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Improved Five-Equation Model Selection

1. Methodology. In my previous post, I relied on ACF and QQ plots to choose whether innovations are IID Gaussian. But I thought this is too informal. Let me instead select the model based on the following. Each series of residuals must have p > 5\% for each of the following 5 statistical tests:

  1. The Jarque-Bera normality test for original values of residuals
  2. The Ljung-Box white noise test for original values of residuals with 5 lags
  3. The Ljung-Box white noise test for original values of residuals with 10 lags
  4. The Ljung-Box white noise test for absolute values of residuals with 5 lags
  5. The Ljung-Box white noise test for absolute values of residuals with 10 lags

2. Results. We accept the following models:

Stock Returns: Q_k(t) = \ln(W_k(t)) - \ln(W_k(t-1)) is modeled as Q_k(t) = a_k - d_k(R(t) - R(t-1)) + b_kV(t) + V(t)Z_k(t) for k = 1, 2 which corresponds to domestic and international geometric stock returns. Also, a sub-model with d_k = 0 (no duration) or a_k = 0 (no volatility as an additive factor) or both.

Bond Returns: Continue this blog post. Adjust them Q_0(t) = W_0(t)/W_0(t-1) - 0.01R(t-1) and regress \ln(Q_0(t)) = -d_0(R(t) - R(t-1)) + V(t)Z_0(t) or Q_0(t) - 1 = -d_0(R(t) - R(t-1)) + V(t)Z_0(t) We accept them but reject the augmented models (for both arithmetic and geometric adjusted returns): with factors a_0 (intercept) and b_0V(t) (volatility as an additive factor).

Volatility: The classic AR(1) model for log volatility \ln V(t) = \alpha + \beta \ln V(t-1) + W_V(t) works.

Bond Rates: Consider autoregression with volatility \ln R(t) = \mu + \gamma \ln R(t-1) + V(t)W_R(t) is accepted. But we reject the one with \delta V(t) instead of \mu or with \mu + \delta V(t). Models with R(t) instead of \ln R(t) are rejected. These are stationary models.

Random walk for logarithms \ln R(t) - \ln R(t-1) = \delta V(t) + V(t)Z_R(t) is also accepted, as well as an augmented model with volatility as an additive factor: \ln R(t-1) - \ln R(t-1) = \mu + \delta V(t) + V(t)Z_R(t). The models without logarithms or volatility or both are rejected. These are non-stationary models.

3. Choice. We pick the following model: Q_k(t) = a_k - d_k(R(t) - R(t-1)) + b_kV(t) + V(t)Z_k(t) for k = 1, 2 and \ln(W_0(t)/W_0(t-1) - 0.01R(t-1)) = -d_0(R(t) - R(t-1)) + V(t)Z_0(t) for stock and bond returns. Next, AR(1) for log volatility and \ln R(t) = \mu + \gamma \ln R(t-1) + V(t)W_R(t) for log rates.

Stock Returns: Q_1(t) = 0.2111 - 0.0107 V(t) - 0.0621(R(t) - R(t-1)) + V(t)Z_1(t) for domestic and Q_2(t) = 0.2684 - 0.0180 V(t) - 0.0390(R(t) - R(t-1)) + V(t)Z_2(t) for international stocks.

Bond Returns: \ln(W_0(t)/W_0(t-1) - 0.01R(t-1)) = -0.0596(R(t) - R(t-1)) + V(t)Z_0(t).

Stock Volatility: \ln V(t) = 0.8569 + 0.6176 \ln V(t-1) + Z_V(t).

Bond Rates: \ln R(t) - \ln R(t-1) = 0.0708 - 0.0411\ln R(t-1) + V(t)Z_R(t).

The covariance matrix for innovations Z_1, Z_2, Z_0, Z_V, Z_R (times 10000) is:

2.218258 0.948039 -0.210064 -6.113641 0.215726
0.948039 2.975298 0.036818 -5.399599 -0.000237
-0.210064 0.036818 1.935904 14.068448 -0.050806
-6.113641 -5.399599 14.068448 1338.685234 2.754026
0.215726 -0.000237 -0.050806 2.754026 0.113497

The correlation matrix for innovations is

1.000000 0.403307 -0.101369 -0.113717 0.459186
0.403307 1.000000 0.014113 -0.083611 -0.000410
-0.101369 0.014113 1.000000 0.275118 -0.097698
-0.113717 -0.083611 0.275118 1.000000 0.217726
0.459186 -0.000410 -0.097698 0.217726 1.000000

We can consider Z_1, Z_2, Z_0, Z_V, Z_R to be multivariate Gaussian independent identically distributed.

See the data and code in GitHub repository which verify the research here.

Published by

Andrey Sarantsev

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