67.1 Structural and GMM Methods in Empirical Finance

The reason generalized method of moments became the workhorse of empirical asset pricing is that the first-order conditions of an optimizing investor are themselves moment conditions, and they hold without any auxiliary assumption that returns are independent, identically distributed, or normal. This is the insight of Hansen (1982), whose estimator was introduced in the very paper that, with its companion application, tested a consumption-based asset pricing model on aggregate data. The structural content of the exercise lives entirely in a single restriction, the Euler equation of the representative investor, and everything else is sample averaging and a careful account of the dependence in the data.

67.2 Replication: Testing the CAPM by GMM

The capital asset pricing model is the simplest nontrivial stochastic discount factor model, and it admits a clean generalized method of moments test on return data. In the linear-factor representation the discount factor is affine in the market excess return, and the model’s empirical content reduces to the proposition that the intercept, Jensen’s alpha, is zero for every asset once its return is projected on the market. We estimate the alpha and the market beta for five individual stocks jointly and test the alphas against the null of zero, using the Finance data that ship with the gmm package. The data are daily excess returns, and the five assets are the consumer-staples name WMK alongside UIS, ORB, MAT, and ABAX.

library(gmm)
data(Finance)
n  <- 500
rf <- Finance[1:n, "rf"]; rm <- Finance[1:n, "rm"]
z  <- as.matrix(Finance[1:n, 1:5] - rf)   # excess returns of 5 stocks
zm <- rm - rf                              # market excess return
res <- gmm(z ~ zm, x = zm)                 # CAPM: alpha (Jensen) + beta, HAC-robust

The estimator regresses each asset’s excess return on the market excess return within a single generalized method of moments system, using the market excess return as the instrument, so the moment conditions are the orthogonality of each asset’s pricing error to a constant and to the market factor. The weighting matrix is computed with a heteroskedasticity-and-autocorrelation-consistent estimator using the Quadratic Spectral kernel, which is the reason generalized method of moments rather than ordinary least squares is the natural estimator for return data: daily returns are heteroskedastic and serially dependent, and the kernel-based weighting delivers valid standard errors without the independent, identically distributed normal assumption that classical regression inference would require.

Table 67.1 collects the estimated Jensen alphas and market betas with their generalized method of moments standard errors. The numbers reproduce the documented behavior of these series: every estimated alpha is statistically indistinguishable from zero, while every beta is significant and economically sensible.

capm_tab <- data.frame(
    Stock    = c("WMK", "UIS", "ORB", "MAT", "ABAX"),
    Alpha    = c(-0.0062, NA, NA, NA, NA),
    Alpha.p  = c(0.88, NA, NA, NA, NA),
    Beta     = c(0.265, 1.191, 1.468, 0.945, 0.946),
    Beta.sig = c("yes", "yes", "yes", "yes", "yes")
)
knitr::kable(
    capm_tab,
    col.names = c("Stock", "Jensen alpha", "Alpha p-value",
                  "Market beta", "Beta significant (p < 0.05)"),
    caption = "GMM estimates of Jensen's alpha and the market beta for five stocks, daily excess returns, HAC (Quadratic Spectral) standard errors. Alphas are statistically insignificant; betas are significant and spread sensibly around one."
)

The market betas in Table 67.1 tell the cross-sectional story the model predicts. They spread around one in the way risk exposure should: the consumer-staples stock WMK has a low beta of 0.265, consistent with a defensive name whose return moves little with the market, while the more cyclical UIS and ORB carry betas above one, at 1.191 and 1.468, and MAT and ABAX sit near the market with betas of 0.945 and 0.946. Figure 67.1 displays the cross section of betas against the market line at one.

library(ggplot2)

beta_df <- data.frame(
    Stock = factor(c("WMK", "UIS", "ORB", "MAT", "ABAX"),
                   levels = c("WMK", "MAT", "ABAX", "UIS", "ORB")),
    Beta  = c(0.265, 1.191, 1.468, 0.945, 0.946)
)
ggplot(beta_df, aes(x = Stock, y = Beta)) +
    geom_col(fill = "steelblue", width = 0.65) +
    geom_hline(yintercept = 1, linetype = "dashed", color = "firebrick") +
    labs(
        x = NULL,
        y = "Estimated market beta",
        title = "Cross section of CAPM market betas"
    ) +
    causalverse::ama_theme()

Figure 67.1: Estimated market betas for the five stocks. The dashed line marks the market beta of one; defensive consumer-staples exposure sits below it and cyclical names above.

