Pacific-Basin Finance Journal 71 (2022) 101683
Contents lists available at ScienceDirect
Pacific-Basin Finance Journal
journal homepage: www.elsevier.com/locate/pacfin
Predicting the Australian equity risk premium
Doureige J. Jurdi
La Trobe University, Department of Economics and Finance, Melbourne, VIC, Australia
A R T I C L E I N F O
A B S T R A C T
JEL classification:
C58
G11
G12
G17
This paper examines the predictive performance of a range of financial, economic, and sentiment
variables that may predict the Australian All Ordinaries index equity risk premium using data for
the last 28 years (1992–2020). The methods employed address a range of potential econometric
biases that affect inference based on the predictive regression. Results show consistent in-sample
and out-of-sample predictability evidence for various predictors, including the dividend yield,
interest rates, and sentiment at selected forecasting horizons ranging from one month to one year.
The analysis reveals new insights about time-varying predictability patterns in the Australian
stock market and identifies phases of predictability in the time series. For several predictors,
results show that mean-variance investors may rely on forecasts generated by the predictive
regression to derive significant utility gains. Additional tests indicate that the predictability ev­
idence is robust to the microstructure bias and variable selection bias for several predictors used
in the analysis.
Keywords:
Return premium predictability
Time-varying predictability
Out-of-sample forecasting
Asset allocation
Econometric bias
Financial and economic predictors
Sentiment
1. Introduction
Predicting the stock market equity risk premium, hereafter the equity premium, is one of the most popular topics in finance.
Understanding predictability has important practical implications for investor asset allocation and portfolio management decisions.
Studies in the literature continuously debate the empirical evidence based on a range of predictors and the econometric methods used
for estimating the predictive regression. Just over two decades ago, Bossaerts and Hillion (1999) had raised concerns about parameter
instability in estimates from the predictive regression. Despite proposing several methods to estimate the predictive regression and
correct various related estimation issues such as endogeneity, data mining, model misspecification, and others (see Stambaugh, 1999;
Ferson et al., 2003; Rapach and Wohar, 2006; Amihud et al., 2010; and others), the existing literature comparing in-sample and out-ofsample predictive ability for a range of predictors reports disparity in the predictability evidence (Welch and Goyal, 2008; Rapach and
Wohar, 2006; Dou et al., 2012). Recently, Devpura et al. (2018) suggest that predictive regressions that allow for time variation in
parameter estimates conveniently capture the predictability and endogeneity dynamics for a range of predictor variables.
This paper investigates the in-sample and out-of-sample predictability of the Australian All Ordinaries index’s equity premium
using a range of financial, economic, and sentiment variables or indicators. It complements and contributes to existing studies that
investigate the predictability evidence in the Australian stock market; for example, Dou et al. (2012), Gray (2008), Alcock and Gray
(2005), and others. I estimate the predictive regression in-sample at the monthly, quarterly, semi-annual, and annual horizons and
examine the robustness of in-sample results by conducting out-of-sample forecasting evaluation tests. The analysis addresses the wellknown endogeneity bias affecting the predictive regression (see Stambaugh, 1999) for accurate inference about predictability.
Furthermore, time-varying predictability is evaluated using a moving sub-sample window utilizing the augmented predictive
E-mail address: d.jurdi@latrobe.edu.au.
https://doi.org/10.1016/j.pacfin.2021.101683
Received 17 April 2021; Received in revised form 14 November 2021; Accepted 20 November 2021
Available online 24 November 2021
0927-538X/© 2021 Elsevier B.V. All rights reserved.
Pacific-Basin Finance Journal 71 (2022) 101683
D.J. Jurdi
regression method (see Amihud et al., 2010). This method addresses the endogeneity bias and potential misspecification in the lagged
equation of the predictor. The empirical analysis is based on a recent dataset covering almost three decades (28 years) for most
predictors. The availability of this dataset makes it convenient to investigate the time-variation in predictability in Australia.
Results show consistency in the predictability evidence in-sample and out-of-sample for several predictors such as the dividend
yield, lagged returns, and interest rate variables such as the risk-free rate and the long-term yield. Other predictors show mixed ev­
idence in their predictive performance and across various forecasting horizons. This paper reveals new insights on time-varying
predictability in Australia and identifies predictability phases in various predictors’ time series. An asset allocation exercise sug­
gests that mean-variance investors improve their economic utility by relying on forecasts generated by the predictive regression for at
least five out of thirteen predictors investigated, depending on the forecasting horizons or the predictive regression model used. The
prowess of predictive regression models and methods used in handling econometric bias is still constrained by other issues such as the
microstructure bias and variable selection problem, which may render predictability results spurious. However, estimation approaches
and statistical tests proposed by Ahn et al. (2002) and Foster et al. (1997) provide the means to address these issues. Indeed, comfort is
gained considering results based on equity index futures which suggest that the predictability evidence in-sample and out-of-sample,
for several predictors, is robust to the microstructure bias. In addition, inference based on the Maximal R2 computed using the Rencher
and Pun (1980) approximation indicates that the results are robust to the variable selection bias. These findings suggest that the
information content of several predictors drives the predictability evidence.
The rest of the paper is organised as follows. Section 2 provides a review of the relevant literature. Section 3 describes the
econometric methods used. Section 4 describes the data and its sources, whereas Section 5 presents the main results. Finally, Section 6
concludes.
