Extended IS-LM model - construction and analysis of behavior

Extended IS-LM model - construction and analysis of behavior David Martinčík Department of Economics and Finance, Faculty of Economics, University of West Bohemia, martinci@kef.zcu.cz Blanka Šedivá Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia, sediva@kma.zcu.cz Abstract The article presents an extended IS-LM model, its construction and modification, an analysis of its behavior and a statistical verification on data of the Czech economy. The extended IS-LM model is used for analyses of efficiency of macroeconomics stabilization policy in Czech Republic under period 1998-2005. IS-LM model in various modifications always presents the standard approach of the mainstream macroeconomics to the efficiency of the stabilization policy. The fundamental diagram of the model was developed by John Richard Hicks in 1937 and it included only 3 market the aggregate market of products and services (real sector of the economy) and the aggregate money market and the aggregate market of others financial assets (monetary sector). The platform of extended model is connection between the real and the monetary sector by way of interest rate (keynesian transmission mechanism). The model describes an internal balance of the economy that is determined by the fiscal policy, the monetary policy and the exchange rate policy; further the policy of inflationary expectation and the characteristics of yield curve. These five determinants of aggregate product are statistically tested by the help of linear regression model including different time lags. Keywords IS-LM model, yield curve, expected inflation, macroeconomic stabilization policy JEL: E12 1. IS-LM and yield curve basic model The IS-LM model includes at the same time two balances, the balance of aggregate market of goods and services (product) and the balance of aggregate market of money and other financial assets. We use the linear and static model. The IS curve describes the balance of aggregate market of product: Y = α (A b i RL + v E D/F *P F /P D ) (1) Y α real gross domestic product (it is income) static multiplier of autonomous expenditure (i.e. influence the autonomous expenditure on real gross domestic product) α = 1 / (1 mpc*(1 mrit)); mpc is marginal propensity to consumption and mrit is marginal rate of income taxation

A b i RL v E D/F *P F /P D autonomous expenditure (private consumption and investment expenditure, public expenditure, foreign demand a import expenditure, all are independent on income, interest rate and real exchange rate) autonomous expenditure dependence on long term real interest rate real long term interest rate autonomous expenditure dependence on real exchange rate real exchange rate (nominal exchange rate multiple by the foreign and domestic price level ratio) The curve LM describes the balance of aggregate market of money and other financial assets: i NS = 1/h (k Y M/P D ) (2) i NS h k M/P D nominal short term interest rate demand for real money dependence on short term nominal interest rate (so-called speculative demand for real money) demand for real money dependence on real gross domestic product (so-called transaction/income demand for real money) real money (nominal quantity of money divide by domestic price level) It is needful to put in the yield curve and expected inflation between the curve IS and LM, for extension of the model. This way we transfer the short term nominal interest rate (which is determinate on aggregate market of money) to long term real interest rate (which influences the private consumption and investment expenditure). The characteristic of the yield curve is following: i NL = i NS + ε + λ + σ (3a) i NL ε λ σ nominal long term interest rate expectation of future nominal short term interest rate development liquidity premium risk premium Through the connection of yield curve and expected inflation we obtain the long term real interest rate: i RL = i NS + ε + λ + σ + π e (3b) is π e expected inflation The LM curve (2) extended of yield curve and expected inflation (3b) is following: i RL = 1/h (k Y M/P D ) + ε + λ + σ + π e (4) We solve equation IS (1) and extended LM (4) and get the determination of gross domestic product (income): Y = β A + β b/h M/P D + β v E D/F *P F /P D + β b π e β b (ε + λ + σ) (5a) β = αh/(h + αbk) we replace

