
Research Article


Forecasting Key Macroeconomic Variables of the South African Economy using
Bayesian Variable Selection


Mirriam Chitalu ChamaChiliba,
Rangan Gupta,
Nonophile Nkambule
and
Naomi Tlotlego



ABSTRACT

This study analyzed the forecasting performances of various multivariate models in predicting 18quartersahead of the growth rate of GDP, the consumer price index inflation rate and the three months Treasury bill rate for South Africa over an outofsample period of 2000:Q12011:Q2, using an insample period of 1960:Q11999:Q4. The study compared the forecasting performances of the classical and the Minnesotatype Bayesian vector autoregressive (VAR) models with those of linear (fixedparameter) and nonlinear (timevarying parameter) VARs involving a stochastic search algorithm for variable selection, estimated using Markov Chain Monte Carlo methods. In general, the study finds that variable selection, whether imposed on a timevarying VAR or a fixed parameter VAR, and nonlinearity in VARs, play an important part in improving predictions when compared to the linear fixed coefficients classical VAR. However, the results does not indicate marked gains in forecasting power across the different Bayesian models, as well as, over the classical VAR model, possibly because the problem of over parameterization in the classical VAR is not that acute in our threevariable system. Hence, future research would aim to look at VAR models that include over 10 variables.





Received: January 05, 2012;
Accepted: April 03, 2012;
Published: June 19, 2012


INTRODUCTION
The vector autoregressive (VAR) model, though ‘atheoretical’ is particularly
useful for forecasting purposes (Korobilis, 2011). This
framework essentially involves a system, whereby equal number of lags of all
the dependent variables enters as regressors in the equation of a specific dependent
variable. One drawback of VAR models is that many parameters are needed to be
estimated, some of which may be insignificant. This problem of overparameterization,
resulting in multicollinearity and loss of degrees of freedom leads to inefficient
estimates and large outofsample forecasting errors (Gupta
and Sichei, 2006; Gupta, 2006, 2007,
2009; Gupta and Das, 2008; Liu
and Gupta, 2007; Liu et al., 2009, 2010;
Balcilar et al., 2011). One solution, often adapted,
is simply to exclude the insignificant lags based on statistical tests. Another
approach is to use near VAR, which specifies unequal number of lags for the
different equations (Gupta and Sichei, 2006; Gupta,
2006, 2007, 2009).
However, an alternative approach to overcome this overparameterization, as
described by Litterman (1981, 1986),
Doan et al. (1984), Todd (1984)
and Spencer (1993), is to use a Bayesian VAR (BVAR)
model. Instead of eliminating longer lags, the Bayesian method imposes restrictions
on these coefficients by assuming that these are more likely to be near zero
than the coefficient on shorter lags. However, if there are strong effects from
less important variables, the data can override this assumption. The restrictions
are imposed by specifying normal prior distributions with zero means and small
standard deviations for all coefficients with the standard deviation decreasing
as the lags increases. Unless the variable is meanreverting or stationary,
the exception to this is, however, the coefficient on the first own lag of a
variable, which has a mean of unity. Generally, following Litterman
(1981), a diffuse prior is used for the constant. This is popularly referred
to as the ‘Minnesota prior’ due to its development at the University
of Minnesota and the Federal Reserve Bank at Minneapolis.
Not surprisingly, in the literature, the BVAR models have been found to produce
the most accurate short and longterm outofsample forecasts relative to both
univariate and multivariate unrestricted classical VAR models^{1}.
In this regard, evidence for South Africa is no different, with a large number
of recent studies showing superior forecasting power of BVAR models relative
to not only classical VAR models but also, Dynamic Stochastic General Equilibrium
(DSGE) models^{2}, in predicting key macroeconomic
variables (Gupta and Sichei, 2006; Gupta,
2006, 2007, 2009; Gupta
and Das, 2008; Liu and Gupta, 2007; Liu
et al., 2009, 2010; Alpanda
et al. (2011) and Balcilar et al. (2011)^{3}.
