Macroeconomic Risk and Asset Pricing: Estimating the APT with Observable Factors

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 448

August 1993

MACROECONOMIC RISK AND ASSET PRICING: ESTIMATING THE APT WITH OBSERVABLE FACTORS

John Ammer

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

This paper develops and applies a new maximum likelihood method for estimating the Arbitrage Pricing Theory (APT) model with observable risk factors. The approach involves simultaneous estimation of the factor loadings and risk premiums and can be appled to return panels with more securities than time series observations per security. Observable economic factors are found to account for 25 to 40 percent of the covariation in U.S. equity returns, and the APT pricing restrictions cannot be rejected for most sample periods. A signiticant "firm size anomaly" is measured, but it may be partly due to

sample selection bias.

MACROECONOMIC RISK AND ASSET PRICING: ESTIMATING THE APT WITH OBSERVABLE FACTORS

John Ammer?

1. Introduction

With models like CAPM and the APT, the finance profession has macle substantial progress in determining how the means of asset returns should be related, taking their variances and covariances as given. We have been somewhat less successful at explaining the sources of risk at the root of those variances and covariances. One strand of the literature has attempted to relate asset return innovations to news about future

variables.

The approach taken in this paper has its earliest antececlents in Chen, Roll, and Ross (1986). These authors treat news about the economy as observable risk in the context

of a factor model. Burmeister and McElroy (1988) go further in

0 The author is a staff economist in the International Finance

Division of the Board of Governors of the Federal Reserve System. Opinions expressed herein do not necessarily concur with those of the Federal Reserve Board or any other employees of the Federal Reserve System. I would like to thank Jianping Mei ancl workshop participants at the Federal Reserve Board and Princetion University for helpful conversations, and Jianping Mei, Tina Sun, and Chris Turner for assistance in obtaining data. However, I made all of the errors.

1 See Fama (1990), Campbell (1990), Campbell and Ammer (1991),

and Campbell and Mei (1991).

this direction, devising a means for estimating the Arbitrage Pricing Theory (APT) model with observable risk factors. This paper makes further technical progress in a new estimation method for the APT with observable factors which can handle a large enough amount of data to do justice to Ross! (1975) concept of no asymptotic arbitrage opportunities. The next section of the paper briefly reviews the APT, and the third section presents a maximum likelihood estimation method for the model. The next section discusses construction of the factor space and preliminary results of an application to U.S. equity return data. The fifth section develops improvements in measuring the observable factors, by incorporating revisions in expectations of future variables. The following section presents estimates and the results of some hypothesis tasts, and the subsequent one undertakes a brief investigation of the

firm size anomaly. The eighth section concludes the paper.

2. The APT Model

Assume that excess returns (over the risk-free rate) on

assets are generated by a linear factor model:

- of. : 1 Ll 1,) >j,t 1,t (1) for i=1, n and t=1, T

where E(f,) = 0, E(w; ) = 0, E(f5w;) = 0, and E(wswi) = 0.

The absence of asymptotic arbitrage opportunities (Ross 1976)

requires that for some vector di: » 2 Aa 2 , 17) J (2)

where the A are factor prices (risk premiums). With this

restriction, (1) can be rewritten: K : = b. .| A. + £. + W. 3 7i,t 7 2) i3| j 7 *4,t | i,t (3) or in matrix form:

Z = tA'B + FB +W (4)

where F is a matrix of (zero mean) random risk factors, B is a matrix of factor loadings, * A is a vector of factor prices, t

is a vector of ones, and the residuals We are independently and

identically distributed with zero mean and diagonal covariance

matrix Q.

2 More general specifications of the APT allow the factor

loadings to vary over time, for the factors and residuals to be heteroscedastic, and for the residuals to have non-zero covariances. However, the more restrictive version of the model we use here is common in the APT literature.

3. Estimation with Observable Factors

If one is to evaluate the importance of particular risk factors in the APT model, it is essential that the estimation method employed have three properties. First, it should be able to accommodate observable factors as inputs. In addition, one would like to be able to obtain consistent estimates and standarcl errors for the factor prices. A third property is also important if any asymptotic hypothesis testing will be done: t:she method should be capable of handling a large amount of data; in particular it should not require there to be more return observations per security than there are securities in

the model.

All of the estimation methods in the published literature on the APT fail to simultaneously satisfy all three of the criteria listed above. The techniques of Ross and Roll (1980) and Mei (1990), treat the factors as unobserved and cannot extract them. The methods presented in Lehmann and Modest (1988), Connor and Korajczyk (1988), and Mei (1991) infer the risk factors (F) from the covariance structure of returns. The factor estimates are returns on particular portfolios (with the means subtracted), and the means of these returns are consistent estimates of the risk premiums (A). Unfortunately,

with these methods, an additional step would be required to

relate an observable risk factor, such as inflation risk, to the extracted factor space. Without such further analysis, there is little that one would be able to say about the nature

of the risk that is priced.

Methods which use observable factors directly are potentially more appealing. Chen, Roll, and Ross (1986) apply the two step "cross-sectional regression" procedure of Fama and MacBeth (1973) to a panel of twenty size-sorted portfolios.° The first step is to obtain estimates of the factor loadings (B) by applying ordinary least squares to (1) with an unrestricted intercept (H;) for each security. Next, for each time period, the cross-section of security excess returns is regressed on the estimated factor loadings, to obtain estimates of the factor prices (A). Unfortunately, this technique suffers from an errors-in-variables problem in the second stage regression, which in general causes the precision of the estimates of the risk premiums (A) to be overstated. * In addition, there is no means for imposing the model rest:rictions

when estimating the factor loadings with this method, so that

it cannot truly be deemed a procedure for estimating the APT.

