Identifying the Monetary Policy Shock Christiano et al. (1999)

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, R t, and that the policy rule it uses takes the form R t = f(x t ) + ε t where X t is a set of variables the central bank responds to (inflation, output,... ), f( ) is a function that summarizes the behavior of the central bank 1 and ε t is a purely exogenous shock. 1 We will deal with one example of such a behavior later on in this chapter. 1

In this context, f(x t ) is the endogenous part of the response. This makes it clear that any shock that affects any of the variable in the vector X t will have an effect on the monetary policy. But only ε t 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 p Y t = A 1 Y t 1 + A 2 Y t 2 +... + A p Y t p + u t = A i Y t i + u t i=1 where Y t is a (n 1) row vector stacking the n variables we want to model. 2 For instance, Y t may take the form (GDP t, π t, R t ) ). u t 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 Y t = (y 1,t, y 2,t ). This VAR actually corresponds to the system of stochastic finite difference equations which rewrites ( ) y1,t y 2,t ( ) y1,t 1 = C + A 1 + y 2,t 1 ( ) u1,t u 2,t y 1,t = C 1 + A 1 (1, 1)y 1,t 1 + A 1 (1, 2)y 2,t 1 + u 1,t y 2,t = C 2 + A 1 (2, 1)y 1,t 1 + A 1 (2, 2)y 2,t 1 + u 2,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)Y t = u t where A(L) is a matrix polynomial that takes the form A(L) = I A 1 L A 2 L 2... A p L p 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 Y t = C + A 1Y t 1 + A 2Y t 2 +... + A py t p + u t where the vector C is the constant. 2

Let us go back for a while to the simple example where Y t takes the form (GDP t, π t, R t ) ) 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 GDP t = A 11 GDP t 1 + A 12 π t 1 + A 13 R t 1 + u 1,t π t = A 21 GDP t 1 + A 22 π t 1 + A 23 R t 1 + u 2,t R t = A 31 GDP t 1 + A 32 π t 1 + A 33 R t 1 + u 3,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, u 3,t would represent a shock to monetary policy. This is actually flawed. Indeed, the covariance matrix of the vector (u 1,t, u 2,t, u 3,t ) is very likely to be non diagonal and will rather take the form σ 11 σ 12 σ 13 Σ = σ 12 σ 22 σ 23 σ 13 σ 23 σ 33 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: u t = Sε t where E(ε t ε t) = I. A direct implication of this assumption is that Σ = SS 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 s 11 s 12 s 13 S = s 21 s 22 s 23 s 31 s 32 s 33 In other words, we have u 1,t = s 11 ε 1,t + s 12 ε 2,t + s 13 ε 3,t u 2,t = s 21 ε 1,t + s 22 ε 2,t + s 23 ε 3,t u 3,t = s 31 ε 1,t + s 32 ε 2,t + s 33 ε 3,t and we would like to qualify one of the three εs, say ε 3,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 ε 3,t as the monetary policy shock. How many restrictions should be placed? The answer to this question 3 But this could have well been ε 1,t or ε 2,t. 3

is dictated by identification restrictions. The S matrix is computed by solving the system of equations SS = 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 (198), which amounts to assume that the system is recursive. More precisely, we will assume that s 12 = s 13 = s 23 =. In other words u 1,t = s 11 ε 1,t u 2,t = s 21 ε 1,t + s 22 ε 2,t u 3,t = s 31 ε 1,t + s 32 ε 2,t + s 33 ε 3,t such that the VAR system can be rewritten as GDP t = A 11 GDP t 1 + A 12 π t 1 + A 13 R t 1 + s 11 ε 1,t π t = A 21 GDP t 1 + A 22 π t 1 + A 23 R t 1 + s 21 ε 1,t + s 22 ε 2,t R t = A 31 GDP t 1 + A 32 π t 1 + A 33 R t 1 + s 31 ε 1,t + s 32 ε 2,t + s 33 ε 3,t This amounts to assume that GDP only responds contemporaneously to ε 1,t which amounts to say that ε 1,t is an output shock. 5 The inflation rate π t responds instantaneously to both the output shock and ε 2,t. In other words, ε 2,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 ε 3,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 SS s 11 s 11 s 21 s 31 σ 11 σ 12 σ 13 = Σ s 21 s 22 s 22 s 32 = σ 12 σ 22 σ 23 s 31 s 32 s 33 s 33 σ 13 σ 23 σ 33 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 n 2 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. 4

which implies that s 2 11 s 11 s 21 s 11 s 31 s 11 s 21 s 2 21 + s2 σ 11 σ 12 σ 13 22 s 21 s 31 + s 22 s 32 = σ 12 σ 22 σ 23 s 11 s 31 s 21 s 31 + s 22 s 32 s 2 31 + s2 32 + s2 33 σ 13 σ 23 σ 33 where only the black part should be solved for the remaining equations begin redundant. We therefore end up with the following system of equations s 2 11 = σ 11 s 11 s 21 = σ 12 s 2 21 + s 2 22 = σ 22 s 11 s 31 = σ 13 s 21 s 31 + s 22 s 32 = σ 23 s 2 31 + s 2 32 + s 2 33 = σ 33 which is easily solved recursively as s 11 = σ 11 s 21 = σ 12 /s 11 s 22 = σ 22 s 2 21 s 31 = σ 13 /s 11 s 32 = σ 23 s 21 s 31 s 33 = s 22 Since the system admits the representation s 11 Y t = AY t 1 + s 21 s 22 s 31 s 32 s 33 σ 33 s 2 31 s2 32 ε 1,t ε 2,t ε 3,t Y t = AY t 1 + SE 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 ε 1,t = ε 2,t = for all t and ε 3,t = 1 for t = 1 and 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 196:1 to 22:4. The choice of variables in Y t 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 Y t could be corrupted by an omitted variable bias. We therefore have to adopt an intermediate empirical strategy. Y t includes the following 1 variables in that particular order: the log of real output (ŷ t ), the log of the consumption output ratio (c t y t ), the log of the investment output 5

ratio (x t y t ), the inflation rate (π t ), a measure of inflation of commodity prices ( CRB t ), the nominal interest rate (i t ), wage inflation (πt w ), a measure of profits (Prof t ), money growth (γ M2,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 Y t : Y t = (ŷ t, c t y t, x t y t, π t, CRB t π t, i t π t, πt w π t, Prof t, γ M2,t π t, (y/h) t ) 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 Y t = C + A 1 Y t 1 + A 2 Y t 2 + A 3 Y t 3 + u t 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, money 9 or productivity. Note that this identification scheme is perfectly disputable and the robustness of the results have to be checked. 1 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 and I c = I c y + I y I x = I x y + I y I i = I i π + I π I π w = I π w π + I π I M2 = I M2 π + I π 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. 1 The results are robust to changes in the ordering of variables provided the first block is maintained. 6

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 (1 draws). 1 x 1 3 Figure 1: Response to a Monetary Policy Shock Output 6 x Prices Interest Rate 1 3.5 8 6 4 2 2 5 1 Non Borrowed Reserves.2 4 2 2 5 1.2 Total Reserves.5 1 5 1 1 x Money (M1) 1 3.1.1.1.1 5.2 5 1.2 5 1 5 5 1 Note: Plain line: average impulse response across 1 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, North- Holland, 1999, chapter 3. Sims, C., Macroeconomics and Reality, Econometrica, 198, 48, 1 48. 8