Optimal Orthogonal Portfolios with Conditioning Information

Abstract

Optimal orthogonal portfolios are a central feature of tests of asset pricing models and are important in active portfolio management problems. The portfolios combine with a benchmark portfolio to form ex ante mean variance efficient portfolios. This paper derives and characterizes optimal orthogonal portfolios in the presence of conditioning information in the form of a set of lagged instruments. In this setting, studied by Hansen and Richard (1987), the conditioning information is used to optimize with respect to the unconditional moments. We present an empirical illustration of the properties of the optimal orthogonal portfolios. From an asset pricing perspective, a standard stock market index is far from efficient when portfolios trade based on lagged interest rates and dividend yields. From an active portfolio management perspective, the example shows that a strong tilt toward bonds improves the efficiency of equity portfolios.

The methodology in this paper includes regression and maximum likelihood parameter estimation, as well as method of moments estimation. We form maximum likelihood estimates of nonlinear functions as the functions evaluated at the maximum likelihood parameter estimates. Our analytical results also provide economic interpretation for test statistics like the Wald test or multivariate F test used in asset pricing research.

Appendices

Appendix 1: Theorems and Proofs

35.1.1 Efficient Portfolio Solutions

Portfolio weights for efficient portfolios in the presence of conditioning information are derived by Ferson and Siegel (2001). They consider the case with no risk-free asset and the case with a fixed risk-free asset whose return is constant over time. In Theorem 1, we generalize to consider the case with a risk-free asset whose return is known at the beginning of the period, and thus is included in the information Z, and may vary over time. We then, in Theorem 2, reproduce the case with no risk-free asset from Ferson and Siegel (2001) for future reference.

Consider N risky assets with returns R. In N × 1 column vector notation, we have

$$ R=\mu (Z)+\varepsilon . $$

The noise term ε is assumed to have conditional mean zero given Z and nonsingular conditional covariance matrix ${\not\Sigma}_{\varepsilon }(Z)$. The conditional expected return vector is μ(Z) = E(R|Z). Let the 1 × N row vector x′(Z) = (x ₁(Z), … , x _N (Z)) denote the portfolio share invested in each of the N risky assets, investing (or borrowing) at the risk-free rate the amount $1-{x}^{\hbox{'}}(Z)\underset{\bar{\mkern6mu}}{1}$, where 1 ≡ (1 ,…,1)′ denotes the column vector of ones. We allow for a conditional risk-free asset returning R _f = R _f (Z). The return on the portfolio is ${R}_s={R}_f+{x}^{\prime }(Z)\left(R-{R}_f\underset{\bar{\mkern6mu}}{1}\right)$, with unconditional expectation and variance as follows:

$$ \begin{array}{l}{\mu}_s=E\left({R}_f\right)+E\left\{{x}^{\prime }(Z)\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]\right\},\\ {}{\sigma}_s^2=E\left({R}_s^2\right)-{\mu}_s^2=E\left[E\left({R}_s^2\Big|Z\right)\right]-{\mu}_s^2,\\ {}{\sigma}_s^2=E\left({R}_f^2\right)+E\left[{x}^{\prime }(Z){Q}^{-1}x(Z)\right]+2E\left\{{R}_f{x}^{\prime }(Z)\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]\right\}-{\mu}_s^2,\end{array} $$

(35.17)

where we have defined the N × N matrix

$$ Q=Q(Z)\equiv {\left\{E\left[\left.\left(R-{R}_f\underset{\bar{\mkern6mu}}{1}\right){\left(R-{R}_f\underset{\bar{\mkern6mu}}{1}\right)}^{\prime}\right|Z\right]\right\}}^{-1}={\left\{\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]{\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]}^{\prime }+{\not\varSigma}_{\varepsilon }(Z)\right\}}^{-1}. $$

Also, define the constants:

$$ \zeta \equiv E\left\{{\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]}^{\prime }Q\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]\right\}, $$

$$ \varphi \equiv E\left\{{R}_f{\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]}^{\prime }Q\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]\right\}, $$

and

$$ \psi \equiv E\left\{{R}_f^2{\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]}^{\prime }Q\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]\right\}. $$

