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ARIMA Model Results

In document MScBA Digital Business (pagina 36-46)

Chapter 7. Results

7.1 ARIMA Model Results

Model selection is an essential element in the modeling process in time series analysis. AIC and BIC are two popular model selection criteria. To find the best model Arima, best.AIC and Arima, best.BIC functions will be used. This study will use Akakie Information Criterion (AIC) first.

This function estimates the model with all possible combinations of model orders but does not remove intermediate lags. After choosing the proper ARIMA model, precise estimations were attempted using the Box and Jenkins approach. Box and Jenkins's methodology is based on the concept that previous events impact future events. ARIMA models are a form of Box-Jenkins methodology.

Figure 9: The original and differenced series

One of the most important statistical measures used in estimation methods is ACF and PACF.

While PACF measures the relationship between two variables, ACF is used to analyse

univariate time series. ACF and PACF coefficients are not statistically significant in non-stationary data. As a result, if the series is not non-stationary, it must be converted to non-stationary using the proper difference procedure. Information about p and q parameters can be provided by drawing ACF and PACF graphs. The autocorrelation (ACF) and Partial Autocorrelation function (PACF) were applied until the 36th lag.

Figure 10: ACF Graph

Figure 11: PACF Graph

ARIMA (7,1,5) could be a sensible mode without lags 3 and 4.

7.1.1 ARIMA (6.1.3) Model

According to ARIMA (6.1.3) model, regression analyses are shown in Table 1.

Variables Coefficients S.E of the regression

Ar1 -0.7922 0.0312

Ar2 -0.9404 0.0231

Ar3 -0.8523 0.0267

Ar4 -0.8528 0.0268

Ar5 -0.8968 0.0204

Ar6 -0.7321 0.0286

Ma1 0.7454 0.0454

Ma2 0.3949 0.0474

Ma3 0.4115 0.0300

Table 3: Regression analysis Results

In this regression analysis, Sigma^2 is estimated at 70931559: log likelihood= -8480.84 AIC= 16981.69 AICc= 16981.96 BIC= 17028.67 (See Appendix 2)

7.1.1.1 White Noise Test for ARIMA (6,1,3) Model

Ljung-Box methodology was used for the White Noise test for 10, 20, and 30 lags.

Lag=10 Lag=20 Lag=30

X-squared 200.77 265.57 290.67

p-values <0.00000000000000022 <0.00000000000000022 <0.00000000000000022 Table 4: Ljung-Box test for ARIMA (6,1,3) Model

According to the table, the null hypothesis cannot be rejected at 5%. Therefore ARIMA (6,1,3) model can be correct.

7.1.2 ARIMA (7,1,5) Model

In this section, ARIMA (6,1,3) model will be compared with ARIMA (7,1,5) model. Z-test of coefficients for the ARIMA (7,1,5) model is shown in Table (See Appendix 3).

7.1.2.1 White Noise Test for ARIMA (7,1,5) Model

Ljung-Box methodology was used for the White Noise test for 10 and 30 lags.

Lag=10 Lag=30

X-squared 20.306 93.638

p-values 0.02649 0.00000001826

Table 5: Ljung-Box test for ARIMA (7,1,5) Model 7.1.3 ARIMA (7,1,5, s7) Model

Estimate Std. Error z value Pr(>|z|) Ar1 0.3984288 0.1200796 3.3180 0.0009065 ***

Ar2 -0.3153338 0.0961161 -3.2808 0.0010353 **

Ar3 -0.0156597 0.0805020 -0.1945 0.8457641 Ar4 -0.1201007 0.0764995 -1.5700 0.1164256 Ar5 0.0018877 0.0656229 0.0288 0.9770510 Ar6 0.1532682 0.0552267 2.7753 0.0055158 **

Ar7 0.3686493 0.0799743 4.6096 0.000004034 ***

Ma1 -0.4973412 0.1235626 -4.0250 0.000056972 ***

Ma2 -0.1417251 0.1050871 -1.3486 0.1774515 Ma3 0.3501919 0.0968525 3.6157 0.0002995 ***

Ma4 -0.2653412 0.1003724 -2.6436 0.0082037 **

Ma5 0.0789868 0.0665927 1.1861 0.2355758

Sma1 -0.8139067 0.0402969 -20.1977 <

0.00000000000000022

***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Table 6: Z-test of coefficients for ARIMA (7,1,5, s7) Model.

7.1.4 Model Comparison

This section will compare all models based on AIC and BIC functions to find the best model.

Models Df AIC BIC

ARIMA (6,1,3) 10 16981.69 17028.67

ARIMA (7,1,5) 13 16861.93 16923.00

ARIMA (7,1,5, s7) 14 16676.92 16742.58

Table 7: Comparison of ARIMA (6,1,3), ARIMA (7,1,5) and ARIMA (7,1,5, s7) Models.

According to the table, it is understood that the ARIMA (7,1,5, s7) model is more suitable because all terms are significant, residuals are white noise, and low information criterion (IC) values.

