Financial Econometrics Short Course Lecture 3 Multifactor Pricing Model

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1 Financial Econometrics Short Course Lecture 3 Multifactor Pricing Model Oliver Linton obl20@cam.ac.uk Renmin University Financial Econometrics Short Course Lecture 3 MultifactorRenmin Pricing University Model 1 / 62

2 The Population Linear Factor model Assume that asset returns are generated by the linear factor model (LFM). For asset i {1,..., N}, such that R i = α i + K k=1 b ik f k + ε i f 1,..., f K are "common factors" b ik are factor loadings (sensitivity of the return on asset i to factor k) ε i denotes idiosyncratic risk (as opposed to systematic risk of the economy-wide factors), and at least E ε i = 0 ; var(ε i ) = σ 2 εi <. cov(f k, ε i ) = 0. Financial Econometrics Short Course Lecture 3 MultifactorRenmin Pricing University Model 2 / 62

3 A (unit cost) portfolio is w 1,..., w N such that N w i = 1 i=1 An (zero cost) arbitrage portfolio is w 1,..., w N such that N w i = 0 i=1 A portfolio that is hedged against factor risk (e.g., market neutral) is such that N w i b ik = 0 for all k i=1 A well-diversified portfolio is such that N wi 2 0 (as N ) i=1 Financial Econometrics Short Course Lecture 3 MultifactorRenmin Pricing University Model 3 / 62

4 Arbitrage Pricing Theory Large N economy. Chamberlain and Rothschild (1983, Econometrica). Consider the well diversified, arbitrage portfolio p, that is hedged against factor risk R p = = N w i R i i=1 N i=1 w i α i + N w i α i i=1 0 K N k=1 i=1 w i b ik f k + N i=1 w i ε i otherwise you make money for nothing. Financial Econometrics Short Course Lecture 3 MultifactorRenmin Pricing University Model 4 / 62

5 Let B be the N K matrix of factor loadings. An arbitrage portfolio p that is hedged against factor risk satisfies w (i, B) = 0, i.e., w is in the null space of (i, B) (N (K + 1) matrix). Many such portfolios exist by standard linear algebra Since the vector α is orthogonal to w it must lie in the space spanned by (i, B), (using (A ) = A). Therefore for some constants ρ, θ 1,..., θ k we have α i = ρ + K k=1 b ik θ k. Financial Econometrics Short Course Lecture 3 MultifactorRenmin Pricing University Model 5 / 62

6 It follows that for any asset i ER i = α i + K k=1 b ik Ef k = ρ + K k=1 b ik (Ef k + θ k ) i.e., (ER 1,..., ER N ) span(i, B) - there exists constants λ 0, λ 1,..., λ K such that ER i = λ 0 + K k=1 b ik λ k, where λ j = Ef j + θ j are risk premia associated with the j factor. Financial Econometrics Short Course Lecture 3 MultifactorRenmin Pricing University Model 6 / 62

7 If the assets are traded factors and there is a risk free rate λ 0 = R f and λ k = Ef k R f ER i R f = K b ik (Ef k R f ) k=1 The CAPM corresponds to the case where K = 1 and f 1 is the return on the market portfolio. The APT theory doesnt say what the factors are. Financial Econometrics Short Course Lecture 3 MultifactorRenmin Pricing University Model 7 / 62

8 When does diversification work? First, averaging reduces variance provided correlation is not perfect. Consider a portfolio with return R w = wx + (1 w)y, w R whose variance is V (w) = w 2 σ 2 X + (1 w) 2 σ 2 Y + 2w(1 w)ρ XY σ X σ Y. Clearly (setting w = 0 or w = 1) min w V (w) min{σ2 X, σ2 Y } max{σ 2 X, σ2 Y } max V (w) w with strict inequality if and only if ρ XY = +1. Suppose that σ 2 X = σ2 Y = 1, then (solving dv (w)/dw = 0) we have w opt = 1 2 ; V (w opt ) = 1 2 (1 + ρ XY ) 1 Financial Econometrics Short Course Lecture 3 MultifactorRenmin Pricing University Model 8 / 62

