Frontiers in Business, Economics and Management ISSN: 2766-824X | Vol. 15, No. 1, 2024 411 A New SVAR Model Based on Standardized Standard Asymmetric Exponential Power Distribution Tianyu Long1, * 1Department of Mathematics, Columbia University, New York, United States *Corresponding author email: tl3083@columbia.edu Abstract: This paper introduces a novel Structural Vector Autoregressive (SVAR) model incorporating non-Normal marginal errors described by the Standardized Standard Asymmetric Exponential Power Distribution (SSAEPD). This innovation allows the model to effectively capture the inherent skewness and kurtosis typically observed in financial data. Utilizing Markov Chain Monte Carlo (MCMC) techniques for simulating error terms and Maximum Likelihood Estimation (MLE) for parameter estimation in MATLAB, the model's robustness is significantly enhanced. We employ Impulse-Response Function (IRF) and Variance Decomposition (VD) for structural analysis to examine the relevance of classic Keynesian theory in today's US economic context. Our empirical findings suggest that the SSAEPD-enhanced SVAR model not only outperforms traditional SVAR models in capturing data characteristics but also aligns more closely with Keynesian principles. The results underscore the continued applicability of the IS-LM framework in understanding contemporary economic dynamics, demonstrating the model's superior performance in variance decomposition and structural analysis. Keywords: Structural Vector Autoregressive Regression (SVAR); Standardized Standard Asymmetric Exponential Power Distribution (SSAEPD); Classic Keynesian Theory, Impulse Response Function (IRF). 1. Introduction The Vector Autoregression Model (VAR) is a cornerstone linear methodology that effectively captures the complex dynamics present in multiple time series. Its utility and flexibility as a statistical tool are well-documented, yet it is inherently limited as a reduced form model. Specifically, the VAR model does not offer explanations for the instantaneous relationships among variables [1, 2]. To overcome this limitation, the Structural Vector Autoregression model (SVAR) was developed. SVAR enhances our understanding by interpreting the instantaneous correlations among variables through a carefully designed set of restrictions. One of the distinct advantages of the SVAR model over other simultaneous equation models is its identification strategy. While other models impose extensive restrictions on autoregressive coefficients—often more than necessary—the SVAR model applies just enough constraints on error terms to identify the system. This minimalist approach is preferable since economic theory frequently offers limited guidance on these coefficients, thus making data-driven determination a more practical choice [3]. Historically, since Sims' original proposal, the SVAR model has evolved into a significant tool for macroeconomic analysis, demonstrating its worth across various empirical studies [4]. Many researchers have focused on implementing restrictions within different SVAR systems guided by economic theory. For instance, restrictions based on traditional Keynesian models have been applied to analyze macroeconomic fluctuations in the US [5]. Moreover, the imposition of restrictions on the long-run effects of shocks has allowed for the exploration of dynamic responses to demand and supply disturbances in bivariate SVAR systems [6]. In conventional applications, structural shocks within SVAR models are assumed to be serially uncorrelated and mutually orthogonal white noise disturbances, typically modeled using a normal distribution. However, this assumption does not hold well in practice where financial data often display skewness, fat tails, and asymmetric kurtosis, attributes poorly captured by normal distributions. Addressing this challenge, we introduce the Standardized Standard Asymmetric Exponential Power Distribution (SSAEPD) as a novel approach to model error terms in SVAR, enhancing the model's capability to account for these data characteristics [7]. Different from previous studies, our approach extends the SVAR model by focusing on improving the fit of residuals rather than merely refining the system’s identification. In this paper, SSAEPD margins are integrated using a Gaussian Copula in what we term the SVAR-GC SSAEPD model. We employ Maximum Likelihood (ML) for parameter estimation and use Markov Chain Monte Carlo (MCMC) for error term simulation [8]. Additionally, we implement the Kolmogorov- Smirnov (KS) test for model diagnostics and utilize Impulse- Response Function (IRF) and Variance Decomposition (VD) for structural analysis. Our research aims to reassess the validity of the Keynesian theory in the current U.S. economic framework under the assumption of non-Normal error terms [9]. The paper is organized as follows: Section 2 presents the model and methodology, Section 3 describes the simulation analysis, Section 4 discusses empirical results, and Section 5 concludes with future research directions. 