2. Literature review
A voluminous body of the literature investigates the equity return premium’s predictability from financial and economic variables,
especially for the US stock market. Several influential studies agree that financial ratios and economic variables have a better pre­
dictive ability in-sample rather than out-of-sample. On the one hand, Welch and Goyal (2008) assess the performance of a compre­
hensive set of predictors of the US stock market equity premium and repost evidence of poor out-of-sample predictive power and
conclude that predictive models would not have helped investors to generate profits based on the predictive regression. On the other
hand, Campbell and Thompson (2008) report improved out-of-sample forecasts of the US equity premium after imposing theoretically
motivated constraints on the predictive regression parameter estimates. These constraints require setting the coefficient of a predictor
variable to be equal to zero if its sign contradicts the theoretically expected sign. Zhang et al. (2019) propose a new economic con­
straints approach to predict US stock market returns, which improves the out-of-sample forecasts relative to existing methods earlier
proposed by Campbell and Thompson (2008) and others. Dai and Zhu (2020) find that mixing forecasting models improve the accuracy
of the US stock market return forecasts and the economic utility derived by mean-variance investors. Dangl and Halling (2012) use a
Bayesian estimation framework that allows for time-varying coefficients of the predictors and report significant improvements in the
out-of-sample predictability of the US stock market equity premium. Devpura et al. (2018) show that some predictors exhibit timevarying predictability and time-varying endogeneity while others do not.1
In Australia, Faff and Heaney (1999) examine the relationship between inflation and equity returns at the aggregate market and
industry levels. They report a negative and statistically significant relationship between inflation and the monthly stock market equity
returns during the anti-inflation sub-period.2 This finding does not hold during the monetary targeting or checklist approach subperiods when the analysis is done using a quarterly time series or at the industry level. Boudry and Gray (2003) use an asset allo­
cation framework to evaluate the economic significance of predictors of the excess return of the AGSM value-weighted market
portfolio. In a univariate analysis, they find that the dividend yield predicts excess return; however, the term spread and short rate do
not. However, they argue that investors’ knowledge about the dividend yield and the term spread impacts a utility-maximizing in­
vestor’s asset allocation decision. Alcock and Gray (2005) examine the economic significance of predicting the Australian All Ordi­
naries index’s excess return by comparing a portfolio switching strategy’s performance based on a predictive model with a passive buy
and hold strategy. Their analysis is based on 12 predictors, including financial and economic indicators and sentiment indices. They
show that economic significance is realized when the predictive model is chosen based on Pesaran and Timmermann (1995) sign and
recursive wealth criteria. Alcock and Gray (2005) show that the portfolio switching strategy outperforms a buy-and-hold market
investment, even after accounting for transaction costs. However, they note that the economic significance of predicting the Australian
All Ordinaries index excess returns vanishes when the predictive model is chosen under alternative statistical model selection criteria
leading to inconclusive evidence. Yao et al. (2005) use a dynamic Bayesian approach to investigate the predictive ability of financial
and economic variables to monthly industrial stock return indices on the Australian Stock Exchange. The authors show that most
industry sector returns are predictable from the unanticipated change of the term structure, short-term interest rates, and aggregate
dividend yield. Using a probit model, Gray (2008) evaluates the directional accuracy of forecasting one-month positive excess returns
1
Recent papers investigating stock return predictability in an international context also show discrepancy in the predictability evidence in-sample
and out-of-sample for various variables. These include Bahrami et al. (2018) who focus on emerging markets, Charles et al. (2017) who provide
evidence from developed, emerging and frontier markets and others.
2
Faff and Heaney (1999) divide the sample used in their study into three sub-periods: A monetary targeting sub-period (July 1976–January
1985), checklist approach sub-period (February 1985–December 1989) and anti-inflation sub-period (January 1990–March 1996).
2
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D.J. Jurdi
on the Australian All Ordinaries index. Gray demonstrates that the performance of a portfolio switching strategy based on a set of seven
financial and economic predictors is not statistically different from a buy and hold market strategy. However, Gray shows evidence of
the portfolio switching strategy’s persistent economic significance in the presence of transaction costs. This evidence, however, is not
consistent across the different sample periods. Dou et al. (2012) conduct a comprehensive study on the equity premium predictability
in Australia using the MSCI Australian stock index and sector returns. The authors estimate the predictive regression using a set of 15
predictors describing stock characteristics, interest rates, and other macroeconomic indicators. Their results show that a combination
forecast approach produces better out-of-sample equity premium forecasts than an individual prediction. These results are stronger at
the annual horizon rather than quarterly. The combination forecast approach produces positive utility gains for risk-averse investors,
mainly when Campbell and Thompson (2008) restrictions are imposed.
3. Econometric methodology
3.1. The predictive regression model
Studies on the predictability of the equity premium often estimate the following predictive regression
(1)
rt,t+h = α + βxt + ut,t+h for t = 1, …, T − h
()
where rt,t+h =
1
h
(rt+1 + ⋯ + rt+h ), rt denotes the equity premium of the stock market index for month t, xt is a potential predictor
variable of the equity premium that follows an autoregressive equation of lag order p = 1, and u is the disturbance term. The predictive
ability of xt is assessed by examining the t-statistic of the least-squares estimate of β. Under the null hypothesis H0 : β = 0, the predictor
xt has no predictive power of rt, t+h. Stambaugh (1999) shows that the least-squares estimates of eq. (1) are biased in the presence of
endogeneity, leading to over-rejecting the null hypothesis of no predictability. The endogeneity bias is observed for highly persistent
predictors in the presence of a significant contemporaneous correlation between the regressor’s innovations and the regressand. Also,
when the forecasting horizon h > 1, the observations of r become overlapping and dependent (Richardson and Stock, 1989); this
induces serial correlation in the innovations of eq. (1) which will require a correction for the standard errors used in computing the tstatistic of the estimate. Despite the possibility of using Newey and West (1987) standard errors, which are robust for hetero­
scedasticity and serial correlations, the t-statistic of the estimated predictor’s coefficient ̂
β is likely to increase with horizon h (Hodrick,
1992; Nelson and Kim, 1993). To avoid size distortion in the t-statistic under the null hypothesis of no predictability, a wild bootstrap
procedure similar to Neely et al. (2014) is used to compute heteroskedasticity and autocorrelation robust t-statistics from a wild
bootstrap procedure that tests the null hypothesis H0 : β = 0 against the one-sided alternative HA : β > 0.3 The t-statistics and their
corresponding p-values are obtained using 1000 bootstrap draws.
3.2. Time-varying predictability
Several studies investigating predictability in the US market suggest that the predictive regression coefficient estimates are timevarying (see Timmermann, 2008; Dangl and Halling, 2012; Bannigidadmath and Narayan, 2016; Devpura et al., 2018 and others).
Recently, Devpura et al. (2018) show that some US stock market return predictors exhibit time variations in their coefficients while
others do not. The authors also show that the endogeneity bias is time-varying for some predictors. While the Stambaugh (1999) bias
correction addresses the problem of endogeneity in predictors, it is only valid when the predictor is of autoregressive order 1. Amihud
et al. (2010) note that an AR (1) predictive regression model is misspecified if the predictor is of a lag order greater than 1. The authors
show that model misspecification results in a significant loss in the efficiency of equity premium forecasts. The authors generalize the
predictive regression by allowing the predictor to be of order p ≥ 1. The predictive regression is given by
(2)
rt,t+h = α0 + β1 xt− 1 + … + βp xt− p + ut,t+h
The predictor follows an autoregressive equation of order p as
(3)
xt,t+h = α1 + ρ1 xt− 1 + … + ρp xt− p + vt,t+h
where the errors (ut, vt) are each serially independent and identically Gaussian distributed (i.i.d) with a zero mean. The error term in
eq. (2) is expressed as ut = ϕ′ vt + et, where et is i.i.d with a zero mean and is independent of ut and vt. Amihud et al. (2010) augment eq.
(2) with the expression for ut and develop a method that produces a reduced-bias point estimate of the predictive coefficients and
derive an appropriate hypothesis testing procedure. In this paper, the lag order (p) in the autoregressive equation of the predictor (eq.
3) is determined by the Akaike information criteria statistic. The null hypothesis of no predictability is H0 : β1 = … = βp = 0. The model
captures the total size effect of the predictor as the sum of the coefficient estimates. Amihud et al. (2010) show that a properly selected
lag order in eq. (3) improves the predictability inference relative to the least square and bias-corrected methods, which restrict the lag
3
A description of the wild bootstrap procedure applied by Neely et al. (2014) is currently available in an online appendix at: http://apps.olin.
wustl.edu/faculty/zhou/Returns_econ_tech_appendix_08-20-2013.pdf
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D.J. Jurdi
order of the autoregressive equation of the predictor to 1.4
3.3. Out-of-sample performance
The significance of the out-of-sample performance of predictors is assessed statistically and economically using the Clark and West
(2007) test and evaluating economic utility.
