Nonparametric cointegration analysis

Lemma 9. For k = 1,2,...,

n

1∙                  τ          l-ʌ T Z,       _λ -X , -,            τ             -,

lim_n →Σ d cos[2kπ (t - 0.5)/n] = -—, and
'          2 s

( A

-L

∑ ^d_t,∙ ^cos[^{2kπ (t} - 0.5)/^n]

Ij¹     )

Thus, the problem is now similar to the previous problem of unconstrained drift, but
the cure is much simpler:

Theorem 8. If we conduct our tests on the basis of moving averages of s adjacent z_t’s, where
s is the number of seasons, using the weight functions F_n,k , then the results of Theorems 1
through 7 carry over to cointegrated systems with seasonal drift.

By taking these moving averages, the lag polynomial C(L) becomes C_s(L) = [(1/s)Σ^ss=^l0L^j ]C(L).
Since C_s(1) = C(1), all our results go through.

8. Empirical and Monte Carlo comparison with Johansen’s approach
8.1. The data

In this section we compare our nonparametric approach with Johansen’s tests, using
the Nelson-Plosser (1982) data, extended by Schotman and Van Dijk (1991) to 1988. This
data set consists of fourteen annual macroeconomic U.S. time series, where the longest series
starts at the year 1860. The series involved are: CPI, GNP deflator, employment,
unemployment, GNP, real GNP, GNP per capita, wage, real wage, index of industrial
production, money, stockprices, velocity of money, and the interest rate. All variables are in
logs, except the interest rate. In order to select candidates among these series for our
cointegration analysis, we have first applied the Phillips-Perron (1988), Bierens-Guo (1993)
and Bierens (1993) unit root and trend stationarity tests to these fourteen series, augmented

25

<<   22   23   24   25   26   27   28   >>

More intriguing information