ESTIMATE: Estimate a BIMETS model

Description

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 Lagrange Multipliers' theory, while the Cochrane-Orcutt method allows accounting 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;
- Structural stability analysis (Chow tests);

Further details on estimation calculus can be found in MDL help page; further details on Chow test can be found below in this section;

Usage

ESTIMATE(model=NULL, 
         eqList=NULL, 
         TSRANGE=NULL,
         forceTSRANGE=FALSE,
         estTech='OLS',
         IV=NULL,
         forceIV=FALSE,
         quietly=FALSE,
         tol=1e-28,
         digits=getOption('digits'),
         centerCOV=TRUE,
         CHOWTEST=FALSE,
         CHOWPAR=NULL,
         avoidCompliance=FALSE,
         ...)

Value

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;

All probabilities lie between [0, 1].

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).

3) residuals_no_error_correction: if an ERROR> structure has been provided in the behavioral definition, the residuals calcucated by using the original dependent and independent variables are stored into this list element.

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;

- matrixX: the regressors matrix;

- vectorY: the dependent variable;

- matrixX_error_corrected: the regressors matrix arising from the Cochrane-Orcutt procedure;

- etc.

Structural Stability - Chow test

One of the main purposes of econometric modeling is its use for forecast and policy evaluation and, to this end, the stability of any behavioral equation parameters over time should be verified. In order to check for structural stability two different procedures, which can be derived from the so-called Chow-tests, are applied.

Given a sample of \(T_{0} = t_{k},...,t_{n}\) observations (i.e. the base TSRANGE) and selecting an arbitrary forward extension in \(T_{1} = t_{k},...,t_{n},...,t_{m}\) observations (i.e. the extended TSRANGE), with \( k < n < m\), in the general case we have the following two regressions:

1) \(Y_{0} = \beta_{0}*X_{0}+\epsilon_{0}, \quad \epsilon_{0} \sim \mathcal{N}(0,\,\sigma_{0}^{2}) \), having time series projected on the base TSRANGE

2) \(Y_{1} = \beta_{1}*X_{1}+\epsilon_{1}, \quad \epsilon_{1} \sim \mathcal{N}(0,\,\sigma_{1}^{2}) \), having time series projected on the extended TSRANGE

In general, a stability analysis is carried on in the following ways:

- comparing the parameter estimates arising from the two regressions: this is known as the covariance analysis;

- checking the accuracy of the forecast for the dependent variable in the extended TSRANGE, using the estimates produced in the base TSRANGE: this is known as the predictive power test.

The first Chow test is calculated as: \(\tau = \frac{SSR_{1}-SSR_{0}}{SSR_{0}} \frac{DoF_{1}}{DoF_{1}-DoF_{0}} \),

with \(SSR_{i}\) as the sum of squared residuals and \(DoF_{i}\) as the number of degrees of freedom in the regression \(i=0,1\).

The test is completed by calculating the following time series on the extended TSRANGE:

- the forecast error;

- the standard error of forecast;

- the t-statistic for the error;

The standard error of the forecast for the \(t_j\) observation in the extended TSRANGE is computed according to:

\(SE_{j} = \sigma_{0} \sqrt{1+x_j^\top * ( X_{0}^\top * X_{0}^{ })^{-1} * x_j} \)

having \(x_j\) as the independent values (i.e. regressors) on the \(t_{j}\) observation in the \(T_{1}\) extended TSRANGE, with \(n < j \leq m\).

The null hypothesis for \(\tau\) is:

\( H^{*} : \beta_{1} = \beta_{0}\), given \(\sigma_{1}^{2} = \sigma_{0}^{2}\)

The test statistic \(\tau\) follows the \(F\) distribution with \( ( DoF_{1}-DoF_{0} ) \) and \(DoF_{1}\) degrees of freedom, and can be performed during the ESTIMATE() function execution by using the CHOWTEST argument set to TRUE.

If CHOWTEST is TRUE, for each behavioral in the eqList the output model will have the following named element:

- ChowTest: it contains the statistics and the time series computed during the last structural analysis performed on the related behavioral.

