The Academy of Economic StudiesDoctoral School of Finance and

Banking

Monetary Policy Rules Evaluation using a Forward

Looking Model for Romania

MSc student Muraraşu Bogdan

Coordinator Professor Moisă Altăr

John Taylor (1998): “researchers first build a structural model of the economy, consisting of mathematical equations with estimated numerical parameter values. They then test out different rules by simulating the model stochastically with different policy rules placed in the model. One monetary policy rule is better than another monetary policy rule if the simulation results show better economic performance.”

I motivate the importance of my topic by the following remark of

CONTENTS• Policy evaluation with a forward

looking model• Estimation (calibration) of the

model• Klein algorithm (generalized

Schur decomposition)• Central bank’s loss function and

the optimization problem• Optimal monetary policy rules

The forward looking model•Coats, Laxton, and Rose (2003) argued that in order to support

the policy decisions necessary to respect a target for inflation, the framework had to be forward-looking and capable of dealing with the process of controlling inflation.

1 1 1 2 2 1

1 1 2 1 2 1

1 2 3 1

( )

[ (1 ) ]

gap gap gap gap gap devt t t t t t t t

dev gap dev devt t t t t t

gap dev gap gapt t t t t

y E y a y a y b r u

y E v

r y r w

•Another specification of the system includes the real effective exchange rate in the IS curve. Taking into consideration that are no great differences between the two cases regarding the methodology and even the main results, I will describe the procedure I follow referring to first model. •This model introduces two layers of complexity: 1. agents’ actions depend upon expected future output and inflation which may cause the existence of zero or many reduced form equations; 2. the system must be solved for simultaneity.

(1)

Estimation vs. Calibration

• Problems: • data are very limited, both in terms of the coverage and the duration

of series • data sample is very short and describes a period of major structural

change in the economy and major change in policy regimes These are reasons to expect very imprecise

identification of the parameters from any estimation.

•Solutions:•I chose a full information method of estimation (3SLS) in order to solve for simultaneity•after estimation I kept the coefficients that were statistically or economically significant •I applied a kind of calibration for the coefficient from the Phillips curve which is statistically and economically inconsistent

Data used in estimation

• The model is fitted to quarterly data for the Romanian economy for 1998Q1 – 2006Q2, subject to the restriction that the coefficients of the policy rule minimize a quadratic loss function.

deviation of inflation from its target (inflation is measured as a percentage change of headline CPI, quarter-over-quarter, at annual rates and is seasonally adjusted using Demetra (Tramo-Seats))

devt

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

10.6

10.7

10.8

10.9

11.0lpibrsa95 Trend_lpibrsa95

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

-0.025

0.000

0.025

Detrended_lpibrsa95

gapty

gaptr

For the interest rate gap I applied a Hodrick-Prescott filter to the data and I computed the gap as a deviation from the trend.

Structural parameters

• quality of the instruments in 3SLS estimation

The stability of the coefficients from the two curves across the interval of variation of

Structural system is written in Klein format as

1

1

1

t t tgap gap

t t t tdev dev

t t t t

X X u

A E y B y C v

E w

'1( , , , )gap dev gap gap

t t t t tX y r y

the forward looking or non-predetermined variables

is a vector of predetermined variables

1 1, gap devt t t tE y E

1 2

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 0

0 0 0 0 0

A

3 2 1

1 2

1 2

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0

1 0 0 0 0 0

0 1

(1 ) 0 0 0 1

B

a b a b

0 0 0

0 0 0

0 0 1

0 0 0

1 0 0

0 1 0

C

1t t tX AX Reduced form of the system (Klein(2000) algorithm)

(1)

Klein algorithm (generalized Schur decomposition)

• solves systems of linear rational expectations • the system need to be solved distinctly for the predetermined

variables (or backward-looking in the language of Klein) and non-predetermined ones ( or forward looking variables)