9 Suppose that E (εε ) = Ω ε ; Σ f = E (ff ). Portfolio variance is the sum of two terms var(w R) = common idiosyncratic {}}{{}}{ w BΣ f B w + w Ω ε w. Key result: Can show that with many assets, portfolio can have zero (idiosyncratic) variance under some very weak conditions. Financial Econometrics Short Course Lecture 3 MultifactorRenmin Pricing University Model 9 / 62

10 Definition We say that the idiosyncratic error is diversifiable if lim N w w = 0 implies that lim N w Ω ε w = 0. We consider the diagonal case first. Suppose that Ω ε = diag{σ 2 1,..., σ2 N }. Then We have var(w ε) = w Ω ε w = N wi 2 σ 2 i i=1 max 1 i N σ2 i N wi 2 σ 2 i. i=1 N wi 2, i=1 and it suffi ces that σ 2 i c < for all i. So if all the variances are bounded then clearly diversification works for well balanced portfolios. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 10 / 62

11 Note that w w = N i=1 wi 2 is the Herfindahl index of the weight vector. Example Equal weighting with w i = 1/N satisfies the property lim N w w = 0 and in fact w w = 1/N 0 quite fast. For this sequence of weights, lim N w Ω ε w = 0 can hold under the weaker condition that e.g, σ 2 i = i α, 0 α < 1. 1 N max 1 i N σ2 i 0, Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 11 / 62

12 The assumption of uncorrelated errors is considered a bit strong and is stronger than is needed for the diversification property to hold as we next show. Theorem Suppose that Ω ε = D 1/2 ΨD 1/2, where D = diag{σ 1,..., σ N } is a diagonal matrix with bounded entries and λ max (Ψ) c < (as N ). Then, lim N w w = 0 implies that lim w Ω ε w = 0. N The proof of this result is immediate. Letting w = D 1/2 w, we have w Ω ε w = w w w Ψ w w w λ max(ψ) N i=1 w 2 i σ 2 i 0 as N. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 12 / 62

13 There are several models of cross sectional correlation that are in use. First, Connor and Koraczyck (1983) assumed that there is some ordering of the cross section such that the process ε i is alpha mixing (Chapter 2). Suppose that cov(ε i, ε j ) = ρ j i for some ρ with ρ < 1. Then 1 ρ ρ 2 ρ N 1. ρ Ω ε = ρ ρ ρ 2,.... ρ 1 ρ ρ N 1 ρ 2 ρ 1 which has bounded eigenvalues. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 13 / 62

14 In fact 1 N i Ω ε i = 1 [ N + 2(N 1)ρ + 2(N 2)ρ ] N = 1 + ρ 1 ρ. j=1 ρ j So that the equally weighted portfolio has zero variance for large N. So diversification eliminates all risk. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 14 / 62

15 In practice, the raw correlations are high. In Figure xx below we show the distribution of corr(r i, R j ) of S&P500 daily returns over the period Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 15 / 62

16 By contrast the idiosyncratic errors have smaller correlations but they are still significant. In Figure xx we show the corr(ε i, ε j ) (market model residuals) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 16 / 62

17 In conclusion, the diversification arguments can hold more generally even in the presence of correlation between the error terms and large variance terms. If diversification works we obtain that the portfolio variance is dominated by the common components, i.e., w Ω ε w 0 = var(w R) common {}}{ w BΩ K B w. We next consider some empirical approaches to measuring diversification. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 17 / 62

18 Solniks diversification curve Given a set of assets the variance of the equally weighted portfolio is ( ) N 1 var N R i = σ 2 i + σ ij i=1 σ 2 i = 1 N N var(r i ) ; σ ij = i=1 2 N(N 1) N j=i+1 N 1 cov(r i, R j ) i=1 Cross covariances more important than own variances. Want to show how this varies with N by taking a subsample of assets, but which subsample? Definition (Solnik) Sample variance S(m) of a randomly selected equally weighted portfolio of m assets for m = 1, 2,..., N Can show that S(m) = 1 m σ2 i + ( 1 1 ) σ ij σ ij as m m Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 18 / 62

19 (Log of) Solniks curve for 5 subperiods Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 19 / 62

20 Global Minimum Variance portfolio Suppose that Σ is the covariance matrix of returns. For weights w mv with i = (1,..., 1) R m we have w GMV = Σ 1 i i Σ 1 i and we achieve variance σ 2 GMV (m) = 1 i Σ 1 i Compute this for assets R 1,..., R L with L = 2,..., N Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 20 / 62

