This study dives deep into the complexities of long-term stock market predictability, utilizing Tobin's Q and dividend yield across a 119-year dataset. By scrutinizing the equilibrium relationship within the US aggregate stock market, the research offers practitioners robust forecasting tools, challenging conventional perspectives on market returns.
This research adopts an approach leveraging Tobin's Q and dividend yield. Spanning over a century of data, the study constructs a comprehensive Vector Error Correction Model (VECM) for the US stock market, providing practitioners with effective forecasting capabilities. The results challenge conventional ways, emphasizing the significance of long-term predictions and offering valuable insights applicable to the landscape of UK price controls.
PJ McCloskey
Candidate number: 13141288
Supervisor:
Dr. Roald Versteeg
September 2019
Department of Economics, Mathematics and Statistics
Birkbeck College, University of London
Stock markets world-wide have rewarded patient investors, hence the common adviceto “buy-and-hold”. Yet, even with a large body of research over a prolonged period,proof remains an onerous exercise for academics. We use Tobin’s Q and the dividendyield to build an equilibrium relationship for the US aggregate stock market using 119years of data. The resulting VECM model supports practitioners making long-horizonpredictions and provides powerful forecasting ability. Our work is directly applicable toUK price controls
Keywords: stock market; expected returns; predictability; valuation ratios; Tobin’s Q; VECM.
* I thank Roald Versteeg, Donald Robertson and Simon Price for their generous support.
The Expected Market Return (EMR) [1] is the real return that equity investors expect from a well-diversified portfolio. Its use in asset pricing models, investment allocation strategies, pension scheme valuations, and the cost of monopoly services such as water and energy utilities makes it a very important value for investors, consumers, and finance professionals.
The EMR is not directly observable, so practitioners are faced with a common challenge: how should we estimate an expectation? Broadly, there are two options: forward-looking predictions or outturn averages.
Most practitioners make forward-looking predictions. Examples include JP Morgan (2019), Schroders (2019), Vanguard (2019), Willis Towers Watson (2019), Aberdeen Standard Investments (2019), SEB Group (2019), UBS (2019), and Aon Hewitt (2018). However, this approach is inherently subjective and assumption intensive.
In contrast, some practitioners focus on outturn averages. Examples include the Competition Commission (CC) (2014) and Ofgem (2019). Although this approach is less subjective, it assumes that the outturn average remains appropriate.
We use the term “predictability” to refer to the accuracy of EMR forecasts over long holding periods, relative to realised returns. In this way, we need not contrast with the Efficient Market Hypothesis (EMH). We assume markets are very efficient in the short run but less efficient in the long run, and that the EMR varies through time.
Some academics argue that the EMR is lower than historical averages imply. Mehra and Prescott (1985) argue that ex-post returns are inexplicable in the context of standard models of risk. Campbell and Shiller (1988) and Shiller (1990) focus on whether there is “excess volatility” relative to the present value of dividends, with Shiller arguing that markets can be irrationally exuberant. Fama and French (1988) assess the power of dividend yield regressions, showing that explanatory power grows with the investment horizon. They appeal to the simple logic that EMR changes are offset by immediate decreases in the current price, and thus mean reversion arises from time variation of expected returns. Blanchard (1993) argues that, relative to risk-free rates, ex-post equity returns are far in excess of what is justified by standard asset-pricing models with reasonable levels of risk aversion.
A common thread in the literature is that, over long horizons, ex-post returns are less uncertain than expected, given the uncertainty of one-year returns. Dimson, Marsh and Staunton (DMS) (2001) show that worldwide stock returns have a degree of long-term consistency. Campbell and Viceira (2001) show how predictability is applied in portfolio allocation strategies. Robertson and Wright (2002b) present a Vector Error Correction Model (VECM), using dividends and asset values, Tobin’s Q, to explain reductions in volatility. In support of Smithers and Wright (2000), Harney and Tower (2003) demonstrate the predictive power of Tobin’s Q, disagreeing with Epstein (2000), who criticised Smithers and Wright. The most forceful advocate of predictability, Siegel (2008), argues that bonds are riskier than stocks in the long term. Cochrane (2008) attempts to address statistical concerns by explaining that the absence of dividend growth predictability supports the argument for return predictability. Tower (2011) develops an updated model for CAPE, Tobin’s Q and CAPER, and argues that Tobin’s Q plays the largest explanatory role.
Kim, Nelson and Startz (1991) argue that mean reversion is entirely a pre-war phenomenon. Epstein (2000) dismisses the use of Tobin’s Q as a value indicator, arguing that value indication is not its intended purpose and that “new economy” and intangible factors are unduly ignored. Other academics challenge the econometric properties, the statistical significance, the power of the statistical tests, and the statistical bias in favour of predictability inference. Examples include Torous, Valkanov and Yan (2004), Campbell and Yogo (2006), Goyal and Welch (2003), and Campbell and Thompson (2008). Boudoukh, Richardson and Whitelaw (2005) argue that the econometric problem is one of overlapping observations and persistence of the predictive variable. Ang and Bekaert (2007) argue that, after accounting for small sample properties of standard tests, excess return predictability by the dividend yield is not statistically significant.
If we believe that the stock market can deviate from its fundamental value for a prolonged period, the current literature is light on detail with regards to Tobin’s Q or other long-run fundamentals, with CAPE found to be second best when both are used. Earnings ratios also lack consistent measurement over time given changes to accounting rules, and are difficult to consolidate at a macroeconomic level for long periods. Further, CAPE ratios can be strongly correlated with Tobin’s Q and may therefore lack added information value; hence, we do not include these. There is also a lack of evidence for low-frequency data, such as annual rather than monthly data, over long horizons of more than five years.