β 1 = β β 2 = β b/h β 3 = β v β 4 = β b β 5 = β b (ε + λ + σ) R = E D/F *P F /P D autonomous expenditure multiplier, it is fiscal policy multiplier real money multiplier, it is monetary policy multiplier real exchange rate multiplier multiplier of expected inflation equation term containing the yield curve real exchange rate so we get the equation: Y = β 1 A + β 2 M/P D + β 3 R + β 4 π e + β 5 (5b) This equation we first multiple by domestic price level (we drop the index, so P D = P) and then date for period t and t-1. These two equations (for date t and t-1) we deduct and obtain the increase version 1 : (YP) = β 1 (AP) + β 2 M + β 3 (RP) + β 4 (π e P)+ β 5 P (5c) YP AP M RP π e P increase nominal gross domestic product (nominal income), i.e. YP = Y * P nominal autonomous expenditure, i.e. AP = A * P nominal quantity of money real exchange rate multiplied by domestic price level, i.e. RP = R * P expected inflation multiplied by domestic price level, i.e. π e P = π e * P The equation (5c) we could read this way: Increase of nominal gross domestic product = fiscal policy + monetary policy + exchange rate policy multiple by price level + expected inflation multiple by price level + yield curve parameters multiple by price level (all terms are multiplied by relevant multipliers) The equation (5c) indicates the determinants of nominal gross domestic product and is the equation of aggregate demand. The different dating of separate variables - determinants of nominal product, is useful for time lag analysis. The date basis was quarterly time series of Czech economy from 1998 to 2005 (period of relatively stable economics development after currency turbulences in 1997). There were 32 768 linear regress equations analyzed in MATLAB. These equations were developed from 8 possibilities by each variable. It was allowed the time lag from 0 to 6 quarters (7 possibilities), and the missing variable (+1). The total number of equations is 8 5 = 32 768 then. The equation (5c) was transformed to econometric form (accidental variable µ) and there was also added the absolute equation term β 0. The estimated linear regress equation is following: (YP t )= β 0 + β 1 (AP t-n ) + β 2 M t-n + β 3 (RP t-n ) + β 4 (π e P t-n ) + β 5 P t-n + µ (6) are the indexes: t current period t-n last period, as n = 0,, 6 1 The increase version is useful in the situation, when we don t know the total autonomous expenditure in the economy. Further we use this simple abstraction: the private and foreign components of autonomous expenditure are constant and then the change of total autonomous expenditure must be caused only by public expenditure. This way the total autonomous expenditure could be replaced by the fiscal policy.

The 25 best solutions according to the determination R 2 are in the table: Interval Variable time lag (n) Multiplier (AP) M (RP) (π e P) P β 0 β 1 β 2 β 3 β 4 β 5 R 2 estimation 5 4 5 2 4 17912 0,66-0,10-76145 -1281-408790 0,86 no 5 1 5 0 4 13251 0,84 0,04-70722 1782-359330 0,85 no 5 4 5 4 4 17114 0,60-0,09-62195 914-362170 0,85 no 4 2 4 4 3 11607 1,29 0,15-125320 -1462-481010 0,85 no 5 4 5 0 4 15894 0,73-0,05-70697 1048-351880 0,84 no 2 4 1 5 4 18777-0,54-0,11-29605 3179-332460 0,84 yes 2 3 6 4 4 19231 0,55-0,19-76768 2850-275370 0,84 yes 5 2 4 6 3 15464-0,62 0,14-92270 -3160-472970 0,84 yes 0 3 6 0 2 17640-0,68-0,25-50684 1786 188260 0,83 yes 5 4 5 1 4 16048 0,56-0,09-62107 780-278090 0,83 no 5 4 5 5 4 16892 0,62-0,09-58465 930-372070 0,83 no 4 2 4 1 3 12057 1,07 0,11-96070 898-438850 0,83 no 4 6 4 5 3 18855 0,84-0,16-53918 2336-473070 0,83 yes 2 4 5 1 4 17022-0,42-0,10-41533 1517-184900 0,83 no 4 2 4 6 3 12827 0,93 0,13-102200 -827-507680 0,82 no 4 2 4 2 3 12595 1,03 0,13-101200 596-500330 0,82 no 4 2 4 3 3 12210 1,17 0,14-111020 -634-508360 0,82 no 2 4 5 5 4 19051-0,51-0,11-29419 1949-363580 0,82 no 5 4 5 6 4 16724 0,68-0,09-69758 234-341230 0,82 no 5 6 5 0 4 14943 0,82-0,02-70673 1914-365770 0,82 no 5 3 5 0 4 14939 0,84-0,02-72676 1680-375160 0,82 no 5 4 5 3 4 16685 0,66-0,09-68893 -77-336320 0,82 no 5 4 5 NaN 4 16709 0,65-0,09-68476 NaN -336220 0,82 no 4 4 5 6 4 19206-0,79-0,11-59733 -2052-259250 0,82 yes 5 5 5 0 4 14654 0,88 0,01-76813 1773-404320 0,82 no (NaN not a number, i.e. a variable having non-numeric value Interval estimation no one or more variables interval estimation contains zero) The different time lags and different multipliers β 1 till β 5 (there are also negative values often) are the effect of unstability of transmissions mechanisms in the Czech economy. The low sizes of multipliers (especially β 1 and β 2 ) detect the low efficiency of stabilization economics policy generally. Next the absolute equation term β 0 is statistically significant and it means, that the growth of Czech economy is the effect of the other factors (long run growth factors), not effect of the terms of the equation of aggregate demand (short run economics policy). The increase of Czech nominal gross domestic product per quarter is between 12 and 19 billions CZK, so the annual long run trend is between 48 and 76 billions CZK and it is out of the control of stabilization policy. 2. First modification dynamic model The first modification of the basic model consists in its transformation in dynamic form. We replace the variables increases by theirs growth rate. The equation (5c) is transformed to dynamic form in the way that both sides of equation we divide by YP t-1, further we multiple each term of right side by suitable one (e.g. the first term we multiple by AP t-1 / AP t-1 ). This way the multipliers are changed to elasticity and the increases to percentual growth rate. The dynamic equation is following: yp = γ 1 ap + γ 2 m + γ 3 rp + γ 4 π e p + γ 5 p (7a)