21 Log of σ 2 GMV for five different subperiods Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 21 / 62

22 When are factors Pervasive? A key assumption in the sequel is that all the included factors are pervasive, which is to say they each affect many assets returns. It is saying that all the factors play an important role in explaining the returns of the assets, essentially, nearly all assets are affected in some way by the factors. For example when K = 1 we might just require that the number of β i = 0, denoted r, is a large fraction of the sample. We next give a formal definition. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 22 / 62

23 Definition We say that the factors are strongly influential or pervasive when 1 N B B M, where M is a strictly positive definite matrix. We have the following result Theorem For all weighting sequences such that B w = 0 and such that i w = 1 and lim N w w = 0, we have lim w BB w λ min (M) > 0. N (wlog set Σ f = I K ) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 23 / 62

24 If the diversification condition is also satisfied, then this says that for such portfolios, the variance of returns is dominated by the common factor, and this term cannot be eliminated. The common component is not diversifiable. Of course there are also portfolios for which B w = 0, and these have already been introduced in Chapter xx, and are called hedge portfolios. It is a classical result in linear algebra that the subspace of R N of hedge portfolios is of dimension N K, and so its complement is of dimension K. The APT tells us that those well diversified hedge portfolios do not make any money over their cost. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 24 / 62

25 There are some concerns about whether the condition holds or at least whether all factors are pervasive, when there are multiple factors. If this condition is not satisfied, then some of the approximations we employ below are no longer valid. Definition (Onatskiy (2012)) We say that the factors are weakly influential when where D is a diagonal matrix. B B D, Under this condition, the contribution of the common components to the variance of the portfolio is of the same magnitude as the idiosyncratic components. This will affect some econometric testing and estimation discussed below. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 25 / 62

26 For example, suppose that B are actually observed (see characteristic based factor model section below) and f t are estimated by a cross sectional regression of returns on B, then the necessary condition for consistency (under iid errors ref) is that as N which is intermediate. λ min (B B), We may find that some factors are strongly influential, whereas others are not, so that in the multifactor case the matrix B may be rank deficient. One way of modelling this is to say that B = (B 1, B 2 ), where B 1 are strong factors, whereas B 2 are weakish factors that are small in the sense that T B 2 satisfies the strong factor condition but B 2 does not, Bryzgalova (2015). Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 26 / 62

27 The Econometric Model We suppose we observe the panel of returns {Z it, i = 1,..., N, t = 1,..., T }. The K-factor model (for returns or excess returns) is Z it = α i + K j=1 b ij f jt + ε it Z t = α + Bf t + ε t, where ε it is an (iid over t) idiosyncratic error term with cov(ε t, f t ) = 0 ; E (ε t ε t ) = Ω ε Can be justified from multivariate normality of returns and the factors. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 27 / 62

28 Different types of Factor Models There are three different types of factor models 1 Observable factor models (b ij are unknown quantities) 1 The factors are returns to traded portfolios (specifically, the returns on portfolios formed on the basis of security characteristic such as size, B/M) 2 The factors are macro variables such as yield spread etc. 2 Statistical factor models (f jt and b ij are unknown quantities) 3 Characteristic based models (b ij are observed characteristics such as industry or country and f jt are unknown quantities) In each case there are slight differences in cases where there is a risk free asset and in cases where there are not. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 28 / 62

29 Multivariate Tests of the Multibeta Pricing Model with Observable factors Multivariate tests are very similar to those for the CAPM, but with β replaced with the matrix B. First suppose that there is a risk-free asset and that the factor returns F are observable. The log likelihood function of the data Z 1,..., Z T conditional on Z K 1,..., Z KT is l(α, B, Ω ε ) = c T 2 log det Ω ε 1 2 T t=1 (Z t α BZ Kt ) Ω 1 ε (Z t α BZ Kt ). Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 29 / 62

30 Letting µ K = 1 T Ω K = 1 T Ω ZK = 1 T The unrestricted MLE of α, B, Ω ε are: T t=1 Z Kt ; µ = 1 T T Z t t=1 T (Z Kt µ K ) (Z Kt µ K ) t=1 T (Z t µ) (Z Kt µ K ) t=1 α = µ B µ K ; B = Ω ZK Ω 1 K. Ω ε = 1 T T t=1 (Z t α BZ Kt )(Z t α BZ Kt ) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 30 / 62