Our approach to test predictability, using a long-horizon weak-form approach, therefore:
• uses physical asset values, Tobin’s Q, and dividends as predictors
• uses annual data, focusing on the US non-financial sector
• tests return variances in 21 countries
• tests stationarity and cointegration, building a VECM model for the US equilibrium
• presents impulse responses
• compares VECM predictions with outturn averages, in-sample and out-of-sample
• applies this work to UK price controls
In doing so, we follow closely Robertson and Wright, The Good News and the Bad News about Long-Run Stock Returns (2002b). We do not attempt to explain the risk premium of stock returns, as discussed by Favero and Gozluklu (2009), as this would, in our view, introduce more difficulty than is strictly necessary.
For the aggregate stock market, Tobin’s Q is the ratio of companies’ market value to the replacement value of companies’ assets. Theoretically, market value should equal asset replacement value – the ratio should never deviate from unity. By extension, a ratio more (less) than unity indicates poor (good) value. Empirically, measurement issues arise for assets – depreciation may be underestimated, intangible assets may not be captured, and/or markets for assets may be non-existent or illiquid. To avoid primary measurement issues, we focus on deviations from the average value of Tobin’s Q, rather than the absolute value. Thus, when the stock market is overvalued (undervalued), Tobin’s Q will be high (low), relative to its average value. A working paper from the Bank of England notes:
“…the common perception that Q is interesting from a theoretical perspective, but of little empirical relevance, is not true. In contrast, it appears to be a rich source of information about real and financial quantities.” (Bank of England, 2006, p. 5)⁴
In levels, equity Q can be defined as:
Pₜ
Qₜ = ─── (1)
Kₜ
Where Pₜ represents stock market value and Kₜ represents capital replacement value. In logarithms,
qₜ = pₜ − kₜ (2)
We define log equity returns following Campbell & Shiller (1988) as follows,
rₜ₊₁ ≡ ln(1 + Rₜ₊₁) = ln ⎛ Pₜ₊₁ + Dₜ₊₁ ⎞
⎝ Pₜ ⎠
≈ φ + Δpₜ + (1 − ρ)(dₜ₊₁ − pₜ₊₁) (3)
Wherein ρ = 1 / (1 + exp(d̄ − p̄)) and φ = ln(1 + exp(d̄ − p̄)) − (1 − ρ)(d̄ − p̄) and d̄ − p̄ is the mean dividend yield. Substituting from (2) into (3) to eliminate pₜ, solving for qₜ, and iterating forwards, subject to the transversality condition limᵢ→∞ ρⁱqₜ₊ᵢ = 0, we obtain:
φ
qₜ ≅ ─────── + ∑ᵢ₌₁∞ ρⁱ⁻¹fₜ₊ᵢ (4)
(1 − ρ)
where fₜ = Δkₜ + (1 − ρ)(dₜ − kₜ) − rₜ
In other words, q is asymptotically equal to a weighted sum of the future values of f, which in turn is
In other words, q is asymptotically equal to a weighted sum of the future values of ƒ, which in turn is the sum of; changes in capital ∆k (i.e. total capital), future profitability, (d − k) and returns, 𝑟.However, this design assumes 100% equity financing; to extend this model further, we include theimpact of leverage, adding 𝐿𝑡to the numerator of (1), and updating (2) and (4) accordingly, to derive
qₜ ≅ ∑ᵢ₌₁∞ ρⁱ⁻¹eₜ₊ᵢ (5)
where fₜ = Δkₜ + (1 − ρ){(1 − ζ)(dₜ − kₜ) + ζ(lₜ − kₜ)} − ζΔlₜ − (1 − ζ)rₜ
This model should be relatively uncontroversial given the following intuition.⁵ Simply, (5) shows that q reflects total capital Δkₜ, adjusted downwards for total liabilities ζΔlₜ, and further downwards for total returns to equity, (1 − ζ)rₜ, wherein ζ represents total leverage, such that equity returns decrease, in levels not proportions, as leverage increases. Further, any upward pressure on q, in fundamental rather than measured terms, must arise from increases in capital, increases in profitability as proxied by (dₜ − kₜ), reductions in liabilities, or reductions in returns.
Thus, this formulation of q serves three very important purposes. First, it confirms an analytical grounding that links q to equity returns r. Second, we can naturally see that if q is mean-reverting, the components on the right-hand side, individually or collectively, must be mean-reverting also. Third, and most significantly, we have confirmed clear theoretical grounds that equity returns are bounded by underlying asset values.
In this sense, “predictability” is an entirely natural occurrence, caused by rational economic forces. Plausibly, it could nonetheless be argued that, if mean-reversion of q is weak, r could still be random, or indistinguishable from random. However, Poterba & Summers (1983) demonstrate that q has little explanatory power for investment. Leaving three other parameters that mean-reversion of q could explain: (i) liabilities, l, which has considerably less uncertainty given natural bondage with capital and its fixed-income nature; (ii) dividends, d, which we address in the following section; and lastly, (iii) equity returns, r.
The stock market should equal the discounted value of future dividends and other distributions to equity. With a constant discount rate and a constant growth rate, increases or decreases in dividends/distributions should be reflected in increases or decreases in Pₜ.⁶ If the dividend/price ratio is higher or lower than average, this may indicate the market is undervalued or overvalued. We therefore use the Dividend Yield (DY), or when including other cashflows to/from equity, the Adjusted Dividend Yield (ADY), as a fundamental indicator of value. The ADY captures all flows between equity investors and corporations, including dividends, stock repurchases, new issues, cash-financed mergers and acquisitions, and private equity issuance. In contrast, the DY captures only dividend payments. ADY is therefore a more comprehensive measure of value — we will demonstrate other differences below.
As is standard with all time-series analysis, we inhabit a log-linear world, following standard notation of using lower case letters to denote logs. To avoid making spurious inferences, q and ady need to be stationary, which we can refer loosely to as “mean-reversion”.
