Table 3. Standard Deviations and Correlations of HP-filtered Output, Labor, and Investment (Means and Root Mean Squared Errors)

Standard Deviations of HP-filtered Series Autocorrelations of HP-filtered Series  Cross-correlations of HP-filtered Series
Output and Output and Labor and
Output  Labor  Investment Output  Labor  Investment  Labor  Investment Investment
True  1.9  2.4  6.9  .70  .69  .69  .89  .91  .92

Restricted SS

Tight  1.9  2.4  6.8  .69  .68  .68  .89  .91  .92
constraints  (.086)  (.13)  (.36)  (.008)  (.011)  (.011)  (.013)  (.012)  (.011)
Modest  1.9  2.4  6.8  .69  .68  .68  .89  .91  .92
constraints  (.097)  (.13)  (.37)  (.011)  (.015)  (.014)  (.015)  (.013)  (.011)
Loose  1.9  2.4  6.8  .69  .68  .68  .89  .91  .92
constraints  (.098)  (.13)  (.38)  (.012)  (.017)  (.016)  (.015)  (.013)  (.011)
Unrestricted SS  1.9  2.3  6.7  .68  .67  .67  .89  .91  .92
(.13)  (.17)  (.47)  (.035)  (.034)  (.035)  (.020)  (.017)  (.014)
VARMA  1.9  2.4  6.9  .70  .69  .69  .89  .91  .92
(.14)  (.16)  (.51)  (.036)  (.029)  (.035)  (.018)  (.018)  (.015)

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 second moments reported in the table. ‘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.