31 The multivariate tests of a = 0 are similar to those for the CAPM. The F test F = (T N K ) (1 + µ K N Ω 1 K µ K ) 1 α Ω 1 ε α is exactly distributed as F (N, T N K ) under the normality assumption. In the absence of normality the statistic is asymptotically (sample size T gets large) chi-squared with N degrees of freedom. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 31 / 62

32 Multifactor Pricing Tests with Macro Factors and Risk Free Rate Suppose instead that the observed factors are macroeconomic shocks rather than portfolio returns. Then the expected returns associated with the factors have to be estimated as additional free parameters. We have R = α + Bf + ε E [f ] = µ K It follows that the APT restrictions are α = Bγ for some γ R K rank n K {( }} ){ I n B(B B) 1 B α = 0. Likelihood ratio test easiest here. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 32 / 62

33 Fama and French Factors Fama and French (1993) "measure" f directly. They construct (6) double sorted portfolios formed on 2 size and 3 book to market. That is, first sort the stocks according to their size (at the given date) and divide into two: large size and small size. Then sort each of these groups according to the BTM and divide each into three further groups. The sorting could equally be done the other way round, and in both ways one obtains six groups of stocks BTM/Size high/large high/small 2 medium/large medium/small 3 low/large low/small Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 33 / 62

34 These are then equally weighted in within each group to produce a portfolio that measures large size and high BTM (denoted (1,1)), etc. The size factor return is proxied by the difference in return between a portfolio of low-capitalization stocks and a portfolio of high-capitalization stocks, adjusted to have roughly equal book-to-price ratios (SMB) 1 {(1, 1) (1, 2) + (2, 1) (2, 2) + (3, 1) (3, 2)}. 3 Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 34 / 62

35 The value factor is proxied by the difference in return between a portfolio of high book-to-price stocks and a portfolio of low book-to-price stocks, adjusted to have roughly equal capitalization (HML) The market factor return is proxied by the excess return to a value-weighted market index (MKT ) Data on the factors is available from Ken French web site Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 35 / 62

36 Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 36 / 62

37 Then form 25 portfolios (on size and value characteristics too!) R pt = α p + b p,smb SMB t + b p,hml HML t + b p,mkt MKT t + ε pt, p = 1,.. They do not reject the APT restrictions in their sample (although they do reject CAPM where SMB and HML are dropped). They explain well the size and value anomalies of the CAPM. Subsequently, it all went pear shaped and APT rejected. Additional factors have been proposed in the literature Momentum factor, Carhart (1997). Own-volatility factor, Goyal and Santa Clara (2003). Liquidity, Amihud and Mendelson (1986), Pastor and Stambaugh (2003) Fama and French 5 factor model (2015) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 37 / 62

38 Explicit Characteristics-Based Models Rosenberg (1974) considered the multifactor regression model where f are unknown "parameters" and B is related to observable (time invariant) stock characteristics such as: industry/country (dummy variables). Other possible characteristics include: size and value. For example, suppose that β i = Dx i, where x i is an observed J 1 vector of characteristics and D is a K J matrix of unknown parameters. Substituing in we obtain Z t = x i D f t + ε t = x i f t + ε t, where ft = D f t is a J 1 vector of characteristic specific factors. It follows that E (Z t X ) = X µ f var (Z t X ) = X Ω f X + Ω ε, where X is the N J matrix of observed characteristics and µ f and Ω f are the mean and variance of the factors. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 38 / 62

39 Connor, Hagmann, and Linton (2012) generalize this idea to allow the beta functions to depend in an unknown fashion on observed characteristics R it = f ut + J j=1 β j (X ji )f jt + ε it They proposed a nonparametric method for estimating the beta functions and then cross-sectional regression to get the factors. They find that the estimated β j have nonlinear shapes. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 39 / 62

40 Macroeconomics factors (Chan et al. (1985) and Chen et al. (1986)). R it = α i + b i f t + ε it f t = m t E t 1 m t where m t are (nonstationary) macroeconomic variables. Monthly data The percentage change in industrial production (led by one period) 2 A measure of unexpected inflation 3 The change in expected inflation 4 The difference in returns on low-grade (Baa and under) corporate bonds and long term government bonds (junk spread) 5 The difference in returns on long-term government bonds and short term Tbills (Term spread) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 40 / 62