Each indicator, but particularly q, is strongly linked to a time-series approach, motivating us to obtain a long time-series of 119 years, to then estimate how far, and for how long, stock prices can stray from underlying fundamentals. Westerlund & Narayan (2014), in contrast, argue that precision may be increased using panel data rather than a long time-series, effectively trading Ns for Ts. However, for our purposes, we are not persuaded. A larger number of Ns, which in our case could be international stock markets, instead of Ts, meaning longer time series, would come at a cost of understanding the maximum disequilibria and/or average equilibrium that a major market is content to bear. Our valuation theory is that equity values cannot stray unbounded from q or ady and that future returns reflect previous values of underlying fundamentals. Following standard practice, we use total real returns throughout.
We collect 119 years of annual data for the US non-financial sector, using data from Wright (2004b) for the years 1900 to 1945, and from FRED, US financial accounts, for the years 1945 to 2018 regarding annual balances⁷ and annual flows.⁸ Wright’s methodology (2004a) and data are published online, and we follow these carefully to build on the original research.⁹ Following Wright, and to avoid a discontinuity in the FRED data regarding land values, we re-create an alternative series that is more consistent over time. Further, we adjust corporate bonds and mortgages to market values on the basis that FRED data on these items are primarily recorded at book values. Without this adjustment, the use of book values causes two problems: first, q would be downward biased, as liabilities represent a significant value of corporate worth; second, a negative value for net liabilities makes it impossible to input log values to our econometric model. We convert from nominal using Shiller’s inflation series¹⁰, noting that it is simply a rebased version of Wright’s data. We publish our dataset including reconciliations to Wright’s original work.¹¹
To supplement this, we obtain an unrivalled dataset from DMS, on equities, bonds, inflation and exchange rates, for 21 countries for the same 119-year period, 1899 to 2018.¹² For the UK, and for the period 1995 to 2018, we derive q from ONS national balance sheet estimates for nonfinancial corporations and use S&P Capital IQ to collect FTSE All-share information for 631 firms’ ADY.¹³ We supplement this with a discontinued series from the Financial Times on FTSE All-share dividend yields from 1963 to 2016. We sought, but unfortunately did not obtain, data from the Bank of England on q.
For 21 countries including the US and UK, we record the variance of ex-post equity returns overdifferent holding periods. Figure 1 shows the results for the US market; as the investment horizonincreases, actual returns are more certain than simulated multi-year returns using 1-year variances. Theintuition here is that patience is rewarded – a 15-year investment horizon is very likely (95%probability) to produce positive returns.
Figure 1: Equity returns in the US, 1900 to 2018, actual compared with simulation, 90% confidence
This intuition is not unique to US stock markets – similar results appear for most countries, with anarrower 90% confidence band for the UK than for the US. This result is consistent with Robertson &Wright (2002b).
Similarly, for 21 countries, Figure 2 compares the h-year variance with the 1-year variance.
Figure 2: Variance ratios for 21 stock markets over 119 years
Relative to the holding variance at the 1-year horizon, the US (UK) 20-year horizon is 22% (31%)lower. Figure 2 also shows that only two countries, Spain and Japan, demonstrate a variance increase,whereas 19 other countries demonstrate a decrease after approximately nine years. Denmark is anoutlier, with the 20-year variance being only 31% of its 1-year variance, implying that patience isrewarded most in Denmark
These are however essentially point-estimates and therefore do not reflect the confidence interval,which also increases with the horizon. Further, for a globally diversified portfolio, the overall varianceratio will reflect stock market weightings and correlations. As at the start of 2019 the world stockmarket is dominated by the US (53.3%), followed by Japan (8.4%) and the UK (5.5%) (Credit Suisse,2019).
These results are consistent with other research, including Poterba & Summers (Mean reversion instock prices: evidence and implications, 1988) and the associated variance ratio tests (see for exampleLo & MacKinlay (1989) and Amélie & Darné (Variance ratio tests of random walk: An overview,2009))
We observe a high degree of persistence in q (Figure 3) with a tendency to cross its mean infrequently.We clearly see the impact of stock market crashes in 1929, 2001 and 2008, each helping to bring qback to its long-term average (as identified by 0% - a proxy for equilibrium).
Figure 3: Measured Tobin’s q, log deviation from its mean15, 1900 to 2018
dy and ady are compared in Figure 4 below – ady has remained in-or-around its long-term average (asidentified by 0% - a proxy for equilibrium).
Figure 4: Dividend Yield (dy) & Adjusted Dividend Yield (ady), log deviations from respective means
The ady has markedly different properties from the dy – more volatile, and more frequently crossing(and recently greater than) its mean. In contrast, dy is less volatile – arguably due to stickiness, with anotable drop below its mean for most of the previous 40 years. The main difference between ady and dy is that payout policies have shifted, from dividends to (material levels of) stock repurchases (for further explanation see Bunn & Shiller (2014)). We note however that other explanations have been used to explain the non-stationarity of dy. For example, Geanakoplos, Magill, & Quinzii (2004) argue that demographic factors, not repurchases, explain the change over time. However, given that stationarity is so easily explained by using a comprehensive measure, demographic explanations seem, in our view, motivated by a problem (non-stationarity) that doesn’t exist.
More formally, a null hypothesis of a unit root (non-stationarity) is rejected for ady at the 1% level over various sub-samples (see Table 1). Note, in contrast, results for q are less conclusive, particularly for 1900 to 2018. This reflects its persistence and the prolonged departure from its average since 1990,weakening the statistical evidence for periods thereafter.
An additional KPSS test (‘confirmatory data analysis’ as referenced by Brooks (2008)) does not provide muchfurther clarification.
