedia fungsi tambahan yang memungkinkan spesifikasi dinamika (melalui d() dan L()) atau pola linier/siklus dengan mudah (melalui trend(), season(), dan harmon()). Semua fungsi formula baru mengharuskan argumennya berupa objek deret waktu (yaitu, "ts" atau "zoo").

#sama dengan model dlm q=1
cons_lm1 <- dynlm(Yt ~ Xt+L(Xt),data = train.ts)
#sama dengan model ardl p=1 q=0
cons_lm2 <- dynlm(Yt ~ Xt+L(Yt),data = train.ts)
#sama dengan ardl p=1 q=1
cons_lm3 <- dynlm(Yt ~ Xt+L(Xt)+L(Yt),data = train.ts)
#sama dengan dlm p=2
cons_lm4 <- dynlm(Yt ~ Xt+L(Xt)+L(Xt,2),data = train.ts)

Ringkasan Model

summary(cons_lm1)

## 
## Time series regression with "ts" data:
## Start = 2, End = 48
## 
## Call:
## dynlm(formula = Yt ~ Xt + L(Xt), data = train.ts)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -3.059e+09 -8.676e+08 -1.184e+08  7.920e+08  5.004e+09 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  6.480e+09  2.117e+09    3.06  0.00376 ** 
## Xt           1.490e+03  5.402e+00  275.85  < 2e-16 ***
## L(Xt)       -1.180e+01  5.618e+00   -2.10  0.04152 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.725e+09 on 44 degrees of freedom
## Multiple R-squared:  0.9999, Adjusted R-squared:  0.9999 
## F-statistic: 1.683e+05 on 2 and 44 DF,  p-value: < 2.2e-16

summary(cons_lm2)

## 
## Time series regression with "ts" data:
## Start = 2, End = 48
## 
## Call:
## dynlm(formula = Yt ~ Xt + L(Yt), data = train.ts)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -3.079e+09 -8.939e+08 -1.348e+08  8.224e+08  5.011e+09 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  6.420e+09  2.141e+09   2.998  0.00445 ** 
## Xt           1.489e+03  5.450e+00 273.309  < 2e-16 ***
## L(Yt)       -7.359e-03  3.834e-03  -1.919  0.06145 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.738e+09 on 44 degrees of freedom
## Multiple R-squared:  0.9999, Adjusted R-squared:  0.9999 
## F-statistic: 1.658e+05 on 2 and 44 DF,  p-value: < 2.2e-16

summary(cons_lm3)

## 
## Time series regression with "ts" data:
## Start = 2, End = 48
## 
## Call:
## dynlm(formula = Yt ~ Xt + L(Xt) + L(Yt), data = train.ts)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -427216010  -53657039     944397   64276418  334261035 
## 
## Coefficients:
##               Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) -8.520e+07  1.659e+08   -0.513     0.61    
## Xt           1.488e+03  3.842e-01 3873.073   <2e-16 ***
## L(Xt)       -1.500e+03  1.596e+01  -93.954   <2e-16 ***
## L(Yt)        1.008e+00  1.081e-02   93.244   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 122400000 on 43 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 2.229e+07 on 3 and 43 DF,  p-value: < 2.2e-16

summary(cons_lm4)

## 
## Time series regression with "ts" data:
## Start = 3, End = 48
## 
## Call:
## dynlm(formula = Yt ~ Xt + L(Xt) + L(Xt, 2), data = train.ts)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -2.782e+09 -7.190e+08 -5.682e+07  7.527e+08  5.285e+09 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  8.146e+09  2.045e+09   3.984 0.000264 ***
## Xt           1.490e+03  4.939e+00 301.669  < 2e-16 ***
## L(Xt)       -2.599e-01  6.777e+00  -0.038 0.969597    
## L(Xt, 2)    -1.350e+01  5.165e+00  -2.614 0.012361 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.576e+09 on 42 degrees of freedom
## Multiple R-squared:  0.9999, Adjusted R-squared:  0.9999 
## F-statistic: 1.342e+05 on 3 and 42 DF,  p-value: < 2.2e-16

SSE

deviance(cons_lm1)

## [1] 1.3086e+20

deviance(cons_lm2)

## [1] 1.328506e+20

deviance(cons_lm3)