Nowadays though, besides the shrinkage approach of the Minnesotatype BVAR
models, there are numerous other efficient methods to prevent the proliferation
of parameters and eliminate parameter or model uncertainty. For example, variable
selection priors (George et al., 2008), steady
state priors (Villani, 2009), Bayesian model averaging
(Anderson and Karlsson, 2008) and factor models (Stock
and Watson, 2005), to name a few popular methods. Against this backdrop,
following the study of Korobilis (2011), this study compares
the forecasting performances of the classical and the Minnesotatype BVAR models
with those of linear (fixedparameter) and nonlinear (timevarying parameter
(TVP). VARs involving a stochastic search algorithm for variable selection,
estimated using Markov Chain Monte Carlo (MCMC) methods^{4}.
The term “stochastic search” simply means that if the model space
is too large to assess in a deterministic manner, the algorithm will look for
only the most probable models. Note that, the two main benefits of using this
approach over the shrinkage methods are: First, variable selection is automatic,
meaning that along with estimates of the parameters we get associated probabilities
of inclusion of each parameter in the best model. This allows one to select
among all possible VAR model combinations, without the need to estimate each
and every one of these models. Second, this form of Bayesian variable selection
is independent of the prior assumptions about the parameters. Note that the
decision to use the stochastic search variable selection algorithm proposed
by Korobilis (2011) over other available ones, such as
those developed by George et al. (2008) and Korobilis
(2008), is that one can apply the current algorithm to variable selection
nonlinear (timevarying) VAR models.
Specifically speaking, this study compared the forecasting performances of
all these models in predicting oneto eightquartersahead of the growth rate
of GDP, the Consumer Price Index (CPI) inflation rate and the three months Treasury
bill rate for South Africa over an outof sample period of 2000:Q12011:Q2,
using an insample period of 1960:Q11999:Q4^{5}.
While the start and endpoints of the sample is determined by data availability,
the decision to use 2000: Q1 as the beginning of the outofsample period is
determined by the fact that South Africa moved to an inflation targeting regime
in the February of 2000. Besides, this choice is also consistent with most of
the studies, mentioned above, that deals with forecasting in South Africa using
BVAR models. The basic idea behind this exercise is to see if we could perform
better than the BVAR models in forecasting key macroeconomic variables for South
Africa by allowing for stochastic search for variable selection imposed on fixed
and timevarying parameter models. In this regard, to the best of our knowledge,
this is a first such attempt for South Africa.
THE ECONOMETRIC METHODS
Variable selection in VAR and the TVPVAR model^{6}:
A reduced form VAR can be written using following linear regression specification:
where, Y_{i+1} is an (mx1) vector of dependent variables at time t = 1, …,T; is a (kx1) vector, which may include lags of the dependent variables, intercept, dummies, trends and exogenous regressors, B is an (mxk) vector of VAR coefficients and ε_{t}~N (0, Σ), where Σ is a (mxm) covariance matrix. Equation 1 can be rewritten as a System of Unrelated Regressions (SUR) as follows, thus allowing for different equations in the VAR to have different explanatory variables:
where, Y_{i+1} and ε_{i} are defined as above in Eq.
