In this section, we specify the family of methods by evaluating the CLMM
(2) at [x.sub.n+i], i = i, v - 1, v + 1, ..., 2v, which is also used to obtain the derivative formula given by
In contrast to the standard MCI, the alternative MCI proposed by CLMM does take into account the endogeneity of the policy instruments.
To understand the intuition for why the MCI by CLMM is a potentially much better indicator of the stance of monetary policy, it is useful to go through a simple static analytical example.
Comparing expressions (15) and (10), it is obvious that the MCIs of BT and CLMM have different weights on the short-term interest rate and house prices.
To see why the MCI of CLMM is a better indicator of the monetary policy stance, it is useful to investigate how the weights in (15) will depend on systematic policy behavior.
Substituting the coefficients [[beta].sub.1] and [[beta].sub.2] in (5) with the coefficients in expression (16), it can be verified that the MCI of CLMM will be equal to zero.
In this case, it can be shown that the MCI of CLMM is given by
This result is very intuitive: When the central bank does not respond to house price shocks and a rise in house prices has a stimulative effect on output, the MCI of CLMM will indicate easy monetary conditions whenever there is a positive shock to house prices.
To illustrate the effect of taking endogeneity of the indicators of stance into account, we also compare the MCI of CLMM (which incorporates the full set of shocks) with the MCI of BT.
Figure 7A shows for the DVAR and 7B for the LVAR the estimated 68 percent probability regions for the MCI of CLMM (blue dotted lines) and the MCI of BT (gray shaded areas) based on one-year-ahead annual output growth (left column) and two-year-ahead annual inflation (right column) using the following indicators of monetary conditions: the federal funds rate (first row); the federal funds rate and the term spread (second row); and the federal funds rate, the term spread, and real house prices (third row).
In Figure 7B, a comparison of the 68 percent posterior probability regions for the MCI of CLMM (blue dotted lines) with those for the MCI of BT (shaded areas) reveals that, although the broad messages of the estimated MCIs are similar, conditioning on the three identified exogenous shocks only (the MCI of BT) gives less-precise estimates.