Abstract:
Distributed Lag Model (DLM) is a major workhorse in dynamic single-equation regression, which
requires stringent assumptions for its validity. One of the critical assumptions of DLM is the
normality of the Error Term (ET) which is often violated in practice and often leads to spurious
inference and poor forecast performance. Violations of other assumptions had been considered in
previous studies but not the Exponentiated Generalised Normal ET (EGNET) of the DLM.
Therefore, this study was designed to develop a Robust DLM (RDLM) that could enhance
inference when the assumption of normality of ET is violated.
Exponentiated Generalised Normal Distribution (EGND) was examined by convoluting the
exponentiated link function; ( ) = [ ( )] ( ), where > 0 is the shape parameter, ( ) and ( ) are the
probability density and distribution functions respectively with the generalised
normal distribution : ( ) =where σ and
are the standard
√
deviation and mean of the distribution, respectively. The DLM was then used in EGND to obtain
the density function of the RDLM. The maximum likelihood method was used to estimate the
parameters and the statistical properties of RDLM. The proposed model was validated with life
and simulated data. Monthly data on Nigeria’s gross domestic product and external reserve from
1981 to 2015 extracted from the Central Bank of Nigeria statistical bulletin were used, while data
of sample sizes 20, 50, 200, 500, 1000, 5000 and 10,000 were simulated and replicated 10,000
times. For each of the simulated data, outliers were injected randomly to obtain non-normally
distributed data. The performance of the proposed model was compared with DLM model with
normal ET using Akaike Information Criteria (AIC), Root Mean Square Error (RMSE) and Mean
Absolute Error (MAE). The lower the value of the performance criteria the better the model.
The developed probability density function of RDLM was:
( ) = , where is the observed data
iii
of the response variable at time t , is the intercept, βi, ( = 1, … , ) and , = 1, 2, … , are the response
rates at the lags of both explanatory and response variables , respectively. The derived properties
of the proposed model confirmed that EGND was a valid distribution. The simulated data of sizes
20, 50, 200, 500, 1000, 5000 and 10,000 showed AIC of 67.18, 151.58, 568.22, 1419.89, 2876.
86, 14156.15, 28220.94, respectively for DLM with normal ET. For DLM with EGNET, the AIC
values were -40.01, -116.66, -282.19, -655.10, -1533.01, -3007.01, 5606.92, -26960.82, and -
5283.44, respectively. For life data, DLM with EGNET performed better than DLM with normal
ET as indicated by AIC values of 1590.08 and1695.19, respectively. Forecast performance
indicated that RDLM was better than DLM for forecasting with lower RMSE and MAE values of
1730.50, 18348.71 and 4325.37, 30839.37, respectively.
The distributed lag model with exponentiated generalised normal error term showed improved
forecasting and inference even when the residual term were not normally distributed. It is therefore
recommended for normally distributed and skewed data sets.
