### 商科代写|计量经济学代写Econometrics代考|ECOM30002

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|T-ARDL Model

Finally, as an extension to the classical approach, we propose the T-ARDL model. ${ }^{4}$ The linear ARDL is a classical method used to capture persistence in time series data, and Pesaran et al. (2001) proposed a bounds test to detect cointegration based on the ARDL. An advantage of this method is its ability to determine the presence of cointegration without prior knowledge of the explanatory variables being stationary $(I(0))$ or non-stationary $(I(1))$. This is a useful feature in studies of bubbles, as economies often experience periods of tranquility and mild bubbles.

Pesaran et al. (2001) proposed five specifications of the ARDL with a different combination of deterministic terms. Here, we use the most popular model in financial research, with an unrestricted constant and no trend. For asset prices $(y)$, we can express this as
$$\Delta y_{t}=a+c y_{t-1}+\boldsymbol{b} \boldsymbol{x}{t-1}+\sum{i=1}^{p-1} \boldsymbol{d}{i} \Delta z{t-i}+\boldsymbol{f}^{\prime} \Delta \boldsymbol{x}{t}+u{t}$$
where $a, c, \boldsymbol{b}, \boldsymbol{d}$, and $\boldsymbol{f}$ are the parameters to estimate by the ordinary least squares (OLS) for time $(t=1, \ldots, T)$, and $u_{t}$ is the residual $\left(u_{t} \sim N\left(0, \sigma^{2}\right)\right)$. $x$ is a matrix of explanatory variables and $z=[y, x]$. The appropriate lag length $(p)$ is determined such that it captures the data generating process of $y$. We can study the cointegrated relationship between $y$ and $\boldsymbol{x}$ by analyzing the time series properties of $c y_{t-1}+\boldsymbol{b} \boldsymbol{x}_{t-1}$, known as the ECM. We can test the null hypothesis of no ECM $(c=0$ and $\boldsymbol{b}=\mathbf{0})$ by the $F$ test or $c=0$ the $t$ test.

As the conventional asymptotic distribution is invalid here, Pesaran et al. (2001) provided critical values based on Monte Carlo simulations for a different dimension of $x$. Because economic variables may be $I(0)$ or $I(1)$, the critical values for these tests have both lower and upper bounds. For the $F$ tests, the lower bound is determined when the data are $I(0)$, and the latter when they are $I(1)$. Test statistics above the upper bound imply evidence of cointegration, and those below the lower bound suggest the absence of cointegration. Test statistics between these bounds are inconclusive. For the $t$ tests, the lower bound is determined when the data are $I(1)$, as the test statistics are expected to he negative. The urper hound is designed for $I(0)$ data.

Howcver, this bounds tssting approach is inappropriate for a study of bubblcs because it investigates the possibilities of both negative and positive bubbles. That is, like the standard unit root tests, it considers bubbles even when housing prices are low. To treat bubbles as high price phenomena, we introduce nonlinearity into the ARDL as follows:
\begin{aligned} \Delta y_{t} &=\alpha_{1} g+\alpha_{2} \tilde{g}+c_{1} g y_{t-1}+c_{2} \tilde{g} y_{t-1}+b_{1} g \boldsymbol{x}{t-1}+b{2} \tilde{g} \boldsymbol{x}{t-1}+\sum{i=1}^{p-1} d_{i} \Delta z_{t-i} \ & \boldsymbol{f} \Delta \boldsymbol{x}{t}+u{t} \end{aligned}
where $g$ and $\tilde{g}$ are dummy variables that distinguish the regimes, and Eq. (15) has two (upper $(g)$ and lower $(\widetilde{g})$ ) regimes. When these regimes are determined by a certain threshold point $(w)$, the dummies are defined as follows:
$$g=\left{\begin{array}{ll} 1 & \text { if } y>w \ 0 & \text { otherwise } \end{array} \text { and } \tilde{g}= \begin{cases}1 & \text { if } y \leq w \ 0 & \text { otherwise }\end{cases}\right.$$

