16 Single Index Model

Updated: October 16, 2020

Copyright © Eric Zivot 2015

Outline

  1. Single Index Model and Portfolio Theory
    1. portfolio optimization using SI covariance matrix
      1. always pd, reduces number of estimated components in matrix.
      2. Compare results with sample covariance matrix
    2. Risk analysis of asset and portfolios: factor risk reports
  2. Statistical Properties of Least Squares Estimates
    1. Bias
    2. Standard errors
    3. Asymptotic distributions and confidence intervals
  3. Hypothesis Testing in the Single Index Model
    1. Tests for coefficients
    2. Tests for model assumptions: check to see if residual correlation matrix is diagonal!
    3. Diagnostics for covariance stationarity
    4. Rolling estimates from the single index model

16.1 Motivation

The CER model for monthly asset returns assumes that returns are jointly normally distributed with a constant mean vector and covariance matrix. This model captures many of the stylized facts for monthly asset returns. While the CER model captures the covariances and correlations between different asset returns it does not explain where these covariances and correlations come from. The single index model is an extension of the CER model that explains the covariance and correlation structure among asset returns as resulting from common exposures to an underlying market index. The intuition behind the single index model can be illustrated by examining some example monthly returns.

Example 16.1 (Monthly returns for single index model examples)

For the single index model examples in this chapter, consider the monthly adjusted closing prices on four Northwest stocks (with tickers in parentheses): Boeing (BA), Nordstrom (JWN), Microsoft (MSFT), Starbucks (SBUX). We will also use the monthly adjusted closing prices on the S&P 500 index (^GSPC) as the proxy for the market index. These prices are extracted from the IntroCompFinR package as follows:

# get data from IntroCompFin package
data(baDailyPrices, jwnDailyPrices, 
     msftDailyPrices, sbuxDailyPrices, sp500DailyPrices) 
baPrices = to.monthly(baDailyPrices, OHLC=FALSE) 
jwnPrices = to.monthly(jwnDailyPrices, OHLC=FALSE) 
msftPrices = to.monthly(msftDailyPrices, OHLC=FALSE) 
sbuxPrices = to.monthly(sbuxDailyPrices, OHLC=FALSE) 
sp500Prices = to.monthly(sp500DailyPrices, OHLC=FALSE) 
siPrices = merge(baPrices, jwnPrices, msftPrices, 
                 sbuxPrices, sp500Prices)

The data sample is from January, 1998 through May, 2012 and returns are simple returns:

smpl = "1998-01::2012-05" 
siPrices = siPrices[smpl] 
siRetS = na.omit(Return.calculate(siPrices, method="simple")) 
head(siRetS, n=3)
##               BA    JWN    MSFT   SBUX   SP500
## Feb 1998  0.1425 0.1298 0.13640 0.0825 0.07045
## Mar 1998 -0.0393 0.1121 0.05570 0.1438 0.04995
## Apr 1998 -0.0396 0.0262 0.00691 0.0629 0.00908

Prices are shown in Figure 16.1, created using:

plot.zoo(siPrices, main="", lwd=2, col="blue")
Monthly closing prices on four Northwest stocks and the S\&P 500 index.

Figure 16.1: Monthly closing prices on four Northwest stocks and the S&P 500 index.

The time plots of prices show some common movements among the stocks that are similar to movements of the S&P 500 index . During the dot-com boom-bust at the beginning of the sample, except for Starbucks, prices rise during the boom and fall during the bust. For all assets, prices rise during the five year boom period prior to the financial crisis, fall sharply after 2008 during the bust, and then rise afterward.

Figure 16.2 shows time plots of the monthly returns on each stock together with the return on the S&P 500 index, created using

par(mfrow=c(2,2))
plot.zoo(siRetS[,c("SP500","BA")], plot.type="single", 
         main="S&P 500 and Boeing", ylab="returns", 
         col=c("blue","orange"), lwd=c(2,2)) 
abline(h=0)
plot.zoo(siRetS[,c("SP500","JWN")], plot.type="single", 
         main="S&P 500 and Nordstom", ylab="returns", 
         col=c("blue","orange"), lwd=c(2,2)) 
abline(h=0)
plot.zoo(siRetS[,c("SP500","MSFT")], plot.type="single", 
         main="S&P 500 and Microsoft", ylab="returns", 
         col=c("blue","orange"), lwd=c(2,2)) 
abline(h=0)
plot.zoo(siRetS[,c("SP500","SBUX")], plot.type="single", 
         main="S&P 500 and Starbucks", ylab="returns", 
         col=c("blue","orange"), lwd=c(2,2)) 
abline(h=0) 
Monthly returns on four Northwest stocks. The orange line in each panel is the monthly return on the stock, and the blue line is the monthly return on the S\&P 500 index.

Figure 16.2: Monthly returns on four Northwest stocks. The orange line in each panel is the monthly return on the stock, and the blue line is the monthly return on the S&P 500 index.

par(mfrow=c(1,1))

Figure 16.2 shows that the individual stock returns are more volatile than the S&P 500 returns, and that the movement in stock returns (orange lines) tends to follow the movements in the S&P 500 returns indicating positive covariances and correlations.

