Forecasting stock returns: A momentum approach
Master Thesis
Robert Schippers
Faculty of economics and business
University of Groningen
January 9, 2020
Supervisor:
Ioannis Souropanis
Abstract
1
Introduction
Forecasting stock returns is notably difficult. Seminal work done by Welch and Goyal (2008) shows that, by using macroeconomic fundamentals, stock returns can be forecast in-sample. Forecasting out-of-sample stock returns is less successful, and the historical average often outperforms forecasting models. Campbell and Thompson (2008) were the first to show that out-of-sample forecasting performance can be significantly improved by imposing economic constraints. In recent years, many scholars have tried to improve out-of-sample forecasting performance by introducing constraints directly on the predictor variables, or on the forecast itself.
Academics in finance are interested in forecasting stock returns, as the ability to forecast is vital in devel-oping better and more realistic asset pricing models to explain the data. There is a common misconception that predictability in the stock market is conflicting with the efficient market hypothesis. The random walk model, popularized by Malkiel (1973), suggests the future stock returns are unpredictable given the currently available information. Even though this model is consistent with the efficient market hypothesis it does not disqualify a predictable return process with exposure to time-varying aggregate risk. Campbell and Shiller (1988) show that deviations from the long-term mean in the dividend-price ratio indicate changes in the expected future dividend growth rate or stock return. Fluctuations in the aggregate risk exposure result in time-varying discount rates, which in turn affect asset prices. This is consistent with the efficient market hypothesis, and only fails when the risk-adjusted expected return is nonzero. In theory, this means that asset prices are a product of the state variables of the real economy. If aggregate risks are related to fluctuations in business-cycle, we would expect fluctuations in expected returns as well. Rapach and Zhou (2013) show that business-cycle has indeed a significant impact on the expected return.
In addition to the business-cycle, Pan et al. (2018) show that momentum can be an essential factor to take in consideration when forecasting financial markets. They argue that variables that exhibit persistent forecasting performance can be expected to outperform the historical average in the future if they also did in the past several months. They call this phenomenon momentum of predictability. In particular, they show that dividend yield, dividend price ratio, earning price ratio, treasury bill rate and long term yield exhibit persistent forecasting performance.
In this paper, we will introduce a simple constraint to further improve the predictive ability of the forecasting model. This new constraint takes into consideration the sign of the past forecast and compares it with the forecast for the next period. If we observe a difference in signs, we set the forecast to 0, i.e., the benchmark of no change (Wang et al., 2018; Pan et al., 2018). Moreover, this approach of setting a value to zero when sign inconsistencies occur is not new and extensively reported in the literature. Campbell and
Thompson (2008) were the first to impose a sign restriction, setting ˆβi,t = 0 when the coefficient has an
Next to introducing a new economic constraint, this paper will contribute to the forecasting literature in two more ways.
First, we will horse race the performance of various economic constraints in the literature and compare their results. This has been done before, but only with the proposed constraints of Campbell and Thompson (2008) and Pettenuzzo et al. (2014). We will contribute by also including Zhang et al. (2019), Wang et al. (2018) and Pan et al. (2018) into the experiment.
Second, we will make use of an extensive and recently updated dataset. This dataset contains the fourteen most-studied economic predictor variables and has data ranging from January 1871 until December 2018. Making use of this dataset can be very interesting as it captures economic developments not yet included in older datasets and studies. One of these economic developments is the post-financial crisis bull market.
We employ a bivariate predictive regression with the fourteen economic predictor variables suggested by Welch and Goyal (2008) to produce excess return forecasts from December 1950 until December 2018. Next to this, we also employ a pooled model and a model based on principal components. We then report both
the in-sample and the out-of-sample results. We report the out-of-sample R2, the performance over time the
economic value of our forecasts.
Our main finding is that the proposed constraint consistently outperforms the historical average bench-mark during regular times and during a time of economic recession. All economic predictors, except inflation, are economically and statistically significant at conventional levels. For economic recessions, the proposed
method largely outperforms the benchmark, with the pooled model having an out-of-sample R2 of 5.05%
and the principal component model enjoying an out-of-sample R2of 8.47%. Both are impressive considering
these are monthly results. Furthermore, the performance over time and economic value of the considered models are consistent with the overall performance, where better forecasting performance leads to a higher economic gain for the mean-variance investor. Finally, we find that our results remain robust when changing the risk-aversion coefficient and considering different business-cycles.
2
Data on economic fundamentals
In this section, we discuss the fourteen established variables used for monthly stock return predictability. These variables were first introduced by Welch and Goyal (2008) and various studies utilize these predictors, see Campbell and Thompson (2008); Pettenuzzo et al. (2014); Pan et al. (2018); Rapach and Zhou (2013) for an example. We use the revised dataset from Welch and Goyal (2008), with a sample period from 1950:12 to
2018:12.1 We require a lengthy in-sample estimation period to make sure we get a reliable and representative
regression estimate at the start of the out-of-sample evaluation period. We decided to follow Neely et al. (2014) and set the length of the initial estimation period to be 15 years. This way, we end up with an out-of-sample evaluation period from 1966:01 to 2018:12.
Our dependent variable is the equity premium. That is, the continuously compounded return of the S&P500 (including dividends) minus the prevailing short-term rate. As for our predictive variables, most fall into three broad categories: (1) economic fundamentals, such as the dividend price ratio or the market-to-book ratio; (2) bond risks and yields, such as the three-month T-bill rate and the default yield spread; (3) estimates of equity, such as the long-term return and stock variance. We use the following fourteen variables:
1. Dividend Price Ratio (DP): difference between the log of dividends and the log of prices. 2. Dividend Yield (DY): difference between the log of dividends and the log of lagged prices. 3. Earnings Price Ratio (EP): difference between log of earnings and log of prices
4. Dividend Payout Ratio (DE): difference between log of dividends and log of earnings. 5. Stock Variance (SVAR): sum of squared daily returns on S&P500 index.
6. Book to Market Ratio (BM): the ratio of book value to market value for the Dow Jones Industrial Average.
7. Net Equity Expansion (NTIS): the ratio of twelve-month moving sums of net issues by NYSE listed stocks divided by the total market capitalization of NYSE stocks.
8. Treasury Bill Rate (TBL): interest rate on a three-month Treasury bill on the secondary market. 9. Long Term Yield (LTY): long-term government bond yield.
