The Nobel Memorial Prize for Robert F. Engle

where z_t ≡ [yt(3)-y_t_1(3), y_t(12)-y_t_1(12), y_t(36)-y_t.₁(36), y_t(60)-y_t.1(60), y_t(120)-y_t_ι(120)]^z.
(8) ECM(1) with one common trend:

^zt.h /1 ^{= c + r z}_t^,

where z_t ≡ [yt(3)-y_t.1(3), y_t(12)-y_t(3), y_t(36) ~y_t(3), y_t(60)~y_t(3), y_t(120) ^t(3)]'∙
(9) ECM(1) with two common trends:

^zt.h /1 ⁼ c ⁺ Γ ^z_t t

where z_t ≡ [yt(3)-y_t. ₁(3), y_t(12)-y_t.1(12), y((36)-y_t(3), y((60)-y_t(3), y_t(120)-y_t(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 y_t,
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 x_t ≡ [x ₁₁, x₂₁, x₃₁]
are then defined by x_tt = qiy_t, 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 y_t ≡ [y_t(3), y((12), y((36), y((60), y((120)]^z as

y_t+_f,∖_t (^τ)⁼ q ɪ(ɪ) ^x1,t₊ht⁺ q 2(^τ) ∖t₊ht⁺ q ɜ(ɪ) ^t_t^,

where q_i(τ) is the element in the eigenvector q_i that corresponds to maturity τ.
We define forecast errors at t+h as y_t+h(τ) ~y_t+hht(τ). Note well that, in each case, the object being
forecast ( y_t+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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