## [1] 6.440083e+17

deviance(cons_lm4)

## [1] 1.043136e+20

Autokorelasi

#durbin watson
dwtest(cons_lm1)

## 
##  Durbin-Watson test
## 
## data:  cons_lm1
## DW = 0.13447, p-value < 2.2e-16
## alternative hypothesis: true autocorrelation is greater than 0

dwtest(cons_lm2)

## 
##  Durbin-Watson test
## 
## data:  cons_lm2
## DW = 0.11426, p-value < 2.2e-16
## alternative hypothesis: true autocorrelation is greater than 0

dwtest(cons_lm3)

## 
##  Durbin-Watson test
## 
## data:  cons_lm3
## DW = 2.37, p-value = 0.8386
## alternative hypothesis: true autocorrelation is greater than 0

dwtest(cons_lm4)

## 
##  Durbin-Watson test
## 
## data:  cons_lm4
## DW = 0.20901, p-value < 2.2e-16
## alternative hypothesis: true autocorrelation is greater than 0

Heterogenitas

bptest(cons_lm1)

## 
##  studentized Breusch-Pagan test
## 
## data:  cons_lm1
## BP = 12.236, df = 2, p-value = 0.002203

bptest(cons_lm2)

## 
##  studentized Breusch-Pagan test
## 
## data:  cons_lm2
## BP = 12.561, df = 2, p-value = 0.001872

bptest(cons_lm3)

## 
##  studentized Breusch-Pagan test
## 
## data:  cons_lm3
## BP = 12.034, df = 3, p-value = 0.007266

bptest(cons_lm4)

## 
##  studentized Breusch-Pagan test
## 
## data:  cons_lm4
## BP = 8.6535, df = 3, p-value = 0.03427

Kenormalan

shapiro.test(residuals(cons_lm1))

## 
##  Shapiro-Wilk normality test
## 
## data:  residuals(cons_lm1)
## W = 0.94555, p-value = 0.02906

shapiro.test(residuals(cons_lm2))

## 
##  Shapiro-Wilk normality test
## 
## data:  residuals(cons_lm2)
## W = 0.94568, p-value = 0.02939

shapiro.test(residuals(cons_lm3))

## 
##  Shapiro-Wilk normality test
## 
## data:  residuals(cons_lm3)
## W = 0.94352, p-value = 0.02427

shapiro.test(residuals(cons_lm4))

## 
##  Shapiro-Wilk normality test
## 
## data:  residuals(cons_lm4)
## W = 0.95509, p-value = 0.07383

Perbandingan Model

akurasi <- matrix(c(mape.koyck, mape.dlm, mape.dlm2, mape.ardl))
row.names(akurasi)<- c("Koyck","DLM 1","DLM 2","Autoregressive")
colnames(akurasi) <- c("MAPE")
akurasi

##                       MAPE
## Koyck          0.039043569
## DLM 1          0.004270489
## DLM 2          0.004273572
## Autoregressive 0.000472012

Berdasarkan nilai MAPE, model paling optimum didapat pada Model Koyck karena memiliki nilai MAPE yang terkecil.

Plot

par(mfrow=c(1,1))
plot(test$Xt, test$Yt, type="b", col="black", ylim=c(9.099e+11,1.6e+12))
points(test$Xt, fore.koyck$forecasts,col="red")
lines(test$Xt, fore.koyck$forecasts,col="red")
points(test$Xt, fore.dlm$forecasts,col="blue")
lines(test$Xt, fore.dlm$forecasts,col="blue")
points(test$Xt, fore.dlm2$forecasts,col="orange")
lines(test$Xt, fore.dlm2$forecasts,col="orange")
points(test$Xt, fore.ardl$forecasts,col="green")
lines(test$Xt, fore.ardl$forecasts,col="green")
legend("topleft",c("aktual", "koyck","DLM 1","DLM 2", "autoregressive"), lty=1, col=c("black","red","blue","orange","green"), cex=0.8)

Berdasarkan plot tersebut, terlihat bahwa plot yang paling mendekati data aktualnya adalah Model koyck, sehingga dapat disimpulkan model terbaik dalam hal ini adalah model regresi koyck

Pengayaan (Regresi Berganda)

Data

data(M1Germany)
data1 = M1Germany[1:144,]