3 It is not clear why Chen, Roll, and Ross use so few assets,

since their methodology is not constrained in this’ dimension. Statistical power is lost by bundling assets into portfolios instead of allowing them to enter the estimation individually.

4 See Shanken (1992).

The APT estimation methods of Burmeister and McElroy (1988) and King, Sentana, and Wadhwani (1990) can handle both observed and unobserved factors, but both techniques require that there to be more return observations per security than there are securities in the model. Since the time series dimension of applications involving observable factors tends to be severely limited by the availability of macroeconomic data,

this shortcoming can be quite constraining.

The estimation method we use is the first to satisfy all three of our criteria above. Note that if all risk factors are observed, and a parameterization is chosen for the distribution of the residuals (W), equation (4) can be estimated directly by numerical methods, choosing the values of B and A which maximize the likelihood function.? we take this approach, allowing observable excess returns on well diversified portfolios to proxy for the unobservable dimensions of the

factor space, ° and assuming that the residuals are drawn from a

5 One negative feature of both our method and that of

Burmeister and McElroy (1988), is that it requires restrictions on the covariance matrix of the residuals. The Chen, Roll, and Ross (1986) paper is not subject to this criticism.

© Burmeister and McElroy (1988) also use returns as proxies for

latent factors. It is important to account for any unobservable risk factors that might be present in returns, because it will be assumed that the residuals will be uncorrelated across assets.

multivariate normal distribution.’ If the returns on these

portfolios have no idiosyncratic risk, ® they can be wrii:ten:

Za = [« Ag! + Fo| Boop + [' A! + F,| Buip (5)

where ut is a T-length vector of ones, the subscripts of F distinguish observed and unobserved factors, and Z_ and Fu are

assumed to have the same dimensions. If By p is nonsingular, ,

the observable factors and diversified portfolio excess returns

will jointly span the underlying factor space:

-1

' ' oy ' Fy + Agt _ I 0 Po Agt (6) Fo' + Ac! B_! B! zZ! u u o,p u,p p

In order to insure that the first matrix on the right side of

(6) is invertible, one should choose portfolios for Z_ with

7 This does not mean that the excess returns are themselves

multivariate normal, as not distributional assumption is made about the risk factors.

8 as discussed by Burmeister and McElroy (1988), non-zero

residuals in the portfolio returns proxying for the latent factors can bias the estimates of the risk premiums to the extent that the sample mean of those residuals differs from zero. In one of their applications, they attempt to avoid this problem by substituting projections of these returns’ onto instruments which are contemporaneous returns on assets’ that are neither in the portfolios or in the data panel on the right side of (4). Wanting as many securities as possible to be included in Z, we chose not to take their approach. As long as residual risk in our portfolio returns is small compared to the

factor risk, the effect on our estimates of A should be negligible.

some distinguishing features; if they have identical factor

loading:3, 25 will not be able to span a multidimensional latent

factor space.

Without loss of generality, we will assume that the unobserved factors are orthogonal to the observed factor space,” that they are mutually orthogonal, and that they have

unit variances. Under those assumptions, equation (5) implies

that _ -1 Boop = (Var(F.)) Cov(F., 24) (7) and ' = - Buip Buip Var (Zz) Cov (2, Fy) Bop (8) and = 1(B 'B Pup Chol (By ‘Bu, p) (9)

where Chol(-) denotes a Cholesky decomposition of a symmetric

matrix into an upper triangular matrix (and its transpose).

As, in some sense, they must be to truly be "unobserved".

Maximum likelihood estimation of (4) is facilitated by the

following shortcut:

Max £(A, B, F, Z) = Max [Max £(B, F, Z | A)] (10) A,B A B

Conditional on A, maximum likelihood estimates of B can be computed by ordinary least squares. Thus only Ky (the number of observable factors) of the Kn+K model parameters need to be involved directly in the numerical maximization of the likelihood function. For a given Aor F+A is computed firom (6). Then the OLS residuals from (3) can be used to compute the likelihood function. Asymptotic standard errors for the AG

estimates?° can be computed from the second derivatives: of

M(A | F, Z) = Max £(B, F, Z | A) (11) B

taken with respect to A.

10 The procedure does not provide standard errors for B, but

most of the hypotheses one would want to test involve only A.

10

4. Simple VAR Residuals as Factors

If financial markets are informationally efficient, asset prices should react to news about relevant economic variables. Accordingly, for the results reported in Tables 2-6, the

observable factors (Fo) are the residuals from a 5-lag vector

autorecression estimated from 1952 to 1990, +1 scaled to unit variance: 5 X = a A, X50 + Foot (12)

Table 1 lists the macroeconomic variables (X) in the VAR

specification.

The returns panel for a given sample period consists of all NYSE and AMEX firms for which there was a complete set of monthly returns on the 1990 CRSP tapes. The sample periods, which were chosen to correspond to Connor and Korajczyk (1988) and Mei. (1991), were 1979-1983, 1984-1988, 1964-1968, 1969-1973, and 1974-1978. An additional 1979-1983 sample was drawn for firms which were listed at the beginning of the

period and had at least 30 monthly return observations. ?? Note

11 the Akaike Information Criterion was applied to choose the lag length.

Allowing missing values in the returns panel substantially

11

that it is necessary to have at least K return observations to

identify the factor loadings for a given asset.

13

The portfolio returns used to identify the latent factors

were the first ten principal components of the individual excess returns panels. 14 These principal components are excess returns on portfolios formed from the securities in the panel. Similar results were obtained when capitalization-based New

York Stock Exchange (NYSE) decile portfolios were used instead.

