Endogenous targeting of a BIMETS model.
The endogenous targeting of econometric models (a.k.a. "renormalization") consists of solving the model while interchanging the role of one or more endogenous variables with an equal number of exogenous variables.
This procedure determines the values for the INSTRUMENT exogenous variables that allow the objective TARGET endogenous values to be achieved, with respect to the constraints given by the model equations (see MDL).
This is an approach to economic and monetary policy analysis, and is based on two assumptions:
there exists a desired level for a set of the n endogenous variables defined as TARGET;
there exists a set of the n exogenous variables defined as INSTRUMENT;
Given these premises, the endogenous targeting process consists in determining the values of the exogenous variables chosen as INSTRUMENT allowing us to achieve the desired values for the endogenous variables designated as TARGET. In other words the procedure allows users to exchange the role of exogenous and endogenous among a set of variables pairs.
Given a list of exogenous INSTRUMENT variables and a list of TARGET endogenous time series, the iterative procedure can be split into the following steps:
Computation of the multipliers matrix MULTMAT of the TARGET endogenous variables with respect to the INSTRUMENT exogenous variables (this is a square matrix by construction);
Solution of the linear system (if any):
MULTMAT TARGET , where are the exogenous variables in the INSTRUMENT list and are the endogenous variables that have a related target in the TARGET list, given the current iteration;
TARGET list. If the convergence condition is satisfied, or the maximum number of iterations is reached, the algorithm will stop, otherwise it will go back to step 1;Users can also declare an endogenous variable as an INSTRUMENT variable. In this case, the constant adjustment (see SIMULATE) related to the provided endogenous variable will be used as the instrument exogenous variable. This procedure is particularly suited for the automatic computation of the add-factors needed to fine tune the model into a baseline path and to improve the forecasting accuracy.
If the convergence condition is satisfied, the RENORM procedure will return the INSTRUMENT time series allowing us to achieve the desired values for the endogenous variables designated as TARGET.
For more realistic scenarios, several advanced econometric exercises on the US Federal Reserve FRB/US econometric model (e.g., dynamic simulation in a monetary policy shock, rational expectations, endogenous targeting, stochastic simulation, etc.) are available in the "US Federal Reserve quarterly model (FRB/US) in R with bimets" vignette.
RENORM(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, renormIterLimit=10, renormConvergence=1e-4, TARGET=NULL, INSTRUMENT=NULL, MM_SHOCK=0.00001, quietly=FALSE, quietlyMULTMATRIX=FALSE, tol=1e-28, 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 SIMULATErenormIterLimit: The value representing the maximum number of iterations to be performed. The iterative renormalization procedure will stop when renormIterLimit is reached or the TARGET variables satisfy the renormConvergence criterionrenormConvergence: The convergence value requested for the iterative renormalization process, that stops when the Euclidean distance between each TARGET time series and the related simulated endogenous variable is less than the renormConvergence valueTARGET: A named list that specifies the target endogenous variables. List names must be equal to the names of the target endogenous variables involved in the renormalization; each list element must contain the time series of the desired target endogenous values; time series must be compliant with the compliance control check defined in is.bimets (see example)INSTRUMENT: A character array built with the names of the instrument exogenous variables involved in the renormalization. User 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 instrument exogenous variable (see example)tol: the tolerance for detecting linear dependencies in the columns of a matrix while an inversion is requested.MM_SHOCK: see MULTMATRIXquietly: If TRUE, information messages related to the RENORM operation will be suppressed.quietlyMULTMATRIX: If TRUE, information messages related to the underlying MULTMATRIX operation will be suppressed. If quietly argument is set to TRUE, then quietlyMULTMATRIX will be set to TRUE.JACOBIAN_SHOCK: see SIMULATEJacobianDrop: see SIMULATEforceForwardLooking: see SIMULATEavoidCompliance: see SIMULATE...: see SIMULATEThis function will add a new named element renorm into the output BIMETS model object.
This new renorm element is a named list that contains the following elements:
INSTRUMENT: a named list that contains the INSTRUMENT exogenous time series that allow the objective TARGET endogenous values to be achieved. This element is populated only if the convergence is reached. List names are equal to the names of the related exogenous variables. 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 instrument exogenous variable, and this INSTRUMENT output list will contains the constant adjustment time series that allow the objective TARGET endogenous values to be achieved (see example);
TARGET: a named list built with the achieved TARGET endogenous time series. List names are equal to the target endogenous variable names;
unConvergedTARGET: the names array of the endogenous TARGET variables that failed the convergence. This element is populated only if the convergence has not been reached;
modelData: the whole model input dataset wherein the INSTRUMENT exogenous variables have been modified accordingly to the RENORM results. This data can be useful in order to refine results or to verify that the model, fed with the proper INSTRUMENT exogenous time series, produces the desired TARGET endogenous values. This element is populated only if the convergence is achieved (see example);
ConstantAdjustment: a modified constant adjustment input list (see SIMULATE) wherein the constant adjustment time series related to a INSTRUMENT endogenous variables have been modified accordingly to the RENORM results. This data can be useful in order to refine results or to verify that the model, fed with the proper INSTRUMENT exogenous time series (therefore with the proper ConstantAdjustment time series, if any), produces the desired TARGET endogenous values (see example). This element is populated only if the convergence is achieved;
__RENORM_PARAMETERS__: a named list that contains the arguments passed to the function call during the latest RENORM run, e.g. TSRANGE, INSTRUMENT, TARGET, renormIterLimit, renormConvergence, ConstantAdjustment, Exogenize, etc.: this data can be useful in order to replicate renorm results.
