The time series regression analysis in evaluating the economic impact of COVID-19 cases in Indonesia

2.1The transfer function model

The transfer function model is a model describing the future predictive value of a time series (output series {Yt}) based on the past values of the series itself and one or more time-series related to the output series (input series {Xt}). This model involves two or more variables consisting of the output variable {Yt}, the variable input {Xt}, and another input called residual {Nt} that has a relationship causality.

Assume {Xt} and {Yt} are stationary series. In a single input case, {Yt} and {Xt} can be written in the form:

where v⁢(B)=ω⁢(B)⁢Bbδ⁢(B) is a transfer function where ω⁢(B)=ω0-ω1⁢B-…-ωs⁢Bs,δ⁢(B)=1-δ1⁢B-…-δr⁢Br, for B is a backshift operator which is defined as Bℓ⁢Xt=Xℓ-1. The b is a delay parameter which shows the time lag when the input series starts to affect the output series, the r indicates the pattern of the impulse response weights, and the s shows the pattern of decay for impulse response weight begins. The {Nt} is the total disturbance sequence (noise series) that affects {Yt} which is independent with {Xt}. The input series {Xt} and noise series {Nt} are assumed to follow the ARIMA model. Then (b,r,s) is the main order in determining the best transfer function model. Since noise series follows the ARIMA Model, then the transfer function model can be written as (Wei, 2006):

where θ⁢(B)=1-θ1⁢B-θ2⁢B2-…-θq⁢Bq and ϕ⁢(B)=1-ϕB-ϕ2⁢B2-…-ϕp⁢Bp. The VARMAX Model is a multivariate version of the transfer function model. Uniquely, the transfer function model is the case specifically from the VARMA Model.

2.2The VARMAX model

According to Wei (2006), the VARMA Model (p,q) can generally be stated in equation as follows:

where 𝚽p⁢(B)= I -k𝚽1B-…-𝚽pBp and 𝚯q⁢(B)= I -k𝚯1B-…-𝚯qBq. The 𝒀t is a vector multivariate time series, 𝚽p⁢(B) is a k×k auto-regressive matrix polynomial of order p, and 𝚯q⁢(B) is a k×k moving-average matrix polynomial of order q.

Then the VARMA (1,1) SISO Model can be written in matrix form as follows:

Suppose Φ1;1,2⁢(B)=Θ1;1,2⁢(B)=Θ1;2,1⁢(B)=0. The VARMA (1,1) SISO Model Equation can be written as:

and

According to Eqs (2) and (3), the future value of Y1,t is only influenced by its past value but not affected by the past value of Y2,t, while the future value of Y2,t is influenced by the future value then from Y1,t and Y2,t.

This also shows that Y1,t results in Y2,t but Y2,t does not result in Y1,t. So, Y1,t is called exogenous variable or it can also be called a predictor. Equation (3) is also called the ARMAX model or transfer function model with the output variable Y2,t and the input variable Y1,t. Equation (3) can also be written as follows:

where

and

Equation (4) can be stated as:

where Nt is a noise series that follows the VARMA Model Ψ1;2,2⁢(B)⁢Nt=Θ1;2,2⁢(B)⁢a2,t. Hence, the ARMAX Model or transfer function model is a special case of the VARMA Model. Vector Autoregressive Moving-Average with Exogenous Regressors or VARMAX (p,q,r) is a special case of the VARMA Model (p,q) with the addition of exogenous/predictor variables. The VARMAX model is a multivariate version of the ARMAX model or the transfer function model (Widyanti, 2020). This model is suitable for multivariate time series without trends or seasonal components with the addition of exogenous variables. According to Spliid (1983), the VARMAX model (p,q,r) can be written as follows:

where, 𝜸r⁢(B)=𝜸0+𝜸1⁢B+…+𝜸r-1⁢Br-1. The 𝒀t is a k×1 multivariate vector time series, 𝑿t is a m×1 predictor vector time series, 𝚽p⁢(B) is a k×k autoregressive matrix polynomial of orde p, 𝚯q⁢(B) is a k×k moving-average matrix polynomial of orde q, 𝒂t is a k×1 vector noise series, and 𝜸r is a k×m matrix polynomial which its variance-covariance matrix ∑ of 𝒂t is a positive definite.

