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金融计量经济学Financial Econometrics的一个基本工具是多元线性回归模型。计量经济学理论使用统计理论和数理统计来评估和发展计量经济学方法。计量经济学家试图找到具有理想统计特性的估计器，包括无偏性、效率和一致性。应用计量经济学使用理论计量经济学和现实世界的数据来评估经济理论，开发计量经济学模型，分析经济历史和预测。

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经济代写|计量经济学代写Econometrics代考|Literature Review

According to Borio (2014), the term “financial cycle” refers to the self-reinforcing interactions among perceptions of value and risk, risk-taking and financing constraints. Typically, rapid increases in credit boost property and asset prices, which in turn increases collateral values and the amount of credit the private sector can obtain, but the process subsequently tends to go into reverse. As highlighted by Borio et al. (2019), this mutually reinforcing interaction between financing constraints and perceptions of value and risks has historically been likely to generate severe macroeconomic imbalances.

The financial cycle can be approximated in different ways. The empirical literature suggests that a reasonable strategy is to capture it through fluctuations in credit and property prices, but also by means of the debt service ratio, defined as interest payments plus amortisation divided by GDP. Drehmann et al. (2018) find a robust relationship between debt accumulation and subsequent debt service (i.e., interest payments plus amortisation), which has a large negative effect on economic growth. All these series may be used individually or combined, as a composite financial cycle proxy similar to that constructed by Drehmann et al. (2012).

Borio et al. (2019) point out that previous literature has identified two important features of the financial cycle. First, its peaks generally coincide with banking crises or considerable financial stress. During expansions, the interaction between asset prices and risk-taking can overstretch balance sheets, making them more fragile and generating the consequent financial tightening. Second, financial cycles can be much longer than business cycles: most of the former have lasted around 15 to 20 years since the early 1980s, whilst the latter have typically lasted up to eight years. Therefore, a financial cycle can span more than one business cycle, which is the reason why peaks in a financial cycle are generally followed by downturns, whilst not all recessions are preceded by one of those peaks.

A number of recent studies have provided more evidence on financial cycles. Specifically, Oman (2019), using a frequency-based filter, details the existence of a Eurozone financial cycle and high-amplitude and low-amplitude national financial cycles. Applying concordance and similarity analysis to business and financial cycles, he provides evidence of several empirical regularities: the aggregate Eurozone credit-to-GDP ratio behaved pro-cyclically in the years preceding euroarea recessions; financial cycles are less synchronised than business cycles; business cycle synchronisation has risen whilst financial cycle synchronisation has decreased; financial cycle desynchronisation was more pronounced between high-amplitude and low-amplitude countries; high-amplitude countries experienced divergent leverage dynamics after 2002. Filardo et al. (2018) explore financial conditions (120 years of data) over time in order to improve our understanding of financial cycles. They find that financial cycles are characterised by recurrent, endogenous swings in financial conditions, which result in booms and busts. Yet the recurrent nature of such swings may not appear so obvious when looking at conventionally plotted time-series data. Using the pioneering framework developed by Stock (1987), they offer a new statistical characterisation of the financial cycle based on a continuous-time autoregressive (AR) model subject to time deformation.

经济代写|计量经济学代写Econometrics代考|The Model

The adopted model is the following:
$$
(1-L)^{d_1}\left(1-2 \cos w_r L+L^2\right)^{d_2} x_t=u_t, \quad t=1,2, \ldots,
$$
where $x_t$ is the observed time series; $d_1$ and $d_2$ are the orders of integration corresponding to the long-run (zero) and the (cyclical) (nonzero) frequency, respectively, and $u_t$ is an $I(0)$ process, defined as a covariance-stationary process with a spectral density function that is positive and finite at all frequencies in the spectrum. The first polynomial in Eq. (1) refers to the standard case of fractional integration or $I(d)$ that basically imposes a singularity or pole in the spectrum at the long-run or zero frequency. The literature includes plenty of papers with such a specification and testing for unit or fractional degrees of differentiation (for the unit root case, see, e.g., Fama and French 1988a, b; Poterba and Summers 1988; for the fractional case see instead Baillie 1996; Gil-Alana and Robinson 1997; Abbritti et al. 2016; and others).

The second polynomial refers to the case of integration at a frequency away from zero and uses Gegenbauer processes, where $w_r=2 \pi r / T$ and $r=T / s$. Thus, $s$ indicates the number of time periods per cycle, whilst $r$ refers to the frequency with a pole or singularity in the spectrum of $x_t$. In this context, if $r=0(s=1)$, the second polynomial in (2) becomes $(1-L)^{2 d_2}$, and therefore, the whole process corresponds to the classical fractional integration model widely studied in the literature. Andel (1986) introduced this process for values of $r$ different from 0 and fractional values of $d_2$ and Gray et al. $(1989,1994)$ showed that, by denoting $\mu=\cos w_r$, one can express the polynomial in terms of the orthogonal Gegenbauer polynomial $C_{j, d_2}(\mu)$, so that, for all $d_2 \neq 0$,
$$
\left(1-2 \mu L+L^2\right)^{-d_2}=\sum_{j=0}^{\infty} C_{j, d_2}(\mu) L^j,
$$
where we can define $C_{j, d_2}(\mu)$ recursively as follows:
$$
C_{0, d_2}(\mu)=1, \quad C_{1, d_2}(\mu)=2 \mu d_2,
$$
and
$$
C_{j, d_2}=2 \mu\left(\frac{d_2-1}{j}+1\right) C_{j-1, d_2}(\mu)-\left(2 \frac{d_2-1}{j}+1\right) C_{j-2 d_2}(\mu), j=2,3, \cdots
$$

