Identifying the Monetary Policy Shock Christiano et al. (1999)
The question we are asking is: What are the consequences of a monetary policy shock —a shock which is purely related to monetary conditions— on the main aggregate variables of the economy? In other words how does the economy respond over time to a monetary policy shock? A first question we should address is why focusing on monetary policy shocks rather than the actions of the monetary policy maker? Indeed, if we are interested in the effects of money on the economy, one may be tempted to to just have a look at the effects of a change in money or in the interest rate on main macroeconomic variables. This however may be misleading. Indeed, consider the following situation: the economy faces an adverse supply shock (an example of such a shock is the 1974 oil price shock), the economy enters a recession while prices raise. The central bank intervenes by either changing its nominal interest rate or its money supply so as to impact on the economy, and manage to bring the economy back to the non–recession state and reduce the inflationary tension. If the econometrician were to infer the effects of a monetary policy shock by just observing the co–movements between output, inflation and the nominal interest rate/money supply in such case, the conclusion he will obtain would be totally spurious. What he would actually identify is the effects of the response of the central bank to a technology shock on the economy! In other words, he would get the effects of monetary policy on the economy in the aftermaths a real shock, which does not tell us anything about monetary shocks. What we want is to know how the economy reacts to a purely exogenous shock to monetary policy. From a technical point of view, this can be described in the following way. In order to keep notations consistent, let us just consider that the central bank conducts monetary policy by manipulating the interest rate, Rt , and that the policy rule it uses takes the form Rt = f (Xt ) + εt where Xt is a set of variables the central bank responds to (inflation, output, . . . ), f (·) is a function that summarizes the behavior of the central bank1 and εt is a purely exogenous shock. 1
We will deal with one example of such a behavior later on in this chapter.
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In this context, f (Xt ) is the endogenous part of the response. This makes it clear that any shock that affects any of the variable in the vector Xt will have an effect on the monetary policy. But only εt should be taken to be the monetary policy shock. Our aim is to come up with a way to identify this shock. In order to do that we will resort to statistical methods and impose some restrictions on what we think is a monetary policy shock. What type of statistical model can we use? Since we are interested in the dynamic effects of a monetary policy shock, this suggests using statistical tools that can handle dynamics. This therefore suggests that we will use a model that will explicitly take dynamics into account —such as an ARMA model. The basic tool is found in Time Series Econometrics. But since we want to be able to study the effects of monetary policy on several variables, we need to use a tool that explicitly deals with dynamic systems of equations. This tool is a Vector Autoregressive Process (VAR). From a formal point of view a VAR is a linear stochastic finite difference equations system which takes the form Yt = A1 Yt−1 + A2 Yt−2 + . . . + Ap Yt−p + ut =
p X
Ai Yt−i + ut
i=1
where Yt is a (n × 1) row vector stacking the n variables we want to model.2 For instance, Yt may take the form (GDPt , πt , Rt )0 ). ut is a vector of innovations with covariance matrix Σ. As an example of VAR, let us consider the case of a VAR(1) —meaning that each equation features 2 lags— of the vector Yt = (y1,t , y2,t )0 . This VAR actually corresponds to the system of stochastic finite–difference equations y1,t y2,t
!
!
y1,t−1 u1,t + y2,t−1 u2,t
= C + A1
!
which rewrites y1,t = C1 + A1 (1, 1)y1,t−1 + A1 (1, 2)y2,t−1 + u1,t y2,t = C2 + A1 (2, 1)y1,t−1 + A1 (2, 2)y2,t−1 + u2,t Standard results on SURE systems teach us that this system can be simply estimated by running an OLS regression for each of the equations of the system. Note that a VAR also admits a representation in terms of the lag operator A(L)Yt = ut where A(L) is a matrix polynomial that takes the form A(L) = I − A1 L − A2 L2 − . . . − Ap Lp 2 Note that for exposition purposes, we assumed that all variables are centered. Should this not be the case, this model should take an extra term into account
Yt = C + A1 Yt−1 + A2 Yt−2 + . . . + Ap Yt−p + ut where the vector C is the constant.
