Compute the multiplier matrix of a BIMETS model
This function computes the matrix of both impact and interim multipliers, for a selected set of endogenous variables (i.e. TARGET) with respect to a selected set of exogenous variables (i.e. INSTRUMENT), by subtracting the results from different simulations in each period of the provided time range (i.e. TSRANGE). The simulation algorithms are the same as those used for the SIMULATE operation.
The MULTMATRIX procedure is articulated as follows:
1- simultaneous simulations are done;
2- the first simulation establishes the base line solution (without shocks);
3- the other simulations are done with shocks applied to each of the INSTRUMENT one at a time for every period in TSRANGE;
4- each simulation follows the defaults described in the SIMULATE help page, but has to be STATIC for the IMPACT multipliers and DYNAMIC for INTERIM multipliers;
5- given MM_SHOCK shock amount as a very small positive number, derivatives are computed by subtracting the base line solution of the TARGET from the shocked solution, then dividing by the value of the base line INSTRUMENT time the MM_SHOCK.
The IMPACT multipliers measure the effects of impulse exogenous changes on the endogenous variables in the same time period. They can be defined as partial derivatives of each current endogenous variable with respect to each current exogenous variable, all other exogenous variables being kept constant.
Given an endogenous variable at time and an exogenous variable at time the impact multiplier is defined as and can be approximated by , with the values fo the simulated endogenous variable at time when is shocked to
The INTERIM or delay-r multipliers measure the delay-r effects of impulse exogenous changes on the endogenous variables in the same time period. The delay-r multipliers of the endogenous variable Y with respect to the exogenous variable X related to a dynamic simulation from time t to time t+r can be defined as the partial derivative of the current endogenous variable Y at time t+r with respect to the exogenous variable X at time t, all other exogenous variables being kept constant.
Given an endogenous variable at time and an exogenous variable at time the impact interim or delay-r multiplier is defined as and can be approximated by , with the values fo the simulated endogenous variable at time when is shocked to
Users can also declare an endogenous variable as the INSTRUMENT variable. In this case, the constant adjustment (see SIMULATE) related to the provided endogenous variable will be used as the INSTRUMENT exogenous variable (see example);
MULTMATRIX(model=NULL, simAlgo='GAUSS-SEIDEL', TSRANGE=NULL, simType='DYNAMIC', simConvergence=0.01, simIterLimit=100, ZeroErrorAC=FALSE, BackFill=0, Exogenize=NULL, ConstantAdjustment=NULL, verbose=FALSE, verboseSincePeriod=0, verboseVars=NULL, TARGET=NULL, INSTRUMENT=NULL, MM_SHOCK=0.00001, quietly=FALSE, JACOBIAN_SHOCK=1e-4, JacobianDrop=NULL, forceForwardLooking=FALSE, avoidCompliance=FALSE, ...)
model: see SIMULATEsimAlgo: see SIMULATETSRANGE: see SIMULATEsimType: see SIMULATEsimConvergence: see SIMULATEsimIterLimit: see SIMULATEZeroErrorAC: see SIMULATEBackFill: see SIMULATEExogenize: see SIMULATEConstantAdjustment: see SIMULATEverbose: see SIMULATEverboseSincePeriod: see SIMULATEverboseVars: see SIMULATETARGET: A character array built with the names of the endogenous variables for which the multipliers are requestedINSTRUMENT: A character array built with the names of the exogenous variables with respect to which the multipliers are evaluated. Users can also declare an endogenous variable as INSTRUMENT variable: in this case the constant adjustment (see SIMULATE) related to the provided endogenous variable will be used as the instrument exogenous variableMM_SHOCK: The value of the shock added to INSTRUMENT variables in the derivative calculation of the multipliers. The default value is 0.00001 times the value of the exogenous variablequietly: see SIMULATEJACOBIAN_SHOCK: see SIMULATEJacobianDrop: see SIMULATEforceForwardLooking: see SIMULATEavoidCompliance: see SIMULATE...: see SIMULATEThis function will add a new element named MultiplierMatrix into the output BIMETS model object.
The new MultiplierMatrix element is a
(NumPeriods * Nendogenous) X (NumPeriods * Nexogenous) matrix,
with NumPeriods as the number of periods specified in the TSRANGE, Nendogeous the count of the endogenous variables in the TARGET array and Nexogenous the count of the exogenous variables in the INSTRUMENT array.
The arguments passed to the function call during the latest MULTMATRIX run will be inserted into the '__SIM_PARAMETERS__' element of the model simulation list (see SIMULATE); this data can be helpful in order to replicate the multiplier matrix results.
Row and column names in the output multiplier matrix identify the variables and the periods involved in the derivative solution, with the syntax VARIABLE_PERIOD (see example).