10. Long Term Return (LTR): return on long-term government bonds.
11. Term Spread (TMS): difference between the long term yield on government bonds and treasury bills. 12. Default Yield Spread (DFY): difference between BAA- and AAA- rated corporate bond yields.
1We thank Amit Goyal for providing us with the updated variables. All variables are available at
13. Default Return Spread (DFR): difference between the return on long-term corporate bonds and returns on the long-term government bonds.
14. Inflation (INFL): inflation is the Consumer Price Index (CPI). To account for delay in releases of the CPI, we lag inflation for an extra month.
Table 1 reports the summary statistics for the equity premium and the fourteen predictor variables used in this study. The sample period is from December 1950 to December 2018. On average, the monthly excess return is 0.50%, which, with a standard deviation of 5.41%, results in a monthly Sharpe ratio of 0.09. In addition, the first-order autocorrelation of the monthly excess return is 0.09, which implies stock returns are very difficult to predict by their past values alone. Most of the fourteen predictor variables are highly persistent with a first-order auto-correlation close to one. Out of all variables, only the default yield spread (DFR) has a negative first-order auto-correlation of -0.12. The summary statistics are consistent with the previously mentioned large body of related literature.
Table 1: Summary Statistics
Variable Mean Std. Dev. Min. Median Max. Skewness Kurtosis ρ(1)
ER (%) 0.45 5.41 -33.93 0.95 34.55 -0.43 10.89 0.09 DP -3.38 0.46 -4.52 -3.35 -1.87 -0.20 2.61 0.99 DY -3.37 0.46 -4.53 -3.35 -1.91 -0.22 2.58 0.99 EP -2.74 0.42 -4.84 -2.79 -1.78 -0.58 5.56 0.99 DE -0.64 0.33 -1.24 -0.63 0.29 1.53 9.04 0.99 SVAR (%) 0.29 0.57 0.01 0.13 7.10 5.81 46.83 0.63 BM 0.57 0.27 0.12 0.54 2.03 0.78 4.42 0.99 NTIS 0.02 0.03 -0.06 0.02 0.18 1.62 10.97 0.98 TBL (%) 3.39 3.09 0.01 2.93 16.30 1.10 4.33 0.99 LTY (%) 5.10 2.79 1.75 4.21 14.82 1.10 3.63 0.99 LTR (%) 0.47 2.45 -11.24 0.31 15.23 0.58 7.54 0.04 TMS (%) 1.71 1.30 -3.65 1.76 4.55 -0.27 3.17 0.96 DFY (%) 1.12 0.69 0.32 0.90 5.64 2.50 11.95 0.98 DFR (%) 0.03 1.37 -9.75 0.05 7.37 -0.37 10.52 -0.12 INFL (%) 0.24 0.53 -2.06 0.24 5.88 1.08 16.84 0.48
3
Methodology
3.1
Bivariate predictive regression model
To explore the issue of the out-of-sample predictive power of our economic variables, we consider the following bivariate predictive regression model:
rt+1= α + βiXi,t+ εi,t+1 (1)
where rt+1is the equity risk premium, the return on the S&P 500 index in excess of the risk-free rate from
period t to t+1; Xi,tis the predictor variable at time t; and εi,t+1is the error term. Under the null-hypothesis
of no predictability, βi becomes 0 and equation (1) reduces to a model of constant excess returns.
To forecast excess returns, we use a recursive estimation window (Campbell and Thompson, 2008; Neely et al., 2014; Rapach et al., 2016). This estimation window expands after every period, beginning right after the initial in-sample estimation period. As previously mentioned in section 2, we decided on an estimation period of 15 years. In this way, we have divided the entire sample into an in-sample part consisting of M observations and an out-of-sample part consisting of Q observations (i.e., M + Q = T ). Based on this design, we estimate the first out-of-sample excess return by
ˆ
ri,t+1 = ˆαi,t+ ˆβi,tXi,t (2)
where ˆri,t+1is the forecast equity risk premium based on the ith predictor, ˆαi,tand ˆβi,tare the ordinary least
square estimates obtained by regressingrt+1
M
t=1 on a constant andXi,t
M −1
t=1 . Repeating this for every
period in our sample, we obtain a series of Q out-of-sample forecasts on the equity premium, ˆri,t
T
t=m+1
We compare the forecasts given by equation (2) with the historical average. Using the historical average as benchmark is a practise repeatedly found in the literature (Welch and Goyal, 2008; Campbell and Thompson,
2008; Neely et al., 2014) and assumes a constant expected risk premium (rt+1= α + εt+1). Welch and Goyal
(2008) show that predictive regression forecasts based on individual macroeconomic variables generally fail to outperform the historical average. The historical average forecast is given by
¯ rt= 1 T T X i=1 ri (3)
where ¯rtis the average return and T is the total amount of considered periods.
Following the convention in return forecasting, we use the out-of-sample R2statistic advocated by
Camp-bell and Thompson (2008). The R2
OS measures the out-of-sample predictive accuracy of the model relative
R2OS= 1 − PQ k=1(rM +k− ˆrM +k)2 PQ k=1(rM +k− ¯rM +k)2 (4)
were rM +k is the actual stock return; ¯rM +kis the historical average return; and ˆrM +k is the return forecast
at month M + k. As for index, M and Q are the length of the initial estimation period and the evaluation
period, respectively. Similar to their in-sample counterparts, the monthly R2
OS appears to be small at first
sight. This hasty conclusion is often incorrect as the R2
OS statistic can be economically significant at levels
as small as 0.5% (Campbell and Thompson, 2008).
3.2
Predictive model based on Principal Components
In addition to the bivariate model, we consider a predictive model based on principle components. For this
model, we let Xt= (X1,t, . . . , XN,t)0 denote the vector of the entire set of fourteen macroeconomic variables
(N = 14) and let ˆFtECON = ( ˆF1,tECON, . . . , ˆFK,tECON) denote the vector comprised of K principle components
extracted from Xt. The principle component predictive regression model is defined as
rt+1= α +
K
X
k
βkFˆkECON+ εt+1. (5)
The first few principle components describe the key movements among the entire set of predictors. This filters out much noise enclosed within individual predictors and prevents in-sample overfitting. Following Neely et al. (2014), individual predictors are standardized prior to computing the principle components. In addition to this, we made use of their scripts created to do so.