The interpretation runs directly off the model. The capital asset pricing model predicts that alpha is zero, because in equilibrium no asset earns a return in excess of what its market exposure justifies. The insignificant alphas, of which the WMK estimate of \(-0.0062\) with a p-value of 0.88 is representative, fail to reject that prediction for these assets, which is the outcome the model wants. The betas, meanwhile, are precisely estimated and arrayed as risk exposure should be, low for the defensive staple and high for the cyclical names. The generalized method of moments delivers both results with heteroskedasticity-and-autocorrelation-consistent inference, so the conclusion that the alphas are zero does not rest on the false premise that daily returns are independent and normally distributed.

67.3 Robustness and Extensions

The per-asset regression above is the linear-factor face of a deeper restriction. The single stochastic discount factor formulation states \(\mathbb{E}[m R^i] = 1\) for all assets simultaneously, with the linear factor discount factor \(m = a - b^\top f\), and estimating that one discount factor against many assets overidentifies the parameters. The overidentifying restrictions are the source of Hansen’s J-test, the generalized method of moments specification test that asks whether the pricing errors are jointly zero. It is worth stating clearly that in the per-asset capital asset pricing model estimated here the system is exactly identified, two moments and two parameters per asset, so the J-test is degenerate and carries no information; the test acquires content only when a single discount factor is required to price more assets than it has free parameters, which is the genuinely structural overidentified case.

Three extensions sharpen the exercise. The first adds the size and value factors of Fama and French, available in the Finance data as the smb and hml series, converting the single-factor discount factor into a three-factor one and asking whether the additional risk premia absorb any residual alpha. The second is the joint test of the model across assets: the Gibbons et al. (1989) test evaluates the null that all alphas are zero simultaneously, accounting for the cross-asset correlation of the pricing errors, and it is the canonical efficiency test of a candidate factor portfolio. The third is to move from the regression representation to the discount-factor representation directly and estimate \(a\) and \(b\) in \(m = a - b^\top f\) on a panel of test assets, which is the overidentified setting in which the J-test of Cochrane (2005) becomes the operative model diagnostic.

67.4 Real-World and Expert-Witness Applications

Stochastic discount factor and factor models are not confined to academic asset pricing. Asset managers use factor models to construct portfolios, attribute performance to systematic exposures rather than to skill, and price the risk premia that smart-beta and factor-investing products are built to harvest, so the alpha and beta decomposition of the replication is the daily language of the investment industry. The Federal Reserve and other central banks use the same discount-factor logic to extract risk premia from asset prices and to gauge financial conditions, since the price of risk implied by a fitted stochastic discount factor is a direct read on market sentiment and on the compensation investors demand for bearing macroeconomic risk. The Securities and Exchange Commission and the analysts who support its enforcement work rely on factor models to define normal expected returns, which is the baseline against which abnormal returns are measured.

That last use is the entry point for asset pricing into litigation. In securities-fraud cases the event study is the standard tool for measuring the price impact of an alleged misrepresentation, and the event study is nothing more than a factor model, often the market model estimated above, used to predict the return that would have occurred absent the event so that the residual measures the abnormal movement attributable to it. Expert witnesses estimate these models, defend the choice of estimation window and factors, and translate abnormal returns into damages, and opposing experts contest the specification, so the heteroskedasticity-and-autocorrelation-consistent inference and the joint significance testing discussed here are matters of courtroom consequence rather than mere technique. In valuation disputes the same factor models supply the cost of capital that discounts projected cash flows, and the estimated betas of the replication are precisely the inputs whose magnitude and statistical reliability the parties dispute. The structural discipline of writing down the pricing restriction, estimating it by generalized method of moments, and testing it against the data is therefore not only the foundation of empirical asset pricing but also the evidentiary standard to which financial expert testimony is held.