Estimates of the model!> for the five sample periods appear in Tables 2-6. The second column of each table contains the estimated factor prices and standard errors from the negative inverse of the estimated Hessian matrix. The third column reports the average effect on the mean of an asset return from

exposure to each risk factor, which is simply the product of

increases the computation time for each evaluation of the likelihood function because the moment matrix of the factors is no longer the same for each return in the data panel.

13 Excess returns are over the one-month treasury bill return from Ibbotson (1991).

14 This is the Connor and Korajczyk factor extraction method.

The portfolio returns are the eigenvectors associated with the largest eigenvalues of 22Z'.

15 We used the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm

provided in the Gauss 2.01 MAXLIK module, with muamerical derivatives. Although analytical derivatives of the likelihood function are tractable, they are computationally very complicated, and some initial experimentation suggested that their use would actually increase the computation time required to achieve convergence.

12

the factcr price (plus the sample mean realization of the.

factor) and the mean factor loading. By choosing an order of orthogonalization, it is possible to decompose the R? of each return into contributions attributable to each of the factors;

these are reported in the rightmost column of the tables.

To interpret the estimated elements of A associated with the observable macroeconomic risk factors, it is helpful to borrow scme rough intuition from intertemporal asset pricing. Generally, assets that perform well when the endowment process (or econcmy) is doing poorly will have some hedging value, which will compensate for a lower mean return. Accordingly, assets with returns that covary positively with good news will be poor hedges, and should pay a premium return to compensate. Thus one would expect to estimate positive risk premiums for

"good news" risk factors and negative prices for "bad news"

factors.

A negative price was estimated for the PROD factor in all five sample periods, which was statistically significant in all but one case, implying that higher industrial production represents bad news for the economy, and is a contingency against which it is valuable to be able to hedge. This counter-intuitive result is the opposite of what was found by

Chen, Roll, and Ross (1986) over a similar sample period.

13

Perhaps more disturiing, in the 1984-1988 sample, the average factor loading for PROD was negative, implying that good news

about industrial production was bad news for the typical U.S.

equity security in the sample.

14

5. Constructing Forward-Looking Factors

There are two obvious problems with using VAR residuals to represent relevant economic factors. First, for the pricing of long-term assets, current innovations in macroeconomic variables may not matter as much as news about their future values. Secondly, if there is information other than lags of these variables which helps to forecast them, some portion of the estimated residuals from the VAR system, which excludes this extra information, reflects old news which should have

already been incorporated into asset prices.

In the results reported in Tables 8-18, these issues are

addressed by measuring the observable factors as

ot = Ge - Eee 37 Lf Xt (13)

ye (xz), (14)

where Zi is the excess return on the NYSE value-weighted market portfolio. Thus measurement of the factors will reflect

information about future realizations of X which have been

15

incorporated into the general level of U.S. equity prices. ?® By choosing to aggregate news about the economic variables for a period zero through twelve months ahead, we have arbitrarily limited (to a one-year horizon) the extent to which investors can be forward-looking. +’ It would be easy to modify the assumptions, by either changing the horizon or using a

discounted sum of forecast revisions.

Table 7 presents correlations of the excess return on the NYSE equal-weighted market portfolio with both the "residuals" factors Fo and the (forward-looking) "news" factors Fo These correlations are generally stronger for the "news" factors, which is a preliminary vindication of the forward-looking factor measurement method. This is in part because of the ability of ZVWM to help predict next month's macroeconomic variables (in the vector autoregression). Apparently investors have a lot of information about very short-term economic

prospects.

16 We implicitly assume that the variables in Y are observed

upon realization. This is not strictly true for data on aggregate sales, production, and consumer prices. However, Huberman and Schwert (1985) found that investors in indexed Israeli government bonds were very good at forecasting the consumer price index data releases to which the bonds were indexed.

17 A horizon of at least one year seemed appropriate because

Fama (1990) has found that market portfolio returns predict production growth up to 12 months ahead.

16

6. Results and Hypothesis Tests

Tables 8-12 contain estimates of (4) for all five sample periods using the "news" factors. Significantly positive risk premiums were estimated for the PROD* factor in the 1979-1983 sample, and for the SALE* factor!® in three of the sample periods. These results suggest that investors demand a higher

mean return from assets with pro-cyclical returns, because they

have poor hedging value.

For three of the samples, a negative risk premium was estimated for the oil price factor. Under the logic of intertemporal asset pricing, this would be consistent with high oil prices; being bad news for the economy of the United States, a net importer. Using similar methodology, Ammer (1992, chapter 3) measured a positive oil factor price for equities

traded in the UK, a net oil exporter.

For four of the periods, a negative risk premium was

estimated for inflation news. +? This result suggests that

18 Our postive risk premium estimates for the SALE* factor in

the 1974-1978 and 1979-1983 periods contrast the negative factor price estimated for a final sales factor by Burmeister

and McElroy (1988) using a sample of 70 securities from 1972 to 1982.

19 Chen, Roll, and Ross (1986) also estimated a negative inflation factor price for the period 1958-1984.

17

securities which are good hedges against inflation are

perceived to have extra value.

Negative risk premiums were estimated for both of tae yield spread factors in most of the periods. For the corporate bond quality spread, this is consistent with default risk increasing during periods of bad economic news. The negative risk premium estimated for the maturity yield curve factor

suggests that relatively high long term interest rates are bad

for the economy.

The contribution of the observable factors to the mean R’* ranges from 25 to 40 percent in these samples, compared to 20 to 30 percent when the "residual" factors were used. Nevertheless, unobservables are still driving the majority of

the covariation in asset returns.