yp ap m rp π e p p γ 1 γ 2 γ 3 γ 4 γ 5 nominal gross domestic product (nominal income) growth rate nominal autonomous expenditure growth rate nominal quantity of money growth rate growth rate of real exchange rate multiplied by domestic price level growth rate of expected inflation multiplied by domestic price level domestic price level growth rate = inflation nominal gross domestic product (yp) elasticity to autonomous expenditure yp elasticity to nominal quantity of money yp elasticity to real exchange rate multiplied by domestic price level yp elasticity to expected inflation multiplied by domestic price level yp elasticity to inflation The estimated linear regress equation is following: yp t = γ 0 + γ 1 ap t-n + γ 2 m t-n + γ 3 rp t-n + γ 4 π e p t-n + γ 5 p t-n + µ (7b) γ 0 absolute equation term µ accidental variable the indexes: t current period t-n last period, as n = 0,, 6 The 25 best solutions according to the determination R 2 are in following table: Interval Variable time lag (n) Multiplier ap m rp π e p p γ 0 γ 1 γ 2 γ 3 γ 4 γ 5 R 2 estimation 6 3 4 3 1 0,02-0,05-0,41-0,16-0,01 0,73 0,90 no 0 3 6 0 0 0,02-0,10-0,29-0,08 0,01 0,57 0,90 no 3 4 4 1 0 0,02-0,06-0,21-0,07 0,01 0,45 0,90 no 3 3 4 1 1 0,02-0,06-0,35-0,13 0,01 0,60 0,89 no 6 3 4 0 1 0,02-0,05-0,38-0,13 0,01 0,66 0,89 no 0 4 4 1 0 0,02 0,05-0,25-0,08 0,01 0,42 0,89 no 1 4 4 1 0 0,02-0,05-0,24-0,07 0,01 0,49 0,89 no 5 4 1 4 0 0,02-0,04-0,25-0,10 0,01 0,51 0,89 no 3 4 1 1 0 0,02-0,07-0,21-0,07 0,01 0,52 0,89 no 3 4 1 4 0 0,02-0,04-0,23-0,10 0,01 0,55 0,89 no 0 3 4 0 1 0,02-0,04-0,40-0,11 0,01 0,62 0,89 no 6 3 4 5 1 0,02-0,05-0,43-0,14 0,01 0,67 0,89 no 1 4 1 4 0 0,02-0,04-0,26-0,09 0,01 0,55 0,89 no 3 3 4 3 1 0,02-0,03-0,41-0,15-0,01 0,71 0,88 no 4 4 1 4 0 0,02-0,03-0,24-0,09 0,01 0,54 0,88 no 4 4 4 1 0 0,02 0,04-0,23-0,08 0,01 0,42 0,88 no 5 4 4 1 0 0,02-0,03-0,23-0,07 0,01 0,44 0,88 no 0 4 1 4 0 0,02-0,03-0,23-0,10 0,01 0,54 0,88 no 6 3 4 1 1 0,02-0,03-0,37-0,13 0,01 0,62 0,88 no 6 4 1 4 0 0,02 0,02-0,24-0,10 0,01 0,53 0,88 no 0 3 6 3 0 0,02-0,11-0,33-0,09 0,00 0,58 0,88 no 3 3 4 0 1 0,02-0,03-0,39-0,13 0,01 0,66 0,88 no 4 3 4 0 1 0,02-0,03-0,40-0,11 0,01 0,68 0,88 no 0 3 4 5 1 0,02-0,05-0,44-0,12 0,01 0,62 0,88 no

