Palgrave Handbook of Econometrics: Applied Econometrics

Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1055

by a no-change model:

st+k−st=εt+k, k=1, 4, 8, 12, 16 quarters, (22.79)

and motivates the use of long-horizon regressions:

st+k−st=ak+βkzt+εt+k, k=1, 4, 8, 12, 16 quarters. (22.80)

The exchange rate is expected to rise (fall) when it is below (above) its fundamental
value. Thus the slope coefficientβkshould be positive and statistically significant.
Formally, the null hypothesis to be tested isH 0 :βk =0 against the alterna-
tiveH 1 :βk>0, or the joint hypothesisH 0 :βk= 0 ∀kversusH 1 :βk>0 for somek.
The evaluation of the out-of-sample performance of the model involves generating
recursive or rolling forecasts based on equations (22.66) and (22.79) and estimat-
ing a forecast evaluation statistic such as Theil’sU-statistic, or theDM-statistic of
Diebold and Mariano (1995).
In evaluating the statistical significance of the results Mark confronted a number
of econometric problems. First, a highly persistent explanatory variable implies
biased OLS estimates of the slope coefficients in finite samples (see Neely and
Sarno, 2002, and the references therein). Second, the fact that thek-period change
instis used as the regressand induces serial correlation in the disturbances of
order(k− 1 ), fork>1. In order to correct for serial correlation Mark used the
Newey–West covariance matrix estimator based on either a fixed truncation lag of
20 or a truncation lag specified by Andrews’ (1991) procedure. Finally, although
theDM-statistic follows the standard normal distribution for non-nested models,
long-horizon regressions nest the RW model and, therefore, the distribution of
theDM-statistic is not known in general (McCracken, 1999). To this end, Mark
proposed a bootstrap procedure:

1. Estimate the following vector autoregression (VAR), where the null of no
   predictability has been imposed:

st=a 0 +us,t, (22.81)

zt=μ+

∑p

j= 1

bjzt−j+uz,t. (22.82)

2. Use the estimates of the fitted model and draw from the bivariate normal dis-
   tribution with mean 0 and covariance matrix equal to the covariance matrix
   of the estimated residuals, in order to recursively generate pseudo observations
   forstandzt. Alternatively, if the error term is not normal, resample from
   the observed residuals.


3. In turn, estimate the long-horizon regression so as to obtain the slope coefficient,

thet-statistics andR^2 for the simulated series and generate forecasts based on

the monetary and the driftless RW model and compute Theil’sU-statistic and

theDM-statistic.