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Slide 10. The Basic Model
$$$$ \mathrm{Equation\ 1}: x_{t}=E_{t}x_{t+1}-\sigma\left(\hat{\imath}_{t}-E_{t}\pi_{t+1}-\hat{r}_{t}^{n}\right) $$$$ $$$$ \mathrm{Equation\ 2}: \pi_{t}=\kappa x_{t} + \beta E_{t}\pi_{t+1} $$$$ $$$$ \mathrm{Equation\ 3}: \hat{\imath}_{t}=\overline{i_{t}}+\phi_{\pi}\left(\pi_{t}-\bar{\pi}\right)+\phi_{x}\left(x_{t}-\bar{x}\right)/4 $$$$
where
$$ x_{t} $$ is the output gap, $$\hat{y}_{t}-\hat{y}_{t}^{n}$$;
$$ \hat{y}_{t}^{n} $$ is an exogenous variable, representing variations in the natural rate of output;
$$ \pi_{t} $$ is the rate of inflation;
$$ \hat{\imath}_{t} $$ is the interest rate;
$$ \overline{i_{t}} $$ is an exogenous, possibly time-varying, intercept of the Taylor Rule.
Slide 11. Figure: Estimates of the Real Natural Rate of Interest from Different Macroeconomic Models
| Period | Minimum estimate | Average estimate | Maximum estimate |
|---|---|---|---|
| 1985:Q3 | 1.75 | 4.26 | 8.20 |
| 1985:Q4 | 1.43 | 4.48 | 7.02 |
| 1986:Q1 | 2.59 | 4.38 | 6.21 |
| 1986:Q2 | -1.51 | 2.35 | 4.89 |
| 1986:Q3 | -4.60 | 1.44 | 3.93 |
| 1986:Q4 | 3.34 | 4.61 | 7.01 |
| 1987:Q1 | 3.63 | 6.31 | 11.62 |
| 1987:Q2 | 3.76 | 4.51 | 4.92 |
| 1987:Q3 | 1.80 | 3.37 | 4.14 |
| 1987:Q4 | 3.14 | 3.97 | 4.91 |
| 1988:Q1 | 3.16 | 5.77 | 8.03 |
| 1988:Q2 | 2.88 | 4.81 | 5.85 |
| 1988:Q3 | 4.32 | 7.21 | 10.87 |
| 1988:Q4 | 4.18 | 5.85 | 6.66 |
| 1989:Q1 | 5.19 | 6.41 | 8.55 |
| 1989:Q2 | 4.86 | 6.08 | 7.06 |
| 1989:Q3 | 3.81 | 6.01 | 6.99 |
| 1989:Q4 | 4.97 | 8.05 | 12.40 |
| 1990:Q1 | 2.13 | 4.95 | 6.30 |
| 1990:Q2 | 5.14 | 8.13 | 12.27 |
| 1990:Q3 | 3.68 | 6.83 | 10.34 |
| 1990:Q4 | 3.26 | 5.00 | 6.38 |
| 1991:Q1 | 2.14 | 4.02 | 6.09 |
| 1991:Q2 | 1.89 | 3.81 | 6.14 |
| 1991:Q3 | 0.55 | 1.80 | 2.50 |
| 1991:Q4 | -1.34 | 0.73 | 2.24 |
| 1992:Q1 | 0.75 | 2.92 | 7.26 |
| 1992:Q2 | -4.96 | -0.64 | 1.43 |
| 1992:Q3 | -1.35 | 1.05 | 3.14 |
| 1992:Q4 | 0.00 | 1.20 | 1.94 |
| 1993:Q1 | -0.44 | 0.38 | 2.17 |
| 1993:Q2 | -0.11 | 1.47 | 2.44 |
| 1993:Q3 | 0.79 | 3.31 | 4.89 |
| 1993:Q4 | -0.46 | 0.43 | 1.31 |
| 1994:Q1 | 1.04 | 2.01 | 2.71 |
| 1994:Q2 | 1.57 | 2.96 | 5.26 |
| 1994:Q3 | 2.32 | 4.67 | 7.23 |
| 1994:Q4 | 2.59 | 3.19 | 3.78 |
| 1995:Q1 | 3.23 | 5.12 | 9.11 |
| 1995:Q2 | 2.71 | 5.69 | 8.99 |
| 1995:Q3 | 2.80 | 4.26 | 5.67 |
| 1995:Q4 | 2.66 | 3.67 | 4.84 |
| 1996:Q1 | 1.97 | 3.97 | 5.79 |
| 1996:Q2 | 1.12 | 2.79 | 3.78 |
| 1996:Q3 | 2.75 | 3.54 | 4.45 |
| 1996:Q4 | 2.86 | 4.52 | 6.30 |
| 1997:Q1 | 3.05 | 4.39 | 5.80 |
| 1997:Q2 | 2.66 | 4.07 | 5.20 |
| 1997:Q3 | 3.98 | 4.95 | 5.97 |
| 1997:Q4 | 3.40 | 4.87 | 5.43 |
| 1998:Q1 | 3.18 | 4.61 | 6.74 |
| 1998:Q2 | 3.50 | 5.32 | 7.32 |
| 1998:Q3 | 3.55 | 4.44 | 5.07 |
| 1998:Q4 | 2.07 | 3.02 | 3.87 |
| 1999:Q1 | 3.00 | 4.68 | 7.73 |
| 1999:Q2 | 2.80 | 3.78 | 4.63 |
| 1999:Q3 | 3.02 | 3.80 | 4.65 |
| 1999:Q4 | 4.19 | 6.35 | 9.47 |
| 2000:Q1 | 3.53 | 5.18 | 7.48 |
| 2000:Q2 | 4.91 | 6.36 | 9.43 |
| 2000:Q3 | 3.92 | 6.35 | 10.40 |
