n-m
V- (n - mq Vm m ∖-^fι(in ɪ /хч
∑l --------- l(it + mq - it + m(q-1) )= α + β(it - it )+ εt (5)
q=1 ∖ n J
Henceforth this equation will be referred to as the Campbell-Shiller equation. The above model is
equivalent to the following:
n-m
[m IX ` m -ttn 4m __. nljn jmʌ ∣ λ ∕rj∖
— l / Efi-tj.n,n — if = α + β∖it — it )+ εt (5 )
t t+mq t t t t
n J q=0
According to Thornton (2004), we label Equation (5) the conventional test. The LHS of (5) is the
theoretical, or perfect foresight, spread. The expectations hypothesis holds if the estimated
coefficients α and β are zero and one respectively, i.e. if the actual spread is equal to the
theoretical spread. The above regression generates n-period overlapping errors, causing OLS
residuals to be serially correlated; residuals thus follow a moving average stochastic process. In
order to deal with the non-spherical disturbances issue, Hansen-Hodrick (1980) and Newey-West
(1987)8 have suggested a consistent estimate of the variance-covariance matrix.
Unfortunately, in the financial literature the expectation hypothesis has found weak empirical
support; the estimated slope coefficient β in (5) is almost always below unity. The presence of a
time-varying risk premium is widely acknowledged to be a potential cause of the EH failure
(Mankiw and Miron, 1986; Fama, 1986; Cook and Hahn, 1989; Lee, 1995; Tzavalis and Wickens,
1997; Hejazi and Li, 2000).
In this paper we estimate a threshold model (Hansen, 2000) in which we use the term premium to
separate regimes. Threshold (THR) models are a special case of Markov switching models in which
the probability of switching regime is known ex ante. A natural approach to modelling non linear
economic relationships seems thus to define different states of the world, or regimes, and to allow
for the possibility that the behaviour of economic and financial variables depends upon the regime
that occurs at any different point in time. Regime switching models à la Markov imply that the
transition between regimes occurs with a certain probability that needs to be estimated. THR
models can be considered a deterministic version of Markov switching models, in which the
transition between regimes occurs whenever the threshold variable assumes a certain identified
value. In this sense, we say that Markov switching models nest threshold models.
8 L. P. Hansen, R. J. Hodrick, 1980, Forward Rates as Optimal Predictors of Future Spot Rates, in Journal of Political
Economy. W. K. Newey, K. D. West, 1987, A Simple, Positive Definite, Heteroscedasticity and Autocorrelation
Consistent Covariance Matrix, in Econometrica.
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