Theorem 1

Given a target unconditional expected return μ _s , N risky assets, instruments Z, and a conditional risk-free asset with rate R _f = R _f (Z) that may vary over time, the unique portfolio having minimum unconditional variance is determined by the weights

$$ \begin{array}{c}{x}_s(Z)=\left(\frac{\mu_s-E\left({R}_f\right)+\varphi }{\zeta }-{R}_f\right)Q\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]\\ {}=\left[\left(c+1\right){\mu}_s+b-{R}_f\right]Q\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right],\end{array} $$

(35.18)

and the optimal portfolio variance is

$$ {\sigma}_s^2=a+2b{\mu}_s+c{\mu}_s^2 $$

where $a=E\left({R}_f^2\right)+\frac{{\left[E\left({R}_f\right)-\varphi \right]}^2}{\zeta }-\psi$, $b=\frac{\varphi -E\left({R}_f\right)}{\zeta }$, and $c=\frac{1}{\zeta }-1$. When the risk-free asset return is constant, then these formulas simplify to Theorem 2 of Ferson and Siegel (2001) with

$$ {x}_s(Z)=\frac{\mu_s-{R}_f}{\zeta }Q\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right] $$

and with optimal portfolio variance ${\sigma}_s^2=\frac{1-\zeta }{\zeta }{\left({\mu}_s-{R}_f\right)}^2$.

Proof

Our objective is to minimize, over the choice of x _s (Z), the portfolio variance Var(R _s ) subject to E(R _s ) = μ _s , where ${R}_s={R}_f+{x}^{\prime }(Z)\left(R-{R}_f\underset{\bar{\mkern6mu}}{1}\right)$ and the variance is given by Eq. 35.17. We form the Lagrangian:

$$ \begin{array}{c}L\left[x(Z)\right]=E\left[{x}^{\prime }(Z){Q}^{-1}x(Z)\right]+2E\left\{{R}_f{x}^{\prime }(Z)\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]\right\}\\ {}+2\lambda E\left\{{\mu}_s-{R}_f-{x}^{\prime }(Z)\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]\right\}\end{array} $$

and proceed using a perturbation argument. Let q(Z) = x(Z) + dy(Z), where x(Z) is the conjectured optimal solution, y(Z) is any regular function of Z, and d is a scalar. Optimality of x(Z) follows when the partial derivative of L[q(Z)] with respect to d is identically zero when evaluated at d = 0. Thus,

$$ 0=E\left({y}^{\prime }(Z)\left\{{Q}^{-1}x(Z)+\left({R}_f-\lambda \right)\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]\right\}\right) $$

for all functions y(Z), which implies that ${Q}^{-1}x(Z)+\left({R}_f-\lambda \right)\left[\mu (Z)-{R}_f\underset{\bar{\mkern6mu}}{1}\right]=0$ almost surely in Z. Solve this expression for x(Z) to obtain Eq. 35.18, where the Lagrange multiplier λ is evaluated by solving for the target mean, μ _s . The expression for the optimal portfolio variance follows by substituting the optimal weight function into Eq. 35.17. Formulas for fixed R _f then follow directly. QED.

When the risk-free asset’s return is time varying and contained in the information set Z at the beginning of the portfolio formation period, the conditional mean variance efficient boundary varies over time with the value of R _f (Z) along with the conditional asset means and covariances . In this case, a zero-beta parameter, γ₀, may be chosen to fix a point on the unconditionally efficient-with-respect-to- Z boundary. The choice of the zero-beta parameter corresponds to the choice of a target unconditional expected return μ _s . For a given value of γ₀, the target mean maximizes the squared Sharpe ratio (μ _s − γ ₀)²/σ ² _s along the mean variance boundary, which implies μ _s = − (a + bγ ₀)/(b + cγ ₀).

When there is a risk-free asset that is constant over time, the unconditionally efficient-with-respect-to-Z boundary is linear (a degenerate hyperbola) and reaches the risk-free asset at zero risk. In this case, we use γ₀ = R _f and can obtain any μ _s larger or smaller than R _f , levering the efficient portfolio up or down with positions in the risk-free asset.

