Multivariate time series analysis of COVID-19 Pandemic and gold price by using Error Corrections Model

: Coronavirus (COVID-19) severe acute respiratory syndrome is an infectious disease that has a direct influence on the world’s population, as is obvious from the prevalence of COVID-19. However, as the COVID-19 virus continues to proliferate, its impact is becoming more and more significant and extensive. Accordingly, the study’s goal is to use the Error Corrections Model (ECM) to examine the relationships between the COVID-19 pandemic and gold prices. An Error Corrections Model was used by the researchers in this investigation. Error Correction Models and long-term equilibrium can be found for non-stationary time series that have been cointegrated together. Two-step estimation can be used to estimate the ECM since a proportion of the imbalance from one period is adjusted in the succeeding period, the system returns to equilibrium. The calculated cointegrating relations are used to create the error correction terms, and the estimation of VAR is done by a first differencing process, and then appears in the model as an explanatory variable

volatility are periods when investors turn to gold as a safety net.As a safe, liquid, and long-term store of value, gold is ideal for investors who want to achieve their primary goals of safety, liquidity, and return.As a result, gold's price may rise in the future (Grima et al., 2020: 1-31).Whether or not the global COVID-19 pandemic has an ideal influence on gold's price remains to be seen.And how will the COVID-19 pandemic impact the gold price going into the future?Because of that.The theoretical framework of the ECM is used in this study, so we ask three questions: (1) How can one investigate the relationship between the COVID-19 pandemic and gold price time series?(2) Do COVID-19 pandemic affect gold price?(3) How consistent and robust are the different regression approaches in assessing the impacts of questions ( 1) and ( 2)?In order to identify the causal effects of the COVID-19 pandemic on gold prices, the researchers set up an Error Corrections Model (ECM) to derive the relationship between the COVID-19 pandemic and gold prices.The study of co-integration has long attracted the attention of econometricians in particular.It is considered to be "cointegrated" when a stationary linear combination of the nonstationary series occurs."The cointegrating equation" refers to an imbalance longrun relationship between the variables, where the imbalance is a stationary state defined by forces that tend to return a system to balance anytime it deviates from the equilibrium point.The error correction model is based on the assumption that a portion of the disequilibrium from a previous period gets corrected in the next.There are cointegration relations in ECM specification that constrain the long-term behavior of endogenous variables while still permitting short-term dynamic adjustment.Cointegration can be further generalized by applying the nonlinear Wang-Phillips in 2009 model in the case where X t is stationary at 1 st difference and U t is the stationary disturbance, which was first proposed by Box and Tiao in 1977 and more thoroughly examined by Engle and Granger in 1987.It incorporates a unitroot nonstationary regressor into the model of Linton et al. in 2008 (Lin et al., 2020: 175-191).The Granger causality test and the cointegration approach based on the vector error correction model (VECM) are used to analyze long and short-term correlations between distinct variables (Lahmiri S., 2017: 181-189).To evaluate data on dam monitoring (Li et al., 2013: 12-20) used the cointegration theory and the model of error correction, which depends on an error correction model suggested to reflect long-run balance and short-run imbalance relationships.Scheiblecker proposed a cumulative error correction model for time series dynamics between cointegration and multi-cointegration (Scheiblecker, M., 2013: 511-517).Apergis and Payne were working on the research about Energy consumption and economic growth in Central America: Evidence from a panel cointegration and error correction model (Apergis & Payne., 2009:211-216).GDP and electricity usage in Pakistan are linked by a number of different factors studied by (Jamil & Ahmad., 2010: 6016-6025) which found that there is a unidirectional causality from real economic activity to electricity consumption.So, the error correction model has been seen as a powerful statistical tool to investigate the relationship between variables, that can be used to analyze the relationship between the COVID-19 pandemic and gold price when the variables were stationary at the first difference I (1).
Since the COVID-19 pandemic began, scientists have begun moving away from theoretical models and toward practical studies.As a result, researchers working to construct a theoretical model that may account for the most recent evidence on the association between the COVID-19 pandemic and gold prices.therefore, the study contributions are: Using the most recent data, we'll examine the relationship between these two variables by modern modeling which is the Error Corrections Model (ECM).Secondly, constructing a multivariate time series model to assess the relationship between those two variables.Because the coronavirus affects the price of gold economically, and this situation has a direct impact on the lives of all humanity.Although, the objective of the study is to analyze the relationships between the COVID-19 pandemic and gold prices during 2020 by using the Error Corrections Model (ECM).In addition, the current study consists of four sections, the second section deals with a summary of the theory about ECM.The other section presents the data and results of econometric methodologies.The conclusions and further discussion of the study are examined in section four.

