Document Type : Research Paper

Authors

• Manijeh Mahmoodi ¹
• Mohammad Reza Salehi Rad ²

¹ Ph.D. Candidate in Statistics, Allameh Tabataba’i University, Tehran, Iran

² Associate Professor, Department of Statistics, Allameh Tabataba’i University, Tehran, Iran

Abstract

Modeling plays a crucial role in economic and financial research, forming the foundation for analysis, decision-making, policy development, and planning. Assumptions made during the modeling process are particularly important for estimation and forecasting, as they can significantly influence the results. One of the most widely used classical time series models is the autoregressive model, in which current values are expressed as a finite linear combination of past values. However, in real-world scenarios, many variables interact with each other. To capture these interdependencies, vector time series models-an important class of multivariate time series models-are employed. The vector autoregressive (VAR) models are commonly used in economic and financial modeling. VAR models are typically formulated assuming that the shocks (or noise terms) follow a normal distribution. However, in economic and financial contexts-particularly in macroeconomics-shocks do not often follow a symmetric distribution. The present article focused on a VAR model in which the shocks follow a multivariate skew normal (MSN) distribution. The expectation conditional maximization (ECM) algorithm were used to estimate the model parameters. Finally, using real-world datasets from Canada and Iran-where the shocks exhibit skewness-the study found that the VAR model with MSN-distributed shocks is more efficient than the VAR model with multivariate normal distribution for shocks.

Introduction

The multivariate normal distribution is commonly used to model shocks in VAR models. However, in fields such as economics, finance, the stock market, and medicine, various factors can introduce skewness (asymmetry) into the shocks, resulting in non-symmetric distributions. In such cases, the normal distribution becomes an inappropriate choice. To address this, the multivariate skew distribution-which accounts for asymmetry-should be used for modeling shocks. Despite its relevance, this approach has received limited attention in previous research. The family of multivariate skew distributions is broad and complex, posing practical challenges. The present study aimed to test the VAR model in which the shocks follow a multivariate skew normal (MSN) distribution, using the real-world datasets from Canada and Iran.

Materials and Methods

Consider the VAR model of order p:
 
 
 
where   ,  o is location parameter,  is the scale parameter, and S is the skew parameter. Its density function is given by:
 
where  , , , and   denotes density function of MN, while  denotes the standard cumulative distribution function.
To find the maximum likelihood estimates of the parameters requires derivatives of the log-likelihood function; however, these derivatives do not have closed-form expressions. Therefore, they must be approximated using numerical methods. The maximum likelihood estimators were obtained via the expectation conditional maximization (ECM) algorithm. Based on the hierarchical representation of the multivariate skew normal distribution, we have:
,
(o, I)
,
The logarithm of the conditional likelihood function in VAR(p) can be formulated as:
 
 
where . The expectation of the logarithm of the conditional likelihood is denoted by , and the steps for the maximizing  are as below:

Step 1: Assuming no skewness, estimate the initial values for the coefficients and scale parameters.
Step 2

 
 
                    

Step 3

 
 

Step 4

 
 
 

Step 5: Repeat steps 2 to 5 until the convergence condition of the algorithm is established:

 

Results and Discussion

The performance of the proposed method was evaluated using two real-world datasets from Canada and Iran. The Dickey-Fuller test was employed to determine the stationarity of the data, while the Akaike Information Criterion (AIC), Hannan-Quinn Criterion (HQC), and Schwartz Bayesian Criterion (BIC) were used to select the order of the VAR model. The Canadian dataset consists of seasonally adjusted employment and unemployment data from 1980 to 2000. According to the Mardia test, the shocks follow a multivariate skew normal (MSN) distribution. Therefore, we estimated the parameters of the VAR(1) model. The AIC and BIC results are presented in Tables 1 and 2, respectively.
 Table 1. The Estimated Parameters of Model for Stationary Differenced Canadian Data




Estimation


 


 




 


0.9103


0.7094




 


0.2139


-.0149




 


-0.2018


-0.4663




 


0.3303


0.0255




 


-


0.0703




 


-


0.1066




 


0.1646


0.1595




 


-0.09106


-0.1003




 


0.09106


-0.1003




 


0.11400


0.0977




Table 2. The AIC and BIC for Data




Distribution


AIC


BIC


-Log-Like




 


282.0266


286.836


139.011




 


260.6781


265.4915


128.339




The collected data on agriculture, forestry, and fishing (AFF) and employment of women (EW) in Iran span the years 1991 to 2021. According to the Mardia test, the shocks follow a multivariate skew normal (MSN) distribution. The parameter estimates for the VAR(1) model, along with the AIC and BIC values, are presented in Table 3.
   Table 3. The Estimated Parameters of Model for Stationary Differenced Iranian Data




Estimation


 


 




 


-


0.1104




 


-


-0.0798




 


1.2397


1.2267




 


0.2748


0.2836




 


0.2748


0.2836




 


0.3855


0.3796




AIC


146.1145


145.9185




BIC


148.8491


148.6531




The fitted model can be formulated as follows:
 
 
where    and     represent  AFF  and EW, respectively.
According to the AIC and BIC criteria presented in Tables 2 and 3 for the Canadian and Iranian data, the VAR model with MSN-distributed shocks is more appropriate than the VAR model with MN-distributed shocks.

Conclusion

Considering the VAR(p) model with shocks following a multivariate skew normal (MSN) distribution, the present study employed the maximum likelihood method and the ECM algorithm to estimate the model parameters. Based on two real-world datasets from Canada and Iran, the findings showed that the VAR model with MSN-distributed shocks provides a better fit than the model with MN shocks when the shocks exhibit skewness.

Keywords

• Vector Autoregressive
• Skewness
• Multivariate Skew Normal
• Maximum Likelihood Estimation
• Expectation Conditional Maximization Algorithm

Main Subjects

• SVAR