### 商科代写|计量经济学代写Econometrics代考|ECON 2517

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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

## 商科代写|计量经济学代写Econometrics代考|The Inflation Puzzle and Money Velocity in the EA

The quantity theory of money (QTM) predicts a positive relationship between monetary growth and inflation. However, even if inflation truly were a monetary phenomenon in the long run, as stated in Friedman (1963) seminal book, a conventional vector cointegration approach does not necessarily identify the long-run relation between money and prices correctly, because it neglects the structural development of the velocity of money. Since excess supply and demand of money are captured by transitory movements of velocity in a world where real and nominal rigidities

prevail, the identification of excess liquidity that endangers price stability is tied to the identification of equilibrium velocity.

In general, evidence from cross-country studies strongly supports the one-to-one correlation of average money growth and average inflation. However, the impact of money on prices is hard to identify within one country. De Grauwe and Polan (2005) have argued that the long-run link between nominal money growth and inflation might be much looser than commonly assumed in countries which have operated in moderate inflation environments as it is the case of the EA.

Flexible inflation targeting implies that the central bank attempts to reach the target gradually in the medium term and not in the immediate period. As recently stressed by Cochrane (2017), existing theories of inflation make straight predictions. The Keynesian school argues that velocity is a highly fluctuating variable which is significantly affected by economic policies. As a result, changes in velocity could nullify the effects of monetary policy. They stress that the velocity of money is severely affected by demand management policies; hence, it is a non-stationary yariable. Furthermore, they argue that the movements of velocity are the opposite of the movement of money supply. Interest rate is also regarded as one of the variables influencing velocity. The opposite forecast is made by monetarist models, who predict that, as provided the velocity is “stable” in the long run, a massive increase in reserves must lead to galloping inflation. Yet, none of these predictions have proved right. This issue was already central to the debates of the 1950 s and 1960 setween Keynesians and monetarists. Keynesians thought that at the zero rates of the Great Depression, money and bonds were perfect substitutes, so monetary policy could do nothing, and advocated fiscal stimulus instead. On the contrary, monetarists held that additional money, even at zero rates, would be stimulative; therefore, the failure to provide additional money was the big monetary policy mistake of that time. The view that inflation is always a monetary phenomenon has a long tradition based on the quantity theory of money. In its simplest form, the QTM states that changes in money supply growth are followed by equal changes in the inflation rate and, through the force of the Fisher effect, in the nominal interest rate.
According to the monetarist doctrine:
$$M V=P Y$$

## 商科代写|计量经济学代写Econometrics代考|State-Space Models and Time-Varying Parameter Models

State-space representation of a linear system constitutes a statistical framework for modeling the dynamics of a $(n \times 1)$ vector of variables observed at regular time intervals $t, y_{t}$, in terms of a possibly unobserved (or state) $(r \times 1)$ vector $\xi_{t}{ }^{10}$ The origin of state-space modeling is intimately linked with the Kalman filter, a recursive algorithm for generating minimum mean square error forecasts in state-space models.
The measurement equation models the dynamics of the observable variables $y_{t}$, possibly measured with noise, that are assumed to be related to the state vector, providing information on $\xi_{t}$. It takes the following general form:
$$\underset{(n \times 1)}{y_{t}}=\underset{(n \times k)}{\mathbf{A}{(k \times 1)}^{\top}} x{t}+\underset{(n \times r)}{\mathbf{H}^{\top}} \xi_{(r \times 1)}+\underset{(n \times 1)}{w_{t}}$$
where $y_{t}$ represents an $(n \times 1)$ vector of variables that are observed at date $t$ and $x_{t}$ represents a $(k \times 1)$ vector of exogenous determinants, their coefficients being included in the $(k \times n)$ matrix $A . H$ is an $(r \times n)$ matrix of coefficients for the $(r \times 1)$ vector of unobserved components $\xi_{r}$. Finally, the measurement or observational error, $w_{t}$, is an $(n x 1)$ vector assumed to be i.i.d. $N(0, R)$, independent of $\xi_{t}$ and $\nu_{t}$ and for $t=1,2, \ldots$, where $$E\left(w_{t} w_{t}^{\top}\right)=\underset{(n \times n)}{R}$$
and variance covariance equal to
$$E\left(w_{t} w_{\tau}^{\top}\right)= \begin{cases}\mathrm{R} & \text { for } t=\tau \ 0 & \text { for } t \neq \tau\end{cases}$$
The state-transition equation describes the evolution of the underlying unobserved states that determine the time series behavior, generated by a linear stochastic difference representation through a first-order Markov process, such as in 11 :
where $F$ denotes an $(r \times r)$ state-transition matrix, which applies the effect of each system state parameter at time $t-1$ on the system state at time $t, \xi_{t}$, and $Z_{t}$ is a $(s \times 1)$ vector containing any control inputs, either deterministic (drift and/or deterministic trend) or stochastic. If present, control inputs affect the state through the $(r \times s)$ control input matrix, $B$, which applies the effect of each control input parameter in the vector on the state vector.

