RDP 2011-06: Does Equity Mispricing Influence Household and Firm Decisions? 3. Estimation Methodology

I use two ideas to identify the effects of equity mispricing. The first is that equity mispricing should only have transitory economic effects (see, for example, Lee (1998)). Such an assumption appears reasonable from a theoretical standpoint, given that many economists have the prior that equity prices are not entirely disconnected from the fundamental processes underpinning the economy. If the converse were true, and mispricing shocks had permanent effects, then equity prices would effectively be indeterminate and have no relationship with the underlying value of the dividend streams they pay.

The second idea is that there is observable information that can be used to distinguish between fundamental and non-fundamental transitory shocks. This reasoning follows a recent literature which argues that there are observables that are correlated with equity market mispricing, and that are uncorrelated with measures of economic fundamentals, see, for example, Diether et al (2002) and Gilchrist et al (2005).

I use these two ideas, in conjunction with a cointegration framework implied by economic theory, to identify the effects of equity mispricing shocks. More specifically, I use five economic relationships to motivate the empirical work in this paper. The first is an accumulation equation for aggregate household wealth

where W_t is beginning of period wealth, C_t is total flow consumption in the period, and is the return to total wealth. This formulation assumes that the market value of human capital is tradeable and included in aggregate wealth. This assumption simplifies exposition, but is an assumption that can be relaxed without substantively affecting any of the analysis that follows (see Lettau and Ludvigson (2004)). The second relationship used is that household wealth can be decomposed into its respective equity, non-equity, and human capital components

where E_t is total equity wealth held by households, N_t is total non-equity wealth (such as housing, consumer durables, and other forms of financial non-equity wealth), and H_t is human capital. The third and fourth relationships used are an accumulation equation for tradeable human capital, and the definition of equity wealth

where is the return to human capital, Y_t is labour income, Q_t is the quantity of equity held, P_t is the ex-dividend price of equity, and D_t is the dividend paid on equity held in period t. The final relationship used is the definition of the return to equity, , where

Using arguments that are similar to those used by Campbell and Mankiw (1989), Lettau and Ludvigson (2004, 2005) and Kishor (2007), I log-linearise these relationships, assuming a balanced growth path, and obtain the following economic system^[7]

where lower case variables denote natural logarithms,^[8] ω_e, ω_n and ω_h are the steady state shares of equity, non-equity and human capital wealth in total wealth respectively, ρ_w is the steady state share of savings in total wealth, ρ_h is one minus the share of labour income in steady state human capital, and ρ_d is the steady state ratio of the ex-dividend equity price to the equity price that includes dividends. It should be noted that the system defined by Equations (1) to (5) contains two variables that are not directly observable, human capital wealth and total household wealth. To account for this, I substitute human capital and total wealth out of the above system to obtain^[9]

Assuming that consumption, the quantity of equity held, non-equity wealth, equity prices, labour income, and dividends are integrated of order one, and that returns to total financial wealth, human capital wealth and equity wealth are stationary, it follows that Equations (6) to (8) make up a cointegrated system with two cointegrating vectors.

It should be made clear that Equations (6) to (8) make up a partially specified economic system. Additional model structure, for example including an Euler equation for consumption or an equity pricing equation, could potentially imply more restrictions or additional cointegrating relationships in this system. I choose not to include such structure given existing disagreement over the ‘correct’ model for either consumption or equity prices. Instead, I use the above framework as a motivation for modelling a system consistent with Equations (6) to (8), and use empirical analysis to determine the number of cointegrating relationships. I do not impose any model-specific restrictions that could otherwise be incorporated.