2
Let us go back for a while to the simple example where Yt takes the form (GDPt , πt , Rt )0 ) and for the sake of the exposition let us assume that the joint dynamics of these three variables can be represented by the VAR(1) and that all variables have been demeaned such that the system writes GDPt = A11 GDPt−1 + A12 πt−1 + A13 Rt−1 + u1,t πt = A21 GDPt−1 + A22 πt−1 + A23 Rt−1 + u2,t Rt = A31 GDPt−1 + A32 πt−1 + A33 Rt−1 + u3,t A monetary policy shock would then be a shock that affect the interest rate, as this would be the only variable the central bank can control. One would obviously be tempted to consider that a shock on the innovation associated to the last equation of the system, u3,t would represent a shock to monetary policy. This is actually flawed. Indeed, the covariance matrix of the vector (u1,t , u2,t , u3,t )0 is very likely to be non–diagonal and will rather take the form
σ11 σ12 σ13 Σ = σ12 σ22 σ23 σ13 σ23 σ33 meaning that the innovations are correlated. A simple interpretation of the later result is that innovations are just linear combinations of otherwise orthogonal shocks, ε. In other words, this amounts to assume that there exists a matrix S such that: ut = Sεt where E(εt ε0t ) = I. A direct implication of this assumption is that Σ = SS 0 We will use this result to identify a monetary policy shock. For the moment, let us keep working with our 3 variables system such that the matrix S takes the form
s11 s12 s13 S = s21 s22 s23 s31 s32 s33 In other words, we have u1,t = s11 ε1,t + s12 ε2,t + s13 ε3,t u2,t = s21 ε1,t + s22 ε2,t + s23 ε3,t u3,t = s31 ε1,t + s32 ε2,t + s33 ε3,t and we would like to qualify one of the three εs, say ε3,t ,3 as a monetary policy shock. To do that, we have to place some restrictions on the matrix S so as to be able to identify ε3,t as the monetary policy shock. How many restrictions should be placed? The answer to this question 3
But this could have well been ε1,t or ε2,t .
3
is dictated by identification restrictions. The S matrix is computed by solving the system of equations SS 0 = Σ The S matrix contains 9 different parameters while the Σ matrix only contains 6 different parameters.4 We therefore have to place 3 restrictions. We will use an identification scheme proposed by Sims (1980), which amounts to assume that the system is recursive. More precisely, we will assume that s12 = s13 = s23 = 0. In other words u1,t = s11 ε1,t u2,t = s21 ε1,t + s22 ε2,t u3,t = s31 ε1,t + s32 ε2,t + s33 ε3,t such that the VAR system can be rewritten as GDPt = A11 GDPt−1 + A12 πt−1 + A13 Rt−1 + s11 ε1,t πt = A21 GDPt−1 + A22 πt−1 + A23 Rt−1 + s21 ε1,t + s22 ε2,t Rt = A31 GDPt−1 + A32 πt−1 + A33 Rt−1 + s31 ε1,t + s32 ε2,t + s33 ε3,t This amounts to assume that GDP only responds contemporaneously to ε1,t which amounts to say that ε1,t is an output shock.5 The inflation rate πt responds instantaneously to both the output shock and ε2,t . In other words, ε2,t is the shock that is purely an inflation shock. The interest rate responds to all shocks. Note that since it responds instantaneously to both the output and inflation shock, this reveals that the central bank attempts to correct for any development happening in these variables. In other words, monetary policy has an endogenous component that tries to correct for the fluctuations in output and inflation. The interest rate finally responds to ε3,t which is exogenous. This shock can therefore be interpreted as an exogenous shock to monetary policy and therefore be interpreted as the monetary policy shock. The attractive feature of this identification scheme is twofold. First, it is very easy to interpret the so–identified shocks — which can easily be related to structural innovations.6 Second it is very easy to compute the S matrix as the system is recursive. From a mathematical point of view, this amounts to take a Cholesky decomposition of the matrix Σ.7 In our simple example, we have
s11 0 0 s11 s21 s31 σ11 σ12 σ13 0 SS = Σ ⇐⇒ s21 s22 0 0 s22 s32 = σ12 σ22 σ23 s31 s32 s33 0 0 s33 σ13 σ23 σ33 4 Remember that this is a covariance matrix and that it is therefore symmetric. More generally, if the matrix is of dimension n, then it has n(n + 1)/2 different parameters, and therefore n2 − n(n + 1)/2 = n(n − 1)/2 restrictions have to be placed. 5 Note that we are not making any statement regarding the exact status of this shock. Is this a supply or a demand shock? a fiscal or a technology shock? We do not know. 6 For instance, standard keynesian models usually place restrictions on the degree of instantaneous reactions of variables to shocks. 7 Loosely speaking the Cholesky decomposition corresponds to taking the “square root” of a matrix.