The sample return covariance and correlation matrices are computed using:

covmatHat = cov(siRetS)
covmatHat
##            BA     JWN    MSFT    SBUX   SP500
## BA    0.00796 0.00334 0.00119 0.00245 0.00223
## JWN   0.00334 0.01492 0.00421 0.00534 0.00339
## MSFT  0.00119 0.00421 0.01030 0.00348 0.00298
## SBUX  0.00245 0.00534 0.00348 0.01191 0.00241
## SP500 0.00223 0.00339 0.00298 0.00241 0.00228
cormatHat = cov2cor(covmatHat)
cormatHat
##          BA   JWN  MSFT  SBUX SP500
## BA    1.000 0.306 0.131 0.251 0.524
## JWN   0.306 1.000 0.340 0.401 0.581
## MSFT  0.131 0.340 1.000 0.314 0.614
## SBUX  0.251 0.401 0.314 1.000 0.463
## SP500 0.524 0.581 0.614 0.463 1.000

The sample correlation matrix is visualized in Figure 16.3 using the corrplot function corrplot.mixed():

corrplot.mixed(cormatHat, upper="ellipse")
Sample correlation matrix of monthly returns.

Figure 16.3: Sample correlation matrix of monthly returns.

All returns are positively correlated and each stock return has the highest positive correlation with the S&P 500 index.

The positive covariance and correlation of each stock return with the market return can also be visualized with scatterplots as illustrated in Figure 16.4, created with:

siRetSmat = coredata(siRetS) 
par(mfrow=c(2,2)) 
plot(siRetSmat[, "SP500"], siRetSmat[, "BA"], 
     main="S&P 500 and Boeing", col="cornflowerblue",
     lwd=2, pch=16, cex=2, xlab="S&P 500", 
     ylab="BA", ylim=c(-0.4, 0.4)) 
abline(h=0,v=0)
plot(siRetSmat[, "SP500"], siRetSmat[, "JWN"], 
     main="S&P 500 and Nordstrom", col="cornflowerblue",
     lwd=2, pch=16, cex=2, xlab="S&P 500", 
     ylab="JWN", ylim=c(-0.4, 0.4)) 
abline(h=0,v=0)
plot(siRetSmat[, "SP500"], siRetSmat[, "MSFT"], 
     main="S&P 500 and Microsoft", col="cornflowerblue",
     lwd=2, pch=16, cex=2, xlab="S&P 500", 
     ylab="MSFT", ylim=c(-0.4, 0.4)) 
abline(h=0,v=0)
plot(siRetSmat[, "SP500"], siRetSmat[, "SBUX"], 
     main="S&P 500 and Starbucks", col="cornflowerblue",
     lwd=2, pch=16, cex=2, xlab="S&P 500", 
     ylab="SBUX", ylim=c(-0.4, 0.4)) 
abline(h=0,v=0)
Monthly return scatterplots of each stock vs. the S\&P 500 index.

Figure 16.4: Monthly return scatterplots of each stock vs. the S&P 500 index.

The scatterplots show that as the market return increases, the returns on each stock increase in a linear way.

16.2 William Sharpe’s SI Model

  • What is SI model used for? Extension of the CER model to capture the stylized fact of a common component in the returns of many assets.
    • Provides an explanation as to why assets are correlated with each other: share an exposure to a common source.
    • Provides a simplification of the covariance matrix of a large number of asset returns.
    • Provides additional intuition about risk reduction through diversification.
  • Example of a factor model for asset returns. Factor models are used heavily in academic theories of asset pricing and in industry for explaining asset returns, portfolio construction and risk analysis. Sharpe’s SI model is the most widely used.
  • Motivate by showing positive correlation between individual asset returns and market index (sp500)
  • Mostly used for simple returns; but can be used for cc returns for risk analysis.

Let Rit denote the simple return on asset i over the investment horizon between times t1 and t. Let RMt denote the simple return on a well diversified market index portfolio, such as the S&P 500 index. The single index (SI) model for Rit has the form:96 Rit=αi+βiRMt+εit,RMtiidN(0,σ2M),ϵitGWN(0,σ2ϵ,i),cov(RMt,ϵis)=0,foralltands,cov(ϵit,ϵjs)=0,forallijandalltands. The SI model (16.1) - (16.5) assumes that all individual asset returns, Rit, are covariance stationary and are a linear function of the market index return, RMt, and an independent error term, ϵit. The SI model is an important extension of the regression form of the CER model. In the SI model individual asset returns are explained by two distinct sources: (1) a common (to all assets) market-wide source RMt; (2) and asset specific source ϵit.

16.2.1 Economic interpretation of the SI model

16.2.1.1 Interpretation of βi

First, consider the interpretation of βi in (16.1). The coefficient βi is called the asset’s market exposure or market “beta”{Market beta}. It represents the slope coefficient in the linear relationship between Rit and RMt. Because it is assumed that RMt and ϵit are independent, βi can be interpreted as the partial derivative RitRMt=RMt(αi+βiRMt+εit)=βi. Then, for small changes in RMt denoted ΔRMt and holding εit fixed, we have the approximation ΔRitΔRMtβiΔRitβi×ΔRMt, Hence, for a given change in the market return βi determines the magnitude of the response of asset is return. The larger (smaller) is βi the larger (smaller) is the response of asset i to the movement in the market return.

