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6.6. NOISE-TRADERS 201implied slope coefficient in a regression of the future depreciation onthe forward premium is negative.Next, if we compute the implied second moments of the deviationfrom uncovered interest parity and the expected depreciationCov([x t − E t (∆s t+1 )],E t (∆s t+1 )) =k(1 − µ)(1 − k(1 − µ))σx 2 − (1 − µ) 2 σu, 2 (6.71)Var(x t − E t (∆s t+1 )) = (1 − µ) 2 [k 2 σx 2 + σ2 u ], (6.72)Var(E t (∆s t+1 )) = Var(x t − E t (∆s t+1 )) + [1 − 2k(1 − µ)]σx 2 . (6.73)We see that 1 − k(1 − µ) < 0alsoimplesthatFama’sp t covariesnegatively with and is more volatile than the rationally expected de- ⇐(125)preciation. The noise-trader model is capable of matching the stylizedfacts of the data as summarized by Fama’s regressions.Matching the Survey Expectations. The survey research on expectationspresents results on the behavior of the mean forecast from a survey ofindividuals. Let ˆµ be the fraction of the survey respondents comprisedof fundamentalists and 1 − ˆµ be the fraction of the survey respondentsmade up of noise traders.Suppose the survey samples the proportion of fundamentalists andnoise traders in population without error (ˆµ = µ). Then the meansurvey forecast of depreciation is ∆ŝ e t+1 = µE t (∆s t+1 )+(1−µ)E t (∆s t+1 )= µ[1−k(1−µ)]x t +µ(µ−1)u t +(1−µ)(1+µk)x t +(1−µ)µu t = x t ,whichpredicts that β 2 = 1. There is no risk premium if ˆµ = µ. In additionto β 2 =1,wehaveβ =1− k(1 − µ) =1− β 1 ,andβ 1 = k(1 − µ), which ⇐(126)amounts to one equation in two unknowns k and µ, sothecoefficientof over-reaction k cannot be identiÞed here.We can ‘back out’ the implied value of over-reaction k if we arewilling to make an assumption about survey measurement error. Ifˆµ 6= µ, then∆ŝ e t+1 = ˆµE t (∆s t+1 )+(1− ˆµ)E t (∆s t+1 )=[1+k(µ −ˆµ)]x t +(µ − ˆµ)u t , which implies, β 2 =1+k(µ − ˆµ), β 1 = k(1 − ˆµ),and β = 1 − k(1 − µ). For given values of ˆµ, β 1 , and β, wehave,k = β 1 /(1 − ˆµ), and µ =(β − 1+k)/k. For example, if we assume thatˆµ =0.5, the 3-month horizon BIC-US results in Table 6.4 imply thatk =11.94 and µ =0.579.