# 经济代写|计量经济学代写Econometrics代考|The Ramsey RESET test for general misspecification

## 经济代写|计量经济学代写Econometrics代考|The Ramsey RESET test for general misspecification

One of the most commonly used tests for general misspecification is Ramsey’s (1969) Regressions Equation Specification Error Test (RESET). As with many tests, this has both an $F$ form and an $L M$ form. Suppose the ‘true’ population model is:
$$Y=\beta_1+\beta_2 X_2+\beta_3 X_2^2+u$$
and we wrongly estimate:
$$Y=\beta_1+\beta_2 X_2+\hat{u}^*$$
where we omit $X_2^2$ because we do not know the real nature of $Y$.
The RESET test for such misspecification is based on the fitted values of $Y$ obtained from the regression in Equation (8.53) as:
$$\hat{Y}=\hat{\beta}_1+\hat{\beta}_2 X_2$$

The RESET test involves including various powers of $\hat{Y}$ as proxies for $X_2^2$ that can capture possible non-linear relationships. Before implementing the test we need to decide how many terms are to be included in the expanded regression. There is no formal answer to this question, but in general the squared and cubed terms have proved to be useful in most applications; so the expanded equation will be:
$$Y=\beta_1+\beta_2 X_2+\delta_1 \hat{Y}^2+\delta_2 \hat{Y}^3+\epsilon$$
Then the situation boils down to a regular $F$-type test for the additional explanatory variables $\hat{Y}^2$ and $\hat{Y}^3$. If one or more of the coefficients is significant this is evidence of general misspecification. A big drawback of the RESET test is that if we reject the null hypothesis of a correct specification, this merely indicates that the equation is misspecified in one way or another, without providing us with alternative models that are correct.

## 经济代写|计量经济学代写Econometrics代考|Tests for non-nested models

To test models that are non-nested the $F$-type test cannot be used. By non-nested models we mean models in which neither equation is a special case of the other; in other words, we do not have a restricted and an unrestricted model.

Suppose, for example, that we have the following two models:
$$\begin{gathered} Y=\beta_1+\beta_2 X_2+\beta_3 X_3+u \ Y=\beta_1+\beta_2 \ln \left(X_2\right)+\beta_3 \ln \left(X_3\right)+\varepsilon \end{gathered}$$
and that we want to test the first against the second, and vice versa. There are two different approaches.

The first is an approach proposed by Mizon and Richard (1986), who simply suggest the estimation of a comprehensive model of the form:
$$Y=\delta_1+\delta_2 X_2+\delta_3 X_3+\delta_4 \ln \left(X_2\right)+\delta_5 \ln \left(X_3\right)+\epsilon$$ then apply an $F$-test for significance of $\delta_2$ and $\delta_3$, having as the restricted model Equation (8.57), or test for $\delta_4$ and $\delta_5$, having as the unrestricted model Equation (8.56). The second approach is proposed by Davidson and MacKinnon (1993), who suggest that if the model in Equation (8.56) is true, then the fitted values of Equation (8.57) should be insignificant in Equation (8.56) and vice versa. Therefore, in order to test Equation (8.56) we need first to estimate Equation (8.57) and take the fitted values of this model, which may be called $\widetilde{Y}$. The test is then based on the $t$-statistic of $\widetilde{Y}$ in the following equation:
$$Y=\beta_1+\beta_2 X_2+\beta_3 X_3+\zeta \tilde{Y}+v$$
where a significant $\zeta$ coefficient will suggest, of course, the rejection of Equation (8.56). A drawback of this test is that the comprehensive Equation (8.58) may not make sense from an economic theory point of view.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|The Ramsey RESET test for general misspecification

$$Y=\beta_1+\beta_2 X_2+\beta_3 X_2^2+u$$

$$Y=\beta_1+\beta_2 X_2+\hat{u}^*$$

$$\hat{Y}=\hat{\beta}_1+\hat{\beta}_2 X_2$$
RESET 测试包括各种权力 $\hat{Y}$ 作为代理 $X_2^2$ 可以捕获可能的非线性关系。在实施测试之 前，我们需要决定扩展回归中要包含多少项。这个问题没有正式的答案，但总的来说， 平方项和立方项在大多数应用中都被证明是有用的；所以扩展方程将是:
$$Y=\beta_1+\beta_2 X_2+\delta_1 \hat{Y}^2+\delta_2 \hat{Y}^3+\epsilon$$

## 经济代写|计量经济学代写Econometrics代考|Tests for non-nested models

$$Y=\beta_1+\beta_2 X_2+\beta_3 X_3+u Y=\beta_1+\beta_2 \ln \left(X_2\right)+\beta_3 \ln \left(X_3\right)+\varepsilon$$

$$Y=\delta_1+\delta_2 X_2+\delta_3 X_3+\delta_4 \ln \left(X_2\right)+\delta_5 \ln \left(X_3\right)+\epsilon$$

$$Y=\beta_1+\beta_2 X_2+\beta_3 X_3+\zeta \tilde{Y}+v$$

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