ESTIMATE {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 estimates equations that are linear in the coefficients, as specified in the behavioral equations of the input model object. Coefficients can be estimated for single equations or blocks of simultaneous equations. Coefficients restriction procedure derives from the theory of Lagrange Multipliers, while the Cochrane-Orcutt method allows to account for residuals autocorrelation.
The estimation function supports:
- Ordinary Least Squares;
- Instrumental Variables;
- Deterministic linear restrictions on the coefficients;
- Almon Polynomial Distributed Lags;
- Autocorrelation of the errors;
Further details on estimation calculus can be found in MDL
help page;
ESTIMATE(model=NULL, eqList=NULL, TSRANGE=NULL, estTech='OLS', IV=NULL, quietly=FALSE, showWarnings=FALSE, tol=.Machine$double.eps, digits=getOption('digits'), centerCOV=TRUE, ...)
model |
The BIMETS model object to be estimated (see also |
eqList |
The |
TSRANGE |
The time range of the estimation, as a four dimensional numerical array, |
estTech |
The estimation technique used in the regression. Ordinary Least Squares |
IV |
The |
quietly |
If |
showWarnings |
If |
tol |
The tolerance for detecting linear dependencies in the columns of a matrix while an inversion is requested. The default is |
digits |
Controls the number of digits to print when printing coefficients and statistics of the estimation. Valid values are 1 to 16 with a default of 7. |
centerCOV |
If |
... |
Backward compatibility. |
If outputText=TRUE
, for each behavioral in the eqList
this function will print out:
- the name of the estimated behavioral;
- the estimation technique used;
- the autocorrelation order of the error, if any, and the iterations count required to achieve the convergence;
- the estimated equation with calculated coefficients and regressor expression; for each coefficient the T-statistic and the significance will be printed out;
- the restriction equations imposed on the coefficients, if any;
- the F-test for the restrictions, including the PDL restrictions, if any;
- the final autocorrelation parameters for the error, along with their standard error, the T-statistic and the significance;
- the R-Squared and the Adjusted R-Squared;
- the Durbin-Watson Statistic;
- the Sum of squares of residuals;
- the Standard Error of Regression;
- the Log of the Likelihood Function;
- the F-statistic and the F-probability;
- the AIC and the BIC;
- the Mean of the Dependent Variable;
- the Number of Observations;
- the Number of Degrees of Freedom;
- the Current Sample, i.e. the TSRANGE
of estimation;
For each behavioral in the eqList
this function will add 4 new named elements into the related behavioral of the output model object:
1) coefficients
: a numerical array built with the estimated coefficients;
2) errorCoefficients
: a numerical array built with the estimated coefficient for the error autoregression, if the ERROR>
structure has been provided in the model MDL
definition;
3) residuals
: the time series of the regression residuals. If an ERROR>
structure has been provided in the behavioral definition, the related residuals will be calculated as described in the Cochrane-Orcutt procedure (see MDL
).
4) statistics
: a list built with the parameters and the statistics of the estimation, e.g.:
- TSRANGE
: TSRANGE requested in the latest estimation of the behavioral;
- estimationTechinque
: estimation technique requested in the latest estimation of the behavioral;
- CoeffCovariance
: coefficients covariance;
- StandardErrorRegression
and StandardErrorRegressionNotCentered
: standard error of the regression (centered and not-centered);
- CoeffTstatistic
: T-statistic of the coefficients;
- RSquared
: R-Squared;
- AdjustedRSquared
: adjusted R-Squared;
- DegreesOfFreedom
: degrees of freedom of the regression;
- CoeffPvalues
: coefficients p-values;
- LogLikelihood
: Log of the Likelihood Function;
- Fstatistics
: F-statistics;
- RhosTstatistics
: rhos T-statistic (if any);
- FtestRestrValue
: F-test value for the restrictions;
- FtestRestrProbability
: F-test probability for the restrictions;
- AIC
: Akaike's Information Criterion;
- BIC
: Schwarz's Information Criterion;
- etc.