412 Table 1. Empirical Applications about SVAR [10]. Author Contents Sample Data Bernanke (1986) Money-income correlation U.S. 1954: Q1-1984: Q4 Blanchard and Quah (1989) Effects of demand and supply on GNP and unemployment U.S. 1950: Q2-1987: Q4 Dalsgaard and Serres(1999) Source of government deficit ratio of 4 economic variances EU 1996.01-1997.12 Monticelli et.al. (1999) Effects of monetary policy on macroeconomy U.S. and 3 European countries 1978: Q1-1997: Q4 Breitung (2000) IS-LM model U.S. 1970: Q1-1997: Q4 Dungey and Pagan (2000) Effects of 5 types of shocks on macroeconomy Australia and U.S. 1980: Q1-1998: Q4 Gottschalk and Zandweghe (2001) Source of output fluctuations Germany 1962: Q1-1998: Q4 Garc´ia et al. (2002) Core inflation EU 1981.01-2000.12 Ho¨ppner (2002) Effects of fiscal policy on macroeconomy Germany 1970: Q1-2000: Q4 Mohr (2002) Effects of fiscal policy on macroeconomy Germany 1970: Q1-2000: Q2 Bankasi (2003) Effects of fiscal policy on macroeconomy Turkish 1987: Q1-2005: Q4 Bruneau and Bandt (2003) Effects of ficsal and moentary policy on macroeconomy EU 1979: Q1-2000: Q2 Djivre and Ribon (2003) Effects of monetary policy on macroeconomy Israeli 1990: Q1-1999: Q4 Giacinto (2003) Regional effects of monetary policy on macroeconomy U.S. 1958: Q2-2000: Q4 Michal (2003) Source of natural interest rate U.S. 1960.01-2002.06 Claeys (2004) Moentary-budgetary policy correlation OECD mid 1960s -2001: Q2 mids 1970s-2001: Q2 Peersman and Straub (2004) GDP-GDP deflator- interest rate- hours EU 1982: Q1-2002: Q4 worked-real wage correlation Giordano (2005) Effects of fiscal policy on macroeconomy Italy 1982: Q2-2003: Q4 Ferna´ndez (2006) Effects of fiscal policy on macroeconomy Spain 1980: Q1-2004: Q4 Mitra (2006) Effects of government investment on private investment India 1969.01-2005.12 Simorangkir (2006) Openness-economic performance correlation Latin America and East Asia 1965.01-1989.12 Sousa and Zaghini (2006) Effects of global monetary policy on macroeconomy G5 countries 1980: Q1-2001: Q4 GUAY (2007) Effects of structural policy settings on macroeconomy OECD 1975: Q1-2004: Q4 Elbourne (2008) Housing market-monetary policy correlation UK 1987.01-2003.05 Mirdala (2008) Effects of macroeconomic shocks on interest rate and GDP EU 1995.01-2007.12 Dungey and Pagan (2009) NK model Austrilia and U.S. 1980:Q1-2006:Q4 Afonso and Sousa (2009) Effects of fiscal policy on macroeconomy Portugal 1979:Q1-2007:Q4 Beetsma et.al. (2009) Effects of government spending on GDP EU 1965:Q1-2004:Q4 Bencˇ´ik (2009) Effects of fiscal policy on business cycle Slovakia 1997.01-2007.12 Estrada and Montero (2009) Effects of R&D investment on GDP Spain and 6 developed countries 1970.01-2006.12 Inoue and Hamori (2009) Source of exchange rate fluctuations India 1999.01-2009.02 Mirdala (2009) Effects of monetary policy on interest rate EMU 1999:Q1-2008:Q3 Partridge and Rickman (2009) Source of labor market fluctuations Canada 1976-2003 Alexandru (2010) Shadow economy-unemlpoyment rate correlation U.S. 1980:Q1-2009:Q2 Loria et.al. (2010) Effects of monetary approach on exchange rate Mexico 1994.01-2007.12 Ravnik and Z˘ilic´ (2010) Effects of fiscal policy on macroeconomy Croatia 2001.01-2009.12 Ahmed and Wadud(2011) Effects of oil price on macroeconomy Malaysian 1986.01-2009.12 Hausman et.al. (2012) Effects of acreage supply on crop prices U.S. 1956.01-2007.12 Zhao (2010) Source of price fluctuation of real estate China 1999:Q1-2009:Q2 Huang (2010) Effects of monetary policy on real estate price China 2005.07-2009.09 Zhao (2008) Source of core inflation China 1986.01-2007.12 Jiang (2009) Regional effects of monetary policy on macroeconomy China 1978-2006 Liang (2011) Effects of exchange rate and export tax rebates on export China 1984.01-2009.12 Lv (2009) Source of demand & supply shocks China 1992:Q1-2008:Q3 413 Table 2. Extensions and Applications of the Normal Distribution [11]. Authors Distributions and their Applications De Moivre (1738) Normal distribution Gauss (1809) Normal applied in astronomy Subbotin (1923) EPD Aitchison J. and Brown J.A.C. (1957) Lognormal distribution Azzalini (1986) SEPD Zolotarev V.M. (1986) Stable distribution Bolleslev (1987) Student-t distribution Fernandez et al. (1995) Modified SEPD Swamee P.K. (2002) Near lognormal distribution Aybeo and Kozubowski (2004) SEPD in finance DiCiccio and Monti (2004) Properties of MLE of the SEPD Zhu and Zinde-Walsh (2009) AEPD Notes: EPD=Exponential Power Distribution; SEPD=Skewed Exponential Power Distribution. AEPD=Asymmetric Exponential Power Distribution. 2. Model and Methodology 2.1. SVAR-MGC SSAEPD Model Based on the AB-type SVAR model, we propose a new SVAR(p) model with non-Normal margins. The structure form of this new model is (denoted as SVAR(p)-MGC SSAEPD): 𝐴𝑌 Γ 𝑌 Γ 𝑌 ⋯ Γ 𝑌 𝐵𝜀 , 𝑡 1,2, … 𝑇 (1) where 𝐴 1 𝑎 𝑎 𝑏 1 𝑏 𝑐 𝑐 1 Γ 𝑎 𝑎 𝑎 𝑏 𝑏 𝑏 𝑐 𝑐 𝑐 , 𝑖 1,2, … , 𝑝 𝐵 𝑎 0 0 0 𝑏 0 0 0 𝑐 𝑌 𝑦 𝑦 𝑦 , 𝜀 𝜀 𝜀 𝜀 𝜀 , 𝜀 , 𝜀 ∼ 𝑀𝐺𝐶 𝑆𝑆𝐴𝐸𝑃𝐷 𝛼 , 𝑝 , 𝑝 , 𝛼 , 𝑝 , 𝑝 , 𝛼 , 𝑝 , 𝑝 , 𝑅 (2) 𝜀 ∼ 𝑆𝑆𝐴𝐸𝑃𝐷 𝛼 , 𝑝 , 𝑝 , 𝛼 ∈ 0,1 , 𝑝 0, 𝑖 1,2,3, (3) where MGC_SSAEPD is the joint probability density function (PDF) of errors 𝜀 . 𝑐 𝜀 , 𝜀 , 𝜀 ; 𝑅 | | / exp (4) where 𝜍 𝜍 , 𝜍 , 𝜍 , 𝜍 Φ 𝐹 𝜀 , 𝑖 1,2,3, For equation (4), the 𝑐 𝜀 , 𝜀 , 𝜀 ; 𝑅 is the probability density function (PDF) of multivariate Gaussian copula (MGC) with correlation matrix 𝑅 [12]. Φ (.) is the cumulative density function (CDF) of standard Normal. 