Dynamic Factor Model and Predictability of Stock Returns : Canadian Evidence · 2013-09-27 · Dynamic Factor Model and Predictability of Stock Returns : Canadian Evidence Alexandre

Dynamic Factor Model and Predictability of

Stock Returns : Canadian Evidence

Alexandre Kopoin∗ Stephane Chretien†

Preliminary version:

[Please do not quote without the authors’ permission]

August, 2013

Abstract

This paper contributes to the existing literature on the ability of information variablesto predict the equity premium by showing that macro-financial variables have importantpredictive power for the Canadian equity premium. We use a dynamic factor analysis tosummarize the information from a large panel of 188 monthly macro-financial series span-ning from 1976 to 2011. Our main results document significant predictable variation in theCanadian equity premium that is associated with macroeconomic activity both in-sample (IS)and out-of-sample(OOS). In addition, using a categorization of data within the dynamic fac-tor analysis, our results show that,pure financial factors are not statistically significant forthe full-sample IS exercise, but pure economic and price factors are statistically significant.Those findings suggest strong business cycle variation in expected Canadian equity premiumthat is not highlighted in the yield curve or in valuation variables like the price dividendratio. Overall, the dynamic factor model appears to be an interesting method to capture therelation between the Canadian equity premium and the information variables.

Keywords: Factors model, principal component, equity premium, forecasting.

J.E.L. Classification: G10, G12, E0, E4.

∗Department of Economics and CIRPEE, Laval University. Pavillon J.-A.-DeSeve, 1025 avenue des Sciences-Humaines, Quebec city, Canada, G1V 0A6. Email: [email protected].†Investors Group Chair in Financial Planning. Associate Professor of Finance, Finance, Insurance and

Real Estate Department. Faculty of Business Administration, Laval University, CIRPEE, GReFA, and LAB-IFUL. Pavillon Palasis-Prince, 2325, rue de la Terrasse, Quebec City, QC, Canada, G1V 0A6. Email:[email protected]

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1 Introduction

Financial economists have long been interested in the empirical relation between the equitypremium and risk factors. The knowledge of this relation, which is central to the development oftheoretical models for explaining the risk-return nexus, remains an important topic for financialinvestors, specifically in portfolio management. In dynamic financial theory, several classes ofmodels have been proposed to conceptualize the underlying relation between the equity premiumand risks factors. The basic idea of these models is to capture the predictive variations in theequity premium through some pre-selected information variables that are able to explain thisrelation. From cross-sectional point of view, the conditional capital asset pricing model (CAPM)first proposed by Merton (1972), and the intertemporal CAPM proposed by Merton (1973) areconsidered as benchmarks among theoretical models that have been standard to motivate thisrelation.

More recently, a large number of empirical studies have proposed multiple regression mod-els and/or combination forecast models to provide additional insights in the equity previsionliterature (Aıt-Sahalia and Brandt (2001); Avramov (2002); Campbell and Yogo (2006); Angand Bekaert (2007); Campbell and Thompson (2008); Campbell (1997); Rapach et al. (2010);Chretien and Coggins (2012) among others). These empirical studies conclude that the equitypremium is forecastable using both financial and economic information variables such as the in-flation rate, unemployment rate, the Treasury bill yield, the term structure of interest rate, thedividend yield ratio and the credit default premium. However, this literature is facing criticismsbased on the choice of information variables as well as the optimal number of lags to be retained.Indeed, in most of these studies, only a few conditioning information variables are used in theeconometric framework.

In practice, several possible reasons may force researchers to choose among a large panel ofpotential predictors, a small number of information variables. The most known reason is thatconventional statistical analyses are quickly exacerbated by the degrees-of-freedom problems asthe number of information variable increases. Unfortunately, the selection process by whichpredetermined information variables are selected for the econometric framework affects substan-tially the estimations of the risk-return relation (Ludvigson and Ng (2007)) and leads to severalproblems such as omitted information, spurious regression, model instability, data mining, smallsample bias and poor predictive ability (Goyal and Welch (2003); Paye and Timmermann (2006);Rapach et al. (2005); Ang and Bekaert (2007); Timmermann (2008); Welch and Goyal (2008)among others).

