Problems

Ex. \(3.18\)

\(d\): -2, 1,-1, 3, 0,-1,-1,-2, 1, 2

\(d^2\): 4, 1, 1, 9, 0, 1, 1, 4, 1, 4

\(\sum {d^2 } = 26\)

Correlation of Spearman: \(r_s = 1 - \frac{{6\sum {d^2 } }}{{n(n^2 - 1)}} = 1 - \frac{{6*26}}{{10(100 - 1)}} = 1 - \frac{{156}}{{990}} = 0.84\)

We see that the coefficient is high b/n students’ midterm and final ranks in statistics. There is a direct relationship b/n them.

Ex. \(3.19\)

a)The value of intercept = 6,68 means that if the X (ratio of the U.S. CPI to German CPI) is 0,a dollar’d exchange on 6,68 German marks. We see in this case there is no economic sense. \(R^2\) shows how variation in index of CPI US to CPI German explain the exchange rate of the German mark to the U.S. dollar.As for the value of slope = -4,31 with an increas on 1 unit in relative prices the Y exchange rate decline on 4,31 units.

\(R^2\) shows to us that variability of values of relative prices explaine variability of values of nominale exchange only on 52,3%. there are some other unreported factors.

b)The negative value of \(X_t\) makes economic sense, i.e. prices of US increase and become bigger than German ones. So people begin to buy German good and products that strengthen German mark.

Ex. \(3.20\)

table3.6 <- read.table("C:/Users/HOME/Documents/Table3.6.txt", header=TRUE)
attach(table3.6)
plot(X1~Y1,main="Business sector", xlab= "output per hour, business sector", ylab="compensation per hour, business sector", pch =20)

plot of chunk unnamed-chunk-1

plot(X2~Y2,main="Nonfarm business sector", xlab= "output per hour, nonfarm business sector", ylab="compensation per hour, nonfarm business sector", pch=20)

plot of chunk unnamed-chunk-1

lm <- lm( X1~Y1)
summary(lm)
## 
## Call:
## lm(formula = X1 ~ Y1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -18.721  -8.337  -0.158   9.852  21.288 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -109.383      9.712   -11.3  1.6e-13 ***
## Y1             2.004      0.118    17.0  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.5 on 37 degrees of freedom
## Multiple R-squared:  0.887,  Adjusted R-squared:  0.884 
## F-statistic:  290 on 1 and 37 DF,  p-value: <2e-16
lm2 <- lm(X2~Y2)
summary(lm2)
## 
## Call:
## lm(formula = X2 ~ Y2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -19.784  -8.399  -0.392  10.499  21.388 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -123.600     11.020   -11.2  1.8e-13 ***
## Y2             2.139      0.131    16.3  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.9 on 37 degrees of freedom
## Multiple R-squared:  0.878,  Adjusted R-squared:  0.874 
## F-statistic:  266 on 1 and 37 DF,  p-value: <2e-16

Ex. \(3.21\)

\(\sum {Y_i } = 1110\),

\(\sum {X_i } = 1700 - (110 + 210) + (120 + 220) = 1680\),

\(\sum {X_i } Y_i = 20550 - (90*120 + 140*220) + (80*110 + 150*210) = 204200\),

\(\sum {X_i } ^2 = 322000 - (120^2 + 220^2 ) + (110^2 + 210^2 ) = 315400\),

\(\sum {Y_i } ^2 = 132100 - (90^2 + 140^2 ) + (80^2 + 150^2 ) = 133300\),

Coeff. of corr = \(\frac{{n\sum {X_i Y_i - (\sum {X_i )(} } \sum {Y_i )} }}{{\sqrt {[n\sum {X_i^2 - (\sum {X_i )^2 ][n\sum {Y_i^2 - (\sum {Y_i )^2 ]} } } } } }}=\frac{{177200}}{{182916,483}} = 0,9687 = 96,87\%\)

Ex. \(3.22\)

table3.7 <- read.table("C:/Users/HOME/Documents/Table3.7.txt", header=TRUE)

attach(table3.7)
plot(YEAR,CPI,type="l",col="green", lwd=6, main="CPI and the NYSE Index")
lines(YEAR,PRICE,col="red", lwd=6)
lines(YEAR,NYSE,col="red", lwd=6)

plot of chunk unnamed-chunk-3

gold <-lm(PRICE~CPI)
summary(gold)
## 
## Call:
## lm(formula = PRICE ~ CPI)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -149.8  -55.0  -17.3   42.0  274.6 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   186.18     125.40    1.48     0.16
## CPI             1.84       1.22    1.52     0.15
## 
## Residual standard error: 105 on 13 degrees of freedom
## Multiple R-squared:  0.15,   Adjusted R-squared:  0.0849 
## F-statistic:  2.3 on 1 and 13 DF,  p-value: 0.153
nyse<-lm(NYSE~CPI)
summary(nyse)
## 
## Call:
## lm(formula = NYSE ~ CPI)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -34.50 -17.44   4.67  17.48  26.71 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -102.06      23.77   -4.29  0.00087 ***
## CPI             2.13       0.23    9.25  4.4e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19.8 on 13 degrees of freedom
## Multiple R-squared:  0.868,  Adjusted R-squared:  0.858 
## F-statistic: 85.5 on 1 and 13 DF,  p-value: 4.43e-07

We can see that New York Stock Exchange is better than Price of gold react to inflation.