𝐹 (.) is cumulative density function (CDF) the of SSAEPD. By premultiplying with A−1, the reduced form of the new SVAR model is: 𝑌 Π 𝑌 Π 𝑌 ⋯ Π 𝑌 𝑢 (5) where Π = A−1 𝐹 . Moreover, the reduced-form disturbances 𝑢 can be modeled as an interdependent system of linear equations of the residuals 𝜀 in its structure form (see equation (6)). 414 𝐴𝑢 𝐵𝜀 (6) To identify the structure form parameters, we must place restrictions on the parameters of 𝐴 and 𝐵 matrices. For this model, a set of n(n + 1)/2 restritions should be imposed, leaving overall 2n2 − n(n + 1)/2 free elements. 2.2. Standardized Standard AEPD (SSAEPD) The probability density function (PDF) of a random variable 𝜀 with Standardized Standard AEPD (i.e., 𝜀 ∼ SSAEPD (𝛼 , 𝑝 , 𝑝 )) is: 𝑓 𝜀 𝛿 ∗ 𝐾 𝑝 exp ∗ , if 𝜀 ⩽ 𝛿 ∗ 𝐾 𝑝 exp ∗ , if 𝜀 (7) And its cummulative density function (CDF) of SSAEPD is: 𝐹 𝜀 𝛼 1 𝐺 | | ∗ ; , if 𝜀 𝛼 1 𝛼 𝐺 | | ∗ ; , if 𝜀 (8) Where 𝜔 1 𝛼 / / 𝛼 / / (9) 𝛿 1 𝛼 / / 𝛼 / / 1 𝛼 / / 𝛼 / / (10) 𝐾 𝑝 / / , 𝐾 𝑝 / / (11) where 𝑝 and 𝑝 are the parameters which control the left tails and right tails respectively. 𝛼 is the skewness parameter of SSAEPD. The SSAEPD can be reduced to many distributions. If 𝛼 = 0.5 and 𝑝 = 𝑝 , it becomes the EPD. If 𝑝 = 𝑝 , the AEPD reduces to Skewed EPD [13, 14]. If 𝛼 = 0.5, the AEPD becomes symmetric EPD [15]. If 𝛼 = 0.5 and 𝑝 = 𝑝 = 1, it reduces to the Laplace distribution. If 𝛼 = 0.5 and 𝑝 = 𝑝 = 2, it becomes the Normal distribution and then the new model will be reduced to the so-called AB-model [16]. 2.3. Maximum Likelihood Estimation Method of Maximum Likelihood Estimation (MLE) is used to estimate this new model. The reduced form residuals 𝑢 corresponding to structure form disturbances 𝜀 have the form 𝑢 = 𝐴 𝐵𝜀 . 𝜀 is assumed to be uncorrelated white noise, 𝜀 , 𝜀 , 𝜀 ∼ MGC_SSAEPD ({𝛼 , 𝑝 , 𝑝 , 𝐼 ). The log likelihood function of the new model is shown as follows. 𝐿 𝑦 , 𝑦 , 𝑦 ; 𝜃 ∑   ln 𝑓 𝑦 , 𝑦 , 𝑦 ln |𝐴| ln |𝐵| ∑   ln 𝑓 𝜀 , 𝜀 , 𝜀 , ln |𝐴| ln |𝐵| ∑   ln 𝑐 𝜍 ; 𝐼 ∑   ln 𝛿 ∗ 𝐾 𝑝 exp ∗ , if 𝜀 ⩽ , 𝛿 ∗ 𝐾 𝑝 exp ∗ , if 𝜀 , ∑   ln 𝛿 ∗ 𝐾 𝑝 exp ∗ , if 𝜀 ⩽ , 𝛿 ∗ 𝐾 𝑝 exp ∗ , if 𝜀 , ∑   ln 𝛿 ∗ 𝐾 𝑝 exp ∗ , if 𝜀 ⩽ , 𝛿 ∗ 𝐾 𝑝 exp ∗ , if 𝜀 . (12) 𝜃 𝑎 , 𝑎 , 𝑎 , 𝑏 , 𝑏 , 𝑏 , 𝑐 , 𝑐 , 𝑐 , 𝛼 , 𝑝 , 𝑝 are the parameters of the model to be estimated. All the numerical computations are run in MatLab. 2.4. Impulse Response Function (IRF) Once the VAR system is estimated, we apply the Impulse- Response Function (IRF) to trace out the effect of exogenous shocks on realizations of the random variable across time. By Wold (1960)'s scheme, VAR (p) in equation (5) can be written as VMA (∞) evaluate the relative importance of error shocks [17]. 𝑌 ∑   Φ 𝑢 (13) Φ ∑   Φ Π (14) The s-period ahead forecast value is [17]: 𝑌 ∑   Φ 𝑢 (15) The (i, j) th elements of the matrices Φ , regarded as a function of s, trace out the expected responses of 𝑦 , to a unit change in 𝑦 , , holding constant all past values of 𝑦 . Equation (16) is called the Impulse-Response Function (IRF) [18]. Φ , , , (16) For the AB-type SVAR model, the relation of dependent variables to the reduced form residuals is given by 𝐴𝑢 𝐵𝜀 . Therefore, the VMA (∞) form of SVAR model can be shown as 𝑌 ∑   Ψ 𝜀 (17) where Ψi = A−1BΦi [17]. The impulse responses in a general SVAR model may be obtained from Equation (18): Ψ , , , (18) 415 2.4.1. Variance Decomposition Variance decomposition is also a popular tool for interpreting SVAR model. By separating the variation of an endogenous variable into error shocks, the variance decomposition provides information about the relative importance of each random shocks in affecting dependent variable. Recall the s-step forecast error of 𝑌 from a VAR model is: 𝑌 𝐸 𝑌 ∑   Ψ 𝐸 (19) The variance of the 𝑌 is: Var 𝑌 ∑   Var Ψ 𝐸 ∑   Ψ 𝜎 (20) Given that the 𝜀 are serially uncorrelated and have unit variance, the s-step forecast error variance of variance i is: 𝜎 𝑠 ∑   ∑   Ψ , (21) Dividing the preceding terms by 𝜎 gives the percentage contribution of variable to the s-step forecast error variance of variable i, 𝜔 → 𝑠 ∑   , ∑   ∑   , (22) 3. Simulation Results In this section, we run the simulation for this new model to check the validity of the MatLab program. Simulation results show our program is valid and can be used for empirical analysis. 3.1. Simulation for Model Estimation The results from both MatLab and EViews (Version 7.2) are compared with the true values. The simulation process is as follows. 1. Select a set of true parameter values for 𝜃 𝑎 , 𝑎 , 𝑎 , 𝑏 , 𝑏 , 𝑏 , 𝑐 , 𝑐 , 𝑐 , 𝛼 , 𝑝 , 𝑝 2. Generate error terms 𝜀 from the joint density of 𝜀 , 𝜀 , 𝜀 ~ 𝑀𝐺𝐶_𝑆𝑆𝐴𝐸𝑃𝐷 𝛼 , 𝑝 , 𝑝 , 𝑅 with the restriction that the error terms 𝑢 are uncorrelated (i.e. the covariance matrix 𝑅 is I3×3). SSAEPD (𝛼 , 𝑝 , 𝑝 ) is used as the proposal density for 𝜀 . Another method is used to get the temporate draws of SSAEPD ( 𝛼, 𝑝 , 𝑝 ) [11]. MCMC is used to draw the tri-variate errors from 𝜀 , 𝜀 , 𝜀 ~ 𝑀𝐺𝐶_𝑆𝑆𝐴𝐸𝑃𝐷 𝛼 , 𝑝 , 𝑝 , 𝑅 . 3. Generate data 𝑌 by equation (23) with the initial values 𝑌 0.1, 0.1, 0.1 , 𝑖 3, 2, 1, 0 . 