2-1. Vector AR(p) Model:
The following steps can be used to develop a VAR model.As an initial step, can be used either the test of M(i) or the criterion of Akaike information to determine the order of the model.(2.6)The expression (2.6) represents the moment equations of a model of VAR(p) which is the version of the multivariate of the Yule-Walker equation.

2-2. Cointegration Theory:
Cointegration is a statistical property possessed by some time series data that is defined by the concepts of stationarity and the order of integration of the series.it describes the relationship between variables.More specifically, it utilizes cointegration as a form of estimation for describing long-run equilibrium between variables.Given that it can estimate the long-run equilibrium, it can also estimate short-run dynamics around the equilibrium to see how a 'pair' of cointegrated variables will revert back to their long-run relationship.Two or more stationary time series can be combined into one linear combination by Granger and Engle (Engle, Granger, 1987: 251-276).Time series that do not follow a stationary linear combination are considered to be cointegrated."The cointegrating equation" refers to an imbalance long run relationship between the variables, where the imbalance is a stationary state defined by forces that tend to return a system to balance anytime it deviates from the equilibrium point.Definitions based on Engle and Engle (Engle, Granger, 1987: 251-276) are included below.An integration of (d) order is defined as a time series with a stationary, invertible, Auto Regressive Moving Average (ARMA) representation after (d) of the differencing process.For one stationary time series that doesn't have any deterministic elements, a finite ARMA process can be used to approximate the infinite moving average representation.The vector autoregressive (VAR) model framework has been widely applied to model cointegration system.In the modeling of cointegrated systems, the determination of the number of cointegrating relations, or the cointegration rank, is the most important decision.Cointegration is said to exist between two or more nonstationary time series if they possess the same order of integration and a linear combination (weighted average) of these series is stationary.For first differencing I(1) process of series, all of the theoretically infinite variance is derived from the series long-run section.As a result, an I(1) time series is smoother than an I(0) time series, with lengthy swings dominating.When Z t is distributed as (d) differencing, next α + βZ t is I(d), where α and β represent constants with the value of β differs from zero.It is obvious that a time series is nonintegrated if it cannot be transformed into a stationary time series by infinitely differencing.

2-3. Cointegration test 2-3-1. Engle and Granger:
Cointegration tests for nonstationary time series can be performed using a variety of methods.Cointegration can be determined by examining the stationarity of the residuals of a regression model, according to Engle and Granger.In Engle and Ganger's technique, variables are constrained by a common element in their dynamic interaction.Regression equation models of the type used to test cointegration are the initial step (Engle, Granger, 1987: 251-276).     (2.7) where there is equal integration order of time series (Z 1t , Z 2t ,..., Z nt ), and the cointegrating vector (ḇ 1 , ḇ 2 , ḇ 3 , …, ḇ n ) is unknown.The estimated residual values which are denoted e t are acquired after the cointegrating vector has been computed from the data.The augmented Dickey-Fuller (ADF) test is used to determine whether is stationary or not as part of the cointegration test.Cointegration exists in the series if is stationary.If is not stationary, the series does not exhibit cointegration (Johansen, 1991(Johansen, : 1551(Johansen, -1580).
In the Engle-Granger analysis, let gold price () and case of covid-19 () be I(1) series which means that  and  are not stationary at the level, but the first difference of the series are stationary.The regression model using  and  series is as follows   (2.8) where , ∈ ℝ and is error term.If is I(0) or ∑ where | | < 1 then  and  are cointegrated.If the series are cointegrated, there is at least one causal relationship between the series.In order for the series to be cointegrated, they must be stationary.The differencing process is applied to ensure stability.However, applying the differencing process causes a loss of long-run information.Therefore, these imbalances are tried to be eliminated by using the error correction model.If there is a long-run relationship between series, an error correction model, which is used to determine the short-run relationship, shows a deviation period from a long-run relationship.The following error correction model (ECM) is used to determine the possible causality relationship between the cointegrated series and to determine the direction.