3.3
Statistical significance
To determine statistical significance in our model, we employ the Clark and West (2007) statistic. This statistic tests the null-hypothesis that the mean squared forecast error (MSFE) of the benchmark is smaller than, or equal to that of the forecasting model of interest. The alternative hypothesis is that the MSFE of the historical benchmark is larger than the model of interest. To calculate the Clark and West (2007) statistic, we first need to compute
ft= (rt− ¯rt)2− (rt− ˆrt)2+ (¯rt− ˆrt)2, (6)
where rt, ˆrtand ¯rt are the equity risk premium, the forecast equity risk premium and the historical mean,
respectively. By now regressingfs
T
s=m+1on a constant, we obtain the Clark and West (2007) test statistic,
3.4
Imposed economic constraints
When estimating a regression over a short period of time, the results can be contradictory to what the literature suggests. This could for example result in a negative coefficient when the literature would suggest a positive one. Seminal work done by Campbell and Thompson (2008) shows that imposing an economic constraint on the predictive model can improve the forecasting performance by reducing parameter estimation uncertainty. In this paper, we consider six economic constraint models. The first five are existing models, the last one is our proposed model. We will let these six models compete to explore how well they perform comparatively.
3.4.1 Non-negative equity premium
The first constraint introduced is from Campbell and Thompson (2008). They propose a non-negativity constraint. The reasoning for this restriction lies in the fact that, in equilibrium, there are no risk-averse investors who would want to hold stocks if their expected compensation were negative. They show that imposing this constraint never worsened, and almost always improved the performance of the out-of-sample predictive regression. Even though this constraint is seemingly simple, it has substantial implications for the estimated parameters of the return prediction model, pinning them down and generating more precise estimates. To efficiently utilize the information enclosed in the non-negativity constraint, the parameters α and β should estimated so that
α + βXi,t≥ 0 for t = M + 1, ..., T (7)
Despite this constraint not being a direct constraint to the model, it affects the parameters as they have to be consistent with equation (7). Moreover, because the constraint has to hold at all times, the number of equity premium constraints grows corresponding to an increase in sample size.
Campbell and Thompson (2008) implement this equity premium constraint in their model by using a truncated forecast, setting the forecast equity premium equal to the largest of the unconstrained OLS forecast and zero:
ˆ
rt+1= max (0, ˆαi,t+ ˆβi,tXi,t) (8)
where ˆαi,t and ˆβi,t are the OLS estimates from equation (1). Even tough this truncating prohibits the
equity premium to become negative, by simply overruling the forecast if it is negative and not constraining the parameters as mentioned in equation (7), their model does not make efficient use of the theoretical constraints.
3.4.2 Beta sign restrictions
can be contradictory to what the literature suggests. For example, obtaining a negative coefficient when this should be positive according to theory. To solve this problem, they recommend a sign restriction on β in
equation (1). If the coefficient has undoubtedly an unexpected sign, we set ˆβi,t = 0. This helps to stabilize
the predictive regression forecast and reduce parameter estimation uncertainty (Campbell and Thompson, 2008; Rapach and Zhou, 2013). Out of the 14 economic predictor variables, we expect all but NTIS, TBL and INFL to be positive (Li and Tsiakas, 2017).
3.4.3 Smooth forecast outliers
The third economic constraint is proposed by Zhang et al. (2019). They argue that rational investors are improbable to trust extremely large or small forecasts and that these extreme values therefore should be smoothed out. To achieve this, they propose a new economic constraint which bounds the return forecast within a rational scope. The scope follow the three-sigma rule and the new, smoothed, return forecasts are defined as ˆ rSmoothi,t+1 =
ri,t+ 3σt, if ˆri,t+1> ri,t+ 3σt
ri,t− 3σt, if ˆri,t+1< ri,t− 3σt
ˆ
ri,t+1, otherwise
(9)
where ˆrSmooth
i,t+1 is the new, smoothed, forecast at month t + 1 based on the ith predictor, ri,t is the
uncon-strained forecast based on equation (1) and σtis the standard deviation of the excess return. Zhang et al.
(2019) only consider the three signma deviation as they found that smaller deviations are too probable and thus not extreme, requiring a higher deviation than three will result in the constraint not having a binding effect as this event is too unusual.
3.4.4 Momentum of return predictability
The fourth constraint is introduced in a recent paper by Wang et al. (2018). The momentum of predictability (MoP) is defined as the property whereby past period predictability can also be found in the current period. When the benchmark is outperformed in past periods, it also achieves lesser forecasts in the current period. During period t, the current predictability is given by
cpt= I
(rt− ˆrt)2− (rt− ¯rt)2< 0
, (10)
where I(.) is a function which returns 1 if the given condition is satisfied and 0 otherwise. rt, ˆrt and ¯rtare
respectively. Comparably, the past predictability is defined as ppt(k) = I Xt−1 j=t−k (rt− ˆrt)2− t−1 X j=t−k (rt− ¯rt)2< 0 , (11)
where k is the chosen look-back period. When ppt(k) equals 1, the R2OS is positive and the model is
outperforming the benchmark in the chosen time-frame. As ppt(k) is highly dependent on the look-back
period, Wang et al. (2018) consider different look-back periods.2
They furthermore argue that, by definition, the dependence of ppt and cpt implies the existence of
the momentum of predictability. To implement their findings, they design a strategy in which forecasting models are alternated between the benchmark model and the model of interest. When a given model beats the benchmark, it is used to predict future returns; otherwise the benchmark of average historical returns is used. This will result in the following forecasts;
ˆ rt+1M oP = ˆ rt+1, if ppt(k) = 1 ¯ rt+1, if ppt(k) = 0 (12)
This strategy is highly dependent on the choice of the look-back period k. To address this problem, an average momentum of predictability (MoP-AVG) is considered. This strategy uses an average of forecasts over all k-values and is given by
ˆ rM oP −AV Gt+1 = 1 N N X k=1 ˆ rt+1M oP(k), (13)
where N is the total number of k-values.