𝐴𝑌 Γ 𝑌 Γ 𝑌 Γ 𝑌 Γ 𝑌 𝐵𝜀 , 𝑡 1,2, … 𝑇 (23) The simulation results are listed in Table 3. We find out that all estimates from MatLab and Eviews are close to the true values. Hence, we may conclude that our program in MatLab is valid and can be used to analyze empirical data. Table 3. Simulation results. Par. T M E T M E T M E T M E α1 0.5 0.4860 - 0.5 0.4956 - 0.5 0.5105 - 0.5 0.5437 - p11 2 1.9291 - 2 1.9380 - 1.5 1.5584 - 1.5 1.5652 - p12 2 2.0351 - 2 1.9203 - 2 1.9373 - 2 1.8305 - a01 0.2 - - 0.2 - - 0.2 - - 0.2 - - a02 0.1 0.0793 0.0441 0.1 0.0910 0.1013 0.1 0.0876 0.0783 0.1 0.1142 0.0982 a03 1 - - 0.8 - - 1 - - 0.7 - - a11 -0.2 -0.1949 -0.2555 -0.2 -0.1672 -0.3567 -0.2 -0.1967 -0.1892 -0.2 -0.1652 -0.1872 a12 0.2 0.2300 0.2750 0.2 0.1950 0.2587 0.2 0.2000 0.1732 0.2 0.2210 0.1979 a13 -0.3 -0.2971 -0.3478 -0.3 -0.3005 -0.3537 -0.3 -0.3139 -0.2793 -0.3 -0.3003 -0.2987 a21 0.2 0.2181 0.2673 0.2 0.2272 0.3023 0.2 0.2089 0.1394 0.2 0.1931 0.1872 a22 -0.2 -0.1990 -0.2541 -0.2 -0.2063 -0.2871 -0.2 -0.1999 -0.1674 -0.2 -0.1908 -0.1689 a23 0.1 0.0873 0.1293 0.1 0.1195 0.1589 0.1 0.0928 0.0784 0.1 0.1000 0.0987 a31 0.1 0.1184 0.1207 0.1 0.0849 0.1490 0.1 0.0713 0.0821 0.1 0.0873 0.0682 416 a32 -0.2 -0.2044 -0.1931 -0.2 -0.1917 -0.1961 -0.2 -0.2279 -0.1782 -0.2 -0.2087 -0.1982 a33 0.2 0.1989 0.1476 0.2 0.1955 0.1676 0.2 0.1952 0.1872 0.2 0.1868 0.1680 a41 -0.1 -0.1072 -0.1201 -0.1 -0.1055 -0.0627 -0.1 -0.1037 -0.0982 -0.1 -0.0961 -0.0892 a42 0.2 0.2096 0.2091 0.2 0.2248 0.2173 0.2 0.1779 0.1572 0.2 0.1974 0.1808 a43 -0.2 -0.2210 -0.1918 -0.2 -0.2116 -0.1853 -0.2 -0.2310 -0.1782 -0.2 -0.2019 -0.1972 α2 0.5 0.4907 - 0.5 0.5374 - 0.6 0.5268 - 0.6 0.5540 - p21 2 1.9873 - 2 2.2210 - 2 1.7920 - 2 1.9074 - p22 2 2.0491 - 2 1.8782 - 1.5 1.7911 - 1.5 1.6124 - b01 0.2 0.2084 0.2154 0.2 0.2070 0.2781 0.2 0.1908 0.2012 0.2 0.1956 0.1782 b02 0.1 0.1023 0.0766 0.1 0.1014 0.1331 0.1 0.1260 0.0982 0.1 0.0994 0.0987 b03 1 1.0034 1.0036 0.8 0.7844 0.7922 1 1.0006 0.9989 0.8 0.8019 0.7972 b11 -0.2 0.2307 0.2165 0.2 0.2128 0.2160 0.2 0.2441 0.1982 0.2 0.1911 0.1672 b12 0.1 -0.3020 -0.3351 -0.3 -0.2578 -0.3699 -0.3 -0.2818 -0.2541 -0.3 -0.3246 -0.2981 b13 -0.1 0.2298 0.2643 0.2 0.2015 0.2752 0.2 0.1993 0.1362 0.2 0.2041 0.2492 b21 -0.2 -0.1788 -0.2202 -0.2 -0.1941 -0.2280 -0.2 -0.2348 -0.1872 -0.2 -0.2062 -0.2301 b22 0.1 0.0873 0.1341 0.1 0.1044 0.1059 0.1 0.1242 0.0792 0.1 0.0945 0.0876 b23 -0.1 -0.0798 -0.0853 -0.1 -0.1097 -0.1000 -0.1 -0.1086 -0.0982 -0.1 -0.0900 -0.0697 b31 -0.1 -0.1093 -0.1101 -0.1 -0.0896 -0.1188 -0.1 -0.0990 -0.1320 -0.1 -0.1145 -0.0980 b32 -0.1 -0.1360 -0.0986 -0.1 -0.1098 -0.0687 -0.1 -0.0783 -0.0324 -0.1 -0.1179 -0.1392 b33 0.2 0.1821 0.1672 0.2 0.2095 0.1556 0.2 0.1787 0.1592 0.2 0.2032 0.1987 b41 -0.1 -0.0698 -0.0948 -0.1 -0.3205 -0.3147 -0.1 -0.3003 -0.3421 -0.1 -0.3025 -0.3092 b42 0.1 0.0873 0.0809 0.1 0.1021 0.0778 0.1 0.1086 0.0932 0.1 0.0850 0.0897 b43 -0.2 -0.183 0.1672 -0.2 -0.2094 -0.1769 -0.2 -0.2042 -0.1872 -0.2 -0.1942 -0.1797 α3 0.5 0.4911 - 0.5 0.4826 - 0.5 0.4785 - 0.5 0.4801 - p31 2 2.0484 - 2 2.0247 - 2.5 2.5050 - 2.5 2.2595 - p32 2 1.9928 - 2 2.0644 - 2 2.0219 - 2 2.0044 - c01 0.1 0.1237 0.1538 0.1 0.1348 0.1504 0.1 0.1178 0.0672 0.2 0.1415 0.1982 c02 -0.2 - - -0.2 - - -0.2 - - -0.2 - - c03 1 0.9889 0.9921 1.5 1.5013 1.5044 1 1.0103 1.0023 1.5 1.5154 1.4982 c11 0.2 0.2111 0.2660 0.2 0.1732 0.2487 0.2 0.1968 0.1409 0.2 0.1703 0.1672 c12 0.1 0.1347 0.0966 0.1 0.1303 -0.0189 0.1 0.1048 0.1092 0.1 0.0873 0.0786 c13 -0.2 -0.2171 -0.0987 -0.2 0.2164 -0.2434 -0.2 -0.2077 -0.1778 -0.2 -0.1979 -0.1798 c21 -0.2 -0.1930 -0.2723 -0.2 -0.1619 -0.3334 -0.2 -0.1950 -0.1872 -0.2 -0.2633 0.2381 417 c22 0.1 0.0897 0.1484 0.1 0.1178 0.1878 0.1 0.0959 0.0787 0.3 0.3036 0.2987 c23 -0.2 -0.1905 -0.2172 -0.2 -0.2108 -0.2781 -0.2 -0.1730 -0.1492 -0.2 -0.2277 -0.1978 c31 -0.1 -0.0838 -0.1311 -0.1 -0.0792 -0.1503 -0.1 -0.1147 -0.0897 -0.1 -0.1209 -0.0978 c32 0.2 0.1821 0.1808 0.2 0.1785 0.1648 0.2 0.1631 0.1583 0.2 0.2026 0.1682 c33 0.1 0.1412 0.1394 0.1 0.1078 0.1265 -0.1 -0.1037 -0.0872 -0.1 -0.0747 -0.0971 c41 0.1 0.0844 0.0964 0.1 -0.1430 -0.1036 0.1 0.1173 0.1492 0.1 0.1053 0.0989 c42 -0.2 -0.1766 -0.2162 -0.2 -0.1948 -0.1963 -0.2 -0.2122 -0.1872 -0.2 -0.2261 -0.2871 c43 0.1 0.0913 0.1014 0.1 0.1179 0.1226 0.1 0.1243 0.0938 0.1 0.1285 0.1682 Notes: Par. = Parameter; T = True values; M = Estimates by MatLab; E = Estimates by EViews. 3.2. Simulation for Structure Analysis To check the validity of Impulse Response Function (IRF) and Variance Decomposition (VD), the results from MatLab are compared with those from EViews (Version 7.2). The results of Table 4 and 5 show the estimates from our MatLab program are very close to those from EViews. That means our MatLab program can be used for empirical data. The simulation process for the Impulse Response Function (IRF) is as follows: 1. Set true parameter values 𝜃 𝑎 , 𝑎 , 𝑎 , 𝑏 , 𝑏 , 𝑏 , 𝑐 , 𝑐 , 𝑐 , 𝛼 , 𝑝 , 𝑝 . 