The coefficient βi in (16.1) has another interpretation that is directly related to portfolio risk budgeting. In particular, from chapter 14 βi=cov(Rit,RMt)var(RMt)=σiMσ2M, and MCRσMi=βiσM. Here, we see that βi is the “beta” of asset i with respect to the market portfolio and so it is proportional to the marginal contribution of asset i to the volatility of the market portfolio. Therefore, assets with large (small) values of βi have large (small) contributions to the volatility of the market portfolio. In this regard, βi can be thought of as a measure of portfolio risk. Increasing allocations to assets with high (low) βi values will increase (decrease) portfolio risk (as measured by portfolio volatility). In particular, when βi=1 asset is percent contribution to the volatility of the market portfolio is its allocation weight. When βi>1 asset is percent contribution to the volatility of the market portfolio is greater than its allocation weight, and when βi<1 asset is percent contribution to the volatility of the market portfolio is less than its allocation weight.

The derivation of (16.6) is straightforward. Using (16.1) we can write cov(Rit,RMt)=cov(αi+βiRMt+εit,RMt)=cov(βiRMt,RMt)+cov(εit,RMt)=βivar(RMt)  (since cov(εit,RMt)=0)βi=cov(Rit,RMt)var(RMt).

16.2.1.2 Interpretation of RMt and ϵit

To aid in the interpretation of RMt and ϵit in (16.1) , re-write (16.3) as εit=RitαiβiRMt. In (16.3), we see that ϵit is the difference between asset is return, Rit, and the portion of asset is return that is explained by the market return, βiRMt, and the intercept, αi. We can think of RMt as capturing “market-wide” news at time t that is common to all assets, and βi captures the sensitivity or exposure of asset i to this market-wide news. An example of market-wide news is the release of information by the government about the national unemployment rate. If the news is good then we might expect RMt to increase because general business conditions are good. Different assets will respond differently to this news and this differential impact is captured by βi. Assets with positive values of βi will see their returns increase because of good market-wide news, and assets with negative values of βi will see their returns decrease because of this good news. Hence, the magnitude and direction of correlation between asset returns can be partially explained by their exposures to common market-wide news. Because ϵit is assumed to be independent of RMt and of ϵjt, we can think of ϵit as capturing specific news to asset i that is unrelated to market news or to specific news to any other asset j. Examples of asset-specific news are company earnings reports and reports about corporate restructuring (e.g., CEO resigning). Hence, specific news for asset i only effects the return on asset i and not the return on any other asset j.

The CER model does not distinguish between overall market and asset specific news and so allows the unexpected news shocks ϵit to be correlated across assets. Some news is common to all assets but some is specific to a given asset and the CER model error includes both types of news. In this regard, the CER model for an asset’s return is not a special case of the SI model when βi=0 because the SI model assumes that ϵit is uncorrelated across assets.

16.2.2 Statistical properties of returns in the SI model

In this sub-section we present and derive the statistical properties of returns in the SI model (16.1) - (16.5). We will derive two types of statistical properties: unconditional and conditional. The unconditional properties are based on unconditional or marginal distribution of returns. The conditional properties are based on the distribution of returns conditional on the value of the market return.

16.2.2.1 Unconditional properties

The unconditional properties of returns in the SI model (16.1) - (16.5) are: E[Rit]=μi=αi+βiμM,var(Rit)=σ2i=β2iσ2M+σ2ϵ,i,cov(Rit,Rjt)=σij=βiβjσ2M,cor(Rit,Rjt)=ρij=βiβjσ2M(β2iσ2M+σ2ϵ,i)(β2jσ2M+σ2ϵ,j),RitiidN(μi,σ2i)=N(αi+βiμM,β2iσ2M+σ2ϵ,i) The derivations of these properties are straightforward and are left as end-of-chapter exercises.

Property (16.8) shows that αi=μiβiμM. Hence, the intercept term αi can be interpreted as the average return on asset i that is in excess of the average return due to the market.

From property (16.9), an asset’s return variance is additively decomposed into two independent components: var(Rit)=σ2i=β2iσ2M+σ2ϵ,i(totalassetivariance=marketvariance+assetspecificvariance) Here, β2iσ2M is the contribution of the market index return RMt to the total variance of asset i, and σ2ϵ,i is the contribution of the asset specific component ϵit to the total return variance, respectively. If we divide both sides of (16.14) by σ2i we get 1=β2iσ2Mσ2i+σ2ϵ,iσ2i=R+(1R2), where R2=β2iσ2Mσ2i is the proportion of asset is variance that is explained by the variability of the market return RMt and 1R2=σ2ϵ,iσ2i is the proportion of asset is variance that is explained by the variability of the asset specific component ϵit.

  • Sharpe’s rule of thumb. A typical stock has R2=0.30. That is, 30% of an asset’s variability is explained by the market movements. - R2 can be interpreted as the fraction of risk that is non-diversifiable, 1R2 gives the fraction of risk that is diversifiable. Come back to this point after discussing the SI model and portfolios below.
  • This is an example of factor risk budgeting

Properties (16.10) and (16.11) show how assets are correlated in the SI model. In particular,

  • σij=0 if βi=0 or βj=0 or both. Assets i and j are uncorrelated if asset i or asset j or both do not respond to market news.
  • σij>0 if βi,βj>0 or βi,βj<0 . Assets i and j are positively correlated if both assets respond to market news in the same direction.
  • σij<0 if βi>0 and βj<0 or if βi<0 and βj>0. Assets i and j are negatively correlated if they respond to market news in opposite directions.