𝑌
where Z t is a 2x1 vector made up of the variables Z t1 (gold price) and Z t2 (covid-19); μ denotes a 2x1 vector of constant terms; the Γ and Π denote a 2x2 matrix of coefficients; and εt denotes a 2x1 white noise error terms vector.The null hypothesis of no cointegration (rank(r) = 0) is tested against an alternative hypothesis of cointegration (r > 0) in this test, which is dependent on maximum likelihood estimation and trace statistics (λ trace ) (Johansen, 1991(Johansen, : 1551(Johansen, -1580)).
The Johansen (1991) technique is adopted to test the long run relationship among variables, The long run relationship could be stated as follows: (2.12)In a model with more than two variables there is possibility of the existence of more than one cointegrating vector.In our present model we have two variables, gold price, and covid-19 daily cases.In our present model, П is a 2x2 matrix containing information about the long run relationship among the vector of variables.Johansen describe two separate procedures namely, trace statistics and maximal eigenvalue test to find out the number of cointegrating vectors.The maximal eigenvalue test is the likelihood ratio test for the null hypothesis, presence of r cointegrating vector against the alternative hypothesis r+1 cointegrating vector which can be stated as follows: 15) The second procedure is also based upon likelihood ratio test.Trace statistic verify the increasing tendency of trace as a result of the addition of more eigenvalues beyond the r th .The null hypothesis based on λtrace verify the presence of cointegration vector is less than or equal to r against the alternative hypothesis, presence of more than r cointegrating vectors.It can be estimated as follows: where ̂ ̂ are smallest estimated eigenvalue.After the conformation of the existence of long run equilibrium relationship.

2-5. Error Correction Model:
Error Correction Models can be used to depict nonstationary time series that are cointegrated.The disequilibrium of the previous period is expected to be redressed in the succeeding one.A key feature of the ECM is that it prevents endogenous variables from converging to their cointegration relations over the long term while still permitting short-term adjustment processes.To repair a long-term imbalance by correcting short-term imbalances, the cointegration term has been dubbed the "error correction term."According to the cointegration relation equation, when the y t and x t are in long-term equilibrium,   (2.17) where defined as a disturbance term that is stationary.Change in one variable can be linked back to earlier equilibrium errors and to previous changes of both variables using a typical ECM, and the analogous ECM is: while    is equal to zero in long-term equilibrium, which is disequilibrium error correction term.The speed with which the error correction term is brought back into equilibrium is used to determine the value of the coefficient.Regarding a system with multiple variables, then, suppose that each time series Z t =(z 1t , z 2t ,..., z nt ́ is stationary at first difference, so that There is no change in each of the time series' component vectors, indicating that the series is nondeterministic a completely stationary process.The backshift operator for a Finite Vector Autoregression (VAR) representation in terms of B is:  (2.19).It is implied that the variables are cointegrated by the existence of their respective levels.If the variables are cointegrated, incorrectly defining a VAR in differences, and a VAR with levels will forget important restrictions.The dependent variable's response to the independent variables' shocks is used to formulate the errorcorrecting behaviors (Scheiblecker, M., 2013: 511-517).