• infinite and finite unstable eigenvalues are treated in a unified way• preferable from a computational point of view to other similar

numerical methods

• for the pair of square matrices from the equation (1)

the orthonormal matrices and the upper triangular matrices exist such that : (2)

• The generalized eigenvalues of the system are the ratios where and are the diagonal elements of and

• The decomposition matrices can be transformed so that the

generalized eigenvalues are arrayed in ascending modulus order (stable eigenvalues come first corresponding to backward looking variables and unstable come next corresponding to forward looking variables)

and S T and Q Z

' ', ' ' and ' 'A Q SZ B Q TZ QQ ZZ I

( , )A B

/ii iiT SiiT iiS T S

Solutions•)

•)

•)

•) (3)

(4)

Reduced form• Now I have the structural system (1) written in the reduced form as:

• is a vector of predetermined variables

1 111 11 11 11 1( )t t tX Z S T Z X Lz

'

1( , , , )gap dev gap gapt t t t tX y r y

•Taking into account equation (5) we can recover the covariance matrix of structural errors from the covariance matrix of reduced form errors with the relationship:

(5)

1111 'L L

Loss function• The central bank chooses the values for the coefficients from

the reaction function that minimize the loss function:

'0

0

tt t

t

Loss E X WX

W is a matrix of policy weights that represent the relative importance to the central bank of stabilizing inflation, output and interest rate (stabilization objectives).

•These weights range between zero and one and sum to one in order to determine whether the performance of the policies is sensitive to policy objectives (represented by the weights assigned to stabilize inflation, output and respectively interest rate).

•By minimizing the loss function I also obtain optimal values for the coefficients of the reaction function

Computation of the loss function

' ' '0 0 0

0 0 0

' '0 0 0 0 0

0

[ ( )] [ ( )]

[ ( ( )( ) ( )( ) ] [ ( )]

t t tt t t t t t

t t t

tt t t t t t

t

Loss E X WX trace WE X X trace W E X X

trace W E X E X X E X E X E X trace W M N

[ ]Loss trace WM

' ' 2 2 ' 1 1 '0 0 0( )( ) ( ) ... ( )k k

k k k kE X E X X E X A A A A A A

'0 0 0

0

2 ' ' 1 1 '

1 2 ' 1 1 '

'0 0

0

( )( )

[ ] ... [ ... ( ) ] ...

(1 ) [ ... ( ) ...]

( )( ) 0.

tt t t t

t

k k k

k k k

tt t

t

M E X E X X E X

A A A A A A

A A A A

N E X E X

Because the reduced form errors are linear combinations of the serially uncorrelated structural errors, they are serially uncorrelated.

Correlograms and serial correlation LM test for the structural errors

• Tests for no autocorrelation of the residual (residual from IS curve)

• Tests for no autocorrelation of the residual (residual from Phillips curve)

Alternative policy rules•The interest rate rules proposed by John Taylor are the most used ones. Taylor Rule with Interest Rate Smoothing:1 2 3 1

gap dev gap gapt t t tr y r

•Original Taylor Rule (Taylor, 1993) assigns exact coefficient values that describe Federal Reserve policy: 1 2 31.5; 0.5; 0.

•Optimal Taylor Rule: but chooses the values for and that minimize the loss function of the central bank

3 0 1 2

•Taylor Backward-Looking Rule, where lagged values of output and inflation replace the current values of the two variables:1 1 2 1

gap dev gapt t tr y

•Full State Rule (respond to all, rather than a subset, of the variables in the state vector): 1 1 2 1 3 1 4 2

gap dev gap gap gapt t t t tr y r y

•Woodford (2002) attributes to Goodhart a simple rule where the central bank responds only to deviations of the inflation rate from its target value: and choosing an optimal value for