计量经济学代考

经济代写|计量经济学代写Econometrics代考|Literature Review

根据Borio(2014)的说法，“金融周期”一词是指价值和风险感知、冒险和融资约束之间自我强化的相互作用。通常情况下，信贷的快速增长会推高房地产和资产价格，这反过来又会增加抵押品价值和私人部门可以获得的信贷数量，但这一过程随后往往会逆转。正如Borio等人(2019)所强调的那样，从历史上看，融资约束与价值和风险认知之间的这种相互加强的相互作用可能会产生严重的宏观经济失衡。

金融周期可以用不同的方式来近似。实证文献表明，一个合理的策略是通过信贷和房地产价格的波动来捕捉它，但也可以通过偿债比率(定义为利息支付加上摊销除以GDP)来捕捉它。Drehmann等人(2018)发现债务积累与随后的偿债(即利息支付加上摊销)之间存在强大的关系，这对经济增长有很大的负面影响。所有这些序列可以单独使用，也可以组合使用，作为类似于Drehmann等人(2012)构建的复合金融周期代理。

Borio等人(2019)指出，以前的文献已经确定了金融周期的两个重要特征。首先，它的峰值通常与银行业危机或相当大的金融压力同时发生。在经济扩张期间，资产价格与冒险行为之间的相互作用可能会使资产负债表过度扩张，使其变得更加脆弱，并由此引发金融紧缩。其次，金融周期可能比商业周期长得多:自上世纪80年代初以来，金融周期大多持续了15至20年左右，而商业周期通常长达8年。因此，一个金融周期可以跨越多个商业周期，这就是为什么金融周期的高峰之后通常是衰退，而不是所有的衰退之前都有一个高峰。

最近的一些研究为金融周期提供了更多证据。具体而言，阿曼(2019)使用基于频率的滤波器，详细说明了欧元区金融周期以及高振幅和低振幅国家金融周期的存在。通过对商业和金融周期的一致性和相似性分析，他提供了几个经验规律的证据:在欧元区衰退之前的几年里，欧元区总信贷与gdp之比表现出顺周期的特征;金融周期不如商业周期同步;商业周期同步性上升，而金融周期同步性下降;金融周期不同步在高振幅和低振幅国家之间更为明显;高振幅国家在2002年后经历了不同的杠杆动态。Filardo等人(2018)随着时间的推移探索金融状况(120年的数据)，以提高我们对金融周期的理解。他们发现，金融周期的特点是金融环境中反复出现的内生波动，这种波动会导致繁荣和萧条。然而，当观察传统绘制的时间序列数据时，这种波动的周期性可能并不那么明显。使用Stock(1987)开发的开创性框架，他们基于受时间变形影响的连续时间自回归(AR)模型，提供了金融周期的新统计特征。

经济代写|计量经济学代写Econometrics代考|The Model

采用的模型如下:
$$
(1-L)^{d_1}\left(1-2 \cos w_r L+L^2\right)^{d_2} x_t=u_t, \quad t=1,2, \ldots,
$$
其中$x_t$为观测时间序列;$d_1$和$d_2$分别是对应于长期(零)和(周期)(非零)频率的积分阶数，$u_t$是一个$I(0)$过程，定义为协方差-平稳过程，其谱密度函数在频谱中的所有频率上都是正的和有限的。Eq.(1)中的第一个多项式指的是分数阶积分或$I(d)$的标准情况，它基本上在长期或零频率下施加频谱中的奇点或极点。文献中包括大量具有这样的规范和单位或分数阶微分测试的论文(对于单位根情况，参见，例如Fama和French 1988a, b;Poterba and Summers 1988;对于分数情况，请参见Baillie 1996;gill – alana and Robinson 1997;Abbritti et al. 2016;以及其他)。

第二个多项式指的是频率远离零的积分情况，并使用Gegenbauer过程，其中$w_r=2 \pi r / T$和$r=T / s$。因此，$s$表示每个周期的时间周期数，而$r$表示$x_t$频谱中具有极点或奇点的频率。在这种情况下，如果$r=0(s=1)$，则(2)中的第二个多项式变为$(1-L)^{2 d_2}$，因此整个过程对应于文献中广泛研究的经典分数阶积分模型。Andel(1986)对$r$不同于0的值和$d_2$的分数值引入了这一过程，Gray等人$(1989,1994)$表明，通过表示$\mu=\cos w_r$，可以将多项式表示为正交Gegenbauer多项式$C_{j, d_2}(\mu)$，因此，对于所有$d_2 \neq 0$，
$$
\left(1-2 \mu L+L^2\right)^{-d_2}=\sum_{j=0}^{\infty} C_{j, d_2}(\mu) L^j,
$$
我们可以像下面这样递归地定义$C_{j, d_2}(\mu)$:
$$
C_{0, d_2}(\mu)=1, \quad C_{1, d_2}(\mu)=2 \mu d_2,
$$
和
$$
C_{j, d_2}=2 \mu\left(\frac{d_2-1}{j}+1\right) C_{j-1, d_2}(\mu)-\left(2 \frac{d_2-1}{j}+1\right) C_{j-2 d_2}(\mu), j=2,3, \cdots
$$

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