3.3.1. Out-of-sample R-squared
Welch and Goyal (2008) and Campbell and Thompson (2008) use the historical average of the equity premium (rt+h ) as a
benchmark to equity premium forecast (̂r t+h ) generated by the predictive regression. I compare the predictive regression forecasts with
the historical average model by calculating the out-of-sample R-squared (OOS-R2) following Campbell and Thompson (2008) as
OOS − R2 = 1 −
MSFEPR
MSFEHA
where MSFEPR is the mean squared error of the out-of-sample forecast from the predictive regression calculated over an out-of-sample
∑
forecast period denoted by s as MSFEPR = 1s st=1 (rt+h − ̂r t+h )2 , and MSFEHA is the mean squared error of the historical sample average
∑
calculated as MSFEHA = 1s st=1 (rt+h − rt )2 . The OOS-R2 explains the proportional reduction in the mean squared error of the out-ofsample forecast of the predictive variable relative to the historical average model.
To evaluate whether the predictive regression provides a statistically significant improvement in the mean squares error relative to
the historical average model, I use the Clark and West (2007) test for evaluating nested models since the predictive regression nests the
historical average model. The null hypothesis of the test is H0 : OOS − R2 ≤ 0 against the alternative hypothesis HA : OOS − R2 > 0. The
Clark and West (2007) test is a mean squared error adjusted test statistic given as.
]
[
̂f t+h = (rt+h − rt+h )2 − (rt+h − ̂r t+h )2 − (rt+h − ̂r t+h )2 ,
where ̂f t+h is regressed on a constant. The inference is based on the statistical significance of the t-statistic of the constant. The Clark
and West (2007) test has one-sided (upper-tail) critical values with p-values computed using the standard normal distribution. The
Newey-West HAC standard errors corrected t-statistic of the constant is used in the analysis.
3.3.2. Asset allocation and economic utility
The OOS-R2 presented in the previous section does not account for investor risk exposure during the out-of-sample period. To assess
the forecast’s economic significance, I evaluate a risk-averse investor’s utility gains following Campbell and Thompson (2008). To this
end, I consider a mean-variance investor who optimally allocates her investment between the equity market and the risk-free bank
accepted bills. At the end of each month t, the investor allocates wt to the equity market during period t + h using the predictive
regression forecast as
)
(
1 ̂r t+h
wPR,t =
γ ̂
σ 2t+h
where γ is the investor’s coefficient of relative risk aversion, ̂r t+1 is the forecast of the equity premium using the predictive regression,
and ̂
σ 2t+h is an estimate of the return variance calculated using a 5-year (60-months) rolling window. The investor realizes a certainty
equivalent return (CER) of
CERPR = ̂
up −
1 2
γ̂
σ
2 p
where ̂
u p and ̂
σ 2p denote the sample mean and variance, respectively, of the portfolio over the forecasted period. I restrict wt to lie
between − 50% and 150% (see Campbell and Thompson, 2008) to allow for short selling and leverage.5
Alternatively, the investor allocates w0, t to the benchmark portfolio,
(
)
1 rt+h
w0,t =
γ σ 2t+h
where rt+h and σ2t+h denote the historical average return and variance calculated using a 5-year rolling window. The realized CER of the
4
See Amihud et al. (2010) for technical details about model construction, related proofs and their implications to estimation.
The analysis allows for short selling throughout the sample period. It is noted that short selling in Australia was subject to a temporary ban from
the opening of the market on Monday, 22 September 2008. This ban restricted the covered and naked short sales of stocks. The ban on covered short
sales was lifted for nonfinancial stocks on 19 November 2008. The ban on short selling for financial stocks was extended to 25 May 2009 (see Li
et al., 2014).
5
4
Pacific-Basin Finance Journal 71 (2022) 101683
D.J. Jurdi
Table 1
Descriptive statistics.
Skewness
Kurtosis
JB p-values
ρ
Panel A: All Ordinaries index equity return premium
ER
0.03%
0.045
Mean
− 0.313
3.525
0.016
0.233
Panel B: Predictors
DY
PE
Lag ER
RVol1
RVol2
RF
LTY
TMS
M3
VOL
CVM
BCS
CSI
1.521
− 0.101
− 0.408
3.806
3.754
− 0.298
0.010
1.144
0.136
− 0.155
8.837
− 0.0829
− 0.217
7.518
3.342
3.356
26.387
25.764
2.309
2.609
5.035
5.933
10.259
90.660
7.406
3.676
0.001
0.297
0.010
0.001
0.001
0.009
0.308
0.001
0.001
0.000
0.000
0.001
0.017
0.946
0.945
0.231
0.592
0.589
0.985
0.984
0.966
0.111
0.782
0.407
0.011
− 0.207
0.038
17.616
− 0.02%
0.038
0.036
0.045
0.051
0.006
0.688
16.578
1.071
0.228
1.001
Stdev
0.612
2.633
0.044
0.022
0.020
0.018
0.022
0.011
0.754
0.455
0.544
11.755
0.053
The table reports the descriptive statistics for the Australian All Ordinaries index’s equity premium in Panel A and for predictor variables in Panel B
over the whole sample period from July 1992 to June 2020 except for the BCS time series, which is available from March 1997 to June 2020 and VOL/
CVM times series which is available from January 2000 to June 2020. The last two columns in the table report the p-values of the Jarque-Berra
normality test and the first-order autocorrelation coefficients.
same investor when the historical average model is used is given by
CERHA = ̂
u HA −
1 2
γ̂
σ
2 HA
where ̂
u HA and ̂
σ 2HA denote the sample mean and variance of returns, respectively, of the benchmark portfolio over the forecasted
period. The net utility gain is the difference between CERPR and CERHA which is expressed as an annualized percentage return.
Economic gains are estimated over the monthly, quarterly, semi-annual, and annual horizons using the predictive regression forecasts
corresponding to these horizons. It is assumed that the investor rebalances her portfolio at the same frequency of the forecast horizon.
4. Data
The sample includes monthly data for the Australian All Ordinaries index (All Ords) - from July 1992 to June 2020 and a set of 13
potential predictors. Financial variables and sentiment survey time-series are obtained from Thomson DataStream. Interest rates and
monetary aggregates data are obtained from the Reserve Bank of Australia. The equity premium (ER) is calculated as the difference
between logarithmic market returns, including dividends and the risk-free rate. The set of predictors used in this study is provided as
follows.
Fundamental financial variables:
▪ Dividend yield (DY) is the total dividend amount for the index expressed as a percentage of the total market value of the index
constituents.
▪ Price-earnings ratio (PE) is the total market value of the index divided by total earnings.
▪ Lagged equity premium (Lag ER) is the one-month lagged return premium.
