266 Claudio Pizzi
the case of homoscedasticity, a constant weight is assigned to each piece of available
sample data. On the contrary, the local approach replaces the model represented by
(1) with the following equation:
Yt=β 0 ,t+β 1 ,tXt+zt. (7)
The estimation of parameters in the local model is achieved through the following
weighted regression:
βˆt=arg min
β∈^2∑T
i= 1[
Yi−β 0 ,t−β 1 ,tXi] 2
wt^2 ,i, (8)wherewt,iindicates the weight associated to theith sample point in the estimate of
the function at pointt. They measure the similarity between the sample pointsXtand
Xiand are defined as follows:
wt,i= k[(
Xt−j−Xi−j)
/h]
, (9)
where is an aggregation operator that sums the similarities between ordered pairs
of observations. Functionkis continuous, positive and achieves its maximum in zero;
it is also known as a kernel function. Amongst the different and most broadly used
kernel functions aqnd the Epanechnikov kernel, with minimum variance, and the
gaussian kernel, which will be used for the application described in the next section.
The kernel function in (9) is dependent on parameterh, called bandwidth, which
works as a smoother: as it increases, the weightwt,iwill be higher and very similar
to each other. Parameterhhas another interpretation, i.e., to measure the model’s
“local” nature: the smallerhis, the more the estimates will be based on few sampling
data points, very similar to the current one. On the other hand, a higherhvalue means
that many sampling data points are used by the model to achieve an estimate of the
parameters. This paper has considered local linear models, but it is also possible to
consider other alternative local models models such as, for example, those based on the
Nearest Neighbours that resort to constant weights and a subset of fixed size of sample
observations. Once the local model has been estimated, a nonlinear cointegration test
can be established considering the model’s residuals and following the two-stage
procedure described by Engle and Granger. Furthermore, with an adaptation from a
global to a local paradigm, similar to the one applied to (1), equation (6) becomes:
Yt=α∗t+βt∗Xt+π 1 ,tYt− 1 +π 2 ,tXt− 1 +vt. (10)
The long-run relationship and the speed of adjustment will also be dependent on
time and no longer constant as they depend on the parametersπ 1 ,tandπ 2 ,tthat are
estimated locally.
In the next section both LECM (equation 7) and uLECM (equation 10) will be
estimated. The former to test the nonlinear cointegration hypothesis and the latter to
estimate the speed of adjustment.