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Forecasts can either be short term, medium term or long term. In this work we considered short term forecast because of the problem of limited data or time series data that is often encounter in time series analysis. This simulation study considered the performances of the classical VAR and Sims-Zha Bayesian VAR for short term series at different levels of collinearity and correlated error terms. The results from 10,000 iteration revealed that the BVAR models are excellent for time series length of T=8 for all levels of collinearity while the classical VAR is effective for time series length of T=16 for all collinearity levels except when ρ = -0.9 and ρ = -0.95. We therefore recommended that for effective short term forecasting, the time series length, forecasting horizon and the collinearity level should be considered.

Forecasts from classical Vector Autoregression (VAR) models and the Bayesian VAR had gained great popularity in the 1980s ([

Forecast can either be short term, medium term or long term. In this present work, our focus is on short term forecast because of the problem of limited data or time series that may be encountered in time series analysis.

Short term forecasting is very useful for decision making in many fields of life. [

This present work is motivated by the work of Johnson, [

Therefore, the aim of this paper is to compare the performances of the classical VAR and Bayesian VAR for time series with collinear variables and correlated error terms in the short term.

The name “multicollinearity” was first introduced by Ragnar Frisch. In his original formulation the economic variables are supposed to be composed of two parts, a systematic or “true” and an “error” component. This problem which arise when some or all the variables in the regression equation are highly intercorrelated and it becomes also impossible to separate their influences and obtain the corresponding estimates of the regression coefficient [

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[_{p} statistic in the presence of a multicollinear regression using simulated data from SAS programme. Their work revealed that the performances of AIC and BIC in choosing the correct model among collinear variables are better when compared with the performances of Mallow’s C_{p}.

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[^{2} in the predictor matrix that is of the same order of magnitude as the R^{2} of the overall model.

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Sources of multicollinearity includes: the data collection method employed; constraints on the model or in the population being sampled; model specification; and overdetermined model. Multicollinearity especially in time series data may occur if the regressors included in the model share a common trend, that is, they all increase or decrease over time.

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1) Although BLUE, the OLS estimators have large variances and covariances making precise estimation difficult.

2) Because of consequence 1, the confidence interval tends to be much wider, leading to the acceptance of the (zero null hypothesis) and the t-ratio of one or more coefficients tends to be statistically insignificant.

3) Although the t-ratio of one or more coefficients is statistically insignificant, R^{2}, the overall measure of goodness-of-fit can be very high.

4) The OLS estimators and their standard errors can be sensitive to small change in the data.

VAR methodology superficially resembles simultaneous equation modeling in that we consider several endogenous variables together. But each endogenous variable is explained by its lagged values and the lagged values of all other endogenous variables in the model; usually, there are no exogenous variables in the model [

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 [

The process is stable if

In that case it generates stationary time series with time invariant means and variance covariance structure.

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

It can be written alternatively as

where

As the explanatory variables are the same in each equation, the multivariate least squares is equivalent to the Ordinary Least Squares (OLS) estimator applied to each equation separately, as was shown by [

In the standard case, the MLE estimator of the covariance matrix differs from the OLS estimator.

OLS estimator for a model with a constant, k variables and p lags, in a matrix notation, gives

Therefore, the covariance matrix of the parameters can be estimated as

In recent times, the BVAR model of [

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-Zha form from the structural model to the reduced form ([

The summary of the Sims-Zha prior is given in

The simulated data will be generated for time series lengths of 8 and 16. The choice of the length chosen is to be able to study the models in the short run [

The simulation procedure is given in the following steps.

Step1: we generated a VAR (2) process that obeys the following form

Our choice for this form model is to obtain a stable process and a VAR process that is not affected by overparameterization [

Step2: let the desired correlation matrix be

Step 3: the VAR and BVAR models of lag length of 2 was be used for modeling and forecasting simulated data to obtain the RMSE and MAE.