2. Generate a group of simulation data. 3. Calculate values for Impulse Response Functions (IRFs) and Variance Decomposition (VD) by MatLab and EViews. Table 4. Simulation of Impulse Response Function (IRF). Lag Response of y1 Ψ11 Ψ12 Ψ13 E M E M E M 1 0.6491 0.6491 -0.2095 -0.2095 -0.3673 -0.3661 2 -0.2856 -0.2855 0.2416 0.2416 -0.4362 -0.4350 3 0.3464 0.3463 -0.4103 -0.4102 0.3816 0.3805 4 0.0036 0.0037 -0.0005 -0.0005 -0.0952 -0.0949 5 -0.0835 -0.0842 0.1359 0.1362 -0.3445 -0.3431 6 0.0079 0.0084 0.0036 0.0033 0.1058 0.1057 7 0.1670 0.1663 -0.1560 -0.1554 0.0845 0.0840 8 -0.1012 -0.1010 0.0952 0.0950 -0.1963 -0.1953 9 -0.0016 -0.0016 0.0102 0.0102 0.0440 0.0440 10 0.0529 0.0529 -0.0599 -0.0599 0.0678 0.0675 Lag Response of y2 Ψ21 Ψ22 Ψ23 E M E M E M 418 1 -0.0767 -0.0767 0.8202 0.8202 -0.0039 -0.0040 2 0.2457 0.2457 -0.3298 -0.3298 0.3340 0.3331 3 -0.2952 -0.2951 0.2826 0.2824 -0.3273 -0.3263 4 0.1228 0.1227 -0.1756 -0.1754 0.5346 0.5329 5 -0.2817 -0.2815 0.2058 0.2057 -0.2588 -0.2579 6 0.0898 0.0896 -0.0959 -0.0957 0.1512 0.1508 7 -0.1524 -0.1521 0.1851 0.1849 -0.0664 -0.0662 8 0.1406 0.1403 -0.1960 -0.1958 0.1667 0.1659 9 -0.1529 -0.1525 0.1734 0.1730 -0.1421 -0.1417 10 0.1010 0.1007 -0.1168 -0.1166 0.1557 0.1551 Lag Response of y3 Ψ31 Ψ32 Ψ33 E M E M E M 1 0.2802 0.2802 0.0686 0.0686 1.4969 1.4925 2 0.1507 0.1507 -0.0706 -0.0706 -0.1563 -0.1557 3 -0.4063 -0.4063 0.4383 0.4383 -0.4034 -0.4023 4 0.1832 0.1831 -0.1399 -0.1398 0.6811 0.6790 5 0.0567 0.0567 -0.2081 -0.2081 0.0668 0.0669 6 -0.2392 -0.2394 0.2714 0.2715 -0.3303 -0.3292 7 0.0066 0.0069 -0.0081 -0.0082 0.2836 0.2826 8 0.1024 0.1021 -0.1338 -0.1336 0.0644 0.0641 9 -0.0782 -0.0780 0.0957 0.0954 -0.1479 -0.1472 10 -0.0394 -0.0393 0.0393 0.0393 0.0630 0.0627 Notes: E=EViews (Version 7.2); M=MatLab; Ψij = a response of yi to a shock of εj. True parameters are: a01 = 0.2; a02 = 0; a03 = 0.7 b01 = 0.2; b02 = 0.1; b03 = 0.8 c01 = 0.2; c01 = −0.2; c03 = 1.5 a11 = −0.2; a12 = 0.2; a13 = −0.3 b11 = 0.2; b12 = −0.3; b13 = 0.2 c11 = 0.2; c12 = 0.1; c13 = −0.2. a21 = 0.2; a22 = −0.2; a23 = 0.1. b21 = −0.2; b22 = 0.1; b23 = −0.1. c21 = −0.2; c22 = 0.3; c23 = −0.2. a31 = 0.1; a32 = −0.2; a33 = 0.2. b31 = −0.1; b32 = −0.1; b33 = 0.2. c31 = −0.1; c32 = 0.2; c33 = 0.1. a41 = −0.1;a42 = 0.2; a43 = −0.2. b41 = −0.3; b42 = 0.1; b43 = −0.2. c41 = 0.1; c42 = −0.2; c43 = 0.1. Since EViews (Version 7.2) can only generate IRF of VAR with Normal error terms, we set α1 = 0.5; p11 = p12 = 2 419 Table 5. Simulation of Variance Decomposition (VD). Lag Variance Decomposition of y1 Ψ11 Ψ12 Ψ13 E M E M E M 1 0.7021 0.7031 0.0073 0.0733 0.2247 0.2236 2 0.5406 0.5417 0.1099 0.1102 0.3495 0.3482 3 0.4565 0.4575 0.1984 0.1987 0.3451 0.3438 4 0.4535 0.4545 0.1970 0.1974 0.3494 0.3481 5 0.4151 0.4161 0.1905 0.1910 0.3944 0.3929 6 0.4120 0.4131 0.1891 0.1896 0.3988 0.3973 7 0.4142 0.4152 0.1973 0.1978 0.3884 0.3870 8 0.4059 0.4069 0.1959 0.1964 0.3982 0.3967 9 0.4054 0.4064 0.1958 0.1962 0.3989 0.3974 10 0.4044 0.4054 0.1966 0.1971 0.3990 0.3975 Lag Variance Decomposition of y2 Ψ21 Ψ22 Ψ23 E M E M E M 1 0.0087 0.0087 0.9913 0.9913 0.0022 0.0000 2 0.0691 0.0691 0.8146 0.8151 0.1163 0.1158 3 0.1244 0.1245 0.6983 0.6990 0.1773 0.1765 4 0.1077 0.1078 0.5700 0.5711 0.3223 0.3211 5 0.1413 0.1415 0.5329 0.5339 0.3258 0.3246 6 0.1427 0.1428 0.5261 0.5271 0.3313 0.3300 7 0.1504 0.1506 0.5270 0.5280 0.3226 0.3214 8 0.1539 0.1541 0.5234 0.5245 0.3226 0.3214 9 0.1599 0.1600 0.5192 0.5203 0.3209 0.3197 10 0.1612 0.1612 0.5137 0.5148 0.3251 0.3239 Lag Variance Decomposition of y3 Ψ31 Ψ32 Ψ33 E M E M E M 1 0.0340 0.0340 0.0020 0.0020 0.9642 0.9640 420 2 0.0426 0.0428 0.0041 0.0041 0.9533 0.9531 3 0.0919 0.0924 0.0697 0.0700 0.8384 0.8376 4 0.0879 0.0883 0.0649 0.0652 0.8473 0.8465 5 0.0875 0.0879 0.0764 0.0768 0.8361 0.8363 6 0.0973 0.0978 0.0913 0.0918 0.8114 0.8105 7 0.0952 0.0957 0.0894 0.0899 0.8154 0.8145 8 0.0971 0.0976 0.0933 0.0938 0.8095 0.8086 9 0.0978 0.0983 0.0948 0.0953 0.8074 0.8065 10 0.0980 0.0985 0.0950 0.0955 0.8069 0.8060 4. Empirical Analysis 4.1. Data Following the work from Breitung et. al., we analyze the traditional Keynesian theory, i.e., IS-LM theory (or IS-LM curves) using quarterly U.S. data from 1959 (i) to 2023 (ii) [19]. The output qt is measured by real GDP growth rate, mt is the real monetary base, and it is the three-month interbank interest rate. The vector of dependent variables is 𝑌 𝑞 , 𝑖 , 𝑚 . All data are downloaded from the database of Federal Reserve Economic Data (FRED) maintained at the Federal Reserve Bank of St. Louis. The descriptive statistics are presented in Table 6. Both interest rate (it) and monetary base (mt) are slightly skewed to the right while GDP (qt) is slightly skewed to the left. All three-time series are a little leptokurtic. The P-values of JB normality test is 0, which means under 5% significant level, all the data are not distributed as Normal. Hence, non-Normal error distribution may be better to describe the data. We perform unit root test using Augmented Dicky-Fuller (ADF) test and Phillips-Perron (PP) test for all variables (see Table 7). A unit root is rejected at 5% significance level for all variables, with and without trends. We also perform a Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test for each variable. Results indicate that trend stationarity cannot be rejected for all variables. Table 6. Descriptive Statistics. Mean St. Dev. Skewness Kurtosis P-value (JB) 𝑞 0.01 0.01 -0.29 4.41 0 𝑖 0.05 0.03 0.70 4.01 0 𝑚 0.02 0.01 0.58 4.55 0 Notes: qt = GDP Growth Rate; it = Interest rate; mt = Monetary Base Growth Rate; St. Dev. = Standard Deviation. H0 of the JB test: Data is distributed as Normal. CV (5%) = Critical Value under 5% Significance Level. Table 7. Unit Root Test. Test Determ. Terms GDP (qt) Interest Rate (it) Monetary Base (mt) Test Statistic P-value Test Statistic P-value Test Statistic P-value ADF Intercept -7.18 0 -6.47 0 -6.89 0 Trend and intercept -10.88 0 -6.60 0 -7.11 0 None -4.52 0 -6.48 0 -4.13 0 PP Intercept -11.04 0 -11.50 0 -6.89 0 Trend and intercept -11.19 0 -11.51 0 -7.11 0 421 None -7.98 0 -11.52 0 -2.43 0.02 KPSS Intercept - 0.40 - 0.22 - 0.48 Trend and intercept - 0.05 - 0.12 - 0.15 Notes: H0 of the ADF test: Data has a unit root. Or data is not stationary. H0 of the PP test: Data has a unit root. H0 of the KPSS test: Data is stationary. 