▪ Monthly realized volatility (RVOL1) is the square root of the monthly realized variance calculated as the sum of squared daily
returns over a month.
▪ Monthly realized volatility (RVOL2) is calculated following Mele (2007, p. 460) and Neely et al. (2014) to address a potential
√̅̅√̅̅̅̅̅̅
outlier problem in the realized volatility time series. RVOL2 = 2π 12 ̂
σt.
Interest rates and monetary aggregate variables:
▪ The risk-free rate (RF) is the 90-day bank accepted bill rate obtained from the Reserve Bank of Australia.
▪ Long-term yield (LTY) is the yield on 10-year Australian government bonds.
▪ Term spread (TMS) is the difference between the yield on a 10-year Australian government bond and the 90-day bank
accepted bills rate.
▪ M3 money supply (M3) is the monthly change in the M3 money supply. M3 is defined as currency plus current bank deposits
from the private non-bank sector plus all other authorized deposit-taking institutions (ADI) deposits from the private non-ADI
sector, plus certificates of deposits issued by banks, less ADI deposits held with one another.
5
D.J. Jurdi
Table 2
Correlations matrix – predictor variables.
6
DY
PE
Lag ER
RVol1
RVol2
RF
LTY
TMS
M3
VOL
CVM
BCS
CSI
DY
PE
Lag ER
RVol1
RVol2
RF
LTY
TMS
M3
VOL
CVM
BCS
CSI
1.000
¡0.724
0.062
0.429
0.428
− 0.397
− 0.387
− 0.136
− 0.059
0.523
− 0.111
− 0.003
0.038
1.000
0.124
− 0.424
− 0.425
− 0.018
0.132
0.311
− 0.120
¡0.628
0.182
0.023
0.030
1.000
− 0.287
− 0.287
− 0.439
− 0.367
− 0.005
− 0.116
0.130
− 0.004
− 0.041
− 0.042
1.000
0.999
− 0.038
− 0.092
− 0.131
0.198
0.357
0.109
− 0.092
− 0.050
1.000
− 0.039
− 0.095
− 0.136
0.195
0.355
0.110
− 0.088
− 0.051
1.000
0.886
0.131
0.206
− 0.312
0.100
− 0.053
− 0.048
1.000
0.573
0.101
− 0.340
0.111
0.012
− 0.013
1.000
− 0.155
0.013
0.026
0.151
0.042
1.000
0.023
0.072
− 0.099
− 0.114
1.000
− 0.320
− 0.033
− 0.013
1.000
0.278
− 0.097
1.000
− 0.152
1.000
The table reports the matrix of pairwise linear correlation coefficients between predictors over the whole sample period. Correlations greater (less) than 0.5 (− 0.5) are reported in bold.
Pacific-Basin Finance Journal 71 (2022) 101683
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D.J. Jurdi
Table 3
In-sample Predictive Regression Estimation Results.
1
h
Monthly
Predictor
β
DY
1.501***
4.160
− 0.441
− 1.33
0.747***
2.443
− 0.002
− 0.005
− 0.005
− 0.012
2.075***
8.115
1.804***
7.528
− 0.239
− 0.756
− 0.723
− 2.485
0.502*
1.534
− 0.485
− 1.537
1.296***
4.450
0.545**
2.134
PE
Lag ER
RVOL1
RVOL2
RF (− )
LTY (− )
TMS
M3
VOL
CVM
BCS
CSI
2
3
4
5
Quarterly
R2 (%)
11.30
0.97
2.79
0.00
0.00
21.25
16.12
0.28
2.62
1.17
1.16
7.89
1.49
β
1.528***
5.213
− 0.413
− 1.166
0.999***
5.183
− 0.056
− 0.160
− 0.065
− 0.190
2.025***
9.175
1.747***
8.477
− 0.238
− 0.662
− 0.627
− 0.262
0.456
1.400
− 0.031
− 0.181
0.338**
1.855
0.279**
1.864
6
Semi-Annual
R2 (%)
25.12
1.85
10.48
0.02
0.03
42.48
32.04
0.62
4.16
1.92
0.01
1.07
0.85
β
1.582***
7.481
− 0.446
− 1.217
0.884***
4.492
0.067
0.183
0.069
0.190
2.028***
10.030
1.767***
8.538
− 0.284
− 0.718
− 0.649
− 2.834
0.431
1.323
− 0.013
− 0.084
0.264*
1.536
0.214*
1.684
7
8
Annual
R2 (%)
37.75
3.05
11.42
0.05
0.05
57.61
44.49
1.24
6.07
2.37
0.01
0.81
0.67
β
1.521***
5.019
− 0.527
− 1.842
0.695**
2.998
0.099
0.272
0.101
0.279
2.158***
12.132
1.813***
7.667
0.412
0.980
0.345
0.891
0.395
1.171
− 0.072
− 0.991
0.085
0.765
0.121
1.073
R2 (%)
44.92
5.45
8.87
0.14
0.14
64.03
55.57
3.32
1.52
2.76
0.08
0.11
0.28
The table reports the ordinary least squares estimates of the coefficient of the predictor β in percentages, the t-statistic beneath coefficient estimates,
and the R2 statistic of univariate predictive regressions in eq. (1). (− ) indicates that the negative of the predictor variable is used in the predictive
regression. Heteroscedasticity and autocorrelation robust t-statistics for testing the null hypothesis of no predictability (H0: β = 0) against (HA: β > 0)
are reported in brackets. The in-sample t-statistics and their corresponding p-values are estimated using a wild bootstrap procedure using 1000 draws.
The predictive regression is estimated at the monthly, quarterly, semi-annual, and annual horizons; ***, **, and * indicate the 1%, 5%, and 10%
significance levels, respectively, according to wild bootstrapped p-values.
Trading activity variables:
▪ Volume (VOL) is the logarithm of the aggregate number of constituent shares traded on the index during a month.
▪ Change in volume traded (CVM) is the monthly change in traded volume.
Sentiment survey-based variables:
▪ Business conditions survey (BCS) is the monthly change in the level of the Business Confidence Index constructed by the
National Australia Bank (NAB). The index evaluates business sentiment using data collected from a survey of around 350
companies. A level above (below) zero indicates improving (worsening) conditions.
▪ Consumer sentiment index (CSI) is the monthly change in the value of the Westpac Consumer Sentiment Index. The index
measures the change in the level of consumer confidence in economic activity. A level above 100.0 indicates optimism; below
indicates pessimism.
Table 1 reports the descriptive statistics for the Australian All Ordinaries index’s equity premium and a set of 13 predictors over the
full sample period. The mean of the equity premium over the whole sample period is 0.03%, with a standard deviation of 4.5%. The
equity premium time series exhibits left skewness and is leptokurtic. The Jarque-Berra test rejects the null hypothesis of normality for
the equity return premium. It also rejects the null hypothesis of normality for all other predictors (except PE and LTY) due to excessive
skewness and kurtosis in the empirical distribution of these variables’ time series. The last column of Table 1 reports the first-order
autocorrelation coefficient of the variables as defined above, which shows that 6 out of 13 predictors are persistent and have auto­
correlation coefficients higher than 0.75. Table 2 reports the Pearson correlation coefficients for the predictor variables in the dataset.