3.4.5 Restrict predictor variable
The fifth economic constraint is proposed in Pan et al. (2018). In this paper, the constraint is applied
directly to the predictor variable. A simple truncation of the variable Xtis discussed, where the non-linear
transformation of the original variable is given by
Xt∗= Xt, if Xt> max(Xt−1, Xt−2, . . . , Xt−n OR Xt< min(Xt−1, Xt−2, . . . , Xt − n) 0, otherwise (14)
where t = n + 1, . . . , T − 1 and n is the lookback period. Applying this constraint, we obtain the following out-of-sample forecasting model
ˆ
r∗t+1(n) = ˆα∗i,t(n) + ˆβi,t∗ (n)Xi,t∗ (n), (15)
where ˆα∗i,t(n) and ˆβi,t∗ (n) are the OLS estimates of α and β respectively and ˆr∗t+1(n) is the forecast equity
premium for t = M, . . . , T .
An important part of this newly defined predictor variable is the range of period used to find the local highs and lows. In general, a longer look-back period (a larger n) will result in more zeros for the predictor
{X∗
t(n)} T −1
t=n+1. We follow the literature and set our look-back period to n = 24 (Pan et al., 2018).
One major drawback of this proposed approach is that we only consider information coming for the abnormal periods. This potentially ignores valuable information contained in the normal periods. To mitigate
this issue, Pan et al. (2018) propose another forecast, which is simply the average between ˆrt+1and ˆrt+1∗ (n),3
ˆ
rt+1# (n) = 0.5ˆrt+1+ 0.5ˆrt+1∗ (n). (16)
3.4.6 Proposed constraint
Finally, we propose a new economic constraint. This constraint is based on sign momentum, more specifically, the sign of the return forecast. If the sign of our forecast at time t + 1 is not equal to the sign of the actual return at time t, we disregard our forecast at time t + 1 and return 0 instead. This approach is very similiar to that of Wang et al. (2015), but instead of oil prices we use S&P500.
Forecasts obtained from a misleading predictive relation could cause a greater loss than simply setting the forecast null. Furthermore, this erratic pattern can be the product of model over-fitting. Accordingly, it is rational for an investor to disregard the forecast when the sign is inconsistent (Wang et al., 2015). Applying this will result in the following forecast;
ˆ rSigni,t+1=
0, if sign(ˆri,t+1) ∼ sign(ri,t)
ˆ
ri,t+1, otherwise
(17)
where ˆri,t+1and ri,tare the forecast and actual return, respectively. This constraint shares similarities with
the momentum of predictability by Wang et al. (2015). Both models assume, to some extent, persistence in the stock returns.This is a reasonable assumption, as persistence in stock returns is extensively discussed in the literature (Campbell, 1990; Singleton and Wingender, 1986). In addition, when examining table ??, we observe that most variables have a first-order autocorrelation very close to 1, indicating persistent values. Moreover, both models make use of the momentum found in the stock returns, which is a central topic in the well acclaimed papers by Carhart (1997) and Fama and French (2012).
4
Empirical findings
First, we report the in-sample results. The in-sample performance is done ex-post and uncovers the historical performance of the independent variables. Since investors and policymakers are more concerned about future performance, we also comprehensively report the out-of-sample results in the next section.
4.1
In-sample results
Table 2 reports the in-sample estimation results for both the bivariate predictive model and the model based on principal components. For each model, the slope coefficients, heteroskedasticity-consistent t-statistics and
R2 statistics are reported. To account for the lag in the predictive regression, the estimation sample ranges
from January 1951 to December 2018, which attributes to a total of 816 observations.
The bivariate predictive model regresses the previously discussed fourteen macroeconomic variables of interest. Out of these fourteen variables, six exhibit statistical significance at conventional levels. These variables are; Dividend Yield (DY), Stock Variance (SVAR), Treasury Bill Rate (TBL), Long Term Yield (LTY), Long Term Return (LTR) and Term Spread (TMS). Campbell and Thompson (2008) argue that,
because of the unpredictable component inherent to monthly stock returns, a monthly R2 statistic as low
as 0.5% can already be economically significant. Four out of fourteen macroeconomic variables satisfy this requirement; SVAR, TBL, LTR and TMS.
Estimation results of equation (5) are given in panel B of table 2. We observe the third principal
component ( ˆF3,tECON) being statistically significant at the 5% level. In addition, the overall model exhibits
a R2 of 0.85%, which is economically significant.
Figure 1 shows the factor loading of the economic predictor variables. We plot three bar graphs in Panels A-C, representing the three respective principal components. When examining panel A, we observe that valuation ratios (DP, DY, EP and BM) along with long-term government bonds (LTY and LTR)
load heavily on our first principle component, ˆFECON
1,t . Furthermore, in panel B we examine an evident
distribution of factor loading where SVAR and DFY load most heavily on our second principle component, ˆ
FECON
Table 2: Predictive regression in-sample estimation results
Predictor Slope coefficient T-statistic P-value R2(%)
Panel A: Bivariate predictive regression
DP 0.62 1.69 0.18 0.38 DY 0.67 1.84 0.03** 0.45 EP 0.41 0.95 0.31 0.17 DE 0.41 0.66 0.27 0.08 SVAR 6.12 2.10 0.02** 0.51 BM 0.37 0.55 0.71 0.05 NTIS -2.36 -0.28 0.62 0.01 TBL -0.11 -2.29 0.99*** 0.71 LTY -0.09 -1.62 0.97 0.34 LTR 0.13 2.17 0.02** 0.76 TMS 0.22 1.97 0.03** 0.52 DFY 0.14 0.32 0.39 0.02 DFR 0.15 0.94 0.19 0.26 INFL -0.22 -0.45 0.67 0.04
Panel B: Principal Components predictive regression ˆ FECON 1 0.02 0.21 0.52 0.85 ˆ FECON 2 0.12 0.97 0.24 ˆ FECON 3 0.25 2.10 0.04**
Notes: Panel A reports the estimation results for the bivariate pre-dictive model defined in equation (1). Panel B reports the estima-tion results for the predictive model based on principal components (equation 5). ∗, ∗∗ and ∗ ∗ ∗ indicate statistical significance at the 10%, 5% and 1% levels, respectively. Values are based on one-sided wild bootstrapped p-values as proposed by Neely et al. (2014). The
Figure 1: Loadings on principal components extracted from 14 macroeconomic variables
DP DY EP DE SVAR
BM
NTIS TBL LTY LTR TMS DFY DFR INFL -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
(A) First principle component
DP DY EP DE SVAR
BM
NTIS TBL LTY LTR TMS DFY DFR INFL -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
(B) Second principle component
DP DY EP DE SVAR
BM
NTIS TBL LTY LTR TMS DFY DFR INFL -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
(C) Third principle component
Notes: Panels A - C illustrate the first three principle components, extracted from the fourteen macroeconomic variables.
