## 统计代写|回归分析作业代写Regression Analysis代考|ST503

2022年10月17日

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## 统计代写|回归分析作业代写Regression Analysis代考|Unbiasedness

The Gauss-Markov (G-M) theorem states that, under certain model assumptions (the premise, ” $\mathrm{A}$ ” of the theorem), the OLS estimator has minimum variance among linear unbiased estimators (that is the consequence, the “condition $\mathrm{B}^{\text {” }}$ of the theorem). To understand the $\mathrm{G}-\mathrm{M}$ theorem, you first need to understand what “unbiasedness” means. Recall the view of regression data shown in Chapter 2, shown again in Table 3.1.

To be specific, please consider the Production Cost data set from Chapter 1. The actual data are shown in Table 3.2, along with the random data-generation assumption of the regression model.

In particular, the value 2,224 is assumed to be produced at random from a distribution of potentially observable Cost values among jobs having 1,500 widgets, the value 1,660 is assumed to be produced at random from a distribution of potentially observable Cost values among jobs having 800 widgets, and so on. If you are having trouble visualizing these different distributions, just have a look at Figure $1.7$ again, and put yourself in the position of the job manager at this company: In two different jobs where the number of widgets is the same, will the costs also be the same? Of course not; see the first and third observations in the data set, for example. There is an entire distribution of potentially observable Cost values when Widgets $=1500$, and this is what is meant by $p(y \mid X=1500)$.

Now, use your imagination. Imagine another collection of 40 jobs, from the same process that produced the data above, with the widgets data exactly as observed, but with specific costs not observed. Further, imagine that the classical model is true so that the distribution $p(y \mid X=x)$ is the $\mathrm{N}\left(\beta_0+\beta_1 x, \sigma^2\right)$ distribution. The specific costs are not observed, but the potentially observable data will appear as shown in Table $3.3$.

In Table $3.3$, the $Y_i$ are random variables, coming from the same distributions that produced the original data. Again, use your imagination: There are infinitely many potentially observable data sets as shown in Table $3.3$, because there are infinitely many sequences of potentially observable values for $Y_1 ;$ infinitely many sequences of potentially observable values for $Y_2, \ldots$; and there are infinitely many sequences of potentially observable values for $Y_{40}$. Again, if you are having a hard time visualizing this, just look at Figure $1.7$ again: There are an infinity of possible values under each of the normal curves shown there. The $n=40 Y_i$ values in Table $3.3$ are one set of random selections from such distributions.

For each of these potentially observable data sets of $n=40$ jobs, you will get different parameter estimates. Because there are infinitely many potentially observable data sets, and because each data set gives different parameter estimates, there are also infinitely many different potentially observable values of $\hat{\beta}_0$ and $\hat{\beta}_1$.

Thus, your task is to use your imagination and view the one data set you actually observed (the one above with 40 observations, for example) as one of infinitely many potentially observable data sets that you could have observed from the same data-generating process. As such, you must also view the particular parameter estimates you actually observed, for example, the OLS estimates $\hat{\beta}_0=55.5, \hat{\beta}_1=1.62$, as one of the infinitely many pairs of parameter estimates that you could have observed.

In this context, unbiasedness of a parameter estimate $\hat{\theta}$ is defined as follows. The Greek letter ” $\theta$ ” refers to any generic parameter or function of parameters, for example $\theta$ might refer to $\beta_1$ (where $\theta=\beta_1$ ) or to $\sigma$ (where $\theta=\sigma$ ) or to a conditional mean such as $\beta_0+\beta_1(15)$ (where $\theta=\beta_0+\beta_1(15)$ ), etc.

## 统计代写|回归分析作业代写Regression Analysis代考|A Simulation Study

To start a simulation study, you must specify the model and its parameter values, which in the case of the classical model will be the $\mathrm{N}\left(\beta_0+\beta_1 x, \sigma^2\right)$ probability distribution, along with the three parameters $\left(\beta_0, \beta_1, \sigma\right)$. These parameters are unknown, so just pick any values that make sense. No matter what values you pick for those parameters, the estimates you get are (i) random, and (ii) when unbiased, neither systematically above nor below those parameter values, in an average sense.

In reality, Nature picks the actual values of the parameters $\left(\beta_0, \beta_1, \sigma\right)$, and you do not know their values. In simulation studies, you pick the values $\left(\beta_0, \beta_1, \sigma\right)$. The estimates $\left(\hat{\beta}_0, \hat{\beta}_1\right.$, and $\left.\hat{\sigma}\right)$ target those particular values, but with error that you know precisely because you know both the estimates and the true values. In the real world, with your real (not simulated) data, your estimates $\hat{\beta}_0, \hat{\beta}_1$, and $\hat{\sigma}$ also target the true values $\beta_0, \beta_1$, and $\sigma$, but since you do not know the true values for your real data, you also do not know the error. Simulation allows you to understand this error, so you can better understand how your estimates $\hat{\beta}_0$, $\hat{\beta}_1$ and $\hat{\sigma}$ relate to Nature’s true values $\beta_0, \beta_1$, and $\sigma$.

In the Production Cost example, the values $\beta_0=55, \beta_1=1.5, \sigma^2=250^2$ produce data that look reasonably similar to the actual data, as shown in Chapter 1 . So let’s pick those values for the simulation. No matter which values you pick for your simulation parameters $\beta_0, \beta_1$, and $\sigma$, the statistical estimates $\hat{\beta}_0, \hat{\beta}_1$, and $\hat{\sigma}$ “target” those values.

To make the abstractions concrete and understandable, run the following simulation code, which produces data exactly as indicated in Table $3.3$.

# 回归分析代写

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## MATLAB代写

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