The name is absent

test. In addition, the Hansen test is not subject to criticism regarding both the CUSUM and CUSUM
of squares proposed by Brown, Durbin, and Evans (1975). In particular, the former has been
criticized for being a trivial test to detect instability in the intercept of a model; while the latter
suffers from poor asymptotic power. The test proposed by Hansen has local optimal power. The

variables in the linear model (5) must be weakly dependent process, that is they cannot contain any
deterministic or stochastic trend¹⁰. A linear model returns efficient estimates when the disturbance
covariance between noise and the explanatory variables is necessary to presume the model is
correctly specified E(x_t^I e_t)= 0 . The Hansen test statistics are based on the cumulative sums of the
aforementioned first-order conditions. The test is used to check for both individual (Li) and joint
(Lc) parameter stability. Asymptotic critical values for the individual stability test are provided by
Hansen (1992)¹¹. The null hypothesis of joint parameter constancy implies that the first-order
conditions are zero-mean, i.e. the cumulative sum tends to be distributed around zero. The non-
standard distribution is tabulated by Hansen (1992)¹².

There are three explanatory variables in the Campbell-Shiller equation (5) including both the
constant and the errors variance. Test results are displayed in Table 4. The calculated statistics
associated to the expectations hypothesis equation are extremely high for any couple of maturities
(n, m); the null hypothesis of parameter constancy is decisively rejected. The Hansen test thus
suggests clear parameter instability.

Table 4

¹⁰ As shown in Section 3 all the variables are covariance stationary.

¹¹ The 5% critical value is 0.47; while the 10% is 0.353. Large values of the test statistics implies a violation of the first-
order conditions, and thus lead to the rejection of the null hypothesis of parameter stability.

¹² At 5% significance level the critical value is 1.01, while the 10% critical value is 0.846. The null hypothesis of joint
parameter stability is rejected if the test statistics exceeds the critical values.

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