### 商科代写|计量经济学代写Econometrics代考|ECON2515

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## 商科代写|计量经济学代写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 halances 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 $\left(Y{\text {pct }}^{p}\right)$ 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.

## 商科代写|计量经济学代写Econometrics代考|Univariate Properties of the Data

Trend breaks appear to be prevalent in macroeconomic time series, and unit root tests therefore need to make allowance for these if they are to avoid the serious effects that unmodeled trend breaks have on power. ${ }^{16}$ Consequently, when testing for a unit

root it has become a matter of regular practice to allow for this kind of deterministic structural change.

In order to avoid this pitfall, we run tests to assess whether structural breaks are present in the series. This testing problem has been addressed by Perron and Yabu (2009), who define a test statistic that is based on a quasi-GLS approach using an autoregression for the noise component, with a truncation to 1 when the sum of the autoregressive coefficients is in some neighborhood of 1 , along with a bias correction. For given break dates, one constructs the $F$-test (Exp $-W_{F S}$ ) for the null hypothesis of no structural change in the deterministic components. The final statistic uses the Exp functional of Andrews and Ploberger (1994). Perron and Yabu (2009) specify three different models depending on whether the structural break only affects the level (Model I), the slope of the trend (Model II) or the level and the slope of the time trend (Model III). The computation of these statistics, which are available in Table 1 , shows that we find more evidence against the null hypothesis of no structural break with Model III.

The analysis shows instabilities in the money velocity for all the countries with two exceptions, Spain and France. Therefore, in a second step, we have computed the unit root test statistics in Carrion-i Silvestre et al. (2009). The unit root tests in Carrion-i Silvestre et al. (2009) allow for multiple structural breaks under both the null and alternative hypotheses which make especially suitable for our purpose, since we have obtained evidence in favor of the presence of structural breaks regardless of their order of integration. The results of all these statistics are reported in Table 3 . As can be seen, the unit root tests proposed by Carrion-i Silvestre et al. (2009) led to the non-rejection of the null hypothesis of a unit root in most of cases at the $5 \%$ level of significance. ${ }^{17}$ Our conclusion is that the income velocity variable for the countries considered has unit roots with breaks both in levels and in most of the cases.

## 商科代写|计量经济学代写Econometrics代考|Linear Cointegration Specification for Wealth Effects

First, we specify the consumption-wealth relationship differently while assessing for the effect of total wealth (Eq. 1) and that of disaggregate wealth (Eq. 2) since households might react differently to shocks on financial assets or on property prices. Indeed, in line with the theoretical framework from Lettau and Ludvigson $(2001$, 2004), we can write the following log-linear model:
$$\begin{gathered} c_{t}=\alpha+\beta_{1} T W_{t}+\beta_{2} y_{t}+\varepsilon_{t} \ c_{t}=\alpha+\beta_{1} F W_{t}+\beta_{2} H W_{t}+\beta_{3} y_{t}+\varepsilon_{t} \end{gathered}$$
where: $c_{t}, W_{t}, \mathrm{FW}{t}, \mathrm{HW}{t}$ and $y_{t}$ refer to consumption, total wealth (TW), financial wealth (FW), housing wealth (HW) and disposable income respectively. All variables are in logarithm.

Considering Lettau and Ludvigson (2001, 2004) in line with the life-cycle approach of wealth effects, Eqs. (1) and (2) are estimated in a cointegration framework. Indeed, Lettau and Ludvigson (2001) used the Campbell and Mankiw (1989) micro-funded model of consumption to show that consumption tends to a stationary fraction of wealth. The so-called cointegration-based approach from Lettau and Ludvigson $(2001,2004)$ lead directly to the estimations of wealth effects elasticities.

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

Bordo 和 Jonung (1987) 提出了代表传统货币需求理论的基准回归，在 Bordo 和 Jonung (1990) 中进行了更新，并由 Bordo 等人使用协整技术重新审视。(1997)。该公式在 Hamilton (1989) 中使用如下等式进行了描述：

## 商科代写|计量经济学代写Econometrics代考|Linear Cointegration Specification for Wealth Effects

C吨=一个+b1吨在吨+b2是吨+e吨 C吨=一个+b1F在吨+b2H在吨+b3是吨+e吨

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## MATLAB代写

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