From (16.11), assets i and j are perfectly correlated (ρij=±1) only if σϵ,i=σϵ,j=0.

Property (16.12) shows that the distribution of asset returns in the SI model is normal with mean and variance given by () and (), respectively.

In summary, the unconditional properties of returns in the SI model are similar to the properties of returns in the CER model: Returns are covariance stationary with constant means, variances, and covariances. Returns on different assets can be contemporaneously correlated and all asset returns uncorrelated over time. The SI model puts more structure on the expected returns, variances and covariances than the CER model and this allows for a deeper understanding of the behavior of asset returns.

16.2.2.2 Conditional properties

The properties of returns in the SI model (16.1) - (16.5) conditional on RMt=rMt are: E[Rit|RMt=rMt]=αi+βirMt,var(Rit|RMt=rMt]=σ2ϵ,i,cov(Rit,Rjt|RMt=rMt]=0,cor(Rit,Rjt|RMt=rMt]=0,Rit|RMtiidN(αi+βirMt,σ2ϵ,i).

Recall, conditioning on a random variable means we observe its value. In the SI model, once we observe the market return two important things happen: (1) an asset’s return variance reduces to its asset specific variance; and (2) asset returns become uncorrelated.

16.2.3 SI model and portfolios

A nice feature of the SI model for asset returns is that it also holds for a portfolio of asset returns. This property follows because asset returns are a linear function of the market return. To illustrate, consider a two asset portfolio with investment weights x1and x2 where each asset return is explained by the SI model: R1t=α1+β1RMt+ϵ1t,R2t=α2+β2RMt+ϵ2t. Then the portfolio return is Rp,t=x1R1t+x2R2t=x1(α1+β1RMt+ε1t)+x2(α2+β2RMt+ε2t)=(x1α1+x2α2)+(x1β1+x2β2)RMt+(x1ε1t+x2ε2t)=αp+βpRMt+εp,t where αp=x1α1+x2α2, βp=x1β1+x2β2, and εp,t=x1ε1t+x2ε2t.

16.2.3.1 SI model and large portfolios

Consider an equally weighted portfolio of N assets, where N is a large number (e.g. N=500) whose returns are described by the SI model. Here, xi=1/N for i=1,,N. Then the portfolio return is Rp,t=Ni=1xiRit=Ni=1xi(αi+βiRMt+εit)=Ni=1xiαi+(Ni=1xiβi)RMt+Ni=1xiεit=1NNi=1αi+(1NNi=1βi)RMt+1NNi=1εit=ˉα+ˉβRMt+ˉεt, where ˉα=1NNi=1αi, ˉβ=1NNi=1βi and ˉεt=1NNi=1εit. Now, var(ˉεt)=var(1NNi=1εit)=1N2Ni=1var(εit)=1N(1NNi=1σ2ϵ,i)=1Nˉσ2 where ˉσ2=1NNi=1σ2ϵ,i is the average of the asset specific variances. For large N, 1Nˉσ20 and we have the Law of Large Numbers result ˉεt=1NNi=1εitE[εit]=0. As a result, in a large equally weighted portfolio we have the following:

  • Rp,tˉα+ˉβRMt: all non-market asset-specific risk is diversified away and only market risk remains.
  • var(Rp,t)=ˉβ2var(RMt)SD(Rp,t)=|ˉβ|×SD(RMt): portfolio volatility is proportional to market volatility where the factor of proportionality is the absolute value of portfolio beta.
  • R21: Approximately 100% of portfolio variance is due to market variance.
  • ˉβ1. A large equally weighted portfolio resembles the market portfolio (e.g., as proxied by the S&P 500 index) and so the beta of a well diversified portfolio will be close to the beta of the market portfolio which is one by definition.97

These results help us to understand the type of risk that gets diversified away and the type of risk that remains when we form diversified portfolios. Asset specific risk, which is uncorrelated across assets, gets diversified away whereas market risk, which is common to all assets, does not get diversified away.

  • (Relate to average covariance calculation from portfolio theory chapter).
  • Related to asset R2 discussed earlier. R2 of an asset shows the portion of risk that cannot be diversified away when forming portfolios.

16.2.4 The SI model in matrix notation

  • Need to emphasize that the SI model covariance matrix is always positive definite. This is an important result because it allows for the mean-variance analysis of very large portfolios.

For i=1,,N assets, stacking (16.1) gives the SI model in matrix notation (R1tRNt)=(α1αN)+(β1βN)RMt+(ϵ1tϵNt), or Rt=α+βRMt+ϵt. The unconditional statistical properties of returns (16.8), (16.9), (16.10) and (16.12) can be re-expressed using matrix notation as follows: E[Rt]=μ=α+βμM,var(Rt)=Σ=σ2Mββ+D,RtiidN(μ,Σ)=N(α+βμM,σ2Mββ+D), where D=var(ϵt)=(σ2ϵ,1000σ2ϵ,2000σ2ϵ,N)=diag(σ2ϵ,1,σ2ϵ,1,,σ2ϵ,N).

The derivation of the SI model covariance matrix (16.19) is var(Rt)=Σ=βvar(RMt)β+var(ϵt)=σ2Mββ+D, which uses the assumption that the market return RMt is uncorrelated will all asset specific error terms in ϵt.

