MULTMATRIX {bimets} | R Documentation |
(Note: this is the html version of the reference manual. Please consider reading the pdf version of this reference manual, wherein there are figures and the mathematical expressions are better formatted than in html.)
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 the shock amount MM_SHOCK
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 variable being kept constant.
Given Y(t)
an endogenous variable at time t
and X(t)
an exogenous variable at time t
the impact multiplier m(Y,X,t)
is defined as m(Y,X,t) = dY(t)/dX(t)
and can be approximated by m(Y,X,t)=(Y_shocked(t)-Y(t))/(X_shocked(t)-X(t))
, with Y_shocked(t)
the values fo the simulated endogenous variable Y
at time t
when X(t)
is shocked to X_shocked(t)=X(t)(1+MM_SHOCK)
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 variable being kept constant.
Given Y(t+r)
an endogenous variable at time t+r
and X(t)
an exogenous variable at time t
the interim or delay-r
multiplier m(Y,X,t,r)
is defined as m(Y,X,t,r) = dY(t+r)/dX(t)
and can be approximated by (m(Y,X,t,r)=Y_shocked(t+r)-Y(t+r))/(X_shocked(t)-X(t))
, with Y_shocked(t+r)
the values fo the simulated endogenous variable Y
at time t+r
when X(t)
is shocked to X_shocked(t)=X(t)(1+MM_SHOCK)
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, ...)
model |
The BIMETS model object for which the multipliers matrix has to be calculated. The operation requires that all the behaviorals in the model have been previously estimated: all the behavioral coefficients (i.e. the estimation coefficients and the autoregression coefficients for the errors, if any) must be numerically defined in the model object. (see also |
... |
Other options to be sent to the underlying simType, the simulation type requested: TARGET: a INSTRUMENT: a MM_SHOCK: the shock value added to variables in the derivative calculation. The default value is simConvergence: the percentage convergence value requested for the iterative process, that stops when the percentage difference of all the feedback variables between iterations is less than simIterLimit: the value representing the maximum number of iterations to be performed. The iterative procedure will stop when ZeroErrorAC: if Exogenize: a named list that specifies the endogenous variables to be exogenized. During the simulation, in the specified time range the exogenized endogenous variables will be assigned their historical values. The list names are the names of the endogenous variables to be exogenized; each element of this list contains the time range of the exogenization for the related endogenous variable, in the form of a 4-dimensional integer array, i.e. start_year, start_period, end_year, end_period. An element of the list can also be assigned ConstantAdjustment: a named list that specifies the constant adjustments (i.e. add factors) to be added to the selected equations of the model. Each constant adjustment can be see as a new exogenous variable added to the equation of the specified endogenous variable. The list names are the names of the endogenous variables involved; each element of this is list contains the time series to be added to the equation of the related endogenous variable. Each provided time series must be compliant with the compliance control check defined in verbose: if quietly. if |
This 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
); that can be useful 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
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 non-Wage Spending '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$MultiplierMatrix) # w2_1 g_1 #cn_1 0.4540346 1.671956 #y_1 0.2532000 3.653260 #Results show that the impact multiplier of "y" #with respect to "g" is +3.65 #If we change Government non-Wage Spending 'g' value in 1941 #from 22.3 (its historical value) to 23.3 (+1) #then the simulated Gross National Product "y" #in 1941 changes from 95.2 to 99, #thusly roughly confirming the +3.65 impact multiplier. #Note that "g" appears only once in the model definition, and only #in the "y" equation, with a coefficient of one. (Keynes would approve) #multi-period interim multipliers myModel=MULTMATRIX(myModel, TSRANGE=c(1940,1,1941,1), INSTRUMENT=c('w2','g'), TARGET=c('cn','y')) #output multipliers matrix (note the zeros when the period #of the INSTRUMENT is greater than the period of the TARGET) print(myModel$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