Thus, the VARMAX Model (p,q,r) Single Input Multi Output (SIMO) (k=2) and 1 predictor (m=1) can be written in the form of:

Then suppose k=1 and m=1 (SISO), then the VARMAX Model (p,q,r) can be stated as:

Note that Eq. (2.2) is similar to the Transfer Function Model equation which has the order of the transfer function (b,r,s) and the order

Therefore, the univariate VARMAX Model (p,q,r) in Eq. (2.2) will be the same as the Transfer Function Model when γr⁢(B)ϕp⁢(B)=v⁢(B) or γr⁢(B)=v⁢(B)⁢ϕp⁢(B). The parameters of this model can be estimated by minimizing the sum square of errors through multivariate generalized least square method. The sum square of errors is defined by utilizing matrix operations that include vectorization and kronecker multiplication (Tsay, 2014).

3.1Data analysis using the VARMAX model

In this section, the impact of COVID-19 cases in Indonesia is used to forecast the USD/IDR exchange rate and the IDX Composite. Suppose daily data of USD/IDR {Y1,t} and IDX Composite {Y2,t}. These two series can be influenced by the increase in the number of COVID-19 cases in Indonesia {Xt}. This paper compares which method is more accurate to forecast the USD/IDR exchange rate and IDX Composite. Previously, here are the characteristics of the data. Suppose 𝒀t=(Y1,t,Y2,t)′, Xt=(Xt)′, k=2, and m=1 (SIMO). In this paper, the data for the period 1 March 2020–29 June 2020 is used as in-sample data while the data in period 30 June 2020–3 July 2020 is used as out-sample data. The two endogenous variables have a value of 0 on every Saturday and Sunday since the market is closed on that day. As a consequence, for Friday data will be also used for both Saturday data and Sunday data.

Figure 1.

USD/IDR exchange rate vs the increase of COVID-19 cases time series plot (a) and IDX composite vs The Increase of COVID-19 Cases time series plot (b). It can be seen that as the number of COVID-19 cases in Indonesia increases, both the USD/IDR exchange rate and the IDX composite are weakening.

According to Table 1, it is known that the lowest USD/IDR exchange rate is Rp 13,085.40. This means that the strongest appreciation of the Rupiah is Rp 13,085.40 while the Rupiah depreciates the strongest at the level of Rp 16,504.80, namely on 24 March 2020 while the increase in the number of COVID-19 cases in Indonesia jumped by 107 cases. As the COVID-19 case in Indonesia has increased, investors have attracted their Rupiah to buy US dollars, as a result, the value of the Rupiah has weakened. The average USD/IDR exchange rate is Rp 14,995.31 with a standard deviation of 808.31.

The average IDX Composite value is Rp 4,725.1 and the highest value of the IDX Composite is Rp 5,650.14 meaning the strongest IDX Composite value is Rp 5,650.14 at 4 March 2020. However, after the strengthening of the IDX Composite on that date, the value of the IDX Composite continues to decline to below Rp 5,000.00 as cases of COVID-19 in Indonesia increase. It is also similar to the USD/IDR exchange rate, investors choose to invest in assets that are safer than placing their assets in shares in Indonesia, causing IDX Composite to touch the lowest level in history. It can be seen that, with the increase in COVID-19 cases in Indonesia, investors are shifting their assets to a safe haven, one of which is US dollars. The following is the movement of the two variables data displayed in the form of a time series plot.

The use of graphics in detecting that a series is stationary needs foresight so that the data stationarity test is performed using the Augmented Dickey-Fuller Test (ADF Test) with null hypotheses is “Data is not stationary” with alternative hypotheses is “Data is stationary”. The results are summarized in Table 2.

According to Table 2, the USD/IDR exchange rate and the increase of COVID-19 cases both are not stationary for α=0.05. As a result, these data need to be differentiated first. In the formation of the VARMAX Model, it is necessary to fulfill the basic assumption, namely stationarity. The next step is to look at the Sample Cross Correlation Matrix Function table and the Partial Autoregression Matrix Function table:

Based on Table 3(a), it can be seen that the (+) sign and the (-) sign that appear simultaneously in each lag shows that the data is not stationary. Then a similar pattern is seen for the two endogenous variables, so it needs to be differentiated for both variables and the Sample Correlation Matrix Function as obtained on Table 3(b).