A general econometric representation that is consistent with Equations (6) to (8) is the structural vector error correction model (SVECM)

where y_t is an n × 1 vector of observables, y_t = [ c_t d_t n_t y_t q_t p_t ]′, A(L) is a lag polynomial of order l, β′ is the matrix of cointegrating vectors, and α^* the matrix of loading coefficients on the cointegrating vectors.^[10] I assume A₀ is non-singular and that α^*β′ has rank r < n so that at least one cointegrating vector exists. The ε_t are the primitive structural shocks. I assume these shocks are independently identically distributed, with E (ε_t) = 0 and

where Ω is a diagonal matrix (with elements that are not necessarily equal).^[11] The ε_t are the underlying structural shocks that I am in interested in identifying. Specifically, I wish to identify the elements of ε_t that only have transitory effects, and in particular, non-fundamental transitory effects.

It should be noted that the structural shocks being serially uncorrelated is not necessarily a restrictive assumption in the current context. In particular, Equation (9) can be viewed as a finite-order approximation of a model in which the structural shocks are serially correlated (see, for example, Lütkepohl (2006)). That is, Equation (9) can be viewed as an approximation of a SVECM with moving average errors,

where Ψ(L) is an infinite-order lag polynomial. In this model, transitory mispricing disturbances in ν_t can be serially correlated with permanent shocks to fundamentals, an assumption that is consistent with the idea that permanent shocks to fundamentals, such as permanent changes in technology, can precede mispricing in the equity market. In the analysis that follows, I focus on estimating Equation (9), which can be interpreted as a finite-order approximation of Equation (11).^[12]

3.1 Identification of Reduced-form Shocks

To distinguish between the reduced-form transitory and permanent shocks in Equation (9), I follow a re-parameterisation of the approach to identification suggested by Pagan and Pesaran (2008). Without loss of generality, I order the permanent and transitory shocks according to

where is a (n – r) × 1 vector of shocks with permanent effects , and is a r × 1 vector of shocks that have transitory effects .^[13] Since I assume that mispricing shocks have only transitory effects on the system, a mispricing shock must be an element of .

I proceed by estimating Equation (9) using limited information methods. The first step is to obtain a consistent estimate of the cointegrating matrix, β (or use the known cointegration matrix in the case that β is known). Importantly, as emphasised by Pagan and Pesaran, only a consistent estimate of the cointegration space – the column space of β, – is required since the instrumental variable (IV) methods described below are invariant to non-singular transformations. A consistent estimate of this space can be obtained, for example, from the Johansen full information maximum likelihood (FIML) estimates of the reduced form of Equation (9) or using alternative system methods discussed by Lütkepohl (2006).

Assuming a consistent estimate of β is available, I partition Equation (9) into a system of n – r equations with permanent shocks, , and r remaining equations with transitory shocks

and I assume, without loss of generality, that A(L) = A₂L. This assumption abstracts from lag dynamics that do not affect the generality of the identification approach proposed.

Since I previously assumed , where Ω is a diagonal matrix, I impose n normalisation restrictions on the main diagonals of and . I further assume that is non-singular. With these assumptions, a simple matrix premultiplication yields

where

The above premultiplication is useful because it allows identification of transitory shocks, without requiring identification of the permanent shocks to the system. That is, I only identify linear combinations of the permanent shocks, , and not the underlying permanent structural shocks, .

Using the result that lagged error correction terms should not be present in the structural permanent equations, = 0,^[14] one can use these restrictions to estimate the first n – r permanent equations in Equation (13). Specifically, this set of restrictions implies that the r × 1 vector ξ_{t – 1} = β′y_{t – 1} can be used as instruments for the vector Δy_2t. And so, the first n – r permanent equations

can be estimated using standard IV methods. This provides consistent estimates of the reduced-form matrices , and the reduced-form permanent shocks, .

To estimate the remaining r transitory equations

I can now use the consistent estimates, , as instruments for the endogenous variables in Δy_1t (see Pagan and Pesaran (2008)). This would enable identification of the reduced-form transitory shocks, , but to identify the structural transitory shocks, , it is clear that additional restrictions are required.