For q, over the time period 1900-2018, all three tests indicate an absence of stationarity. In contrast,over the same time period, dy has conflicting results, with two tests (ADF & KPSS) indicating anabsence of stationarity. The ady however, provides a clear message, with all three tests indicatingstationarity over all periods, with the ADF test from 1900-1990 the only exception, and one of onlytwo results that differ notably from Robertson & Wright (2002b). Overall, in light of our theoreticalpriors and explation for Tobin’s q test results, we focus hereafter on ady and q.
These two indicators, ady and q, have mostly, but not always, provided the same signal in terms ofvalue. To see this, Figure 5 below presents inverse-ady (iady) alongside 𝑞, such that stock market valueis the numerator for both ratios. Therefore, when each ratio is above (below) 0% it indicates a high(low) price, and thus poor (good) value, relative to its respective average. By extension, when bothratios are above 0%, it indicates a consistent signal: poor value by both measures. The previous 20years represent a predominantly mixed signal – with 𝑞 indicating poor value (too high a price relativeto underlying capital value) while iady indicates good value (most recent year aside).
Figure 5: Two value indicators, relative to mean values
In our econometrics, we follow this approach, combining q with the iady (in contrast with Robertson& Wright (2002b) who combine q with ady).
Cointegration is an econometric technique that helps us link our theory on fundamentals with observed returns. Fabozzi provides the following intuition:
“Cointegration is one of the key concepts of modern econometrics. Let’s startby giving an intuitive explanation of cointegration and its properties. Two ormore processes are said to be cointegrated if they stay close to each other evenif they “drift about” as individual processes. A colorful illustration is that ofthe drunken man and his dog: Both stumble about aimlessly but never drift toofar apart.” (Fabozzi, 2006, p. 373)
It is easy to understand that stock market levels (p) are more likely to be predictable when jointlyconsidered with assets/capital (k) and dividends to equity (d). Adding corporate liabilities (l) we havefour variables with four processes, each with individual freedom, but with, we anticipate, one or morecointegrating relationships between them. Our theoretical priors imply there are two such cointegratingrelations, q and iady. In Figure 6, all variables appear non-stationary in log levels. We see notable stockmarket crashes in 1929, 2001 and 2008, as downward dips in p.
Figure 6: Log levels of p, d, k, & l, in constant price terms
In vector form, define the following system:
Standard tests for lag length indicate two lags, using AIC criteria, hence we estimate a VAR(2) asfollows:
𝑿𝑡 = 𝚽(𝑿𝑡−1+ 𝑿𝑡−2) + 𝝐𝑡(7)
Where Φ is a (4 x 4) matrix of coefficients, and 𝜖𝑡is the error term. For all three sample periods,residuals are not autocorrelated.16 Therefore we conduct a Johansen test (lag order 1, retaining interceptin VAR) to establish whether, and how many (denoted ‘r’ for rank), cointegrating relationships thereare. On this basis, and in line with Robertson & Wright (2002a), we find clear evidence (at 5% or 1%level) of one cointegrating relationship between the four variables, across all three sample periods.
The Johansen test however does not confirm the source of cointegration. We therefore test our theoryby restricting the system, creating a VECM and in doing so we difference the VAR(2), reflecting now
one less lag, such that:
Δ𝑿𝑡 = 𝚽Δ𝑿𝑡−1 + 𝜶(𝜷′𝑿𝑡−1 + 𝜿) + 𝝐𝑡(8)
Where 𝜶 and 𝜷 are 4 x r, where r is the number of cointegrating relations and 𝜿 is a vector of rcointegrating constants (mean value of the r cointegrating relations).
We now restrict17 the VECM to test whether q is a cointegrating relation,
𝜷1𝐴′ = (1 − 𝜁 0 −1 𝜁 ) (9)
to find
Table 4 shows that, with an intercept in VAR, our restriction and hence our theory on q is rejected –not supported by the data. If, however, we believe there is no intercept in the VAR, the results are lessstrong.
Similarly, we restrict18 the VECM to test whether iady is a cointegrating relation,
𝜷1𝐵′ = (1 −1 0 0 ) (10)
to find:
Indicating support for iady – for the period 1900 to 2018 we cannot reject the restriction of nocointegration.
Now, we test a double restriction19 on the VECM to reflect two simultaneous cointegrating relationships, (p+l)/k and k/(d+l), modelling the dividend yield as a function of debt and capital20
to find:
Indicating that, a combined restriction, using q and iady, gives strong support for our theoretical discussions in a combined system. We therefore proceed on this basis. We recognise that in doing so we imply two cointegrating relationships where the Johansen test found only one, but we do not consider this is an issue given; Johansen may lack power, the persistence in q, and our theoretical priors.
6.7. Granger Causality and VECM Interpretation
Granger Causality can confirm whether the values of fundamentals are correlated with future returns. This is important because cointegration does not, in isolation, necessarily indicate whether “causality” runs to or from the fundamental. Brooks notes:
“the word causality is somewhat of a misnomer, for Granger-causality really means only a correlation between the current value of one variable and the past values of others; it does not mean that movements of one variable cause movements of another” (Brooks, 2008, p. 298)
For the period 1900 to 2018, Table 7 presents our estimated VECM model including only the statistically significant variables, at 90% confidence.²¹
The equation for Δl𝑡is not presented because none of its explanatory variables are statisticallysignificant. Further, Δ𝑝𝑡−1 is an explanatory variable for Δk𝑡 only, but the coefficient of 3%, and thoseof the other explanatory variables for Δk𝑡, are each economically insignificant, in line with the puzzlethat q cannot predict investment (see Oliner, Rudebusch, & Sichel (1995)). The omission of Δ𝑝𝑡−1from Δ𝑝𝑡supports the random walk theory, within our model of long-run predictability. As expected,Δ𝑝𝑡 has a negative relationship with q𝑡−1 and iady𝑡−1, in line with theory.