4.2. Model Specification: Lag Length Selection The lag length for a SVAR (p) model may be determined using model selection criteria. The general approach is to fit a SVAR (p) model with orders p = 1, 2, ..., pmax and choose the value of p which minimizes some model selection criteria. The three most common criteria are Akaike Information Criterion (AIC), Schwarz-Bayesian Information Criterion (BIC) and Hannam-Quinn Information Criterion (HQ) [20]. Table 8 shows that all three criteria are minimized when the lag length is determinded as 4. That means, the SVAR (4) model can fit our data better. Table 8. Results of Model Selection Criteria. Model AIC BIC HQ Rank SVAR (1)-MGC_SSAEPD -21.46(2) -21.04(1) -21.50(2) 2.67 SVAR (2)-MGC_SSAEPD -19.95(4) -19.39(4) -20.00(4) 4 SVAR (3)-MGC_SSAEPD -20.59(3) -19.89(3) -20.65(3) 3 SVAR (4)-MGC_SSAEPD -21.59(1) -20.75(2) -21.67(1) 1.33 4.3. Model Estimation In this section, we estimate our new model for empirical data. To identify the structural form parameters, some restrictions must be placed on the parameter matrices A and B. Based on related studies [21], we use a small macro system from Keynesian arguments to specify the following relations between the reduced form residuals and the structural innovations: 𝑢 𝑎 𝑢 𝑏 𝜀 (IS curve) 𝑢 𝑎 𝑢 𝑎 𝑢 𝑏 𝜀 (inverse LM curve) 𝑢 𝑏 𝜀 (money supply rule) 4.3.1. Empirical Results The estimates results are reported in Table 9. For the output (qt), the fat-tail phenomenon is strong since both tail parameters (pˆ11 = 1.4912, pˆ12 = 1.5230) are much lower than 2. However, the skewness is not obvious since the estimate of the skewness parameter α1 is around 0.5. Same is true for both interest rate (it) (αˆ2 = 0.5082, pˆ21 = 0.9866, pˆ22 = 0.8218) and monetary base (mt) (αˆ3 = 0.5448, pˆ31 = 1.4211, pˆ32 = 1.0774). Compared to Normal, SSAEPD can capture the skewness and asymmetric kurtosis of the sample data. For comparison, we also estimate SVAR (4)-MGC Normal model by setting pi1 = pi2 = 2 and αi = 0.5 (i = 1, 2, 3), which is the traditional SVAR model advocated by some studies from Sims [3] and Bernanke [1]. We can find out that the estimates in SVAR are quite different from those in new model, especially for those in structural model. This discovery implies that the specification of margins is an important determinant for SVAR estimation. Table 9. Empirical Estimates. SVAR (4)-MGC_SSAEPD SVAR(4)-MGC_Normal qt it mt qt it mt α1 0.5243 α2 0.5082 α3 0.5448 α1 α2 α3 p11 1.4912 p21 0.9866 p31 1.4211 p11 p21 p31 p12 1.5230 p22 0.8218 p32 1.0774 p12 p22 p32 a01 -0.0302 b01 -0.0937 c01 a01 0.2653 b01 -0.3709 c01 a02 - b02 0.2357 c02 a02 b02 0.2490 c02 a03 0.0075 b03 0.0058 c03 0.0060 a03 0.0081 b03 0.0068 c03 0.0063 a11 0.2320 b11 0.0716 c11 -0.0054 a11 0.2842 b11 -0.0254 c11 -0.0193 a12 0.0329 b12 1.1917 c12 -0.1358 a12 0.3391 b12 0.9972 c12 -0.1442 a13 0.1654 b13 0.1680 c13 0.8157 a13 0.1496 b13 0.0907 c13 0.7633 a21 0.2455 b21 0.0305 c21 0.0449 a21 0.2967 b21 0.1096 c21 0.0306 a22 -0.1839 b22 -0.1320 c22 0.2613 a22 -0.2702 b22 -0.1341 c22 0.2438 a23 0.1569 b23 0.0301 c23 -0.1328 a23 0.2692 b23 0.0345 c23 -0.1094 422 a31 -0.0609 b31 0.0685 c31 0.0668 a31 -0.0578 b31 -0.0115 c31 0.0617 a32 0.1749 b32 -0.0568 c32 -0.0604 a32 0.2890 b32 0.2022 c32 -0.0642 a33 -0.0716 b33 -0.0184 c33 0.1428 a33 -0.1512 b33 -0.0252 c33 0.1675 a41 -0.0127 b41 0.0186 c41 -0.0765 a41 0.0304 b41 0.0442 c41 -0.0959 a42 -0.0583 b42 -0.0313 c42 -0.0222 a42 -0.1239 b42 -0.0941 c42 0.0159 a43 0.0077 b43 0.0265 c43 0.0007 a43 0.0121 b43 0.0231 c43 -0.0002 Notes: qt= GDP Growth Rate; it= Interest rate; mt= Real Monetary Base. 4.3.2. Model Diagnostics In this subsection, we first apply Kolmogorov-Smirnov (KS) test to check the residuals in both models. For the new model, all Kolmogorov-Smirnov (KS) test accept the null hypothesis under 5% significance level while for the traditional SVAR model, two null hypotheses are rejected (see Table 10). As expected, SSAEPD can give better fitness compared to Normal. Second, we apply Likelihood Ratio test (LR) to run Normality tests. The rejection of the null hypothesis in column T22 means that the joint distribution of errors does not follow multi-variate Normal under 5% significance level. For output (qt), the normality assumption is accepted, which is same as the result of KS test. The null for skewness parameter is not rejected in monetary base (mt) while for interest rate (it), both nulls for tail parameters are accepted. This result confirms that the interest rate has fat tailness and the monetary base has skewness. Hence, we conclude that there