# Problema 2.10.1 del libro: Hill, C; Grinths, W and
Lim,
# G. (2011). Principle of Econometric. United States
of America. Fourth edition.
2.10.1 PROBLEMS
2.1 Consider the following five observations. You are
to do all the parts of this exercise
using
only a calculator.
> x<-c(0,1,2,3,4)
> y<-c(6,2,3,1,0)
(a)
Complete the entries in the table. Put the sums in the
last row. What are the sample means x and y?
> sum(x-mean(x))
[1] 0
> sum((x-mean(x))^2)
[1] 10
> sum(y-mean(y))
[1] 4.440892e-16
> sum((x-mean(x))*(y-mean(y)))
[1] -13
(b)
Calculate b1 and b2 using (2.7) and (2.8) and state
their interpretation.
> # b. Calcula beta mediante sumatoria
> b1<-sum((x-mean(x))*(y-mean(y)))/sum((x-mean(x))^2)
> b0<-mean(y)-b1*mean(x)
> c(b0,b1)
[1] 5.0 -1.3
Para
validar los resultados anteriores:
> lm(y~x)
Call:
lm(formula = y ~ x)
Coefficients:
(Intercept) x
5.0 -1.3
(c)
Compute sum(x^2) Using these numerical values, show
that:
> sum(x^2)
[1] 30
> sum(x*y)
[1] 11
>
> sum((x-mean(x))^2)==(sum(x^2)-(N<-length(x))*(mean(x)^2))
[1] TRUE
> sum((x-mean(x))*(y-mean(y)))==sum(x*y)-length(x)*mean(x)*mean(y)
[1] TRUE
(d)
Use the least squares estimates from part (b) to
compute the fitted values of y, and complete the remainder of the table below.
Put the sums in the last row.
> yhat=b0+b1*x
> e=y-yhat
> sum(e^2)
[1] 4.3
> sum(x*e)
[1] 1.776357e-15
>
> cbind(x,y,yhat,e,e^2,x*e)
x y yhat e
[1,] 0 6 5.0 1.0 1.00 0.0
[2,] 1 2 3.7 -1.7 2.89 -1.7
[3,] 2 3 2.4 0.6 0.36 1.2
[4,] 3 1 1.1 -0.1 0.01 -0.3
[5,] 4 0 -0.2 0.2 0.04 0.8
(e)
On graph paper, plot the data points and sketch the
fitted regression line.
> plot(x,y)
> abline(b0,b1)
(f)
On the sketch in part (e), locate the point of the
means (x; y). Does your fitted line pass through that point? If not, go back to
the drawing board, literally.
> mean(y)==b0+b1*mean(x)
[1] TRUE
(g)
Show that for these numerical values:
> mean(y)==b0+b1*mean(x)
[1] TRUE
(h)
Show that for these numerical values:
> mean(y)==mean(yhat)
[1] TRUE
(i)
Compute
> sigmahat<-sum(e^2)/(length(x)-2)
> sigmahat
[1] 1.433333
(j)
Compute var(b2)
> sigmahat<-sum(e^2)/(length(x)-2)
> sigmahat
[1] 1.433333
(k) Adicional. estudie la significancia
de los coeficientes:
> seb0<- sigmahat*(sum(x^2)/(length(x)*sum((x-mean(x))^2)))
> seb0
[1] 0.86
> seb1<- sigmahat/sum((x-mean(x))^2)
> seb1
[1] 0.1433333
>
> c(sqrt(seb0),sqrt(seb1))
[1] 0.9273618 0.3785939
>
> t1<-b1/sqrt(seb1)
> t0<-b0/sqrt(seb0)
> c(t0,t1)
[1] 5.391639 -3.433759
>
> gl<-length(e)-2
>
> #logical; if TRUE (default), probz P[X = x], otherwise, P[X > x].
> pvalor1<-pt(abs(t1), gl, lower.tail = FALSE)*2
> pvalor2<-pt(abs(t0), gl, lower.tail = FALSE)*2
>
> c(pvalor1,pvalor2)
[1] 0.04142418 0.01250200
> #r2
> r2<-sum((yhat-mean(y))^2)/sum((y-mean(y))^2)
> r2
[1] 0.7971698
>
> # residual error
> sqrt(sum(e^2)/3)
[1] 1.197219
>
> #verificar sean los valores correctos.
> summary(lm(y~x))
Call:
lm(formula = y ~ x)
Residuals:
1 2 3 4 5
1.0 -1.7 0.6 -0.1 0.2
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.0000 0.9274 5.392 0.0125 *
x -1.3000 0.3786 -3.434 0.0414 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.197 on 3 degrees of freedom
Multiple R-squared: 0.7972, Adjusted R-squared: 0.7296
F-statistic: 11.79 on 1 and 3 DF, p-value: 0.04142