DLM

#Run the search over finite DLMs according to AIC values
finiteDLMauto(formula = logprice ~ interest+logm1,
              data = data.frame(data1), q.min = 1, q.max = 5,
              model.type = "dlm", error.type = "AIC", trace = FALSE)

##   q - k    MASE       AIC       BIC   GMRAE    MBRAE R.Adj.Sq Ljung-Box
## 5     5 1.77163 -463.1393 -422.0566 1.43662 -1.60494  0.98836         0

#model dlm berganda
model.dlmberganda = dlm(formula = logprice ~ interest + logm1,
                data = data.frame(data1) , q = 5)
summary(model.dlmberganda)

## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.095761 -0.028610 -0.000012  0.029496  0.102597 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -7.81759    0.11384 -68.669  < 2e-16 ***
## interest.t  -1.75616    0.80358  -2.185 0.030707 *  
## interest.1   1.38935    1.22707   1.132 0.259679    
## interest.2   0.40776    1.23726   0.330 0.742273    
## interest.3   1.23130    1.20752   1.020 0.309830    
## interest.4  -0.08718    1.20869  -0.072 0.942616    
## interest.5   3.06850    0.89380   3.433 0.000808 ***
## logm1.t      0.43219    0.20876   2.070 0.040474 *  
## logm1.1      0.42190    0.19807   2.130 0.035109 *  
## logm1.2      0.20943    0.12883   1.626 0.106532    
## logm1.3      0.22053    0.13011   1.695 0.092567 .  
## logm1.4      0.05513    0.21457   0.257 0.797633    
## logm1.5      0.03042    0.19192   0.159 0.874296    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.04343 on 126 degrees of freedom
## Multiple R-squared:  0.9894, Adjusted R-squared:  0.9884 
## F-statistic: 977.9 on 12 and 126 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##         AIC       BIC
## 1 -463.1393 -422.0566

model.dlmberganda2 = dlm(formula = logprice ~ interest + logm1,
                        data = data.frame(data1) , q = 1)
summary(model.dlmberganda2)

## 
## Call:
## lm(formula = as.formula(model.formula), data = design)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.134002 -0.044697  0.006407  0.036962  0.113063 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -7.77917    0.13299 -58.492  < 2e-16 ***
## interest.t  -3.22103    0.94184  -3.420 0.000824 ***
## interest.1   6.52775    0.94501   6.908 1.66e-10 ***
## logm1.t      0.73918    0.08419   8.780 5.61e-15 ***
## logm1.1      0.63330    0.08429   7.513 6.55e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.05443 on 138 degrees of freedom
## Multiple R-squared:  0.9832, Adjusted R-squared:  0.9828 
## F-statistic:  2025 on 4 and 138 DF,  p-value: < 2.2e-16
## 
## AIC and BIC values for the model:
##         AIC       BIC
## 1 -419.7575 -401.9805

ARDL

#Mencari orde lag optimum model ARDL
ardlBoundOrders(data = data1 , formula = logprice ~ interest + logm1,
                ic="AIC")

