10
"But if any of these control variables are endogenous to some omitted variable, doesn't this contaminate the unbiasedness of ALL the independent variables?"
I don't want to emphasize this too much, but it's worth mentioning that this is not true in general. The following derivation will hopefully provide some understanding of the "contamination" you mention....
9
Welcome to the wonderful world of econometrics! Most introductory econometrics courses will extend Ordinary Least Squares (OLS) by considering binary outcomes models such as the logit and probit.
Whilst OLS is typically restricted to modelling continuous outcomes bound between $-\infty$ and $\infty$, in research one will often come across data where this is ...
9
The $\mathbf M = \mathbf I-\mathbf X(\mathbf X'\mathbf X)^{-1}\mathbf X'$ matrix is the "annihilator" or "residual maker" matrix associated with matrix $\mathbf X$. It is called "annihilator" because $\mathbf M\mathbf X =0$ (for its own $X$ matrix of course). Is is called "residual maker" because $\mathbf M \mathbf y =\mathbf {\hat e}$, in the regression $\...
9
You can use code like the following (making use of the as_latex function) to output a regression result to a tex file but it doesn't stack them neatly in tabular form the way that outreg2 does:
import pandas as pd
import statsmodels.formula.api as smf
x = [1, 3, 5, 6, 8, 3, 4, 5, 1, 3, 5, 6, 8, 3, 4, 5, 0, 1, 0, 1, 1, 4, 5, 7]
y = [0, 1, 0, 1, 1, 4, 5, 7,0, ...
9
Because $\Bbb E[\varepsilon \mid x]= 0$ is one of the key assumptions for the estimation.
8
Disclaimer: this answer comes from a microeconomic research perspective. Time series / macroeconomic specialists will likely have other perspectives.
There is no general rule for what's too low across the entire field of economics. Yes, microeconomic models (i.e., individual-level observations) will tend to give low R-squared values (often in single ...
6
All is too strong, but probably some. This problem is called "smearing". Take a look at the proof in Greene's lecture notes on slide 5.
Emily Oster has a nice working paper (and Stata command psacalc) that can help bound the bias.
6
The incorporation of a price elasticity in your regression requires that your dependent variable, quantity, be logged as well.
Take an example of a basic demand side equation including two independent variables
$$Q_d=\beta_0+\beta_1 P+\beta_2 S+\mu$$
where $Q_d$ is quantity demanded, $P$ is the price of the good in question and $S$ is the price of a ...
6
There are numerous directions to go which start moving you beyond ordinary least squares (OLS), linear regression. The universe of statistical methods is large!
Two books that I particularly enjoyed are Econometrics by Hayashi and Elements of Statistical Learning by Hastie et. al. Looking back at your question, these books may be too advanced. But maybe ...
6
Using corruption is part of it but a bit restrictive way to measure government "quality". You may use aggregate indicators as the one developed by the Worldwide Governance Indicators (WGI) project from the World Bank. They reports aggregate and individual governance indicators for over 200 countries and territories over the period 1996–, for six dimensions ...
5
In the benchmark hedonic price analysis, we assume a utility function of the general form
$$U = U(x, z_1,...,z_n)$$
where "$x$" stands for the composite good, and $(z_1,...,z_n)$ are the characteristics of good $y$ that are valued by the consumer. Assume for simplicity (as is usually done in the literature, and as is the OP case), that the consumer will ...
5
This is an example of what statistician Andrew Gelman calls "the fallacy of controlling for an intermediate outcome". Here is his description of this fallacy popping up when researchers ask if having more daughters changes your politics. The decision to have a second child is necessarily conditional on the previous decision to have the first child, and so ...
5
In the context of Least-squares estimation, the way we have to (attempt to) deal with possible endogeneity of regressors is through Instrumental Variables estimation. This approach does not depend on having just one endogenous regressor -you may have many. In such a case of course you need to find more instruments which make things harder -but in principle, ...
5
This is a somewhat "dated" subject in introductory econometrics, I suspect because, in econometrics the models come from theories and arguments that try to a priori establish causality and not just association.
Anyway, the issue is analyzed in Maddala's Econometrics textbook.
In the 2001 3d ed. the issue for simple regression is presented in ch. 3. ...
5
Use -areg- in Stata, and the standard errors will come out as in the textbook. Specifically, the command
areg lpassen lfare ldist ldistsq y98 y99 y00, absorb(id) vce(robust)
will produce the desired result.
-xtreg- with fixed effects and the -vce(robust)- option will automatically give standard errors clustered at the id level, whereas -areg- with -vce(...