The introduction of stochastic control inputs is common practice in the literature on control engineering where this concept was coined. Basically, the idea is to simulate the effect of changes in the control variable on a system, namely the state vector. ${ }^{11}$ Despite their many potential uses, empirical economic research generally has employed simple state-transition equations, where the unobserved component evolves as a random walk process and no control inputs are present.

## 商科代写|计量经济学代写Econometrics代考|A Panel Time-Varying State-Space Extension

In this subsection, we extend the previous time-varying parameter model to a panel setting. Our main goal is to explore the use of the state-space modelization and the Kalman filter algorithm as an effective method for combining time series in a panel. This flexible structure allows the model specification to be affected by different potential sources of cross-sectional heterogeneity. This approach can be a superior alternative to the estimation of the model in unstacked form, commonly employed when there is a small number of cross sections.
The general model can be written as follows:
$$y_{i, t}=x_{i, t}^{\top} \bar{\beta}+x_{i, t}^{\top} \xi_{i, t}+\omega_{t}$$
or in matrix form:
$$\underset{(n \times i)}{y}=\underset{(n \times n * k)}{\mathbf{A}^{\top}} \times \underset{(n \times k \times t)}{x}+\underset{(n \times r)}{\mathbf{H}^{\top}(x) \times} \underset{(r \times i)}{\xi}+\underset{(n \times i)}{w}$$
representing the measurement equation for a $y_{t} \in \mathbb{R}^{n}$ vector containing the dependent variable for a panel of countries. $x_{t} \in \mathbb{R}^{k \times n}$ is a vector of $k$ exogenous variables, including either (or both) stochastic or deterministic components. The unobserved vector $\xi_{t} \in \mathbb{R}^{r}$ influences the dependent variable through a varying $H^{\top}\left(x_{t}\right)(n \times r)$ matrix, whose simplest form is $H^{\top}\left(x_{t}\right)=x_{t}$. Finally, $w_{t} \in \mathbb{R}^{n}$ represents the $(n \times 1)$ vector of $N$ measurement errors.

The specification of the model in Eq. (19) relies on a mean-reverting-type modelization of the measurement equation, which also allows for the inclusion of fixed parameters, in matrix $\mathbf{A}$. Each of the fixed parameters can be modeled, either as a common parameter for all the agents in the panel, $\bar{\beta}$, or, alternatively, as a countryspecific coefficient, $\bar{\beta}{i}$. The model also includes time-varying parameters $\left(\xi{t}\right)$ for some of the regressors that eventually can be interpreted as deviations from the mean parameters $\left(\left(\beta_{i t}-\bar{\beta}{i}\right)=\xi{t}\right)$.

The measurement equation for each $i$ th element in the $t$ th period $\left(y_{i, t}\right)$ in the vector of the dependent variable can be expressed as follows:
$y_{i, t}=\sum_{\mathrm{ks}=\mathrm{ksmin}}^{\mathrm{ksmax}} \bar{\beta}{\mathrm{ks}, i} x{\mathrm{ks}, i, t}+\sum_{\mathrm{kc}=\mathrm{kcmin}}^{\mathrm{kmmax}} \bar{\beta}{\mathrm{kc}} x{\mathrm{kc}, i, t}$
$+\sum_{\mathrm{kv}=\mathrm{kvmin}}^{\operatorname{kvmax}} \xi_{\mathrm{kv}, i t} x_{\mathrm{kv}, i, t}+h_{i} \xi_{r, i t}+w_{(n \times 1)}$

## 商科代写|计量经济学代写Econometrics代考|State-Space Models and Time-Varying Parameter Models

：F表示一个(r×r)状态转移矩阵，它在时间应用每个系统状态参数的影响吨−1在当时的系统状态吨,X吨， 和从吨是一个(s×1)包含任何控制输入的向量，无论是确定性的（漂移和/或确定性趋势）还是随机的。如果存在，控制输入通过(r×s)控制输入​​矩阵，乙，它将向量中每个控制输入参数的影响应用于状态向量。

## 商科代写|计量经济学代写Econometrics代考|A Panel Time-Varying State-Space Extension

$y_{i, t}=\sum_{\mathrm{ks}=\mathrm{ksmin}}^{\mathrm{ksmax}}\bar{\beta} {\mathrm{ks}, i} x {\mathrm{ks},i,t}+\sum_{\mathrm{kc}=\mathrm{kcmin}}^{\mathrm{kmmax}} \bar{\beta } {\math{kc}} x {\math{kc}, i, t}+\sum_{\mathrm{kv}=\mathrm{kvmin}}^{\operatorname{kvmax}} \xi_{\mathrm{kv}, it} x_{\mathrm{kv}, i, t}+h_{ i} \xi_{r, it}+w_{(n \times 1)}$

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