Likelihood ratio (LR) tests of the APT pricing restrictions can be performed by comparing the likelihood values reported in the tables to those obtained from unrestricted ordinary least squares estimation of equation (1).

We fail to reject at the five percent level in four out of five

cases.

In addition, variation in factor loadings typically

18

accounts for between 35 and 40 percent of the cross-sectional variation in returns in a given time period. Variations in the sensitivity of returns to the six observable factors usually accounts for about 10 percent of this cross-sectional

variation.

Table 13 presents estimates of Ay which have been restricted to be the same across all five samples. 7° A likelihood ratio test leads to strong rejection of the

restriction, suggesting that factor prices may not be stable

over time.

All of the results discussed so far are for panels of returns with no missing values. In other words, the samples have excluded companies for which listing on the NYSE or AMEX was discontinued sometime during the period. If such firms are fundamentally different from the survivors, the results may be biased. Table 14 presents estimates for 1979-1983 for all firms with at least thirty (not necessarily consecutive) return observations. The estimated factor prices are very close to

the results reported in Table 8.

20 Note that it would not be appropriate to Similarly restrict Aue because the portfolios from which the unobservable factors

are constructed vary across sample periods.

19

Tables 15 and 16 contain results for 1979-1983 in which one of the 16 factors has been dropped from the specification reported in Table 8. [In Table 15, the tenth latent factor is omitted, which has little effect on the estimated risk premiums for the observable factors. Nevertheless, a likelihood ratio test leads to strong rejection of the null hypothesis that the sixteenth factor is redundant. Table 16 reports estimates for a 15-dimensional factor space which excludes ROIL*. Because this model is nested by the 16-factor model, it appears at first blush that an LR test statistics would have an asymptotic v2 distribution when the restricted model is the true mcdel. This is not the case. When all of the (n) loadings for a particular factor are restricted to be zero, the price cf that factor becomes unidentified. 7+ However, the results of Monte Carlo simulations (see Appendix A) suggest that when n is large, the 95% quantile of the statistic is fairly close to that of of the xv? (n+1) distribution. With a test statistic of about 2744, one can confidently reject the null hypothesis of exclusion of the ROIL* factor. Yet an application of the Akaike Information Criterion, which would be more oriented to parsimonious model choice than LR tests on large numbers of

restrictions, would lead to dropping the oil factor.

21 Note that this is a feature of the APT model in general, not

of this estimation method. Garcia and Perron (1990) document a similar problem for testing for the number of discrete states in a regime switching model.

20

7. The Firm Size Anomaly

This estimation framework can also be used to test the APT against specific alternatives which nest it. Some of the interesting alternatives are anomalies against which the older CAPM moclel was rejected, such as the "small firm effect", under which the equities of firms with lower ex ante market capitalizations have higher mean returns than larger firms, even after adjusting for market risk. 2 One could test for a

security-specific effect by augmenting the APT equation (4) to Z = cAS' + (ct A' + F)B + W (15)

where S is a vector of security characteristics and A is a

scalar t:o be estimated.

Under this alternative model, the excess returns on well

diversified portfolios would be

= ' ' ' Z, = LASS + [e4, + Fo|Bo, p + [es + Fa} Bu,p (16) where Sy = G'S and G is the matrix of portfolio weights.

22 This was first discovered by Banz (1981).

21

If G is known, estimation can proceed as above. However, when the portfolios are principal components of the return panel (Z), extracting the portfolio weights can be computationally expensive.?? For the application presented here, we chose the alternative of estimating So (along with A

and A.) by allowing it to enter as parameters to the numerical

likelihood maximization.

Table 17 presents estimates of (15) in which S is the difference between the firm's log capitalization at the beginning of the sample (January 1, 1979) and the mean log capitalization at that time for all firms in the sample. A significant size effect is measured here, but it could be due to sample selection bias. Smaller firms are much more likely to drop out of the sample (from ceasing to be listed on the exchange). If, as seems reasonable, the casualties tend to be securities that have "performed" poorly, the ex post mean

returns for survivors will overstate their ex ante means.

The sample selection bias problem is reduced in the results reported in Table 18. Assets are included in the sample if they made it at least half of the way through. The

estimated size effect coefficient is only slightly smaller, and

23 In particular, it involves computing eigenvectors of ann by n matrix.

22

translates to two percent being either added to or subtracted

from the annual return on a typical security in the sample.

This result contrasts what was found by Chan, Chen, and Hsieh (1985), who applied the Fama and MacBeth (1973) cross-sectional regression methodology to the same data set used by Chen, Roll, and Ross (1986). In particular, they used the excess returns on twenty size-sorted portfolios as Z in (1). They found that high correlation between their portfolio of the smallest firms and a factor similar to RISK, for which they had estimated a large negative risk premium, explained the high mean return on this portfolio. However, because of the statistical power which is sacrificed by bundling securities into portfolios, it is hard to know whether the assets in this portfolio which cause its return to be correlated with the default: risk spread are the same ones which cause the mean return to be high. For example, some of the firms in the sample may have equity returns which are negatively correlated with RISK simply because their debt securities are constituents

of the measured BAA bond yield, while others have a high mean

return for unrelated reasons.

23

8. Conclusions

This paper develops a new maximum likelihood method for estimating the Arbitrage Pricing Theory (APT) model with observable risk factors. The use of observable factors in the APT enables greater economic interpretation of the systematic risk which is (or is not) priced. The technique produces consistent standard errors for the factor prices, it allows one to impose (and test) the APT model pricing restrictions, and can be applied to panels of return data with more securities

than time series observations per security.