7. 2 VAR Model

To develop the VAR model, all datasets need to be merged by using merge.xts () function.

The first five rows of the merged dataset can be shown below

Date FRA_COV19 GER_COV19 POL_COV19

2020-03-04 73 157 1

2020-03-05 138 186 0

2020-03-06 190 187 4

2020-03-07 336 142 1

2020-03-08 177 105 5

2020-03-09 286 347 6

Table 8: The Merged Dataset

Figure 12: Confirmed Covid-19 Cases in Germany, France, and Poland.

Figure 13: Plotting Confirmed Covid-19 Cases in Germany, France, and Poland Separately.

From these graphs, it can be concluded that there is an impact between the three countries in terms of rising and fall periods of the Covid-19 pandemic. The effect between these countries will be investigated in this section by using Granger Causality Test.

7.2.1 ADF Test

Adf test was applied to the dataset from France. The value of the test statistic is -2.0313, and the second test statistic is 26.6692. Critical values for the test statistics can be seen in the table.

1pct 5pct 10pct

tau2 -3.43 -2.86 -2.57

phi1 6.43 4.59 3.78

Table 9: Critical values for the test statistics for the FRA_COV19 series.

tau2 term stands for the null hypothesis, which is δ = 0. Phi1 term stands for a combined null hypothesis named a joint hypothesis where c and δ are equal to 0. According to the table, it can be concluded that tau2 is -2.86, which means that it is higher than the value of the test statistic, which is -2.0313. Therefore, the null hypothesis cannot be rejected. It can be presumed that there is a unit root. In other words, the variable is non-stationary.

Data differencing by first order can be used to solve non-stationarity. Therefore, the ADF test should be repeated in the first order. As a result of the first-order difference, the Value of the test statistic is -7.3033. Critical values for the test statistics can be seen in Table 12. The value of the test statistic is lower than the 5% critical value, which is -2.86. Therefore, the null hypothesis is rejected. In addition to that joint hypothesis is also rejected second test statistic is higher than its critical value, which is 4.59. After differentiating the data by the first order, variables became stationary.

7.2.2 Granger Causality Test

GER_COV19 is selected as the dependent variable to determine if POL_COV19 is a Granger cause of GER_COV19. To perform Granger Causality Test, the granger test () function was used. Firstly, the Granger causality test will be done for 3 lags.

Model 1: GER_COV19 ~ Lags (GER_COV19, 1:3) + Lags (POL_COV19, 1:3) Model 2: GER_COV19 ~ Lags (GER_COV19, 1:3)

Model 1 is the unrestricted model, which returns with lags, and Model 2 is the restricted model, which returns without the lags.

Res.Df Df F Pr(>F)

1 802

2 805 -3 0.6137 0.6063

Table 10: Granger-Causality Test Results with 3 lags (GER_COV19 as the dependent)

As it can be concluded p-value is 0.6063, which is greater than 0.05. Hence, the null hypothesis cannot be rejected. Thus, it is found that Covid-19 cases in Poland do not affect the Covid-19 cases in Germany.

Granger- Causality test was repeated for 4 lags.

Model 1: GER_COV19 ~ Lags (GER_COV19, 1:4) + Lags (POL_COV19, 1:4) Model 2: GER_COV19 ~ Lags (GER_COV19, 1:4)

Res.Df Df F Pr(>F)

1 799

2 803 -4 24.58 < 0.00000000000000022 ***

Table 11: Granger-Causality Test Results with 4 lags (GER_COV19 as the dependent)

When the test was repeated with 4 lags, it can be seen that the p-value was less than 0.05.

Hence, the null hypothesis can be rejected, and it is found that POL_COV19 is a Granger cause of the GER_COV19 series.

The Granger-Causality test is conducted for the case where POL_COV19 is the dependent variable with 3 lags.

Model 1: POL_COV19 ~ Lags (POL_COV19, 1:3) + Lags (GER_COV19, 1:3) Model 2: POL_COV19 ~ Lags (POL_COV19, 1:3)

Res.Df Df F Pr(>F)

1 802

2 805 -3 67.606 < 0.00000000000000022 ***

Table 12: Granger-Causality Test Results with 3 lags (POL_COV19 as a dependent)

The Granger-Causality test is conducted for the case where POL_COV19 is the dependent variable with 4 lags.

Model 1: POL_COV19 ~ Lags (POL_COV19, 1:4) + Lags (GER_COV19, 1:4) Model 2: POL_COV19 ~ Lags (POL_COV19, 1:4)

Res.Df Df F Pr(>F)

1 799

2 803 -4 53.642 < 0.00000000000000022 ***

Table 13: Granger-Causality Test Results with 4 lags (POL_COV19 as a dependent)

In this case, the null hypothesis was rejected. Therefore, it is understood that GER_COV19 is a Granger cause of POL_COV19, both with lags 3 and 4.

In document MScBA Digital Business (pagina 36-46)

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