| 2000:Q4 | 3.77 | 6.32 | 9.29 |
| 2001:Q1 | 1.34 | 5.23 | 6.89 |
| 2001:Q2 | -0.21 | 3.26 | 4.98 |
| 2001:Q3 | -1.32 | 1.65 | 3.14 |
| 2001:Q4 | -5.71 | -0.47 | 4.28 |
| 2002:Q1 | -3.08 | -0.96 | 1.32 |
| 2002:Q2 | -2.58 | 0.06 | 2.88 |
| 2002:Q3 | -3.96 | -0.83 | 2.80 |
| 2002:Q4 | -2.20 | 0.94 | 2.80 |
| 2003:Q1 | -3.23 | -0.86 | 2.18 |
| 2003:Q2 | -6.69 | -2.12 | 0.59 |
| 2003:Q3 | -4.68 | -1.99 | -0.11 |
| 2003:Q4 | -2.69 | -1.48 | 0.81 |
| 2004:Q1 | -1.88 | -0.29 | 1.44 |
| 2004:Q2 | -3.70 | -1.82 | 0.11 |
| 2004:Q3 | -1.75 | -0.26 | 1.52 |
| 2004:Q4 | -1.27 | 0.01 | 1.36 |
| 2005:Q1 | -2.32 | -0.37 | 1.30 |
| 2005:Q2 | -0.94 | 0.78 | 2.76 |
| 2005:Q3 | -1.88 | 0.64 | 2.86 |
| 2005:Q4 | 0.09 | 2.32 | 3.66 |
| 2006:Q1 | 0.04 | 1.57 | 3.48 |
| 2006:Q2 | 0.70 | 3.48 | 5.34 |
| 2006:Q3 | 0.83 | 3.13 | 5.36 |
| 2006:Q4 | 1.83 | 5.13 | 6.66 |
| 2007:Q1 | 1.76 | 4.29 | 5.89 |
| 2007:Q2 | 1.92 | 2.85 | 4.48 |
| 2007:Q3 | 2.39 | 3.55 | 4.75 |
| 2007:Q4 | 1.31 | 3.19 | 4.67 |
| 2008:Q1 | 0.81 | 3.41 | 4.56 |
| 2008:Q2 | -0.49 | 1.61 | 4.13 |
| 2008:Q3 | -2.37 | 0.11 | 2.42 |
| 2008:Q4 | -2.82 | 0.64 | 3.17 |
| 2009:Q1 | -9.00 | -2.32 | 2.19 |
| 2009:Q2 | -10.69 | -3.92 | 0.99 |
| 2009:Q3 | -13.26 | -6.03 | -2.59 |
| 2009:Q4 | -8.73 | -5.43 | -3.18 |
| 2010:Q1 | -8.73 | -4.76 | -1.70 |
| 2010:Q2 | -10.79 | -5.22 | -2.34 |
| 2010:Q3 | -6.68 | -3.56 | -1.25 |
| 2010:Q4 | -5.74 | -2.82 | -0.38 |
| 2011:Q1 | -3.74 | -2.27 | -0.93 |
| 2011:Q2 | -4.95 | -3.07 | -1.41 |
| 2011:Q3 | -3.89 | -2.77 | -0.49 |
| 2011:Q4 | -10.65 | -5.28 | -2.59 |
| 2012:Q1 | -5.37 | -2.93 | -1.21 |
| 2012:Q2 | -6.86 | -3.62 | -1.78 |
| 2012:Q3 | -5.80 | -3.20 | -1.67 |
| 2012:Q4 | -5.50 | -3.06 | -1.05 |
| 2013:Q1 | -5.99 | -3.19 | -1.06 |
| 2013:Q2 | -4.49 | -2.86 | -1.16 |
| 2013:Q3 | -5.22 | -3.19 | -1.48 |
| 2013:Q4 | -3.47 | -1.35 | 0.31 |
| 2014:Q1 | -3.18 | -2.34 | -1.55 |
| 2014:Q2 | -4.37 | -2.41 | -0.96 |
| 2014:Q3 | -3.05 | -1.88 | -0.45 |
| 2014:Q4 | -0.73 | 0.32 | 2.44 |
| 2015:Q1 | -2.23 | -0.96 | 0.49 |
| 2015:Q2 | -0.99 | -0.46 | -0.08 |
| 2015:Q3 | -0.87 | -0.35 | -0.07 |
*This exhibit was drawn from figure 1 in Janet Yellen, "The Economic Outlook and Monetary Policy," remarks at the Economic Club of Washington, December 2.
Note: The shaded bars indicate periods of business recession as defined by the National Bureau of Economic Research: July 1990-March 1991, March 2001-November 2001, and December 2007-June 2009.
Source: The estimates are drawn from four models: (1) a dynamic stochastic general equilibrium (DSGE) model developed by the staff of the Federal Reserve Board and described in Kiley (2013); (2) a DSGE model developed by the staff of the Federal Reserve Bank of New York and described in Del Negro and others (2013); (3) a DSGE model developed by the staff of the Federal Reserve Board based on Christiano, Motto, and Rostagno (2014); and (4) a DSGE model developed by the staff of the Federal Reserve Board based on Guerrieri and Iacoviello (2013 [rev. 2014]).