When there is no risk-free asset, we define portfolio s by letting x′ = x′(Z) = [x ₁(Z), …, x _N (Z)] denote the shares invested in each of the N risky assets, with the constraint that the weights sum to 1.0 almost surely in Z. The return on this portfolio, R _s = x′(Z)R, has expectation and variance as follows:

$$ {\mu}_s=E\left[{x}^{\prime }(Z)\mu (Z)\right], $$

$$ {\sigma}_s^2=E\left\{{x}^{\prime }(Z){\Lambda}^{-1}x(Z)\right\}-{\mu}_s^2, $$

where we have defined the N × N matrix

$$ \Lambda =\Lambda (Z)\equiv {\left\{E\left[\left.R{R}^{\prime}\right|Z\right]\right\}}^{-1}={\left[\mu (Z){\mu}^{\prime }(Z)+{\not\varSigma}_{\varepsilon }(Z)\right]}^{-1}. $$

Also, define the constants:

$$ {\delta}_1=E\left(\frac{1}{{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda \underset{\bar{\mkern6mu}}{1}}\right), $$

$$ {\delta}_2=E\left(\frac{{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda \mu (Z)}{\underset{\bar{\mkern6mu}}{1}\hbox{'}\Lambda \underset{\bar{\mkern6mu}}{1}}\right), $$

and

$$ {\delta}_3=E\left[{\mu}^{\hbox{'}}(Z)\left(\Lambda -\frac{\Lambda \underset{\bar{\mkern6mu}}{1}{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda}{{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda \underset{\bar{\mkern6mu}}{1}}\right)\mu (Z)\right]. $$

Theorem 2

(Ferson and Siegel 2001, Theorem 3) Given N risky assets and no risk-free asset, the unique portfolio having minimum unconditional variance and unconditional expected return μ _s is determined by the weights:

$$ \begin{array}{c}{x}_s^{\prime }(Z)=\frac{\underset{\bar{\mkern6mu}}{1^{\hbox{'}}}\Lambda}{{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda \underset{\bar{\mkern6mu}}{1}}+\frac{\upmu_s-{\updelta}_2}{\updelta_3}{\upmu}^{\hbox{'}}(Z)\left(\Lambda -\frac{\Lambda \underset{\bar{\mkern6mu}}{1}{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda}{{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda \underset{\bar{\mkern6mu}}{1}}\right)\\ {}=\frac{{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda}{{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda \underset{\bar{\mkern6mu}}{1}}+\left[\left(c+1\right){\upmu}_s+b\right]{\upmu}^{\hbox{'}}(Z)\left(\Lambda -\frac{\Lambda \underset{\bar{\mkern6mu}}{1}{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda}{{\underset{\bar{\mkern6mu}}{1}}^{\hbox{'}}\Lambda \underset{\bar{\mkern6mu}}{1}}\right),\end{array} $$

(35.19)

and the optimal portfolio variance is

$$ {\sigma}_s^2=a+2b{\mu}_s+c{\mu}_s^2, $$

where a = δ ₁ + δ ²₂ /δ ₃, b = − δ ₂/δ ₃, and c = (1 − δ ₃)/δ ₃.

The efficient-with-respect-to-Z boundary is formed by varying the value of the target mean return μ _s in Eq. 35.19. Note that the second term on the right-hand side of Eq. 35.19 is proportional to the vector of weights of an excess return, or zero net investment portfolio (post multiplying that term by a vector of ones implies that the weights sum to zero). The first term in Eq. 35.19 is the weight of the global minimum conditional second moment portfolio. Thus, Eq. 35.19 illustrates two-fund separation: any efficient-with-respect-to-Z portfolio can be found as a combination of the global minimum conditional second moment portfolio and some weight on the unconditionally efficient excess return described by the second term.

35.1.2 Proofs of Propositions

35.1.2.1 Proof of Propositions 2 and 3

We begin with Proposition 3. To see that Eqs. 35.15 and 35.16 are equivalent, use the fact that efficiency of R _s implies that ${\mu}_p={\gamma}_0+\frac{\sigma_{ps}}{\sigma_s^2}\left({\mu}_s-{\gamma}_0\right)$ and thus $\frac{\mu_s-{\gamma}_0}{\sigma_s^2}=\frac{\mu_p-{\gamma}_0}{\sigma_{ps}}$. Next, given any portfolio R _q ≠ R _c , we will show that α ² _q /σ ² _q < α ² _c /σ ² _c . Beginning with Eq. 35.16, we compute:

$$ {\sigma}_{cs}=\frac{\sigma_p^2{\sigma}_s^2-{\sigma}_{ps}^2}{\sigma_p^2-{\sigma}_{ps}} $$

and

$$ {\sigma}_c^2=\frac{\sigma_p^2\left({\sigma}_p^2{\sigma}_s^2-{\sigma}_{ps}^2\right)}{{\left({\sigma}_p^2-{\sigma}_{ps}\right)}^2}. $$

