### 商科代写|计量经济学代写Econometrics代考|ECON 2504

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• Statistical Inference 统计推断
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• 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|Classical Approaches

Left-tailed integration tests are classical approaches. Indeed, the majority of previous analyses used this conventional approach; in particular, the relationship between housing prices and their economic fundamentals was investigated by a cointegration method (Engle and Granger 1987). The concept of cointegration is widely accepted by economists who established a theoretical framework to identify economic equilibrium conditions and led to Prof. Granger receiving a Nobel Prize (2003). Today, many applied studies used this concept to analyze housing markets worldwide (Hendry 1984; Meese and Wallace 2003; McGibany and Nourzad 2004; Gallin 2006; Adams and Fuss 2010; Oikarinen 2012; De Wit et al. 2013). Because most economic and financial data, including real estate prices and their economic fundamentals, follow a non-stationary process (e.g., Nelson and Plosser (1982), cointegration was considered appropriate to test their long-run relationship and bubbles.

The concept of cointegration can be summarized by rewriting it as a dynamic bivariate relationship. More specifically, to derive the long-run relationship between housing prices $(y)$ and covariates $(x)$, for the period $(t=1, \ldots, T)$, consider the following dynamic equation:
$$y_{t}=\alpha_{0}+\rho_{1} y_{t-1}+\beta_{0} x_{t}+\beta_{1} x_{t-1}+u_{t}$$
where the residual $u$ is normally distributed $\left(u_{t} \sim N\left(0, \sigma^{2}\right)\right)$. Both $x$ and $y$ are in natural the logarithmic form and are assumed to exhibit persistence, in line with many economic and financial variables. Then, we can transform Eq. (7) as follows:
$$\Delta y_{t}=\alpha_{0}+\beta_{0} \Delta x_{t}+\left(\rho_{1}-1\right)\left(y_{t-1}+\frac{\beta_{0}+\beta_{1}}{\alpha_{1}-1} x_{t-1}\right)+u_{t}$$
or simply
$$\Delta y_{t}=a+b \Delta x_{t}+c_{1}\left(y_{t-1}+f x_{t-1}\right)+u_{t}$$
where $\Delta$ is the difference parameter and $c_{1}=\rho_{1}-1$. When $y$ is a housing price, $\Delta y_{t}=y_{t}-y_{t-1}$ represents housing price inflation. Parameters $a, b, c_{1}$, and $f$ need to be estimated. The parameter $b$ measures the short-term sensitivity of $y$ to $x$, and $c_{1}$ measures the speed of the return to the long-run path. The parameter $f$ is a vector of cointegrating parameters that summarize the long-run relationship between $x$ and $y$, and $y_{t-1}+f x_{t-1}$ is the error correction mechanism (ECM). It is stationary, $I(0)$, in the presence of cointegration; in this case, the adjustment parameter $c_{1}$ will be $-1<c_{1}<0$ according to the Granger representation theorem (Engle and Granger 1987). A value of parameter $c_{1}$ close to $-1$ indicates a quick return to the long-run path, and a value close to 0 indicates a slow adjustment. In contrast, when there is no cointegration, $c_{1}$ will not lie within this theoretical range, which implies that there are significant deviations of prices from economic fundamentals, which provides evidence of a bubble. Because financial bubbles are unobservable and are considered leftover (i.e., residuals) in Eq. (9), bubble analyses are sensitive to what comprises economic fundamentals.

## 商科代写|计量经济学代写Econometrics代考|Explosive Unit Root Tests

Financial bubbles are expected to occur occasionally and be recurrent (Blanchard and Watson 1982); furthermore, housing prices may be more chaotic and integrated of an order higher than one. In these cases, the classical unit root and cointegration tests possess only a weak statistical power for detecting bubbles (Evans 1991). To address these shortcomings, Phillips and Yu (2011) proposed conceptually different statistical methods based on Bhargava (1986) and Diba and Grossman (1998). Their tests are right-tailed and aim to examine a high level of a non-stationary process based on Eq. (11). They are designed to trace the orientation and collapse of bubbles, and thus to find chaotic moments (i.e., explosive bubbles) in financial markets. These statistical tests do not aim to determine tranquil periods.