Arguments

model: The BIMETS model object to be estimated (see also LOAD_MODEL).
eqList: The character array of behavioral names to be estimated. If it is NULL then all the behaviorals of the model will be estimated.
TSRANGE: The time range of the estimation, as a four dimensional numerical array,
i.e. TSRANGE=c(start_year, start_period, end_year, end_period). The TSRANGE provided in the behavioral MDL definition takes precedence over this function argument.
forceTSRANGE: If TRUE, the TSRANGE defined in the previous argument takes precedence over the TSRANGE provided in the behavioral MDL definition.
estTech: The estimation technique used in the regression. Ordinary Least Squares OLS and Instrumental Variables estimation IV are supported.
IV: The character array built with the Instrumental Variable expressions, in the case of Instrumental Variables estimation (see example).
forceIV: If TRUE, the IV defined in the previous argument takes precedence over the IV provided in the behavioral MDL definition.
quietly: If TRUE, information messages will be suppressed, e.g. results and regression statistics.
tol: The tolerance for detecting linear dependencies in a matrix's columns when its inverse is requested.
digits: Controls the number of digits to print when printing coefficients and statistics of the estimation. Valid values are 1 to 22 with a default of 7.
centerCOV: If TRUE, the function subtracts the mean from the residuals before calculating the residual covariance matrix.
CHOWTEST: If TRUE, the structural stability analysis will be performed.
CHOWPAR: Indicates the last year-period where the stability test is performed. If NULL it will be automatically calculated by using all available time series data. It must be provided as an integer array, e.g. c(YEAR,PERIOD).
avoidCompliance: If TRUE, compliance control check of model time series will be skipped. See is.bimets
...: Backward compatibility.

Examples

Run this code


#define model
myModelDefinition<-
"MODEL

COMMENT> Modified Klein Model 1 of the U.S. Economy with PDL,
COMMENT> autocorrelation on errors, restrictions 
COMMENT> 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. 12.13083    ***
#
#                    +   0.3442816   p
#                        T-stat. 3.533253    **
#
#                    +   0.03443117  TSLAG(p,1)
#                        T-stat. 0.3937881   
#
#                    +   0.6993905   (w1+w2)
#                        T-stat. 14.0808     ***
#
#ERROR STRUCTURE:  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             : 0.9785644   
#Durbin-Watson Statistic        : 1.966609    
#Sum of squares of residuals    : 9.273455    
#Standard Error of Regression   : 0.9181728   
#Log of the Likelihood Function : -18.97047   
#F-statistic                    : 147.0844    
#F-probability                  : 1.090551e-09
#Akaike's IC                    : 51.94093    
#Schwarz's IC                   : 57.77343    
#Mean of Dependent Variable     : 55.71765    
#Number of Observations         : 17
#Number of Degrees of Freedom   : 11
#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


#################################################
#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)'))
#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                  =   18.07073    
#                        T-stat. 11.72958    ***
#
#                    +   0.2530483   p
#                        T-stat. 1.583881    
#
#                    +   0.08631646  TSLAG(p,1)
#                        T-stat. 0.7556204   
#
#                    +   0.7363227   (w1+w2)
#                        T-stat. 13.11572    ***
#
#ERROR STRUCTURE:  AUTO(2) 
#
#AUTOREGRESSIVE PARAMETERS:
#Rho             Std. Error      T-stat.         
#0.01559806      0.343195        0.04544955      
#-0.1196327       0.283432       -0.422086        
#
#
#STATs:
#R-Squared                      : 0.9843186   
#Adjusted R-Squared             : 0.9771907   
#Durbin-Watson Statistic        : 1.917329    
#Sum of squares of residuals    : 9.867739    
#Standard Error of Regression   : 0.9471363   
#Log of the Likelihood Function : -19.49844   
#F-statistic                    : 138.0938    
#F-probability                  : 1.532807e-09
#Akaike's IC                    : 52.99689    
#Schwarz's IC                   : 58.82938    
#Mean of Dependent Variable     : 55.71765    
#Number of Observations         : 17
#Number of Degrees of Freedom   : 11
#Current Sample (year-period)   : 1925-1 / 1941-1
#
#
#Signif. codes:   *** 0.001  ** 0.01  * 0.05  
#
#
#...ESTIMATE OK

#define model
myShortModelDefinition<-
  "MODEL
  COMMENT> Consumption with IV
  BEHAVIORAL> cn
  TSRANGE 1925 1 1941 1
  EQ> cn = a1 + a2*p + a3*TSLAG(p,1) + a4*(w1+w2)
  COEFF> a1 a2 a3 a4
  IV> 1
  IV> TSLAG(y)
  IV> TSLAG(w1)*pi+0.5
  IV> exp(w2)
  END
  " 


#load model 
myShortModel<-LOAD_MODEL(modelText=myShortModelDefinition)

#load data into the model
myShortModel<-LOAD_MODEL_DATA(myShortModel,myModelData,showWarnings = TRUE)

