where zt ≡ [yt(3)-yt_1(3), yt(12)-yt_1(12), yt(36)-yt.1(36), yt(60)-yt.1(60), yt(120)-yt_ι(120)]z.
(8) ECM(1) with one common trend:
zt.h /1 = c + r zt,
where zt ≡ [yt(3)-yt.1(3), yt(12)-yt(3), yt(36) ~yt(3), yt(60)~yt(3), yt(120) ^t(3)]'∙
(9) ECM(1) with two common trends:
zt.h /1 = c + Γ zt t
where zt ≡ [yt(3)-yt. 1(3), yt(12)-yt.1(12), y((36)-yt(3), y((60)-yt(3), yt(120)-yt(3)]z.
(10) Direct regression on three AR(1) principal components
We first perform a principal components analysis on the full set of seventeen yields yt,
effectively decomposing the yield covariance matrix as QΛQ T, where the diagonal
elements of Λ are the eigenvalues and the columns of Q are the associated eigenvectors.
Denote the largest three eigenvalues by λ1, λ2, and λ3, and denote the associated
eigenvectors by q 1, q2, and q3. The first three principal components xt ≡ [x 11, x21, x31]
are then defined by xtt = qiyt, i = 1,2, 3. We then use a univariate AR(1) model to
produce h-step-ahead forecasts of the principal components:
∖t+h/1 = c, + Y,x,t, i = 1, 2, 3,
and we produce forecasts for yields yt ≡ [yt(3), y((12), y((36), y((60), y((120)]z as
yt+f,∖t (τ)= q ɪ(ɪ) x1,t+ht+ q 2(τ) ∖t+ht+ q ɜ(ɪ) ^tt,
where qi(τ) is the element in the eigenvector qi that corresponds to maturity τ.
We define forecast errors at t+h as yt+h(τ) ~yt+hht(τ). Note well that, in each case, the object being
forecast ( yt+h (τ)) is a future yield, not a future Nelson-Siegel fitted yield. We will examine a number of
descriptive statistics for the forecast errors, including mean, standard deviation, root mean squared error
(RMSE), and autocorrelations at various displacements.
Our model’s 1-month-ahead forecasting results, reported in Table 4, are in certain respects
humbling. In absolute terms, the forecasts appear suboptimal: the forecast errors appear serially
correlated. In relative terms, RMSE comparison at various maturities reveals that our forecasts, although
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