The ability to handle large cross-sections appear to have allowed us to estimate the risk premiums more precisely. We estimate factor prices significantly different from zers (using a test size of five percent) more often than do Burmeister and McElroy (1988). In addition, note that the standard errors reported by Chen, Roll, and Ross (1986) overstate the precision of their estimates because they do not correct for the errors-in-variables problem that arises in their multi-stage estimation procedure. If the APT model is true, then our method, in which one can impose the APT restrictions should

produce more precise estimates, as well as consistent standard

errors.

24

Our technique is applied to several large panels of U.S. equity returns. Observable economic factors are found to account for 25 to 40 percent of the common variation in excess returns over the risk-free rate. A number of factor prices are found to be significantly different from zero, but estimates do not appear to be stable over different sample periods. The APT

pricing restrictions cannot be rejected for most of the sample

periods.

A significant "firm size anomaly" is measured, which

appears not to be entirely due to sample selection bias.

25

Appendix A: Monte Carlo Distribution of LR Statistic

For each simulation, the APT was estimated with zero and one factors on 60 observations on 1797 returns, where the true model: was white noise (zero factors). Likelihood ratio

statistics were computed for 1000 simulations.

95% quantile of empirical distribution: 971.123 95% quantile of x? (1798) distribution: 948.738 x? quantile of empirical critical value: 0.009 empirical quantile of x? critical value: 0.186

26

References

Ammer, J.(1992), "Noise Trader Risk, Management Expenses, Takeovers and Liquidations, and the Closed End Fund Discount Puzzle: New Evidence from the United Kingdom", Ph.D Dissertation, Princeton University.

Banz, R. (1981), "The Relationship between Return and Market Value of Common Stocks", Journal of Financial Economics 9(1):3-18.

Burmeister, E. and M. McElroy (1988), "Joint Estimation of Factor Sensitivities and Risk Premia for the Arbitrage Pricing Theory", Journal of Finance 43(3):721-735.

Campbell, J. (1990), "A Variance Decomposition for Stock Returns", the H.G. Johnson Lecture to the Royal Economic Society, Economic Journal 101:157-179.

Campbell, J. and J. Ammer (1991), "What Moves the Stock and Bond Markets? A Variance Decomposition for Long-Term Asset Returns", NBER Working Paper 3760 (June).

Campbell, J. and J. Mei (1991), "Where do Betas Come From? Asset Frice Dynamics and the Sources of Systematic Risk", Working Paper S-91-52, NYU Stern School of Business (November) .

Chen, N. and J. Ingersoll (1983), "Exact Pricing in Linear Factor Models with Infinitely Many Assets", Journal of Finance 38:985-988.

Chan, K., N. Chen, and D. Hsieh (1985), "Qn Exploratory Investigation of the Firm Size Effect", Journal of Financial Economics 14:451-471.

Chen, N., R. Roll, and S. Ross (1986), “Economic Forces and the Stock Markets", Journal of Business 59:383-403.

Connor, G. and R. Korajczyk (1988), "Risk and Return in an

Equilibrium APT: Application of a New Test Methodology", Journal of Financial Economics 21(2):255-289.

Fama, EF. (1990), "Stock Returns, Expected Returns, and Real Activity", Journal of Finance 45(4):1089-1108.

Fama, E. and J. MacBeth (1973), "Risk, Return, and Equilibrium: Empirical Tests", Journal of Political Economy 81:607-636.

Garcia, R. and P. Perron (1990), "An Analysis of the Real Interest Rate Under Regime Shifts", unpublished paper,

27

Princeton University (August).

Huberman, G. (1986), "A Review of the Arbitrage Pricing Theory", Working Paper 166, Graduate School of Business, University of Chicago (May).

Huberman, G. and W. Schwert (1985), "Information Aggrecation, Inflation, and the Pricing of Indexed Bonds", Journal of Political Economy 93(1):92-114.

Ibbotson Associates (1991), Stocks, Bonds, Bills, and Inflation.

King, M., E. Sentana, and S. Wadhwani (1990), "A Heteroscedastic Factor Model of Asset Returns and Risk Premia with Time-Varying Volatility: An Application to Sixteen World Stock Markets", LSE Financial Markets Group Discussion Paper 80 (May).

Lehmann, B. and D. Modest, "The Empirical Foundations of the

Arbitrage Pricing Theory", Journal of Financial Economics 21(2):213-254.

Mei, J. (1990), "New Methods for Testing the Arbitrage Pricing Theory and the Present Value Model", Ph.D Dissertation, Princeton University.

Mei, J. (1991), “Extracted Factors and Time-Varying Conditional Risk Premiums -- A New Approach to the APT", Working Paper S~91-43, NYU Stern School of Business (August).

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28

Table 1

Construction of Factor Space for APT Models

Observable Factors are based on the residuals from a VAR(5) :

eee PROD growth in industrial production (seasonally adjusted)

ROIL log relative (to CPI) wholesale price of oil (SA)

GCPI Consumer Price Index (all items) inflation (SA)

SALE growth in total real final retail sales (SA)

RISK yield spread between BAA bonds and AAA corporate bonds

TERM yield spread between 10-year and 3-month treasury securities

For the results reported in tables 8-18, the following variable

is added to the VAR system to improve forecasting power:

ZVWM excess return (over the 1l-month treasury bill) on the New York Stock Exchange Value-Weighted Market Index

In addition, the factors used there are news about the average values of the VAR variables for the current and next 12 months

. , 22 Pot = (fe - Ee) a LE Xe;

instead of the VAR residuals used for tables 2-6

Po t = (Ee - Ee) % -

The Latent Factor space is comprised by the components of excess returns (over the 1-month treasury bill) on well-diversified portfolios which are orthogonal to the observable factor space.