## $p
##    interest logm1
## 65        0     4
## 
## $q
## [1] 4
## 
## $Stat.table
##            q = 1     q = 2     q = 3     q = 4     q = 5     q = 6     q = 7
## p = 1  -760.1786 -757.9195 -846.8342 -975.2079 -965.7536 -958.9072 -956.7315
## p = 2  -760.0433 -759.3090 -843.6247 -971.2514 -961.7929 -955.2809 -953.4890
## p = 3  -753.7746 -753.7746 -841.2485 -970.4543 -961.4343 -953.7173 -950.0412
## p = 4  -829.8076 -832.6436 -832.6436 -971.0837 -962.1804 -955.0429 -953.4667
## p = 5  -749.4144 -753.2292 -962.9290 -962.9290 -961.7063 -954.3406 -951.7660
## p = 6  -742.2103 -742.9945 -891.6195 -952.3771 -952.3771 -952.2461 -950.1105
## p = 7  -728.9374 -733.0286 -851.2943 -945.7445 -944.6879 -944.6879 -949.3720
## p = 8  -747.9277 -746.2948 -812.4289 -937.9446 -938.9491 -937.3393 -937.3393
## p = 9  -722.6891 -724.5786 -863.2734 -928.9215 -927.2914 -926.8716 -936.6432
## p = 10 -714.8175 -714.5658 -816.3319 -918.5218 -918.6350 -916.9076 -921.1246
## p = 11 -703.1807 -705.3383 -794.0772 -909.6457 -908.8225 -906.9542 -912.9605
## p = 12 -716.7111 -714.7403 -774.0127 -910.0315 -910.6834 -908.7146 -909.6612
## p = 13 -697.7175 -698.1931 -793.4602 -895.5927 -894.9273 -893.5995 -897.7589
## p = 14 -686.5600 -685.7967 -766.5292 -886.0709 -885.4341 -885.2283 -890.1638
## p = 15 -676.7280 -678.3689 -753.2854 -875.6392 -874.1257 -874.3117 -879.2727
##            q = 8     q = 9    q = 10    q = 11    q = 12    q = 13    q = 14
## p = 1  -954.3375 -946.6293 -936.5328 -927.7728 -920.6435 -917.5463 -918.3110
## p = 2  -951.1470 -943.9360 -933.7047 -924.7949 -917.5334 -913.6213 -914.4063
## p = 3  -948.4683 -941.1039 -930.8509 -922.0563 -914.5728 -910.5351 -913.4996
## p = 4  -948.2330 -941.8238 -931.5689 -923.2663 -916.2063 -911.6023 -913.9345
## p = 5  -947.5994 -939.3767 -929.0155 -920.4475 -913.5968 -909.0781 -911.6312
## p = 6  -945.5758 -937.4076 -927.2439 -919.3949 -911.9537 -907.7394 -910.2890
## p = 7  -945.5181 -937.1826 -926.9640 -917.9619 -910.2774 -905.9449 -907.8712
## p = 8  -941.9617 -933.5959 -923.3691 -914.6251 -907.0608 -902.2187 -903.9255
## p = 9  -936.6432 -935.7172 -925.2881 -917.0877 -911.6973 -903.9027 -904.6405
## p = 10 -926.6891 -926.6891 -924.6986 -917.0904 -911.4197 -903.4313 -903.0612
## p = 11 -917.9145 -918.2328 -918.2328 -919.2867 -913.3674 -904.8733 -903.6541
## p = 12 -916.1321 -914.4362 -914.4610 -914.4610 -912.5159 -904.2394 -901.6216
## p = 13 -905.4744 -903.7559 -902.4406 -902.2530 -902.2530 -902.9434 -901.2363
## p = 14 -896.2370 -896.2620 -894.2896 -897.5711 -899.1407 -899.1407 -902.2350
## p = 15 -884.5637 -886.8221 -884.9832 -890.5665 -893.2335 -891.6220 -891.6220
##           q = 15
## p = 1  -908.0863
## p = 2  -904.1665
## p = 3  -903.3006
## p = 4  -903.9256
## p = 5  -901.6220
## p = 6  -900.1824
## p = 7  -897.9867
## p = 8  -894.1031
## p = 9  -894.7387
## p = 10 -893.6199
## p = 11 -893.6060
## p = 12 -892.4805
## p = 13 -892.5115
## p = 14 -893.6214
## p = 15 -891.3741
## 
## $min.Stat
## [1] -977.2745
## 
## $Stat.p
##     interest logm1      Stat
## 65         0     4 -977.2745
## 1          0     0 -976.5191
## 2          1     0 -976.2558
## 17         0     1 -975.9606
## 66         1     4 -975.6027
## 18         1     1 -975.2079
## 49         0     3 -974.4859
## 3          2     0 -974.4275
## 33         0     2 -974.0166
## 50         1     3 -973.7500
## 67         2     4 -973.6028
## 34         1     2 -973.2324
## 19         2     1 -973.2188
## 68         3     4 -972.5992
## 4          3     0 -972.4875
## 51         2     3 -971.7743
## 20         3     1 -971.3872
## 35         2     2 -971.2514
## 69         4     4 -971.0837
## 5          4     0 -970.5114
## 52         3     3 -970.4543
## 81         0     5 -969.9284
## 53         4     3 -969.5311
## 21         4     1 -969.4756
## 36         3     2 -969.3907
## 82         1     5 -968.6783
## 37         4     2 -967.4756
## 83         2     5 -966.8835
## 84         3     5 -965.6393
## 85         4     5 -963.9662
## 86         5     5 -962.9290
## 70         5     4 -961.2547
## 54         5     3 -960.9580
## 97         0     6 -960.7402
## 6          5     0 -960.6858
## 22         5     1 -959.8419
## 98         1     6 -959.6604
## 38         5     2 -957.8547
## 99         2     6 -957.7528
## 100        3     6 -956.7875
## 101        4     6 -955.2416
## 71         6     4 -954.8953
## 87         6     5 -954.6855
## 102        5     6 -954.3662
## 103        6     6 -954.0973
## 7          6     0 -954.0615
## 113        0     7 -953.9160
## 55         6     3 -953.2860
## 23         6     1 -953.1080
## 114        1     7 -952.6540
## 39         6     2 -951.1356
## 115        2     7 -950.6562
## 116        3     7 -949.6038
## 88         7     5 -949.2090
## 72         7     4 -948.5194
## 117        4     7 -947.7999
## 104        7     6 -947.7424
## 56         7     3 -947.6915
## 8          7     0 -947.5092
## 120        7     7 -947.3660
## 24         7     1 -947.0094
## 118        5     7 -946.9631
## 119        6     7 -946.8080
## 40         7     2 -945.0123
## 129        0     8 -943.9035
## 130        1     8 -942.6627
## 131        2     8 -940.6818
## 145        0     9 -940.0114
## 132        3     8 -939.6913
## 89         8     5 -939.1878
## 73         8     4 -938.5330
## 146        1     9 -938.2680
## 133        4     8 -937.8368
## 105        8     6 -937.6834
## 57         8     3 -937.6370
## 9          8     0 -937.5705
## 121        8     7 -937.5351
## 136        7     8 -937.3948
## 25         8     1 -937.0088
## 134        5     8 -936.9393