It is useful to examine the SI covariance matrix (16.19) for a three asset portfolio. In this case, we have Rit=αi+βiRMt+εit, i=1,2,3σ2i=var(Rit)=β2iσ2M+σ2ε,iσij=cov(Rit,Rjt)=σ2Mβiβj The 3×3 covariance matrix is Σ=(σ21σ12σ13σ12σ22σ23σ13σ23σ23)=(β21σ2M+σ2ε,1σ2Mβ1β2σ2Mβ1β3σ2Mβ1β2β22σ2M+σ2ε,2σ2Mβ2β3σ2Mβ1β3σ2Mβ2β3β23σ2M+σ2ε,3)=σ2M(β21β1β2β1β3β1β2β22β2β3β1β3β2β3β23)+(σ2ε,1000σ2ε,2000σ2ε,3). The first matrix shows the return variance and covariance contributions due to the market returns, and the second matrix shows the contributions due to the asset specific errors. Define β=(β1,β2,β3). Then σ2Mββ=σ2M(β1β2β3)(β1β2β3)=σ2M(β21β1β2β1β3β1β2β22β2β3β1β3β2β3β23),D=diag(σ2ε,1,σ2ε,2,σ2ε,3)=(σ2ε,1000σ2ε,2000σ2ε,3), and so Σ=σ2Mββ+D.

The matrix form of the SI model (16.18) - (16.20) is useful for portfolio analysis. For example, consider a portfolio with N×1 weight vector x=(x1,,xN). Using (16.17), the SI model for the portfolio return Rp,t=xRt is Rp,t=x(α+βRMt+ϵt)=xα+xβRMt+xϵt=αp+βpRMt+ϵp,t, where αp=xα, βp=xβ and ϵp,t=xϵt.

16.3 Monte Carlo Simulation of the SI Model

To give a first step reality check for the SI model, consider simulating data from the SI model for a single asset. The steps to create a Monte Carlo simulation from the SI model are:

  1. Fix values for the SI model parameters α, β, σϵ, μM and σM
  2. Determine the number of simulated returns, T, to create.
  3. Use a computer random number generator to simulate T iid values of RMtN(μM,σ2M) and ϵtN(0,σ2ϵ), where RMt is simulated independently from ϵt.
  4. Create the simulated asset returns ˜Rt=α+β˜RM+˜ϵt for t=1,,T

The following example illustrates simulating SI model returns for Boeing.

Example 16.2 (Simulating monthly returns from the SI model for Boeing)

Consider simulating monthly returns on Boeing from the SI model where the S&P 500 index is used as the market index. The SI model parameters are calibrated from the actual monthly returns on Boeing and the S&P 500 index using sample statistics, as discussed in the next section, and are given by: αBA=0.005,βBA=0.98,σϵ,BA=0.08,μM=0.003,σM=0.048. These values are set in R using:

alpha.BA = 0.005 
beta.BA = 0.98 
sd.e.BA = 0.08 
mu.sp500 = 0.003 
sd.sp500 = 0.048

The simulated returns for the S&P 500 index and Boeing are created using:

n.sim = nrow(siRetS) 
set.seed(123) 
sp500.sim = rnorm(n.sim, mu.sp500, sd.sp500) 
e.BA.sim = rnorm(n.sim, 0, sd.e.BA) 
BA.sim = alpha.BA + beta.BA*sp500.sim + e.BA.sim 
BA.sim = xts(BA.sim, index(siRetS)) 
sp500.sim = xts(sp500.sim, index(siRetS)) 
colnames(BA.sim) = "BA.sim" 
colnames(sp500.sim) = "SP500.sim"
Simulated SI model monthly returns on the S\&P 500 index and Boeing.

Figure 16.5: Simulated SI model monthly returns on the S&P 500 index and Boeing.

Simulated SI model monthly returns on the S\&P 500 index and Boeing. The orange line is the equation $R_{BA,t}=\alpha_{BA}+\beta_{BA}R_{Mt}=0.005+0.98\times R_{Mt}.$

Figure 16.6: Simulated SI model monthly returns on the S&P 500 index and Boeing. The orange line is the equation RBA,t=αBA+βBARMt=0.005+0.98×RMt.

The simulate returns are illustrated in Figures 16.5 and 16.6. The time plot and scatterplot of the simulated returns on the S&P 500 index look much like the corresponding plots of the actual returns shows in the top left panels of Figures 16.2 and 16.4, respectively. Hence, the SI model for the S&P 500 index and Boeing passes the first step reality check. In Figure 16.6, the line αi+βiRMt=0.005+0.98RMt is shown in orange. This line represents the predictions of Boeing’s return given the S&P 500 return. For example, if RMt=0.05 then the predicted return for Boeing is RBA=0.005+0.98×(0.05)=0.054. The differences between the observed Boeing returns and (blue dots) and the predicted returns (orange line) are the asset specific error terms. These are the random asset specific news components. The standard deviation of these components, σϵ,BA=0.08, represents the typical magnitude of these components. Here, we would not be too surprised if the observed Boeing return is 0.08 above or below the orange line. How close the Boeing returns are to the predicted returns is determined by the R2 value given by (16.15). Here, σ2BA=β2BAσ2M+σ2ϵ,BA=(0.98)2(0.048)2+(0.08)2=0.00861 and so R2=(0.98)2(0.048)2/0.00861=0.257. Hence, 25.7% of the variability (risk) of Boeing returns are explained by the variability of the S&P 500 returns. As a result, 74.3% (1R2=0.743) of the variability of Boeing returns are due to the random asset specific terms.

