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统计代写|线性回归分析代写linear regression analysis代考|Do those in the reference group experience a lower-intensity

Suppose you wanted to estimate the following:
$$
\text { Postural problems }=\beta_0+\beta_1 \times(\text { whether had } 5+\text { hours on cell phone })+\varepsilon
$$
The key-X variable is a dummy variable for whether the person spends 5-or-more hours per day on their cell phone. The reference group would be those who spend $<5$ hours on their cell phone.
In this situation, those in the reference group would have some lower-intensity effects of the treatment. They could have 4.99 hours on their cell phone, on average, and yet be considered part of the reference group in this model. Any harmful effects of too much use of cell phones on posture might be already ingrained in a body from 3 hours per day. At the same time, perhaps those having 0 hours of cell phone use might not be a good reference group, as that would not be the counterfactual for someone if they did not spend $5+$ hours on their phone.

Thus, estimating equation (6.15) could understate the impact. This strategy could be used by someone researching a pharmaceutical drug (or cell phone use) if they wanted to show that the side effects of the drug (the pharmaceutical one or the phone) are minimal.

The direction of the bias for this sub-question is more straightforward than the others, as it would be a muted estimated effect (downward in magnitude). This use of improper reference groups could be common in health-related studies.
Perhaps step-wise effects should be estimated. For instance, the variables could be:

• 0-1 hours per day (the excluded category)
• At least 1 hour per day
• At least 3 hours per day
• At least 5 hours per day.
  Based on the discussion from Section 3.1 on the highest degree and income, the coefficient on “At least 5 hours” would now be a comparison to the “At least 3 hours” group. All that said, it would be difficult to know what the likely counterfactual (range of hours using the cell phone) would be for someone who chooses not to use the cell phone for at least 5 hours per day, and generally what the groups (variables) should be. It could be useful to examine the data to determine if there were any natural breaks in the data in the distribution of average daily cell phone use. Perhaps the better approach is to just use actual hours per day as the key- $\mathrm{X}$ variable.

统计代写|线性回归分析代写linear regression analysis代考|Common cases in which groups could get over-weighted

Basically, this bias could occur in any situation in which the variance of the key-X variable could be different across groups. Just due to natural variation, there will always be some difference in variance across groups, causing some over-weighting of certain groups. The question is whether the differences across groups in the variance of the key-X variable and the differences in the causal effects are large enough to meaningfully bias the estimated effects of the key-X variable.

There is a multitude of cases in which this could occur. A generic case that I believe has a great risk for such bias would be policy analysis at the state or local level. Many studies estimating policy effects will use city- or state-level data. For example, from PITFALL #5, there are studies that attempt to estimate the effects of state income tax rates on state economic growth. Other studies examine the effects of state minimum-wage laws, medical-marijuana laws, drunk-driving laws, welfare laws, and more. These studies typically would use panel data with either pooled cross-sections or aggregate state-level data across numerous years. And the studies would typically control for the state with a set of dummy variables or fixed effects.

Let me note that such studies would also typically weight states or cities, to some extent, by population size. This would either be by having more individual-level observations for the larger states or using sample weights related to state populations for aggregate-level analyses. The over-weighting discussed here would be over- and under-weighting beyond differences in group weights based on the population size.

Controlling for the state (or city) would be essential to avoid omitted-factors bias. However, there would certainly be large differences in the variance of the key- $\mathrm{X}$ (policy) variable across states. For example, some states have no income-tax rate or have a rate that had not changed during a period of analysis. Those states would be under-weighted in the model; in fact, they would not contribute at all to the overall estimated tax-rate effect because a coefficient estimate on the tax rate could not be estimated for those states. The states that had the larger changes in the tax rate would be over-weighted.
The same would apply to evaluating state minimum-wage laws. The states with the larger changes would have greater variances in the minimum wage and consequently be over-weighted. And, if states that choose the larger minimum-wage increases are those that could handle the larger changes without too much employment loss, then there would be a bias in the estimated effects of minimum-wage increases towards zero.

This could also occur for cases in which a policy is implemented, represented as a dummy variable. For example, about half of all states have implemented medical-marijuana laws, allowing people to legally use marijuana to address a medical condition. If there were a study on how such laws affect some outcomes (say, overall marijuana use) over the 2000-2020 time period, then the states with the law implemented closer to the middle of the period (2010) would have a larger variance in the policy variable and would consequently be over-weighted. To demonstrate this, in a sequence of six numbers:

• The sequence of $(0,0,0,1,1,1)$ – equivalent to implementing the policy halfway through a sixyear period – has a variance of 0.30 .
• The sequence of $(0,1,1,1,1,1)$ or $(0,0,0,0,0,1)$ – equivalent to implementing the policy before the second year or the last year – has a variance of 0.17 , and would have just over one-half the weight of the state implementing the law half-way through the period.

