Econometrics

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Shrinkage and the James-Stein Estimator

October 5, 2010 in Econometrics | No comments

Is an average always the best estimate?  Let us say that we are evaluating physician quality.  Does a physician’s average score across patients (or episodes of care) best represent their true quality level?  Stein’s paradox says that when we are evaluating the true quality value for a number of doctors, we can do better than the average.  To do this, we can use a shrinkage methodology.

Shrinkage methodologies in essence estimate the physician’s true quality score as a weighted average of the individual physicians average score and the average score for all physicians.  The intuition behind this is that if we observe a high quality doctor, there is some probability that the are actually a high quality doctor and some probability that they are an average doctor, but they just happened to perform above average when treating the patients in the sample.  Similarly, for low quality doctors, there is some probability that the individual is actually a low quality doctor and some probability that they are an average doctor who just scored poorly on the patients in the sample.

How to calculate the James-Stein Shrinkage Estimator

Because of an abundance of statistics, let us move from evaluating physician quality to evaluating the quality of quarterbacks (QB) in the NFL.  Let yi be the average QB rating for an individual quarterback i, and Y be the average QB rating for all quarterbacks. Then in this case, given we observe yi and Y, we can calculate the James-Stein estimator as:

  • yJS=Y+c(yi-Y)

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Risk Adjustment Models

September 9, 2010 in Econometrics | 1 comment

Risk adjustment is important for many aspects of health care.  Medicare uses risk adjustment to modify payments to Medicare Advantage (Part C) plans based on the health of the beneficiaries they cover.  Private insurance companies can use risk adjustment to fine-tune capitation payments to physicians or determine a potential enrollee’s premium.  Providers can use risk adjustment to identify likely high cost patients and how to adjust their likely treatment pattern accordingly.

There is little doubt that risk adjustment is important, but determining which risk adjustment model is ideal is difficult.  A paper by the Society of Actuaries (2007) examines this topic.  The ideal risk adjustment method will depend on a number of factors.   These include:

  • Ease of use of the software;
  • Specificity of the model to the population to which it is being applied;
  • Cost of the software;
  • Transparency of the mechanics and results of the model;
  • Access to data of sufficient quality;
  • Underlying logic or perspective of a model that makes it best for a specific application;
  • Whether the model provides both useful clinical as well as financial information;
  • Whether the model will be used mostly for payment to providers and plans or for underwriting, rating and/or case management;
  • Reliability of the model across settings, over time or with imperfect data (models that are calibrated and tested on a single data set and population may or may not perform well on different data sets/populations);
  • Whether the model is currently in use in the market or organization; and
  • Susceptibility of the model to gaming or upcoding.

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Econometric Methods for Fractional Response Variables

August 31, 2010 in Econometrics | No comments

Oftentimes, people use the following rule of thumb: if the dependent variable is continuous, use OLS; if binary use a logit or probit.  But what should you do if your dependent variable is fraction between 0 and 1.  To use a logit or probit one would have to unnecessarily transform the dependent variable into binary form.  If one would use OLS, the estimation of the coefficients would likely be incorrect.  Because the dependent variable is bounded between 0 and 1, the effenct of any explanatory variably xj cannot be constant through its entire range. Additionally, the predicted values from an OLS regression often produce figures outside the range of 0 to 1.

A paper by Papke and Wooldridge (1996) examines potential econometric alternatives when your dependent variable is fractional.

LOG-ODDS RATIO

One option to estimate a fractional response variable is to transform the dependent variable into a a log-odds ratio.  For instance:

  • E(log[y/(1-y)]|x) = xβ

This model is simple and can be estimated with OLS techniques onces the depenent variable is transformed.  It only works, however, when the dependent variable is strictly between 0 and 1. [If y=0 the you have the log(0) and if y=1 then you get the log(1/0) which is ∞].   Additionally, using this framework, it is difficult to recover E(y|x).  Under the model specified above:

  • E(y|x)=∫ {exp(xβ+ν)/[1+exp(xβ+ν)]} * f(ν|x)dν

If the residuals are independent of the explanatory variables (i.e., ν⊥x), one can use Duan’s (1983) smearing technique to estimate f(•).   If not, one must make functional form assumptions regarding the distribution of the error terms.