16.4 Estimation of SI Model

Consider the SI model (16.1) - (16.5). The asset specific parameters to be estimated are αi, βi and σ2ϵ,i, (i=1,,N), and the market parameters to be estimated are μM and σ2M. These parameters can be estimated using the plug-in principle, linear regression, and maximum likelihood. All methods give essentially the same estimators for the SI model parameters.

16.4.1 Plug-in principle estimates

Let {(rit,rMt)}Tt=1 denote a sample of size T of observed returns on asset i and the market return which are assumed to be generated from the SI model. (16.1) - (16.12). Recall, the plug-in principle says to estimate population model parameters using appropriate sample statistics. For the market parameters, the plug-in principle estimates are the same as the CER model estimates ˆμM=1TTt=1rMt,ˆσ2M=1T1Tt=1(rMtˆμM)2. From (16.13) - (16.6) we see that αi and βi are functions of population parameters αi=μiβiμM,βi=cov(Rit,RMt)var(RMt)=σiMσ2M. The corresponding plug-in principle estimates are then: ˆαi=ˆμiˆβiˆμM,ˆβi=ˆσiMˆσ2M, where ˆμi=1TTt=1rit,ˆσiM=1T1Tt=1(ritˆμi)(rMtˆμM).

Given the plug-in principle estimates ˆαi and ˆβi, the plug-in principle estimate of ϵit is ˆεit=ritˆαiˆβirMt,t=1,,T. Using (16.23), the plug-in principle estimate of σ2ϵ,i is the sample variance of {ˆϵit}Tt=1 (adjusted for the number of degrees of freedom): ˆσ2ε,i=1T2Tt=1ˆε2t=1T2Tt=1(ritˆαiˆβirMt)2.

Plug-in principle estimates of R2 based on (16.15) can be computed using ˆR2=ˆβ2iˆσ2Mˆσ2i=1ˆσ2ε,iˆσ2i.

Example 16.3 (Computing plug-in principle estimators for SI model parameters)

Consider computing the plug-in principle estimates for αi, βi and σ2ϵ,i from the example data using the formulas (16.21), (16.22) and (16.24), respectively. First, extract the sample statistics ˆμi, ˆσiM, ˆμM, and ˆσ2M:

assetNames = colnames(siRetS)[1:4] 
muhat = colMeans(siRetS) 
sig2hat = diag(covmatHat)
covAssetsSp500 = covmatHat[assetNames, "SP500"]

Next, estimate ˆβi using

betaHat = covAssetsSp500/sig2hat["SP500"]
betaHat
##    BA   JWN  MSFT  SBUX 
## 0.978 1.485 1.303 1.057

Here, we see that ˆβBA and ˆβSBUX are very close to one and that ˆβJWN and ˆβMSFT are slightly bigger than one. Using the estimates of ˆβi and the sample statistics ˆμi and ˆμM the estimates for ˆαi are

alphaHat = muhat[assetNames] - betaHat*muhat["SP500"] 
alphaHat
##      BA     JWN    MSFT    SBUX 
## 0.00516 0.01231 0.00544 0.01785

All of the estimates of ˆαi are close to zero. The estimates of σ2ϵ,i can be computed using:

sig2eHat = rep(0, length(assetNames)) 
names(sig2eHat) = assetNames 
for (aName in assetNames) {   
     eHat = siRetS[, aName] - alphaHat[aName] - betaHat[aName]*siRetS[, "SP500"]   
     sig2eHat[aName] = crossprod(eHat)/(length(eHat) - 2) 
} 
sig2eHat
##      BA     JWN    MSFT    SBUX 
## 0.00581 0.00994 0.00646 0.00941

Lastly, the estimates of R2 can be computed using

R2 = 1 - sig2eHat/sig2hat[assetNames] 
R2
##    BA   JWN  MSFT  SBUX 
## 0.270 0.334 0.373 0.210

16.4.2 Least squares estimates

The SI model representation (16.1) shows that returns are a linear function of the market return and an asset specific error term Rit=αi+βiRMt+ϵit, here αi is the intercept and βi is the slope. Least squares regression is a method for estimating αi and βi by finding the “best fitting” line to the scatterplot of returns where Rit is on the vertical axis and RMt is on the horizontal axis.

{[}Insert Figure here{]} To be completed…

To see how the method of least squares determines the “best fitting” line, consider the scatterplot of the sample returns on Boeing and the S&P 500 index illustrated in Figure xxx. In the figure, the black line is a fitted line with initial guess ˆαBA=0 and ˆβBA=0.5. The differences between the observed returns (blue dots) and the values on the fitted line are the estimated errors ˆϵBA,t=rBA,tˆαBAˆβBArMt=rBA,t00.5×RMt. Some estimated errors are big and some are small. The overall fit of the line can be measured using a statistic based on all t=1,,T of the estimated errors. A natural choice is the sum of the errors Tt=1ˆϵt. However, this choice can be misleading due to the canceling out of large positive and negative errors. To avoid this problem, it is better to measure the overall fit using Tt=1^|ϵt| or Tt=1ˆϵ2t. Then the best fitting line can be determined by finding the intercept and slope values that minimize Tt=1^|ϵt| or Tt=1ˆϵ2t.

