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Licensed Unlicensed Requires Authentication Published by De Gruyter July 12, 2018

A nonlinear model of asset returns with multiple shocks

  • Hannu Kahra , Vance L. Martin and Saikat Sarkar EMAIL logo
From the journal Studies in Nonlinear Dynamics & Econometrics
https://doi.org/10.1515/snde-2017-0064

Abstract

A nonlinear model of asset returns allowing for multiple shocks is specified. The nonlinear features of the model are demonstrated graphically using a 3-dimensional diagram referred to as the mean impact surface. A new class of nonlinearity tests is also developed which is compared with existing testing methodologies. Applying the framework using excess returns on US and world equities the empirical results provide strong statistical evidence that domestic and foreign shocks have nonlinear effects on expected returns in the US with the effects being determined by the sign and the size of shocks. In contrast, the effects on world expected returns from shocks in the US and the world are found to react more smoothly. The empirical nonlinearities identified are also shown to be robust to alternative choices of risk factors and distributional assumptions.

Keywords: multivariate mean impact surface; shock spillovers; testing
JEL Classification: C12; C51; C58; G15

Acknowledgement

We would like to thank the editor and two anonymous referees for their extremely helpful comments and suggestions.

Appendix

A Derivations of the MIS statistics

Let the conditional means of the asset excess returns μit+1 and μjt+1, be given by

(26)μit+1=αi+δi1rit+βi1σit+12+λi1ritσit+12+δi2rjt+βi2σjt+12+λi2rjtσjt+12,μjt+1=αj+δj1rit+βj1σit+12+λj1ritσit+12+δj2rjt+βj2σjt+12+λj2rjtσjt+12,

where the variables are defined in Section 2 and the conditional variances σit+12,σjt+12, and conditional covariance σijt+1, have MGARCH specifications. Defining the asset excess returns equations as

(27)rit+1=μit+1+uit+1rjt+1=μjt+1+ujt+1,

and assuming bivariate conditional normality, the joint conditional distribution of (rit+1, rjt+1), is

(28)f(rit+1,rjt+1|It)=12πσit+1σjt+11−ρt+1×exp⁡[−12(1−ρt+12)((rit+1−μit+1σit+1)2+(rjt+1−μjt+1σjt+1)2−2ρt+1(rit+1−μit+1σit+1)(rjt+1−μjt+1σjt+1))],

where It represents the information set conditional on information at t and ρt+1=σijt+1/(σit+1σjt+1) is the time-varying conditional correlation. The log-likelihood is for a sample of t = 1, 2, ⋯ , T observations is

(29)log⁡L=1T∑t=1Tlog⁡f(rit+1,rjt+1|It)=−log⁡2π−1T∑t=1Tlog⁡(σit+1σjt+11−ρt+12)+1T∑t=1T[−12(1−ρt+12)((rit+1−μit+1σit+1)2+(rjt+1−μjt+1σjt+1)2−2ρt+1(rit+1−μit+1σit+1)(rjt+1−μjt+1σjt+1))].

To construct a general test of nonlinearity for each of the excess returns, consider rit+1. Under the null hypothesis the mean impact surface of μit+1, is flat with respect to the shocks in the system, this implies the following restrictions on the rit+1 equation

(30)δi1=βi1=λi1=δi2=βi2=λi2=0.

To derive the nonlinearity test for rit+1, the first order derivatives of the log-likelihood in (29) with respect to the conditional mean parameters in μit+1 of equation (26), are

∂log⁡L∂αi=1T∑t=1T(rit+1−μit+1σit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))∂log⁡L∂δi1=1T∑t=1T(rit+1−μit+1σit+1(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))rit∂log⁡L∂βi1=1T∑t=1T(rit+1−μit+1σit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))σit+12∂log⁡L∂λi1=1T∑t=1T(rit+1−μit+1σit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))σit+12rit∂log⁡L∂δi2=1T∑t=1T(rit+1−μit+1σit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))rjt∂log⁡L∂βi2=1T∑t=1T(rit+1−μit+1σit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))σjt+12∂log⁡L∂λi2=1T∑t=1T(rit+1−μit+1σit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))σjt+12rjt.

Evaluating the derivatives of the log-likelihood function under the null hypothesis in (30), and rearranging terms gives

∂log⁡L∂αi|H0=1T∑t=1T(rit+1−αiσit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))∂log⁡L∂δi1|H0=1T∑t=1T(rit+1−αiσit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))rit∂log⁡L∂βi1|H0=1T∑t=1T(rit+1−αiσit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))σit+12∂log⁡L∂λi1|H0=1T∑t=1T(rit+1−αiσit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))σit+12rit∂log⁡L∂δi2|H0=1T∑t=1T(rit+1−αiσit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))rjt∂log⁡L∂βi2|H0=1T∑t=1T(rit+1−αiσit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))σjt+12∂log⁡L∂λi2|H0=1T∑t=1T(rit+1−αiσit+12(1−ρt+12)−ρt+1(1−ρt+12)σit+1(rjt+1−μjt+1σjt+1))σjt+12rjt.

