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# 经济代写|计量经济学代写Introduction to Econometrics代考|A Panel Time-Varying State-Space Extension

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## 经济代写|计量经济学代写Introduction to 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:
$$\underset{(n \times t)}{y}=\underset{(n \times n * k)}{\mathbf{A}^{\top}} \times \underset{(n * k \times t)}{x}+\underset{(n \times r)}{\mathbf{H}^{\top}(x)} \times \underset{(r \times t)}{\xi}+\underset{(n \times t)}{w}$$
representing the measurement equation for $\mathrm{a} y_t \in \mathbb{R}^n$ 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.

## 经济代写|计量经济学代写Introduction to Econometrics代考|A Time-Varying Parameter Model for the M3 Velocity

In modern economies, neglecting what happens to money velocity leads to large relative errors in estimating inflation and output. Moreover, velocity, or its twin sibling, the demand for money, turns out to be highly volatile, difficult to model and hard to measure. Hence, movements in $P$ end up being dominated by unexplained movements in $V$ rather than in $M$.

Traditional theories of money demand identify income as the principal determinants of velocity. As highlighted in Friedman and Schwartz (1963), if money demand elasticity to income is greater than one, then economic growth would induce a secular downward trend in velocity, inflation and interest rates. The theoretical literature (see Orphanides and Porter 2000) also posits that velocity fluctuates with the opportunity cost of money, driven by inflation and interest rates.

A benchmark regression representing the traditional theories of money demand is presented in Bordo and Jonung (1987), updated in Bordo and Jonung (1990) and revisited using cointegration techniques by Bordo et al. (1997). This formulation is described in Hamilton (1989) using an equation such as:
$$\log V_{i, t}=\beta_{0, i}+\xi_{i, t}+\lambda_i f_t+\beta_{1, i} i_t+\beta_{2, i} \pi_t^e+\beta_{3, i} \log Y_{p c_{i, t}}+\beta_{4, i} \log Y_{p c_{i, t}}^p+\varepsilon_{i, t}$$

The above model expresses the log of velocity $\left(V_t\right)$ as a function of the opportunity cost of holding money balances in terms of an appropriate nominal interest rate $\left(i_t\right)$, expected inflation $\left(\pi_t^e\right)$, proxied by the fitted values of a univariate autoregression for actual inflation, the log of real GNP per capita $\left(Y_{\mathrm{pc}t}\right)$ and its smoothed version $\left(Y{\mathrm{pc}_t}^p\right)$ interpreted as permanent real GNP per capita. The velocity formulation is strongly based on economic theory of permanent income hypothesis (Friedman and Schwartz 1963). We expect a positive sign for permanent income as any increase in it will rise the number of transactions in the economy affecting the velocity positively. Transitory income with a positive coefficient but less than one would indicate that velocity moves pro-cyclically, which would be in line with Friedman’s permanent income hypothesis. Over the cycle, the transitory income would increase the demand for money, because cash balances serve as buffer stock, and therefore, in the long run these transitory balances would disappear, returning the coefficient to unity. As for the real interest rate, it is expected to have a positive sign as an increase in it would decrease the demand for real money balances and thus a raise in the velocity for a given level of income. Finally, the impact of inflation on velocity is ambiguous depending upon its relative influence on money balances and income growth.

## 经济代写|计量经济学代写Introduction to Econometrics代考|A Panel Time-Varying State-Space Extension

$$y_{i, t}=x_{i, t}^{\top} \bar{\beta}+x_{i, t}^{\top} \xi_{i, t}+\omega_t$$

$$\underset{(n \times t)}{y}=\underset{(n \times n * k)}{\mathbf{A}^{\top}} \times \underset{(n * k \times t)}{x}+\underset{(n \times r)}{\mathbf{H}^{\top}(x)} \times \underset{(r \times t)}{\xi}+\underset{(n \times t)}{w}$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|A Time-Varying Parameter Model for the M3 Velocity

$$\log V_{i, t}=\beta_{0, i}+\xi_{i, t}+\lambda_i f_t+\beta_{1, i} i_t+\beta_{2, i} \pi_t^e+\beta_{3, i} \log Y_{p c_{i, t}}+\beta_{4, i} \log Y_{p c_{i, t}}^p+\varepsilon_{i, t}$$

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