2-6. Error Correction Model Estimation:
Estimating the error correction model can be done using a two-step estimator (Johansen, 1991(Johansen, : 1551(Johansen, -1580)), an easy operation.Initially least squares regression procedure is used to estimate the cointegrating vector's parameters on each variable level.The non-cointegration hypothesis is investigated.Long-term relationships can be estimated using cointegrating vector parameter estimations.In the second stage, the regressors show a lagged value of the cointegrating regression's residuals, and this is taken into account in the dynamic specification.Using the calculated cointegrating relations, error correction terms are generated, and a VAR is estimated using the error correction terms as regressors.For both phases, just a single equation least squares approach is necessary because all parameters are consistent.An advantage of an estimator is that it does not necessitate specifying the dynamics until the error correction structure is computed (Johansen, 1991: 1551-1580).

2-7. Impulse Response Functions of Structural Analysis:
The dependent variables' responsiveness of vector autoregressive models to shocks can be traced using impulse responses.A one standard deviation positive shock helps to determine the behavior of the variable when the error terms are given a positive shock, as well as how the variables react to each other.It is possible that the original variable can feed back into itself if a shock is delivered to one of the series.Researchers can use the impulse response function method to see how one parameter changes in reaction to a rapid shift in another parameter's expected behavior in the future.This might be referred to as a shock, innovation, or unexpected shift in a variable (Engle, Granger, 1987: 251-276).It's usually referred to as an impulse, which reflects the idea of a one-time shock that takes place at some point along the timeline (Inoue, A., Killian, L., 2013: 1-13).
The The error vector 's covariance matrix has been estimated to be a diagonal matrix.Structural errors refer to the non-correlated errors (Engle, Granger, 1987: 251-276).According to orthogonal errors , the Wold representation of X t can be calculated as follows: (2.25)Where (As can be seen from the equation (2.23), B is the lower triangular matrix of Bi, j. and 1 is the number of B's diagonal elements).Orthogonal shocks' impulse responses are , where define as (i,j) th element of .The relationship between 's plot and s is known as the orthogonal impulse response function of X i with regard to (Inoue, A., Killian, L., 2013: 1-13).

2-8. Forecasting:
If the fitted model is sufficient, it can be utilized to provide forecasts.The same principles used in univariate analysis can be applied to forecasting.The following steps can be taken to generate forecasts and standard deviations of the corresponding forecast errors.The 1-step ahead forecast at time origin h for a VAR(p) model is  ∑  , and the corresponding forecast error is  .Σ represents the covariance matrix of the forecast error.If X t is weakly stationary, then as the forecast horizon rises, the 1-step ahead forecast  converges to its mean vector μ.If X t is poorly stationary, then the 1step ahead forecast  converges to its mean vector μ as the forecast horizon increases (Aziz, et al., 2023: 531-543; Tsay, & R.S., 2001: 312-318).

