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# 经济代写|计量经济学代写Introduction to Econometrics代考|Linear Cointegration Specification for Wealth Effects

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## 经济代写|计量经济学代写Introduction to 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 $(\mathrm{FW})$, housing wealth $(\mathrm{HW})$ 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.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Threshold ECM Specification for Wealth Effects

In a second time, we consider potential thresholds into the ECM framework and we extend the LECM by introducing nonlinearity in the adjustment and consideration of a Nonlinear ECM. In particular, we propose two different regressions for the wealth-consumption short-term adjustment dynamics: TAR-ECM and M-TARECM models in order to capture different forms of nonlinearities in the adjustment. This extension aims to capture further asymmetry and nonlinearity in the wealthconsumption adjustment relationship that may escape the LECM. It is thus a more flexible methodology to investigate cointegration. The difference between the TAR and MTAR models is in the definition of the Heaviside indicator function, which is based on the level value of the threshold of the indicator variable in the TAR model, while it is based on the difference of the indicator variable for the MTAR model.
In practice, TAR-ECM and M-TAR-ECM are estimated using the Enders and Siklos (2001) methodology which generalized the Engle and Granger procedure to allow for nonlinear adjustments. Indeed, the estimation of the TAR-ECM and M-TAR-ECM processes is based on the estimated residuals from the long-run relationship between the two variables, consumption and wealth [models (1) and (2)]. This also requires the existence of a single cointegration relationship (see for instance Marquez et al. 2013 for the UK case).
The M-TAR-ECM model can be written as:
$$\Delta \hat{u}t=I_t \rho_1 \hat{u}{t-1}+\left(1-I_t\right) \rho_2 \hat{u}{t-1}+\sum{i=1}^k \gamma_i \Delta \hat{u}{t-1}+\varepsilon_t$$ with the indicator function $I_t$ defined as: $I_t=\left{\begin{array}{l}1 \text { if } \Delta \hat{u}{t-1} \geq 0 \ 0 \text { if } \Delta \hat{u}{t-1}<0\end{array}\right}$. A TAR-ECM corresponds to: $$\Delta \hat{u}_t=I_t \rho_1 \hat{u}{t-1}+\left(1-I_t\right) \rho_2 \hat{u}{t-1}+\sum{i=1}^k \gamma_i \Delta \hat{u}{t-1}+\varepsilon_t$$ with the indicator function $I_t$ defined as: $I_t=\left{\begin{array}{l}1 \text { if } \hat{u}{t-1} \geq 0 \ 0 \text { if } \hat{u}_{t-1}<0\end{array}\right}$.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Linear Cointegration Specification for Wealth Effects

$$\begin{gathered} c_t=\alpha+\beta_1 T W_t+\beta_2 y_t+\varepsilon_t \ c_t=\alpha+\beta_1 F W_t+\beta_2 H W_t++\beta_3 y_t+\varepsilon_t \end{gathered}$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Threshold ECM Specification for Wealth Effects

M-TAR-ECM模型可以写成:
$$\Delta \hat{u}t=I_t \rho_1 \hat{u}{t-1}+\left(1-I_t\right) \rho_2 \hat{u}{t-1}+\sum{i=1}^k \gamma_i \Delta \hat{u}{t-1}+\varepsilon_t$$，其中指标函数$I_t$定义为:$I_t=\left{\begin{array}{l}1 \text { if } \Delta \hat{u}{t-1} \geq 0 \ 0 \text { if } \Delta \hat{u}{t-1}<0\end{array}\right}$。一个TAR-ECM对应:$$\Delta \hat{u}t=I_t \rho_1 \hat{u}{t-1}+\left(1-I_t\right) \rho_2 \hat{u}{t-1}+\sum{i=1}^k \gamma_i \Delta \hat{u}{t-1}+\varepsilon_t$$，其中指标函数$I_t$定义为:$I_t=\left{\begin{array}{l}1 \text { if } \hat{u}{t-1} \geq 0 \ 0 \text { if } \hat{u}{t-1}<0\end{array}\right}$。

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