29

Table 2

1797 returns

ln(£) = 137379.753 average R® = 0.551. name mean average of A annual variance factor A boost share i PROD -0.052 -0.003 0.025 (0.031) ROIL -0.212 -0.005 0.015 (0.024) GCPI -0.285 0.010 0.023 (0.038) SALE 0.026 0.007 0.022 (0.028) RISK 0.029 -0.001 0.018 (0.063) TERM 0.017 -0.002 0.014 (0.070) total: 0.116 with APT restrictions relaxed: In(£) = 138230.339 x? P-value for LR test: 0.911

Notes: 10 latent factors were derived from the first 10

principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

Table 3

Estimates of 16 Factor APT Model, 1/84-12/88

1529 returns

in(£) = 124137.187 average R* = 0.576 nanie mean average of A annual variance factor A boost share eee PROD -0.096 0.016 0.032 (0.024) ROIL -0.208 0.010 0.019 (0.054) GcPI 0.054 -~0.001 0.012 (0.044) SALE -0.339 0.026 0.020 (0.049) RISK -0.038 -0.004 0.013 (0.037) TERM 0.051 0.001 0.012 (0.076) eee total: 0.106

a nen

with APT restrictions relaxed: in(£) = 124934.547

x2 P-value for LR test: 0.118 a eee Notes: 10 latent factors were derived from the first 10 principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance, The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

31

Table 4

Estimates of 16 Factor APT Model, 1/64-12/68

1529 returns

In(2) = 122654.691 average R? = 0.506 name mean average of n annual variance factor A boost share eee PROD -0.064 0.000 0.013 (0.032) ROIL 0.123 0.000 0.016 (0.006) GCPI 0.103 0.000 0.019 (0.028) SALE 0.434 -0.031 0.016 (0.052) RISK -0.079 0.051 0.026 (0.018) TERM -0.024 -0.002 0.014 (0.009) a total: 0.104

with APT vestricticns relaxed: Iin(#) = 123324.675 «" P-value fer LR test: 9.999 Motes: io caters factors were derived from the first 10 principal xcwponents of the excess return panel. Standard errors are in parentheses. Factors are caled to unit , . . 2 variance. The contributions to R are calculated by

orthogona «zing the observable factors in their order of appearance.

32

Table 5

Estimates of 16 Factor APT Model, 1/69-12/73

1800 returns

ln(2) = 141107.095 average R? = 0.608 name mean average of A” annual variance factor A boost share eee PROD -0.251 -0.045 0.037 (0.028) ROIL 0.251 0.001 0.016 (0.009) GCPI -0.046 -0.009 0.024 (0.037) SALE -0.235 -0.000 0.013 (0.032) RISK -0.081 0.003 0.028 (0.016) TERM -0.019 -0.035 0.015 (0.009)

total: 0.133

ee with APT restrictions relaxed: In(2) = 141975.141

x P-value for LR test: 0.788

LSS SSeS hoe pss Ss

Notes: 10 latent factors were derived from the first 10

principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

Table 6

1808 returns

in(2) = 142455.091 average R? = 0.651 name mean average of A annual variance factor A boost share ee PROD -0.121 -0.005 0.017 (0.039) ROIL -0.066 0.010 0.028 (0.018) GCPI -0.378 0.049 0.043 (0.037) SALE 0.059 0.027 0.031 (0.038) RISK 0.311 -0.033 0.016 (0.029) TERM -0.201 0.050 0.013 (0.012) a total: 0.149 ie with APT restrictions relaxed: In(£) = 143473.602 x? P-value for LR test: 0.000

serene

Notes: 10 latent factors were derived from the first 10

principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

34

Table 7

Correlation of Factors and Equal-Weighted NYSE, 1/64 - 2/88

factors which are contemporary residuals

PROD ROIL GCPI SALE RISK 1.00 0.05 0.06 0.29 -0.13 1..00 0.16 0.05 -0.02

1.00 -0.07 0.05

1.00 -0.18

1.00

TERM

-0.13

-0.06

-0.05

-0.00

ZEWM

PROD

ROIL

GCPI

SALE

RISK

TERM

ZEWM

factors which are news about mean ef variable 0-12 months ahead

OO eee OE “St

PROD* ROIL* GCPI* SALE* RISK* 1.00 -0.04 -0.19 0.76 -0.69 1.00 0.52 -0.21 0.26

1.00 -0.38 0.16

1.00 -0.44

1.00

35

TERM*

0.23

~0.54

-0.36

ZEWM

0.48

0.07

-0.18

PROD*

ROIL*

GCPI*

SALE*

RISK*

TERM*

ZEWM

Table 8 Estimates of 16 Factor APT Model, 1/79-12/83

1797 returns

In(£) = 137282.520 average R? = 0.551 name mean average of n annual variance factor A boost share ees PROD* 0.119 -0.022 0.041 (0.037) ROIL* ~0.216 -0.001 0.016 (0.030) GCPI* -0.278 0.002 0.025 (0.037) SALE* 0.204 0.036 0.023 (0.036) RISK* -0.118 0.006 0.020 (0.061) TERM* 0.032 0.024 0.019 (0.068) _ eee total: 0.146