The efficiency of R _s implies that ${\mu}_c={\gamma}_0+\frac{\sigma_{cs}}{\sigma_s^2}\left({\mu}_s-{\gamma}_0\right)$, ${\mu}_p={\gamma}_0+\frac{\sigma_{ps}}{\sigma_s^2}\left({\mu}_s-{\gamma}_0\right)$, and ${\mu}_q={\gamma}_0+\frac{\sigma_{q\ s}}{\sigma_s^2}\left({\mu}_s-{\gamma}_0\right)$. Substituting these expressions and using the fact that σ _cp = 0, which follows from (35.16), we compute:

$$ \begin{array}{c}\frac{\alpha_c^2}{\sigma_c^2}-\frac{\alpha_q^2}{\sigma_q^2}=\frac{{\left({\mu}_c-\left[{\gamma}_o+\left({\mu}_p-{\gamma}_o\right){\sigma}_{cp}/{\sigma_p}^2\right]\right)}^2}{\sigma_c^2}-\frac{{\left({\mu}_q-\left[{\gamma}_o+\left({\mu}_p-{\gamma}_o\right){\sigma}_{qp}/{\sigma_p}^2\right]\right)}^2}{\sigma_q^2}\\ {}={\left({\mu}_s-{\gamma}_0\right)}^2\left(\frac{\sigma_{cs}^2}{\sigma_s^4{\sigma}_c^2}-\frac{{\left({\sigma}_{qs}{\sigma}_p^2-{\sigma}_{ps}{\sigma}_{qp}\right)}^2}{\sigma_s^4{\sigma}_p^4{\sigma}_q^2}\right)\\ {}={\left({\mu}_s-{\gamma}_0\right)}^2\left(\frac{\sigma_p^2{\sigma}_s^2-{\sigma}_{ps}^2}{\sigma_s^4{\sigma}_p^2}-\frac{\left({\sigma}_{qs}^2{\sigma}_p^4+{\sigma}_{ps}^2{\sigma}_{qp}^2-2{\sigma}_{qs}{\sigma}_p^2{\sigma}_{ps}{\sigma}_{qp}\right)}{\sigma_s^4{\sigma}_p^4{\sigma}_q^2}\right)\\ {}=\frac{{\left({\mu}_s-{\gamma}_0\right)}^2}{\sigma_s^4{\sigma}_p^4{\sigma}_q^2}\left({\sigma}_p^4{\sigma}_s^2{\sigma}_q^2-{\sigma}_p^2{\sigma}_q^2{\sigma}_{ps}^2-{\sigma}_{qs}^2{\sigma}_p^4-{\sigma}_{ps}^2{\sigma}_{qp}^2+2{\sigma}_p^2{\sigma}_{qs}{\sigma}_{ps}{\sigma}_{qp}\right).\end{array} $$

We now use the fact that σ ² _sp < σ ² _s σ ² _p (because, by assumption, R _p and R _s are not perfectly correlated) to see that

$$ \begin{array}{c}\frac{\alpha_c^2}{\sigma_c^2}-\frac{\alpha_q^2}{\sigma_q^2}\ge \frac{{\left({\mu}_s-{\gamma}_0\right)}^2}{\sigma_p^4{\sigma}_q^2{\sigma}_s^4}\left[{\sigma}_p^4{\sigma}_q^2{\sigma}_s^2-{\sigma}_p^2{\sigma}_q^2{\sigma}_{sp}^2-{\sigma}_p^4{\sigma}_{qs}^2-{\sigma}_{qp}^2\left({\sigma}_s^2{\sigma}_p^2\right)+2{\sigma}_p^2{\sigma}_{qs}{\sigma}_{qp}{\sigma}_{ps}\right]\\ {}=\frac{{\left({\mu}_s-{\gamma}_0\right)}^2}{\sigma_p^2{\sigma}_q^2{\sigma}_s^4}\left({\sigma}_p^2{\sigma}_q^2{\sigma}_s^2-{\sigma}_q^2{\sigma}_{sp}^2-{\sigma}_p^2{\sigma}_{qs}^2-{\sigma}_{qp}^2{\sigma}_s^2+2{\sigma}_{qs}{\sigma}_{qp}{\sigma}_{ps}\right)\ge 0\end{array} $$

where the final inequality follows from recognizing that the variance-covariance terms in parentheses are equal to the determinant of the (necessarily nonnegative definite) covariance matrix of (R _p, R _q, R _s ). This establishes the maximal property of R _c .