It can be seen that the number of the (+) signs that appear simultaneously for the two endogenous variable in Sample Correlation Matrix Function Table is significant at lag 0. The (+) sign denotes a value that is more than 2 times the standard error by observing 121 data, the (-) sign denotes a value less than 2 times the standard error by observing 121 data, while the (.) sign denotes a value that is at the interval -2 times the standard error up to 2 times the standard error which can be calculated as:

It can be said that the (+) sign is greater than 0.18, which means there is a correlation between each variable with the variable shown in lag. Furthermore, it can be seen in the Partial Autoregression Matrix Function Table as follows:

It can be said that the value is between ± 0.18 which means there is no correlation between each variable with the variable shown in lag. The number of (+) signs that appear simultaneously in the Partial Autoregression Matrix Function Table is significant in the first lag. Order selection is also considered through the smallest AICC information on Minimum Information Criterion as follows:

Based on Table 5, model candidate can be chosen based on the smallest AICC value, namely VARMAX(1, 1, 1) Model with a value of 20.05. After obtaining the desired VARMAX Model, the next step is to estimate all the model parameters as follows:

The above equation can be formed in a mathematical equation:

Since the unit of the response variables (USD/IDR exchange rate and IDX composite) are in IDR meanwhile the predictors (right-side of the equation) have various units, then the parameters coefficient will take the roles to adjust the unit. For example, the estimated parameter γ^0,1=0.08 and γ^0,2=-0.003 for number of COVID-19 cases (Xt, in person unit) has unit is IDR/person. The same way applies to other models. Notice that for both USD/IDR exchange rate and IDX composite are influenced by the common predictors with opposite sign, except for the one-step-behind of USD/IDR exchange rate, which both have positive sign. It means that the USD/IDR exchange rate in previous day will strengthen the current’s rate in the same direction (positive coefficient).

After the VARMAX Model is selected, it will be checked whether the model is feasible or not to be predicted using the diagnostic test and the multivariate normal distribution at the residual.

After obtaining the VARMAX model, then the next process is to predict USD/IDR exchange rate and the IDX Composite using the VARMAX(1, 1, 1) model for some time to come with T=121 observations that have been known by the realization value of Y^1,121⁢(l) and Y^2,121⁢(l). The prediction of the USD/IDR exchange rate and the IDX composite will be carried out for the next 4 steps, namely 30 June–3 July 2020 are summarized in Table 7.

Based on Table 7(a) and (b), the prediction results for both USD/IDR exchange rate and IDX composite are still between the respective 95% lower limit and the 95% upper limit. The results of the absolute difference between the predicted value and the actual value divided by the actual value quite small for the next 4 days. This shows that the VARMAX (1, 1, 1) model for predicting the USD/IDR exchange rate and IDX composite is equally good. Likewise in Fig. 2a and b, the respective comparison plot looks quite close in value. In addition, its MAPE value is very good at 1.04% and 0.95% respectively. The predictive ability for the VARMAX(1, 1, 1) model is very good since the MAPE value is less than 10%.

Figure 2.

Comparison of Predicted Value and Actual Value for (a) The USD/IDR exchange rate, and (b) The IDX composite prediction. Both plots show the quite close of predicted and actual values.

3.2Data analysis using the transfer function model

An analysis of the impact of COVID-19 cases in Indonesia on the USD/IDR exchange rate using the transfer function model is being conducted here. Using the characteristic from the previous study above, the next step is to identify the ARIMA Model for the exogenous variable, namely, the increase of the number of COVID-19 cases in Indonesia. In the formation of the transfer function mModel, the basic assumption is stationary. Take a look at the ACF and PACF plots of the increase of the number of COVID-19 cases in Indonesia on the following Fig. 3.

Figure 3.

ACF and PACF Plot for the Exogenous Variable: (a) ACF and (b) PACF of original series, (c) ACF and (b) PACF of differenced series. Those plots are used to identify the suitable ARIMA Model.

The two figures above show that the data is not stationary in the average since in Fig. 3a shows exponential decay. Therefore, to make the data is stationary in the average, it needs to be differenced first whose ACF and PACF are given by (c) and (d). From these figures, the ARIMA model can be determined based on the significant lag in the ACF and PACF plots. The ACF and PACF cut off at the first lag, thus the possible ARIMA Model for the exogenous variable are the ARIMA(1, 1, 1) model and the ARIMA(0, 1, 1) model.

The next step is the estimation and evaluation of both models parameters. To test the parameters significance, the null hypotheses is “exogenous variable is not significant” versus “exogenous variable is significant”. The estimators and its significance are presented on Table 8. In Table 8, parameter significance test has been performed for two possible models. The number in bold indicates that the parameter has p-value is more than 1% ⩽α⩽ 5%, so the parameter is not significant in the model. Note that the ARIMA(1, 1, 1) model has one insignificant parameter and the AIC’s model is 1488.69. Then by eliminating the insignificant parameter from the previous model, the AIC value become 1487.12 which is smaller than the previous one. Since the ARIMA(0, 1, 1) Model has a significant parameter and has the smallest AIC value, thus it is a suitable model for the exogenous variable.