TT eeeeeeseeesess—=* with APT restrictions relaxed: In(£) = 138129.939

v* P-value for LR test: 0.927

Notes: 10 latent factors were derived from the first 10

principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

36

Table 9

1529 returns

in(£) = 124065.147 average R? = 0.576 name mean average of A annual variance factor A boost share PROD* -0.054 -0.028 0.070 (0.029) ROIL* -0.300 -0.014 0.016 (0.057) GCPI* -0.046 0.013 0.016 (0.047) SALE* -0.166 0.008 0.023 (0.043) RISK* -0.174 0.011 0.012 (0.037) TERM* 0.206 -0.027 0.018 (0.077) total: 0.156 with APT restrictions relaxed: ln(#) = 124860.991 v2 P-value for LR test: 0.129

Notes: 10 latent factors were derived from the first 10

princir’1 components of the excess return panel. Standard errors ure in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

37

Table 10

1529 returns

ln(2) = 122828.669 average R? = 0.509 name mean average of A annual variance factor A boost share PROD* 0.008 0.002 0.039 (0.023) ROIL* 0.124 -0.000 0.016 (0.009) GCPI* 0.168 -0.012 0.029 (0.028) SALE* 0.193 -0.016 0.016 (0.032) RISK* -0.022 0.097 0.041 (0.016) TERM* -0.067 0.024 0.023 (0.009) total: 0.164 with APT restrictions relaxed: In(£) = 123537.948 v2 P-value for LR test: 0.959

Notes: 10 latent factors were derived from the first 10

principal components cf the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

38

Table 11

Hstimates of 16 Factor APT Model, 1/69-12/73

1800 returns

In(£) = 141102.873 average R? = 0.609 name mean average of A annual variance factor A boost share PROD* -0.174 0.014 0.146 (0.029) ROIL* 0.175 0.004 0.016 (0.014) GCPI* -0.046 0.002 0.012 (0.035) SALE* -0.189 -0.037 0.014 (0.034) RISK* -0.016 -0.022 0.039 (0.017) TERM* -0.039 0.027 0.014 (0.010) -_—_———— eee total: 0.240

-_——— esses with APT restrictions relaxed: in(£) = 142003.209

x? P-value for LR test: 0.387

errr eee Notes: 10 latent factors were derived from the first 10

principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance,

39

Table 12

1808 returns

In(#2) = 142515.480 average R2 = 0.652 name mean average of A annual variance factor A boost share PROD* -0.020 -0.005 0.128 (0.034) ROIL* -0.086 0.000 0.029 (0.021) GCPI* -0.281 0.020 0.026 (0.037) SALE* 0.117 0.071 0.022 (0.031) RISK* 0.178 -0.043 0.035 (0.025) TERM* -0.205 -0.020 0.021 (0.011) total: 0.261 with APT restrictions relaxed: ln(£) = 143580.814 v2 P-value for LR test: 0.000

Notes: 10 latent factors were derived from the first 10

principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

40

Table 13

Estimates of 16 Factor APT Model, 1/64-12/88 factor prices restricted to be the same for 5 sub-periods

In(2) = 667193.625

name of A factor A

PROD* -0.005

(0.015)

ROIL* 0.104

(0.007)

GCPI* 0.052

(0.019)

SALE* 0.064

(0.017)

RISK* -0.005

(0.011)

TERM* -0.068 (0.008)

with A restrictions relaxed: ln(2) = 667794.640

x? P-value for LR test: 0.000

Sass SSS ss Schepf

Notes: 10 latent factors were derived from the first 10

principal components of each excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance.

41

Table 14

Estimates of 16 Factor APT Model, 1/79-12/83

2123 returns

In(£) = 152587.326 average R? = 0.595 name mean average of A annual variance factor A boost share eee PROD* 0.120 -0.023 0.042 (0.035) ROIL* -0.229 -0.002 0.017 (0.031) GCPI* -0.305 0.003 0.028 (0.038) SALE* 0.215 0.042 0.026 (0.038) RISK* -0.122 _ 0.006 . 0.021 (0.059) TERM* 0.023 0.025 0.020 (0.072) total: 0.155 ©

—_—_—e'oooee

with APT restrictions relaxed: 1n(£) = 153720.179

v2 P-value for LR test: 0.012 —_—_-eOOQ error Notes: 10 latent factors were derived from the first 10

principal components of the excess’ return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance. Assets with missing returns were included if they had at least 30 observations in the 60 month sample period.

42

Table 15

Estimates of 15 Factor APT Model, 1/79-12/83

1797 returns

In(2) = 135324.253 average R? = 0.535 name mean average of A annual variance factor A boost share PRO)D* 0.118 -0.021 0.041 (0.038) ROI‘L* -0.213 -0.000 0.016 (0.030) GCPI* -0.270 0.002 0.025 (0.036) SALIE* 0.202 0.035 0.023 (0.037) RISK* -0.120 0.006 0.020 (0.063) TERM* 0.039 0.023 0.019 (0.069) eee total: 0.146

————————————

with APT restrictions relaxed: in(£) = 136142.601 x2 P-value for LR test: 0.993 y P-value for adding 16th factor: 0.000 — eee Notes: 2 latent factors were derived from the first 9 principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

43

Table 16

1797 returns

ln(2) = 135910.605 average R? = 0.539 name mean average of A annual variance factor A boost share — PROD* 0.102 -0.017 0.041 (0.036) GCPI* -0.317 -0.004 0.021 (0.038) SALE* 0.219 0.045 0.023 (0.037) RISK* -0.071 -0.002 0.014 (0.060) TERM* 0.015 0.035 0.031 (0.067) eee total: 0.131

rr rrr ree sneennecneneeere,

with APT restrictions relaxed: in(2) = 136745.636 x? P-value for LR test: 0.971 x? P-value for adding 16th factor: 0.000

eee Notes: 10 latent factors were derived from the first 10

principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

44

Table 17

Estimates of 16 Factor APT Model, 1/79-12/83, allowing for firm size effect on mean return

