# rm(list=ls())
options(OutDec = ",") 
#===================================================================
# Multicolinearidade
# O modelo: 
# y representa emprego.
# y = beta0+beta1*ano+beta2*preco+beta3*PIB+beta4*militar+epsilon
#===================================================================
dataxy   <- read.table("../dados/exemplo_09.txt",header=T)
print(cor(dataxy))

          Employ     Price       GNP     Armed      Year
Employ 1,0000000 0,9708985 0,9835516 0,4573074 0,9713295
Price  0,9708985 1,0000000 0,9915892 0,4647442 0,9911492
GNP    0,9835516 0,9915892 1,0000000 0,4464368 0,9952735
Armed  0,4573074 0,4647442 0,4464368 1,0000000 0,4172451
Year   0,9713295 0,9911492 0,9952735 0,4172451 1,0000000

#===================================================================
# Ajuste com 15 dados.
#===================================================================
ajuste1 <- lm(Employ~Year+Price+GNP+Armed,data=dataxy[1:15,])
print(summary(ajuste1))

Call:
lm(formula = Employ ~ Year + Price + GNP + Armed, data = dataxy[1:15, 
    ])

Residuals:
    Min      1Q  Median      3Q     Max 
-760,49 -281,76  -16,70   82,24 1182,69 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)  1,459e+06  7,142e+05   2,043  0,06825 . 
Year        -7,218e+02  3,700e+02  -1,951  0,07965 . 
Price       -1,811e+02  1,355e+02  -1,336  0,21101   
GNP          9,107e-02  2,026e-02   4,495  0,00115 **
Armed       -7,494e-02  2,611e-01  -0,287  0,77999   
---
Signif. codes:  0 ‘***’ 0,001 ‘**’ 0,01 ‘*’ 0,05 ‘.’ 0,1 ‘ ’ 1

Residual standard error: 561,6 on 10 degrees of freedom
Multiple R-squared:  0,9798,	Adjusted R-squared:  0,9717 
F-statistic:   121 on 4 and 10 DF,  p-value: 2,006e-08

#===================================================================
# Ajuste com 16 dados.
#===================================================================
ajuste2 <- lm(Employ~Year+Price+GNP+Armed,data=dataxy)
summary(ajuste2)

Call:
lm(formula = Employ ~ Year + Price + GNP + Armed, data = dataxy)

Residuals:
   Min     1Q Median     3Q    Max 
-905,8 -342,7 -107,6  216,8 1437,7 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)  1,169e+06  8,359e+05   1,399  0,18949   
Year        -5,765e+02  4,335e+02  -1,330  0,21049   
Price       -1,977e+01  1,389e+02  -0,142  0,88940   
GNP          6,439e-02  1,995e-02   3,227  0,00805 **
Armed       -1,015e-02  3,086e-01  -0,033  0,97436   
---
Signif. codes:  0 ‘***’ 0,001 ‘**’ 0,01 ‘*’ 0,05 ‘.’ 0,1 ‘ ’ 1

Residual standard error: 667,3 on 11 degrees of freedom
Multiple R-squared:  0,9735,	Adjusted R-squared:  0,9639 
F-statistic: 101,1 on 4 and 11 DF,  p-value: 1,346e-08

#===================================================================
# Verificando o número de condição de XtX. 
#===================================================================
X        <- cbind(rep(1,16),as.matrix(dataxy[,2:5]))
XtX      <- t(X)%*%X
for(i in 1:5) XtX[,i] <- XtX[,i]/sqrt(sum(XtX[,i]^2))
raizcar  <- svd(XtX)$d
ncond    <- sqrt(max(raizcar)/min(raizcar))
print(ncond)

[1] 5570620

#===================================================================
# Fim
#===================================================================