semibruin
  • Member for 8 years, 6 months
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To some extent, I think we are interested in stationary distribution of an AR process for the same reason why we are interested in the distribution of an iid process. Provided the stationary distribution exists, it is then the marginal distribution of $X_{t}$, which tells us its mean and variance or confidence interval in general. Indeed, I am only interested in covariance stationary process, because with the CLT, I know the asymptotic distribution as long as I know the mean and variance.

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@doubled In case you are unfamiliar with 'linear projection' in econometrics, check out the first few chapters of Jeff Wooldridge's book 'cross-seciton and panel data analysis'.

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@doubled I would write $Y = X\beta + W\gamma + U$, rather than $Y = \hat{X}\beta + W \gamma + D$. Writing $Y = X\beta + W\gamma + U$ inserts $U$ into your arguments. This is helpful, since endogeneity is defined as the correlation between $U$ and explanatory variables. Using $U$ makes the idea more transparent. Also $Y = \hat{X}\beta + W \gamma + D$ is unclear and not necessarily true. What is $D$ anyway? I think in your mind, you want to write $Y$ as a linear projection of $\hat{X}$ and $W$, i.e. $Y = X b + W c + V$, but $b$ and $c$ are different from $\beta$ and $\gamma$ without exogeneity.

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If you want the effect of "national spending" to vary across local branch. You may consider panel model with random slope. Think about $y_{it} = x_{it}'\alpha + s_{t}b_{i} + c_{i} + u_{it}$. $b_{i}$ is the partial effect of spending for branch $i$. Under additional exogeneity condition (e.g. $b_{i}$ is mean independent of $(x_{it}, s_{t})$), you can still use FE estimator to estimate a model like $y_{it} = x_{it}'\alpha + s_{t}\beta + c_{i} + v_{it}$, but FE will converge to $E(b_{i})$. So $\beta$ is the average, not branch-specific, partial effect of spending.

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@EricFail I replaced -c(1:5) with regular expression. It is more generic now. In general, you would like to use grepl to match patterns in the presence of a lot of variables.

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use regular expression to match variable names.

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Don't think F test would work here, since your current two models (female and male) are not nested. Why not include run plm with interaction terms between female and explanatory variables, e.g. plm(math ~ Female * (x1 + x2)). To test the first null hypothesis, you just run F test for all coefficients associated with Female:x1, Female:x2. To test the second null, you just need t test the parameter associated with Female:year1.5.

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It seems that it should $var(S \vert \lambda, \theta) = E(N^2 \vert \lambda) var(X \vert \theta) + var(N \vert \lambda) {E(X \vert \theta)} ^ 2$. Was there a typo ---- $E(N ^ 2 \vert \lambda)$ rather than $E(N \vert \lambda)$?

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If $W$ is nonrandom (since you said it is known), you in general need $E(X_{i} \varepsilon_{i*}) = 0$. Then the statement follows from the LLN.

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Theorem 2.4 is on page 8.

awarded
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format math notation in latex

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Approve
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Yes, it is true. c.f. van der Vaart's asymptotic statistics (theorem 2.4).

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You may want to check out Theorem 4.2.1 and 4.2.2 of Herman Bierens' book "Topics in advanced econometrics". They are standard M estimator results.

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