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which implies that
s211 σ11 σ12 σ13 s11 s21 s11 s31 2 + s2 s s s s s + s s σ = 11 21 12 σ22 σ23 21 31 22 32 21 22 s11 s31 s21 s31 + s22 s32 s231 + s232 + s233 σ13 σ23 σ33 where only the black part should be solved for—the remaining equations begin redundant. We therefore end up with the following system of equations s211 = σ11 s11 s21 = σ12 s221 + s222 = σ22 s11 s31 = σ13 s21 s31 + s22 s32 = σ23 s231 + s232 + s233 = σ33 which is easily solved recursively as s11 =
√
σ11
s21 = σ12 /s11 s22 =
q
σ22 − s221
s31 = σ13 /s11 σ23 − s21 s31 s32 = s22 s33 =
q
σ33 − s231 − s232
Since the system admits the representation
ε1,t s11 0 0 Yt = AYt−1 + s21 s22 0 ε2,t ⇐⇒ Yt = AYt−1 + SEt s31 s32 s33 ε3,t This system can actually exactly be thought of as a state–space model where we actually only use the state equation. We can therefore compute the impulse response function to a monetary shock very easily, by setting ε1,t = ε2,t = 0 for all t and ε3,t = 1 for t = 1 and 0 in all following periods. This is what we will do now. We consider an extended VAR model which we estimate using US quarterly data. The sample runs from 1960:1 to 2002:4. The choice of variables in Yt implies a trade–off. On the one hand, we would like to include as many variables as possible. However, this would imply estimating a very large number of parameters in a finite sample, thus yielding very imprecise estimates of impulse responses. On the other hand, a regression featuring too few variables in Yt could be corrupted by an omitted variable bias. We therefore have to adopt an intermediate empirical strategy. Yt includes the following 10 variables in that particular order: the log of real output (ybt ), the log of the consumption–output ratio (ct − yt ), the log of the investment–output 5
ratio (xt − yt ), the inflation rate (πt ), a measure of inflation of commodity prices (∆CRBt ), the nominal interest rate (it ), wage inflation (πtw ), a measure of profits (Proft ), money growth (γM2 ,t ) and productivity growth (∆(y/h)t ). Real output is detrended by fitting a linear trend on the log of real GDP.8 The consumption–output ratio is measured as the ratio of nominal consumption expenditures (including nondurables, services and government expenditures) to nominal GDP. The investment–output ratio is defined as the ratio of nominal expenditures on consumer durables and private investment to nominal GDP. We measure inflation using the growth rate of the GDP deflator, obtained as the ratio of nominal to real GDP. The commodity price is taken from the CRB. Wage inflation is measured as the growth rate of hourly compensation in the Non Farm Business (NFB) sector. The nominal interest rate is the Federal fund rate. The rate of profits is defined as the ratio of after tax corporate profits to nominal GDP. Money growth is the growth rate of M2. The labor productivity is obtained by dividing GDP by hours worked in the Non–Farm Business sector. We adopt the following specification for Yt : Yt = (ybt , ct − yt , xt − yt , πt , ∆CRBt − πt , it − πt , πtw − πt , Proft , γM2 ,t − πt , ∆(y/h)t )0 Note that because we will adopt a recursive identification scheme, the ordering of variables is important. Standard likelihood tests indicate that 3 lags are sufficient to characterize the dynamics of the system, such that the VAR takes the form Yt = C + A1 Yt−1 + A2 Yt−2 + A3 Yt−3 + ut We then adopt the same recursive scheme as the one discussed above. This implies that monetary policy is assumed to respond instantaneously and endogenously to any developments in output, consumption, investment, inflation and commodity prices. It however responds with at least one lags to changes in wage inflation, profits, money9 or productivity. Note that this identification scheme is perfectly disputable and the robustness of the results have to be checked.10 Once the identification of shocks is obtained, we build the impulse responses for all variables. In particular to get the response,I· , of consumption and investment, we compute Ic = Ic−y + Iy Ix = Ix−y + Iy and Ii = Ii−π + Iπ Iπw = Iπw −π + Iπ IM2 = IM2 −π + Iπ 8
