direct multi-step estimation and forecasting - OFCE - Sciences Po

Direct multi-step estimation and forecasting
The significant aspect is that the maximum likelihood estimate of α is that of α 2 too.
Johnston, Klein, and Shinjo compare several forecasting methods for the AR(1) with different
parameter values and apply their techniques to the Wharton model. Their idea is to compute
estimators and resulting forecasts which incorporate the dynamic structure of the data generating
process. The hypothesis is that ‘systems using up to pth order generated lag values as instruments
or regressors will perform best in p period point prediction’. In general, the DLS estimator for a
model such as
A (L) y t = ɛ t ,
where A (L) is a matrix polynomial and L the lag operator, is that which minimizes the criterion:
tr ∑ T
t=1
(A (L) −1 ɛ t
) (
A (L) −1 ɛ t
) ′
.
In the univariate AR(1) case from (4), this is:
∑ T (
˜α = argmin yt − y 0 α t) 2
. (5)
α t=1
The procedure used by the authors for actual minimization is a grid search. The results of Monte
Carlo simulations with fixed or stochastic initial values and various stationary values of α show that
the variance of the DLS estimator is higher than that of OLS when the initial value is stochastic,
but lower for a fixed initial value. In terms of mean-square forecast error (MSFE), their results are
that for small samples (either 20 or 50 observations and the forecast horizons, respectively, of 5 or
10 periods) DLS outperforms OLS when the initial value is fixed, but when the latter is stochastic,
the forecast loss is lower for DLS only for very small samples.
The authors then use Two-Stage Least Squares estimators for the Wharton model. They match
the values of the lag used for the endogenous variable as an instrument and the horizon at which
it is desired to forecast. Unfortunately, their results for out-of-sample prediction are somewhat
inconclusive. Some gains are obtained at short horizons and seem to improve with the lead in the
forecast for Personal Income and Total Consumption but not for the G.N.P. deflator, Investment
in Non-farm inventories and Unemployment rate.
The G.N.P. deflator is the only variable for
which the within-sample residual standard error and the post-sample root-MSFE are of the same
magnitude. The latter is much larger than the former as regards the other variables.
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