Material for the Course on Inequality

Here is the video that we discussed during the first course,

References for the course are the following,

For similar course lectures notes, see Emmanuel Flachaire’s ECON-473 webpage, as well as Michel Lubrano’s notes.

For the project, some references in French,

and in English

There is also the following report (one can focus only on a chapter of the report)

 

Graduate Course on Advanced Methods in Econometrics

I will give a short graduate course for PhD students, in Rennes, on Thurday mornings, in March (2nd, 9th, 23rd and 30th). The agenda will be

  1. Nonlinear Regression Models and Smoothing Techniques

  2. Bootstrapping and Regression

  3. Penalized Regression Models and LASSO

  4. Quantile Regression and Expectiles

There will be slides available by the end of February.

 

Third Actuarial Pricing Game

With the support of ACTINFO Chair and the (French) Institute of Actuaries, our Third Actuarial Pricing Game starts today ! There is a toolbox file available online, with

  • a description of the game : the rules, the dates, and a description of the datasets
  • 3 datasets : one underwriting and one claims databases, for year 0 (training data) and one underwriting dataset to enter the game

Anyone can play. Students from various programs around the world, as well as practitioners are welcome to play. It can be by teams, and there are no limit on the size. And there is no registration: to start playing, teams have to submit a dataset before the deadline (end of February), to pricing-game@univ-rennes1.fr.

Teasing for the Third Actuarial Pricing Game

We will launch within the next few days the Third Actuarial Pricing Game. The goal will be to replicate the behavior of insurance markets over time. There will be a first stage (January-February) where players will have to build a pricing model for motor insurance policies on the basis of 50,000 contracts observed Year 0 (characteristics of contracts, policyholder, and claims datasets). Proposals for premiums for Year 1 will have to be provided for the same contracts (updated for the age, the location, the car model, etc.). Players will use only the underwriting datasets to provide premiums for Year 1.

Between February 25th (deadline for the proposal for premiums) and March 1st, we will replicate an insurance market by creating competition among insurers (players), and by setting simple rules to match drivers and insurers (randomly among the cheapest at the beginning of the game, and then, as time goes by, we will add inertia, i.e. a tendency to stay with the same insurer if the latter is not too expensive). Once the drivers and the playees are matched we will provide claims information for the insured that each player has won. A new premium proposal must then be submitted at the end of Year 1 (i.e. at the end of March). Etc.

We will here try to replicate the a car insurance market over four years. The goal (for players) will be to maximize the profit of the insurance company over those four years. To make the game more realistic, insurers are assumed to have capital, and they can remain in the game only if their yearly loss ratio (claims over premium) is less than 150%. More information next week (rules, and training datasets).

Forecasting Natural Catastrophes (is rather difficult)

Following my previous post, I wanted to spend more time, on the time series with “global weather-related disaster losses as a proportion of global GDP” over the time period 1990-2016 that Roger Pilke sent me last night.

db=data.frame(year=1990:2016,
ratio=c(.23,.27,.32,.37,.22,.26,.29,.15,.40,.28,.14,.09,.24,.18,.29,.51,.13,.17,.25,.13,.21,.29,.25,.2,.15,.12,.12))

In my previous post, I spend some time explaining that we should provide some sort of ‘confidence interval’ when we try to predict a pattern. That was what we call ‘model uncertainty’. But there are two (important) issues that I did not mention. (1) it is a time series, so why not use techniques dedicated to time series objects ? (2) we do not really care actually about ‘model uncertainty’ (unless we want to assess if a decreasing trend is significant, or not), and we care more about real prediction uncertainty: in the next ten years, what could be the range for the this ratio, with some given probability (say 95%)? Could we say that with 95% chance the global weather-related disaster losses as a proportion of global GDP should be (each year) within 0 and 0.35 or 0 and 0.7?

A first idea might be to use exponential smoothing techniques (without a seasonal component here).

ratio=ts(db$ratio,start=1990,frequency=1)
plot(ratio,xlim=c(1990,2030))
hw=HoltWinters(ratio,gamma=FALSE)
phw=predict(hw,n.ahead=15,prediction.interval = TRUE)
plot(hw,phw,xlim=c(1990,2030))
polygon(c(2017:2031,rev(2017:2031)), c(phw[,2],rev(phw[,3])),border=NA,col=rgb(0,0,1,.2))

The decreasing trend is coming from the fact that exponential smoothing is here a linear regression, with weight exponentially decaying with time (the older, the smaller the weight). But we cannot use that prediction, since the ratio cannot (obviously) be negative. So why not consider, here, the logarithm of the ratio

plot(db$year,db$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio")
hw=HoltWinters(log(ratio),gamma=FALSE)
phw=predict(hw,n.ahead=15,prediction.interval = TRUE)
abline(v=2016,lty=2,col="grey")
lines(2017:2031,exp(phw[,2]),col="blue")
lines(2017:2031,exp(phw[,3]),col="blue")
lines(c(1992:2016,2017:2031),c(exp(hw$fitted[,1]),exp(phw[,1])),col="red")
polygon(c(2017:2031,rev(2017:2031)),exp(c(phw[,2],rev(phw[,3]))),border=NA,col=rgb(0,0,1,.2))
abline(h=0,lty=2)