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

We use three left-tailed unit root tests (the ADF, Phillips-Perron, and DF-GLS) that are popular univariate tests in economic and financial research. These tests investigate the null hypothesis that the price-to-rent ratio in levels follows the unit root process $(I(1))$, and a rejection of this null provides evidence of stationarity in this ratio, and thus cointegration between housing prices and rents. Therefore, a failure to reject

this null hypothesis indicates that rents cannot explain the long-term housing price movements, thereby suggesting the presence of mild bubbles. We conduct these tests for the ratios in levels and first differences in order to check the order of integration.
Table 3 summarizes the test statistics for the Euro area, Japan, the UK, and the USA. The results suggest that these ratios follow the unit rovl process. Using the $5 \%$ critical values, we often fail to reject the null hypothesis for the data in levels, but can do so for the differenced data. Therefore, we conclude that mild bubbles existed in all countries, suggesting that rental increases are not always associated with housing price inflation, and there must be some periods when housing prices deviate substantially from the trend in rentals. Obviously, these tests preclude a possibility of explosive bubbles, and moreover, we need to pay attention to the composition of economic fundamentals. However, these outcomes are consistent with our expectations that all housing markets experienced chaotic moments during our sample period.
The potential non-stationary periods identified by the classical method are shaded in Fig. 3. We present two graphs for each country, and the upper figures (denoted as OLS estimates) are obtained from the classical method. As explained earlier, a drawback of this approach is the lack of statistical power to differentiate between hypotheses and that it allows for negative bubbles. For consistency with the standard phenomenon of financial bubbles, we should consider only the positive bubbles (above the horizontal line) as relevant to financial bubbles. Thus, only positive bubble periods are highlighted in gray in this figure and are potential bubble periods because these classical tests are not designed to identify the exact periods of bubbles while they give us evidence of mild bubbles during the sample period. ${ }^{5}$

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

Next, we conduct explosive unit root tests for each country and for a group of countries. From the classical tests, we already know that the price-to-rent ratio is nonstationary and, in fact, implies the presence of mild bubbles. However, when it is not known, we propose the following general steps to reach a conclusion. In short, explosive unit root tests should be conducted if and only if the classical approaches show the data $(y)$ to be a non-stationary process.

Failing to reject the null hypothesis that price-to-rent ratios are $I$ (1) by the classical tests, we eliminated the possibility of market tranquility and thus conduct explosive unit root tests for each market. Table 4 summarizes the results from the right-tailed tests (RADF, SADF, and GSADF) for each country. The null hypothesis of these tests is consistent with our finding of a random walk price-to-rent ratio from the classical tests. The explosive test results differ somewhat by test type, but the null hypothesis is rejected frequently using the $p$-values obtained from 1000 replications, which is evidence in favor of explosive bubbles in all markets. The results from the GSADF are also depicted in Fig. 4. GSADF statistics greater than $95 \%$ critical values suggest the presence of explosive bubbles, which are also shaded in this figure, and generally

identify explosive bubble periods when housing prices are high. The timing and duration of explosive bubbles differ among countries, but many countries seemed to experience explosive bubbles before the Lehman Brothers collapse in September 2018. The presence of real estate markets is consistent with the results from the classical approaches, but here we have evidence of explosive bubbles, which the classical approach cannot capture.

Given the cross-sectional dependence in international housing markets, we also conduct a multi-country analysis using two methods. First, we calculate panel GSADF statistics using the data of 12 countries: Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, the Netherlands, Switzerland, the UK, and the USA. Second, in order to check the robustness of the findings from the panel explosive tests, we conduct the explosive unit root tests for a price-to-rent index that covers OECD countries. ${ }^{6}$ These analyses help us identify explosive bubbles in the global housing market.

## 商科代写|计量经济学代写Econometrics代考|T-ARDL Model

$$\Delta y_{t}=a+c y_{t-1}+\boldsymbol{b} \boldsymbol{x} {t-1}+\sum {i=1}^{ p-1} \boldsymbol{d} {i} \Delta z {ti}+\boldsymbol{f}^{\prime} \Delta \boldsymbol{x} {t}+u {t}$$

Δ是吨=一个1G+一个2G~+C1G是吨−1+C2G~是吨−1+b1GX吨−1+b2G~X吨−1+∑一世=1p−1d一世Δ和吨−一世 FΔX吨+在吨

$$g=\left{ 1 如果 是>在 0 否则 \text { 和 } \tilde{g}= {1 如果 是≤在 0 否则 \正确的。$$

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