The majority of the pairwise correlations between variables are relatively low, indicating that these variables’ information content is
substantially different. Few pairwise correlations are larger than 0.5 or less than − 0.5. These correlations are highlighted in bold.
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5. Main results
5.1. In-sample stock market return predictability
Before estimating the predictive regression in eq. (1), I normalize each predictor by dividing each observation by the predictor’s
standard deviation over the full sample period. This allows for comparing the coefficient estimates of predictors, as indicated by Baker
and Wurgler (2000) and Rapach and Wohar (2006).6 I estimate the predictive regression in eq. (1) in-sample for each predictor at the
monthly, quarterly, semi-annual, and annual horizons. Columns labeled (1) to (8) of Table 3 report the coefficient estimate β of each
predictor, the corresponding t-statistic, and the in-sample R2 from predictive regressions estimates over the monthly, quarterly, semiannual, and annual horizons.
Table 3 shows that four predictors are significant in-sample predictability at all horizons at the 1% significance level. Two of these
predictors are fundamental financial variables (DY and Lag ER), and the other two are interest rate variables (RF and LTY). The trading
volume (VOL) shows weak evidence of in-sample predictability at the monthly horizon. The BCS and CSI coefficient estimates are
statistically significant at the 10% level or lower at various horizons. Changes in sentiment survey scores predict the equity premium
in-sample at the monthly, quarterly, and semi-annual horizons.
Since the predictors are normalized, predictability is interpreted as the change in the equity premium given one standard deviation
change in the predictor. For example, a one standard deviation increase in DY leads to a 1.501% increase in the equity premium. The
effect size of other estimates is interpreted similarly. By comparing the coefficients of various predictors, RF has the most considerable
effect on the equity premium across all horizons. Results show that the in-sample R2 of DY, RF, and LTY are significantly higher than
the R2 of other predictors.
Some of the findings discussed above are similar to results from existing Australian studies investigating predictability using
different datasets or sample periods. For example, Dou et al. (2012) show that the dividend yield is a good predictor of the Australian
MSCI Index equity premium at the quarterly and annual horizons and long-term government yield at the annual horizon. Alcock and
Gray (2005) use several model-selection criteria to select a predictive model for the 1-month-ahead excess market returns Australian
All Ordinaries Index. The authors find that lagged excess return and the short rate are consistently identified as good predictors by Rsquared criteria. Comparing the findings reported in Table 3 to existing studies that use different sample periods may suggest that some
variables’ predictive ability has changed over time. This is investigated in the next section.
5.2. Time-varying predictability
In this section I examine whether the All Ords equity premium’s predictability is time-varying, considering the recent evidence
reported for the US stock market (see Section 2). To this end, I use a moving sub-sample window of 60 months for estimating the
predictive regression.7 Before estimating the predictive regression, however, I test for the presence of endogeneity and persistence
following Devpura et al. (2018) and find that both exist during many phases of the time series for many predictors.8 Therefore, it is
convenient to use Amihud et al. (2010) predictive regression specification that is capable of correcting the endogeneity bias in subsample windows it is present.9 More importantly, the model addresses potential model misspecification errors since it allows for a
data-driven autoregressive order in the predictor’s equation. To this end, the lag order of the predictor within each sub-sample window
is determined using the Akaike information criteria. Fig. 1 reports the p-values from testing the null hypothesis of no predictability.
Results show that the predictability of the Australian equity premium is time-varying. The null hypothesis of no predictability is
rejected during various phases of the time series by each predictor. Valuation ratios (DY and PE) predict the equity premium 47% and
43% of the time during the sample period, respectively. Other financial variables such as lagged return, RVOL1, and RVOL2 predict the
equity premium by approximately 23% of the time. Interest rate and monetary aggregate variables RF, LTY, TMS, and M3 predict the
equity premium by approximately 71%, 70%, 58%, and 55% of the time, respectively. Trading activity variables VOL and CVM predict
the equity premium by 22% and 12% of the time, respectively, whereas sentiment variables BCS and CSI predict the equity premium by
36% and 63% of the time, respectively. Overall, RF, LTY, and TMS depict longer phases of predictability relative to other variables. The
BCS plots show that this variable was useful for predicting the equity premium during various phases of the time series before 2009.
After 2009, BCS shows a few episodes of predictability. In contrast,CSI appears to contain useful information to predict the equity
6
Rapach and Wohar (2006) note that normalizing the time-series of a predictor has no effect on in-sample and out-of-sample statistical inference.
Results reported in this section are based on estimations using a moving monthly sub-sample window of 5-years. Tests using a different window
size show robust results.
8
Following Devpura et al. (2018), I test for the presence of endogeneity and persistence using a moving sub-sample window approach for each of
the predictors in the dataset. These tests are based on estimating the size of the coefficient in the autoregressive equation of the predictor and testing
the null hypothesis of no endogeneity H0: γ = 0, using the regression ut = γvt + ηt where ut and vt are the error terms of eq.(1) and from the
autoregressive equation of the predictor, respectively. Plots are not reported for brevity.
9
Stambaugh (1999) addresses the problem of endogeneity in a predictive regression, where the autoregressive lag order of the predictor is equal
S
̂
σ
to 1, by proposing a bias-corrected estimator for β as ̂
β =̂
β + σu,v2 (1+3n ρ 1 ). In a sample of size n, the bias corrected estimator is a function of
v
persistence in the autoregressive equation of the predictor ̂
ρ 1 and the correlation between the error terms of the predictive regression u and the
predictor autoregressive equation error term v. Amihud et al. (2010) propose a method for reduced-bias parameter estimation and hypothesis testing
when the autoregressive order of the predictor is equal to or greater than 1. This method is used in modeling time-varying predictability.
7
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Fig. 1. Predictability hypothesis tests.
Plots in Fig. 1 depict the p-values of the null hypothesis of the no-predictability test based on the predictive regression in eq.(2) and eq. (3) over a
moving sub-sample window of 60 months. The dotted horizontal lines plot p-values of 5% and 10%.
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Fig. 1. (continued).
premium during and after the global financial crisis (GFC), 2008, until recent time series phases. To form a view about the dynamics in
the magnitude of predictability, I plot the effect size for each predictor and the corresponding confidence intervals in Fig. 2. It is clear
from the plots that each predictor’s effect size varies substantially during different phases of the time series. This finding is consistent
with the behavior of several predictors reported in US studies, for example, by Welch and Goyal (2008) and Dangl and Halling (2012).
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Fig. 1. (continued).
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Fig. 1. (continued).
However, it is different to a certain extent from findings reported by Devpura et al. (2018), who show mixed evidence of time-variation
in predictability for a range of predictors.