1; z_{t} = I_{m}
x’_{t} is a (mxn) matrix vector; while β = vec (B) is an (nx1)
matrix. When there are no parameter restrictions, Eq. 2 can
be called an unrestricted VAR model. Bayesian variable selection therefore,
will be incorporated in Eq. 2 by embedding indicator variables:
γ = (γ_{1} = γ_{n}) such that β_{i}
= 0 if γ_{i} = 0 and β_{i} ≠ 0 if γ_{i}
= 1. Note that the indicator variables are treated as random variables by assigning
a prior on them and allowing the data likelihood to determine their posterior
values. These indicator variables can be explicitly inserted multiplicatively
in the VAR model using the form:
where, θ = Γβ, Γ is an (nxn) diagonal matrix with Υ_{JJ} = γ_{J} (j = 1,2...., n) elements on its main diagonal and for Υ_{JJ} = γ_{J} θ_{j} = Γ_{jj}β_{j} = 0 where, θ_{j} is restricted while for Υ_{JJ} = 1 θ_{j} = Γ_{jj}β_{j} =β_{j}, so that all possible 2^{n} specifications can be explored and variable selection is equivalent to model selection in this case. Gibbs sampling can be used to estimate these parameters by conditioning on the data and Γ. Assuming the socalled independent NormalWishart prior, the densities of β and Σ are of standard form. The restriction indices γ add one more block to the Gibbs sampler of the unrestricted VAR model, and if needed, for the restriction indicators the n element in the column vector γ = (γ_{t},...., γ_{n})’ is sampled and the diagonal matrix Γ = diag{γ_{t},...., γ_{n}} is recovered. Derivations are however, simplified if indicators γ_{j} are independent of each other. The priors in particular can be defined as below: where, b_{0} is (nx1), V_{0} is (nxn), π_{0} = (π’_{01},...., π’_{0n}) is (nx1), Ω is (mxm) matrix and α is a scalar. It is argued that this form of variable selection may be adopted in many nonlinear extensions of the VAR as compared to stochastic variable selection algorithms for VAR models. Adopting variable selection in TVPVAR model therefore, is a simple extension of the VAR model with constant parameters, where Eq. 7 is replaced with Eq. 3 and variables are as explained in Eq. 3 while priors are as explained by Eq. 4 through 6 (except now β∼N_{n} (b_{0}, V_{0}).
Modern macroeconomic applications increasingly involve the use of VARs with
mean regression coefficients and covariance matrices which are timevarying,
in the process implying a nonlinear VAR model. Note that, a timevarying parameter
VAR with constant variance (Homoscedastic VAR) takes the form:
where, z_{t}, x_{t}^{′}, Σ and ε_{t }are
defined as before in Eq. 1; β_{t} is an (nx1)
vector of t = 1,....T parameters, η_{t}∼N(0, Q) with Q as a
(nxn) covariance matrix. The implied prior for β_{1} to β_{t}
are of the form β_{t}β_{t1}, Q~N (β_{t1},
Q) and the covariance matrix Q is considered to be unknown hence will have its
own prior of the form Q^{1}∼Wishart (ξ, R^{1}). In
order to avoid the explosive behaviour (which might affect forecasting negatively)
of the random walk assumption on the evolution of β_{t}, it is
of importance to restrict its covariance Q. As such, to get a tight prior we
subjectively choose the hyperparameters for the initial condition β_{0}
and the covariance matrix Q. It is worth noting that the performance of variable
selection is influenced by the hyperparameters which affect the mean and variance
of the mean coefficients β. For the VAR case, when γ_{j} =
0 and β_{j} is restricted, a draw is taken from each prior implying
that the prior variance V_{0} cannot be very large^{7}
since it would mean no predictors are selected. Variable selection is also affected
by the hyperparameter of the Bernoulli prior of γ_{j}.
Alternative forecasting models and forecast evaluation metric: Specifically, the priors that we use for the restricted VAR i.e., VAR with variable selection (VARVS) are: γ_{j}γ_{\j}∼Bernoulli (1, 0.5) for all j = 1,...,N and β_{J}∼n(0, 10^{2}) if β_{J} is an intercept, and β_{J}∼N(0, 3^{3}) otherwise. For the benchmark VAR, the priors are the same as the VARVS, except that we restrict γ_{j} = 1 for all j. As far as the BVAR based on the Minnesota prior (VARMIN) is concerned, the means and variances of the Minnesota prior for β takes the form β∼N(0, 03^{3}) where:
with g_{1}/p and g_{3}xs^{2}_{i} applying to
parameters on own lags and for intercepts, respectively, while g_{3}xs^{2}_{i}/(pxs^{2}_{i})
is for parameters j on variable l ≠ i, l, i = 1,.., m. s^{2}_{i}
is the residual variance from the plag univariate autoregression for variable
i. After experimenting to produce the best possible forecast, the hyperparameters
were set to the following values: g_{1} = 0.09, g_{2} = 0.0225,
and g_{3} = 100. Since the variables used in the forecasting exercise
is transformed to induce stationarity, the prior mean vector b_{MIN}
is set equal to zero for parameters on the lags of all variables including the
first own lag (Banbura et al., 2010).