MDL
LOAD_MODEL
ESTIMATE
SIMULATE
STOCHSIMULATE
MULTMATRIX
OPTIMIZE
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) #we want an arbitrary value of 66 on Consumption 'cn' in 1940 and 78 in 1941 #we want an arbitrary value of 77 on GNP 'y' in 1940 and 98 in 1941 kleinTargets<-list( cn = TSERIES(66,78,START=c(1940,1),FREQ=1), y = TSERIES(77,98,START=c(1940,1),FREQ=1) ) #Then, we can perform the model endogenous targeting by using Government Wage Bill 'w2' #and Government Expenditure 'g' as #INSTRUMENT in the years 1940 and 1941: myModel<-RENORM(myModel ,INSTRUMENT = c('w2','g') ,TARGET = kleinTargets ,TSRANGE = c(1940,1,1941,1) ,simIterLimit = 100 ) with(myModel,TABIT(modelData$w2, renorm$INSTRUMENT$w2, modelData$g, renorm$INSTRUMENT$g)) # Date, Prd., modelData$w2, renorm$INSTRUMENT$w2, modelData$g, renorm$INSTRUMENT$g # etc. # 1938, 1 , 7.7, , 13, # 1939, 1 , 7.8, , 14.4, # 1940, 1 , 8, 7.41333, 15.4, 16.1069 # 1941, 1 , 8.5, 9.3436, 22.3, 22.6599 #So, if we want to achieve on Consumption 'cn' #an arbitrary simulated value of 66 #in 1940 and 78 in 1941, and if we want #to achieve on GNP 'y' an arbitrary #simulated value of 77 in 1940 and 98 in 1941, #we need to change exogenous 'w2' from 8 to 7.41 #in 1940 and from 8.5 to 9.34 in 1941, #and we need to change exogenous 'g' #from 15.4 to 16.1 in 1940 and from 22.3 to 22.66 in 1941 #Let's verify: #create a new model kleinRenorm<-myModel #get instruments to be used newInstruments=myModel$renorm$INSTRUMENT #change exogenous by using new instruments kleinRenorm$modelData<-within(kleinRenorm$modelData, { w2[[1940,1]]=newInstruments$w2[[1940,1]] w2[[1941,1]]=newInstruments$w2[[1941,1]] g[[1940,1]] =newInstruments$g[[1940,1]] g[[1941,1]] =newInstruments$g[[1941,1]] } ) #users can also replace last two commands with: #kleinRenorm$modelData<-kleinRenorm$renorm$modelData #simulate the new model kleinRenorm<-SIMULATE(kleinRenorm ,TSRANGE=c(1940,1,1941,1) ,simConvergence=0.00001 ,simIterLimit=100 ) #Simulation: 100.00% #...SIMULATE OK #verify targets are achieved with(kleinRenorm$simulation, TABIT(cn,y) ) #Date, Prd., cn , y #1940, 1 , 66.01116 , 77.01772 #1941, 1 , 78.02538 , 98.04121 ###################################################################### #now use 'i' endogenous variable as an instrument #first, define the related exogenous constant adjustment myCA<-list(i = myModel$modelData$i*0+0.1) #run renorm with endogenous 'i' as instrument myModel<-RENORM(myModel ,INSTRUMENT = c('w2','i') ,TARGET = kleinTargets ,TSRANGE = c(1940,1,1941,1) ,simIterLimit = 100 ,ConstantAdjustment = myCA ) #get the values of the constant adjustment for the endogenous 'i' #in 1940-1941 that allow achieving the target values for 'cn' and 'y' myModel$renorm$ConstantAdjustment #$i #Time Series: #Start = 1920 #End = 1941 #Frequency = 1 # [1] 0.1000000 0.1000000 0.1000000 ... #[20] 0.1000000 0.7069039 0.4388811 #these values are also reported in the INSTRUMENT output list myModel$renorm$INSTRUMENT$i #Time Series: #Start = 1940 #End = 1941 #Frequency = 1 #[1] 0.7069039 0.4388811