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CPS1947 Hsein K. et al.
these predictor variables are highly persistent and are often integrated of
order one. However, they did not consider the case in which the predictor
T
could potentially cointegrated; that is −1 ~ (0) . We thus extend the
linear multivariate predictive regression model, focusing on predictors that
can plausibly be modelled as cointegrated and to allow the possibility that
stock return depends in a nonlinear way on predictors.
We use Goyal and Welch updated quarterly data over the 1927-2017
sample period. The dataset were obtained from Amit Goyal's website at
http://www.hec.unil.ch/agoyal. Their dataset is one of the most widely used
datasets in research on stock return predictability. The dependent variable, ,

is the US equity premium, which is defined as the log return on the S&P 500
index including dividends minus the log return on a risk-free bill.

2. Methodology
To allow for potential non-linearity and cointegration among the
predictors, we consider a semiparametric single index model of the form

T
= ( −1 ) + ℯ

0
0

T
where = ( , . . . , , ) is a vector of d-dimensional potential integrated
1,

predictors, (. ) is an unknown nonlinear integrable function and is often
called the link-function in the literature, is the single index parameter such
0
T
that −1 is stationary and ℯ is a martingale difference sequence. Thus our
0

model allows for the presence of cointegration among the integrated
predictors. Also our model includes the linear parametric multivariate
predictive model as a special case since function (. ) can take a linear form.
In the case of a univariate integrated predictor with = 1, Kasparis, Andreou
0
and Phillips (2015) established the statistical theory for the estimation of the
(. ) function. Following the estimation procedure discussed in Dong, Gao and
Tjostheim (2016), a profile approach is used to derive the estimators of the
unknown link-function and the unknown single index parameters.
Among the 14 financial and macroeconomic variables that Goyal and
Welch (2008) use to predict the equity premium, we consider the following
four pairs of potentially cointegrated variables: (a) dp and ep; (b) 3-month T-
bill rate (tbl) and long-term yield (lty); (c) baa and aaa rated corporate bond
yields; and (d) dp and dividend yield (dy). Goyal and Welch (2008) provide the
definitions and sources of these predictors.

3. Result
For initial illustration, Figure 1 plots those four pairs of variables using
quarterly data in the subperiod 1952--2017 and demonstrates that the two
series in each of the four pairs considered are positively correlated and they
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