^{1}

^{*}

^{1}

^{1}

In time series literature, many authors have found out that multicollinearity and autocorrelation usually afflict time series data. In this paper, we compare the performances of classical VAR and Sims-Zha Bayesian VAR models with quadratic decay on bivariate time series data jointly influenced by collinearity and autocorrelation. We simulate bivariate time series data for different collinearity levels ( ﹣0.99, ﹣0.95, ﹣0.9, ﹣0.85, ﹣0.8, 0.8, 0.85, 0.9, 0.95, 0.99) and autocorrelation levels ( ﹣0.99, ﹣0.95, ﹣0.9, ﹣0.85, ﹣0.8, 0.8, 0.85, 0.9, 0.95, 0.99) for time series length of 8, 16, 32, 64, 128, 256 respectively. The results from 10,000 simulations reveal that the models performance varies with the collinearity and autocorrelation levels, and with the time series lengths. In addition, the results reveal that the BVAR4 model is a viable model for forecasting. Therefore, we recommend that the levels of collinearity and autocorrelation, and the time series length should be considered in using an appropriate model for forecasting.

There are various objectives for studying time series. These include the understanding and description of the generated mechanism, the forecasting of future value and optimum control of a system [

The aim of this study is to examine the performances of the classical VAR and Sims-Zha Bayesian VAR model in the presence of collinearity and autocorrelated error terms.

Given a set of k time series variables,

provide a fairly general framework for the Data General Process (DGP) of the series. More precisely this model is called a VAR process of order p or VAR(p) process. Here _{u} and the A_{i} are (k × k) coefficient matrices. The process is easy to use for forecasting purpose though it is not easy to determine the exact relations between the variables represented by the VAR model in Equation (1) above [

Also, polynomial trends or seasonal dummies can be included in the model.

The process is stable if

In that case it generates stationary time series with time invariant means and variance covariance structure. The basic assumptions and properties of a VAR processes is the stability condition. A VAR(p) processes is said to be stable or fulfills stability condition, if all its eigenvalues have modulus less than 1 [

Therefore To estimate the VAR model, one can write a VAR(p) with a concise matrix notation as:

Then the Multivariate Least Squares (MLS) for B yields

In recent times, the BVAR model of Sims and Zha [

Given the reduced form model

The matrix representation of the reduced form is given as:

We can then construct a reduced form Bayesian SUR with the Sims-Zha prior as follows. The prior means for the reduced form coefficients are that B_{1} = I and

This representation translates the prior proposed by Sims and Zha form from the structural model to the reduced form ([

The summary of the Sims-Zha prior is given in

The simulation procedure is as follows:

Step 1: We generated an artificial two-dimensional (Bivariate data) VAR (2) process that obeys the following form:

such that

Step 2: We then use the Cholesky Decomposition to apply to the data generated in Step 1 in order to create a bivariate time series data so that y_{1} and y_{2} have the desired correlation level [

The combination of Step 1 and 2 therefore produce a bivariate time series such that y_{1} and y_{2} are jointly influenced by multicollinearity and autocorrelation.

The simulated data assumed time series lengths of 8, 16, 32, 64, 128 and 256. A sample of simulated data is presented in

The time series were generated data using a VAR model with lag 2. The choice here is to obtain a bivariate time series with the true lag length. While the VAR and BVAR models of lag length of 2 was used for modeling and forecasting purpose.

Parameter | Range | Interpretation |
---|---|---|

[0, 1] | Overall scale of the error covariance matrix | |

>0 | Standard deviation around A_{1} (persistence) | |

=1 | Weight of own lag versus other lags | |

>0 | Lag decay | |

≥0 | Scale of standard deviation of intercept | |

≥0 | Scale of standard deviation of exogenous variable coefficients | |

µ_{5} | ≥0 | Sum of coefficients/Cointegration (long-term trends) |

µ_{6} | ≥0 | Initial observations/dummy observation (impacts of initial conditions) |

v | >0 | Prior degrees of freedom |

Source: Brandt and Freeman, [

ρ = 0.9 | Residuals | d = 0.9 | ||||
---|---|---|---|---|---|---|

y_{1} | y_{2} | [U, 1] | [U ,2] | −0.36417791 | −1.05632564 | |

1 | 4.635822 | 8.562592 | 0.55065249 | 0.3621112 | −2.63216817 | −2.31257463 |

2 | 8.276017 | 9.102328 | 0.90145401 | 0.6982855 | −0.97256527 | −1.03760062 |

3 | −1.176551 | 2.678471 | 1.46132912 | 1.0779790 | 0.19179026 | 0.85192775 |

4 | −1.324869 | 1.132437 | −8.63500727 | −5.8296621 | 0.07215923 | −0.24242029 |

5 | −1.217209 | 2.879577 | −0.02956038 | −1.4306597 | −0.43158786 | −0.92545664 |

6 | 1.841691 | 4.163813 | 0.11858453 | 1.7689057 | −1.16868494 | −1.66643992 |

7 | 1.244165 | 4.437567 | 3.15291385 | 1.3667331 | −2.00270147 | −2.87661209 |

8 | 3.418945 | 5.086300 | −0.30830834 | 0.4905413 | ||

2.45848960 | 0.8676871 | |||||

Estimated correlation = 0.9449642 | Estimated correlation = 0.925 | Estimated value = 0.616 |

For the BVAR model with Sims-Zha prior, we considered the following range of values for the hyperparameters given below and the Normal-inverse Wishart prior was employed.

We considered two tight priors and two loose priors as follows:

where nµ is prior degrees of freedom given as m + 1 where m is the number of variables in the multiple time series data. In work nµ is 3 (that is two (2) time series variables plus 1 (one)).

Our choice of Normal-inverse Wishart prior for the BVAR models follow the work of Kadiyala & Karlsson, [

The following are the criteria for Forecast assessments used:

1) Mean Absolute Error (MAE) has a formular

series in absolute terms, and measures how much the forecast is biased. This measure is one of the most common ones used for analyzing the quality of different forecasts.

2) The Root Mean Square Error (RMSE) is given as _{i} is the time series data and y^{f} is the forecast value of y [

For the two measures above, the smaller the value, the better the fit of the model [

In this simulation study,

In this study three procedures in the R package will be used. They are: Dynamic System Estimation (DSE) [

The Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) were obtained for the various models in the work and are presented in Appendix A, while the ranks are presented in Appendix B. But here the preferred models with respect to their rank are presented. The preferred model for the time series length of 8, 16, 32, 64, 128 and 256 are presented in Tables 3-8 respectively.

In

In

In