4.2
Out-of-sample results
Table 3 reports the out-of-sample performance for the unconstrained model and for the models which take in consid-eration the constraints proposed by Campbell and Thompson (2008) [NN, CT], Zhang et al. (2019) [ZWMY], Wang et al. (2018) [WLMD], Pan et al. (2018) [PPW] and this paper [Sign Momentum], respectively. We measure the
out-of-sample performance using the R2OS, where a positive value corresponds to the appropriate economic predictor
variable outperforming the benchmark of the historical mean. CT, NN, ZWMY, WLMD, PPW and the proposed model outperform the unconstrained counterpart with 4, 10, 2, 14, 13 and 14 out of the 14 predictor variables,
respec-tively. Furthermore, the proposed constraint produces a positive R2OS for 11 of these, outperforming the benchmark.
components. CT appears to be the only model which both outperforms the unconstrained POOL and falls behind the unconstrained PC, interestingly.
Table 3: Out-of-sample performance
Unconstrained CT NN ZWMY WLMD PPW Sign Momentum
DP -0.23 -0.23 0.07* -1.20 0.37** 0.19 0.81** DY -0.18 -0.18 0.11* -1.17 0.40** 0.25 0.79** EP -0.57 -0.57 -0.10 -1.51 0.34** -0.14 0.77** DE -0.83 -0.66 -0.14 -0.69 0.34** 0.06 -0.34 SVAR -0.05* -0.05* -0.01* -0.68 0.57*** 0.17 0.66** BM -1.21 -1.02 -1.21 -1.85 0.39** -0.53 0.64** NTIS -0.90 -0.91 -0.76 -1.77 0.24* -0.32 -0.01* TBL -0.85** -0.85** 0.48** -0.96** 0.16 0.56** 0.46** LTY -0.75** -0.75** 0.76** -0.81** 0.51*** 0.37** 0.38** LTR 0.32** 0.32** 0.19** -0.52* 0.00 0.43** -0.33* TMS -0.81** -0.81** -0.81** -1.33** 0.52*** 0.84*** -0.27** DFY -0.62 -0.78 -0.60 -1.44 0.34** -0.38 0.56** DFR -0.44 0.62** -0.46 -1.32 0.31** -0.47 0.60** INFL -0.36 -0.36 -0.05 -0.72 0.26** -0.12 0.08 POOL 1.09*** 1.19*** 0.72*** 0.41 0.39*** 0.91*** 1.08** PC -1.03*** -2.64** 0.51*** -0.73*** 0.99*** 0.31** 0.17***
Notes: This table reports the out-of-sample R2 for the predictive regression described in equation (4).
Unconstrained refers to the initial forecast in which no economic constraints are imposed. CT and NN correspond to the sign-restriction constraint and the non-negativity constraint first proposed by Campbell and Thompson (2008), respectively; ZWMY is based on the smoothed forecast advocated by Zhang et al. (2019); WLMD takes in consideration the momentum of return predictability and imposes constraints ac-cordingly (Wang et al., 2018); PPW restricts the predictor variable, only taking into consideration local highs and lows Pan et al. (2018). Finally, the results of our newly proposed economic constraint (Sign Momentum) are reported. ∗, ∗∗ and ∗ ∗ ∗ indicate statistical significance at the 10%, 5% and 1% levels, re-spectively. Values are based on the Clark and West (2007) test. The initial estimation period ranges from 1950:12-1965:12 and the out-of-sample period ranges from 1966:01-2018:12.
Table 4 reports the performance in terms of predicting the sign of the actual values, where a value of 1 corresponds to a 100% success rate. The methodology used is based on a paper by Liu and Pan (2019); however, we slightly adjusted it to make up for the large series of zeros due to the employed constraints. When the forecast is zero, we will measure the correctness of the forecast by comparing this period‘s actual value against the previous period‘s actual value. If the actual value exceeds its predecessor, we have a mean-reverting process and categorize the forecast as correct. Alternatively, we compare the sign of the actual value with the sign of the forecast value. We find that the proposed model makes a more accurate sign prediction on all variables, beating the unconstrained model. This makes sense as the proposed constraint is built around the sign momentum, disregarding forecasts with an ‘invalid‘ sign. CT, NN, ZWMY, WLMD, PPW outperform the sign prediction of the unconstrained model 3, 10, 1, 12 and 4 times out the 14, respectively.
Table 4: Out-of-sample sign prediction
Unconstrained CT NN ZWMY WLMD PPW Sign Momentum
DP 0.51 0.51 0.57 0.50 0.59 0.57 0.58 DY 0.51 0.51 0.57 0.51 0.59 0.56 0.58 EP 0.58 0.58 0.58 0.57 0.59 0.57 0.63 DE 0.59 0.59 0.61 0.59 0.59 0.58 0.64 SVAR 0.58 0.58 0.58 0.57 0.59 0.58 0.63 BM 0.58 0.59 0.58 0.58 0.59 0.58 0.64 NTIS 0.57 0.57 0.58 0.57 0.59 0.57 0.62 TBL 0.59 0.59 0.62 0.59 0.59 0.59 0.65 LTY 0.59 0.59 0.63 0.60 0.59 0.58 0.65 LTR 0.56 0.56 0.59 0.56 0.58 0.57 0.63 TMS 0.59 0.59 0.60 0.59 0.59 0.59 0.65 DFY 0.58 0.58 0.58 0.58 0.59 0.58 0.63 DFR 0.58 0.63 0.58 0.58 0.59 0.58 0.64 INFL 0.58 0.58 0.59 0.58 0.59 0.59 0.63 POOL 0.58 0.58 0.58 0.58 0.59 0.58 0.58 PC 0.54 0.59 0.60 0.55 0.59 0.54 0.61
Notes: This table reports the correctness of the sign forecast. The labels represent the respec-tive papers (Campbell and Thompson, 2008; Zhang et al., 2019; Wang et al., 2018; Pan et al., 2018). The initial estimation period ranges from 1950:12-1965:12 and the out-of-sample period ranges from 1966:01-2018:12.
3 panels B - F report the cumulative squared error difference between the benchmark and models based on Campbell and Thompson (2008); Zhang et al. (2019); Wang et al. (2018); Pan et al. (2018), respectively. Panel A shows the results for when no constraints are employed. 3A - 3F are shown in the appendix.