This paper contributes to the existing literature on the ability of information variables topredict the equity premium by showing that macro-financial variables have important predictivepower for the Canadian equity premium. To do so, we use dynamic factor analysis to summarizethe information from a large number of macroeconomic series. The advantage of such modelrelies on the fact that they allow to summarize in the first few principal components a significantfraction of the overall covariation among the potential information variables in the panel. Basedon the seminal work of Stock and Watson (1989, 2002, 2004a,b) , Bai and Ng (2011) and Boivinand Ng (2005), the dynamic factor model assumes that the data are driven by a small numberof factors which are a linear combination of several hundred variables (Bernanke and Boivin(2003), Bai and Ng (2009), Ludvigson and Ng (2010a,b)). By summarizing the information

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from a large number of financial series in a few estimated factors, we eliminate the arbitrarychoice of a small number of exogenous predictors to estimate the empirical predictive relation.Hence, our empirical procedure contributes to alleviate criticisms on the standard regressionmodels.

Using the monthly equity premium on the Canadian market from 1976 to 2011 as dependentvariable and a comprehensive database of 188 information variables classified into a financialblock, an economic block an price block, our results are twofold: First, in contrast to some of theexisting empirical literature (Welch and Goyal (2008), among others), we find strong predictablevariation both in sample (IS) and out-of-sample (OOS) in the Canadian equity premium that isassociated with macroeconomic activity. The adjusted R2 of the main predictive model are 4.4%in sample and 3.4% out of sample. We can learn more on the economic interpretation of thepredictability by using a categorization of the information variables within the dynamic factoranalysis, although the IS performance is lower with such a grouping technique. Our resultsshow that pure financial factors appear to be not statistically significant for the full-sample ISexercise. However, pure economic and price factors are statistically significant in the IS exercise.This suggests strong business cycle variation in expected Canadian equity premium that is nothighlighted in the yield curve or in valuation variables like the price dividend ratio. Overall, thedynamic factor model appears to be an interesting method to capture the relation between theCanadian equity premium and the informations variables

The rest of the paper is organized as follows. Section 2 presents the related literature andsection 3 gives the set up of the econometric framework. In Section 4, we present the data usedand some summary statistics. Section 5 presents our empirical results and section 6 gives aneconomic interpretation of extracted factors. Section 7 concludes.

2 Related literature

The connection between equity returns, especially the equity premium, and information variableshas been one of the most heavily researched topics in empirical finance over the past twenty years.Hundreds of scholarly papers have been written to assess the predictive power of informationvariables in forecasting equity returns. In early contribution, Fama and French (1988) andCampbell and Shiller (1988) present evidence that valuation ratios such as the dividend yieldpredict the U.S. equity risk premium. Additionally, other empirical papers such as Breen et al.(1989), and Fama and French (1989) have highlighted predictive ability for nominal interestrates, the default and term spreads.

More recently, empirical evidence suggests that the default premium, the term premium, theterm structure of interest rates, the dividend ratio and some measures of stocks variability havepredictive power in the forecasting of stocks returns (Carmichael and Samson (2003), Goyaland Welch (2003), Campbell and Vuolteenaho (2004), Cochrane (2001), Pastor and Stambaugh(2009), Chretien and Coggins (2012)). Similarly, Cochrane and Piazzesi (2005) showed thata linear combination of five forward spreads explains between 30% and 35% of the variationin next year’s excess returns on bonds with maturities ranging from two to five years. Inthe same line, Brandt and Wang (2003) argue that the risk premium is driven by shocks toaggregate consumption as well as inflation. Goyal and Welch (2003), using out-of-sample tests,

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examine the predictive ability of the dividend price ratio for the Center for Research in SecurityPrices (CRSP) value-weighted annual excess returns over the 1926-2000 period. They showthat although the dividend price ratio exhibits limited out-of-sample predictive ability, it hasa strong in-sample predictive power. In other empirical paper, Welch and Goyal (2008) showthat a long list of predictors from the literature is unable to deliver consistently superior out-of-sample forecasts of the U.S. equity premium relative to a simple forecast based on the historicalaverage (a constant expected equity premium model).