For the timevarying parameters model with variable selection (TVPVAR VS), a prior on the initial condition is of the form β∼N(04^{2} V_{MIN}), with γ_{j}γ_{\j}∼Bernoulli (1, 05) The timevarying VAR without variable selection (TVPVAR) uses prior as in TVPVAR VS, with the restriction γ_{j} = 1 for all j = 1,...,n imposed. The covariance Q of the timevarying coefficients in the TVPVAR VS has the prior Q^{1}∼Wishart (ξ, R)where ξ = n+1 and R^{1}, where VMIN is the matrix defined in (9).
To evaluate the forecast accuracy, this study computes and compares the Mean
Squared Forecast Error (MSFE) of onethrough eightquartersahead recursive
outofsample forecasts for the period 2000:Q1 to 2011:Q2 in all the models.
The covariance was integrated out using an uninformative prior of the form
which is equivalent to prior defined by equation and an additional restriction
is that α = 0 and S^{1} = 0_{mxm}. All models are based
on a run of 20,000 draws from the posterior, discarding the first 10,000 draws.
The MSFE is computed as:
where,
is the time t+h prediction of the variable i created using data available up
to time t.
is the observed value at time t+h. For the TVP models^{8},
Averages over the full forecasting period 2000:Q1 to 2011:Q2 are presented using
the formula:
where, τ_{0} is 2000:Q1 and τ_{1 }is 2011:Q2. Data: This study estimated the different models for the South African economy using quarterly data for the period 1960:Q2 to 2011:Q2. The macroeconomic variables of interest were: GDP growth rate (quarter on quarter percentage growth rate of the seasonally adjusted Gross Domestic Product at 2005 constant prices), the inflation rate (the quartertoquarter percentage change in the consumer price index) and the interest rate (yield on three month treasury bill rate). The data on treasury bill rate and consumer price index were obtained from the International Financial Statistics of International Monetary Fund, while GDP data was obtained from the Quarterly Bulletin of the South African reserve bank. Figure 1 shows the graphs of the three variables used in our forecasting exercise.
Since, the interest rate and inflation rate were found to be nonstationary
(based on standard unit root tests)^{9}, the analysis
uses the first difference of these variables, unlike the growth rate of the
real GDP.

Fig. 1(ac): 
Transformed key macroeconomic variables of the South African
economy: 1960:Q2 to 2011:Q2 (a) GPD growth rate, (b) Δ inflation rate
and (c) Δ interest rate 
After the transformations, one quarter was lost at the beginning of the sample,
and hence, the insample contains data from 1960:Q2 to 1999:Q4, while the onethrough
eightquartersahead outof sample forecast is obtained from the outofsample
period of 2000:Q1 to 2011:Q2, by, recursively estimating each of the six models,
namely, the random walk (RW), VAR, VARMIN, VAR VS, TVPVAR and TVPVAR VS.
The appropriate lag length was selected using the Akaike information criterion
(AIC), which, in turn, yielded 3 lags^{10}. Hence,
the period 1960:Q21961:Q1 was used to feed the lags in the alternative VAR
models.
RESULTS
The findings emanating from the forecasting evaluation exercise, as presented
in the Table 1 can be summarized as follows. Note, in line
with the series of studies involving BVAR models for the South African economy
reported in the introduction, the models are compared in terms of the average
relative MSFE, i.e., the MSFE of a specific model with respect to the MSFE of
the RW model^{11}:
• 
The results show that with distant forecasts, the naive RW
model performs worse than all the models whether restricted or unrestricted.