5.3. Out-of-sample performance evaluation
Next, I examine the out-of-sample performance of predictors in the dataset by calculating the out-of-sample R2 and applying the
Clark and West (2007) test. I split the total sample of size T into in-sample observations R and out-of-sample observations P = T - R. The
predictive regression in eq. (1) is estimated by first using R observations (60 months) to obtain a forecast of the equity premium ̂r for
period R + h. The predictive regression is estimated using an expanding window over the remaining out-of-sample period P- h.
Table 4 reports the results, which show that eight predictors (DY, Lag ER, RF, LTY, M3, CVM, BCS, and CSI) have a statistically
significant and positive OOS-R2 at various forecasting horizons. The statistical significance of the Clark and West (2007) test indicates
that the predictive regression provides a more accurate forecast of the equity premium than the historical average benchmark model.
In other words, the null hypothesis that the historical average mean squared forecast error is less than or equal to the predictive
regression mean squared forecast error is rejected at various statistical significance levels (10% or less).
Several variables are consistent in their in-sample and out-of-sample predictive ability (DY, Lag ER, RF, LTY, BCS, and CSI) across
some or all forecasting horizons; however, other variables show discrepancies (VOL, CVM, and M3). This finding is consistent with
evidence from the US market (Rapach and Wohar, 2006; Welch and Goyal, 2008) and the Australian stock market (Dou et al., 2012)
which shows disparity between in-sample and out-of-sample predictive ability for many predictors. Last, the results in Table 4 show
that the sign of the OOS-R2 of two predictors (DY and M3) is consistent with findings reported by Dou et al. (2012) for sample periods
investigated in their study up to 2010.
To compare the economic significance of alternative predictors, I estimate the predictive regression model in eqs. (2) and (3) to
generate h-step ahead forecasts (h = 1, 3, 6, 12) from monthly observations using a moving sub-sample window of 60 observations (5years) as explained in Section 5.2. Utility gains are calculated for a mean-variance investor with γ = 3 using methods explained in
Section 3.3. I impose realistic portfolio constraints that allow for short selling and leverage such that the portfolio weights are confined
between − 50% and 150%. Table 5 reports the results, which show positive utility gains for many variables at various forecasting
horizons. A mean-variance investor can realize significant utility gains by relying on the predictive regression using DY, interest rate
variables, M3, and BCS as predictors. These gains in utility could be interpreted as management fees the investor is willing to pay to
access forecasts based on the predictive regression (Campbell and Thompson, 2008). It is worth noting that the statistical and economic
out-of-sample performance metrics are consistent for RF and LTY at all forecasting horizons. However, inconsistencies are observed for
other predictors due to differences between a statistical-based measure (OOS-R2) and a risk-adjusted performance measure (utility
gain).
These inconsistencies are not surprising since the utility gain and OOS-R2 metrics are not strongly correlated and may provide
different information (Cenesizoglu and Timmermann, 2012). Similar inconsistencies between utility gains and OOS-R2 have been
reported in many other studies on the Australian and international stock markets (Dou et al., 2012; Welch and Goyal, 2008; Bahrami
et al., 2018; and others).
5.4. Out-of-sample multivariate analysis
To assess the out-of-sample performance for all predictors, I estimate a multivariate predictive regression or a kitchen sink model
using the least-squares method by including all potential predictors, except RVOL2, in one regression equation.10 While a kitchen sink
model may address, to a certain degree, the potential misspecification bias from omitting predictors, it may result in low out-of-sample
R2 due to overfitting (Welch and Goyal, 2008).
10
The model is estimated by excluding RVOL2 due to its high correlation with RVOL1.
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Fig. 2. Time-varying effect size.
Plots in Fig. 2 depict the predictor’s effect size based on the predictive regression in eq. (2) and eq. (3) over a moving sub-sample window of 60months. The solid black line and the dotted lines plot the effect size and the 95% confidence intervals, respectively.
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Fig. 2. (continued).
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Fig. 2. (continued).
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Fig. 2. (continued).
Table 4
Out-of-Sample R2.
Out-of-Sample R2 (%)
Predictor
Monthly
Quarterly
Semi-annual
Annual
DY
PE
Lag ER
RVOL1
RVOL2
RF
LTY
TMS
M3
VOL
CVM
BCS
CSI
11.240***
− 0.007
1.959*
− 1.787
− 1.783
23.379***
15.533***
− 0.704
2.452**
0.968
1.461**
8.196***
1.410*
20.122***
− 3.411
10.844***
− 2.667
− 2.561
45.102***
31.499***
− 1.656
4.043**
− 0.781
0.675
0.112
0.987**
30.513***
− 6.880
11.826***
− 7.739
− 7.608
59.145***
44.327***
− 1.812
6.160***
− 4.755
0.436
− 1.502
1.332**
43.165***
− 3.189
7.503***
− 11.178
− 11.060
64.420***
54.517***
− 0.064
3.561**
− 6.623
1.412
− 3.083
0.150
The table reports the estimates of the out-of-sample R2 for the predictive regression in eq. (1) at the monthly, quarterly, semi-annual, and annual
horizons. Statistical significance is based on Clark and West (2007) test statistic. ***, **, and * indicate the 1%, 5% and 10% significance levels
respectively.
Table 5
Asset allocation and economic utility.
Annualized utility gain (%)
Predictor
Monthly
Quarterly
Semi-annual
Annual
DY
PE
Lag ER
RVOL1
RVOL2
RF
LTY
TMS
M3
VOL
CVM
BCS
CSI
1.182
− 1.768
− 1.393
− 1.837
− 1.501
0.566
3.014
2.442
2.665
− 1.936
− 1.887
3.653
− 0.878
0.640
− 1.051
− 0.197
− 0.158
− 0.164
0.748
1.225
0.717
− 0.020
− 1.918
− 0.001
0.555
− 0.001
− 0.411
− 2.044
− 0.001
− 1.266
− 1.316
1.119
2.047
0.464
0.000
− 0.305
1.275
− 0.277
0.000
− 0.105
− 0.425
− 0.001
− 0.193
− 0.204
0.157
0.501
0.140
0.000
− 0.014
0.000
0.067
0.000
The table reports the annualized utility gains derived from forecasting the equity premium using the predictive regression relative to the historical
average.
Results reported in Table 6 show that the kitchen sink model generates positive OOS-R2 only at the monthly forecasting horizon.
This finding may suggest that the model provides more accurate forecasts than the historical average model when forecasting the
monthly equity premium. However, Clark and West (2007) tests are not statistically significant for any of the reported OOS-R2.
Although the kitchen sink model performs poorly in terms of the OOS-R2 at all forecasting horizons other than the monthly horizon,
utility gains are positive at every horizon, decreasing with longer forecasting horizons. The out-of-sample performance of the kitchen
sink model is generally in line with the results reported in other studies. For example, Dou et al. (2012) find negative OOS-R2 using
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Table 6
Kitchen sink model.
