Let's first find the solution for (a), that is, for the bootstrap estimate of variance. Such simulation of the standard error of the sample variance uses the following approach. Denote with\({R^{(i)}},i = 1,2, \ldots ,\nu \)the bootstrap sample correlation coefficients. To estimate the variance, one uses the sample variance
\(Z = \frac{1}{\nu }\sum\limits_{i = 1}^\nu {{{\left( {{R^{(i)}} - \bar R} \right)}^2}} \)
The goal is now to find the simulation variance of this $Z$. The delta method can be used. By denoting
\({W^{(i)}} = {R^{(i)2}}\)
the sample variance Z may be written as
\(Z = \frac{1}{\nu }\sum\limits_{i = 1}^\nu {{{\left( {{R^{(i)}} - \bar R} \right)}^2}} = \bar W - {\bar Y^2}\)
Let V be the sample variance of\({W^{(i)}},i = 1,2, \ldots \ldots ,\nu .\)And lastly, the required covariance between the initial bootstrap sample\({R^{(i)}},i = 1,2, \ldots ,\nu \)and the squared\({W^{(i)}},i = 1,2, \ldots ,\nu \) is given by
\(C = \frac{1}{\nu }\sum\limits_{i = 1}^\nu {\left( {{R^{(i)}} - \bar R} \right)} \left( {{W^{(i)}} - \bar W} \right)\)
Finally, the variance of the necessary Z is
\(\hat \sigma _Z^2 = \frac{1}{\nu }\left( {4{{\bar Y}^2}Z - 4\bar YC + V} \right)\)
This is the formula used in the code in the end. Note that the standard deviation is the square root of the variance. The result given by the code below is $0.0031$.
The solution for part (b) is more straightforward, and it is only required to find the sample variance of\({R^{(i)}},i = 1,2, \ldots ,\nu \)minus the sample correlation coefficient as given in the last line of code of part (b). The standard simulation error is 0.0086.
#Read data
\(data = read.delim(file{ = ^{\prime \prime }}C:\)
Users
Exercise. txt",
Year1970 = matrix (unlisted(data (1)))
Year1980 = matrix (unlisted (data (2))
nu =1000
R = numeric(nu)
W = numeric(nu)
#Generate sample correlation
for (i in (1: n u)) {
#Generate the bootstrap sample with replacement
Sample.1970 = sample (Year1970, length (Year1970), replace =T)
Sample.1980 = sample (Year1980, length (Year1980), replace =T)
#Correlation
\(\begin{aligned}{l}R(i) &= cor(Sample.1970,Sample.1980)\\W(i) &= R{(i)^ \wedge }2\end{aligned}\)
}
\(\begin{aligned}{l}Z &= mean(W) - mean{(R)^ \wedge }2\\V &= var(W)\end{aligned}\)
\(C.for.sum = numeric(nu)\)
for ( i in (1: n u)) {
\(C.for.sum(i) = (R(i) - mean(R))*(W(i) - mean(W))\)
\(C = mean(C.for.sum)\)
\(Bootstrap Eestimate Variance = \left( {4*mean{{(R)}^ \wedge }2*Z - 4*mean(R)*C + V} \right)/nu\)
\(Bootstrap Eestimate SD = sqrt(Bootstrap Eestimate Variance)\)