The equation for Δlₜ is not presented because none of its explanatory variables are statistically significant. Further, Δpₜ₋₁ is an explanatory variable for Δkₜ only, but the coefficient of 3%, and those of the other explanatory variables for Δkₜ, are each economically insignificant, in line with the puzzle that q cannot predict investment, as noted by Oliner, Rudebusch, & Sichel (1995).
The omission of Δpₜ₋₁ from Δpₜ supports the random walk theory, within our model of long-run predictability. As expected, Δpₜ has a negative relationship with qₜ₋₁ and iadyₜ₋₁, in line with theory.
The coefficients for qₜ₋₁ and iadyₜ₋₁ support our theory that both Granger Cause returns, given the marginal predictive significance for Δpₜ. Similarly, to confirm this, we restrict the relevant coefficients of the VECM, obtaining the results in Table 8.
Note also the short-run Granger Causality from Δdₜ₋₁ and Δlₜ₋₁ to Δpₜ. In the short run, consistent with the Dividend Irrelevancy Theorem, Miller & Modigliani (1961), an increase in dividends is partially, 0.12%, offset by a decrease in equity returns in the next period. Similarly, an additional short-run feature is that increases in liabilities tend to increase equity returns in the next period, implying leverage and risk benefits.
In the long run, however, in contrast with the short-run impact, as dₜ₋₁ increases or decreases, which by definition means adyₜ₋₁ increases or decreases and iadyₜ₋₁ decreases or increases, our model shows that Δpₜ increases or decreases:
↑ dₜ₋₁ ⟹ ↑ adyₜ₋₁ = ↓ iadyₜ₋₁ ⟹ ↑ Δpₜ (15)
This means that periods of higher or lower dividends are usually followed by periods of higher or lower returns. The same intuition is reinforced by Shiller:
“As a matter of historical fact, times when dividends have been low relative to stock prices have not tended to be followed by higher stock price increases in the subsequent five or ten years. Quite the contrary: times of low dividends relative to stock prices in the stock market as a whole tend to be followed by price decreases, or smaller-than-usual increases, over long horizons, and so returns have tended to take a double hit at such times, from both low dividend yields and price decreases. Thus, the simple wisdom – that when one is not getting much in dividends relative to the prices one pays for stocks, it is not a good time to buy stocks – turns out to have been right historically.” (Shiller, 2016, p. 206)
Intuitively, dividends must provide information to investors, in contrast with the underpinning assumption of the Dividend Irrelevancy Theorem, of perfect symmetrical information. See Dividend Signalling by Bhattacharya (1979) and Miller & Rock (1985).
Before further inference, we undertake diagnostic checks. We confirm that the VECM model does not suffer from serial autocorrelation for any of the three sub-sample periods, based on Portmanteau and LM tests, see Table 9. For VECM Jarque-Bera normality, however, only Δkₜ is found to contain normal residuals, using Urzua, Lutkepohl and Doornik-Hansen tests. There is, unsurprisingly, evidence of heteroskedasticity – a common issue with all financial time-series. We therefore re-estimate the model under various ARCH designs, noting that the primary coefficients of interest, for qₜ₋₁ and iadyₜ₋₁, do not materially change.
We revisit the original VAR to see how variables respond to each other absent any restrictions for ourcointegrating theory. Figure 7 below shows that each variable is primarily impacted by its own laggedvalue, as demonstrated in the diagonal graphs. Note also that changes in price are statisticallysignificant, and positive, to a change in liabilities. Intuitively, we can interpret this to mean thatinvestors see increased liabilities as a good indicator, but not necessarily increases in capital,presumably because increasing liabilities has leverage/risk-transfer benefits, whereas spending capitalcould be inefficient or empire building. Adjusted dividends also increase by a similar 10% proportion,to changes in liabilities. The bottom row of graphs show that the response of liabilities is notstatistically significant to changes in; price, adjusted dividends, or capital.
Figure 7: Accumulated response, one standard deviation shocks ±2 standard errors, 1900 to 2018
Accumulated Response to Cholesky One S.D. (d.f. adjusted) Innovations ± 2 S.E.
Impulse responses for shorter samples give a similar impression, hence we do not present those results.Of all the responses, capital takes longest to settle - approximately 6-7 years for shocks from itself andfrom liabilities. Intuitively, this is because capital projects can take significant time to complete, suchas construction projects and investment programs for growth.
Using the VECM, we find a similar pattern. Again, each variable responds primarily to itself (Figure8, diagonal graphs). These VECM impulses are shown in levels in contrast with VAR impulses whichare shown in differences. Now, applying the cointegrating vectors, the response of price to adjusteddividends is more visible, and positive as we expected from theory and from equation 12. Notehowever, Figure 8 implies that liabilities decrease in response to price increases. This impressionhowever is misleading; the response of liabilities to price shocks is statistically insignificant.Companies would not be expected to modify liability levels in response to stock price shocks – doingso would, to say the least, be quite a notable treasury policy.
Figure 8: Accumulated response to one standard deviation shocks, 1900 to 1990
Accumulated Response to Cholesky One S.D. (d.f. adjusted) Innovations
Using the variance decomposition method, Figure 9 below reinforces the interaction between the variables, to show that, as the horizon extends, an increasing proportion of the p error variance is explained by Adjusted Dividends.
Figure 9: VECM variance decomposition, 1900 to 1990
Variance Decomposition using Cholesky (d.f. adjusted) Factors
Of further note is the variance decomposition of Adjusted Dividends, which shows a cross-over pointnear the 14-year horizon, where the error variance is equally explained by Adjusted Dividend shocks and liability shocks, growing to a near 60% liability explanation at the 25-year horizon. However, this impression is, in fact, specific to the sample period 1900 to 1990. For the longer sample 1900 to 2018(Figure 10 below), incorporating the dot-com bust 2000-2002 and the Global Financial Crisis 2007-2008, the variance decomposition of Adjusted Dividends is no longer explained, to any not able proportion, by changes in liabilities. Arguably, this was evident from Figure 6 – and readily understand able given the Global Financial Crisis and subsequent quantitative easing.
