4

Let $\hat{\beta}$ be the OLS estimator of $\beta$ in $$y_t = \beta x_t + u_t$$ Let $\tilde{\beta}$ be the OLS estimator of $\beta$ in $$\dfrac{y_t}{\sqrt{k_t}} = \beta \dfrac{x_t}{\sqrt{k_t}} + \dfrac{u_t}{\sqrt{k_t}}$$ $\text{Var}(\hat{\beta}) = \dfrac{\sum x_t^2\sigma_t^2}{\left(\sum x_t^2\right)^{2}} $ $\text{Var}(\tilde{\beta}) = \dfrac{\sigma^2_u}{\sum ...


4

The question should be: Does there appear to be enough heteroskedasticity so that not taking it into account would lower the quality of inference? And this is because "taking heteroskedasticity into account" (even if only for robust standard errors) is not without costs -with small sample sizes it may worsen the reliability of results. And it becomes even ...


3

Is there evidence for time varying second moments in annual economic data? Yes, although not that much in finance in particular but in economics in general resounding yes. For example, the highly cited Engle (2001), GARCH 101: The use of ARCH/GARCH models in applied econometrics, besides examples with daily data refers also to some examples with quarterly ...


2

I'm using the 4th edition, but the content should be the same. It sounds like you are asking how he got from $$ Var(u|\boldsymbol{x}) = \sigma^2exp(\delta_0 +\delta_1x_1+\delta_2x_2+...+\delta_kx_k) $$ to $$ u^2=\sigma^2exp(\delta_0 +\delta_1x_1+\delta_2x_2+...+\delta_kx_k)v $$ As you correctly noted, $Var(u|\boldsymbol{x}) = E(u^2|\boldsymbol{x})$ ...


2

You can regress residual squares (from RE or FE depending on your estimation) on $X_{it} \hat\beta$ and its square using the clustered standard errors (the vce(cl id) option), and read the F statistic and the associated p value. This is basically the same as Het test for cross sectional models (White's simplified test). xtreg y x1 x2, re predict uhat, ue ...


1

The Ω matrix is the matrix of the variance of the error term for each observation. Since we do not observe the true error term, we cannot find the true Ω, but we can try to estimate it. There are different ways of estimating Ω, and there is some debate in econometrics as to which method is the best but the most common method I think would be to take the ...


1

Let us write $\mathbf{x}\delta = \delta_0 + \delta_1 x_1 + \cdots + \delta_k x_k$ for notational brevity. If $u^2 = \sigma^2 \exp(\mathbf{x}\delta) v$, where $E(v|\mathbf{x})=1$, then it is indeed true that $$E(u^2|\mathbf{x}) = \sigma^2 \exp(\mathbf{x}\delta) E(v|\mathbf{x}) = \sigma^2 \exp(\mathbf{x}\delta).$$ The LHS is $Var(u|\mathbf{x})$ by definition ...


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