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# 经济代写|计量经济学代写Introduction to Econometrics代考|ECON335 Moment Estimators

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

We want to estimate the coefficient $\beta$ defined in (3.1) from the sample of observations. Notice that $\beta$ is written as a function of certain population expectations. In this context an appropriate estimator is the same function of the sample moments. Let’s explain this in detail.

To start, suppose that we are interested in the population mean $\mu$ of a random variable $Y$ with distribution function $F$
$$\mu=\mathbb{E}[Y]=\int_{-\infty}^{\infty} y d F(y) .$$
The expectation $\mu$ is a function of the distribution $F$. To estimate $\mu$ given $n$ random variables $Y_i$ from $F$ a natural estimator is the sample mean
$$\widehat{\mu}=\bar{Y}=\frac{1}{n} \sum_{i=1}^n Y_i .$$
Notice that we have written this using two pieces of notation. The notation $\bar{Y}$ with the bar on top is conventional for a sample mean. The notation $\hat{\mu}$ with the hat ” $\wedge$ ” is conventional in econometrics to denote an estimator of the parameter $\mu$. In this case $\bar{Y}$ is the estimator of $\mu$, so $\widehat{\mu}$ and $\bar{Y}$ are the same. The sample mean $\bar{Y}$ can be viewed as the natural analog of the population mean (3.5) because $\bar{Y}$ equals the expectation (3.5) with respect to the empirical distribution – the discrete distribution which puts weight $1 / n$ on each observation $Y_i$. There are other justifications for $\bar{Y}$ as an estimator for $\mu$. We will defer these discussions for now. Suffice it to say that it is the conventional estimator.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Least Squares Estimator

The linear projection coefficient $\beta$ is defined in (3.3) as the minimizer of the expected squared error $S(\beta)$ defined in (3.4). For given $\beta$, the expected squared error is the expectation of the squared error $\left(Y-X^{\prime} \beta\right)^2$. The moment estimator of $S(\beta)$ is the sample average:
$$\widehat{S}(\beta)=\frac{1}{n} \sum_{i=1}^n\left(Y_i-X_i^{\prime} \beta\right)^2=\frac{1}{n} \operatorname{SSE}(\beta)$$
where
$$\operatorname{SSE}(\beta)=\sum_{i=1}^n\left(Y_i-X_i^{\prime} \beta\right)^2$$
is called the sum of squared errors function.
Since $\widehat{S}(\beta)$ is a sample average we can interpret it as an estimator of the expected squared error $S(\beta)$. Examining $\widehat{S}(\beta)$ as a function of $\beta$ is informative about how $S(\beta)$ varies with $\beta$. Since the projection coefficient minimizes $S(\beta)$ an analog estimator minimizes (3.6).
We define the estimator $\widehat{\beta}$ as the minimizer of $\widehat{S}(\beta)$.

## 经济代写|计量经济学代写计量经济学导论代考|力矩估计器

$$\mu=\mathbb{E}[Y]=\int_{-\infty}^{\infty} y d F(y) .$$

$$\widehat{\mu}=\bar{Y}=\frac{1}{n} \sum_{i=1}^n Y_i .$$

## 经济代写|计量经济学代写计量经济学介绍代考|最小二乘估计器

. .

$$\widehat{S}(\beta)=\frac{1}{n} \sum_{i=1}^n\left(Y_i-X_i^{\prime} \beta\right)^2=\frac{1}{n} \operatorname{SSE}(\beta)$$

$$\operatorname{SSE}(\beta)=\sum_{i=1}^n\left(Y_i-X_i^{\prime} \beta\right)^2$$

## MATLAB代写

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