## 135        6     8 -936.8904
## 147        2     9 -936.3875
## 148        3     9 -936.3159
## 137        8     8 -935.5389
## 41         8     2 -935.0088
## 149        4     9 -934.3458
## 150        5     9 -934.1858
## 152        7     9 -934.0733
## 151        6     9 -932.9538
## 153        8     9 -932.3338
## 154        9     9 -930.9065
## 161        0    10 -929.8056
## 90         9     5 -929.2731
## 74         9     4 -928.5254
## 162        1    10 -928.1257
## 10         9     0 -927.9853
## 58         9     3 -927.9744
## 122        9     7 -927.9061
## 106        9     6 -927.6344
## 26         9     1 -927.4482
## 164        3    10 -926.5271
## 163        2    10 -926.2965
## 138        9     8 -926.1307
## 42         9     2 -925.4484
## 165        4    10 -924.5287
## 168        7    10 -924.2716
## 166        5    10 -924.0521
## 167        6    10 -922.7596
## 169        8    10 -922.5928
## 155       10     9 -921.2169
## 170        9    10 -921.1777
## 177        0    11 -920.2608
## 171       10    10 -920.0124
## 91        10     5 -919.0182
## 178        1    11 -918.7342
## 75        10     4 -918.4135
## 11        10     0 -917.8597
## 59        10     3 -917.7711
## 123       10     7 -917.6569
## 107       10     6 -917.3861
## 27        10     1 -917.2925
## 179        2    11 -916.9417
## 180        3    11 -916.8682
## 193        0    12 -916.1477
## 139       10     8 -915.9643
## 92        11     5 -915.3201
## 43        10     2 -915.2941
## 156       11     9 -915.0851
## 181        4    11 -914.8854
## 194        1    12 -914.4423
## 124       11     7 -914.3141
## 184        7    11 -914.1880
## 76        11     4 -914.1395
## 182        5    11 -914.0440
## 108       11     6 -913.4052
## 140       11     8 -913.3026
## 195        2    12 -913.1680
## 172       11    10 -913.0914
## 60        11     3 -912.7714
## 183        6    11 -912.7548
## 196        3    12 -912.5820
## 185        8    11 -912.5636
## 12        11     0 -912.2009
## 28        11     1 -912.0389
## 186        9    11 -911.1737
## 157       12     9 -911.1513
## 188       11    11 -911.1189
## 93        12     5 -910.7693
## 198        5    12 -910.7434
## 197        4    12 -910.6154
## 125       12     7 -910.5873
## 141       12     8 -910.0719
## 44        11     2 -910.0439
## 187       10    11 -909.9928
## 200        7    12 -909.4197
## 173       12    10 -909.2473
## 77        12     4 -909.1913
## 109       12     6 -908.7753
## 199        6    12 -908.7635
## 201        8    12 -908.1609
## 61        12     3 -908.0357
## 29        12     1 -907.8613
## 209        0    13 -907.6473
## 13        12     0 -907.6158
## 205       12    12 -907.5931
## 204       11    12 -907.5525
## 202        9    12 -907.3633
## 189       12    11 -907.3200
## 210        1    13 -906.1005
## 45        12     2 -905.9070
## 203       10    12 -905.7653
## 211        2    13 -904.7293
## 212        3    13 -903.9077
## 214        5    13 -902.0824
## 158       13     9 -901.9574
## 213        4    13 -901.9144
## 94        13     5 -901.6338
## 126       13     7 -901.3766
## 142       13     8 -900.9367
## 216        7    13 -900.5676
## 225        0    14 -900.5066
## 174       13    10 -900.1413
## 215        6    13 -900.1102
## 78        13     4 -900.0282
## 110       13     6 -899.6703
## 226        1    14 -899.0967
## 217        8    13 -899.0866
## 62        13     3 -898.8589
## 30        13     1 -898.7940
## 190       13    11 -898.4409
## 221       12    13 -898.4110
## 220       11    13 -898.3058
## 218        9    13 -898.2568
## 14        13     0 -898.2039
## 206       13    12 -897.9014
## 227        2    14 -897.3889
## 46        13     2 -896.8637
## 219       10    13 -896.6244
## 222       13    13 -896.4458
## 228        3    14 -896.2512
## 230        5    14 -895.1320
## 95        14     5 -894.6021
## 229        4    14 -894.3023
## 159       14     9 -894.2497
## 127       14     7 -893.9663
## 143       14     8 -893.6932
## 231        6    14 -893.4037
## 79        14     4 -893.1343
## 232        7    14 -893.1064
## 111       14     6 -892.6253
## 175       14    10 -892.5085
## 63        14     3 -891.9131
## 191       14    11 -891.1895
## 233        8    14 -891.1877
## 234        9    14 -891.1729
## 31        14     1 -890.7573
## 236       11    14 -890.5576
## 241        0    15 -890.5500
## 15        14     0 -890.3449
## 237       12    14 -890.1854
## 235       10    14 -889.8957
## 207       14    12 -889.7107
## 242        1    15 -889.0419
## 47        14     2 -888.9410
## 238       13    14 -888.1867
## 223       14    13 -887.7488
## 239       14    14 -887.6659
## 243        2    15 -887.3088
## 244        3    15 -886.0691
## 246        5    15 -884.7479
## 96        15     5 -884.2869
## 245        4    15 -884.1417
## 160       15     9 -883.9364
## 128       15     7 -883.6409
## 144       15     8 -883.4503
## 247        6    15 -883.0158
## 80        15     4 -882.8148
## 248        7    15 -882.7881
## 112       15     6 -882.3106
## 176       15    10 -882.2093
## 64        15     3 -881.6497
## 253       12    15 -881.4274
## 252       11    15 -881.3077
## 250        9    15 -881.1831
## 192       15    11 -880.9028
## 249        8    15 -880.8964
## 32        15     1 -880.5983
## 251       10    15 -880.2736
## 16        15     0 -880.2468
## 254       13    15 -879.4467
## 208       15    12 -879.4364
## 255       14    15 -879.2846
## 48        15     2 -878.8432
## 224       15    13 -877.4985
## 240       15    14 -877.4570