—_—_—_—_—_—<<—SS OE I

1797 returns

in(2) = 137322.520 average R? = 0.551 annualized (relative log) size effect: -0.014 (0.002) mean absolute value of size effect on mean: 0.021 effect on smallest firm: 0.064 effect on largest firm: -0.089 name mean average of A annual variance factor A boost share PROD* 0.153 0.002 0.041 (0.036) ROJ:L* ~0.232 -0.000 0.016 (0.029) GCPI* -0.385 0.002 0.025 (0.039) SALE* 0.296 -~0.002 0.023 (0.037) RISK* -0.107 -0.000 0.020 (0.059) TERM* 0.120 -0.001 0.020 (0.067) eee total: 0.145

_ ees

Notes: 10 latent factors were derived from the first 10

principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance.

45

Table 18

Estimates of 16 Factor APT Model, 1/79-12/83, effect on mean return

OO Oe EO tO

2123 returns

ln(£) = 152644.653 average R* = 0.595 annualized (relative log) size effect: -0.013 (0.002) mean absolute value of size effect on mean: 0.020

effect on smallest firm: 0.063 effect on largest firm: -0.084

name mean average of ~ annual variance factor A boost share PROD* 0.135 0.001 0.042 (0.035) ROIL* -0.224 -0.000 0.017 (0.031) GCPI* -0.356 0.001 0.028 (0.038) SALE* 0.284 -0.001 0.026 (0.038) RISK* -0.080 0.000 0.021 (0.056) TERM* 0.094 -0.000 0.020 (0.072) a total: 0.155

Notes: 10 latent factors were derived from the first 10

principal components of the excess return panel. Standard errors are in parentheses. Factors are scaled to unit variance. The contributions to R? are calculated by

orthogonalizing the observable factors in their order of appearance. Assets with missing returns were included if they had at least 30 observations in the 60 month sample period.

46

IFDP NUMBER

448

447

446

445

444

443

442

441

440

439

438

437

436

Please address requests for co Division of International Finance, Reserve System, Washington, D.C.

International Finance Discussion Papers

TITLES

1993

Macroeconomic Risk and Asset Pricing: Rstimating the APT with Observable Factors

Near observational equivalence and unit root processes: formal concepts and =mplications

Market Share and Exchange Rate Pass-Through in World Automobile Trade

Industry Restructuring and Export Performance: Evidence on the Transition in Hungary

Exchange Rates and Foreign Direct Investment: A Note

Global versus Country-Specific Productivity Shocks and the Current Account

The GATT'’s Contribution to Economic F.ecovery in Post-War Western Europe

4. Utility Based Comparison of Some Models of Exchange Rate Volatility

Cointegration Tests in the Presence of Structural Breaks

1992

Life Expectancy of International Cartels: 4n Empirical Analysis

[Daily Bundesbank and Federal Reserve Intervention and the Conditional Variance Tale in DM/$-Returns

War and Peace: Recovering the Market's Probability Distribution of Crude Oil Futures Prices During the Gulf Crisis

Growth, Political Instability, and the Defense Burden

20551.

47

AUTHOR (s)

John Ammer

Jon Faust

Robert C. Feenstra Joseph E. Gagnon Michael M. Knetter Valerie J. Chang Catherine L. Mann

Guy V.G. Stevens

Reuven Glick Kenneth Rogoff

Douglas A. Irwin

Kenneth D. West Hali J. Edison Dongchul Cho

Julia Campos Neil R. Ericsson David F. Hendry

Jaime Marquez

Geert J. Almekinders Sylvester C.W. Eijffinger

William R. Melick Charles P. Thomas

Stephen Brock Blomberg

pies to International Finance Discussion Papers, Stop 24, Board of Governors of the Federal

IFDP NUMBER

435

434

433

432

431

430

426

424

423

422

421

International Finance Discussion Papers TITLES

1992

Foreign Exchange Policy, Monetary Policy, and Capital Market Liberalization in Korea

The Political Economy of the Won: U.S.-Korean Bilateral Negotiations on Exchange Rates

Import Demand and Supply with Relatively Few Theoretical or Empirical Puzzles

The Liquidity Premium in Average Interest Rates

The Power of Cointegration Tests

The Adequacy of the Data on U.S. International Financial Transactions: A Federal Reserve Perspective

Whom can we trust to run the Fed? Theoretical support for the founders views

Stochastic Behavior of the World Economy under Alternative Policy Regimes

Real Exchange Rates: Measurement and implications for Predicting U.S. External imbalances

Central Banks’ Use in East Asia of Money Market Instruments in the Conduct of Monetary Policy

Purchasing Power Parity and Uncovered Interest Rate Parity: The United States 1974 - 1990

Fiscal Implications of the Transition from Planned to Market Economy

Does World Investment Demand Determine U.S. Exports?

The Autonomy of Trade Elasticities: Choice and Consequences

German Unification and the European Monetary System: A Quantitative Analysis

48

AUTHOR(s)

Deborah J. Lindner

Deborah J. Lindner

Andrew M. Warner

Wilbur John Coleman II Christian Gilles Pamela Labadie Jeroen J.M. Kremers Neil R. Ericsson Juan J. Dolado

Lois E. Stekler Edwin M. Truman Jon Faust

Joseph E. Gagnon Ralph W. Tryon

Jaime Marquez

Robert F. Emery

Hali J. Edison William R. Melick

R. Sean Craig Catherine L. Mann

Andrew M. Warner

Jaime Marquez

Gwyn Adams Lewis Alexander Joseph Gagnon