MDL
LOAD_MODEL
ESTIMATE
SIMULATE
STOCHSIMULATE
RENORM
TIMESERIES
BIMETS indexing
BIMETS configuration
#define model myModelDefinition<- "MODEL COMMENT> Klein Model 1 of the U.S. Economy COMMENT> Consumption BEHAVIORAL> cn TSRANGE 1921 1 1941 1 EQ> cn = a1 + a2*p + a3*TSLAG(p,1) + a4*(w1+w2) COEFF> a1 a2 a3 a4 COMMENT> Investment BEHAVIORAL> i TSRANGE 1921 1 1941 1 EQ> i = b1 + b2*p + b3*TSLAG(p,1) + b4*TSLAG(k,1) COEFF> b1 b2 b3 b4 COMMENT> Demand for Labor BEHAVIORAL> w1 TSRANGE 1921 1 1941 1 EQ> w1 = c1 + c2*(y+t-w2) + c3*TSLAG(y+t-w2,1)+c4*time COEFF> c1 c2 c3 c4 COMMENT> Gross National Product IDENTITY> y EQ> y = cn + i + g - t COMMENT> Profits IDENTITY> p EQ> p = y - (w1+w2) COMMENT> Capital Stock IDENTITY> k EQ> k = TSLAG(k,1) + i END" #define model data myModelData<-list( cn =TIMESERIES(39.8,41.9,45,49.2,50.6,52.6,55.1,56.2,57.3,57.8,55,50.9, 45.6,46.5,48.7,51.3,57.7,58.7,57.5,61.6,65,69.7, START=c(1920,1),FREQ=1), g =TIMESERIES(4.6,6.6,6.1,5.7,6.6,6.5,6.6,7.6,7.9,8.1,9.4,10.7,10.2,9.3,10, 10.5,10.3,11,13,14.4,15.4,22.3, START=c(1920,1),FREQ=1), i =TIMESERIES(2.7,-.2,1.9,5.2,3,5.1,5.6,4.2,3,5.1,1,-3.4,-6.2,-5.1,-3,-1.3, 2.1,2,-1.9,1.3,3.3,4.9, START=c(1920,1),FREQ=1), k =TIMESERIES(182.8,182.6,184.5,189.7,192.7,197.8,203.4,207.6,210.6,215.7, 216.7,213.3,207.1,202,199,197.7,199.8,201.8,199.9, 201.2,204.5,209.4, START=c(1920,1),FREQ=1), p =TIMESERIES(12.7,12.4,16.9,18.4,19.4,20.1,19.6,19.8,21.1,21.7,15.6,11.4, 7,11.2,12.3,14,17.6,17.3,15.3,19,21.1,23.5, START=c(1920,1),FREQ=1), w1 =TIMESERIES(28.8,25.5,29.3,34.1,33.9,35.4,37.4,37.9,39.2,41.3,37.9,34.5, 29,28.5,30.6,33.2,36.8,41,38.2,41.6,45,53.3, START=c(1920,1),FREQ=1), y =TIMESERIES(43.7,40.6,49.1,55.4,56.4,58.7,60.3,61.3,64,67,57.7,50.7,41.3, 45.3,48.9,53.3,61.8,65,61.2,68.4,74.1,85.3, START=c(1920,1),FREQ=1), t =TIMESERIES(3.4,7.7,3.9,4.7,3.8,5.5,7,6.7,4.2,4,7.7,7.5,8.3,5.4,6.8,7.2, 8.3,6.7,7.4,8.9,9.6,11.6, START=c(1920,1),FREQ=1), time =TIMESERIES(NA,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10, START=c(1920,1),FREQ=1), w2 =TIMESERIES(2.2,2.7,2.9,2.9,3.1,3.2,3.3,3.6,3.7,4,4.2,4.8,5.3,5.6,6,6.1, 7.4,6.7,7.7,7.8,8,8.5, START=c(1920,1),FREQ=1) ) #load model and model data myModel<-LOAD_MODEL(modelText=myModelDefinition) myModel<-LOAD_MODEL_DATA(myModel,myModelData) #estimate model myModel<-ESTIMATE(myModel) #calculate impact multipliers of Government Expenditure 'g' and #Government Wage Bill 'w2' with respect of Consumption 'cn' and #Gross National Product 'y' in the Klein model on the year 1941: myModel<-MULTMATRIX(myModel, symType='STATIC', TSRANGE=c(1941,1,1941,1), INSTRUMENT=c('w2','g'), TARGET=c('cn','y')) #Multiplier Matrix: 100.00% #...MULTMATRIX OK print(myModel$MultiplierMatrix) # w2_1 g_1 #cn_1 0.4540346 1.671956 #y_1 0.2532000 3.653260 #Results show that the impact multiplier of "y" #with respect to "g" is +3.65 #If we change Government Expenditure 'g' value in 1941 #from 22.3 (its historical value) to 23.3 (+1) #then the simulated Gross National Product "y" #in 1941 changes from 95.2 to 99, #thusly roughly confirming the +3.65 impact multiplier. #Note that "g" appears only once in the model definition, and only #in the "y" equation, with a coefficient equal to one. (Keynes would approve) #multi-period interim multipliers myModel<-MULTMATRIX(myModel, TSRANGE=c(1940,1,1941,1), INSTRUMENT=c('w2','g'), TARGET=c('cn','y')) #output multipliers matrix (note the zeros when the period #of the INSTRUMENT is greater than the period of the TARGET) print(myModel$MultiplierMatrix) # w2_1 g_1 w2_2 g_2 #cn_1 0.4478202 1.582292 0.0000000 0.000000 #y_1 0.2433382 3.510971 0.0000000 0.000000 #cn_2 -0.3911001 1.785042 0.4540346 1.671956 #y_2 -0.6251177 2.843960 0.2532000 3.653260 #multiplier matrix with endogenous variable 'w1' as instrument #note the ADDFACTOR suffix in the column name, referring to the #constant adjustment of the endogneous 'w1' myModel<-MULTMATRIX(myModel, TSRANGE=c(1940,1,1941,1), INSTRUMENT=c('w2','w1'), TARGET=c('cn','y')) #Multiplier Matrix: 100.00% #...MULTMATRIX OK myModel$MultiplierMatrix # w2_1 w1_ADDFACTOR_1 w2_2 w1_ADDFACTOR_2 #cn_1 0.4478202 0.7989328 0.0000000 0.0000000 #y_1 0.2433382 0.4341270 0.0000000 0.0000000 #cn_2 -0.3911001 -0.4866248 0.4540346 0.8100196 #y_2 -0.6251177 -0.9975073 0.2532000 0.4517209