Figure 10: VECM variance decomposition, 1900 to 2018
Variance Decomposition using Cholesky (d.f. adjusted) Factors
In a simple world, theory suggests information (and funds) will flow to equity investors in the follow ingorder: the non-financial corporate raises debt (liabilities) then invests (capital), then pays Adjusted Dividends, which are ultimately reflected in stock prices. We therefore test an alternate ordering (l, k,d, p) of the variables, re-running the analysis displayed in the previous section – we confirm that wedo not get a notably different impression.
Figure 9 shows, for the period 1900 to 1990, that the variance of p reduces by approximately 30% as the horizon extends to 25 years, and that majority of this (approximately 25%) is explained by d .Similarly, for the period 1900 to 2018, Figure 10 shows that as the horizon extends to 25 years, the variance of p reduces by an even greater amount (approximately 34%), and that an even greater proportion (approximately 33%) is explained by d. Thus, indicating the degree to which the variance of long-horizon returns is lower, demonstrating predictability, compared to short-horizon returns. These decomposition results support our analysis on variance ratios (see 6.2).
We compare the forecasting ability of the VECM against a counterfactual of using an historical average, using three periods; 1967 to 1986, 1987 to 2006 and 2007 to 2018. These periods reflectseveral aims. First, we need enough time to allow forces to operate. Second, to avoid biased results,we intentionally include a low return period (1967 to 1986) and a high return period (1987 to 2006).Third, we leave enough data outside these periods to allow us to contrast in-sample and out-of-sampletests. Thus, these periods provide us with (six observations of) a fair two-horse race
Using information from 1899 to 2018 we produce forecasts of p, d, k & l for each sample period, comparing these with actual outturn values, as shown in Figure 11.
Figure 11: VECM forecasts compared to outturn actuals (in-sample)
Source: Author’s calculations using Eviews, VECM model as presented at Table 7, and data from Wright, FRED & ShillerY-axis = log levels as per Figure 6, X-axis top left panel = 1960 to 1986, X-axis top right panel = 1980 to 2006, X-axis bottom left panel, = 2000 to 2018LN =log, DT = adjusted dividends, K = capital, L = liabilities, P = price
We use d & p to derive returns for each year. Table 10 below shows that, for all three periods, theforecast error is smaller when using the VECM model (see Table 7)
Our results are consistent with Campbell & Thompson (2008) who also show that predictive regressions beat the historical average return.
For each period we rebuild the VECM using only information available at the forecast date. In thisway, we equalize the information available to VECM with that used in the historical average. Figure12 shows the results.
Source: Author’s calculations using Eviews, VECM from Table 7 re-built using only data up to forecast date, and data from Wright, FRED & ShillerY-axis = log levels as per Figure 6, X-axis top left panel = 1960 to 1986, X-axis top right panel = 1980 to 2006, X-axis bottom left panel, = 2000 to 2018LN =log, DT = adjusted dividends, K = capital, L = liabilities, P = price
A priori, we expect results to be less impressive. However, Table 11 shows that in two out of three periods the VECM provides lower forecast errors than the use of historical averages. Arguably, the one period where VECM fails to impress is too short (12-years) to provide a fair test, and we note that the VECM forecast is materially larger given the boom period of the 1990s, relative to the in-sample equivalent (10.57% versus 8.12%).
Our analysis shows that VECM forecasts are more accurate than historical averages – justifying attempts by practitioners to predict EMR over long horizons.
Predictability is important for infrastructure regulators and investors, as seen in the following advice to regulators, from Wright et al, dated 2003 and 2018:
“Our central estimate of the cost of equity capital, derived from a wide rangeof markets, is around 5.5% (geometric average), and thus 6.5% to 7.5%(arithmetic average).” (Wright, Mason, & Miles, 2003, p. 59)22“We would, however, argue that the case for an adjustment to arithmeticaverages as large as 2 percentage points… is distinctly weakened if regulatorswish to set returns on a consistent basis at a relatively long (e.g., 10-year)horizon, given that long-horizon returns have distinctly lower volatility thanwould be the case in a random walk stock market.” (Wright, Burns, Mason, &Pickford, 2018, p. 125)23
Our work addresses this in two ways. First, given q and iady, we can test if the market is above (below)average, and therefore whether future returns are likely to be, in order to restore equilibrium, lower(higher) than outturn averages. Second, we can estimate the difference between geometric andarithmetic returns for a variety of holding periods using several different models of predictability.On an annualized basis, given a certain holding period (h), the common approach to estimate the EMR,Ε(𝑅ℎ), is to add the expected log (or alternatively, geometric) return Ε(𝑟ℎ) to the (non-annualised)variance 𝜎2(𝑟ℎ) divided by twice the holding period, 2ℎ.
As the holding period increases, the variance increases, but so too does the denominator 2h. Thus, if log returns are unpredictable, the variance for a 10-year holding period will simply be ten times the variance of a 1-year holding period.
The holding period variance can be inferred from historical returns and from the VECM model. Giventhe ability of q and iady to predict returns, the VECM variance and hence the uplift is understandablylower, relative to observed variances for longer holding periods, as reported in Table 12.
Annually compounded (geometric) and continuously compounded (log) returns, on a nominal and realbasis, are presented in Table 13.
Geometric rates are appropriate if compounding occurs once annually. Log rates are appropriate ifcompounding occurs continuously – arguably more appropriate for utility price controls as customerspay monthly, although the associated implementation model also matters. A more material issue isoutturn inflation – given the BoE’s inflation targeting duties, its estimation carries more weight thanDMS.