QUASI-LIKELIHOOD METHODS

Papke and Wooldridge support using quasi-likelihood methods. Assume the following relationship:

  • E(y|x) = G(xβ)

where 0<1 for all z∈ℜ. The most popular choice for G(z) is the logistic function where G(z)=exp(z)/[1+exp(z)]. In this model, one can estimate the parameters β using the following Brenoulli log-likelihood function:

  • li(β) ≡ yilog[G(xiβ)] + (1-yi)log[1-G(xiβ)]

This method has several advantages.  First, it is fairly easy to estimate.  Secondly, the equation above is a member of the linear exponential family thus the quasi MLE method will produce a consistent estimator of β where β is normally distributed.  Assuming a logit function for G(z) produces the following variance:

  • Var(yi|xi) = σ2 * G(xiβ)[1-G(xiβ)]

The Papke and Wooldridge (1996) also describe how to compute the asymptotic variance of the estimator β.

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Statistical Power

August 3, 2010 in Econometrics | No comments

What is power?  Merriam Webster defines power as the “possession of control, authority, or influence over others.”  The power I will talk about today, however, is statistical power.  Statistical power measures the ability of a statistical test to determine whether the null hypothesis is false.  For instance, in the U.S. judicial system, the null hypothesis is that the defendant is innocent.  Trials that can more accurately determine when the defendant is in fact guilty have more power.

In statistics, there are two types of errors: Type I and Type II. The probability of a Type II error, a false negative, is represented by the symbol β.  Thus, the probability of correctly rejecting the null (i.e., the power) is 1-β.

The larger the magnitude of the hypothesized effect, the higher the power.  It is much easier to detect a large effect than a small effect.  Also, as the size of the sample increases, so does a test’s statistical power.

The more variation that exists in the data, however, the lower the power.  If there is a lot of variation in the data, it is difficult to determine if null hypothesis is false or if observing a phenomenon that contradicts the null is simply due to the excessive amount of variability in the data.  On the hand, if the variability (i.e., standard deviation) is low, then one can generally conclude that that the null hypothesis is false, since the low variability indicates that the anomaly is not caused by normal variation in the data.

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Generalized Linear Models (GLM)

July 20, 2010 in Econometrics | 1 comment

The generalized linear model (GLM) is a flexible generalization of ordinary least squares regression. OLS restricts the regression coefficients to have a constant effect on the dependent variable. GLM allows for the this effect to vary along the range of the explanatory variables.

The basic structure of GLM estimator is as follows:

  • g(Y) = Xβ + ε
  • E(Y) = μ = g-1(Xβ)

To estimate the model, one needs three components:

  1. Random component, specifying the conditional distribution of the
    response variable, given the explanatory variables.  Typically, this distribution is from the exponential family.
  2. A linear predictor which is a linear function of the regressors: η = β0 + β1X1 +…+ βkXk = Xβ
  3. A link function which transforms the expectation of the response to the linear predictor.  In other words, the link function describes the relationship between the linear predictor and the mean of the distribution function.  The link function must be invertible.

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How to construct a Price Index

February 8, 2010 in Econometrics, Economics - General | No comments

Price indices are useful for calculating inflation over time.  The consumer price index (CPI) measures changes in prices for the overall economy.  Researchers can also use price indices to understand the evolution of the price of health care over time.  For instance, the Bureau of Labor Statistics also calculates a CPI for Medical Care and Medical Care Services.

The question of how to calculate a price index is far from trivial however.  To calculate the change in the price of any good between years 1 and T, one could simply use the following formula:

  • Psimple=piT/pi1

However, a price index indicates the change in prices for a basket of goods.  If you are considering the change in price of 10 medical services, how much weight to you give to each one?

Economists have generally come up with the solution: the goods that make up a large share of total expenditures should be weighed more than those that make up a small share.  For instance, let us imagine a simple example where you have two expenses: food and medical care.  The price of food goes up by 10% and the price of medical care goes up by 20%.  Let us assume that food makes up a larger share of your budget than medical expenses and that the initial value of the price index is 1.0 (i.e., T=1).  Thus, if 80% of your income goes to food and 20% of your income goes to medical expenses, than the value of the price index one year from now would be would be 80%*1.1+20%*1.2=1.12.

Sounds easy right?  Not so fast.