To show uniqueness, note further that the inequality will be strict (and we will have α ² _c /σ ² _c − α ² _q /σ ² _q > 0) unless we have both of the following conditions corresponding to the two inequalities in the final calculation: (1) σ _qp = 0 so that R _q and R _p are orthogonal, and (2) the covariance matrix of (R _p, R _q, R _s ) is singular so that R _q is a linear combination of R _p and R _s . However, there is only one portfolio orthogonal to R _p that can be formed as a linear combination λR _p + (1 − λ)R _s , and this solution is R _c . This establishes Proposition 3, which holds in the case of both Theorem 1 and Theorem 2, that is, whether or not there is a conditionally risk-free asset.

The expressions in Proposition 2 for the optimal weights, x _c (Z), follow from substituting portfolio weights from Theorem 1 (if there exists a conditional risk-free asset that may be time varying) or Theorem 2 (otherwise) into Eq. 35.15 and noting that the constants A and B represent the combining portfolio weights implied by (35.15) as R _c = AR _s + BR _p . Substituting, we see that Eq. 35.15 implies that the portfolio weight function Ax _s (Z) + Bx _p (Z) generates returns R _c , completing the proof of Proposition 2. QED.

35.1.2.2 Proof of Proposition 4

Let R _s denote the efficient-with-respect-to-Z portfolio corresponding to zero-beta rate γ₀. The efficiency of R _s implies that we may substitute ${\mu}_p-{\gamma}_0=\left({\mu}_s-{\gamma}_0\right)\frac{\sigma_{ps}}{\sigma_s^2}$ and ${\mu}_c-{\gamma}_0=\left({\mu}_s-{\gamma}_0\right)\frac{\sigma_{cs}}{\sigma_s^2}$ to find

$$ {S}_p^2+{S}_c^2=\frac{{\left({\mu}_p-{\gamma}_0\right)}^2}{\sigma_p^2}+\frac{{\left({\mu}_c-{\gamma}_0\right)}^2}{\sigma_c^2}=\frac{{\left({\mu}_s-{\gamma}_0\right)}^2}{\sigma_s^4}\left(\frac{\sigma_{ps}^2}{\sigma_p^2}+\frac{\sigma_{cs}^2}{\sigma_c^2}\right). $$

Next, substituting for σ _cs and σ ² _c from the above expressions, we verify that this expression reduces to S ² _s . QED.

Appendix 2: Methodology

We estimate the conditional mean functions, μ(Z), by ordinary least squares regressions of the returns on the lagged values of the conditioning variables. On the assumption that the conditional mean returns are linear functions of Z, these are the optimal generalized method of moments (GMM, see Hansen 1982) estimators. The covariance matrix of the residuals is used as the estimate of ${\not\varSigma}_{\varepsilon }(Z)$, which is assumed to be constant. These are the maximum likelihood estimates (MLE) under joint normality of (R,Z). In general, the conditional covariance matrix of the returns given Z will be time varying as a function of Z, as in conditional heteroskedasticity. Ferson and Siegel (2003) model conditional heteroskedasticity in alternative ways and find using parametric bootstrap simulations that this increases the tendency of the efficient-with-respect-to-Z portfolio weights to behave conservatively in the face of extreme realizations of Z.

The optimal orthogonal portfolio weights in Table 35.1 and Fig. 35.1 are estimated from Eqs. 35.7 to 35.8 in the text where, in the time-varying risk-free rate case, the Treasury bill return is assumed to be conditionally risk-free in Eq. 35.8. The benchmark portfolio x _p is a vector with a 1.0 in the place of the market index and zeros elsewhere. The matrix Q is estimated by using the MLE estimates of μ(Z) and ${\not\varSigma}_{\varepsilon }(Z)$ in the function given by Eq. 35.13. The parameter μ _p is estimated as the sample excess return on the market index, and γ₀ is the sample mean of the Treasury return, 3.8 %.