Implicit in this procedure is the assumption that the central banker perfectly observes the GDP trend. This assumption may be questioned on the grounds that policy mistakes in the seventies are sometimes viewed as the result of not knowing the correct trend in output. 9 In fact if we assume that the central bank conducts its monetary policy through changes in the nominal interest, the money supply just accommodates the changes to clear the money market. 10 The results are robust to changes in the ordering of variables provided the first block is maintained.
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for the nominal interest rate, wage inflation and the rate of growth of money. Figure 1 reports the impulse response functions of the main macroeconomic aggregates to a negative monetary policy shock. The size of the shock is one standard deviation. The shaded area corresponds to the 95% confidence band obtained from Monte–Carlo simulations of the model (1000 draws). Figure 1: Response to a Monetary Policy Shock −3
10
x 10
Output
−3
6
x 10
Prices
Interest Rate 0.5
8 4 6
0
4
2
2
−0.5 0
0 −2 0
5 Quarters
10
−2 0
Non−Borrowed Reserves
5 Quarters
10
−1 0
Total Reserves
0.02
0.02
0.01
0.01
10
5 Quarters
10
−3 x 10 Money (M1)
5 0
0
−0.01
−0.01
0
−0.02 0
5 Quarters
10
−0.02 0
5 Quarters
10
−5 0
5 Quarters
10
Note: Plain line: average impulse response across 1000 simulations. Shaded area: 95% confidence band obtained from Monte–Carlo simulations.
Several observations are in order. First of all, following an expansionary monetary policy we observe that 1. the nominal interest rate decreases. 2. a persistent decline in real GDP, consumption, investment, profits; 3. prices and wages (inflation) are almost non responsive in the very short–run but then increase and reach a peak after 12 quarters; 4. money growth increases on impact and decreases quickly toward its initial level. (1) together with (2) and (4) constitute the so–called liquidity effect (see e.g. Christiano (1991)), while (2) together with (3) constitute the monetary transmission mechanism. Finally (1) together with (3) corresponds to standard money market equilibrium. 7
The liquidity effect refers to a situation where by fostering money supply growth, the central bank provides the money market with more liquidities. This reduces the price of money (as the amount of funds in the economy is large) and therefore the nominal interest rate. By reducing the nominal interest rate, the price of credit decreases which favors investment, consumption and output. This effect mainly transits through demand side effects, and is usually associated with a Keynesian view of fluctuations. The monetary transmission mechanism corresponds to the fact that by loosening its monetary policy, the central bank creates a positive demand side effect which raises demand and therefore puts upward pressure on the inflation rate.
References Christiano, L.J., Modelling the Liquidity effect of a Money Shock, Federal Reserve Bank of Minneapolis Quarterly Review, 1991, Winter 91, 3–34. , M. Eichenbaum, and C.L. Evans, Monetary Policy Shocks: What Have we Learned and to What End?, in M. Woodford and J. Taylor, editors, Handbook of Macroeconomics, NorthHolland, 1999, chapter 3. Sims, C., Macroeconomics and Reality, Econometrica, 1980, 48, 1–48.
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