Figure 2: Performance over time - Sign momentum constraint.
5
Economic evaluation
Following the vast body of literature, we measure the economic benefit of stock return predictability from a perspective of asset of allocation. We calculate the certainty equivalent return (CER) for a mean-variance investor. This investor can allocate its money between stocks and risk-free bills. To achieve maximum certainty equivalent return, the investor should allocate the weight of stocks during month t + 1 in such a way that,
wt= 1 γ ˆ rt+1 ˆ σ2 t+1 , (18)
where γ is the investor’s risk aversion coefficient, ˆrt+1 is the return forecast and ˆσ2t+1 is the forecast stock return
variance. Naturally, the share of risk-free bills is given by wt− 1, so that the monthly portfolio return during month
t + 1 is given by
Rp,t+1= wtrt+1+ Rf ree,t+1. (19)
Following Campbell and Thompson (2008), we estimate the return volatility based on a five-year moving window
of past monthly returns. Furthermore, we limit wt to range between 0 and 1.5 as this rules out short selling and
prevents leverage higher than 50%. This will result in a portfolio which exhibits an average CER defined as
CER = ¯Rp− 0.5γσ2p, (20)
where ¯Rpdenotes the mean portfolio return, σ2pis the portfolio variance and γ is the risk aversion coefficient. The
CER can be understood as the risk-free return an investor is willing to accept instead of investing in risky assets (Brennan et al., 1997). The CER gain is calculated as the difference between the CER for the investor when using the return forecasts, and the CER when the investor uses the predominant benchmark. Subsequently, the CER gain can be interpreted as the portfolio management fee that the mean-variance investor is willing to pay to have access to the predictive forecasts instead of only the historical average forecasts (Zhang et al., 2019).
Table 5 reports the certainty equivalent return (CER) gains, along with the Sharpe ratio (SR) for every considered constraint. The constraints introduced by Campbell and Thompson (2008) seem to have no impact on both the CER and the SR. The model subject to ZWMY does actually seem to perform worse than the unconstrained model. WLMD and PPW show a slight improvement. The proposed Sign Momentum constraint shows considerable improvements in the CER gains and a small increase in the Sharpe ratio.
Table 5: Portfolio performance
Unconstrained CT NN ZWMY WLMD PPW Sign Momentum
CER SR CER SR CER SR CER SR CER SR CER SR CER SR
DP -0.39 0.03 -0.39 0.03 -0.39 0.03 -0.80 0.02 -0.07 0.06 -0.30 0.03 2.02 0.10 DY -0.01 0.04 -0.01 0.04 -0.01 0.04 -0.48 0.02 -0.01 0.06 -0.04 0.04 2.04 0.10 EP 0.07 0.05 0.07 0.05 0.07 0.05 -0.44 0.04 -0.19 0.06 -0.01 0.05 2.35 0.11 DE -1.1 0.03 -1.07 0.03 -1.10 0.03 -1.31 0.02 -0.25 0.06 -0.60 0.05 0.74 0.05 RVOL -2.03 0.06 -2.03 0.06 -2.03 0.06 -1.95 0.06 -0.03 0.07 -1.19 0.05 1.42 0.09 BM -2.30 0.02 -1.61 0.03 -2.30 0.02 -2.42 0.02 -0.09 0.06 -1.11 0.04 2.08 0.10 NTIS -4.27 0.04 -4.21 0.04 -4.27 0.04 -4.76 0.04 -0.29 0.06 -2.65 0.04 -0.25 0.06 TBL -0.68 0.06 -0.68 0.06 -0.68 0.06 -0.99 0.06 -0.50 0.05 0.33 0.07 0.63 0.06 LTY 0.51 0.07 0.51 0.07 0.51 0.07 0.11 0.06 0.13 0.06 0.59 0.07 1.06 0.06 LTR -1.92 0.08 -1.92 0.08 -1.92 0.08 -2.30 0.08 -0.76 0.05 -0.84 0.07 -0.72 0.06 TMS -1.19 0.10 -1.19 0.10 -1.19 0.10 -1.32 0.10 0.17 0.07 -0.39 0.10 1.56 0.10 DFY -2.67 0.03 -2.87 0.03 -2.67 0.03 -3.00 0.03 -0.27 0.06 -2.63 0.03 1.17 0.08 DFR -1.88 0.05 1.04 0.07 -1.88 0.05 -2.39 0.04 -0.32 0.06 -2.15 0.03 1.16 0.07 INFL -1.39 0.05 -1.39 0.05 -1.39 0.05 -1.74 0.04 -0.35 0.06 -0.66 0.05 1.18 0.07 POOL 1.21 0.08 1.42 0.08 0.25 0.06 0.93 0.07 -0.13 0.06 0.79 0.07 1.88 0.09 PCA -0.86 0.08 -7.74 0.04 -0.86 0.08 -0.80 0.08 1.02 0.09 -1.94 0.04 0.27 0.07
6
Robustness checks
In this section, we further asses the statistical and economical performance of the considered model by conducting two robustness tests. First, we examine the performance during different business-cycles and second we look at the impact of a higher risk aversion coefficient for the mean-variance investor.
6.1
Business cycles
It could prove very interesting to analyse the R2
OS during business cycles. To distinct between business cycles, we
use NBER based recession indicators provided by the FRED4. We follow the related literature (Rapach et al., 2010;
Neely et al., 2014; Li and Tsiakas, 2017; Zhang et al., 2019), and compute two separate R2OS statistics,
R2OS,c= 1 − PQ k=1I c M +k(rM +k− ˆrM +k)2 PQ k=1I c M +k(rM +k− ¯rM +k)2
for c = EXP, REC (21)
where R2OS,EXP and R
2
OS,REC are the out-of-sample R 2
statistics during a NBER expansion and recession period,
respectively. IM +kc is an indicator that takes a value of one during expansion and zero during a period of recession.
Table 6 reports the out-of-sample forecasting performance during times of economic recession. We observe that our proposed model outperforms the unconstrained model on all fourteen economic predictor variables. Furthermore,
the pooled model and the model based on principal components perform exceptionally well, generating a R2OS,REC
statistics of 5.05% and 8.48% respectively. If we compare our proposed constraint with the other considered con-straints, our proposed constraint substantially outperform the competition.