35.2.1 The Parametric Bootstrap

The parametric bootstrap is a special case of the simple, or nonparametric, bootstrap, itself an example of a resampling scheme. Introduced by Efron (1979), the bootstrap is useful when we wish to conduct statistical inferences, but when we either don’t have an analytical formula for the sampling variation of a statistic, don’t wish to assume normality or some other convenient distribution that allows for an analytical formula or have a sample too small to trust asymptotic distribution theory. The basic idea is to build a sampling distribution by resampling from the data at hand. In the simplest example, we have some statistic that we have estimated from a sample, and we want to know its sampling distribution. We resample from the original data, randomly with replacement, to generate an artificial sample of the same size, and we compute the statistic on the artificial sample. Repeating this many times, the histogram of the statistics computed on the artificial samples is an estimate of the sampling distribution for the original statistic. This distribution can be used to estimate standard errors, confidence intervals, etc. We can think of the bootstrap samples as being related to the original sample as the original sample is to the population. There are many variations on the bootstrap, and a good overview is provided by Efron and Tibshirani (1993).

In the simple, or nonparametric, bootstrap, no assumptions are made about the form of the distribution. It is assumed, however, that the sample accurately reflects the underlying population distribution, and this is critical for reliable inferences. For example, suppose that the true distribution was a uniform on [0, M]. In a sample drawn from this distribution, the maximum value is likely to be smaller than M, so that the bootstrap will likely understate the true variability of the data. This problem is obviously worse if the original sample has fewer observations. If the data are contaminated with measurement errors, in contrast, the extent of the true variability can be overstated. Even with large sample sizes, the bootstrap can be unreliable. For example, if the true distribution has infinite variance, the bootstrap distribution for the sample mean is inconsistent (Athreya 1987).

With a parametric bootstrap, we can sometimes do better than with a nonparametric bootstrap, where “do better” means, for example, obtain more accurate confidence intervals (e.g., Andrews et al. 2006). The idea of the parametric bootstrap is to use some of the parametric structure of the data. This might be as simple as assuming the form of the probability distribution. For example, assuming that the data are independent and normally distributed, we can generate artificial samples from a normal distribution using the sample mean and variance as the parameters. This is not exactly the right thing to do, because we should be sampling from a population with the true parameter values, not their estimated values. But, if the estimates of the mean and variance are good enough, we should be able to obtain reliable inferences.

To illustrate and further suggest the flexibility of the parametric bootstrap, consider an example, similar to the setting in our paper, where we have a regression of stock returns on a vector of lagged instruments, Z, which are highly persistent over time. Obviously, sampling from the Z’s randomly with replacement would destroy their strong time-series dependence. Time-series dependence can be accommodated in a nonparametric way by using a block bootstrap. Here, we sample randomly a block of consecutive observations, where the block length is set to capture the extent of memory in the data.

In order to capture the time-series dependence of the lagged Z in a parametric bootstrap, we can model the lagged instruments as vector AR(1), for example, retaining the estimator of the AR(1) coefficient and the model residual, which we call the shocks, U _z . Regressing the future stock returns on the lagged instruments, we retain the regression coefficient and the residuals, which we call the shocks, U _r . We generate a sample of artificial data, with the same length as the original sample, as follows. We concatenate the shocks as v = (U _z ,U _r ). Resampling rows from v, randomly with replacement, retains the covariances between the stock return shocks and the instrument Z shocks. This can be important for capturing features like the lagged stochastic regressor bias described by Stambaugh (1999). Drawing an initial row from v, we take the U _z shock and add it to the initial value of the Z (perhaps, drawn from the unconditional sample distribution) to produce the first lagged instrument vector, Z _t–1. We draw another row from v and construct the first observation of the remaining data as follows. The stocks’ returns are formed by adding the U _r shock to β z _t–1, where β is the “true” regression coefficient that defines the conditional expected return. The contemporaneous values of the Zs are formed by multiplying the VAR coefficient by Z _t–1 and adding the shock, U _z . The next observation is generated by taking the previous contemporaneous value of the Z as Z _t–1 and repeating the process. In this way, the Z values are built up recursively, which captures their strong serial correlation.