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# 统计代写|回归分析代写Regression Analysis代考|The Classical Model and Its Consequences

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## 统计代写|回归分析代写Regression Analysis代考|The Classical Model and Its Consequences

The classical regression model assumes normality, independence, constant variance, and linearity of the conditional mean function, and is (once again) stated as follows:
$$Y_i \mid X_i=x_i \quad \sim_{\text {independent }} \mathrm{N}\left(\beta_0+\beta_1 x_i, \sigma^2\right) \text {, for } i=1,2, \ldots, n .$$
Whether you like it or not, this model is also what your computer assumes when you ask it to analyze your data via standard regression methods. The parameter estimates you get from the computer are best under this model, and the inferences ( $p$-values and confidence intervals) are exactly correct under this model. If the assumptions of the model are not true, then the estimates are not best, and the inferences are incorrect. You might think we are saying that assumptions must be true in order to use statistical methods that make such assumptions, but we are not. As we noted in Chapter 1, it is not necessarily a problem that any or all of the assumptions of the model are wrong, depending on how badly violated is the assumption. And the easiest way to understand whether an assumption is violated “too badly” is to use simulation.

We have found that students in statistics classes often resist learning simulation. After all, the data that researchers use is usually real, and not simulated, so the students wonder, what is the point of using simulation? Here are some answers:

• Simulation shows you, clearly and concretely, how to interpret the regression analysis of your real (not simulated) data.
• Simulation helps you to understand how a regression model can be useful even when the model is wrong.
• Simulation models help you to understand the meaning of the regression model parameters.
• Simulation models help you to understand the meaning of the regression model assumptions.
• Simulation models help you to understand the meaning of a “research hy pothesis.”
• Simulation helps you to understand how to interpret your data in the presence of chance effects.
• Simulation helps you to understand all the commonly misunderstood concepts in statistics, like “unbiasedness,” “standard error,” ” $p$-value,” and “confidence interval.” $^{\prime \prime}$
• Simulation methods are commonly used in the analysis of real data; examples include the bootstrap and Markov Chain Monte Carlo.

## 统计代写|回归分析代写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 B” of the theorem). To understand the G-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.

## 计代写|回归分析代写Regression Analysis代考|The Classical Model and Its Consequences

$$Y_i \mid X_i=x_i \quad \sim_{\text {independent }} \mathrm{N}\left(\beta_0+\beta_1 x_i, \sigma^2\right) \text {, for } i=1,2, \ldots, n .$$

## MATLAB代写

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