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宏观经济学Macroeconomics研究的主题包括GDP（国内生产总值）、失业（包括失业率）、国民收入、价格指数、产出、消费、通货膨胀、储蓄、投资、能源、国际贸易和国际金融。宏观经济学和微观经济学是经济学中最普遍的两个领域。联合国可持续发展目标17有一个目标，即通过政策协调和一致性来加强全球宏观经济稳定，这是2030年议程的一部分。
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经济代写|宏观经济学作业代写Macroeconomics代考|Under-parameterisation
Given the following DGP:
$$
\mathbf{y}=\mathbf{X}{1} \beta{1}+\mathbf{X}{2} \beta{2}+\mathbf{u}
$$
for which hypotheses $(1.12)-(1.20)$ hold, the following model is estimated :
$$
\mathbf{y}=\mathbf{X}{1} \beta{1}+\nu
$$
Therefore, the OLS estimates are given by the following expression
$$
\hat{\boldsymbol{\beta}}{1}^{u p}=\left(\mathbf{X}{1}^{\prime} \mathbf{X}{1}\right)^{-1} \mathbf{X}{1}^{\prime} \mathbf{y}
$$
while the OLS estimates which would have been obtained by estimation of the DGP would have been:
$$
\widehat{\boldsymbol{\beta}}{1}=\left(\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1}\right)^{-1} \mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{y}
$$
The estimates in (1.39) are Best Linear Unbiased Estimators by construction, while the estimates in (1.38) are biased unless the correlation between $\mathbf{X}{1}$ and $\mathbf{X}{2}$ is zero. To show this point consider that:
$$
\begin{aligned}
\widehat{\boldsymbol{\beta}}{1} &=\left(\mathbf{X}{1}^{\prime} \mathbf{X}{1}\right)^{-1}\left(\mathbf{X}{1}^{\prime} \mathbf{y}-\mathbf{X}{1}^{\prime} \mathbf{X}{2} \widehat{\boldsymbol{\beta}}{2}\right) \ &=\widehat{\boldsymbol{\beta}}{1}^{u p}+\widehat{\mathbf{D}} \widehat{\boldsymbol{\beta}}{2} \end{aligned} $$ where $\widehat{\mathbf{D}}$ is the vector of coefficients in the regression of $\mathbf{X}{2}$ on $\mathbf{X}{1}$, and $\widehat{\boldsymbol{\beta}}{2}$ is the OLS estimator obtained by fitting the DGP.
To provide further interpretation of these results note that if we have:
$$
\begin{aligned}
E\left(\mathbf{y} \mid \mathbf{X}{1}, \mathbf{X}{2}\right) &=\mathbf{X}{1} \boldsymbol{\beta}{1}+\mathbf{X}{2} \boldsymbol{\beta}{2} \
E\left(\mathbf{X}{1} \mid \mathbf{X}{2}\right) &=\mathbf{X}{1} \mathbf{D} \end{aligned} $$ then $$ E\left(\mathbf{y} \mid \mathbf{X}{1}\right)=\mathbf{X}{1} \boldsymbol{\beta}{1}+\mathbf{X}{1} \mathbf{D} \boldsymbol{\beta}{2}=\mathbf{X}{1} \boldsymbol{\alpha} . $$ Therefore the OLS estimator in the underparameterised model is a biased estimator of $\beta{1}$, but it is an unbiased estimator of $\alpha$. Then, if the objective of the model is forecasting and $\mathbf{X}{1}$ is more easily observed than $\mathbf{X}{2}$, than the undeparameterised model can be safely used. On the other hand, if the objective of the model is to test specific predictions on parameters (as it is the case with the Solow’s growth model), than the use of the under-parameterised model will deliver biased results. When we are interested in the effect of $\mathbf{X}_{1}$ on $\mathbf{y}$, independently from other factors, it is crucial to control for the effects of omitted variables.