#estimation of Consumption "cn" with arbitrary IVs
#and error autocorrelation
myShortModel<-ESTIMATE(myShortModel,
                 eqList = 'cn',
                 estTech = 'IV')


#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                   =   34.517544    
#                        T-stat. 1.264388   
#
#                    +   0.3216326   p
#                        T-stat. 0.8648297   
#
#                    +   0.6783672   TSLAG(p,1)
#                        T-stat. 1.824043    
#
#                    -   0.2475568   TSLAG(k,1)
#                        T-stat. -1.842520   
#
#RESTRICTIONS:
#b2+b3=1
#
#RESTRICTIONS F-TEST:
#F-value            : 2.465920       
#F-prob(1,15)       : 0.137190
#
#
#STATs:
#R-Squared                      : 0.805773   
#Adjusted R-Squared             : 0.781494   
#Durbin-Watson Statistic        : 0.940534   
#Sum of squares of residuals    : 48.50580  
#Standard Error of Regression   : 1.741152    
#Log of the Likelihood Function : -35.86365   
#F-statistic                    : 33.18894    
#F-probability                  : 2.025229e-06
#Akaike's IC                    : 79.72731     
#Schwarz's IC                   : 83.50506    
#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  
#
#
#...ESTIMATE OK

##############################################################
#CHOW TEST on w1

#base TSRANGE set to 1925 / 1935
myModel<-ESTIMATE(myModel,
                  eqList='w1',
                  TSRANGE=c(1925,1,1935,1),
                  forceTSRANGE=TRUE,
                  CHOWTEST=TRUE)

#Estimate the Model myModelDefinition:
#the number of behavioral equations to be estimated is 1.
#The total number of coefficients is 6.
#
#_________________________________________
#
#BEHAVIORAL EQUATION: w1
#Estimation Technique: OLS
#
#w1                = -   4.48873    
#                        T-stat. -2.47402   *
#
#                    +   0.545102   (y+t-w2)
#                        T-stat. 15.3462    ***
#
#                    +   c3         TSLAG(y+t-w2,1)
#                        PDL
#
#                    +   0.292018   time
#                        T-stat. 5.58588    **
#
#PDL:
#c3 1 3
#
#Distributed Lag Coefficient: c3
#Lag     Coeff.        Std. Error    T-stat.       
#0        0.0413985     0.0336676     1.22963      
#1        0.0493551     0.00742323    6.64873      ***
#2        0.0573116     0.0265487     2.15873      
#SUM      0.148065      0.0222697    
#
#RESTRICTIONS F-TEST:
#F-value            : 3.35954       
#F-prob(1,5)        : 0.126295      
#
#
#STATs:
#R-Squared                      : 0.995931   
#Adjusted R-Squared             : 0.993219   
#Durbin-Watson Statistic        : 2.43313    
#Sum of squares of residuals    : 0.737093   
#Standard Error of Regression   : 0.350498   
#Log of the Likelihood Function : -0.742173  
#F-statistic                    : 367.183    
#F-probability                  : 2.68564e-07
#Akaike's IC                    : 13.4843    
#Schwarz's IC                   : 15.8717    
#Mean of Dependent Variable     : 34.9909    
#Number of Observations         : 11
#Number of Degrees of Freedom   : 6
#Current Sample (year-period)   : 1925-1 / 1935-1
#
#
#Signif. codes:   *** 0.001  ** 0.01  * 0.05  
#
#
#
#STABILITY ANALYSIS:
#Behavioral equation: w1 
#
#Chow test:
#Sample (auto)      : 1936-1 / 1941-1 
#F-value            : 15.3457       
#F-prob(6,12)       : 5.34447e-05    
#
#Predictive Power:
#
#Date, Prd., Actual        , Predict       , Error         , Std. Error    , T-stat         
#
#1936, 1   ,  36.8         ,  38.439       , -1.63901      ,  0.547471     , -2.99378      
#1937, 1   ,  41           ,  40.824       ,  0.176033     ,  0.630905     ,  0.279017     
#1938, 1   ,  38.2         ,  39.6553      , -1.4553       ,  0.672192     , -2.165        
#1939, 1   ,  41.6         ,  45.0547      , -3.45466      ,  0.834433     , -4.14012      
#1940, 1   ,  45           ,  49.0118      , -4.01179      ,  0.966472     , -4.15096      
#1941, 1   ,  53.3         ,  56.6727      , -3.37275      ,  1.23486      , -2.73127   
#     
#
#...ESTIMATE OK

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