The confidence band is huge, here. What if we consider some ARIMA model here?

fit=auto.arima(ratio)
farma=forecast(fit,15)
farma=cbind(as.numeric(farma$fitted)[1:15],as.numeric(farma$lower[,1]),as.numeric(farma$upper[,1]),as.numeric(farma$lower[,2]),as.numeric(farma$upper[,2]))
plot(db$year,db$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio")
abline(v=2016,lty=2,col="grey")
lines(2017:2031,farma[,4],col="blue")
lines(2017:2031,farma[,5],col="blue")
lines(2017:2031,farma[,1],col="red")
polygon(c(2017:2031,rev(2017:2031)),c(farma[,4],rev(farma[,5])),border=NA,col=rgb(0,0,1,.2))
abline(h=0,lty=2)

Here, there is an intercept, but no dynamics for the time series (which is considered, here, as a pure white noise). We get exactly the same if we consider the average value of the series

fit=lm(ratio~1,data=db)
s=summary(fit)$sigma
plot(db$year,db$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio")
abline(v=2016,lty=2,col="grey")
ndb=data.frame(year=2017:2031)
pf=predict(fit,newdata=ndb)
farma=cbind(pf,pf-1.96*s,pf+1.96*s)
lines(2017:2031,farma[,2],col="blue")
lines(2017:2031,farma[,3],col="blue")
lines(1990:2031,c(predict(fit),farma[,1]),col="red")
polygon(c(2017:2031,rev(2017:2031)),c(farma[,2],rev(farma[,3])),border=NA,col=rgb(0,0,1,.2))
abline(h=0,lty=2)

Here, we get back to my previous post, if we want to consider a possible trend (and not only an intercept)

fit=lm(ratio~year,data=db)
s=summary(fit)$sigma
plot(db$year,db$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio")
abline(v=2016,lty=2,col="grey")
ndb=data.frame(year=2017:2031)
pf=predict(fit,newdata=ndb)
farma=cbind(pf,pf-1.96*s,pf+1.96*s)
lines(2017:2031,farma[,2],col="blue")
lines(2017:2031,farma[,3],col="blue")
lines(1990:2031,c(predict(fit),farma[,1]),col="red")
polygon(c(2017:2031,rev(2017:2031)),c(farma[,2],rev(farma[,3])),border=NA,col=rgb(0,0,1,.2))
abline(h=0,lty=2)

Again, the confidence region is not based on inference related error, but on model uncertainty: we try to visualize where future observations might be with (say) 95% chance. Note we can also consider (why not?) a quadratic regression

fit=lm(ratio~poly(year,2),data=db)
s=summary(fit)$sigma
plot(db$year,db$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio")
abline(v=2016,lty=2,col="grey")
ndb=data.frame(year=2017:2031)
pf=predict(fit,newdata=ndb)
farma=cbind(pf,pf-1.96*s,pf+1.96*s)
lines(2017:2031,farma[,2],col="blue")
lines(2017:2031,farma[,3],col="blue")
lines(1990:2031,c(predict(fit),farma[,1]),col="red")
polygon(c(2017:2031,rev(2017:2031)),c(farma[,2],rev(farma[,3])),border=NA,col=rgb(0,0,1,.2))
abline(h=0,lty=2)

I am usually not a huge fan of those polynomial regression, but recently, I’ve seen that a lot in economic papers (like “if it’s not linear, add a squared version of the explanatory variable”, which is a rather odd strategy, I’ll publish some posts on that issue this year).

Here again, it might be more clever to consider a logarithmic transformation of the ratio, to insure that the ratio remains positive

fit=lm(log(ratio)~year,data=db)
s=summary(fit)$sigma
plot(db$year,db$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio")
abline(v=2016,lty=2,col="grey")
ndb=data.frame(year=2017:2031)
pf=predict(fit,newdata=ndb)
farma=cbind(exp(pf+s^2/2),exp(pf-1.96*s),exp(pf+1.96*s))
lines(2017:2031,farma[,2],col="blue")
lines(2017:2031,farma[,3],col="blue")
lines(1990:2031,c(exp(predict(fit)+s^2/2),farma[,1]),col="red")
polygon(c(2017:2031,rev(2017:2031)),c(farma[,2],rev(farma[,3])),border=NA,col=rgb(0,0,1,.2))
abline(h=0,lty=2)

Observe that future trend is mainly driven by the three latest observations, that were rather low (compared with older observations). What if we remove them?

