INTEREST-RATE RULES: Framework for Analysis 7

The resulting historical time series for the two disturbances st could then be identified with the residuals of the model’s structural equations further.

If the model fit the properties of the U.S. time series perfectly, the vector st constructed in this way would be orthogonal to the identified monetary policy disturbance In practice, the right hand side of (1.26) does depend upon the first element of the vector of VAR residuals etl which we identify as the monetary policy disturbance. Perhaps more troubling is the observation that if the real disturbances st are generated by a law of motion of the kind implied by conjoining (1.26) with equation (1.25) for the evolution of Zu then we should not expect all three of the independent structural disturbances et that matter for the evolution of st to be revealed by data on the three variables in Zt alone. (This is because one of the VAR innovations corresponds to the monetary policy shock, so that only the other two orthogonal innovations can reveal information about the real disturbances.)

INTEREST-RATE RULES: Framework for Analysis 6  where the vector Zt is the transpose of [Z[, Zft_1, Z*_2], and U is a lower triangular matrix with ones on the diagonal and nonzero off-diagonal elements only in the first three rows, the off-diagonal elements of which are estimated so as to make the residuals in et orthogonal to one another. The first three rows of the vectors et contain the VAR residuals e^, e^t and while the other elements are zero. The number of lags included in our VAR is sufficient to eliminate nearly all evidence of serial correlation in the disturbances itat on.

INTEREST-RATE RULES: Framework for Analysis 5

To complete our model specification, we posit that interest rates are set according to a Here rt is the continuously-compounded nominal interest rate (identified with log Rt in terms of our theoretical model, and with the Federal funds rate in our empirical implementation of the model), r* is the steady-state value of r implied by the policy rule, and 7r* is the steady-state inflation rate implied by the rule read more.

In equilibrium, the steady state nominal interest rate r* must equal the sum of the equilibrium steady state real interest rate p and the steady state inflation rate 7Г*. Thus, if p is independent of the monetary policy rule (as our model implies8), the monetary authority’s choice of 7r* implies a value for r*. Thus the pair of values 7Г* and r* represent only a single free parameter in the specification of the policy rule, which we shall treat in the subsequent discussion as the choice of 7Г*.

INTEREST-RATE RULES: Framework for Analysis 4 We can think of as representing variation in the “natural” or “potential” level of output, since it is expected deviations Y — Ys, rather than deviations in the level of output relative to trend, that results in a desire by price-setters to increase the relative price of their goods, which in equilibrium requires inflation of the average level of prices. (An equilibrium in which no prices are ever changed is consistent with (1.18) as long as Yt = Yts at all times, and interest rates vary so as to ensure that t — 0 at all times. Note that the latter condition ensures that (1.10) and hence (1.11) are also satisfied at all times.)

INTEREST-RATE RULES: Framework for Analysis 3 The source of the real effects of monetary policy in our model is that prices do not adjust immediately to shocks. Following Calvo (1983), we assume that prices are changed at exogenous random intervals.4 Specifically, a fraction (1 — a) of sellers get to choose а new price at the end of any given period, wThereas the others must continue using their old prices. Of those who get to choose a new price, a fraction 7 start charging the new price at the beginning of the next period. The remaining fraction (1 — 7) must wait until the following period to charge the new price or, put differently, they must post their prices one quarter in advance. These assumed delays explain why no prices respond in the quarter of the monetary disturbance and why the largest response of inflation to a monetary shock takes place only twro quarters after the shock so.

INTEREST-RATE RULES: Framework for Analysis 2

For our numerical work, we rely upon log-linear approximations to the model’s structural equations. We assume an equilibrium in which the economy always stays near a steady state path, which represents a stationary, deterministic equilibrium in the case of no exogenous disturbances (ft = 0 and Gt — G at all times) and a monetary policy consistent with stable prices. In this steady state, output is constant at a level Y (defined below), and consumption is constant at the level С = Y — G.1 It follows that the marginal utility of reai income.

AtPu is also constant, at the value A = uc{C\ 0). We log-linearize the structural equations of the model around these steady-state values. Percentage deviations in the marginal utility of consumption uc{Ct; ft) around the steady-state value uc{C\ 0) can be written as — a(Ct — Ct), where Ct = log(Ct/C), C% is an exogenous shift variable (a certain linear combination of the elements of ft),2 and a = —uccCjuc, where the partial derivatives are evaluated at the steady-state level of consumption. With this substitution, the log-linear approximations to (1.5) and (1.6) are given by   