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# 经济代写|计量经济学代写Introduction to Econometrics代考|ECO400 Solving for Least Squares with Multiple Regressors

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Solving for Least Squares with Multiple Regressors

We now consider the case with $k>1$ so that the coefficient $\boldsymbol{\beta}$ is a vector.
To illustrate, Figure 3.2(a) displays a scatter plot of 100 triples $\left(y_i, x_{1 i}, x_{2 i}\right)$. The regression function $\boldsymbol{x}^{\prime} \boldsymbol{\beta}=x_1 \beta_1+x_2 \beta_2$ is a 2-dimensional surface and is shown as the plane in Figure 3.2(a).

The sum of squared errors $\operatorname{SSE}(\boldsymbol{\beta})$ is a function of the vector $\boldsymbol{\beta}$. For any $\boldsymbol{\beta}$ the error $y_i-\boldsymbol{x}i^{\prime} \boldsymbol{\beta}$ is the vertical distance between $y_i$ and $\boldsymbol{x}_i^{\prime} \boldsymbol{\beta}$. This can be seen in Figure 3.2(a) by the vertical lines which connect the observations to the plane. As in the single regressor case these vertical lines are the errors $e_i=y_i-$ $\boldsymbol{x}_i^{\prime} \boldsymbol{\beta}$. The sum of squared errors is the sum of the 100 squared lengths. The sum of squared errors can be written as $$\operatorname{SSE}(\boldsymbol{\beta})=\sum{i=1}^n y_i^2-2 \boldsymbol{\beta}^{\prime} \sum_{i=1}^n \boldsymbol{x}i y_i+\boldsymbol{\beta}^{\prime} \sum{i=1}^n \boldsymbol{x}_i \boldsymbol{x}_i^{\prime} \boldsymbol{\beta} .$$
As in the single regressor case this is a quadratic function in $\boldsymbol{\beta}$. The difference is that in the multiple regressor case this is a vector-valued quadratic function. To visualize the sum of squared errors function Figure 3.2(b) displays $\operatorname{SSE}(\boldsymbol{\beta})$. Another way to visualize a 3-dimensional surface is by a contour plot. A contour plot of the same $\operatorname{SSE}(\boldsymbol{\beta})$ function is shown in Figure ??. The contour lines are points in the $\left(\beta_1, \beta_2\right)$ space where $\operatorname{SSE}(\boldsymbol{\beta})$ takes the same value. The contour lines are elliptical.

The least-squares estimator $\widehat{\boldsymbol{\beta}}$ minimizes $\operatorname{SSE}(\boldsymbol{\beta})$. A simple way to find the minimum is by solving the first-order conditions. The latter are
$$0=\frac{\partial}{\partial \boldsymbol{\beta}} \operatorname{SSE}(\widehat{\boldsymbol{\beta}})=-2 \sum_{i=1}^n \boldsymbol{x}i y_i+2 \sum{i=1}^n \boldsymbol{x}_i \boldsymbol{x}_i^{\prime} \widehat{\boldsymbol{\beta}}$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Illustration

We illustrate the least-squares estimator in practice with the data set used to calculate the estimates reported in Chapter 2. This is the March 2009 Current Population Survey, which has extensive information on the U.S. population. This data set is described in more detail in Section 3.22. For this illustration we use the sub-sample of married (spouse present) black female wage earners with 12 years potential work experience. This sub-sample has 20 observations.

In Table 3.1 we display the observations for reference. Each row is an individual observation which are the data for an individual person. The columns correspond to the variables (measurements) for the individuals. The second column is the reported wage (total annual earnings divided by hours worked). The third column is the natural logarithm of the wage. The fourth column is years of education. The fifth and six columns are further transformations, specifically the square of education and the product of education and $\log ($ wage). The bottom row are the sums of the elements in that column.

Putting the variables into the standard regression notation, let $y_i$ be log wages and $\boldsymbol{x}i$ be years of education and an intercept. Then from the column sums in Table 3.1 we have $$\sum{i=1}^n \boldsymbol{x}i y_i=\left(\begin{array}{c} 995.86 \ 62.64 \end{array}\right)$$ and $$\sum{i=1}^n x_i x_i^{\prime}=\left(\begin{array}{cc} 5010 & 314 \ 314 & 20 \end{array}\right)$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|lllustration

$$\sum i=1^n x i y_i=(995.8662 .64)$$

$$\sum i=1^n x_i x_i^{\prime}=\left(\begin{array}{lll} 5010 & 314314 & 20 \end{array}\right)$$

## MATLAB代写

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