dbna=db
db$ratio[25:27]=NA
fit=lm(ratio~1,data=dbna)
s=summary(fit)$sigma
plot(db$year,db$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio")
abline(v=2016-3,lty=2,col="grey")
ndb=data.frame(year=2014:2031)
pf=predict(fit,newdata=ndb)
farma=cbind(pf,pf-1.96*s,pf+1.96*s)
lines(2014:2031,farma[,2],col="blue")
lines(2014:2031,farma[,3],col="blue")
lines(1990:2031,c(predict(fit)[1:24],farma[,1]),col="red")
polygon(c(2014:2031,rev(2014:2031)),c(farma[,2],rev(farma[,3])),border=NA,col=rgb(0,0,1,.2))
abline(h=0,lty=2)

More funny, if we consider a quadratic regression, we obtain an increasing trend for the future

fit=lm(ratio~poly(year,2),data=dbna)
s=summary(fit)$sigma
plot(db$year,db$ratio,type="l",xlim=c(1990,2030),ylim=c(-.2,.7),xlab="year",ylab="ratio")
abline(v=2016-3,lty=2,col="grey")
ndb=data.frame(year=2014:2031)
pf=predict(fit,newdata=ndb)
farma=cbind(pf,pf-1.96*s,pf+1.96*s)
lines(2014:2031,farma[,2],col="blue")
lines(2014:2031,farma[,3],col="blue")
lines(1990:2031,c(predict(fit)[1:24],farma[,1]),col="red")
polygon(c(2014:2031,rev(2014:2031)),c(farma[,2],rev(farma[,3])),border=NA,col=rgb(0,0,1,.2))
abline(h=0,lty=2)

As we can see, it is rather difficult to get relevant prediction for the future, based on 25 observations…. If anyone has a suggestion, comments are open…

 

What is a Linear Trend, by the way?

I had a very strange discussion on twitter (yes, another one), about regression curves. I think it started with a tweet based on some xkcd picture (just for fun, because it was New Year’s Day)

There were comments on that picture, by econometricians, mainly about ‘significant’ trends when datasets are very noisy. And I mentioned a graph that I saw earlier, a couple of days ago

Let us reproduce that graph (Roger kindly sent me the dataset)

db=data.frame(year=1990:2016,
ratio=c(.23,.27,.32,.37,.22,.26,.29,.15,.40,.28,.14,.09,.24,.18,.29,.51,.13,.17,.25,.13,.21,.29,.25,.2,.15,.12,.12))
library(ggplot2)

The graph is here (with the same aesthetic conventions as Roger’s initial graph, i.e. using some sort of barplot)

ggplot(db, aes(year, ratio)) +
geom_bar(stat="identity") +
stat_smooth(method = "lm", se = FALSE)

My point was that we miss the ‘confidence band’ of the regression

In R, at least, it is quite natural to get (and actually, it is the default version of the graph function)

ggplot(db, aes(year, ratio)) +
geom_bar(stat="identity") +
stat_smooth(method = "lm", se = TRUE)

It is hard to claim that the ‘regression line’ is significant (in the sense “significantly non horizontal”). To be more specific, if we look at the output of the regression model, we get

summary(lm(ratio~year,data=db))

Coefficients:
Estimate    Std. Error t value Pr(>|t|)
(Intercept) 9.158531 4.549672  2.013 0.055 .
year       -0.004457 0.002271 -1.962 0.061 .
---
Signif. codes: 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(which is exactly what Roger used in his graph to plot his red straight line). The p-value of the estimator of the slope, in a linear regression model is here 6%. But I found Roger’s point puzzling

See also

First of all, let us get back to a more standard graph, with a scatterplot, and not bars,

ggplot(db, aes(year, ratio)) +
stat_smooth(method = "lm") +
geom_point()

Here, we observe points https://latex.codecogs.com/gif.latex?\{y_{1990},y_{1991},\cdots,y_{2016}\}. In order to draw that blue line, we assume (Econometrics 101, actually) that those observations are realizations of random variables https://latex.codecogs.com/gif.latex?\{Y_{1990},Y_{1991},\cdots,Y_{2016}\}. Randomness here does not come from a survey, or from ‘balls in an urn’. Randomness is because hurricanes and floods are themselves seen are realizations of random events. Yes, there might be measurement errors, but that’s not where randomness comes from (here). When we talk about ‘randomness’, it should be related to ‘model error’ i.e. the error we make if we consider a linear model (here), that is

https://latex.codecogs.com/gif.latex?Y_t=\beta_0+\beta_1t+\varepsilon_t

Even if observations are not obtained from balls in an urn, there is some kind of randomness here. Randomness means that we might have errors (random errors) around the estimated value (that is on the blue curve), https://latex.codecogs.com/gif.latex?y_t=\widehat{y}_t+\widehat{\varepsilon}_t. One might consider a nonlinear model to reduce the error,

ggplot(db, aes(year, ratio)) +
geom_point() +
geom_smooth()

but in the case, the danger is to overfit

So yes, when we fit a linear model, there is always some kind of randomness, and it is possible to get a ‘confidence band’, that will be very useful for predictions (e.g. for reinsurance purpose here).