Ex. \(3.23\)

data3.8 <- read.table("C:/Users/HOME/Documents/Table3.8.txt", header=TRUE)
attach(data3.8)

par(mfrow=c(1,2))
plot(RGDP~Year, type="l", col="blue", lwd=5, lty=7)
plot(NGDP~Year, type="l", col="blue", lwd=5, lty=7)

plot of chunk unnamed-chunk-4

curr <- lm(NGDP~Year)
summary(curr)
## 
## Call:
## lm(formula = NGDP ~ Year)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
##   -830   -576   -123    456   1292 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -3.96e+05   1.83e+04   -21.6   <2e-16 ***
## Year         2.02e+02   9.27e+00    21.8   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 651 on 37 degrees of freedom
## Multiple R-squared:  0.928,  Adjusted R-squared:  0.926 
## F-statistic:  475 on 1 and 37 DF,  p-value: <2e-16
cons <-lm(RGDP~Year)
summary(cons)
## 
## Call:
## lm(formula = RGDP ~ Year)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -378.2  -74.5   18.5   84.8  339.6 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.50e+05   3.89e+03   -64.3   <2e-16 ***
## Year         1.29e+02   1.97e+00    65.5   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 138 on 37 degrees of freedom
## Multiple R-squared:  0.991,  Adjusted R-squared:  0.991 
## F-statistic: 4.29e+03 on 1 and 37 DF,  p-value: <2e-16

In current-dollar GDP model:

\(\beta _1 = {\rm 396500}\),

\(\beta _2 = 202\)

Un constant-dollar GDP model:

\(\beta _1 = {\rm - 250200}\)

\(\beta _2 = 128,8\)

The slope shows the rate of change of GDP (curr. and const.) per year

we can make a conclusion that nominal GDP increses faster than real GDP. ###Ex. \(3.24\)

dataI.1 <- read.table("C:/Users/HOME/Documents/TableI.1.txt", header=TRUE)
attach(dataI.1)
## The following object is masked from data3.8:
## 
##     Year
mod_GDP <- lm(PCE~GDP)
summary(mod_GDP)
## 
## Call:
## lm(formula = PCE ~ GDP)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -39.33  -8.60   1.76  14.77  31.31 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.84e+02   4.63e+01   -3.98   0.0016 ** 
## GDP          7.06e-01   7.83e-03   90.25   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 20.3 on 13 degrees of freedom
## Multiple R-squared:  0.998,  Adjusted R-squared:  0.998 
## F-statistic: 8.14e+03 on 1 and 13 DF,  p-value: <2e-16

So, we verified the equation given in Introduction.

\(\beta _1 = - 181\),

\(\beta _2 = 0,7064\),

\(\R^2 = 99,84\%\)

Ex. \(3.25\)

data2.9 <- read.table("C:/Users/HOME/Documents/Table2.9.txt", header=TRUE)
attach(data2.9)
## The following object is masked from dataI.1:
## 
##     Year
## 
## The following object is masked from data3.8:
## 
##     Year
par(mfrow=c(1,2))
plot(Fem_v~Year, xlab="Year", ylab="Female verbal score", pch=20, col="red", type="l")
plot(Mal_v~Year, xlab="Year", ylab="Male verbal score", pch=20, col="red",type="l")

plot of chunk unnamed-chunk-6

fem<-lm(Fem_v~Mal_v)
summary(fem)
## 
## Call:
## lm(formula = Fem_v ~ Mal_v)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -4.719 -2.103  0.245  1.562  6.153 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -198.1258    25.2112   -7.86  7.9e-08 ***
## Mal_v          1.4364     0.0572   25.10  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.27 on 22 degrees of freedom
## Multiple R-squared:  0.966,  Adjusted R-squared:  0.965 
## F-statistic:  630 on 1 and 22 DF,  p-value: <2e-16

So, in our regression model (where dependent variable is femal verbal score and independent variable is male verbal score) we get next values of parameters:

\(\beta _1 = {\rm - 198}{\rm ,13}\),

\(\beta _2 = {\rm 1}{\rm ,44}\),

\(\R^2 = 96,63\%\)

As for causality we cannot establish it.

Ex. \(3.26\)

par(mfrow=c(1,2))
plot(Fem_m~Year, xlab="Year", ylab="Female math score", pch=20, col="purple", type="l")
plot(Mal_m~Year, xlab="Year", ylab="Male math score", pch=20, col="purple",type="l")

plot of chunk unnamed-chunk-7

mal<-lm(Fem_m~Mal_m)
summary(mal)
## 
## Call:
## lm(formula = Fem_m ~ Mal_m)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -4.810 -1.669  0.048  1.761  4.048 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -189.0572    40.9270   -4.62  0.00013 ***
## Mal_m          1.2854     0.0818   15.71  1.9e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.68 on 22 degrees of freedom
## Multiple R-squared:  0.918,  Adjusted R-squared:  0.914 
## F-statistic:  247 on 1 and 22 DF,  p-value: 1.94e-13

So, in our regression model (where dependent variable is femal math score and independent variable is male math score) we get next values of parameters:

\(\beta _1 = {\rm - 189}{\rm ,057}\),

\(\beta _2 = {\rm 1}{\rm ,285}\),

\(R^2 = 91,81\%\)