Table 7 reports on the out-of-sample performance during expansion. Consistent with the results of Rapach
et al. (2010); Pan et al. (2018), the unconstrained return prediction model is outperformed by the no-predictability benchmark. This is true for both the bivariate prediction model as well as the pooled model and the principal component model. Moreover, when employing the constraints, we observe similar results. During economic expansion only dividend yield and the principal component model perform slightly better when our proposed constraint is employed. The constraint by Wang et al. (2018) performs remarkable well, outperforming the benchmark as well as every competing model. In addition, all predictors, except the long-term rate, are statistically significant.
6.2
Alternative risk aversion
In the economic evaluation section above, we assume that the mean-variance investor has a risk aversion coefficient of five. However, when examining equation (18), we observe that the optimal portfolio weights are dependent on the amount of risk the investor is willing to take. Therefore, we consider another risk aversion coefficient and report the impact of risk aversion on the performance of the portfolio.
Table 8 reports the portfolio performance for the alternative risk aversion coefficient of 10. We observe that, even though the overall CER gains are higher with the higher risk aversion coefficient, the relative model performance stays unaltered. The CER results are robust to an increase in the risk aversion coefficient. Moreover, the Sharpe ratio stays constant, which is expected.
Table 6: Out-of-sample performance during recession
Unconstrained CT NN ZWMY WLMD PPW Sign Momentum
DP 0.87 0.87 0.86 -2.89 -0.10 0.88 4.52*** DY 1.67* 1.67* 1.66* -1.91 0.02 0.83 4.64*** EP -1.40 -1.40 -0.23 -4.95 -0.01 -0.26 3.89** DE 0.20 0.81 0.97 0.53 -0.00 -0.05 3.44 SVAR 0.63 0.63 0.63 -1.98 0.28*** 0.24 4.80*** BM -3.33 -3.57 -3.33 -5.66 0.00 -1.61 3.84*** NTIS -3.03 -2.78 -2.72 -6.41 0.02 -1.49 3.11** TBL 0.90 0.90 0.15 0.80* -0.15 1.81 5.19** LTY 0.34 0.34 1.04 0.27 -0.14 4.16** 4.47* LTR 5.80*** 5.80*** 3.58*** 2.44* 0.55 3.75*** 6.10*** TMS 3.67** 3.67** 2.56** 1.38 1.00** 2.51* 7.06*** DFY -0.86 -1.45 -0.82 -3.89 0.25 -0.62 4.29*** DFR -1.35 0.76 -1.26 -4.66 -0.19 -1.48 3.11** INFL -0.98 -0.98 -0.64 -2.21 -0.20 -0.46 2.65 POOL 1.83** 1.97** 0.73* -0.71 0.12* 1.52** 5.05*** PC 4.53** -3.04 2.13* 4.75** 1.12** 3.46* 8.48***
Notes: This table reports R2OS,REC for the predictive regression, described in Equation 21, in times of
economic recession. Labels represent the considered constraints. ∗, ∗∗ and ∗ ∗ ∗ indicate statistical sig-nificance at the 10%, 5% and 1% levels, respectively. The initial estimation period ranges from 1950:12-1965:12 and the out-of-sample period ranges from 1966:01-2018:12.
Table 7: Out-of-sample performance during expansion
Unconstrained CT NN ZWMY WLMD PPW Sign Momentum
DP -0.66 -0.66 -0.25 -0.53 0.56** -0.08 -0.66 DY -0.91 -0.91 -0.51 -0.88 0.55** 0.02 -0.73 EP -0.24 -0.24 -0.05 -0.15 0.47** -0.10 -0.46 DE -1.24 -1.24 -0.57 -1.18 0.48** 0.10 -1.84 SVAR -0.32 -0.32 -0.27 -0.17 0.69*** 0.15 -0.97 BM -0.37 -0.01 -0.37 -0.33 0.54** -0.10 -0.63 NTIS -0.05* -0.16 0.01* 0.07 0.33* 0.14 -1.25 TBL -1.54** -1.54** 0.61** -1.65* 0.28* 0.07** -1.42 LTY -1.18* -1.18* 0.65** -1.25* 0.77*** -1.14 -1.24 LTR -1.85 -1.85 -1.15 -1.70 -0.22 -0.89 -2.88 TMS -2.59* -2.59* -2.14 -2.41* 0.33* 0.19** -3.18 DFY -0.52 -0.52 -0.52 -0.46 0.38* -0.29 -0.92 DFR -0.08 0.56** -0.13 0.01 0.51** -0.07 -0.40 INFL -0.12 -0.12 0.19 -0.13 0.43** 0.02 -0.94 POOL 0.80*** 0.88*** 0.72*** 0.85 0.50** 0.67*** -0.50 PC -3.23** -2.48*** -0.14** -2.91** 0.94*** -0.94* -3.12
Notes: This table reports R2OS,RECfor the predictive regression, described in Equation 21, in times of
Table 8: Portfolio performance when risk aversion γ = 10
Unconstrained CT NN ZWMY WLMD PPW Sign Momentum
CER SR CER SR CER SR CER SR CER SR CER SR CER SR
DP 0.28 0.03 0.28 0.03 0.28 0.03 0.08 0.02 0.44 0.06 0.33 0.03 1.47 0.10 DY 0.47 0.04 0.47 0.04 0.47 0.04 0.24 0.02 0.47 0.06 0.46 0.04 1.48 0.10 EP 0.52 0.05 0.52 0.05 0.52 0.05 0.27 0.04 0.38 0.06 0.48 0.05 1.64 0.11 DE -0.08 0.03 -0.06 0.03 -0.08 0.03 -0.18 0.02 0.35 0.06 0.17 0.05 0.84 0.05 RVOL -0.54 0.06 -0.54 0.06 -0.54 0.06 -0.50 0.06 0.46 0.07 -0.12 0.05 1.17 0.09 BM -0.66 0.02 -0.32 0.03 -0.66 0.02 -0.72 0.02 0.43 0.06 -0.07 0.04 1.50 0.10 NTIS -1.67 0.04 -1.64 0.04 -1.67 0.04 -1.91 0.04 0.34 0.06 -0.85 0.04 0.33 0.06 TBL 0.14 0.06 0.14 0.06 0.14 0.06 -0.01 0.06 0.23 0.05 0.64 0.07 0.79 0.06 LTY 0.73 0.07 0.73 0.07 0.73 0.07 0.53 0.06 0.54 0.06 0.77 0.07 1.00 0.06 LTR -0.50 0.08 -0.50 0.08 -0.50 0.08 -0.69 0.08 0.10 0.05 0.04 0.07 0.10 0.06 TMS -0.13 0.10 -0.13 0.10 -0.13 0.10 -0.19 0.10 0.56 0.07 0.27 0.10 1.24 0.10 DFY -0.86 0.03 -0.96 0.03 -0.86 0.03 -1.02 0.03 0.34 0.06 -0.84 0.03 1.04 0.08 DFR -0.45 0.05 0.99 0.07 -0.45 0.05 -0.71 0.04 0.32 0.06 -0.59 0.03 1.04 0.07 INFL -0.23 0.05 -0.23 0.05 -0.23 0.05 -0.40 0.04 0.31 0.06 0.14 0.05 1.06 0.07 POOL 1.07 0.08 1.18 0.08 0.60 0.06 0.93 0.07 0.41 0.06 0.86 0.07 1.40 0.09 PCA 0.01 0.08 -3.42 0.04 0.01 0.08 0.05 0.08 0.98 0.09 -0.52 0.04 0.59 0.07
Notes: This table reports the certainty equivalent return (CER) gains along with the Sharpe ratio (SR) for every considered constraint. The CER gain is calculated using equation (20) when γ = 10 and the Sharpe ratio is calculate using the methodol-ogy outlined in Sharpe (1994). The initial estimation period ranges from 1950:12-1965:12 and the out-of-sample period ranges from 1966:01-2018:12.