The method of least squares regression defines the “best fitting” line by finding the intercept and slope values that minimize the sum of squared errors SSE(ˆαi,ˆβi)=Tt=1ˆϵ2it=Tt=1(ritˆαiˆβirMt)2. Because SSE(ˆα,ˆβ) is a continuous and differential function of ˆαi and ˆβi, the minimizing values of ˆαi and ˆβi can be determined using simple calculus. The first order conditions for a minimum are: 0=SSE(ˆαi,ˆβi)ˆαi=2Tt=1(ritˆαiˆβirMt)=2Tt=1ˆεit,0=SSE(ˆαi,ˆβi)ˆβi=2Tt=1(ritˆαiˆβirMt)rMt=2Tt=1ˆεitrMt. These are two linear equations in two unknowns which can be re-expressed as ˆαiT+ˆβiTt=1rMt=Tt=1rit,ˆαiTt=1rMt+ˆβiTt=1r2Mt=Tt=1ritrMt. Using matrix algebra, we can write these equations as: (TTt=1rMtTt=1rMtTt=1r2Mt)(ˆαiˆβi)=(Tt=1ritTt=1ritrMt), which is of the form Ax=b with A=(TTt=1rMtTt=1rMtTt=1r2Mt),x=(ˆαiˆβi),b=(Tt=1ritTt=1ritrMt). Hence, we can determine ˆαi and ˆβi by solving x=A1b. Now,98 A1=1det(A)(Tt=1r2MtTt=1rMtTt=1rMtT),det(A)=TTt=1r2Mt(Tt=1rMt)2=TTt=1(rMtˆμM)2,ˆμM=1TTt=1rMt. Consequently, (ˆαiˆβi)=1TTt=1(rMtˆμM)2(Tt=1r2MtTt=1rMtTt=1rMtT)(Tt=1ritTt=1ritrMt) and so ˆαi=Tt=1r2MtTt=1ritTt=1r2MtTt=1ritrMtTTt=1(rMtˆμM)2,ˆβi=TTt=1ritrMtTt=1rMtTt=1ritTTt=1(rMtˆμM)2. After a little bit of algebra (see end-of-chapter exercises) it can be shown that ˆαi=ˆμiˆβiˆμM,ˆβi=ˆσiMˆσ2M, which are plug-in estimates for ˆαi and ˆβi determined earlier. Hence, the least squares estimates of ˆαi and ˆβi are identical to the plug-in estimates.

The solution for the least squares estimates in (16.30) has an elegant representation using matrix algebra. To see this, define the T×1 vectors ri=(ri1,,riT), rM=(rM1,,rMT) and 1=(1,,1). Then we can re-write (16.29) as (111rM1rMrMrM)(ˆαiˆβi)=(1rirMri) or XXˆγi=Xri where X=(1rM)is a T×2 matrix and ˆγ=(ˆαi,ˆβi). Provided XX is invertible, solving (16.33) for ˆγi gives the least squares estimates in matrix form: ˆγi=(XX)1Xri. The matrix form solution (16.34) is especially convenient for computation in R.

The least squares estimates of ϵt, σ2ϵ,i and R2 are the same as the plug-in estimators (16.23), (16.24) and (16.25), respectively. In the context of least squares estimation, the estimate ˆσϵ,i=ˆσ2ϵ,i is called the standard error of the regression and measures the typical magnitude of ˆϵt (difference between observed return and fitted regression line).

16.4.3 Simple linear regression in R

  • don’t do regression examples until statistical theory is discussed
  • computing least squares estimates using matrix algebra formulas
  • computing least squares estimates using lm()
    • See discussion from my regression chapter in MFTSR
    • describe structure of lm() function, extractor and method functions
  • Do analysis of example data To be completed…

16.4.4 Maximum likelihood estimates

The SI model parameters can also be estimated using the method of maximum likelihood, which was introduced in chapter (GARCH estimation chapter). To construct the likelihood function, we use property () of the SI model that conditional on RMt=rMt the distribution of Rit is normal with mean αi+βirMt and variance σ2ϵ,i. The pdf of Rit|RMt=rmt is then f(rit|rmt,θi)=(2πσ2ε,i)1/2exp(12σ2ε,i(ritαi+βirMt)2),t=1,,T, where θi=(αi,βi,σ2ϵ,i). Given a sample {(rit,rMt)}Tt=1={ri,rM} of observed returns on asset i and the market return, which are assumed to be generated from the SI model, the joint density of asset returns given the market returns is f(ri|rm)=Tt=1(2πσ2ε,i)1/2exp(12σ2ε,i(ritαi+βirMt)2)=(2πσ2ε,i)T/2exp(12σ2ε,iTt=1(ritαi+βirMt)2)=(2πσ2ε,i)T/2exp(12σ2ε,iSSE(αi,βi)). where SSE(αi,βi) is the sum of squared residuals (16.26) used to determine the least squares estimates. The log-likelihood function for θi is then lnL(θi|ri,rM)=T2ln(2π)T2ln(σ2ε,i)12σ2ε,iSSE(αi,βi). From (16.35), it can be seen that the values of αi and βi that maximize the log-likelihood are the values that minimize SSE(αi,βi). Hence, the ML estimates of αi and βi are the least squares estimates.

