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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|经济统计代写Economic Statistics代考|State- Space Model of Employment

Payroll employment growth is one of the most reliable business cycle indicators. Each postwar recession in the United States has been characterized by a year-on-year drop in payroll employment as measured by CES and, outside of these recessionary declines, the year-on-year payroll employment growth has always been positive. Thus, if one knew the “true” underlying payroll employment growth, this would help enormously in assessing the state of the economy in real time. In this section, we present results from a state-space model to infer the “true” underlying payroll employment growth. ${ }^{25}$

Let $\Delta E M P_t^U$ denote the unobserved “true” change in private payroll employment (in thousands of jobs), which is assumed to follow an AR(1) process:
$$\Delta E M P_t^U=\alpha+\rho \Delta E M P_{t-1}^U+\varepsilon_t^U .$$
$\triangle E M P_t^U$ is a latent variable for which we have two observable noisy measures, that is $\mathrm{CES}\left(\Delta E M P_t^{\mathrm{CES}}\right)$ and $\mathrm{ADP}-\mathrm{FRB}\left(\Delta E M P_t^{\mathrm{ADP}-\mathrm{FRB}}\right.$ ). Both are monthly changes in thousands of jobs. The observed values of CES and ADP-FRB employment gains are a function of the underlying state according to the following measurement equations:
$$\left[\begin{array}{l} \Delta E M P_t^{\mathrm{ADP}-\mathrm{FRB}} \ \Delta E M P_t^{\mathrm{CES}} \end{array}\right]=\left[\begin{array}{l} \beta_{\mathrm{ADP}-\mathrm{FRB}} \ \beta_{\mathrm{CES}} \end{array}\right] \Delta E M P_t^U+\left[\begin{array}{l} \varepsilon_t^{\mathrm{ADP}-\mathrm{FRB}} \ \varepsilon_t^{\mathrm{CES}} \end{array}\right] .$$
Without loss of generality, we can assume that $\beta_{\mathrm{CES}}=1$. This assumption only normalizes the unobserved state variable to move one-for-one (on average) with CES. We make the assumption in our baseline specification but leave $\beta_{\text {ADP-FRB }}$ unrestricted. ${ }^{26}$

We assume that all shocks are Gaussian and that $\varepsilon_t^U$ is orthogonal to the observation errors ( $\left.\varepsilon_t^{\text {ADP-FRB }}, \varepsilon_t^{C E S}\right)$. However, we do allow the observation errors $\left(\varepsilon_t^{\mathrm{ADP}-\mathrm{FRB}}, \varepsilon_t^{\mathrm{CES}}\right.$ ) to be contemporaneously correlated, with variancecovariance matrix $\sum$ :
$$\Sigma=\left[\begin{array}{ll} \sigma_{A D P-F R B}^2 & \sigma_{A D P-F R B, C E S}^2 \ \sigma_{A D P-F R B, C E S}^2 & \sigma_{C E S}^2 \end{array}\right]$$

## 统计代写|经济统计代写Economic Statistics代考|Characterization of the State

The estimates for the model above are collected in the first column of table 5.4. Interestingly, the estimate of $\beta_{\text {ADP-FRB }}$ is precise and not statistically different from unity. Somewhat surprisingly, the covariance of the observation errors $\sigma_{\mathrm{ADP}-\mathrm{PBB}, \mathrm{CES}}^2$ is negative, though it is not statistically different from zero. Specification 2 further generalizes the model, allowing for the ADP-FRB observation equation to have its own intercept $\alpha_{\text {ADP-FRB }}$. This modification makes little difference, and the point estimates are essentially unchanged from the baseline. Specification 3 imposes a unit factor loading in the ADPFRB equation and a diagonal $\Sigma$. Again, these alterations do not significantly change the point estimates, though the variances of the observation errors are inflated somewhat. Finally, specification 4 assumes that the unobserved state follows a random walk. All the qualitative features of specification 1 carry through to this model as well.

As discussed above, BLS produces estimates of the sampling error of CES. These estimates are based on the observed cross-sectional variation in employment growth and knowledge of the stratified sampling scheme. The estimated standard error for the change in private CES employment is about 65,000 jobs, which is remarkably close to our estimates of $\sigma_{\mathrm{CES}}$; the square root of $\sigma_{\mathrm{CES}}^2$ reported in table $5.4$ ranges between 61,000 and 69,000 jobs. In our state-space model, $\sigma_{\mathrm{CES}}$ captures all sampling and nonsampling error in the CES series, so it is reassuring that our error estimates align so closely with those of BLS.

Given that both the CES and the ADP-FRB series have been benchmarked to the QCEW, it may not be surprising that the model tends to treat them symmetrically. It is possible that most of the identification is coming Irom year-over-year variation, which would be dominaled by the QCEW. We address this concern in specification 5, which uses an unbenchmarked ADP-FRB series. The results are remarkably similar to the other specifications, indicating that the QCEW benchmark is not, in fact, dominating our estimates.

# 经济统计代考

## 统计代写|经济统计代写经济统计代考|状态空间模型-就业

$$\Delta E M P_t^U=\alpha+\rho \Delta E M P_{t-1}^U+\varepsilon_t^U .$$
$\triangle E M P_t^U$是一个潜在变量，我们有两个可观察的噪声度量，即$\mathrm{CES}\left(\Delta E M P_t^{\mathrm{CES}}\right)$和$\mathrm{ADP}-\mathrm{FRB}\left(\Delta E M P_t^{\mathrm{ADP}-\mathrm{FRB}}\right.$)。这两个数字都是数千个工作岗位的月度变化。根据以下测量方程，CES和ADP-FRB就业收益的观测值是基础状态的函数:
$$\left[\begin{array}{l} \Delta E M P_t^{\mathrm{ADP}-\mathrm{FRB}} \ \Delta E M P_t^{\mathrm{CES}} \end{array}\right]=\left[\begin{array}{l} \beta_{\mathrm{ADP}-\mathrm{FRB}} \ \beta_{\mathrm{CES}} \end{array}\right] \Delta E M P_t^U+\left[\begin{array}{l} \varepsilon_t^{\mathrm{ADP}-\mathrm{FRB}} \ \varepsilon_t^{\mathrm{CES}} \end{array}\right] .$$

$$\Sigma=\left[\begin{array}{ll} \sigma_{A D P-F R B}^2 & \sigma_{A D P-F R B, C E S}^2 \ \sigma_{A D P-F R B, C E S}^2 & \sigma_{C E S}^2 \end{array}\right]$$

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