经济代写|宏观经济学作业代写Macroeconomics代考|Over-parameterization
Given the following DGP:
$$
\mathbf{y}=\mathbf{X}{1} \beta{1}+\mathbf{u}
$$
for which hypotheses $(1.12)-(1.20)$ hold, the following model is estimated :
$$
\mathbf{y}=\mathbf{X}{1} \boldsymbol{\beta}{1}+\mathbf{X}{2} \boldsymbol{\beta}{2}+\mathbf{v}
$$
The OLS estimator of the over-parameterized model is
$$
\widehat{\boldsymbol{\beta}}{1}^{o p}=\left(\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1}\right)^{-1} \mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{y}
$$
while, by estimating the DGP, we obtain:
$$
\widehat{\boldsymbol{\beta}}{1}=\left(\mathbf{X}{1}^{\prime} \mathbf{X}{1}\right)^{-1} \mathbf{X}{1}^{\prime} \mathbf{y} .
$$
By substituting $y$ from the DGP it is immediately shown that both estimators are unbiased. The difference is now made by the variance. In fact we have:
$$
\begin{gathered}
\operatorname{var}\left(\widehat{\boldsymbol{\beta}}{1}^{o p} \mid \mathbf{X}{1}, \mathbf{X}{2}\right)=\sigma^{2}\left(\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1}\right)^{-1} \
\operatorname{var}\left(\widehat{\boldsymbol{\beta}}{1} \mid \mathbf{X}{1}, \mathbf{X}{2}\right)=\sigma^{2}\left(\mathbf{X}{1}^{\prime} \mathbf{X}{1}\right)^{-1} \end{gathered} $$ It can be shown that the estimator derived from the correct model is more efficient. In fact, the difference between the two variance-covariance matrices is a positive semi-definite matrix. To show this result remember that if two matrices $\mathbf{A}$ and $\mathbf{B}$ are positive definite and $\mathbf{A}-\mathbf{B}$ is positive semi-definite, then also the matrix $\mathbf{B}^{-1}-\mathbf{A}^{-1}$ is positive semi-definite. Then we have to show that $\mathbf{X}{1}^{\prime} \mathbf{X}{1}-$ $\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1}$ is a positive semi-definite matrix. Such result is almost immediately shown:
$$
\begin{aligned}
\mathbf{X}{1}^{\prime} \mathbf{X}{1}-\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1} &=\mathbf{X}{1}^{\prime}\left(\mathbf{I}-\mathbf{M}{2}\right) \mathbf{X}{1} \
&=\mathbf{X}{1}^{\prime} \mathbf{Q}{2} \mathbf{X}{1}=\mathbf{X}{1}^{\prime} \mathbf{Q}{2} \mathbf{Q}{2} \mathbf{X}_{1}
\end{aligned}
$$
We can then conclude that overparameterization impact on the efficiency of estimators and on the power of tests of hypotheses.
宏观经济学代写
经济代写|宏观经济学作业代写MACROECONOMICS代考|UNDER-PARAMETERISATION
给定以下 DGP：
$$
\mathbf{y}=\mathbf{X} {1} \beta {1}+\mathbf{X} {2} \beta {2}+\mathbf{u}
F这r在H一世CHH是p这吨H和s和s$(1.12)−(1.20)$H这ld,吨H和F这ll这在一世nG米这d和l一世s和s吨一世米一种吨和d:
\mathbf{y}=\mathbf{X} {1} \beta {1}+\nu
$$
因此，OLS 估计由以下表达式给出
$$
\hat{\boldsymbol{\beta}}{1}^{u p}=\left(\mathbf{X}{1}^{\prime} \mathbf{X}{1}\right)^{-1} \mathbf{X}{1}^{\prime} \mathbf{y}
$$
while the OLS estimates which would have been obtained by estimation of the DGP would have been:
$$
\widehat{\boldsymbol{\beta}}{1}=\left(\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1}\right)^{-1} \mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{y}
$$
The estimates in (1.39) are Best Linear Unbiased Estimators by construction, while the estimates in (1.38) are biased unless the correlation between $\mathbf{X}{1}$ and $\mathbf{X}{2}$ is zero. To show this point consider that:
$$
\begin{aligned}
\widehat{\boldsymbol{\beta}}{1} &=\left(\mathbf{X}{1}^{\prime} \mathbf{X}{1}\right)^{-1}\left(\mathbf{X}{1}^{\prime} \mathbf{y}-\mathbf{X}{1}^{\prime} \mathbf{X}{2} \widehat{\boldsymbol{\beta}}{2}\right) \ &=\widehat{\boldsymbol{\beta}}{1}^{u p}+\widehat{\mathbf{D}} \widehat{\boldsymbol{\beta}}{2} \end{aligned} $$ where $\widehat{\mathbf{D}}$ is the vector of coefficients in the regression of $\mathbf{X}{2}$ on $\mathbf{X}{1}$, and $\widehat{\boldsymbol{\beta}}{2}$ is the OLS estimator obtained by fitting the DGP.