3-1. Data Description:
In this study, we employ daily observations of the COVID-19 pandemic, for which only case-containing records are included in the statistical modeling method and which are seasonality adjusted.These observations are acquired from the WHO website (covid19.who.int) and an additional variable, gold price, which is also seasonally adjusted and retrieved from the International Energy Agency website (www.gold.org).
The sample duration is eleven months, beginning in February 2020 and ending in December 2020.Figure 1 depicts the plots of the original time series of two series of the sample study.In order to construct a suitable model, all series included in the study must have been stationary; consequently, the researchers must examine the structure of unit-root of the data.Even though the following graph offers us a general sense of the stationarity structure of the series, to test for unit roots, we applied the Augmented Dickey-Fuller test to the series.Table 1 displays the results of the ADF test applied to the levels and the first differences of the series.The ADF test findings show that all variables are not stationary through not rejecting the hypothesis of unit-root at all levels, they were all found to be stationary after first differencing.Figure 2 shows the differenced series in time series plots.
Figure (2): Plot of series of the differenced variables (R-Studio v.4.5.1) 3-2.Lag order selection: In accordance with Lutkepohl (Lutkepohl, H., 1991: 241-283), using a higher order lag length than the actual lag lengths increase the mean square forecast errors of the VAR, whilst selecting a smaller order lag length than the actual lag lengths often result in autocorrelated errors.Therefore, the precision of forecasts given by VAR models is highly dependent on the choice of actual lag lengths.Using penalty selection techniques such as the log likelihood (LogL), sequential modified LR, Final prediction error (FPE), Akaike Information Criterion (AIC), Schwarz information (SC), and the Hannan-Quinn (HQ), we found a VAR(p) and ECM model for the analysis (HQ).Table 2  The table 2 provides for the selection of an acceptable lag for the model, the above table reveals that majority of lag selection criteria identified the optimum lag to be 2.Although AIC, FPE, HQ and LR indicating the optimum lag to be consider as 2, the SC criteria suggest lag 1 as the optimum.Based on the majority view we have selected lag 2 for further analysis, this indicates that the best model for our data is order 2.

3-3. Co-integration test:
The next stage, following the selection of the ideal lag duration, is to determine the existence of a long-term relation among variables.To test for co-integration, the co-integration test developed by Johansen (1998) is used to identify the long-term relationship among the variables.Table 3 displays the outcomes of the Johansen multivariate co-integration test for both equation systems.In terms of t-value, the parameter of lagged residual is negative and statistically significant, indicating a long-run causal relationship between covid-19 and gold price.This coefficient suggests that daily changes in the COVID-19 variable may correct 17.6% of the imbalance in the gold price, which is another proof that the variables are co-integrated and that the model is accurate: We can use the Wald test to check short-run causality which follows chi-square distribution and the results the flowing below: From Table 6, it is clear that the test is non-significant and cannot reject the null hypothesis which states that the coefficients of (Δ COVID_19) are equal, meaning that there is no short-term causality running from COVID-19 to gold price.3-5.Diagnostic checks for residuals: After estimating a suitable model for the variables, this phase of the study focuses on diagnostic checking.There are numerous approaches for controlling the robustness of the model; for diagnostic checks, we have employed statistical tests for the residuals.Table 7 shows the outcomes of the serial correlation and heteroskedasticity tests on the residuals.From Figure 3, it can be inferred that when one standard deviation shock is applied to the gold price, the response of the gold price to its own shock is a dramatic decline from period one to period two, followed by nearuniformity from period three to the end.Although the response of gold price to a one-standard-deviation shock to covid-19 instances is a significant, moderate increase, it remains the same from period two to the conclusion.

3-7. Results of Variance Decomposition:
The variance decomposition reveals the forecast error variance.In the short term, for example over two periods, the impulse or shock to gold price accounts for 99.72 percent of the variation in the gold price in the variance decomposition of gold price (own shock), but only 0.11 percent in the variance decomposition of covid-19 (shock of another variable).Table 8 (b) demonstrates that 99.86 percent of the variance in covid-19 explains 0.14 percent of the variance in the gold price during the fifth period.0.24 percent of the volatility in gold price during the tenth lag period is explained by 99.76 percent of the variance in COVID-19, This explains that there is an insignificant short-term causal relationship between COVID-19 and gold price, as indicated by the Wald test results.Depending on the outcomes of the Granger causality tests, Table (8) reveals that Test 1, F-test is equal to 5.22 with a P-value equal to 0.0059; therefore, the null hypothesis can be rejected, and we conclude that the gold price is influenced not only by itself but also by the Covid-19 pandemic.Although.For Test 2, the F-test is 0.0196 with a p-value equal to 0.980; therefore, the null hypothesis cannot be rejected, and we infer that the Covid-19 pandemic is exclusively influenced by itself and not by gold price.