7
Conclusion
In this paper, we contribute to the literature by developing a new economic constraint that truncates the forecasting model based on sign momentum. Not only do we calculate the out-of-sample performance of the proposed constraint, but we also compare it with five existing economic constraints proposed by Campbell and Thompson (2008), Zhang et al. (2019), Wang et al. (2018), and Pan et al. (2018). Results show that sign momentum constraint has superior performance, especially during times of economic downturn. The only exception is during an economic expansion; here, the constraint is outperformed by constrained proposed by Wang et al. (2018).
The main finding of this study is that sign momentum can be effectively utilized to improve out-of-sample forecasting performance and hence asset allocation. This performance stays robust over business-cycles and after a change of risk appetite for the investor.
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A
Figures
(A) Performance over time - Unconstrained.
1970 1980 1990 2000 2010 -0.5 0 0.5 1 DP-Unconstrained 1970 1980 1990 2000 2010 -1 -0.5 0 0.5 1 DY-Unconstrained 1970 1980 1990 2000 2010 -1 -0.5 0 0.5 EP-Unconstrained 1970 1980 1990 2000 2010 -1 0 1 2 DE-Unconstrained 1970 1980 1990 2000 2010 -0.5 0 0.5 1 SVAR-Unconstrained 1970 1980 1990 2000 2010 -1.5 -1 -0.5 0 0.5 BM-Unconstrained 1970 1980 1990 2000 2010 -1 -0.5 0 0.5 NTIS-Unconstrained 1970 1980 1990 2000 2010 -2 -1 0 1 2 TBL-Unconstrained 1970 1980 1990 2000 2010 -2 -1 0 1 2 LTY-Unconstrained 1970 1980 1990 2000 2010 0 0.5 1 1.5 LTR-Unconstrained 1970 1980 1990 2000 2010 -2 -1 0 1 2 TMS-Unconstrained 1970 1980 1990 2000 2010 -0.5 0 0.5 DFY-Unconstrained 1970 1980 1990 2000 2010 -0.5 0 0.5 DFR-Unconstrained 1970 1980 1990 2000 2010 -0.5 0 0.5 1 1.5 INFL-Unconstrained 1970 1980 1990 2000 2010 0 0.5 1 1.5 POOL-Unconstrained 1970 1980 1990 2000 2010 -2 0 2 4 PCA-Unconstrained
(B) Performance over time - Campbell and Thompson (2008) (CT) constraint.
(C) Performance over time - Campbell and Thompson (2008) (NN) constraint. 1970 1980 1990 2000 2010 0 0.5 1 DP-NN 1970 1980 1990 2000 2010 -0.5 0 0.5 1 DY-NN 1970 1980 1990 2000 2010 -0.5 0 0.5 EP-NN 1970 1980 1990 2000 2010 -0.5 0 0.5 1 DE-NN 1970 1980 1990 2000 2010 -0.5 0 0.5 1 SVAR-NN 1970 1980 1990 2000 2010 -1.5 -1 -0.5 0 0.5 BM-NN 1970 1980 1990 2000 2010 -1 -0.5 0 0.5 NTIS-NN 1970 1980 1990 2000 2010 0 0.5 1 1.5 TBL-NN 1970 1980 1990 2000 2010 0 0.5 1 1.5 LTY-NN 1970 1980 1990 2000 2010 0 0.5 1 LTR-NN 1970 1980 1990 2000 2010 -1 0 1 2 TMS-NN 1970 1980 1990 2000 2010 -0.5 0 0.5 DFY-NN 1970 1980 1990 2000 2010 -0.5 0 0.5 DFR-NN 1970 1980 1990 2000 2010 0 0.5 1 INFL-NN 1970 1980 1990 2000 2010 0 0.5 1 POOL-NN 1970 1980 1990 2000 2010 0 1 2 3 PCA-NN
(D) Performance over time - Zhang et al. (2019) (ZWMY) constraint.
(E) Performance over time - Wang et al. (2018) (WLMD) constraint. 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 DP-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 DY-WLMD 1970 1980 1990 2000 2010 0 0.1 0.2 0.3 0.4 EP-WLMD 1970 1980 1990 2000 2010 0 0.1 0.2 0.3 0.4 DE-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 0.8 SVAR-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 BM-WLMD 1970 1980 1990 2000 2010 0 0.1 0.2 0.3 0.4 NTIS-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 TBL-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 0.8 LTY-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 LTR-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 0.8 TMS-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 DFY-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 DFR-WLMD 1970 1980 1990 2000 2010 0 0.1 0.2 0.3 0.4 INFL-WLMD 1970 1980 1990 2000 2010 0 0.2 0.4 0.6 POOL-WLMD 1970 1980 1990 2000 2010 0 0.5 1 1.5 PCA-WLMD
(F) Performance over time - Pan et al. (2018) (PPW) constraint.