To provide further interpretation of these results note that if we have:
$$
\begin{aligned}
E\left(\mathbf{y} \mid \mathbf{X}{1}, \mathbf{X}{2}\right) &=\mathbf{X}{1} \boldsymbol{\beta}{1}+\mathbf{X}{2} \boldsymbol{\beta}{2} \
E\left(\mathbf{X}{1} \mid \mathbf{X}{2}\right) &=\mathbf{X}{1} \mathbf{D} \end{aligned} $$ then $$ E\left(\mathbf{y} \mid \mathbf{X}{1}\right)=\mathbf{X}{1} \boldsymbol{\beta}{1}+\mathbf{X}{1} \mathbf{D} \boldsymbol{\beta}{2}=\mathbf{X}{1} \boldsymbol{\alpha} . $$ Therefore the OLS estimator in the underparameterised model is a biased estimator of $\beta{1}$, but it is an unbiased estimator of $\alpha$. Then, if the objective of the model is forecasting and $\mathbf{X}{1}$ is more easily observed than $\mathbf{X}{2}$, than the undeparameterised model can be safely used. On the other hand, if the objective of the model is to test specific predictions on parameters (as it is the case with the Solow’s growth model), than the use of the under-parameterised model will deliver biased results. When we are interested in the effect of $\mathbf{X}_{1}$ on $\mathbf{y}$, 独立于其他因素，控制遗漏变量的影响至关重要。
经济代写|宏观经济学作业代写MACROECONOMICS代考|OVER-PARAMETERIZATION
给定以下 DGP：
$$
\mathbf{y}=\mathbf{X} {1} \beta {1}+\ mathbf {u}
$$
(1.12)−(1.20)持有，估计以下模型：
$$
\mathbf{y}=\mathbf{X}{1} \boldsymbol{\beta}{1}+\mathbf{X}{2} \boldsymbol{\beta}{2}+\mathbf{v}
$$
The OLS estimator of the over-parameterized model is
$$
\widehat{\boldsymbol{\beta}}{1}^{o p}=\left(\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1}\right)^{-1} \mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{y}
$$
while, by estimating the DGP, we obtain:
$$
\widehat{\boldsymbol{\beta}}{1}=\left(\mathbf{X}{1}^{\prime} \mathbf{X}{1}\right)^{-1} \mathbf{X}{1}^{\prime} \mathbf{y} .
$$
By substituting $y$ from the DGP it is immediately shown that both estimators are unbiased. The difference is now made by the variance. In fact we have:
$$
\begin{gathered}
\operatorname{var}\left(\widehat{\boldsymbol{\beta}}{1}^{o p} \mid \mathbf{X}{1}, \mathbf{X}{2}\right)=\sigma^{2}\left(\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1}\right)^{-1} \
\operatorname{var}\left(\widehat{\boldsymbol{\beta}}{1} \mid \mathbf{X}{1}, \mathbf{X}{2}\right)=\sigma^{2}\left(\mathbf{X}{1}^{\prime} \mathbf{X}{1}\right)^{-1} \end{gathered} $$ It can be shown that the estimator derived from the correct model is more efficient. In fact, the difference between the two variance-covariance matrices is a positive semi-definite matrix. To show this result remember that if two matrices $\mathbf{A}$ and $\mathbf{B}$ are positive definite and $\mathbf{A}-\mathbf{B}$ is positive semi-definite, then also the matrix $\mathbf{B}^{-1}-\mathbf{A}^{-1}$ is positive semi-definite. Then we have to show that $\mathbf{X}{1}^{\prime} \mathbf{X}{1}-$ $\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1}$ is a positive semi-definite matrix. Such result is almost immediately shown:
$$
\begin{aligned}
\mathbf{X}{1}^{\prime} \mathbf{X}{1}-\mathbf{X}{1}^{\prime} \mathbf{M}{2} \mathbf{X}{1} &=\mathbf{X}{1}^{\prime}\left(\mathbf{I}-\mathbf{M}{2}\right) \mathbf{X}{1} \
&=\mathbf{X}{1}^{\prime} \mathbf{Q}{2} \mathbf{X}{1}=\mathbf{X}{1}^{\prime} \mathbf{Q}{2} \mathbf{Q}{2} \mathbf{X}_{1}
\end{aligned}
$$
然后我们可以得出结论，过度参数化会影响估计器的效率和假设检验的能力。
经经济代写|宏观经济学作业代写Macroeconomics代考 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。
