P Square Do As I Do Instrumental Download
Lorean Hoefert
I mean the the R-squared calculated such as in $R^2=1-\fracRSSTSS$ when you use the $RSS$ from the original structural model and not recalculation that you should do in order to do an F test. With said $R^2$, you will not have a proper interpretation for the $R^2$ statistic, as I understand. So why report it? I am familiar with Stata reporting it in commands such as ivreg2 and I think other software packages do it too.
It's true that $R^2$ in instrumental variables regressions is not useful. Since one of the explanatory variables $x$ is correlated with the error $\epsilon$ we can't decompose the variance of the outcome $y$ into $\beta^2 Var(x) + Var(\epsilon)$, so the obtained $R^2$ neither has a natural interpretation nor can it be used for computation of F-tests for joint rejection. Also $R^2$ in instrumental variables regression can be negative and for this point it makes not difference for whether you use$$R^2 = \fracMSSTSS \quad \textor \quad R^2 = 1- \fracRSSTSS$$because when $RSS>TSS$, then we also have that $MSS = TSS - RSS < 0$. In general the two expressions are the same so there should be no reason for why one would be more popular than the other. The issue is discussed in more length on the Stata website resources and support FAQs (link).
In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment.[1] Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term (endogenous), in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable (is correlated with the endogenous variable) but has no independent effect on the dependent variable and is not correlated with the error term, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.
Explanatory variables that suffer from one or more of these issues in the context of a regression are sometimes referred to as endogenous. In this situation, ordinary least squares produces biased and inconsistent estimates.[2] However, if an instrument is available, consistent estimates may still be obtained. An instrument is a variable that does not itself belong in the explanatory equation but is correlated with the endogenous explanatory variables, conditionally on the value of other covariates.
For example, suppose a researcher wishes to estimate the causal effect of smoking (X) on general health (Y).[5] Correlation between smoking and health does not imply that smoking causes poor health because other variables, such as depression, may affect both health and smoking, or because health may affect smoking. It is not possible to conduct controlled experiments on smoking status in the general population. The researcher may attempt to estimate the causal effect of smoking on health from observational data by using the tax rate for tobacco products (Z) as an instrument for smoking. The tax rate for tobacco products is a reasonable choice for an instrument because the researcher assumes that it can only be correlated with health through its effect on smoking. If the researcher then finds tobacco taxes and state of health to be correlated, this may be viewed as evidence that smoking causes changes in health.
First use of an instrument variable occurred in a 1928 book by Philip G. Wright, best known for his excellent description of the production, transport and sale of vegetable and animal oils in the early 1900s in the United States,[6][7] while in 1945, Olav Reiersl applied the same approach in the context of errors-in-variables models in his dissertation, giving the method its name.[8]
Wright attempted to determine the supply and demand for butter using panel data on prices and quantities sold in the United States. The idea was that a regression analysis could produce a demand or supply curve because they are formed by the path between prices and quantities demanded or supplied. The problem was that the observational data did not form a demand or supply curve as such, but rather a cloud of point observations that took different shapes under varying market conditions. It seemed that making deductions from the data remained elusive.
After much deliberation, Wright decided to use regional rainfall as his instrumental variable: he concluded that rainfall affected grass production and hence milk production and ultimately butter supply, but not butter demand. In this way he was able to construct a regression equation with only the instrumental variable of price and supply.[9]
Formal definitions of instrumental variables, using counterfactuals and graphical criteria, were given by Judea Pearl in 2000.[10] Angrist and Krueger (2001) present a survey of the history and uses of instrumental variable techniques.[11] Notions of causality in econometrics, and their relationship with instrumental variables and other methods, are discussed by Heckman (2008).[12]
While the ideas behind IV extend to a broad class of models, a very common context for IV is in linear regression. Traditionally,[13] an instrumental variable is definedas a variable Z that is correlated with the independent variable X and uncorrelated with the "error term" U in the linear equation
Consider for simplicity the single-variable case. Suppose we are considering a regression with one variable and a constant (perhaps no other covariates are necessary, or perhaps we have partialed out any other relevant covariates):
IV techniques have been developed among a much broader class of non-linear models. General definitions of instrumental variables, using counterfactual and graphical formalism, were given by Pearl (2000; p. 248).[10] The graphical definition requires that Z satisfy the following conditions:
Since U is unobserved, the requirement that Z be independent of U cannot be inferred from data and must instead be determined from the model structure, i.e., the data-generating process. Causal graphs are a representation of this structure, and the graphical definition given above can be used to quickly determine whether a variable Z qualifies as an instrumental variable given a set of covariates W. To see how, consider the following example.
Finally, suppose that Library Hours does not actually affect GPA because students who do not study in the library simply study elsewhere, as in Figure 4. In this case, controlling for Library Hours still opens a spurious path from Proximity to GPA. However, if we do not control for Library Hours and remove it as a covariate then Proximity can again be used an instrumental variable.
The parameter vector β \displaystyle \beta is the causal effect on y i \displaystyle y_i of a one unit change in each element of X i \displaystyle X_i , holding all other causes of y i \displaystyle y_i constant. The econometric goal is to estimate β \displaystyle \beta . For simplicity's sake assume the draws of e are uncorrelated and that they are drawn from distributions with the same variance (that is, that the errors are serially uncorrelated and homoskedastic).
As long as Z T e = 0 \displaystyle Z^\mathrm T e=0 in the underlying process which generates the data, the appropriate use of the IV estimator will identify this parameter. This works because IV solves for the unique parameter that satisfies Z T e = 0 \displaystyle Z^\mathrm T e=0 , and therefore hones in on the true underlying parameter as the sample size grows.
This expression collapses to the first when the number of instruments is equal to the number of covariates in the equation of interest. The over-identified IV is therefore a generalization of the just-identified IV.
One computational method which can be used to calculate IV estimates is two-stage least squares (2SLS or TSLS). In the first stage, each explanatory variable that is an endogenous covariate in the equation of interest is regressed on all of the exogenous variables in the model, including both exogenous covariates in the equation of interest and the excluded instruments. The predicted values from these regressions are obtained:
This method is only valid in linear models. For categorical endogenous covariates, one might be tempted to use a different first stage than ordinary least squares, such as a probit model for the first stage followed by OLS for the second. This is commonly known in the econometric literature as the forbidden regression,[15] because second-stage IV parameter estimates are consistent only in special cases.[16]
The resulting estimator of β \displaystyle \beta is numerically identical to the expression displayed above. A small correction must be made to the sum-of-squared residuals in the second-stage fitted model in order that the covariance matrix of β \displaystyle \beta is calculated correctly.
In linear analysis, there is no test to falsify the assumption the Z \displaystyle Z is instrumental relative to the pair ( X , Y ) \displaystyle (X,Y) . This is not the case when X \displaystyle X is discrete. Pearl (2000) has shown that, for all f \displaystyle f and g \displaystyle g , the following constraint, called "Instrumental Inequality" must hold whenever Z \displaystyle Z satisfies the two equations above:[10]
The exposition above assumes that the causal effect of interest does not vary across observations, that is, that β \displaystyle \beta is a constant. Generally, different subjects will respond in different ways to changes in the "treatment" x. When this possibility is recognized, the average effect in the population of a change in x on y may differ from the effect in a given subpopulation. For example, the average effect of a job training program may substantially differ across the group of people who actually receive the training and the group which chooses not to receive training. For these reasons, IV methods invoke implicit assumptions on behavioral response, or more generally assumptions over the correlation between the response to treatment and propensity to receive treatment.[18]
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