MDL
LOAD_MODEL
SIMULATE
MULTMATRIX
RENORM
TIMESERIES
BIMETS indexing
BIMETS configuration
summary
#define model myModelDefinition= "MODEL COMMENT> Modified Klein Model 1 of the U.S. Economy with PDL, COMMENT> autocorrelation on errors, restrictions and conditional evaluations COMMENT> Consumption BEHAVIORAL> cn TSRANGE 1925 1 1941 1 EQ> cn = a1 + a2*p + a3*TSLAG(p,1) + a4*(w1+w2) COEFF> a1 a2 a3 a4 ERROR> AUTO(2) COMMENT> Investment BEHAVIORAL> i TSRANGE 1923 1 1941 1 EQ> i = b1 + b2*p + b3*TSLAG(p,1) + b4*TSLAG(k,1) COEFF> b1 b2 b3 b4 RESTRICT> b2 + b3 = 1 COMMENT> Demand for Labor BEHAVIORAL> w1 TSRANGE 1925 1 1941 1 EQ> w1 = c1 + c2*(y+t-w2) + c3*TSLAG(y+t-w2,1)+c4*time COEFF> c1 c2 c3 c4 PDL> c3 1 3 COMMENT> Gross National Product IDENTITY> y EQ> y = cn + i + g - t COMMENT> Profits IDENTITY> p EQ> p = y - (w1+w2) COMMENT> Capital Stock with switches IDENTITY> k EQ> k = TSLAG(k,1) + i IF> i > 0 IDENTITY> k EQ> k = TSLAG(k,1) IF> i <= 0 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 myModel=LOAD_MODEL(modelText=myModelDefinition); #load data into the model myModel=LOAD_MODEL_DATA(myModel,myModelData,showWarnings = TRUE); ################################################# #OLS case #estimate the model myModel=ESTIMATE(myModel); #HERE BELOW THE OUTPUT OF THE ESTIMATION (COMMENTED OUT): #.CHECK_MODEL_DATA(): warning, there are undefined values in time series "time". # #Estimate the Model myModelDefinition: #the number of behavioral equations to be estimated is 3. #The total number of coefficients is 14. # #_________________________________________ # #BEHAVIORAL EQUATION: cn #Estimation Technique: OLS #Autoregression of Order 2 (Cochrane-Orcutt procedure) # #Convergence was reached in 9 / 20 iterations. # # #cn = 19.01352 # T-stat. 13.1876 *** # # + 0.3442816 p # T-stat. 3.841051 ** # # + 0.03443117 TSLAG(p,1) # T-stat. 0.4280928 # # + 0.6993905 (w1+w2) # T-stat. 15.30744 *** # #ERROR: AUTO(2) # #AUTOREGRESSIVE PARAMETERS: #Rho Std. Error T-stat. # 0.05743131 0.3324101 0.1727725 # 0.007785936 0.2647013 0.02941404 # # #STATs: #R-Squared : 0.985263 #Adjusted R-Squared : 0.979595 #Durbin-Watson Statistic : 1.966609 #Sum of squares of residuals : 9.273455 #Standard Error of Regression : 0.8445961 #Log of the Likelihood Function : -20.14564 #F-statistic : 173.8271 #F-probability : 1.977107e-11 #Akaike's IC : 54.29129 #Schwarz's IC : 60.90236 #Mean of Dependent Variable : 55.71765 #Number of Observations : 19 #Number of Degrees of Freedom : 13 #Current Sample (year-period) : 1925-1 / 1941-1 # # #Signif. codes: *** 0.001 ** 0.01 * 0.05 # # # #_________________________________________ # #BEHAVIORAL EQUATION: i #Estimation Technique: OLS # #i = 2.868104 # T-stat. 0.3265098 # # + 0.5787626 p # T-stat. 4.456542 *** # # + 0.4212374 TSLAG(p,1) # T-stat. 3.243579 ** # # - 0.09160307 TSLAG(k,1) # T-stat. -2.11748 # #RESTRICTIONS: #b2+b3=1 # #RESTRICTIONS F-TEST: #F-value : 8.194478 #F-prob(1,15) : 0.0118602 # # #STATs: #R-Squared : 0.8928283 #Adjusted R-Squared : 0.8794319 #Durbin-Watson Statistic : 1.173106 #Sum of squares of residuals : 26.76483 #Standard Error of Regression : 1.293368 #Log of the Likelihood Function : -30.215 #F-statistic : 66.64659 #F-probability : 1.740364e-08 #Akaike's IC : 68.43001 #Schwarz's IC : 72.20776 #Mean of Dependent Variable : 1.310526 #Number of Observations : 19 #Number of Degrees of Freedom : 16 #Current Sample (year-period) : 1923-1 / 1941-1 # # #Signif. codes: *** 0.001 ** 0.01 * 0.05 # # # #_________________________________________ # #BEHAVIORAL EQUATION: w1 #Estimation Technique: OLS # #w1 = 1.12869 # T-stat. 0.6479266 # # + 0.4398767 (y+t-w2) # T-stat. 12.01268 *** # # + c3 TSLAG(y+t-w2,1) # PDL # # + 0.1368206 time # T-stat. 3.373905 ** # #PDL: #c3 1 3 # #Distributed Lag Coefficient: c3 #Lag Coeff. Std. Error T-stat. #0 0.1076812 0.04283967 2.513586 * #1 0.05074557 0.01291231 3.930015 ** #2 -0.00619005 0.03110492 -0.1990055 #SUM 0.1522367 0.03873693 # #RESTRICTIONS F-TEST: #F-value : 0.06920179 #F-prob(1,11) : 0.7973647 # # #STATs: #R-Squared : 0.9890855 #Adjusted R-Squared : 0.9854474 #Durbin-Watson Statistic : 2.174168 #Sum of squares of residuals : 6.392707 #Standard Error of Regression : 0.7298805 #Log of the Likelihood Function : -15.80848 #F-statistic : 271.8645 #F-probability : 1.172284e-11 #Akaike's IC : 43.61697 #Schwarz's IC : 48.61625 #Mean of Dependent Variable : 37.69412 #Number of Observations : 17 #Number of Degrees of Freedom : 12 #Current Sample (year-period) : 1925-1 / 1941-1 # # #Signif. codes: *** 0.001 ** 0.01 * 0.05 # # #...ESTIMATE OK #get residuals of 'cn' myModel$behaviorals$cn$residuals #Time Series: #Start = 1925 #End = 1941 #Frequency = 1 # [1] -0.88562504 0.25109884 0.66750111 ... #[17] -1.41795908 #get residuals of 'i' myModel$behaviorals$i$residuals #Time Series: #Start = 1923 #End = 1941 #Frequency = 1 # [1] 1.464518775 -1.469763968 0.078674017 ... #[16] -2.425079127 -0.698071507 -1.352967430 -1.724306054 #get estimation coefficients of 'cn' and 'w1' myModel$behaviorals$cn$coefficients # [,1] #a1 19.01352476 #a2 0.34428157 #a3 0.03443117 #a4 0.69939052 myModel$behaviorals$cn$errorCoefficients # [,1] #RHO_1 0.057431312 #RHO_2 0.007785936 myModel$behaviorals$w1$coefficients # [,1] #c1 1.12869024 #c2 0.43987666 #c3 0.10768118 #c3_PDL_1 0.05074557 #c3_PDL_2 -0.00619005 #c4 0.13682057 #get latest estimation parameters myModel$estimate_parameters #$estTech #[1] "OLS" # #$eqList #[1] "cn" "i" "w1" ################################################# #IV case #estimation of Consumption "cn" with arbitrary IVs #and error autocorrelation myModel=ESTIMATE(myModel, eqList = 'cn', estTech = 'IV', IV=c('1', 'TSLAG(y)', 'TSLAG(w1)*pi+0.5', 'exp(w2)')); #.CHECK_MODEL_DATA(): warning, there are undefined values in time series "time". # #Estimate the Model