exists non-Normality in the sample data. Last, from the graphes by method of "eye-rolling", we may also conclude the residuals of new model fit data better. That is, we compare the probability density functions (PDFs) of the estimated residuals εˆit (i = 1, 2, 3) with those of SSAEPD (αˆi, pˆi1, pˆi2) (i = 1, 2, 3). We find out these curves are very close to each other (see Figure 1). Then, we compare the PDFs of the residuals with that of Normal (0, 1). We find out that there are big differences between these curves. Hence, graph results show the residuals are more likely distributed as SSAEPD. Table 10. Test for the Normality of Residuals. qt T1 T4 T7 T10 T13 T16 T19 - 0.4899 0.4066 -5 -1.2 6* 7.6* 4 - it T2 T5 T8 T11 T14 T17 T20 - 0.3488 0.0026 -2.6 44.4* 44.4* 55.8* 56.8* - mt T3 T6 T9 T12 T15 T18 T21 T22 0.3086 0.0272 35.2* -3 -12 25.4* 35.2* 155.6* χ20.05 - - 3.84 3.84 3.84 5.99 7.82 12.59 Note: The null hypothesis of KS test for Tl-T3 is H0: Residuals follow SSAEPD (αˆ, pˆ1, pˆ2) The null hypothesis of KS test for T4-T6 is H0: Residuals follow Standard Normal T7 means H0: α1 = 0.5. T8 means H0: α2 = 0.5. T9 means H0: α3 = 0.5. Tl0 means H0: p11 = 2. Tll means H0: p12 = 2. Tl2 means H0: p21 = 2. Tl3 means H0: p22 = 2. Tl4 means H0: p21 = 2. Tl5 means H0: p22 = 2. Tl6 means H0: p11 = p12 = 2 Tl7 means H0: p21 = p22 = 2 Tl8 means H0: p31 = p32 = 2 Tl9 means H0: α1 = 0.5, p11 = p12 = 2. T2O means H0: α2 = 0.5, p21 = p22 = 2. T2l means H0: α3 = 0.5, p31 = p32 = 2. T22 means H0: α1 = α2 = α3 = 0.5, p11 = p12 = p21 = p22 = p31 = p32 = 2 * Means the parameter restriction is statistically significant. CV (5%) means the Critical Value under 5% Significance Level. 423 Figure 1. Comparisions of PDFs for Residual εˆt. (a) PDFs of εˆ1t & SSAEPD (αˆ1,pˆ11,pˆ12) (up) and PDFs of εˆ1t & Normal (1,0) (down); (b) PDFs of εˆ2t & SSAEPD (αˆ2,pˆ21,pˆ22) (up) and PDFs of εˆ2t & Normal (1,0) (down); (c) PDFs of εˆ1t & SSAEPD (αˆ3,pˆ31,pˆ32) (up) and PDFs of εˆ3t & Normal (1,0) (down). 4.3.3. Structural Analysis Impulse-Response Function: We use the Impulse- Response Function (IRF) to evaluate the relative importance of error shocks. For comparison, the IRFs of the traditional SVAR model are also calculated. The graphical results are plotted in Figure 2. According to the new model, the spending shock (εIS) increase output immediately, but the increase dies out within three years (see Figure (2a)). In response to a positive money demand shock (εLM), both output and interest rate experience a positive effect. However, the shock decreases the output after three years while the interest rate still increases during all periods (see Figure (2b) and (2e)). The relationship between output and interest rate is strong and negative at longer horizons, which conforms to the Keynesian theory. Different from the results from Breitung, the positive response of output follows a positive money supply shock (εm) in this system (see Figure (c)) [22]. The positive correlation between output and money supply proves the view of IS-LM theory. A positive shift in the IS curve (εIS) increases interest rates with a raising response whereas real money decreases gradually after two quarters (see Figure (2d) and (2g)). Similarly, a LM shock (εLM) leads to an increase in interest rates and a decrease in real money base (see Figure(2e) and (2h)). These effects are predicted by the IS-LM theory. Finally, a positive money supply shock (εm) leads to an immediate raise in both interest rate and real money (see Figure (2f) and (2i)). This implies the "liquidity trap" which is also an important consequence of the standard IS-LM model. For the traditional SVAR model, the IRF plots are different from those in the new model. The IRFs of the new model are more consistent with the IS-LM theory than those of the SVAR model. As expected from Keynesian theory, real money (mt) reacts to a money demand shock (εLM) negatively at longer horizons in the new model while the responses are positive in SVAR model. Moreover, the new model can capture the positive relation between output and interest rate, while the SVAR model cannot. Figure 2. Impulse-Response Functions. Responses of GDP, Interest Rate and Monetary Base are shown from left to right. Solid line is from the SVAR-MGC_SSAEPD model. Dash line is from the SVAR-MGC_Normal model. 4.3.4. Variance Decomposition (VD) To assess the importance of the three different shocks for the system variables, the forecast error variances of the variables are decomposed with respect to the shocks. The results for different forecast horizons are presented in Table 11. It turns out that the money demand shock contributes a small fraction to the forecast error variance of output. This result proves that the innovation in money does not seem to be an accurate indicator for monetary policy. IS shocks clearly dominate the behavior of the output series but with respect to a longer forecast horizon, IS shocks become less important. Finally, money supply shocks play a minor role in the short run. However, with an increasing forecast horizon, the money supply shocks become more and more important. The relative contribution of the shocks to the forecast error 424 variance of interest