model.ardlDlmberganda = ardlDlm(formula = logprice ~ interest + logm1,
                        data = data.frame(data1) , p = 4 , q = 4)
summary(model.ardlDlmberganda)

## 
## Time series regression with "ts" data:
## Start = 5, End = 144
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0290527 -0.0075965  0.0005726  0.0072745  0.0304486 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.0145022  0.1822785   0.080  0.93671    
## interest.t   0.0067985  0.2135315   0.032  0.97465    
## interest.1   0.6093502  0.3240545   1.880  0.06238 .  
## interest.2   0.0798544  0.3221168   0.248  0.80461    
## interest.3  -0.3638172  0.3238873  -1.123  0.26347    
## interest.4   0.2084240  0.2447331   0.852  0.39604    
## logm1.t      0.0828689  0.0457486   1.811  0.07248 .  
## logm1.1     -0.0092841  0.0399079  -0.233  0.81642    
## logm1.2     -0.1166129  0.0390732  -2.984  0.00342 ** 
## logm1.3      0.0007016  0.0389297   0.018  0.98565    
## logm1.4      0.0447857  0.0425474   1.053  0.29455    
## logprice.1   0.3274245  0.0651574   5.025  1.7e-06 ***
## logprice.2   0.1323801  0.0684485   1.934  0.05537 .  
## logprice.3  -0.1448245  0.0674268  -2.148  0.03365 *  
## logprice.4   0.6730871  0.0636443  10.576  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01132 on 125 degrees of freedom
## Multiple R-squared:  0.9993, Adjusted R-squared:  0.9992 
## F-statistic: 1.273e+04 on 14 and 125 DF,  p-value: < 2.2e-16

