Cointegration and Tests of Present Value Models

Questions & answers

Q1. What is a “present value model,” and what three problems in the existing literature does the paper set out to fix?

A present value model states that Y_t = θ(1-δ)·Σ δ^i·E_t[y_{t+i}] + c, so that Y is a linear function of the discounted present value of expected future y (eq. 1) – a form that nests the expectations theory of the term structure (Y the long rate, y the short rate), the present-value model of stock prices (Y the price, y the dividend), and, with modification, the permanent-income theory of consumption. The authors identify three sources of the “surprising degree of controversy” over such models: it is unclear how the different test procedures in the literature (single-equation regressions, VAR cross-equation restriction tests, variance-bounds tests) relate to one another; a statistical rejection may have little economic significance, since the model could still explain most of the variation in Y even if formally rejected at the 5 percent level; and the usual practice of removing a deterministic linear trend from y and Y “can bias test procedures against the model” if the variables are instead stationary only in first differences.

Q2. What is the “spread,” and why does the present value model imply it is stationary even when y and Y are not?

Defining the spread S_t ≡ Y_t - θy_t, the authors show the present value model implies S_t = δ/(1-δ)·E_t[Σδ^i Δy_{t+1+i}], a discounted sum of expected future changes in y, which is stationary whenever Δy is stationary (eqs. 3-4) – even though y and Y themselves need only be stationary in first differences (i.e., integrated of order one). This is the paper’s cointegration result: “St is proportional to the optimal forecast of S*, the present value of future Δy, and also to the optimal forecast of ΔY.” For the term structure St is literally the spread between short and long rates; for stocks it is the (scaled) difference between the stock price and a multiple of dividends.

Q3. How does the paper turn the present value model into a testable restriction on a VAR, and what two complementary ways does it use to evaluate the model’s fit?

The authors estimate a bivariate VAR in Δy and S_t, then show the present value model implies specific linear cross-equation restrictions on the VAR’s companion matrix (eq. 8), restrictions numerically equivalent to a Wald test that S_t - δS_{t-1} + θΔy_t is unpredictable given lagged Δy and S. Beyond this formal statistical test, the VAR lets them construct the “theoretical spread” S*_t – the VAR’s own unrestricted optimal forecast of the present value of future Δy given the full VAR information set – and compare its time-series behavior, standard deviation, and correlation with the actual spread S_t; a large and persistent gap between S_t and S*_t, rather than mere statistical rejection, is their preferred metric of “economically important deviations from the model,” since, as they note, “predictable excess returns must be persistent as well as variable” for a deviation to be economically large by this standard.

Q4. What do the term-structure (bond) results show, and why do the authors read the statistical rejection as overstated?

Formal tests of the expectations-theory restrictions reject strongly – at less than the 0.005 percent level in the 1959-78 sample and the 0.03 percent level in the 1959-83 sample – yet the VAR-implied theoretical spread S_t tracks the actual spread S_t closely: their standard deviations are 1.060 versus 1.067 in the short sample (correlation 0.978) and 0.717 versus 1.320 in the long sample (correlation 0.956).* The innovations variance ratio Var(δ)/Var(δ’), which should equal one under the model and exceed one if long bond returns “move too much,” is estimated at 1.160 in the short sample and only 0.502 in the long sample, with large standard errors in both cases, so the hypothesis that the ratio equals one cannot be rejected. The authors attribute part of the sharp statistical rejection to the sensitivity of ordinary regression tests to small amounts of noise added to the model, illustrated with an algebraic example in which a regression of ΔY on S can flip sign even when a regression of S on ΔY, their preferred specification, remains well behaved.

Q5. What do the stock-price results show, and how does the choice of discount rate affect the conclusion?

Using 1871-1986 annual U.S. data, the authors test the model at two discount rates: the 8.2 percent sample mean stock return, and a 3.2 percent rate implied by their own cointegrating regression of the stock price level on the dividend level. At the 8.2 percent rate, the full set of model restrictions is rejected at the 7.2 percent level and the dynamic restrictions alone at the 4.7 percent level, and the innovations variance ratio is estimated at 11.27 (standard error 4.49) – a statistically significant rejection of “move too much.” At the 3.2 percent rate, by contrast, the full restrictions are rejected more strongly, at 1.1 percent, but the innovations variance ratio falls to 1.41 (standard error 0.44) and the dynamic-restrictions-only test no longer rejects (21.8 percent). The authors conclude the stock-market evidence against the present value model is real but “the strength of the evidence…depends sensitively on the discount rate assumed in the test,” and flag that the low 3.2 percent discount rate itself may be an artifact of the postwar boom in stock prices dominating the cointegrating regression, since over 1871-1946 alone the spread is negatively correlated (-0.7) with lagged dividends under that same discount-rate assumption.

Q6. How does the paper’s overall verdict differ between bonds and stocks?

Both the term-structure and stock-price models are statistically rejected at conventional significance levels, “with much stronger evidence for bonds”; but by the authors’ informal fit metric the two asset classes diverge sharply – the bond spread “moves quite closely” with the present value of expected short-rate changes and deviations “do not appear to be highly persistent,” whereas for stocks “the spread between stock prices and dividends moves too much, and…deviations from the present value model are quite persistent.” A supplementary regression-based check (following Scott 1985) of S* on S, which should yield a coefficient of one under the model, estimates coefficients of 0.77-0.81 for bonds but -0.06 to 0.16 (effectively zero or negative) for stocks, reinforcing that the present value model fits the bond data comparatively well despite its formal rejection, while fitting the stock data poorly even though the stock rejection is not overwhelmingly strong at every discount rate.