Step 4: Step 1 to Step 3 was repeated for 10,000 times, and the averages of the criteria were used to access the preferred model. A sample of simulated data is presented in

Parameter Range Interpretation | 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 | 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, [

T | ρ = −0.99 | Residuals | ||
---|---|---|---|---|

y_{1} | y_{2} | [U, 1] | [U, 2] | |

1 | 6.0853378 | −4.699854321 | ||

2 | 8.3361587 | −7.662052368 | −0.7440477 | 0.7748210 |

3 | −1.4119188 | 2.549041262 | 0.2270471 | −0.3013146 |

4 | 1.6662033 | −0.878217866 | 0.5898282 | −0.6190964 |

5 | 1.9049119 | −0.455346148 | 0.6080660 | −0.6421680 |

6 | 4.5498919 | −3.60183345 | 1.4032863 | −1.5356158 |

7 | 1.836577 | −0.801168974 | 0.2527919 | −0.1563110 |

8 | 1.7007428 | −0.544318103 | 0.1370333 | −0.1499999 |

9 | 2.2602923 | −1.158178434 | 0.4050407 | −0.3836562 |

10 | 2.0551198 | −0.897890179 | 0.2085433 | −0.1174189 |

11 | 2.0660043 | −0.891296203 | 0.7917755 | −0.7295352 |

12 | 2.5693203 | −1.430130241 | −1.4718452 | 1.3755778 |

13 | 0.1955522 | 0.800465005 | −0.2213798 | 0.2439514 |

14 | 0.6950544 | 0.494571004 | −1.4442621 | 1.4103745 |

15 | 1.1474605 | 0.006271441 | −0.7418776 | 0.8303912 |

16 | 1.043101 | 0.203232356 | ||

Estimated correlation r = −0.9970061 | Estimated correlation of the residual r = −0.996 |

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 will be used for modeling and forecasting purpose.

For the BVAR model with Sims-Zha prior, we will consider the following range of values for the hyperparameters given below and the Normal-Inverse Wishart prior.

We consider 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)).

The choice of Normal-inverse Wishart prior for the BVAR models follow the work of [

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 results from the analysis are presented in Tables 3-5. The values of the RMSE and MAE are presented in