41 They estimate b i by time series regressions and then do cross sectional regressions on b i to estimate factor risk premia; 20 portfolios are used on the basis of firm size at the beginning of the period They find that average factor risk premia are statistically significand over the entire sample period for industrial production, unexpected inflation, and junk. They also include a market return but find it is not significant Recent developments: Macroeconomic policy endogeneity to level of asset prices. Event study approach. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 41 / 62

42 Statistical Factor Models: Identifying the Factors in Asset Returns Definition The statistical factor model for observed returns where neither B nor F are observed. For each time period write R t = µ + Bf t + ε t., Definition Strict factor structure - idiosyncratic error is cross-sectionally uncorrelated so E ε t ε t = Ω ε = D, where D is a diagonal matrix (containing the idiosyncratic variances). Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 42 / 62

43 We treat B as fixed quantities and f t as random variables where Ef t = 0 without loss of generality. We also assume cov(f t, ε t ) = 0. Then where Σ f = var(f t ). Ω N N = var(r t ) = BΣ f B + D, The RHS has NK + K (K + 1)/2 + N parameters which is less than N(N + 1)/2 on LHS. A big reduction in dimensionality However, in this case where factors are unknown there is an identification issue. Can t uniquely identify (B, Σ f ) or (B, f t ) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 43 / 62

44 Identification Issue One can write for any nonsingular matrix L, f t = Lf t ; Σ f = LΣ f L so that Bf t = BL 1 Lf t = B f t BΣ f B = B Σ f B One solution is to restrict Σ f = I K. In this case, Ω = BB + D. Then B, D are unique (B upto sign) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 44 / 62

45 Can be estimated by maximum likelihood factor analysis provided N << T (small N and large T ). l(µ, B, D) = c T 2 log det Ω(B, D) 1 2 T t=1 The MLE for µ is still the sample average µ = 1 T We then substitute in to obtain T R t. t=1 (R t µ) Ω(B, D) 1 (R t µ). l( µ, B, D) = c T 2 log det Ω(B, D) 1 2 tr ( ΩΩ(B, D) 1) Ω = 1 T T t=1 (R t µ)(r t µ) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 45 / 62

46 Total of NK + N parameters in Ω(B, D). Solve nonlinear first order conditions (for i = 1,..., N and k = 1,..., K ) l ( µ, B, D) b ik = 0 l ( µ, B, D) = 0 σ 2 εi Iterative nonlinear procedure such as Newton Raphson Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 46 / 62

47 Given B, D, the period by period factor realizations can be estimated by cross-sectional regression, i.e., OLS or GLS f t = ( B B) 1 B (R t µ) f t = ( B D 1 B) 1 B D 1 (R t µ) Consistency requires B B (pervasive factors). Estimated factors. Replacing B with B creates an errors in variables problem, affects standard errors at least Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 47 / 62

48 Factor Mimicking Portfolios Portfolios that hedge or mimic the factors are the basic components of various portfolio strategies. The mimicking portfolio for a given factor is the portfolio with the maximum correlation with the factor If all assets are correctly priced, then each investor s portfolio should be some combination of cash and the mimicking portfolios Other portfolios have the same level of expected return and sensitivities to the factor but greater variance. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 48 / 62

49 We can write f jt = ŵ j (R t µ), where ŵ j = (( B B) 1 B ) j or ŵ j = (( B D 1 B) 1 B D 1 ) j. Can show that these portfolio weights solve the following problem (taking Ω = D or Ω = I ) min w w j Ω w j j subject to w j b h = 0 h = j ; w j b j = 1 The set of portfolios that are hedged against factors h, h = j and have unit exposure to factor j is of dimension N K. We are finding the one with smallest idiosyncratic variance. Can normalize the weights ŵ j to sum to one, so that they are portfolio weights. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 49 / 62

50 The value of the factor model is in dimensionality reduction. It is important in portfolio choice and asset allocation, which usually involves an inverse covariance matrix that has to be estimated. Note that when Ω = BB + D, we have by the Sherman, Morrison, Woodbury formula Ω 1 = D 1 D 1 B(I K + B D 1 B) 1 B D 1, which only involves inverting the K K matrix I K + B D 1 B and the elements of the diagonal matrix D. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 50 / 62