To generate a test statistic for rit+1 that is straightforward to implement which does not require estimating a separate model for rjt+1, two simplifying assumptions are made. The first is that the first order conditions evaluated at H0 are approximated at ρt+1 = 0. The second is that the conditional variances σit+12 and σjt+12, are assumed to be linear functions of their respective (squared) excess returns so σit+12=f(rit2) and σjt+12=f(rjt2). These assumptions have the advantage that a nonlinearity test of rit+1 can be constructed that is not conditional on specifying a model for rjt+1, thereby making the test statistic robust to misspecification of the model for rjt+1. Using these assumptions in the first order conditions suggests that the nonlinearity test simplifies to a estimating a weighted least squares regression where rit+1/σit+1 is the dependent variable and the regressors consist of a third order polynomial in the lagged returns rit and rjt, weighted by the conditional standard deviation σit+1. Under the null hypothesis in (30) the statistic TR2 is χ62, where R2 is the coefficient of determination from the weighted least squares regression.

B Additional simulation results

B.1 Simulated size distributions for Experiments A to E

Figure 14: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model A using equation (16) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 14:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model A using equation (16) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 15: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model A using equation (16) based on Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 15:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model A using equation (16) based on Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 16: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model A using equation (16) based on skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 16:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model A using equation (16) based on skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 17: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model B using equation (17) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 17:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model B using equation (17) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 18: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model B using equation (17) based on Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 18:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model B using equation (17) based on Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 19: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model B using equation (17) based on Skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 19:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model B using equation (17) based on Skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 20: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model C using equation (18) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 20:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model C using equation (18) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 21: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model C using equation (18) based onStudent t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 21:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model C using equation (18) based onStudent t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 22: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model C using equation (18) based on skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 22:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model C using equation (18) based on skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 23: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model D using equation (19) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 23:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model D using equation (19) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 24: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model D using equation (19) based on Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 24:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model D using equation (19) based on Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 25: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model D using equation (19) based on skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 25:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model D using equation (19) based on skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 26: Simulated empirical size (percentage) of the MIS for alternative sample sizes.The data generating process is based on Model E using equation (20) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 26:

Simulated empirical size (percentage) of the MIS for alternative sample sizes.

The data generating process is based on Model E using equation (20) based on normal disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 27: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model E using equation (20) based on Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 27:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model E using equation (20) based on Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

Figure 28: Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.The data generating process is based on Model E using equation (20) based on skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.
Figure 28:

Simulated empirical size (percentage) of the MIS tests for alternative sample sizes.

The data generating process is based on Model E using equation (20) based on skewed Student t disturbances. The asymptotic distribution as given by the continuous line, is the Chi-squared distribution with 6 degrees of freedom.

B.2 Simulated power functions for Experiments A to E

Figure 29: Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes, with a nominal significance level of 5%.Based on Model A using the data generating process in equation (16) and equation (22), where the parameter ω controls the power of the test.
Figure 29:

Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes, with a nominal significance level of 5%.

Based on Model A using the data generating process in equation (16) and equation (22), where the parameter ω controls the power of the test.

Figure 30: Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes with a nominal significance level of 5%.Based on Model B using the data generating process in equation (17) and equation (22), where the parameter ω controls the power of the test.
Figure 30:

Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes with a nominal significance level of 5%.

Based on Model B using the data generating process in equation (17) and equation (22), where the parameter ω controls the power of the test.

Figure 31: Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes with a nominal significance level of 5%.Based on Model C using the data generating process in equation (18) and equation (22), where the parameter ω controls the power of the test.
Figure 31:

Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes with a nominal significance level of 5%.

Based on Model C using the data generating process in equation (18) and equation (22), where the parameter ω controls the power of the test.

Figure 32: Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes with a nominal significance level of 5%.Based on Model D using the data generating process in equation (19) and equation (22), where the parameter ω controls the power of the test.
Figure 32:

Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes with a nominal significance level of 5%.

Based on Model D using the data generating process in equation (19) and equation (22), where the parameter ω controls the power of the test.

Figure 33: Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes with a nominal significance level of 5%.Based on Model E using the data generating process in equation (20) and equation (22), where the parameter ω controls the power of the test.
Figure 33:

Size adjusted simulated powers (percentage) of the MIS tests for alternative sample sizes with a nominal significance level of 5%.

Based on Model E using the data generating process in equation (20) and equation (22), where the parameter ω controls the power of the test.

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Published Online: 2018-07-12

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