2 4 1 4 0 0,02 0,01-0,25-0,10 0,01 0,52 0,88 no (Interval estimation no one or more variables interval estimation contains zero) This model shows better results (determination R 2 ) than the basic model. The sizes of elasticities are to near to zero and the interval estimations are unreliable then. There are often negative values and it shows the unstability of transmissions mechanisms as well as the basic (statical) model. The stable estimation value shows only the absolute term the long run growth of economy. 3. Second modification time lag in product (convergent geometrical series of multipliers efficiency) The efficiency of the economics policy is spread out the time and is going through the private consumption and investment expenditures. Therefore, the stabilization policy is exogenous in the model and its efficiency describes the sum of geometrical series of the private expenditure increase. The impact of stabilization policy on gross domestic product proceeds in exponential form: ω k = ν k+1 (8) ν < 1 We transform the equation (5c) to the form: 4 k= 0 ω k (YP t+k ) = β 1 (AP t-n ) + β 2 M t-n + β 3 (RP t-n ) + β 4 (π e P t-n ) + β 5 P t-n (9) the indexes: t current period t-n last period, as n = 0,, 2 t+k future period 2, as k = 0,, 4 Estimated econometric equation is following: 4 k= 0 ν k+1 (YP t+k ) = β 0 + β 1 (AP t-n ) + β 2 M t-n + β 3 (RP t-n ) + β 4 (π e P t-n ) + β 5 P t-n + µ (10) The 25 best solutions according to the determination R 2 are in following table: Interval Variable time lag (n) Multiplier (AP) M (RP) (π e P) P β 0 β 1 β 2 β 3 β 4 β 5 R 2 estimation 0 2 1 0 0 9758-0,71-0,05-36359 571 393990 0,81 no 0 2 1 1 0 9627-0,58-0,07-24332 876 404800 0,81 no 1 2 0 1 0 8298-0,23-0,05-31611 1872 410210 0,80 no 0 1 1 0 0 9177-0,71 0,03-42967 1191 289730 0,80 no 0 2 0 1 0 8832-0,27-0,07-15927 1924 392340 0,80 no 0 2 1 2 0 9533-0,65-0,06-35707 228 429010 0,80 no 0 2 1 NaN 0 9674-0,70-0,06-36676 NaN 429200 0,80 no 1 2 0 1 0 10040-0,25-0,07-34751 2016 436290 0,80 no 0 2 2 1 0 9260-0,35-0,09-6361 1858 384870 0,79 no 0 0 1 0 0 9113-0,69 0,01-40573 1053 325430 0,79 no 0 2 NaN 1 0 9249-0,41-0,09 NaN 1915 391200 0,79 no 2 If the total size of multiplier is 2, then in five round (this is period t+4) is the efficiency 1,9375. So it is acceptable to ignore the period t+5 and further.

1 1 0 1 0 7319-0,23 0,03-41333 1911 351750 0,79 no 0 2 0 1 0 10527-0,26-0,09-19161 2097 415250 0,79 no 2 2 0 1 0 8356-0,08-0,05-27931 2145 375940 0,79 no 0 2 1 1 0 11389-0,59-0,10-23161 1097 426820 0,79 no 0 2 1 0 0 11549-0,75-0,07-38237 690 414840 0,79 no 1 2 0 1 0 12320-0,26-0,10-37909 2224 457290 0,79 no 1 0 0 1 0 7369-0,24 0,01-40056 1758 382390 0,79 no 2 2 0 1 0 10068-0,07-0,07-30492 2305 399810 0,78 no 0 2 1 2 0 11341-0,69-0,09-37886 173 457240 0,78 no 0 2 1 NaN 0 11447-0,73-0,09-38620 NaN 457380 0,78 no 0 2 2 1 0 11040-0,37-0,11-6079 2032 407820 0,78 no 2 1 0 1 0 7361-0,10 0,03-38665 2218 308890 0,78 no 0 2 NaN 1 0 11030-0,42-0,11 NaN 2086 413880 0,78 no 1 2 2 1 0 8871-0,17-0,07-19359 1844 373050 0,78 no (NaN not a number, i.e. a variable having non-numeric value Interval estimation no one or more variables interval estimation contains zero) In this model is the number of equations lower the in the previous. It was allowed the time lag from 0 to 2 quarters and it gives only four possibilities (incl. missing of variable), the number of equation is then 4 5 =1024. This impacts the decreasing of attained determination R 2. It is fully according to the theory on the average higher time lag by monetary policy ( M) then by the fiscal policy ( AP). They are often contained the unstable or negative sizes of multipliers and this validate the unstability of transmissions mechanisms and low efficiency of stabilization policy over again. Only the absolute term, that includes the long run growth factors, shows stable estimates again. References [1] BERNANKE, B. S., WOODFORD, M., Inflation Forecasts and Monetary Policy. NBER Working Paper, No. 6157, 1997. [2] DORNBUSCH, R., FISCHER, S., Makroekonomie. Praha, SPN, 1994. [3] HENDRY, D. F., Dynamic Econometrics. Oxford, Oxford University Press, 2001. [4] HU, Z., The Yield Curve and Real Activity. IMF Staff Paper, Vol. 40, No.4, International Monetary Fund, December 1993. [5] PESARAN, M. H., WICKENS, M. R., Handbook of Applied Econometrics, volume I: Macroeconomics. Oxford, Blackwell Publisher Inc., 1999.