For 20-year horizons, the inference for UK price controls is that, if the equity market is in equilibrium(neither overvalued nor undervalued), the best estimate of EMR is 5.7%, (outturn log returns, 5.0%,plus the VECM model uplift as developed for the UK, 0.7% as per Table 12). Alternatively, a moreconservative EMR, given the lack of long-run UK data for q and iady, is 5.9% (outturn log returns,5.0%, plus UK outturn variance uplift, 1.3%, minus the VECM-explained variance in the US equitymarket (1.6%-1.1%). On the other hand, ignoring VECM, the long-run average for the UK is 6.3%.These three estimates are lower than recent estimates by UK regulators of 6.5%.
Further, if the US equity market is not, in fact, in equilibrium, and is instead thought to be overvalued(given Figure 3) real equity returns in the US and for closely associated markets such as the UK, areexpected to be lower than long-run averages (for approximately 11 years).24
When viewed over long time periods, stock markets exhibit bubbles and crashes, justifying the popularadvice that investing in stocks is a long-term decision. Possibly, this popular advice acts as a selffulfilling prophecy in favour of predictability theories. We link stock markets to indicators of value to provide an objective reference, such that we can better understand the underlying risks
In most countries (19 of 21), variance ratios provide support that returns are predictably safer in thelong run than the short run. Using Adjusted-Dividend-Yields and Tobin’s Q, we show that returns areexplainable in a VECM model. This VECM model provides superior forecasting ability, both in sample and out-of-sample, relative to using historical averages
We avoid, and therefore do not attempt to address, the criticisms of non-VECM inferences. Our worksupports practitioners’ revealed preference to predict long-horizon EMR. Esoteric critiques byeconomists offering no better option appear narrow.
For further strength;
• our work could be replicated for the UK stock market using data for Tobin’s Q and Dividends from 1900 onwards,
• bayesian forecasts and Markov switching models may provide further insight,
• other fundamentals such as interest rate spreads could have explanatory power.
Aberdeen Standard Investments. (2019). Global Outlook. Retrieved from Aberdeen Standard Investments.
Amélie, C., & Darné, O. (2009). Variance ratio tests of random walk: An overview. Journal of Economic Surveys, 503-527.
Ang, A., & Bekaert, G. (2007). Stock Return Predictability: Is It There? Review of Financial Studies, 651-707.
AON. (2018). Capital Market Assumptions. Retrieved from AON.
Bank of England. (2006). Returns to equity, Investment and Q: evidence from the United Kingdom. Working paper no. 310.
Bank of England. (2018). A millennium of macroeconomic data.
Bhattacharya, S. (1979). Imperfect information, dividend policy, and the “bird in the hand fallacy”. Bell Journal of Economics, 259-270.
Blanchard, O. (1993). Movements in the Equity Premium. Brookings Papers on Economic Activity.
Blume, M. (1974). Unbiased Estimators of Long-Run Expected Rates of Return. Journal of the American Statistical Association, 69(347), 634-638.
Boudoukh, J., Richardson, M., & Whitelaw, R. (2005). The Myth of Long-Horizon Predictability.
Brooks, C. (2008). In Introductory Econometrics for Finance (2nd ed.).
Bunn, O., & Shiller, R. (2014). Changing Times, Changing Values: A Historical Analysis of Sectors within the US Stock Market 1872-2013. National Bureau of Economic Research working paper.
Campbell, J., & Shiller, R. (1988). The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors. Review of Financial Studies, 195-227.
Campbell, J., & Thompson, S. (2008). Predicting Excess Stock Returns out of Sample: Can Anything Beat the Historical Average? The Review of Financial Studies, 21(4), 1509-1531.
Campbell, J., & Viceira, L. (2001). Strategic Asset Allocation: Portfolio Choice for Long-Term Investors.
Campbell, J., & Yogo, M. (2006). Efficient Tests of Stock Return Predictability. Journal of Financial Economics, 27-60.
Cochrane, J. (2008, July). The Dog That Did Not Bark: A Defense of Return Predictability. The Review of Financial Studies, 21(4), 1533-1575.
Competition Commission. (2014). Northern Ireland Electricity price determination, Approach to measuring historical returns of a market index.
Credit Suisse. (2019). Summary Edition Credit Suisse Global Investment Returns Yearbook.
Dimson, E., Marsh, P., & Staunton, M. (2001). Triumph of the optimists: 101 years of global investment returns.
Dimson, E., Marsh, P., & Staunton, M. (2019). Credit Suisse Global Investment Returns Yearbook.
Epstein, G. (2000). Smithers’ Contention About Q Is All Fouled Up. Barron’s, 80(20).
Fabozzi, F. (2006). Financial Econometrics. Wiley.
Fama, E., & French, K. (1988). Dividend Yields and Expected Stock Returns. Journal of Financial Economics, 22(1), 3-25.
Favero, C., & Gozluklu, A. (2009). Long-Run Factors and Fluctuations in Dividend/Price. Financial and Real Activity. Paris.
Geanakoplos, J., Magill, M., & Quinzii, A. (2004). Demography and the Long Run behaviour of the Stock Market. Brookings Papers on Economic Activities, 241-325.
Goyal, A., & Welch, I. (2003). Predicting the Equity Premium with Dividend Ratios. Management Science, 49(5), 639-54.
Harney, M., & Tower, E. (2003). Rational Pessimism: Predicting Equity Returns using Tobin’s q and Price/Earnings Ratios. The Journal of Investing, 12(2), 58-69.
Jacquier, E., Kane, A., & Marcus, A. (2005). Optimal Estimation of the Risk Premium for the long run and asset allocation: a case of compounded estimation risk. Journal of Financial Econometrics, 3(1), 37-55.
JP Morgan. (2019). Long-Term Capital Market Assumptions.
Kim, M., Nelson, C., & Startz, R. (1991). Mean Reversion in Stock Prices? A Reappraisal of the Empirical Evidence. National Bureau of Economic Research.