It is always ranked sixth in terms of average MSFE for the three key variables
of our concern 
• 
For the GDP growth rate, on average, the VARVS model performs the best,
followed closely by the TVPVARVS. The TVPVAR model comes in third, while,
the VARMIN and the VAR model ends up being the fourth and fifth best performer 
• 
For the change in the inflation rate, the VARVS model is again the best
performer, as was the case with the growth rate. The TVPVAR ranks a close
second, while, the VARMIN, follows closely on the heels. The TVPVARVS
and the VAR comes in fourth and fifth to round off the list 
• 
As far as the change in the shortterm interest rate is concerned,
the TVPVARVS outperforms all the other models. The VARMIN comes in second
followed by the TVPVAR, VARVS and the VAR models 
• 
As observed in the literature (Gupta and Sichei, 2006;
Gupta, 2006, 2007, 2009;
Gupta and Das, 2008; Liu and
Gupta, 2007; Liu et al., 2009, 2010;
Balcilar et al., 2011) of forecasting with
BVAR model based on the Minnesota prior (VARMIN), the model tends to outperform
the classical VAR in our case as well 
• 
Variable selection, whether imposed on a timevarying VAR or fixed parameter
VAR, is found to play a role in improving forecast performances. Thus, highlighting
that there could be gains in using other forms of efficient methods in solving
the overparameterization problem of the classical VAR, besides the standard
Minnesotapriorbased shrinkage approach 
• 
Nonlinearity, modelled through the TVP VARs, also clearly play an important
role in improving predictions when compared to the linear fixed coefficients
classical VAR 
• 
Having said that, the results do not suggest marked gains in terms of
the relative average MSFE across the different Bayesian models. In fact,
the improvement of the average relative MSFE over the classical VAR model
made by its Bayesian counterparts is not significantly large (3.33, 2.67
and 7.11%, respectively for the GDP growth rate and the first differences
of the inflation rate and the interest rate). However, note the classical
VAR is outperformed by the best performing Bayesian VAR for a specific variable
for each of the one to eightquartersahead forecasts 
• 
One reason behind the result that one does not see significant gains by
using Bayesian variants of the VAR model over the classical VAR could be
because of the fact that the problem of over parameterization is not that
acute for the system in this study. The current system has 30 parameters
to be estimated in all, involving one constant and three lags each for the
three variables, implying 10 parameters for each of the 3 equations. It
is likely that the gains would be bigger for largescale models involving
more than 10 to 15 variables, as observed by Korobilis
(2011) 
Table 1: 
Oneto EightQuartersAhead OutofSample MSFE (2000:Q12011:Q2) 

Models as defined in the text; The third column reports the
MSFE from the Random Walk (RW) model, while columns 4 to 8 presents the
ratio of the MSFE of a specific model relative to the MSFE of the RW model;
h denotes the forecasting horizon; Average denotes the average MSFE of the
RW model and the relative MSFE of the other models for h = 1 to 8 for a
specific variable. For averages, we report up to four decimal places to
distinguish between the models, since some of the models produce the same
average at three decimal places. Bold entries indicate the model with the
lowest average relative MSFE 
CONCLUSION
The Vector Autoregressive (VAR) model, though ‘atheoretical’ is particularly
useful for forecasting purposes. One drawback of VAR models is that many parameters
are needed to be estimated, some of which may be insignificant. This problem
of overparameterization, resulting in multicollinearity and loss of degrees
of freedom leads to inefficient estimates and large outofsample forecasting
errors. One of the most common approaches to overcome this overparameterization
is based on using Bayesian shrinkage, popularly called the Minnesotapriorbased
Bayesian VAR (BVAR). Not surprisingly, in the literature, the BVAR models have
been found to produce the most accurate short and longterm outofsample forecasts
relative to both univariate and multivariate unrestricted classical VAR models.
In this regard, evidence for South Africa is no different, with a large number
of recent studies showing superior forecasting power of BVAR models relative
to classical VAR models. Nowadays though, besides the shrinkage approach of
the Minnesotatype BVAR models, there are numerous other efficient methods to
prevent the proliferation of parameters and eliminate parameter or model uncertainty,
based on stochastic search algorithm for variable selection.