To find the ML estimate for σ2ϵ,i, plug the ML estimates of αi and βi into (16.35) giving lnL(ˆαi,ˆβi,σ2ϵ,i|ri,rM)=T2ln(2π)T2ln(σ2ε,i)12σ2ε,iSSE(ˆαi,ˆβi). Maximization with respect to σ2ϵ,i gives the first order condition lnL(ˆαi,ˆβi,σ2ϵ,i|ri,rM)σ2ϵ,i=T2ˆσ2ϵ,i+12(ˆσ2ϵ,i)2SSE(ˆαi,ˆβi)=0. Solving for ˆσ2ϵ,i gives the ML estimate for σ2ϵ,i: ˆσ2ϵ,i=SSE(ˆαi,ˆβi)T=1TTt=1ˆϵ2t, which is plug-in principle estimate (16.24) not adjusted for degrees-of-freedom.

16.5 Statistical Properties of SI Model Estimates

To determine the statistical properties of the plug-in principle/least squares/ML estimators ˆαi, ˆβi and ˆσ2ϵ,i in the SI model, we treat them as functions of the random variables {(Ri,t,RMt)}Tt=1 where Rt and RMt are assumed to be generated by the SI model (16.1) - (16.5).

16.5.1 Bias

In the SI model, the estimators ˆαi, ˆβi and ˆσ2ϵ,i (with degrees-of-freedom adjustment) are unbiased: E[ˆαi]=αi,E[ˆβi]=βi,E[ˆσ2ϵ,i]=σ2ϵ,i. To shows that ˆαi and ˆβi are unbiased, it is useful to consider the SI model for asset i in matrix form for t=1,,T: Ri=αi1+βiRM+ϵi=Xγi+ϵi, where Ri=(Ri1,,RiT), RM=(RM1,,RMT), ϵi=(ϵi1,,ϵiT), 1=(1,,1), X=(1RM), and γi=(αi,βi). The estimator for γi is ˆγi=(XX)1XRi. Pluging in Ri=Xγi+ϵi gives ˆγi=(XX)1X(Xγi+ϵi)=(XX)1XXγi+(XX)1Xϵi=γi+(XX)1Xϵi. Then E[ˆγi]=γi+E[(XX)1Xϵi]=γi+E[(XX)1X]E[ϵi](becauseϵitisindependentofRMt)=γi(becauseE[ϵi]=0). The derivation of E[ˆσ2ϵ,i]=σ2ϵ,i is beyond the scope of this book and can be found in graduate econometrics textbooks such as Hayashi (1980).

16.5.2 Precision

Under the assumptions of the SI model, analytic formulas for estimates of the standard errors for ˆαi and ˆβi are given by: ^se(ˆαi)ˆσε,iTˆσ2M1TTt=1r2Mt,^se(ˆβi)ˆσε,iTˆσ2M, where “” denotes an approximation based on the CLT that gets more accurate the larger the sample size. Remarks:

  • ^se(ˆαi) and ^se(ˆβi) are smaller the smaller is ˆσε,i. That is, the closer are returns to the fitted regression line the smaller are the estimation errors in ˆαi and ˆβi.
  • ^se(ˆβi) is smaller the larger is ˆσ2M. That is, the greater the variability in the market return RMt the smaller is the estimation error in the estimated slope coefficient ˆβi. This is illustrated in Figure xxx. The left panel shows a data sample from the SI model with a small value of ˆσ2M and the right panel shows a sample with a large value of ˆσ2M. The right panel shows that the high variability in RMt makes it easier to identify the slope of the line.
  • Both ^se(ˆαi) and ^se(ˆβi) go to zero as the sample size, T, gets large. Since ˆαi and ˆβi are unbiased estimators, this implies that they are also consistent estimators. That is, they converge to the true values αi and βi, respectively, as T.
  • In R, the standard error values (16.36) and (16.37) are computed using the summary() function on an “lm” object.

There are no easy formulas for the estimated standard errors for ˆσ2ϵ,i, ˆσε,i and ˆR2. Estimated standard errors for these estimators, however, can be easily computed using the bootstrap.

Example 16.4 (Computing estimated standard errors for ˆαi and ˆβi in R)

to be completed

16.5.3 Sampling distribution and confidence intervals.

Using arguments based on the CLT, it can be shown that for large enough T the estimators ˆαi and ˆβi are approximately normally distributed: ˆαiN(αi,^se(ˆαi)2),ˆβiN(βi,^se(ˆβi)2), where ^se(ˆαi) and ^se(ˆβi) are given by (16.36) and (16.37), respectively.

16.6 Further Reading: Single Index Model

To be completed …

16.7 Problems: Single Index Model

16.8 Solutions to Selected Problems

{Ruppert} Ruppert, D.A.. , Springer-Verlag.

{MartinSchererYollin}Martin, R.D., Scherer, B., and Yollin, G. (2016).


  1. The single index model is also called the* market model* or the single factor model.↩︎

  2. The S&P 500 index is a value weighted index and so is not equal to an equally weighted portfolio of the stocks in the S&P 500 index. However, since most stocks in the S&P 500 index have large market capitalizations the values weights are not too different from equal weights.↩︎

  3. The matrix A is invertible provided det(A)0. This requires the sample variance of RMt to be non-zero. ↩︎