I said that 80% of the person’s budget was made up by food, but does that figure refer to your budget expenditures in the first time period or the second time period?  Let us assume the following:

  • Pfood,1=$1; Qfood,1=800; Efood,1=$800;
  • Pfood,2=$1.1; Qfood,2=800; Efood,2=$880;
  • Pmed,1=$100; Qmed,1=2; Emed,1=$200;
  • Pmed,2=$120; Qmed,2=3; Emed,2=$360;

Above, P, Q and E refers to price, quantity and expenditures respectively; the first subscript in the formulas above refers to the good (food or medicine) and the second subscript refers to the time period (1 or 2).  In the example, 80% of the person’s budget in period 1 is for food and 20% is for medical supplies.  If we use the budget shares in the first period to weight the price changes, then we could calculate the price index as:

  • (800*$1.1+2*$120)/(800*$1+2*$100)=1.120

This method is known as the Laspeyres price index.  The general formula is: [Σ pitqi0]/[Σ pi0qi0].

An alternative measure is the Paasche  price index.  In this case, we weight the price changes depending on the bundle of goods in the last time period under consideration.  In the example, our price index would be:

  • (800*$1.1+3*$120)/(800*$1+3*$100)=1.127

The price index is higher now.  Why?  In the last period, the quantity of medical care we purchased increase (for 2 to 3) compared to the quantity of food purchased (stayed the same at 800).  This means that the Paasche price index will put relatively more weight on the price changes for medicine.  Since the price of medicine increased faster than the price of food, the overall price index level be higher in this example than in the case of the Laspeyres price index.  The general formula for the Paasche price index is: [Σ pitqiT]/[Σ pi0qiT].

However, both the Laspeyres and Paasche indices do not take into account substitution effects between goods. Goods are weighed statically based on the quantity purchased in either the first period (Laspeyres) or last period (Paasche). To solve this problem, one can use the Fisher price index. This index does account for individuals substituting across different types of goods. To calculate the Fischer index, one simply takes the geometric mean of the Laspeyres and Paasche indices. According to the example above, this means the price index would be:

  • Pf=(Pp*Pl)0.5=(1.120*1.127)0.5=1.123

One can also chain the Fisher index calculations from each year in order to produce a chain-weighted Fisher price index, but I’ll save that explanation for another day.

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The Ecological Fallacy

November 27, 2009 in Econometrics | No comments

Are foreign-born individuals more likely to be literate (in English) than native born Americans?  One would think not, but consider the following information:

Robinson (1950) computed the following two pieces of information: the percent of the population who are foreign-born, and the percent who are literate.  Robinson observes that states with a high percentage of foreign-born individuals have higher literarcy rates.  There is a 0.56 correlation between a state’s proportion of foreign born individuals and a state’s proportion of individuals who are literate. Does this mean that being foreign-born causes an increase in literacy?

Actually no. This correlation is an ‘ecological’ correlation,  because the unit of analysis is not an individual person but a group of people—the residents of a state. In reality, the association is negative: the correlation computed at the individual level is -0.11.

These figures can be explained as follows.  Let us assume that high income states have high levels of literacy.  Also assume that foreign born individuals have low levels of literacy.  Because immigrants are drawn to states with high income and the potential for economic growth, one could see a a positive correlation between literacy and foreign born individuals on a state level.  High income states have high literacy among natives, but many foreign born individuals.  Low income states have low literacy levels among natives, but also few immigrants.  Thus, one could see a positive correlation between proportion of foreign born individuals and literacy on an aggregate level, but a negative correlation on an individual level.  

This is the problem of ecological inference.  As David Freeman explains, ecological inference occurs when inferences about individual behavior drawn from data about aggregates.  Stereotypes are another example of ecological inference.  In this case, one assumes that individual members of a group have the average characteristics of the group at large.

Almost Ideal Demand System (AIDS)

October 15, 2009 in Econometrics | No comments

How does one model a demand system? In general, researchers only observe the equilibrium prices and quantities of goods over time. Changing prices or quantities could be due to shifts in either the demand or supply curve. Thus, modeling demand systems is difficult.

Deaton and Muellbauer (1980) propose one method: the Almost Ideal Demand System (AIDS). Today I will review this demand estimation strategy.

Origin

The origin of the AIDS system comes from the piglog model. Piglog models allow researchers to treat treat aggregate consumer behavior as if it were the outcome of a single maximizing
consumer. One must assume that in equilibrium, the marginal propensity to consume is the same across households.

One could add sophistication to the model by including parameters–kh in the paper–which measure household size, age composition, and other household characteristics. In general, Deaton and Muellbauer assume that all the kh parameters are equal to one; thus implying that all household have similar preferences.