Kitchen sink regressions
2
OOS-R (%)
Utility gain (%)
Monthly
Quarterly
Semi-annual
Annual
0.156
0.327
− 0.614
0.095
− 0.595
0.039
− 0.191
0.014
The table reports OOS predictive evaluation tests based on least-squares estimates from a multivariate regression, including k predictors. The pre­
∑
dictive regression model (rt,t+h = α + kj=1 βj xj,t + ut,t+h ) is estimated using a set of k = 12 predictors from January 2000 to June 2020.
Table 7
In-sample Predictive Regression Estimation Results – Equity index futures.
1
h
Monthly
Predictor
β
DY
1.186***
2.614
− 0.446
− 1.386
1.133***
2.709
− 0.118
− 0.263
− 0.138
− 0.306
2.073***
5.84
1.758***
5.059
1.011**
2.367
− 1.069
− 2.757
− 1.012
− 3.183
− 0.263
− 2.085
1.469***
4.406
0.909***
2.858
PE
Lag ER
RVOL1
RVOL2
RF (− )
LTY (− )
TMS
M3
VOL
CVM
BCS
CSI
2
3
4
Quarterly
R2 (%)
6.81
0.96
6.34
0.06
0.09
21.01
15.11
5.10
5.53
4.93
0.33
9.61
3.97
β
0.961
1.851
− 0.312
− 0.962
0.941**
2.375
− 0.616
− 1.632
− 0.62
− 1.668
2.056***
6.194
1.776***
5.507
0.865*
1.783
− 0.939
− 2.771
− 0.301
− 1.365
− 0.211
− 1.616
0.381*
1.697
0.443**
2.068
5
6
Semi-Annual
R2 (%)
9.25
0.97
8.81
2.81
2.85
41.59
30.99
7.83
8.48
0.89
0.44
1.24
1.76
β
0.885
1.545
− 0.323
− 0.966
1.036***
3.49
− 0.482
− 0.992
− 0.481
− 0.998
2.072***
6.136
1.864***
5.549
0.722
1.304
− 0.929
− 2.769
− 0.208
− 0.926
− 0.291
− 2.524
0.281
1.195
0.320**
1.765
7
8
Annual
R2 (%)
10.75
1.44
13.5
2.42
2.42
57.12
45.54
7.69
10.86
0.57
1.17
0.06
1.264
β
0.780
1.391
− 0.348
− 1.162
0.791**
2.495
− 0.273
− 0.611
− 0.273
− 0.612
1.960***
6.537
1.935***
5.98
0.423
0.792
− 0.766
− 2.311
− 0.083
− 0.421
− 0.441
− 3.709
0.077
0.526
0.229
1.357
R2 (%)
11.60
2.31
10.73
1.07
1.08
65.26
60.61
3.67
10.01
0.12
1.71
0.08
0.89
The table reports the ordinary least squares estimates of the coefficient of the predictor β in percentages, the t-statistic beneath coefficient estimates,
and the R2 statistic of univariate predictive regressions in eq. (1). (− ) indicates that the negative of the predictor variable is used in the predictive
regression. The results are based on estimates of the equity premium using equity index futures (ASX SPI200). Heteroscedasticity and autocorrelation
robust t-statistics for testing the null hypothesis of no predictability (H0: β = 0) against (HA: β > 0) are reported in brackets. The in-sample t-statistics
and their corresponding p-values are estimated using a wild bootstrap procedure using 1000 draws. The predictive regression is estimated at the
monthly, quarterly, semi-annual, and annual horizons; ***, **, and * indicate the 1%, 5%, and 10% significance levels, respectively, according to wild
bootstrapped p-values.
estimates based on a range of predictors of the quarterly and annual Australian equity premium using a sample from 1985 to 2010 and
in sub-sample periods and report mixed signs for utility gains in kitchen sink regressions. Bahrami et al. (2018) report inconsistencies
in the signs of the OOS-R2 and utility gains for several developing markets (see Table 6, p. 746). Welch and Goyal (2008) show that the
kitchen sink performs poorly out-of-sample using a long US data history.
5.5. Microstructure bias and predictability
One important concern with the predictability inference is that the results could be spurious when forecasting the equity premium
because equity indices are not tradable assets; however, their constituents are. The price adjustment process of these constituents may
exhibit different response times to information arrival, and they often trade non-synchronously, which induces positive autocorre­
lations in the equity index return. Ahn et al. (2002) find statistically and economically significant positive autocorrelations in equity
indices return that are not present in their corresponding futures contracts. The authors indicate that microstructure-type biases such
as stale prices are responsible for higher positive autocorrelations in equity indices return relative to their corresponding futures
contracts.
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Table 8
Out-of-sample R2 – equity index futures.
Out-of-Sample R2 (%)
Predictor
Monthly
Quarterly
Semi-annual
Annual
Panel A: Individual predictors
DY
PE
Lag ER
RVOL1
RVOL2
RF
LTY
TMS
M3
VOL
CVM
BCS
CSI
3.872***
− 1.098
4.842**
− 2.993
− 2.960
21.657***
13.086***
4.370*
4.471**
4.518***
− 3.607
9.103**
3.463**
− 11.003
− 11.444
5.967**
− 4.729
− 4.386
38.861***
24.356***
2.083
7.453***
− 0.007
− 0.284
0.555
1.064*
− 24.506
− 23.795
8.937***
− 23.498
− 23.066
43.269***
26.251***
− 6.754
8.336***
− 4.862
− 1.101
− 0.974
− 0.232
2.259**
− 30.289
5.247*
− 40.022
− 38.991
36.193***
31.270***
− 15.913
6.647**
− 4.802
− 4.385
− 1.505
− 1.318
Panel B: Kitchen sink model
Multi-predictor
− 0.257
0.505
0.509
0.601
The table reports the estimates of the out-of-sample R2 for the predictive regression in eq. (1) at the monthly, quarterly, semi-annual, and annual
horizons. The results are based on estimates of the equity premium using equity index futures. Estimates based on the predictive regression using
individual predictors and the kitchen sink model are reported in Panels A and B. Statistical significance is based on Clark and West (2007) test statistic.
***, **, and * indicate the 1%, 5% and 10% significance levels respectively.
Hence, assessing whether predictors contain information about predicting the equity premium using equity index futures may
address concerns about the microstructure bias. However, one limitation of this analysis is that there are no future contracts available
on the All Ords index. Instead, some comfort could be gained by using ASX SPI200 index futures contracts on the corresponding
underlying equity index, ASX200, since several predictors are interest rates, aggregate monetary variables, or sentiment variables.
Besides, fundamental financial predictors and trading activity variables are available for the ASX200 index.11 Therefore, a dataset is
obtained from the Thomson Reuters Eikon (Datastream) database from August 2001 to June 2020 for the analysis.12
Estimates of the predictive regression in-sample and out-of-sample are reproduced. Table 7 reports the in-sample results from
univariate predictive regressions. At large, these results confirm that the microstructure bias does not drive the predictability evidence
reported in Table 3. For example, the coefficient estimates for three predictors (Lag ER, RF, LTY) remain statistically significant for all
forecasting horizons. The coefficient estimates for sentiment variables (BCS, CSI) remain statistically significant at the monthly and
quarterly forecasting horizons and CSI at the semi-annual forecasting horizon, whereas the coefficient estimate of DY is statistically
significant only at the monthly forecasting horizon. Unlike results reported in Table 3, TMS emerges as a predictor of equity premium
at the monthly and quarterly forecasting horizons at the 10% significance level.