5
There is now a Python version of the well known stargazer R package, which does exactly this.
See also this related question: https://stackoverflow.com/q/35051673/2858145
5
So basically the question is:
If I know the average ($\hat{\mu}$) of the daily temperatures ($y_i$) of last year, does that tell me anything about how many people were born ($x_i$) each day?
Unsurprisingly the answer is no.
The most you can get is the average of the $x_i$ series if you have the parameters of an unbiased regression between $x_i$ and $y_i$.
5
It's important to consider what exactly the population is about which an inference is being drawn. It's easy to overlook the time aspect in this context.
Suppose for example that the aim is to forecast the next two years' GDP for each country in the world. Then the population of interest is a set of pairs of the form "country, year". It isn't simply "all ...
5
The "LPM" label refers to the structure of the equation, not to the estimator. LPM models can be estimated not only by least-squares methods but also by maximum-likelihood for example.
As regards the nature of the dependent variable, we are talking about an affine transformation here. Let a model be with a binary dependent variable and a single ...
5
Imagine that I am trying to determine whether eating corn has any effect on your height. I see that in the US, total corn consumption is 20 million tons per year (made up number, all others will be made up as well) and adding up the heights of the 300 million citizens we get 500 million meters. Similar statistics for France are 4 million tons per year and ...
5
A degenerate joint normal is distribution is one in which you cannot find a PDF for the distribution. They assume you can. (The covariance matrix is invertible).
Let $f(s_1,s_2\dots,s_n,v)$ be the distribution.
If I was to exchange $s_1$ for $s_2$, $f(s_2,s_1,\dots,s_n,v) = f(s_1,s_2\dots,s_n,v)$, the distribution does not change. And you can exchange as ...
4
It won't help much with the high frequency variation of interest to you but this is a famous and important paper on this topic:
I process satellite-generated data on terrain elevation and presence
of water bodies to precisely estimate the amount of developable land
in U.S. metropolitan areas. The data show that residential development
is effectively ...
4
I'm still not sure if I'm doing something wrong. However, it is useful to note that I get the same results in R.
library(foreign)
library(plm)
library(lmtest)
df <- read.dta("airfare.dta")
fe.out <- plm(lpassen ~ lfare + ldist + ldistsq + y98 + y99 + y00,
data=df, index = c("id", "year"),
method = "within", effect = "individual")
...
4
For a list of methods used in applied econometrics you can take a look at the ReplicationWiki (that I work on). Many of them have data and code so you can easily try them out. (The example is with instrumental variables, replace this method in the search form with any other from the list to search for examples for them.)
4
This is more a related thought than anything else, but I thought it might be useful for people looking at this question in the future. Herr K. should write a short answer and have it marked as accepted.
Let $X$ be an $n \times k$ matrix with linearly independent columns. Let $S \equiv \text{span}(X)$.
Regression can be thought of as the following problem: ...
4
I had initially flagged this question for moderators to examine whether it would be better to migrate over to the statistics SE site Cross Validated. But since the OP introduced a very specific econometrics example, I believe the (very deep) concept of "population/sample" can be usefully discussed for the purposes of this example.
A first issue is that ...
4
Linear regression, despite its simplicity, it is actually a very powerful tool. That's why it's everywhere in econometrics, to give you an example you're maybe familiar with, consider an auto-regressive model, turns out you can write the future state of a variable that follows this model as a linear combination of previous states
$$
X_t = C + \phi_1 X_{t-1} ...
4
As you can see from the plot, they are not the same. Infact, the 'rule of thumb', i.e. your first formula, only holds in approximation. The error gets larger for large coefficient values.
4
Well, I will try to answer 4.
We know that the asset liquidation value $\tilde{v}$ is an affine function of the singals thus we have that
$$\tilde{v}=\bar{v}+\sum_{i=1}^{N}\tilde{s}^i\Rightarrow \tilde{v}=\bar{v}+N\underbrace{\frac{\sum_{i=1}^{N}\tilde{s}^i}{N}}_{\tilde{s}^i}\Rightarrow\tilde{v}=\bar{v}+N\tilde{s}^i$$
where the $\tilde{s}^i$ is the average ...
3
To understand the issue let's review what is the so call robust variance-covariance matrix estimates (VCE) and the implied "robust" standard errors. The robustness is meant to allow for violations of homoscedasticity in the cross-sectional dimension or heteroscedasticity. There are various heteroscedastic robust VCE which are known as the Sandwich estimators ...
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