线性回归代考

统计代写|线性回归分析代写linear regression analysis代考|Do those in the reference group experience a lower-intensity

假设您想估计以下内容:
Postural problems $=\beta_0+\beta_1 \times($ whether had $5+$ hours on cell phone $)+\varepsilon$
key-X 变量是一个虚拟变量，表示此人是否每天在手机上花费 5 小时或更长时间。参考组 是那些花费 $<5$ 小时在他们的手机上。
在伩种情况下，参考组中的那些人会受到一些较低强度的治疗效果。他们平均有 4.99 小时 的手机通话时间，但仍被视为该模型中参考组的一部分。从每天 3 小时起，过度使用手机 对姿势的任何有害影响可能已经根深蒂固。与此同时，那些手机使用时间为 0 小时的人可 能不是一个很好的参考群体，因为如果他们不花钱，这就不会成为反事实 $5+$ 小时在他们的 电话上。
因此，估计方程式 (6.15) 可能低估了影响。如果研究药物（或手机使用）的人想证明药物 (药物或手机) 的副作用很小，则可以使用此策略。
这个子问题的偏差方向比其他问题更直接，因为它是一个减弱的估计效应 (幅度向下)。 这种使用不当参考组的做法在健康相关研究中可能很常见。 也许应该估计逐步影响。例如，变量可以是:

• 每天0-1小时 (排除类别)
• 每天至少1小时
• 每天至少3小时
• 每天至少5小时。
  根据第 3.1 节关于最高学位和收入的讨论，”至少 5 小时”的系数现在将与”至少 3 小时” 组进行比较。综上所述，对于选择每天至少 5 小时不使用手机的人来说，很难知道可能 的反事实 (使用手机的时间范围) 是什么，以及一般情况下这些群体（变量) 应该。检 查数据以确定平均每日手机使用的分布数据中是否存在任何自然中断可能很有用。也许 更好的方法是只使用每天的实际小时数作为关键- $\mathrm{X}$ 多变的。

统计代写|线性回归分析代写linear regression analysis代考|Common cases in which groups could get over-weighted

基本上，这种偏差可能发生在 key-X 变量的方差在不同组之间可能不同的任何情况下。仅 仅由于自然变异，组间方差总会存在一些差异，导致某些组的权重过高。问题是 key-X变 量方差的组间差异和因果效应的差异是否大到足以对 key-X 变量的估计效应产生有意义的 偏差。
在很多情况下都可能发生这种情况。我认为存在这种偏见风险的一般案例是州或地方层面 的政策分析。许多估计政策影响的研究将使用城市或州级数据。例如，从 PITFALL #5 开 始，有一些研究试图估计州所得税率对州经济增长的影响。其他研究考察了州最低工资 法、医用大麻法、酒后驾车法、福利法等的影响。这些研究通常会使用具有汇总横截面或 多年汇总州级数据的面板数据。这些研究通常会用一组虚拟变量或固定效应来控制状态。
让我指出，此类研究通常也会在某种程度上根据人口规模对州或城市进行加权。这可以通 过对较大的州进行更多的个体层面观察，或者使用与州人口相关的样本权重进行总体层面 的分析。这里讨论的超重将是超出基于人口规模的群体权重差异的超重和轻度权重。
控制州 (或城市) 对于避免遗漏因素偏差至关重要。但是，密钥的方差肯定会有很大差异$\mathrm{X}$ (政策) 各州之间的变量。例如，一些州没有所得税率，或者其税率在分析期间没有变 化。这些州在模型中的权重不足；事实上，它们根本不会对总体估计税率影响做出贡献， 因为无法估计这些州的税率系数估计值。税率变化较大的州将被高估。
这同样适用于评估州最低工资法。变化较大的州在最低工资方面的差异较大，因此权重过 高。而且，如果选择大幅提高最低工资的州是那些能够应对更大变化而不会造成太多就业 损失的州，那么最低工资增长的估计影响将偏向于零。
这也可能发生在实施政策的情况下，表示为虚拟变量。例如，大约一半的州实施了医用大 麻法，允许人们合法使用大麻来解决医疗问题。如果研究此类法律如何影响 2000 年至 2020 年期间的某些结果（例如，整体大麻使用)，那么在接近该期间中期（2010 年) 实 施该法律的州将有更大的差异政策变量，因此将被过度加权。为了证明这一点，在六个数 字的序列中:

• 的顺序 $(0,0,0,1,1,1)$ – 相当于在六年期间实施政策的一半 – 方差为 0.30 。
• 的顺序 $(0,1,1,1,1,1)$ 或者 $(0,0,0,0,0,1)$ – 相当于在第二年或最后一年之前实施该政 策-方差为 0.17 ，并且将超过该期间中途实施法律的州的一半。

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