51 Asymptotic Principal Components An alternative to maximum likelihood factor analysis is asymptotic principal components (small T and large N). In the population we can write the T 1 excess return vector as where F is T K. R i µ i i T = Fb i + ε i, Asssume that F is a fixed matrix (or if random we can condition on its realization) and that b i are random variables from a common distribution with E (b i b i ) = Σ b. Let σ 2 = 1 N N σ 2 εi i=1 Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 51 / 62

52 It follows that Ψ = 1 N N [ ] E (R i ER i ) (R i ER i ) F i=1 Σ b (N ) σ 2 (N ) {( }} ){{( }} ){ N 1 = F N E (b i b i ) F N 1 + i=1 N σ 2 εi I T, i=1 = F Σ b F + σ 2 I T. The right hand side has TK + K (K + 1)/2 + 1 parameters which is less than LHS which has T (T + 1)/2. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 52 / 62

53 Identification problem There is an identification problem: for any nonsingular matrix L Fb i = FLL 1 b i = F b i In this case, assume that (with γ 1... γ K ) Σ b = diag{γ 1,..., γ K } = Γ ; F F = I K. Ψ = F ΓF + σ 2 I T, Then F, Γ, σ 2 are unique (F upto sign) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 53 / 62

54 For any symmetric T T matrix Ψ we have the unique eigendecomposition where eigenvectors Q satisfy and eigenvalues Ψ = QΛQ = ordered from largest to smallest. T λ t q t q t t=1 QQ = Q Q = I T Λ = diag{λ 1,..., λ T } Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 54 / 62

55 Let G be a T T K matrix such that Q = (F T K, G T T K ) satisfies Q Q = Q Q = I T. Provided F has full rank (K ) this Q exists and is unique. We have Ψ = F ΓF + σ 2 I [ T Γ 0 = (F, G ) 0 0 ] (F, G ) + σ 2 I T = Q [ Γ + σ 2 I K 0 0 σ 2 I T K ] Q. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 55 / 62

56 So Q = Q by uniqueness of the eigendecomposition. Furthermore, the eigenvalues of Ψ are spike {}}{ γ 1 + σ 2,..., γ K + σ 2, σ 2,..., σ 2. It follows that F are the eigenvectors corresponding to the K largest eigenvalues of Ψ F = eigvec K [Ψ] This shows unique identification of F. The Γ, σ 2 are also uniquely identified by this. Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 56 / 62

57 APC estimates (in the first pass) factor returns rather than factor betas using the above identification argument 1 The sample covariance matrix Ψ = ( 1 N N i=1 (R it R t )(R is R s ) ) T s,t=1, R t = 1 N which is a T T matrix of excess return cross-products. 2 Do the empirical eigendecomposition and take F = eigvec K [ Ψ] N R it i=1 3 Given this estimate of F, the factor betas can be estimated by time-series OLS regression b i = ( F F ) 1 F (R i µ i i T ) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 57 / 62

58 The main problem with this approach is that it assumes time series homoskedasticity for the idiosyncratic error, which is not a good assumption. Jones (2001, JFE) has extended the estimation problem to allow for time varying average idiosnycratic variance. Ψ = F ΓF + D, D = diag{σ 2 1,..., σ 2 T } Iterative application of APC. That is, given first round estimates, calculate the time series heterosekdasticity D = diag{ σ 2 1,..., σ 2 T } and then recompute the factors F = eigvec K [ Ψ D] Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 58 / 62

59 Eigenvalues of Ω T for daily SP500 returns (N = 441, T = 2732) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 59 / 62

60 Eigenvalues of Ψ N for Monthly SP500 returns (N = 441, T = 124) Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 60 / 62

61 Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 61 / 62

62 The case where both N and T are large can yield further results Definition Bai and Ng (2002). Solve the following min B,F T t=1 N i=1 {R it b i f t } 2 subject to the identification constraint either that B B/N = I K or F F = I K The procedure can be understood as iterative least squares (cross-section regression then time series regression then etc). Equivalent (upto normalization) to APC when T is fixed They show consistency of this procedure when N and T are large They also propose model selection method to determine the number of factors K Large literature now on estimating large covariance matrices with shrinkage Financial Econometrics Short Course Lecture 3 Multifactor Renmin PricingUniversity Model 62 / 62