Collinearity Levels (ρ) | Models | T = 8 | T = 16 | ||
---|---|---|---|---|---|

RMSE | MAE | RMSE | MAE | ||

−0.99 | VAR(2) | N/A | N/A | 2.822276 | 1.865384 |

BVAR1 | 3.480238 | 2.459911 | 2.924542 | 2.070996 | |

BVAR2 | 3.475351 | 2.461614 | 2.915648 | 2.064497 | |

BVAR3 | 3.487892 | 2.47295 | 2.8539 | 1.969234 | |

BVAR4 | 3.520224 | 2.502497 | 2.846255 | 1.958829 | |

−0.95 | VAR(2) | N/A | N/A | 2.92741 | 1.923175 |

BVAR1 | 3.513841 | 2.461495 | 2.955763 | 2.077187 | |

BVAR2 | 3.524945 | 2.48011 | 2.93884 | 2.058831 | |

BVAR3 | 3.463551 | 2.424384 | 2.808864 | 1.903786 | |

BVAR4 | 3.477056 | 2.428536 | 2.779634 | 1.878388 | |

−0.9 | VAR(2) | N/A | N/A | 2.839813 | 1.892972 |

BVAR1 | 3.527144 | 2.461289 | 2.957319 | 2.080047 | |

BVAR2 | 3.510038 | 2.455105 | 2.946401 | 2.06848 | |

BVAR3 | 3.453288 | 2.416771 | 2.80427 | 1.932373 | |

BVAR4 | 3.423209 | 2.39757 | 2.785012 | 1.918003 | |

−0.85 | VAR(2) | N/A | N/A | 2.803335 | 1.871187 |

BVAR1 | 3.503068 | 2.448057 | 2.959128 | 2.098831 | |

BVAR2 | 3.512622 | 2.462979 | 2.967651 | 2.114981 | |

BVAR3 | 3.422623 | 2.410443 | 2.846845 | 2.006635 | |

BVAR4 | 3.406495 | 2.409279 | 2.831411 | 2.009857 |

−0.8 | VAR(2) | N/A | N/A | 2.794558 | 1.871746 |
---|---|---|---|---|---|

BVAR1 | 3.499985 | 2.469094 | 3.003398 | 2.172383 | |

BVAR2 | 3.485945 | 2.4724 | 3.023571 | 2.216076 | |

BVAR3 | 3.421835 | 2.443648 | 2.917568 | 2.109152 | |

BVAR4 | 3.407584 | 2.454034 | 2.893364 | 2.10613 | |

0.8 | VAR(2) | N/A | N/A | 2.319124 | 1.560363 |

BVAR1 | 3.048499 | 2.207239 | 2.551886 | 1.87176 | |

BVAR2 | 3.045309 | 2.218768 | 2.575611 | 1.904332 | |

BVAR3 | 3.291741 | 2.559531 | 2.927194 | 2.326211 | |

BVAR4 | 3.345142 | 2.619884 | 2.879254 | 2.269554 | |

0.85 | VAR(2) | N/A | N/A | 2.383968 | 1.599955 |

BVAR1 | 3.088592 | 2.237249 | 2.593269 | 1.898944 | |

BVAR2 | 3.115247 | 2.269976 | 2.607954 | 1.921699 | |

BVAR3 | 3.331697 | 2.580985 | 2.953197 | 2.340492 | |

BVAR4 | 3.391991 | 2.652839 | 2.908163 | 2.285598 | |

0.9 | VAR(2) | N/A | N/A | 2.446215 | 1.638628 |

BVAR1 | 3.159379 | 2.282306 | 2.653275 | 1.934046 | |

BVAR2 | 3.159299 | 2.287451 | 2.657339 | 1.953146 | |

BVAR3 | 3.372884 | 2.588348 | 2.970293 | 2.33445 | |

BVAR4 | 3.435369 | 2.66265 | 2.955572 | 2.308289 | |

0.95 | VAR(2) | N/A | N/A | 2.526335 | 1.683815 |

BVAR1 | 3.239628 | 2.325308 | 2.715885 | 1.968942 | |

BVAR2 | 3.255642 | 2.353053 | 2.726984 | 1.993325 | |

BVAR3 | 3.423552 | 2.582054 | 2.986378 | 2.305256 | |

BVAR4 | 3.480364 | 2.651436 | 2.97899 | 2.290409 | |

0.99 | VAR(2) | N/A | N/A | 2.613225 | 1.739059 |

BVAR1 | 3.346564 | 2.381411 | 2.796383 | 2.012807 | |

BVAR2 | 3.35575 | 2.405958 | 2.820216 | 2.040734 | |

BVAR3 | 3.457429 | 2.546992 | 2.954774 | 2.197399 | |

BVAR4 | 3.531113 | 2.624224 | 2.965086 | 2.200766 |

In

In

Also for time series length and forecasting horizon of T = 16, the classical VAR model is preferred for all the collinearity levels except for ρ = −0.95 and ρ = −0.9 where the BVAR model with loose prior are preferred.

The results from this study revealed that the BVAR models were excellent for time series length of T = 8 for all levels of collinearity while the classical VAR was effective for time series length of T = 16 for all collinearity