As indicated in Campbell (1997) and Campbell and Cochrane (1999), macro variables couldbe key state variables in intertemporal asset-pricing models and represent priced factors in theArbitrage Pricing Model. Hjalmarsson (2008) confirms equity risk premium predictability basedon macroeconomic variables across countries. Under the conventional view that asset pricesequal expected discounted cash flows, Cochrane (2001) indicates how this literature profoundlyshifts the emphasis from expected cash flows to discount rates, reshaping modern asset pricingtheory.

While, using macro-financial variables for equity premium predictability seems to be natu-ral1, the selection process by which macro-financial variables are selected for the econometricframework affects substantially the estimations of the risk-return relation and is not a consen-sus. The lack of consistent out-of-sample evidence indicates the need for improved forecastingmethods to better establish the empirical reliability of equity premium predictability. In thispaper, we propose a large-dimensional dynamic factor model to explore the predictive power ofa large set of macro-financial variables both in sample and out of sample.

The paper is related to the following dynamic factor model studies: Stock and Watson (2002,2004a,b), Bai and Ng (2011), Campbell and Thompson (2008), and Ludvigson and Ng (2007,2010a,b). For example, Stock and Watson (2002) use the factor-augmented vector autoregressive(FAVAR) framework to forecast the U.S. GDP growth. Ludvigson and Ng (2007) use a factordecomposition for risk premia prediction on the bonds market. In the same spirit, Ludvigson andNg (2010b) analyze the forecasting performance of latent common components for stock marketreturns and volatility. In contrast to the previous studies that examine the factor analysisapproach to explaining the U.S. stock market excess returns, this is the first study to use a widearray of macro-financial variables to investigate both in sample and out of sample the Canadianequity premium predictability.

3 Econometric framework

3.1 Predictive regressions for equity premium

For t = 1, 2, · · · , T , consider the predictive regression model

EQPt+1|t = α+ β′(L)Zt + γ′(L)EQPt−p+1 + et+1, et+1 ∼ N(0, 1) (3.1)

where EQPt denotes the Canadian Equity Premium defined as the excess return of the Canadianstock market index over the risk-free interest rate and L is the lag operator. Zt is a set of potential

1Recent literature points out the propagation mechanism of the 2007-2009 financial turmoil over the realeconomy as an overwhelming proof of this empirical evidence.

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predictors and p is the number of lags to be included for the Canadian equity premium. Therandom walk theory assumes that the set of information variables has no predictive powerfor excess stocks returns. However, empirical tests often reject the null hypothesis that theparameter vector β is zero (Fama and French (1989), Welch and Goyal (2008), Ludvigson andNg (2010a,b), Chretien and Coggins (2012), among others). A standard approach to assess thepredictability of information variables is to choose a set of K preselected conditional variablesat time t such as the Treasury bill yield, the term structure of interest rate, the dividend yieldor dividend-price ratio, the credit default premium, the inflation rate and the unemploymentrate. As defined, the set of the K potential predictors is given by the K × 1 vectors Zt andthe forecasting of the equity premium is estimated by a linear equation using a least squaresestimation method. However, such a procedure may be not feasible when the number of potentialpredictors is too large for the period for which the data are observed (K >> T ). Unless followinga method for ordering the importance of the variables in selecting potential predictors or a subsetof variables, the standard econometric method fails in the estimation of the aforementionedequation. For instance with a N × T data set, there are potentially 2N possible combinationsto be considered.

The approach that we consider to assess the predictability of the Canadian equity premiumis to assume that the whole data have a factor structure. Technically, if these common factorswere observed (Ft), a nice way to estimate the forecasting equation is to replace equation (3.1)by the following factor augmented vector regression.