myModelDefinition: #the number of behavioral equations to be estimated is 1. #The total number of coefficients is 4. # #_________________________________________ # #BEHAVIORAL EQUATION: cn #Estimation Technique: IV #Autoregression of Order 2 (Cochrane-Orcutt procedure) # #Convergence was reached in 7 / 20 iterations. # # #cn = 17.93884143014611 # T-stat. 11.35605608264105 *** # # + 0.1594074567906318 p # T-stat. 0.7524421962971953 # # + 0.1277506382165825 TSLAG(p,1) # T-stat. 0.9491296499358723 # # + 0.7596945016016718 (w1+w2) # T-stat. 13.00631014447263 *** # #ERROR: AUTO(2) # #AUTOREGRESSIVE PARAMETERS: #Rho Std. Error T-stat. # 0.1195212459504091 0.3378062168740853 0.3538160045022491 #-0.2124449434554128 0.3005397305932941 -0.7068780657919213 # # #STATs: #R-Squared : 0.981280540184345 #Adjusted R-Squared : 0.9740807479475546 #Durbin-Watson Statistic : 1.852330110333749 #Sum of squares of residuals : 11.77949537517467 #Standard Error of Regression : 0.9519007452773581 #Log of the Likelihood Function : -22.41808505770927 #F-statistic : 136.2928967825036 #F-probability : 9.311151849544785e-11 #Akaike's IC : 58.83617011541855 #Schwarz's IC : 65.44724296958363 #Mean of Dependent Variable : 55.71764705882353 #Number of Observations : 19 #Number of Degrees of Freedom : 13 #Current Sample (year-period) : 1925-1 / 1941-1 # # #Signif. codes: *** 0.001 ** 0.01 * 0.05 # # #...ESTIMATE OK #estimation of Investment "i" with arbitrary IVs #and coefficient restrictions myModel=ESTIMATE(myModel, eqList = 'i', estTech = 'IV', IV=c('1', 'TSLAG(w2)', 'TSLAG(w1)*pi+0.5', 'exp(w2)')); #.CHECK_MODEL_DATA(): warning, there are undefined values in time series "time". # #Estimate the Model myModelDefinition: #the number of behavioral equations to be estimated is 1. #The total number of coefficients is 4. # #_________________________________________ # #BEHAVIORAL EQUATION: i #Estimation Technique: IV # #i = 34.51754474586971 # T-stat. 1.264388701353744 # # + 0.3216327065477544 p # T-stat. 0.8648297095197013 # # + 0.6783672934522457 TSLAG(p,1) # T-stat. 1.824043940185745 # # - 0.2475568598269819 TSLAG(k,1) # T-stat. -1.842520302814061 # #RESTRICTIONS: #b2+b3=1 # #RESTRICTIONS F-TEST: #F-value : 2.465920754930322 #F-prob(1,15) : 0.1371905708637959 # # #STATs: #R-Squared : 0.8057731392963399 #Adjusted R-Squared : 0.7814947817083824 #Durbin-Watson Statistic : 0.9405346241683215 #Sum of squares of residuals : 48.50580729347795 #Standard Error of Regression : 1.741152766371283 #Log of the Likelihood Function : -35.86365535226765 #F-statistic : 33.18894766160036 #F-probability : 2.025229876911894e-06 #Akaike's IC : 79.7273107045353 #Schwarz's IC : 83.50506662120105 #Mean of Dependent Variable : 1.310526315789474 #Number of Observations : 19 #Number of Degrees of Freedom : 16 #Current Sample (year-period) : 1923-1 / 1941-1 # # #Signif. codes: *** 0.001 ** 0.01 * 0.05 # # #...ESTIMATE OK