#model p interest 0 p logm1 4 
rem.p = list(interest = c(1,2,3,4))
remove = list(p = rem.p)
model.ardlDlmberganda2 = ardlDlm(formula = logprice ~ interest + logm1,
                        data = data.frame(data1) , p = 4 , q = 4 ,
                        remove = remove)
summary(model.ardlDlmberganda2)

## 
## Time series regression with "ts" data:
## Start = 5, End = 144
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0290369 -0.0083445  0.0009024  0.0079199  0.0303652 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.174838   0.133708   1.308  0.19333    
## interest.t   0.448826   0.098736   4.546 1.24e-05 ***
## logm1.t      0.056659   0.043836   1.293  0.19849    
## logm1.1     -0.017025   0.039159  -0.435  0.66446    
## logm1.2     -0.118413   0.037399  -3.166  0.00193 ** 
## logm1.3     -0.006454   0.038112  -0.169  0.86580    
## logm1.4      0.060220   0.040337   1.493  0.13789    
## logprice.1   0.319059   0.062107   5.137 1.00e-06 ***
## logprice.2   0.111794   0.066101   1.691  0.09320 .  
## logprice.3  -0.122129   0.065114  -1.876  0.06297 .  
## logprice.4   0.699061   0.062611  11.165  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01149 on 129 degrees of freedom
## Multiple R-squared:  0.9993, Adjusted R-squared:  0.9992 
## F-statistic: 1.73e+04 on 10 and 129 DF,  p-value: < 2.2e-16

Proses selanjutnya sama dengan pemodelan menggunakan peubah tunggal.

Pertemuan 3 - Regresi dengan Peubah Lag

Siti Arbaynah

2024-09-09

Packages

Impor Data

Pembagian Data

Model Koyck

Pemodelan

Peramalan dan Akurasi

Regression with Distributed Lag

Pemodelan (Lag=2)

Peramalan dan Akurasi

Lag Optimum

Model Autoregressive

Pemodelan

Peramalan dan Akurasi

Lag Optimum

Pemodelan DLM & ARDL dengan Library `dynlm`

Ringkasan Model

SSE

Autokorelasi

Heterogenitas

Kenormalan

Perbandingan Model

Plot

Pengayaan (Regresi Berganda)

Data

DLM

ARDL

Pertemuan 3 - Regresi dengan Peubah Lag

Siti Arbaynah

2024-09-09

Packages

Impor Data

Pembagian Data

Model Koyck

Pemodelan

Peramalan dan Akurasi

Regression with Distributed Lag

Pemodelan (Lag=2)

Peramalan dan Akurasi

Lag Optimum

Model Autoregressive

Pemodelan

Peramalan dan Akurasi

Lag Optimum

Pemodelan DLM & ARDL dengan Library dynlm

Ringkasan Model

SSE

Autokorelasi

Heterogenitas

Kenormalan

Perbandingan Model

Plot

Pengayaan (Regresi Berganda)

Data

DLM

ARDL

Pemodelan DLM & ARDL dengan Library `dynlm`