Anon

208 The Basics of financial economeTrics

of the French stock index at time t; y 2 t is the log of the German stock index
at time t; and y 3 t is the log of the Netherlands stock index at time t). We use
logs of the stock market indices to smooth the series. A 0 and A 1 are n × n
matrices of parameters and ut is the n × n error matrix.
The next step is to estimate the model. This means fitting equation
(10.9). We incorporated a linear time trend, hence the inclusion of the
matrix A 0. Since there are restrictions across the equations, the procedure
uses a maximum likelihood estimation procedure and not OLS. (We explain
the maximum likelihood estimation method in Chapter 13.) The focus of
this estimation is not on the parameters of the A matrices. Few software
programs present these estimates, rather the emphasis is on the matrix B
which is estimated to determine the number of cointegrating vectors.
The estimates are presented in Table 10.6. We want to establish whether
i indices are cointegrated. Thus, we test the null hypothesis that the stock
indices lack cointegration. To accomplish this, the λ trace test statistic,
denoted by λtrace(0), is calculated (0 is included to indicate that there are
zero cointegrating vectors). Table 10.6. also provides this statistic. To insure
comprehension of this important statistic, we detail its calculation.
We have 96 usable observations.

λtrace(0) = –T[ln(1− λi*) + ln(1− λ 2 *) + ln(1− λ 3 *)]

= –96[ln(1 − 0.227) + ln(1 − 0.057) + ln(1 − 0.028)] = 33.05

As reported in Table 10.6, this exceeds the critical value for 5% significance
of 29.80^16 and has a p-value of 0.02. Thus, the null hypothesis at a 5% level
of significance is rejected with the evidence consistent with at least one coin-
tegrating vector. Next we can examine λtrace (1) to test the null hypothesis of
at most one cointegrating vector against the alternative of two cointegrating
vectors. Table 10.6 shows that λ 1 at 8.33 is less than the critical value of

table 10.6 Cointegration Test

Hypothesized No.
of Cointegrating
Vectors

Characteristic

Roots

Trace

Statistics

λtrace

5%

Critical

Value p-Value

Max-

Statistic

λmax

5%

Critical

Value p-Value

None 0.227 33.05 29.80 0.02 24.72 21.13 0.01

At most 1 0.057 8.32 15.49 0.43 5.61 14.26 0.66

At most 2 0.028 2.72 3.84 0.10 2.72 3.84 0.10

(^16) The critical values for this cointegration method are obtained from Johansen,
Likelihood-Based Inference in Cointegrated Vector Autoregressive Models.