Against this backdrop, this study compared the forecasting performances of the classical and the Minnesotatype BVAR models with those of linear (fixedparameter) and nonlinear (timevarying parameter [TVP]) VARs involving a stochastic search algorithm for variable selection, estimated using Markov Chain Monte Carlo (MCMC) methods. Specifically speaking, this study analyzes the forecasting performances of all these models in predicting one to eightquartersahead of the growth rate of GDP, the Consumer Price Index (CPI) inflation rate and the three months Treasury bill rate for South Africa over an outofsample period of 2000:Q12011:Q2, using an insample period of 1960:Q11999:Q4.
The results suggest that, the VAR based on variable selection performs the
best for forecasting output growth and inflation, while the timevarying VAR
is the best model in forecasting the interest rate. In general, the study finds
that variable selection, whether imposed on a timevarying VAR or a fixed parameter
VAR, is found to play a role in improving forecast performances. Nonlinearity
modelled through the TVPVARs also play an important part in improving predictions
when compared to the linear fixed coefficients classical VAR. Similar results
were also obtained by Korobilis (2011). However, the results
do not indicate marked gains in forecasting power across the different Bayesian
models, as well as, over the classical VAR model. One reason behind the result
could be because of the fact that the problem of over parameterization in the
classical VAR is not that acute for the smallsystem in this study. It is likely
that the gains would be bigger for largescale models involving more than 10
to 15 variablesan area of research, which is for the future.
^{1}Refer to Banbura et al.
(2010) and Koop and Korobilis (2010) for further
details.
^{2}An exception to this is the recent study by Gupta
and Das (2010) who shows that when one develops a sophisticated DSGE model
involving a variety of nominal and real rigidities, it is possible to outperform
the BVAR model based on the Minnesota prior.
^{3}At the same time, there is also evidence for South
Africa that, large scale BVAR models or factor models, which involve over two
hundred variables, tend to outperform both classical and smallscale BVAR models,
essentially involving three to six variables Gupta and Kabundi
(2010, 2011a, b). In addition,
allowing for nonlinearity in the data generating process through logistic and
exponential smooth transition autoregressive models, are also found to forecast
better than smallscale VAR and BVAR models, as observed for South Africa by
Balcilar et al. (2011).
^{4}For use of other linear, nonlinear and nonparametric
methods in forecasting various types of variables, by Aydin
(2009), Assis et al. (2010), Samsudin
et al. (2010), Khin et al. (2011)
and Yaziz et al. (2011).
^{5}For use of other linear, nonlinear and nonparametric
methods in forecasting various types of variables, see the recent studies by
Aydin (2009), Assis et al.
(2010), Samsudin et al. (2010), Khin
et al. (2011) and Yaziz et al. (2011).
^{6}The choice of these three variables are in line
with the monetary policy literature. Gungor and Berk (2006),
Agbeja (2007), Saibu and Oladeji
(2007), Berument et al. (2009) and Krishnapillai
and Thompson, (2012).
^{7}This section relies heavily on the discussion
available by Korobilis (2011). The readers are referred
to this paper for further details.
^{8}For a recent empirical application using a TVPVAR
based on stochastic volatility for forecasting key macroeconomic variables of
the US economy, see D’Agostino et al. (2011).
^{9}The unit root tests, namely the AugmentedDickeyFuller
(ADF), the DickeyFuller with GLS detrending (DFGLS), the Kwiatkowski, Phillips,
Schmidt and Shin (KPSS) and the PhillipsPerron (PP) tests, are available upon
request from the authors.
^{10}Using 2 lags based on the Schwarz information
criterion (SIC), did not change our results qualitatively. These results are
available upon request from the authors.
^{11}Results based on the mean absolute forecast error (MAFE)
yielded similar conclusions. Also, when a longer outofsample period starting
in 1981:Q1 was used, we obtained results similar to those reported in Table
1. Both sets of results are available upon request from the authors

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