Lo, A., & MacKinlay, A. (1989). The size and power of the variance ratio test in finite samples: A Monte Carlo investigation. Journal of Econometrics, 203-238.
Mehra, R., & Prescott, E. (1985). The equity premium: A puzzle. The Journal of Monetary Economics, 15(2), 145-161.
Miller, M., & Modigliani, F. (1961). Dividend Policy, Growth, and the Valuation of Shares. Journal of Business.
Miller, M., & Rock, K. (1985). Dividend policy under asymmetric information. Journal of Finance, 1031-1051.
Ofgem. (2019, May). RIIO-2 Sector Specific Methodology Decision – Finance.
Oliner, S., Rudebusch, G., & Sichel, D. (1995). New and old models of business investment: comparison of forecasting performance. Journal of Money, Credit and Banking, 27, 806-26.
Poterba, J., & L. S. (1988). Mean reversion in stock prices: evidence and implications. Journal of Financial Economics, 27-59.
Poterba, J., & Summers, L. (1983). Dividend taxation, corporate investment and ‘q’. Journal of Public Economics(22), 247-273.
Robertson, D., & Wright, S. (2002a, May). What Does q Predict.
Robertson, D., & Wright, S. (2002b, June). The Good News and the Bad News about Long-Run Stock Returns. Cambridge Working Papers in Economics.
Schroders. (2019). 30-year return forecasts (2019-48).
SEB Group. (2019). Nordic Outlook.
Shiller, R. (1990). Market Volatility.
Shiller, R. (2016). Irrational Exuberance. Princeton University Press.
Siegel, J. (2008). Stocks For The Long Run - The Definitive Guide to Financial Market Returns and Long-Term Investment Strategies (4th ed.).
Smithers, A., & Wright, S. (2000). Valuing Wall Street: Protecting Wealth in Turbulent Markets. McGraw-Hill Education.
Torous, W., Valkanov, R., & Yan, S. (2004). On Predicting Stock Returns with Nearly Integrated Explanatory Variables. The Journal of Business, 937-966.
Tower, E. (2011). Tobin’s q versus CAPE versus CAPER: Predicting Stock Market Returns Using Fundamentals and Momentum.
UBS. (2019). Capital Market Assumptions Update.
Vanguard. (2019). Vanguard economic and market outlook for 2019: Down but not out.
Westerlund, J., & Narayan, P. (2014). A random coefficient approach to the predictability of stock returns in panels. Journal of Financial Econometrics.
Willis Towers Watson. (2019). Five-Year Capital Market Outlook - 2019 Europe.
Wright, S. (2004a). Measures of Stock Market Value and Returns for the US Non Financial Corporate Sector, 1900-2002. Review of Income and Wealth, 561-584.
Wright, S. (2004b). Measures of Stock Market Value and Returns for the US Non Financial Corporate Sector, 1900-2002: dataset.
Wright, S., Burns, P., Mason, R., & Pickford, D. (2018, March). Estimating the Cost of Capital for Implementation of Price Controls by U.K. Regulators.
Wright, S., Mason, R., & Miles, D. (2003). A Study Into Certain Aspects of the Cost of Capital for Regulated Utilities in the U.K..
Explore MCC’s 2025 view on DSM - how utilities can use flexible demand to integrate more solar, cut system costs, and relieve grid constraints.
2025
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur.
Block quote
Ordered list
Unordered list
Bold text
Emphasis
Superscript
Subscript
Explore how the UK Infrastructure Bank is shaping the UK’s green and economic transformation. This article reviews UKIB’s mandate, funding capacity, early investments, and risks, and assesses its potential to drive long-term, sustainable infrastructure growth.
2023
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur.
Block quote
Ordered list
Unordered list
Bold text
Emphasis
Superscript
Subscript
Empower UK government departments with structured case studies, optimizing public resources effectively.
2023
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur.
Block quote
Ordered list
Unordered list
Bold text
Emphasis
Superscript
Subscript
I was delighted that MCC's work was completed on time, and within budget, helping us deliver important changes and improvements, to the benefit of our stakeholders. MCC's report is published on the CCC website.
I am delighted to recommend MCC Economics. Specifically, I worked closely with PJ, who helped us with our Nuclear and CCUS projects. PJ helped us develop new policies and answer questions from our stakeholders. His support helped us deliver important changes and improvements, to the benefit of our stakeholders.
MCC Economics has helped us better understand the most important issues for our stakeholders, including: charges, shareholder returns, debt payments and inflation impacts.
I am delighted to recommend PJ and his team at MCC Economics. We've been working together on National Policy Statements to help meet net zero targets for 2030 and 2050. We initially appointed MCC Economics to support us on offshore wind consultation analysis and have recently reappointed MCC Economics to undertake a larger consultation analysis role across all sectors, including hydrogen, CCUS and networks. I can confirm that PJ and his team have shown excellent spreadsheet skills, alongside very good project management, planning and analysis skills, helping us deliver important changes, and continuous improvements, to the benefit of our stakeholders.
I am delighted to recommend PJ and his team from MCC Economics. They helped us with our price controls for Heathrow airport and for NATS (En Route) plc (the air traffic services provider). Specifically, the MCC team helped us deliver important changes and improvements to our financial models and supporting policy documents, to the benefit of our stakeholders.
I am delighted to confirm that I worked with PJ on a retail project in 2015. The project helped stakeholders understand electricity costs and charges. Specifically, the project helped us explain to stakeholders, internally and externally, why electricity charges differed across the regions (GB, NI & Ireland). PJ was a key member on the project team, which helped deliver changes and improvements in the understanding of energy retail.
.png)
Evidence-based analysis for transparent, defensible, and effective decision-making.
© All rights reserved – MCC Economics 2026.
.png)