Regarding the out-of-sample predictability evidence, the results reported in Table 8 show consistency between the evidence re­
ported from the equity index and the futures market across the four forecasting horizons in the cases of lagged returns (Lag ER) and
interest rates and monetary aggregates (RF, LTY, and M3). Other predictors show disparity in the evidence from univariate predictive
regressions and the kitchen sink model when the equity premium is calculated from equity index futures returns rather than the equity
index itself.
5.6. Variable selection problem and maximal R2
The variable selection problem and related data snooping bias have always been a concern in the predictability literature. Lo and
Mackinlay (1990) indicate that the data snooping bias can be substantial, leading to spurious results. The authors state that “tests of
financial asset pricing models may yield misleading inferences when properties of the data are used to construct the test statistics.” In some
cases, variable selection results in inflated R2, leading to inaccurate inference when classical methods of assessing goodness of fit are
used (Granger and Newbold, 1974). In a predictive regression estimated using the least-squares method, the R2 is beta distributed
under the null hypothesis of no predictability as,
(
)
k t − (k + 1)
Beta ,
2
2
11
CVM is the log difference in trading volume.
Equity index futures in Australia are available for selected underlying equity indexes. The broadest equity index for which futures are available is
the S&P/ASX 200. The timeseries for ASX SPI200 equity index futures is available from Thomson Reuters Eikon (Datastream) from August 2001. It is
noted that the All Ords and S&P/ASX 200 are benchmark indexes tracking the performance of the top 500 and 200 companies listed on the
Australian Stock Exchange (ASX), respectively. See the ASX website for more information (https://www2.asx.com.au/)
12
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Table 9
Critical R-squared cutoff values at the 95% level.
Number of potential predictors (m)
Number of selected predictors (k)
Univariate model
Kitchen sink model
Panel A: Equity index (Sample size T = 336)
13
20
50
0.021
0.022
0.026
0.079
0.117
0.176
Panel B: Index futures (Sample size T = 222)
13
20
50
0.031
0.034
0.039
0.117
0.174
0.255
The table reports the 95% cutoff values for the predictive regressions assuming that k predictors are chosen out of m for a
given sample size T.
where k and t denote the number of regressors and sample size, respectively. In general, the R2 measure is used to assess the signif­
icance of the regression. However, Foster et al. (1997) note that using classical cutoff values for the R2 leads to erroneous inference due
to variable selection, which is a typical problem in predictability studies, since investigators choose the best predictors out of a set of
potential predictors. Foster et al. (1997) recommend adjusting the critical goodness of fit values by bounding their distribution and
considering the potential number of regressors k chosen out of m that maximize a regressions’ goodness of fit. The authors propose two
techniques for assessing the goodness of fit of regressions using the Bonferroni test and Rencher and Pun (1980) approximation of the
distribution of the maximal R2. They show using numerical analysis and a re-evaluation of past studies that the variable selection bias is
not trivial, leading to incorrect inference.
To address this bias, I use the Rencher and Pun (1980) approximation, which relies on the extreme value theory to derive the
distribution of the maximal R2. In the presence of dependence between possible regression combinations, the γ percent cutoff level of
the maximum R2 is,
[
]
ln(γ)
R2γ = F − 1 1 +
d
ln(N)cN
)
m
is the binomial coefficient, c = 1.8 and d = 0.04.
k
Table 9 reports the critical R2 values corresponding to the number of predictors (k) chosen from potential regressors (m) and sample
size (T). These critical values are used to test the null hypothesis that all predictor coefficients are equal to zero when k predictors are
chosen from m potential predictors. The alternative hypothesis is that at least one of the coefficients is not zero. Since the number of
potential regressors may well exceed the numbers of predictors used in the dataset, additional critical values are provided for m =
(20, 50).
Comparing the critical R2 cutoff values with the R2 values reported in Table 3, the regression R2 for models where the predictor’s
coefficient is statistically significant exceed the critical values reported in Table 9 for all values of m, except for VOL at the monthly
forecasting horizon, CSI at monthly, quarterly, semi-annual forecasting horizons and BCS at the quarterly and semi-annual forecasting
horizons. Hence, an exhaustive search for potential predictors may not produce a better prediction model for several predictors in the
dataset.13 The evidence is qualitatively similar for predictive regressions using equity index futures and kitchen sink models.14
where F− 1 is the inverse of the Beta cumulative distribution function, N =
(
6. Conclusion
In this paper, I investigate the equity risk premium’s predictability in the Australian All Ords stock market using a dataset of
financial, economic, and sentiment variables. Parameter estimates are obtained using methods that address potential biases associated
with predictors such as endogeneity, overlapping observations, and model misspecification. Using a moving sub-sample window
approach, I also examine time-varying predictability and address model misspecification issues in the predictor’s autoregressive
equation. Results identify the prowess of various predictors in-sample and out-of-sample for various forecasting horizons (monthly,
quarterly, semi-annual, and annual). There is strong evidence of time-varying predictability for a range of predictors used in the
analysis. Kitchen sink results show that the predictive regression improves mean-variance investors’ utility gains relative to the
13
Ferson et al. (2003) advocate considering a more stringent cutoff of 1% rather than the 5% rule of thumb for statistical inference for better
inference in a testing environment that is subject to data mining, which is unavoidable in this line of research, and potential for spurious results
when predictors that are highly autocorrelated and when the equity premium is moderately autocorrelated. Tests using a 1% cutoff were conducted
using the Rencher and Pun (1980) approximation result in similar qualitative inference at large to the one reported in Table 9.
14
R2 estimates from the kitchen sink model with 12 predictors are 40% and 25.07% when using the equity index and equity index futures,
respectively, to predict the equity premium at the monthly horizon. The R2 estimates for other forecasting horizons exceed the monthly estimates.
19
Pacific-Basin Finance Journal 71 (2022) 101683
D.J. Jurdi
historical average model. The out-of-sample evidence confirms the predictive regression’s usefulness in realizing utility gains for
investors for various predictors in the dataset.
While the analysis is conducted in an environment that is unavoidably subject to microstructure bias and variable selection bias,
estimates based on equity index futures returns, given available data, indicate that the predictability evidence for several predictors is
robust to the microstructure bias. Furthermore, inference based on critical R2 cutoff values computed using the Rencher and Pun
approximation indicates that, in many cases, the predictive regression fit is not a result of exhaustive variable selection. Hence, the
results suggest that several predictor variables considered in this paper contain useful information for predicting the Australian equity
risk premium.
Declaration of Competing Interest
I declare that I am the sole author of this manuscript. It has not been published before and is not currently being considered for
publication elsewhere. I confirm that I have no conflict of interest with anyone. I also confirm that there has been no funding provided
for this work and there has been no funding that could have influenced its outcome.
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