2 3 0 3

•Clarida, Gali and Gertler (1998) suggest that forecast-based rules are optimal for a central bank with a quadratic objective function: 1 1[ ]gap

t t tr E

1

1

1

t t tgap gap

t t t tdev dev

t t t t

X X u

A E y B y C v

E w

1 2

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 0

0 0 0 0 0

A

3 2 1

1 2

1 2

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0

1 0 0 0 0 0

0 1

(1 ) 0 0 0 1

B

a b a b

'1( , , , )gap dev gap gap

t t t t tX y r y

Results• Table 1 reports the policy rule that achieved the lowest loss level for

each set of policy objective weights considered.

gapyW

devW

Taylor Rule with Interest Rate Smoothing 1 2 3 1gap dev gap gapt t t tr y r

Goodhart Rule 1gapt tr

Expected Inflation Rule 1 1gapt t tr E

In the case where NBR gives an important weight to inflation stabilization, as this is its primary objective and output represents an important but secondary objective, the Taylor Rule with Interest Rate Smoothing is the best rule to adopt.

Relative performance of the rules The figure shows that the Taylor Rule with Interest Rate Smoothing performs at all times better than the Taylor Backward Looking Rule.

When the NBR is preoccupied by the stability of output then it has to respond currently to output gap and not with a lag.

Taylor with Interest Rate Smoothing vs. Full State Rule and Goodhart Rule

The figure shows the superiority of the Taylor rule against the rule which takes into consideration the entire state vector. This rule performs better than the Taylor type rule only when the stability of inflation is the only objective of the central bank.

The figure shows that this simple rule can perform better than the Interest Rate Smoothing Rule when the output weight is small and also that the performance of this rule is not sensitive to weight assigned to interest rate stabilization.

Full State Rule vs. Expected Inflation and Taylor Backward Looking vs. Optimal Taylor

• the central bank should not adopt a policy rule in which the nominal rate of interest responds only to changes in the current expectation of future inflation

• the conclusion is that the central bank performs better if it conditions its policy on current rather than lagged economic variables

Impulse responses to positive demand shock for four policy rules, namely: Taylor Rule with Interest Rate Smoothing; Full State Rule; Backward Looking Rule and Goodhart Rule • Taylor Rule with Interest

Rate Smoothing • Full State Rule

• Backward Looking Rule • Goodhart(interest conditioned on current inflation)

Impulse responses to positive demand shock of expected inflation

and output • Taylor Rule with Interest Rate Smoothing; • Full State Rule

• Backward Looking Rule; • Goodhart Rule

•It is clear that the Taylor Rule with Interest Rate Smoothing achieves a much more stable output gap and inflation, in spite of a relatively small increase in the nominal interest rate. This is achieved by credibly committing to a fixed coefficient rule that conditions the short-term interest rate to current economic variables and to lagged interest rate.

Conclusions

•Taylor Rule with Interest Rate Smoothing responds better to economic conditions in Romania

•A central bank like ours, which takes care mostly about stabilizing inflation and is concerned about the economic stability, should control the interest rate using a Taylor Rule with Interest Rate Smoothing. •Paper provides evidence on the practical importance to a central bank of analyzing the performance of the commitment mechanism •In future work, I intend to compare the performance of fixed coefficients rules to unconstrained optimal commitment policy and discretionary policy, two alternatives proposed by Clarida, Gali and Gertler (1999).

Reference• Anderson, E. W., L. P. Hansen, E. R. McGrattan, and T. J. Sargent, (1996),

“Mechanics of forming and estimating dynamic linear economies,” in Amman, H. M., David A., Kendrick, and J. Rust, eds., Handbook of Computational Economics 1, Handbooks in Economics 13, Elsevier Science, North-Holland, Amsterdam, 171-252.

• Batini, N., and A. Haldane, (1998), “Forward-Looking rules for monetary policy”, Presented at the NBER Conference on Monetary Policy Rules.

• Batini, N., R. Harrison, and S. P. Millard, (2002), “Monetary policy rules for an open economy”, The Bank of England’s working paper.

• Bernanke, B., M. Gertler, and S. Gilchrist, (1998), “The financial accelerator in a quantitative business cycle framework”, NBER Working Paper 6455.