COLLINEARITY LEVELS (ρ) | Models | T = 8 | T = 16 | ||
---|---|---|---|---|---|

RMSE | MAE | RMSE | MAE | ||

−0.99 | VAR(2) | N/A | N/A | 1 | 1 |

BVAR1 | 2 | 1 | 5 | 5 | |

BVAR2 | 1 | 2 | 4 | 4 | |

BVAR3 | 3 | 3 | 3 | 3 | |

BVAR4 | 4 | 4 | 2 | 2 | |

−0.95 | VAR(2) | N/A | N/A | 3 | 3 |

BVAR1 | 3 | 3 | 5 | 5 | |

BVAR2 | 4 | 4 | 4 | 4 | |

BVAR3 | 1 | 1 | 2 | 2 | |

BVAR4 | 2 | 2 | 1 | 1 | |

−0.9 | VAR(2) | N/A | N/A | 3 | 1 |

BVAR1 | 4 | 4 | 5 | 5 | |

BVAR2 | 3 | 3 | 4 | 4 | |

BVAR3 | 2 | 2 | 2 | 3 | |

BVAR4 | 1 | 1 | 1 | 2 | |

−0.85 | VAR(2) | N/A | N/A | 1 | 1 |

BVAR1 | 3 | 3 | 4 | 4 | |

BVAR2 | 4 | 4 | 5 | 5 | |

BVAR3 | 2 | 2 | 3 | 2 | |

BVAR4 | 1 | 1 | 2 | 3 | |

−0.8 | VAR(2) | N/A | N/A | 1 | 1 |

BVAR1 | 4 | 3 | 4 | 4 | |

BVAR2 | 3 | 4 | 5 | 5 | |

BVAR3 | 2 | 1 | 3 | 3 | |

BVAR4 | 1 | 2 | 2 | 2 | |

0.8 | VAR(2) | N/A | N/A | 1 | 1 |

BVAR1 | 1 | 1 | 2 | 2 | |

BVAR2 | 2 | 2 | 3 | 3 | |

BVAR3 | 3 | 3 | 5 | 5 | |

BVAR4 | 4 | 4 | 4 | 4 | |

0.85 | VAR(2) | N/A | N/A | 1 | 1 |

BVAR1 | 1 | 1 | 2 | 2 | |

BVAR2 | 2 | 2 | 3 | 3 | |

BVAR3 | 3 | 3 | 5 | 5 | |

BVAR4 | 4 | 4 | 4 | 4 | |

0.9 | VAR(2) | N/A | N/A | 1 | 1 |

BVAR1 | 2 | 1 | 2 | 2 | |

BVAR2 | 1 | 2 | 3 | 3 | |

BVAR3 | 3 | 3 | 5 | 5 | |

BVAR4 | 4 | 4 | 4 | 4 |

0.95 | VAR(2) | N/A | N/A | 1 | 1 |
---|---|---|---|---|---|

BVAR1 | 1 | 1 | 2 | 2 | |

BVAR2 | 2 | 2 | 3 | 3 | |

BVAR3 | 3 | 3 | 5 | 5 | |

BVAR4 | 4 | 4 | 4 | 4 | |

0.99 | VAR(2) | N/A | N/A | 1 | 1 |

BVAR1 | 1 | 1 | 2 | 2 | |

BVAR2 | 2 | 2 | 3 | 3 | |

BVAR3 | 3 | 3 | 4 | 4 | |

BVAR4 | 4 | 4 | 5 | 5 |

COLLINEARITY LEVELS (ρ) | T = 8 | T = 16 | ||
---|---|---|---|---|

RMSE | MAE | RMSE | MAE | |

−0.99 | BVAR2 | BVAR1 | VAR(2) | VAR(2) |

−0.95 | BVAR3 | BVAR3 | BVAR4 | BVAR4 |

−0.9 | BVAR4 | BVAR4 | BVAR4 | VAR(2) |

−0.85 | BVAR4 | BVAR4 | VAR(2) | VAR(2) |

−0.8 | BVAR4 | BVAR3 | VAR(2) | VAR(2) |

0.8 | BVAR1 | BVAR1 | VAR(2) | VAR(2) |

0.85 | BVAR1 | BVAR1 | VAR(2) | VAR(2) |

0.9 | BVAR2 | BVAR1 | VAR(2) | VAR(2) |

0.95 | BVAR1 | BVAR1 | VAR(2) | VAR(2) |

0.99 | BVAR1 | BVAR1 | VAR(2) | VAR(2) |

levels except when ρ = −0.9 and ρ = −0.95. Therefore, we recommended that for effective short term forecasting, the time series length, forecasting horizon and the collinearity level should be considered.

We wish to thank TETFUND Abuja-Nigeria for sponsoring this research work. Our appreciation also goes to the Rector and the Directorate of Research, Conference and Publication of the Federal Polytechnic Bida for giving us this opportunity to undergo this research work.

M. O.Adenomon,V. A.Michael,O. P.Evans, (2015) Short Term Forecasting Performances of Classical VAR and Sims-Zha Bayesian VAR Models for Time Series with Collinear Variables and Correlated Error Terms. Open Journal of Statistics,05,742-753. doi: 10.4236/ojs.2015.57074