Discussions:
The global economy and finances in the world are faced the effect of coronavirus especially during 2020, whereas, the result of this virus's proliferation, the impact of the virus on human existence is growing increasingly widespread.As a result, investors may continue to seek sanctuary in gold for a considerable amount of time if the COVID-19 pandemic causes a global recession.Consequently, the goal of this study is to use the Error Correction Model (ECM) to assess and examine the relation between the COVID-19 pandemic and the gold price, Error correction models can be used to depict long-run equilibrium of cointegrated nonstationary time series.This sample analysis utilized the daily number of COVID-19 pandemic cases and the daily gold price in the world.The results indicate that COVID-19 and gold prices are cointegrated over the long term.Despite this, the residual lagged coefficient in the model is negative and statistically significant, indicating that COVID-19 is causally related to gold prices in the long run.This coefficient suggests that 17.6 percent of the imbalance in the gold price will be adjusted daily by changes in the COVID-19 variable.In addition, it is possible to conclude that when one standard deviation shock is applied to the gold price, the response to its own shock (by the gold price) is a decline in gold price over time.Although a one-standard-deviation shock to covid-19 cases results in a large increase in gold price, this trend continues from period two until the end.In the fifth period, 99.86 percent of the volatility in covid-19 explains 0.14 percent of the variance in the gold price, therefore, we argued that there is no causal relationship between covid-19 and gold price in the short term, as indicated by the Wald test results in table (6).Finally, the Granger causality results show that the gold price is influenced not only by itself but also by the Covid-19 pandemic.Future studies will be able to employ a variety of factors in many fields and uncover their integration.Incorporating tourism demand and industry variables in modeling is another approach that might be examined.

Figure ( 1 )
Figure (1): Time Series Plots of the Variables (R-Studio v.4.5.1)In order to construct a suitable model, all series included in the study must have been stationary; consequently, the researchers must examine the structure of unit-root of the data.Even though the following graph offers us a general sense of the stationarity structure of the series, to test for unit roots, we applied the Augmented Dickey-Fuller test to the series.Table1displays the results of the ADF test applied to the levels and the first differences of the series.Table (1): Unit root results of lag variables Variables Lags Test Value p-value , and we can argue that the gold price model could have been a good model because that reduces serial correlation.3-6.Impulse Response Function Results: Impulse response functions (IRFs) illustrate how a system responds to exogenous shocks (Inoue, & Kilian, 2013: 1-13).Figures 3 depict the estimated COVID-19 and gold price response functions, respectively.the lines represent point estimates of the IRFs.

Figure ( 3 )
Figure (3): Impulse Response Function (R-Studio v.4.5.1)FromFigure3, it can be inferred that when one standard deviation shock is applied to the gold price, the response of the gold price to its own shock is

.doi.org/10.25130/tjaes.19.64.1.36 560
2.19) where ∑ , X t = (x 1t , x 2t ,..., x dt ́ is an exogenous vector of variables, and c = (c 1 , c 2 , ___, c n ́ is a vector of constant.a = (a 1 , a 2 ,..., a d ́ is a cointegrating vectors matrix, and Prepared by researchers based on the statistical program. Table (1): Unit root results of lag variables illustrates the results of the selection criterion.Table(2): Order selection criteria of model After identifying an order Lag and cointegration test, we proceed to model estimate.The model estimation findings are presented in the table below.
Table (3): Test of Johansen's maximum likelihood for relationships of multiple co-integrating 3-4.Estimation of ECM model:

Table ( 7
): residuals diagnostic checks Prepared by researchers based on the statistical program.The heteroskedasticity test results in Table7indicate that the application of the model for the series is homoskedasticity

Table (
Prepared by researchers based on the statistical program.Finally, Granger causality is employed to examine two tests.The first is to examine the null hypothesis that the COVID-19 pandemic does not Granger Cause gold prices.The second experiment tests the hypothesis that the gold price does not Cause the COVID-19 pandemic.Prepared by researchers based on the statistical program.