Table 4. Variance Decomposition of HP-filtered Output, Labor, and Investment (Means and Root Mean Square Errors)

What Fraction of Variance of  What Fraction of Variance of  What Fraction of Variance of
HP-filtered Series is Due to ε_{z}?  HP-filtered Series is Due to ε_{l}?  HP-filtered Series is Due to ε_{x}?
Output  Labor  Investment  Output  Labor  Investment  Output  Labor  Investment
True  33  2.1  11  47  67  30  23  33  62

Restricted SS

Tight  32  1.9  10  47  67  30  24  34  62
constraints  (3.0)  (0.7)  (2.2)  (8.2)  (12)  (10)  (8.6)  (12)  (12)
Modest  31  1.6  10  50  70  35  21  30  58
constraints  (4.8)  (1.0)  (2.9)  (12)  (15)  (16)  (11)  (15)  (16)
Loose  33  2.6  11  48  67  35  22  32  56
constraints  (8.5)  (3.3)  (6.0)  (19)  (23)  (24)  (14)  (21)  (22)
Unrestricted SS  29  14  18  42  50  34  31  37  50
(2.2)  (26)  (25)  (29)  (37)  (30)  (27)  (31)  (34)
VARMA  32  28  25  –  –  –  –  –  –
(28)  (37)  (29)

NOTES: For each model, parameters are estimated by the method of maximum likelihood. This is done for 1000 datasets of length 200 periods. The estimated parameters are used to compute the variance decompositions reported in the table. Totals do not necessarily sum to 100 percent because a two-sided filter is applied to the time series. ‘SS’ indicates state space model and ‘VARMA’ indicates vector autoregressive moving average model of order (1,1). For the ‘Tight constraints’ case of the restricted state space model, only ψ, σ, and the stochastic processes of the exogenous shocks are estimated. For the ‘Modest constraints,’ all parameters are estimated but the parameters are constrained to be economically plausible. For the ‘Loose constraints’ case, the only restriction imposed is that an equilibrium can be computed. The numbers in parentheses are the root mean square errors.