• Blanchard, O. J. and C. M. Kahn, (1980), “The solution of linear difference models under rational expectations”, Econometrica 48, 1305-11.

• Chadha, J. S., and L. Corrado, (2006), “Sunspots and Monetary Policy”, Centre for Dynamic Macroeconomic Analysis working papers series.

• Clarida, R., J. Gali, and M. Gertler, (1998), “Monetary policy rules and macroeconomic stability: Evidence and some theory”, NBER Working Paper 6442.

• Clarida, R., J. Gali, and M. Gertler, (1999a), “The science of monetary policy: a new Keynesian perspective”, Journal of Economic Literature XXXVII, 1661-1707.

• (1999b), “Inflation dynamics: a structural econometric analysis”, Journal of Monetary Economics 44, 195-222.

• Coats, W., D. Laxton, and D. Rose, (2003), “The Czezh National Bank’s Forecasting and Policy Analysis System”, The Czech National Bank’s working paper.

• Edwards, S., (2006), “The relationship between exchange rates and inflation targeting revisited”, NBER Working Paper 12163.

• Fic, T., M. Kolasa, A. Kot, K. Murawski, M. Rubaszek, and M. Tarnicka, (2005), “ECMOD Model of the Polish Economy”, The National Bank of Poland’s working paper.

• Giannoni, M. P., (2006), “Robust Optimal Policy in a Forward-Looking Model with Parameter and Shock Uncertainity”, NBER Working Paper 11942.

• Givens, G., 2002, “Optimal monetary policy design: solutions and comparisons of commitment and discretion”, University of North Carolina.

• Hansen, L. P. and T. J. Sargent, (1980), “Formulating and estimating dynamic linear rational expectations models”, Journal of Economic Dynamics and Control 2, 7-46.

• Klein, P., (2000), “Using the generalized Schur form to solve a multivariate linear rational expectations model”, Journal of Economic Dynamics and Control, 24, 1405-23.

• Kydland, F. E., and E. C. Prescott, (1977), “Rules rather than discretion: The inconsistency of optimal plans”, Journal of Political Economy, 85, 473-491.

• Lubik, T. A., and F. Schorfheide, (2003), “Computing sunspot equilibria in linear rational expectations models“Journal of Economic Dynamics & Control 28, 273 – 285.

• Ljungqvist, L. and T. J. Sargent, (2000), “Recursive macroeconomic theory”, MIT Press, Cambridge, MA.

• Mohanty, M. S., and M. Klau, (2004), “Monetary policy rules in emerging market economies: issues and evidence”, BIS working paper no. 149.

• Onatski, A., and N. Williams, (2004), “Empirical and Policy Performance of a Forward-Looking Monetary Model”, Princeton University working paper.

• Przystupa, J., and E. Wrobel, “Looking for an Optimal Monetary Policy Rule: The Case of Poland under IT Framework”, National Bank of Poland.

• Salemi, M. K., (1995), “Revealed preference of the Federal Reserve: using inverse-control theory to interpret the policy equation of a vector autoregression”, Journal of Business and Economic Statistics 13, 419-433.

• Soderlind, Paul, (1999), “Solution and estimation of RE macromodels with optimal policy”, European Economic Review, 43, 813-23.

• Soderlind, Paul, (2003), “Lectures Notes for Monetary Policy (PhD course at UNSIG)”, University of St. Gallen and CEPR.

• Svensson, L. E. O., (2000), “Open-economy inflation targeting”, Journal of International Economics, 50, 155-83.

• Taylor, John B., (1993), “Discretion versus policy rules in practice”, Carnegie-Rochester Conferences Series on Public Policy, 39, 195-214.

• Taylor, John B., (1998), “Applying academic research on monetary policy rules: an exercise in translational economics”, The H. G. Johnston Lecture, Macro, Money, and Finance Research Group Conference, Durham University, Durham England, revised.

• Woodford, M., (2002), “Interest and Prices”, Princeton University Press.