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# 经济代写|计量经济学代写Introduction to Econometrics代考|Sub-models and Comparison with Other Models Used in the Literature

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Sub-models and Comparison with Other Models Used in the Literature

By imposing alternative restrictions on the parameters of the model, one can retrieve different types of models used in the literature to estimate potential GDP and growth.

• Model 1. Standard HP filter model with two state variables being potential growth and GDP
It is defined by Eqs. (2), (4) and (5) with the constraint $D_c^{p_c}(L)=D_s^{p_s}(L)=$ $D_w^{p_w}(L)=0$.
• Model 2. Multivariate semi-structural model with no financial variables, Phillips and IS curves and exogenous interest rate gap.

It is defined by Eqs. (2), (4), (5), (9), and (11) with the constraint $D_c^{p_c}(L)=$ $D_s^{p_s}(L)=0$

• Model 3. Multivariate semi-structural model in which the output-gap and the natural interest rate are determined jointly.
This is Model 2 augmented with Eq. (1).
• Model 4. Multivariate HP filter with financial variables and Taylor rule with exogenous natural rate
It is defined by Eqs. (2), (4), (8), (5), (9) and (11)
We, now, compare our general model with four models which have been used with many instances in the literature.
Laubach and Williams (2003)
• no financial variables in their model,
• no Taylor rule,
• instead of Okun law and unemployment rate, the hours worked are signals of cyclical signals in the labor market,
• they consider inflation expectations to define the ex-ante real interest rate in the Phillips curve, similar dynamics for the output-gap and potential GDP.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Estimation Method

Since we have some unobserved variables, the starting point is to write the model using a state-space representation. The parameters and unknown variables are then estimated using Kalman filter method. The state-space representation of the model is as follows:
\begin{aligned} & X_t=A X_{t-1}+Z_t+F_t W_t: \text { state equation } \ & Y_t=\mu_t+C_t^{\prime} X_t+V_t: \text { measurement equation } \end{aligned}
$X_t$ is the vector of $k_1$ state variables (unobserved), $Y_t$ is the vector of $k_2$ observed variables, A is a $k_1 \times k_1$ matrix, $Z_t$ is a $k_1 \times 1$ vector of deterministic terms, $W_t$ is a $r_1 \times 1$ vector of residuals, $F_t$ is a $k_1 \times r_1$ matrix, $\mu_t$ is the product of a $k_2 \times n \operatorname{expl}$ matrix of coefficients by a vector of nexpl explanatory variables. $C_t$ is a matrix of dimension $k_2 \times k_1$ and $V_t$ is a vector of $r_2$ residual terms.
To estimate the model, we adopt a sequential approach based on five steps.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Estimation Method

＄＄

& X_t =间{t – 1} + Z_t + F_t W_t:{状态方程}\ \文本
& Y_t=\mu_t+C_t^{\素数}X_t+V_t: \text{测量方程}

＄＄
$X_t$是$k_1$状态变量(未观测)的向量，$Y_t$是$k_2$观测变量的向量，A是$k_1 \乘以k_1$矩阵，$Z_t$是$k_1 \乘以1$确定性项的向量，$W_t$是\$r_

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