rates and real money can be interpreted in a similar manner. Compared to the new model, the traditional SVAR model shows that the money demand shock contributes a higher fraction to the forecast error variance of output while the money supply becomes less important to interest rate. For the new model, the percentage of the effect from own shocks decreases over time, as the lagged variable's effect starts kicking in. However, for the SVAR model, the rising effect of other shocks can be observed only for the variance of real money. This discovery implies that the use of SSAEPD can improve the performance of variance decomposition. Table 11. Simulation of Variance Decomposition (VD). Lag Variance Decomposition of y1 Ψ11 Ψ12 Ψ13 E M E M E M 1 99.94 95.03 0 4.71 0 0.25 4 87.65 78.54 1.28 10.47 11.07 10.99 8 82.48 60.89 1.49 23.85 16.03 15.26 12 81.62 41.28 1.51 34.57 16.87 16.67 16 81.16 45.78 1.66 41.66 17.18 17.05 20 80.63 36.59 1.89 46.30 17.48 17.12 Lag Variance Decomposition of y2 Ψ21 Ψ22 Ψ23 E M E M E M 1 1.37 15.64 93.10 80.10 5.53 4.26 4 6.44 19.10 90.93 79.25 2.63 1.65 8 13.10 26.17 68.13 70.48 18.77 3.35 12 13.44 28.20 52.06 64.57 34.50 7.23 16 12.65 28.37 43.94 59.20 43.41 10.46 20 12.05 28.12 39.63 61.18 48.32 12.67 Lag Variance Decomposition of y3 Ψ31 Ψ32 Ψ33 E M E M E M 1 0 0 0 0 1 1 4 0.94 0.84 1.16 1.06 97.90 98.10 8 1.25 0.77 3.87 2.77 94.89 96.45 12 2.41 1.33 8.40 6.94 89.19 92.67 16 3.90 2.78 13.70 10.02 83.38 87.20 425 20 5.25 5.00 16.39 14.71 78.36 80.49 4.3.5. IS-LM Theory Still Alive According to the estimates, the structural model of this new model (i.e. SVAR (4)-MGC SSAEPD) is: 𝑢 0.0075𝜀 0.0302𝑢 (IS curve) 𝑢 0.0058𝜀 0.0937𝑢 0.2357𝑢 (inverse LM curve) 𝑢 0.0060𝜀 (money supply curve) (24) The first equation represents a traditional IS curve with a negative parameter for the interest rate innovation u2ti. However, it turns out that the estimated coefficient a01 = −0.0302 is statistically insignificant. According to our results, an IS shock (ε1tIS) increases output, which is the same as that predicted by IS-LM theory. The second equation reflects a Keynesian LM curve with a relationship between money demand innovation u2ti and interest rate innovation u3tm. The parameters of the inverse LM curve have expected sign, and all parameters are significant at conventional level. An LM shock (ε2tLM) leads to an increase in the output and a decrease in the monetary base. The third equation postulates money supply rule. The parameter of this equation is positive and significant, which means innovations of the monetary base u3tm are driven by exogenous money supply shocks ε3tMS. Hence, we conclude the classic IS-LM theory is still alive in the U.S. 4.4. Model Comparisons We apply AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion) and HQ (Hannan-Quinn) to compare the new model with other 9 models. Similar to other previous studies, we rank the models according to these criterions and select the best model based on the mean ranks [23]. The results are listed in Table 12. For our sample, the fat-tailness is a strong phenomenon while the skewness is not an obvious character. Hence, a SVAR model with a SEPD margin is the best compromise for precise in-sample fit. Although our new model does not have the best in-sample fit, it still performs better than the classic SVAR model (i.e., M2). Table 12. Model Comparisons. Models M1 M2 M3 M4 M5 AIC -21.6524(4) -21.0213(10) -21.7483(2) -21.7732(1) -21.6871(3) BIC -20.8140(6) -20.3226(10) -20.9565(2) -21.0279(1) -20.8953(4) HQ -21.7308(4) -21.0866(10) -21.8223(2) -21.8428(1) -21.7612(3) Rank 4.67 10 2 1 3.33 Models M6 M7 M8 M9 M10 AIC -21.6132(6) -21.6358(5) -21.4144(8) -21.5268(7) -21.0680(9) BIC -20.9146(3) -20.8285(5) -20.6071(8) -20.7195(7) -20.3228(9) HQ -21.6786(6) -21.7113(5) -21.4899(8) -21.6023(7) -21.1376(9) Rank 5 5 8 7 9 Notes: Ml means SVAR(4)-GC-SSAEPD. M2 means SVAR(4)-GC-Normal. M3 means SVAR (4)-GC-Skewed EPD M4 means SVAR(4)-GC-SEPD. M5 means SVAR(4)-GC-EPD M6 means SVAR(4)-GC-Laplace. M7 means SVAR(4)-GC-SSAEPD with p11 = p21 = 2 M8 means SVAR(4)-GC-SSAEPD with p12 = p22 = 2 M9 means SVAR(4)-GC-SSAEPD with p13 = p23 = 2 MlO means SVAR(4)-GC-SSAEPD with p11 = p12 = p21 = p22 = 2. 5. Conclusions and Future Extensions In this paper, we have extended the SVAR model by incorporating error terms that follow the Standardized Standard Asymmetric Exponential Power Distribution (SSAEPD). We utilized Markov Chain Monte Carlo (MCMC) techniques for generating error terms and applied the Method of Maximum Likelihood Estimation (MLE) for model estimation. Essential tools like the Impulse-Response Function (IRF) and Variance Decomposition (VD) were employed for structural analysis. Using the same time series data as previously studied, our findings demonstrate that the new model with SSAEPD residuals can effectively capture the skewness, fat tails, and asymmetric kurtosis of sample data. Compared to the traditional SVAR model, our enhanced model shows better fitness. Moreover, the IRFs derived from our model align more closely with IS-LM theory, indicating that the SSAEPD significantly improves the performance of variance decomposition. 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