(Weighted Least Squares) Suppose that in the model yi=β0+β1 xi+ei, the errors have mean zero and

step 1 Alright, so we have E of delta sub i equal to 0, whereas V of delta sub i is equal to sigma squa

(Weighted Least Squares) Suppose that in the model yi=β0+β1...

(Weighted Least Squares) Suppose that in the model $y_{i}=\beta_{0}+\beta_{1} x_{i}+e_{i},$ the errors have mean zero and are independent, but $\operatorname{Var}\left(e_{i}\right)=\rho_{i}^{2} \sigma^{2},$ where the $\rho_{i}$ are known constants, so the errors do not have equal variance. This situation arises when the $y_{i}$ are averages of several observations at $x_{i}$; in this case, if $y_{i}$ is an average of $n_{i}$ independent observations, $\rho_{i}^{2}=1 / n_{i}$ (why?). Because the variances are not equal, the theory developed in this chapter does not apply; intuitively, it seems that the observations with large variability should influence the estimates of $\beta_{0}$ and $\beta_{1}$ less than the observations with small variability. The problem may be transformed as follows: $$ \rho_{i}^{-1} y_{i}=\rho_{i}^{-1} \beta_{0}+\rho_{i}^{-1} \beta_{1} x_{i}+\rho_{i}^{-1} e_{i} $$ or $$ z_{i}=u_{i} \beta_{0}+v_{i} \beta_{1}+\delta_{i} $$ where $$ u_{i}=\rho_{i}^{-1} \quad v_{i}=\rho_{i}^{-1} x_{i} \quad \delta_{i}=\rho_{i}^{-1} e_{i} $$ a. Show that the new model satisfies the assumptions of the standard statistical model. b. Find the least squares estimates of $\beta_{0}$ and $\beta_{1}$ c. Show that performing a least squares analysis on the new model, as was done in part (b), is equivalent to minimizing $$ \sum_{i=1}^{n}\left(y_{i}-\beta_{0}-\beta_{1} x_{i}\right)^{2} \rho_{i}^{-2} $$ This is a weighted least squares criterion; the observations with large variances are weighted less. d. Find the variances of the estimates of part (b).

(Weighted Least Squares) Suppose that in the model $y_{i}=\beta_{0}+\beta_{1} x_{i}+e_{i},$ the errors have mean zero and are independent, but $\operatorname{Var}\left(e_{i}\right)=\rho_{i}^{2} \sigma^{2},$ where the $\rho_{i}$ are known constants, so the errors do not have equal variance. This situation arises when the $y_{i}$ are averages of several observations at $x_{i}$; in this case, if $y_{i}$
is an average of $n_{i}$ independent observations, $\rho_{i}^{2}=1 / n_{i}$ (why?). Because the variances are not equal, the theory developed in this chapter does not apply; intuitively, it seems that the observations with large variability should influence the estimates of $\beta_{0}$ and $\beta_{1}$ less than the observations with small variability. The problem may be transformed as follows:
$$
\rho_{i}^{-1} y_{i}=\rho_{i}^{-1} \beta_{0}+\rho_{i}^{-1} \beta_{1} x_{i}+\rho_{i}^{-1} e_{i}
$$
or
$$
z_{i}=u_{i} \beta_{0}+v_{i} \beta_{1}+\delta_{i}
$$
where
$$
u_{i}=\rho_{i}^{-1} \quad v_{i}=\rho_{i}^{-1} x_{i} \quad \delta_{i}=\rho_{i}^{-1} e_{i}
$$
a. Show that the new model satisfies the assumptions of the standard statistical model.
b. Find the least squares estimates of $\beta_{0}$ and $\beta_{1}$
c. Show that performing a least squares analysis on the new model, as was done
in part (b), is equivalent to minimizing
$$
\sum_{i=1}^{n}\left(y_{i}-\beta_{0}-\beta_{1} x_{i}\right)^{2} \rho_{i}^{-2}
$$
This is a weighted least squares criterion; the observations with large variances are weighted less.
d. Find the variances of the estimates of part (b).

Key Concepts

Variance of Estimators

The variance of the estimated parameters in a regression model is a measure of their precision. When using weighted least squares, the transformation of the model and the subsequent adjustment of the weightings lead to variances of the parameter estimates that reflect the differential reliability of the observations. This is important because it influences the confidence intervals and hypothesis testing for the regression parameters.

Least Squares Estimation

Least squares estimation is a method used to estimate the parameters of a regression model by minimizing the sum of squared residuals. In the context of weighted least squares, the minimization is performed on a weighted sum of squared residuals, ensuring that observations with smaller variances have a larger influence on the estimated parameters.

Model Transformation

A common technique to deal with heteroscedasticity involves transforming the original regression model into a new model where the error terms have constant variance. This transformation typically involves multiplying the entire equation by a function of the known variances, thereby converting the problem into one that satisfies the standard regression assumptions.

Weighted Least Squares (WLS)

Weighted Least Squares is a modification of ordinary least squares used when heteroscedasticity is present. By assigning weights to observations inversely proportional to their variances, WLS reduces the impact of observations with large variability. This results in more efficient and reliable parameter estimates by effectively stabilizing the variance across the data.

Heteroscedasticity

Heteroscedasticity refers to the condition in a regression model where the variance of the errors is not constant across all observations. This violates one of the key assumptions of the classical linear regression model and can lead to inefficient estimates and misleading inferences if not properly addressed.

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