EQPt+1 = α+ β′(L)Ft−q + γ′(L)EQPt−p+1 + εt, εt ∼ N(0, 1) (3.2)

where Ft is a set of r factors whose dimension is much smaller than the initial data set buthas good predictive power for the EQP . In equation (3.2), q and p denote the number of lagsretained for the factors and the equity premium, respectively. Since the common factors areunobserved, to implement the regression given by (3.2), we need to estimated Ft using a matrixreduction technique, like Principal Component Analysis (PCA), and then isolate those factorsthat have predictive power for our targeted variable, EQP . The advantage of such a methodrelies on the fact that it allows to summarize in the first few principal components a significantfraction of the overall covariation among the series in the panel. Based on the seminal workof Stock and Watson (2002, 2004a,b) for the US, Massimiliano et al. (2002) and Cristadoroet al. (2005) for the euro area, the dynamic factor model assumes that the data are driven bya small number of factors which are a linear combination of several hundreds of variables. Thisforecasting methodology is useful for economic and financial institutions, and is widely appliedto monitor hundreds of macroeconomic indicators (see Giannone et al. (2008), Ludvigson andNg (2010a,b)) and for the analysis of monetary shocks (see Bernanke and Boivin (2003)).

3.2 Dynamic factor model for equity premium

In this section, we consider a large data set of macroeconomic and financial variables. We wantto investigate the contribution of these macro-financial information variables to the forecastingof the Canadian equity premium, given by the excess return of the Canadian stock market overthe risk free rate. Our approach to forecast this excess return is to use the dynamic factor model.As formalized by Stock and Watson (2002), the dynamic factor model is a state-space model

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that handles the information contained in a large dataset in parsimonious way. This model usesas input arguments, the T time series observations for N cross-section units of stock returns andmacroeconomic variables, which we denote by Zt = [Z1t · · ·Znt · · ·ZNt]′. In this formalization,Zt is an N−dimensional multiple time series (N ×1 vector) variables, observed for t = 1, · · · , T .Since our goal is to forecast the equity premium in the Canadian stock market index, let Yt bea vector of the Canadian equity premium to be forecasted and YT+1|T , the 1-step-ahead out-

of-sample forecast of a target variable. For notation, let Z = [Z1 · · ·Zt · · ·ZT ]′ be the T × Nmatrix of observable variables known as information variables. Although, both the static factorand the dynamic factor model are used in this forecasting exercise, we focus only on the lastone, since the static factor representation is a special case of the dynamic factor representation.

Following Gweke (1977), Sargent and Sims (1977)2 and Cristadoro et al. (2005)3, the dynamicfactor model has the following representation:

Zit = λ′i0ft + λ′i1ft−1 + · · ·+ λ′iqft−q + eit =

q∑s=0

λ′isft−s + eit, eit ∼ N(0, 1) (3.3)

where Zit is the observed data for the ith cross-section at time t (i = 1, · · · , N) and (t = 1, · · · , T )and ft is a vector (r × 1) of common factors. λi is a vector (r × 1) of factor loadings and eit isthe idiosyncratic component associated with Zit. r and q are respectively the estimated numberof the factors and the number of lags. The static factor representation is obtained by setting qto 0. In the form of an N -dimensional time series with T observations, equation (3.3) is givenby :

Zt = ΛFt + et (t = 1, 2, · · · , T ), (3.4)

where Λ = [Λ0,Λ1, · · · ,Λq], et = (e1t, e2t, · · · , eNt)′ and Λj is given by:

Λj =[(λj1)′, (λj2)′, · · · , (λjN )′

]=

λj11 λj12 · · · λj1rλj21 λj22 · · · λj2r

......

. . ....

λjN1 λjN2 · · · λjNr

∈MN×r(R) 1 ≤ j ≤ q.

In the same way, Ft is given by:

Ft = [ft, ft−1, · · · , ft−q]′ =

(f1t f1t−1 · · · f1t−q)

′

(f2t f2t−1 · · · f2t−q)′

......

...(frt frt−1 · · · frt−q)

′

∈ Vr×(q+1)(R).

We also use the matrix notation to set the representation of the dynamic factor for conve-nience. The matrix form of the dynamic factor model is given by:

2Gweke (1977) and Sargent and Sims (1977) analyze frequency domain methods for the estimation of thedynamic factor model using a small sets of variables.