Phillips et al. (2011) is based on the right-tailed test. Their motivation is (Phillips et al. 2011, p. 206), who state that “In the presence of bubbles, $p_{t}$ is always explosive and hence cannot comove or be integrated with $d_{t}$ if $d_{t}$ is itself not explosive.” Here, $p_{t}$ is the log price, and $d_{t}$ represents the log economic fundamentals. This is a subtle difference from the view of economists who pay most attention to cointegration between prices and economic fundamentals. Whether or not prices and economic fundamentals follow a unit root or explosive process is not their major interest. Economists claim evidence of tranquility as long as prices and economic fundamentals are cointegrated, regardless of the order of integration for each variable.
This can be seen in the alternative hypotheses of statistical tests. With the same null hypothesis as that of the classical $\operatorname{ADF}\left(c_{2}=0\right)$, Phillips and Yu (2011) suggested evaluating the right-tailed alternative of an explosive unit $\left(c_{2}>0\right)$. Therefore, compared with the classical unit root tests that define bubbles as $I(1)$ under the null hypothesis, this alternative hypothesis has an implication for stronger bubbles. Thus, the explosive unit root test is conceptually different from the traditional test that looks for cointegration, that is, tranquil periods, and assumes the prevalence of financial bubbles in the market.

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

A single-country analysis can be extended to a study of financial bubbles in a multivariate context. Panel data estimation approaches often exploit cross-sectional information and increase statistical power. A multi-country analysis may be more appropriate because housing prices are highly correlated, particularly among advanced countries (see next section).

Pavlidis et al. (2016) extended the GADF statistics originally developed for singlecountry analyses by following Im et al. (2003), who proposed a left-tailed panel unit root test by extending the conventional univariate ADF test. In their approach, test statistics calculated for each country are pooled to create a single statistic that can be used to assess the statistical hypotheses in a panel context. For country $k$ $(k=1, \ldots, K)$, we can express the panel data version of Eq. (11) as follows:
$$\Delta y_{k, t}=\alpha_{k}+c_{k} y_{k, t-1}+\sum_{i=1}^{p} \theta_{k, i} \Delta y_{k, t-i}+\epsilon_{k, t}$$
where $\epsilon_{k, t} \sim N\left(0, \sigma_{\epsilon_{k}}^{2}\right)$. The null hypothesis is $c_{k}=0, \forall k$ against the alternative of explosive behaviors, $c_{k}>0$ for some $k$. The noble feature of this approach is that it allows heterogeneity (i.e., $c$ ). However, a conclusion from this test becomes somewhat unclear, as the alternative hypothesis states. In other words, a rejection of the null does not necessarily mean that financial bubbles existed in all countries under investigation, but did in at least one country. To obtain a country-specific conclusion, country-wise analyses are required, as we summarized in the previous subsections.
For example, the panel GSADF can be constructed as the supremum of the panel backward SADF (BSADF). The panel BSADF is in turn obtained as the average of the SADF calculated backwardly for individual countries.
\begin{aligned} \text { Panel GSADF }\left(r_{0}\right)=& \text { sub } \quad \text { Panel } \operatorname{BSADF}{\mathrm{r}{2}}\left(\mathrm{r}{0}\right) \ \mathrm{r}{2} \in\left[\mathrm{r}_{0}, 1\right] \end{aligned}
Given the possible cross-country dependence, we follow the calculation method in Pavlidis et al. (2016) closely and use a sieve bootstrap approach. The panel approach using cross-sectional information may be useful to understand a general trend in real estate prices in global markets.

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

Δ是吨=一个0+b0ΔX吨+(ρ1−1)(是吨−1+b0+b1一个1−1X吨−1)+在吨

Δ是吨=一个+bΔX吨+C1(是吨−1+FX吨−1)+在吨

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

Δ是ķ,吨=一个ķ+Cķ是ķ,吨−1+∑一世=1pθķ,一世Δ是ķ,吨−一世+εķ,吨

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