计量经济学econometrics代写|网课代修

计量经济学（英语：Econometrics），又译经济计量学，是以数理经济学和数理统计学为方法论基础，对于经济问题，试图以理论上的数量接近和经验（实证研究）上的数量接近这两者进行综合，而产生的经济学分支。

传统的经济学研究通过观察经济现象、建立经济模型，来分析单个或多个经济变量与其他经济变量之间的相互关系，这种研究方法的局限在于，很难对经济现象进行量化分析，而计量经济学的产生，使得经济学对于经济现象从以往的定性研究，扩展到同时可以进行定量研究的新阶段^[1]。

“计量”的意思是“以统计方法做定量研究”，“量”字为名词，构成动宾结构，这从其英文metric的含义亦可看出（与数学名词“度量空间metric space”情况类似），所以“量”字应读作“亮”（中国大陆《现代汉语辞典》2012年6月第6版“计量”条）。设若“计量”的“量”字读为“良”，则是两个动词词素的并列结构，含义略简。另如测智力的斯坦福一比奈智力量表（Stanford–Binet Intelligence Scale），按其内涵则应读“量”字为“良”，此亦可从英文scale的含义窥得。

1.
$$
\begin{aligned}
& y=\beta_1+\beta_2 x+\varepsilon \rightarrow \beta_2=\frac{\sum_i\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)}{\sum_i\left(x_i-\bar{x}\right)^2} \
& y=\tilde{\beta}_2 x+\varepsilon \rightarrow \tilde{\beta}_2=\frac{\sum_i x_i y_i}{\sum_i x_i^2}
\end{aligned}
$$
2.
$$
\begin{aligned}
\operatorname{Var}\left(\beta_2\right) & =\frac{\sigma^2}{\sum_i\left(x_i-\bar{x}\right)^2} \
\operatorname{Var}\left(\tilde{\beta}_2\right) & =\frac{\sigma^2}{\sum_i x_i^2}
\end{aligned}
$$

3. The ratio of these two variances is
   $$
   \begin{aligned}
   \frac{\operatorname{Var}\left(\tilde{\beta}_2\right)}{\operatorname{Var}\left(\beta_2\right)}= & \frac{\frac{\sigma^2}{\sum_i x_i^2}}{\frac{\sigma^2}{\sum_i\left(x_i-\bar{x}\right)^2}} \
   = & \frac{\sum_i\left(x_i-\bar{x}\right)^2}{\sum_i x_i^2} \
   & \sum\left(x_i-\bar{x}\right)^2=\sum\left(x_i^2-2 x_i \bar{x}+\bar{x}\right)=\sum x_i^2-2 n \bar{x} \bar{x}+n \bar{x}^2=\sum x_i^2-n \bar{x}^2 \
   = & \frac{\sum_i x_i^2-n \bar{x}^2}{\sum_i x_i^2} \
   = & 1-\frac{n \bar{x}^2}{\sum_i x_i^2} \leq 1
   \end{aligned}
   $$
   It follows that fitting the constant term when it is unnecessary inflates the variance of the least squares estimator if the mean of the regressor is not zero.

Solution

1. If the individual effects are strictly uncorrelated with the regressors then a random effects model is the appropriate model. However, if a fixed effect model is estimated the estimates will be consistent but not efficient.
2. Breush-Pagan LM test: Test statistic is 453.82 , the critical value from the chi-squared table is 3.84 , so the null hypothesis that random effects are not needed can be rejected.
   Hausman Test: Test statistic is 1.27 , the critical value from the chisquared table is 5.99 , so the null hypothesis of the random effects model cannot be rejected.

1. $\Delta y_t=y_t-y_{t-1}=\varepsilon_t$, white noise, no more autocorrelation
2. $\Delta y_t=y_t-y_{t-1}=\beta_1+\varepsilon_t-\varepsilon_{t-1}$. This is an MA(1) process with autocorrelation $\frac{\theta}{1+\theta^2}=\frac{1}{1+1}=\frac{1}{2}$.
3. $\Delta y_t=y_t-y_{t-1}=\beta^{\prime}\left(x_t-x_{t-1}\right)+v_t$, where $v_t=\varepsilon_t-\varepsilon_{t-1}$.
   $$
   \begin{aligned}
   \operatorname{Var}\left(\varepsilon_t\right) & =\frac{\sigma_u^2}{1-\rho^2} \
   \operatorname{Var}\left(v_t\right) & =\operatorname{Var}\left(\varepsilon_t-\varepsilon_{t-1}\right)=\operatorname{Var}\left(\rho \varepsilon_{t-1}-\varepsilon_t+u_t\right) \
   & =\operatorname{Var}\left[(\rho-1) \varepsilon_{t-1}+u_t\right]=(\rho-1)^2 \frac{\sigma_u^2}{1-\rho^2}+\sigma_u^2=\frac{2 \sigma_u^2}{1+\rho} \
   \operatorname{Cov}\left[v_t, v_{t-1}\right] & =\operatorname{Cov}\left[\varepsilon_t-\varepsilon_{t-1}, \varepsilon_{t-1}-\varepsilon_{t-2}\right] \
   & =E\left[\varepsilon_t \varepsilon_{t-1}-\varepsilon_{t-1}^2-\varepsilon_t \varepsilon_{t-2}+\varepsilon_{t-1} \varepsilon_{t-2}\right] \
   & =\rho \frac{\sigma_u^2}{1-\rho^2}-\frac{\sigma_u^2}{1-\rho^2}-\rho^2 \frac{\sigma_u^2}{1-\rho^2}+\rho \frac{\sigma_u^2}{1-\rho^2}=\frac{\sigma_u^2\left(2 \rho-1-\rho^2\right)}{1-\rho^2} \
   & =\frac{\sigma_u^2(\rho-1)^2}{(\rho-1)(\rho+1)}=\frac{\sigma_u^2(\rho-1)}{\rho+1} \
   \frac{\operatorname{Cov}\left[v_t, v_{t-1}\right]}{\operatorname{Var}\left[v_t\right]} & =\frac{\rho-1}{2}
   \end{aligned}
   $$
4. Compare the two autocorrelations:
5. $$
6. \begin{aligned}
7. |\rho| & >\left|\frac{\rho-1}{2}\right| \quad \quad \text { Assume } \rho>0 \text { and }|\rho|<1 \\ \rho & >-\frac{\rho-1}{2} \rightarrow \rho>\frac{1}{3}
8. \end{aligned}
9. $$
10. If the original autocorrelation is greater than 1/3 (For economic data, this is likely to be fairly common.) the differenced process has a smaller autocorrelation.