uncertainty on trade dynamics the volatility measure is not subject to this model reduction
procedure. From this selection procedure we obtain the following (dynamic) regression model
which is nested in (7):
∆ykt = Xktφk + ⅛. (8)
Note that the applied selection strategy can be expected to avoid unnecessarily strong re-
strictions in the sense that a test of the restrictions implied by (8) jointly against the model
in (7) is likely to support the former at a conventional significance level of 5%, say.
To concentrate on the relationship between volatility and trade growth, we first adjust
both processes for the linear impact of the right hand side variables in (8) by means of partial
regression techniques.30,31 For this purpose let yk denote the vector of stacked observations
of the dependent variable in (8) and define similarly vk as the vector of the stacked volatility
estimates vkt. Moreover, Xk is a matrix containing all explanatory variables of the respective
equation. Then a compact representation of (8) is
yk = -Xkφk + '"k . (9)
Now, let Xk = Xk \ vk denote the set of all explanatory variables in (8) other than volatility
and define
yk = (I — Xk(XkXk)-1Xk) yk and vk = (I - Xk(XkXk)-1Xk) vk, (10)
where I is the (T × T) identity matrix. The partial linear impact of FX uncertainty on trade
is then obtained from a bivariate regression model of the form
ykt = ck + vktθk + εkt, ck = 0. (11)
Although (11) is an equivalent representation of the regression (8) in the sense that θk = λ1k,
the partial linear model may be more intuitive when generalizing the impact of volatility on
trade towards a nonlinear relationship.
3.3 A semiparametric model
In the light of the theoretical discussion concerning the relation between FX uncertainty and
trade growth one may doubt the adequacy of a basically linear specification. Therefore a
semiparametric approach is discussed next, which is able to nest a wide range of relations
between ykt and vkt. Combined with suitable tools for inference such a framework is conve-
nient to detect both, local or global deviations from the so far postulated linear relationship.
The semiparametric regression model is given as follows:
y kt = E [ykt∣v = vkt] + ekt
= ak(v) + ekt. (12)