Estimation

The AIDS estimations strategy is attractive because it is simple to estimate and–under certain assumptions–avoids the need for non-linear estimation. Further, it can also test homogeneity and symmetry restrictions through assumptions on the parameter values in the estimation.

Deaton and Muellbauer (1980) describe how one can begin with primitive, individual utility functions and aggregate them to form the following estimation framework:

  • wi=(αi+βiα0) + Σj γijlog pj + βi*{log x – Σkαk *log pk – .5*ΣkΣj (γkj*log pk*log pj)}

In this equation, wi represents the budget share of good i. The index j indexes the good. Prices are represented by p, total expenditures are represented by x, and w represents budget shares. One can estimate all parameters using a maximum likelihood methodology. In general, the following three restrictions are imposed to simplify estimation:

  1. Adding up restriction: Σi αi=1; Σi γij=0; Σi βi=0;
  2. Homogeneity restriction: Σj γij=0;
  3. Slutsky Symmetry restriction: γij=γji

One can simplify this estimation in situations where prices are closely collinear. Instead of using an exact price index, P, one could calculate an approximate price index P*. One candidate recommended by Deaton and Muellbauer is Stone’s (1953) index: log P*=Σk (wk*log pk). If P* is a good approximation, then one can use the following equation as an approximation of the full estimation above:

  • wi=(αi+βilog φ) + Σj γijlog pj + βilog(x/P*)

Testing Restrictions

In order to test the homogeneity restriction, one can leaves out a single pj term and instead focus use relative prices (pj/pn).

  • wi = αi* +Σj=1n-1 {γij*log(pj/pn)} + βilog(x/P*)

An F-ratios are calculated for each of the i equations to determine if the homogeneity restriction holds. Next the paper also gives the steps needed to test the symmetry restriction as well. In order to conduct the symmetry test, one must calculate the price index as follows:

  • log P = α0 + Σk αk*log pk + 0.5*ΣkΣj (γkj*log pk*log pj)}

In order to calculate this “correct” price index, one must choose an appropriate value for α0.

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Non linearities and the Black Swan

September 25, 2009 in Books, Econometrics | 1 comment

The book Black Swan by Nassim Nicholas Taleb is an interesting book about probability outside of the traditional Gaussian framework and how paradigm changing often arise.  The highlight of the book is its philosophy of the black swan, and its unknown unknown.  The book also includes discussion of behavioral economics and tries to discredit Gaussian statistics.  The book is interesting but rambles somewhat.  Further, Taleb writes in a condescending manner disparaging other intellectuals and experts.  Although Taleb does make some good points but the negative tone does become tiresome.  

The Turkey Problem

The crux of the book can be understood by looking at the following series.  This series represents the weight of the turkey over 30 days.

Assume you are a turkey, what would you predict would happen to your weight over then next 15 days. Using ordinary least squares, one would predict that the turkey would continue to grow at 1/4 pound per day. Let us see what happened in reality.

We see that a “black swan” event has a occurred; one that was outside the paradigm one would establish based on past data. We see that on day number 41, the turkey is slaughtered. This is a huge paradigm shift from the point of view of the turkey. One can see that relying on past data to predict the future will be highly inaccurate in the presences of these black swans.

Other Non-linearities

Let us look at another seemingly linerar series.  

 

How would you predict the series would continue into the future?  Using linear extrapolation techniques, one would predict the series would increase linearly ad infinitum. However, let us examine the true data generating process.

We can see that the data come from a sine function.

The key insight of Taleb’s book is that these non-linearities, paradigm shifts and black swans occur all the time. Further, they are responsible for most of the innovatiations and important events in history. Thus, ignoring black swans can be perilious. Taleb’s message is one of humility.  It is exceedingly difficult to predict the future.  A sure thing is rarely ever such.  Thus, we should view expert opinion with some skepticism and embrace–rather than reject–uncertainty.

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What I learned in my Macroeconomics Classes

July 24, 2009 in Econometrics | No comments

The Economist has an interesting article about the failure of macroeconomics to predict the latest downturn and what it means for the future of the profession of economics.  As one who has little faith in macroeconomics, I certainly can commiserate with the opinions of the following Ph.D. students:

According to David Colander, who has twice surveyed the opinions of economists in the best American PhD programmes, macroeconomics is often the least popular class. “What did you learn in macro?” Mr Colander asked a group of Chicago students. “Did you do the dynamic stochastic general equilibrium model?” “We learned a lot of junk like that,” one replied.

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