3.2 Identification of Structural Transitory Shocks

Focusing on the transitory equations in Equation (15), recall that I have already imposed r normalisation (unity) restrictions on the main diagonal of . Since I have previously assumed that transitory shocks are uncorrelated (see Equation 10), this implies that an additional r (r – 1)/2 additional restrictions are required to be able to identify the structural shocks, . Although one could proceed by imposing additional restrictions on the elements of , or using restrictions on any of , ,^[15] in some applications such restrictions may not be appealing on theoretical grounds. This is the case, for example, when attempting to distinguish between fundamental and non-fundamental transitory shocks as considered in the empirical application below.

Instead, I assume there exists additional observable information available to the researcher that allows identification of , or at least some of the elements in this transitory shock vector. Specifically, I assume that Equation (15) can be partitioned in a form that is consistent with the presence of an (r – 1) × 1 vector of fundamental transitory shocks, , and a single non-fundamental transitory shock, ,^[16]

I further assume there exists an observable instrument (or set of instruments) vector (κ ≥ 1), with the properties that,

That is, there exists one or more instruments for equity prices growth that are correlated with mispricing shocks, and contemporaneously uncorrelated with either fundamental transitory or permanent shocks.^[17] Assuming is non-singular, again using a simple premultiplication of Equation (16) yields

where

This system can be estimated using a method analogous to that used for the permanent equations. Specifically, one can estimate the first r – 1 transitory equations using and Z_t as instruments for Δy_1t and Δy_22t respectively. The residuals and can then be used as instruments for Δy_21t and Δy_1t when estimating the final transitory equation.

In sum, this procedure enables identification of the structural mispricing shock, , including associated impulse response functions and forecast error variance decompositions that identify the effects of this shock. If alternative instruments, or valid restrictions can be imposed to identify the effects of fundamental transitory shocks, then these too can be used. However, such restrictions are not required to identity the effects of the mispricing shock.

In the empirical application that follows I eliminate Equation (7) from Equations (6) to (8) and order the vector of observables such that y_t = [c_t d_t n_t y_t q_t p_t]′, and so y_1t = [c_t d_t n_t y_t]′ and y_2t = [q_t p_t]. That is, there are two cointegrating vectors (transitory shocks) in the system (n = 6, r = 2),^[18] and I assume that these shocks have direct effects on the quantity and price of equity held by households, and indirect effects on consumption, dividends, non-equity worth and labour income. The latter variables are also those directly perturbed by permanent fundamental shocks.

Footnotes

In taking these approximations, I assume that each variable in the system can be normalised by an appropriate trend (for example, the level of productivity or another variable that captures the long-run growth rate of the economy), and that limit terms associated with iterating these relationships forwards are small (of second-order). I omit linearisation constants and growth rates in unobserved trends in the above approximations. [7]

Note for returns I use the normalisation r_t+i Ξ 1n(1+r_t+i). [8]

Without loss of generality, I assume when making these substitutions that ρ_w = ρ_h. [9]

Note I substitute e_t out of the system in Equations (6) to (8) in the analysis that follows. [10]

Rather than assuming impose normalisation (unity) restrictions on the main diagonal of A₀. [11]

Lütkepohl (2006) provides a review of the regularity conditions under which such an approximation will be valid. [12]

The fact that the number of transitory shocks is equal to the number of cointegrating vectors is an implication of the Granger representation theorem. Lütkepohl (2006) provides a useful review. [13]

This result is derived in Pagan and Pesaran (2008) with respect to Equation (12), and is consistent with the ordering of permanent and transitory shocks. [14]

This is the approach followed by Pagan and Pesaran (2008) after fully identifying the effects of permanent shocks. [15]

I use the notation that . A similar partition is used with respect to α^*. [16]

To be clear, only the first two conditions are required for identification. I use the stronger requirement since the proxies for mispricing have desirable properties when used as instruments in estimating the permanent equations. [17]

This is confirmed by cointegration matrix rank tests (see Appendix B). [18]