After obtaining the suitable model, it is necessary to do a diagnostic test by conducting a white noise test and a residual normality test. The null hypotheses is “the residual is white noise” and vise versa for the alternative hypotheses. Based on the Ljung-Box test, it is found that only at 5’th lag the p-value is greater than 1% ⩽α⩽ 5%, means that the null hypotheses is not rejected only at 5’th lag. As a result, the residual does not fulfill the white noise assumption.

Hereafter, the normality assumption of the ARIMA(0, 1, 1) residuals be checked through the QQ-Plot, as can be seen on Fig. 4. It shows that quite a lot of data values are along the normal line, so it can be said that the residuals are normally distributed.

Figure 4.

QQ-plot of residual ARIMA(0, 1, 1) model. It can be seen that the ARIMA(0, 1, 1) model meets the normality assumption because of the large amount of data values are along the lines.

Since by seeing the QQ-Plot is considered to be less objective, another test is carried out to check whether the residuals are normally distributed using the Jarque Bera test with the null hypothesis is “residual is not normally distributed”. It is obtained that the p-value is less than 1% ⩽α⩽ 5%, means null hypotheses is rejected. Therefore, the residuals are normally distributed.

Based on the estimated parameter values in Table 8, the model can be written as:

The pre-whitening process for the exogenous series is

such that the pre-whitening process for the output endogenous series is,

The pre-whitening step in the exogenous series must also be executed in the endogenous series. This is for maintaining the integrity of the functional relationship of the transfer function maps Xt into Yt.

In order to obtain the transfer function model, the Crosscorrelation Function (CCF) should be investigated. The CCF is used to determine the value of the main order of the transfer function, namely order (b,r,s). Before forming the CCF Plot, the standard deviations are calculated for {αt^} and {βt^}, which are 115.19 and 277.82 consecutively. The CCF Plot of the pre-whitened exogenous and pre-whitened endogenous variable is as shown on Fig. 5 below. Note in Fig. 5, the possible main order of the transfer function is (b,r,s)=(6,0,2), then the transfer function for the exogenous variable is:

Figure 5.

CCF plot of the pre-whitened exogenous and pre-whitened endogenous variable. It can be seen that the main order of the transfer function is (b= 6, r= 0, s= 2).

The estimated parameters of the transfer function with order (6,0,2) are ω0^=0.01, ω1^=-0.18, and ω2^=-0.26. Thus, the initial transfer function can be formed as:

Next, the ARMA modeling will be performed on the noise series so that the white noise residual will be obtained. The ACF and PACF plot of residuals are presented in Fig. 6. Based on this plot, the possible model obtained is MA(1).

Thereafter, all parameters of the transfer function model are estimated and obtained that θ1^=0.09, ω0^=0.003, ω1^=0.17, and ω2^=-0.24. Thus, the final transfer function model can be formed as:

and be simplified into the following form:

After obtaining the transfer function model, it is necessary to do a diagnostic test between {a^t} and {αt}. The first is autocorrelation test with null hypotheses “{a^t} is not significantly correlated”. By using Ljung-Box test, it is obtained that for all lag times, the p-values are greater than 1% ⩽α⩽ 10%. It means that do not reject the null hypotheses. Thus the residuals meet the white noise assumption.

Further, the crosscorrelation test is carried out using the null hypotheses is “{a^t} and {αt} are not independent”. By still using Ljung-Box test, it is obtained that all p-values are greater than 1% ⩽α⩽ 10%, so null hypotheses is not rejected. Therefore, residuals are independent. In addition to the diagnostic test, the residual normality can also be tested through the following QQ-Plot on Fig. 7.

Figure 6.

ACF dan PACF plot of the residual. The identified time series model from the ACF and PACF plot is MA(1).

Figure 7.

The residual normality test. The graph shows that the residuals are normally distributed.

Figure 7 shows that quite a lot of data values are along the normal line, so it can be said that the residuals are normally distributed. Thus, the next step is to predict the USD/IDR exchange rate with T= 121 observations that have been known with the realization value is Y^121⁢(l). Table 9 shows the USD/IDR exchange rate prediction will be carried out for the next 4 days, namely on 30 June 2020–3 July 2020.

Based on Table 9, the prediction results are still between the 95% lower limit and the 95% upper limit and also the results of the absolute difference between the predicted value and the actual value divided by the actual value quite small for the next 4 days. This shows that the transfer function model for predicting the USD/IDR exchange rate is quite good. Likewise in Fig. 8, the comparison plot looks quite close in value. In addition, its MAPE value is very good at 0.86%. The predictive ability for the transfer function model is very good since the MAPE value is lower 10%.

Figure 8.

Comparison of predicted value and actual value. The USD/IDR exchange rate prediction are quite close from the actual value.