Suppose you ONLY have the access to a statistical software which can only report
ID: 3358524 • Letter: S
Question
Suppose you ONLY have the access to a statistical software which
can only report estimated coefficients and does not report the standard error of the estimated
coefficient. Leveraging this software, please design a bootstrap method to estimate variance
of the estimated coefficients. It is sufficient to just state the procedure in detail and you are
not required to implement it.
Explanation / Answer
1. Estimate the regression coefficients A, B1,... ,Bk for the original sample, and calculate the fitted value and residual for each observation: Y i = A + B1xi1 +···+ Bkxik Ei = Yi Y i 2. Select bootstrap samples of the residuals, e b = [E b1, E b2,... ,E bn] , and from these,1. Estimate the regression coefficients A, B1,... ,Bk for the original sample, and calculate the fitted value and residual for each observation: Y i = A + B1xi1 +···+ Bkxik Ei = Yi Y i
2. Select bootstrap samples of the residuals, e b = [E b1, E b2,... ,E bn] , and from these,1. Estimate the regression coefficients A, B1,... ,Bk for the original sample, and calculate the fitted value and residual for each observation: Y i = A + B1xi1 +···+ Bkxik Ei = Yi Y i 2. Select bootstrap samples of the residuals, e b = [E b1, E b2,... ,E bn] , and from these,
1. Estimate the regression coefficients A, B1,... ,Bk for the original sample, and calculate the fitted value and residual for each observation: Y
i = A + B1xi1 +···+ Bkxik Ei = Yi – Yi
2. Select bootstrap samples of the residuals, e b = [E b1, E b2,... ,E bn] , and from these, calculate bootstrapped Y values, y b = [Y b1, Y b2,... ,Y bn] , where Y bi = Y
i + E bi.
3. Regress the bootstrapped Y values on the fixed X values to obtain bootstrap regression coefficients. *If, for example, estimates are calculated by least-squares regression, then b b = (X
X)1X y b for b = 1,... ,r.
4. The resampled b b = [A b, B b1,... ,B bk]
can be used in the usual manner to construct bootstrap standard errors and confidence intervals for the regression coefficients.
1. Estimate the regression coefficients A, B1,... ,Bk for the original sample, and calculate the fitted value and residual for each observation: Y
i = A + B1xi1 +···+ Bkxik Ei = Yi – Yi
2. Select bootstrap samples of the residuals, e b = [E b1, E b2,... ,E bn] , and from these, calculate bootstrapped Y values, y b = [Y b1, Y b2,... ,Y bn] , where Y bi = Y
i + E bi.
3. Regress the bootstrapped Y values on the fixed X values to obtain bootstrap regression coefficients. *If, for example, estimates are calculated by least-squares regression, then b b = (X
X)1X y b for b = 1,... ,r.
4. The resampled b b = [A b, B b1,... ,B bk]
can be used in the usual manner to construct bootstrap standard errors and confidence intervals for the regression coefficients.
1. Estimate the regression coefficients A, B1,... ,Bk for the original sample, and calculate the fitted value and residual for each observation: Y
i = A + B1xi1 +···+ Bkxik Ei = Yi – Yi
2. Select bootstrap samples of the residuals, e b = [E b1, E b2,... ,E bn] , and from these, calculate bootstrapped Y values, y b = [Y b1, Y b2,... ,Y bn] , where Y bi = Y
i + E bi.
3. Regress the bootstrapped Y values on the fixed X values to obtain bootstrap regression coefficients. *If, for example, estimates are calculated by least-squares regression, then b b = (X
X)1X
y b for b = 1,... ,r.
4. The resampled b b = [A b, B b1,... ,B bk]
can be used in the usual manner to construct bootstrap standard errors and confidence intervals for the regression coefficients.
1. Estimate the regression coefficients A, B1,... ,Bk for the original sample, and calculate the fitted value and residual for each observation: Y
i = A + B1xi1 +···+ Bkxik Ei = Yi – Yi
2. Select bootstrap samples of the residuals, e b = [E b1, E b2,... ,E bn] , and from these, calculate bootstrapped Y values, y b = [Y b1, Y b2,... ,Y bn] , where Y bi = Y
i + E bi.
3. Regress the bootstrapped Y values on the fixed X values to obtain bootstrap regression coefficients. *If, for example, estimates are calculated by least-squares regression, then b b = (X
X)1X
y b for b = 1,... ,r.
4. The resampled b b = [A b, B b1,... ,B bk]
can be used in the usual manner to construct bootstrap standard errors and confidence intervals for the regression coefficients.
1. Estimate the regression coefficients A, B1,... ,Bk for the original sample, and calculate the fitted value and residual for each observation: Y
i = A + B1xi1 +···+ Bkxik Ei = Yi – Yi
2. Select bootstrap samples of the residuals, e b = [E b1, E b2,... ,E bn] , and from these, calculate bootstrapped Y values, y b = [Y b1, Y b2,... ,Y bn] , where Y bi = Y
i + E bi.
3. Regress the bootstrapped Y values on the fixed X values to obtain bootstrap regression coefficients. *If, for example, estimates are calculated by least-squares regression, then b b = (X
X)1X y b for b = 1,... ,r.
4. The resampled b b = [A b, B b1,... ,B bk]
can be used in the usual manner to construct bootstrap standard errors and confidence intervals for the regression coefficients.
1. Estimate the regression coefficients A, B1,... ,Bk for the original sample, and calculate the fitted value and residual for each observation: Y
i = A + B1xi1 +···+ Bkxik Ei = Yi – Yi
2. Select bootstrap samples of the residuals, e b = [E b1, E b2,... ,E bn] , and from these, calculate bootstrapped Y values, y b = [Y b1, Y b2,... ,Y bn] , where Y bi = Y
i + E bi.
3. Regress the bootstrapped Y values on the fixed X values to obtain bootstrap regression coefficients. *If, for example, estimates are calculated by least-squares regression, then b b = (X
X)1X y b for b = 1,... ,r.
4. The resampled b b = [A b, B b1,... ,B bk]
can be used in the usual manner to construct bootstrap standard errors and confidence intervals for the regression coefficients.