3Cristadoro et al. (2005) suggested a feasible econometric approach for the analysis of a large set of variablesby proposing a new forecasting model based on the information set from a large panel of time series.

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Z = FΛ′ + e, e ∼ N(0, IN ) (3.5)

where Λ and F (both unknowns) are respectively N × r(q+ 1) matrix and T × r(q+ 1) matrix.These unobservable common factors are often interpreted as the driving forces in the financialmarket. Under a functional form of the dynamic factor model, each dataset (T ×N matrix) canbe represented as a sum of two components. The first element is the common component and thelast one is an idiosyncratic component. The common components are driven by a few numberof factors which are common to all variables in our dataset and the idiosyncratic componentis specific to each variable. These common factor could be interpreted as a vector of systemicrisk. Likewise, the loadings may be interpreted as the exposure to the risk factors. After therepresentation step, we estimate ft using the method of asymptotic PCA developed by Connorand Korajzcyk (1986) for small T and data-rich environment. In the next section, we presentthe estimation method and technique used to select the optimal number of factors.

3.3 Estimation method and optimal number of factors

Our interest in using the dynamic factor model is to assess the predictability of the equitypremium in the presence of a large number of information variables. For this purpose, afterthe selection of the set of the potential predictors, the common factors matrix and the loadingsmatrix should be estimated. Stock and Watson (2002), Cristadoro et al. (2005) and severalother authors used the PCA method to estimate the loadings matrix and the common factors.Stock and Watson (2002) showed that the common factors can always be estimated by using theasymptotic PCA in presence of a large number of observations. In our framework, the size ofavailable data is relatively large and therefore, following the result in Stock and Watson (2002,2004a,b), we use a PCA-OLS estimator for estimating the factors and the loadings. In such case,the maximal number of factors which can be estimated by using this method is then min{N,T},with N , the number of variables, and T , the numbers observations. This method involves theestimation of the eigenvalues and the eigenvectors decomposition of the spectral density matrixof Zt. The spectral density matrix of Zt, which is estimated using the frequencies −π < ω < πcan be decomposed into the spectral of the unobservable common factors and the idiosyncraticcomponent. The decomposition of spectral density matrix of Zt is given by:

Σ(ω) = ΣZ(ω)⊕ Σe(ω) (3.6)

Where ΣZ(ω) = λ(e−iω)ΣFλ(e−iω)′ is the spectral density matrix of the unobservable com-mon factors. The PCA-OLS estimator of the factors and the loadings Λ is the matrix thatminimizes the following residual sum of squares:

V (s) = minΛ,F

{N∑i=1

T∑t=1

(Zit − λ′iFt)2 F ′F/T = Is where s = r(q + 1)

}, (3.7)

where r and q denote the optimal number of common factors and the number of lags, respec-tively4. The PCA-OLS is computationally convenient, even for very large N . Moreover, it can

4q and r are selected according to the Bayesian Information Criterion (BIC).

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be generalized for some data irregularities like missing observations by using the estimation mo-ment algorithm (EM). According to Stock and Watson (1989), the system could be expressedin a state-space form and estimate using the Kalman filter in presence of missing observations.The estimated factors are a linear combination of the variables in a selected subspace of relevantvariables where the coefficients can be positive or negative, reflecting in a sense the correlationbetween the equity premium and each factor. In practice, factors are extracted in a sequentialway, with first the factor that explains the most variability in the data. The second factor is theone that explains the second most variability in the data, and so on.

After the estimation of the common factors by the asymptotic PCA, the predictive regressionequation is obtained by replacing Ft with an estimated value that we denote by Ft. Technically,for each t, Ft is a linear combination of each element of the N × 1 vector Zt = (Z1t, · · · , ZNt)′,where the linear combination is chosen to solve the above-mentioned minimization. As shown inBai and Ng (2011) and Stock and Watson (2002), the space spanned by the latent factors Ft isconsistently estimated by Ft when N,T →∞. As the common factors are a linear combination ofeach element, a common criticism of the PCA method is that the factors may be not interpreted.However, as shown in Ludvigson and Ng (2007), the common factors could be interpreted byorganizing the data set into several blocks corresponding to the economic sectors. In that case,each estimate could represent the driving force of each economic sector. Other method suchas varimax may be used to interpret the common factors. Since the goal of this paper is toassess the contribution of macro-financial information in the forecasting of the Canadian equitypremium, we consider the interpretation of the common factors by organizing our data into 3blocks (financial block, economic block and price block). With the estimated common factorsas well as its lags, the one period ahead predictive equation is given by:

EQPt+1|t = α+ γ1(L)F 1t−q1 + γ2(L)F 2

t−q2 + · · ·+ γq(L)F rt−qr + β′(L)EQPt−p+1, (3.8)

where α, β and γ are estimated from equation (3.2) and qi is the estimated lags of the factori, 1 ≤ i ≤ r. As shown in Bai and Ng (2011), the difference between the estimated factors, Ft,and the space spanned by Ft vanishes at rate min{N,T}. An advantage of the method relies onthe fact that little structure is imposed on the estimation and the resulting factor augmentedregression is robust to the choice of the estimator.

4 Data and summary statistics

In this empirical investigation, our dependent variable is the Canadian Equity Premium definedas the excess return of the Canadian stock market index over the risk-free interest rate. As inChretien and Coggins (2012), we first consider the information variables that have already beenused or could make sense in a Canadian context, or that are common in U.S. studies. However,some of these potential information variables are not available. For instance, the sample rangeof some variables are shorter than others. As in Chretien and Coggins (2012), we construct someof the information variables from the combination of other underlying variables to obtain longertime series. Our data set includes monthly Canadian financial and macroeconomics indicatorsfrom the Canadian Financial Markets Research Centre (CFMRC) and CANSIM. Our spanningperiod is 1976m1 to 2011m12.

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The data set is then completed by a balanced panel of numerous economic and financial timeseries. Overall, the series are selected to represent broad categories of macroeconomic variables:real output and income, employment and hours, real retail, manufacturing and sales data, inter-national trade, consumer spending, housing starts, inventories and inventory sales ratios, ordersand unfilled orders, compensation and labor costs, capacity utilization measures, price indexes,interest rates and interest rate spreads, stock market indicators, and foreign exchange measures.The complete list of series is given in the panels A to C of a table 9, where Trans. indicatesthe transformation applied to standardize the variables and Gr., denotes the block associatedwith the variable. Since our panel contains financial, macroeconomic and price variables, theestimated factors could not be interpreted as pure macroeconomics, financial and price fac-tors. In fact, the estimated factors are defined as a linear combination of both financial andmacroeconomic indicators. Construction details of the macroeconomic and financial indicatorsare below.

4.1 The equity premium

The equity market returns are the total returns (including dividends) of the S&P/TSX Com-posite Index (previously known as the TSE Composite Index) from the CFMRC database. Therisk-free rate is the one-month return on the three-month Government of Canada Treasury bills,taken from the CFMRC database. The equity premium is the difference between the equitymarket return and the risk-free rate. Figure 1 shows the monthly EQP from January 1976to June 2012. As well documented in Chretien and Coggins (2012), we can easily locate thesharp decline associated with high interest rates and inflation concerns of the early 1980s, theOctober 1987 crash, the Russian debt default and associated Long Term Capital Managementbankruptcy of August 1998, the burst of the tech bubble at the end of 2000 and the start of2001 (lead by the decline in Nortel Networks Inc.), the September 2001 terrorism attack andthe intensification of the subprime crisis in September-October 2008.

4.2 Information variables

4.2.1 Macroeconomic variables

Our macroeconomic information variables are selected judgmentally to represent the main cate-gories of macroeconomic time series. The set of data contains real activity indicators, monetaryand financial indicators, national account data, industrial production data, productivity andlabour market indicators, retail trade indicators and price indicators. In this subsection weprovide a description the information variables used. More details about the macroeconomicinformation variables are given in appendix.

For instance, our data set includes the Inflation rate which is the monthly growth in theConsumer Price Index (CPI) obtained from the CANSIM database as series V41690973, theIndustrial Production Growth which is the monthly growth in the Industrial Production Index(IPI) extracted from the CANSIM database. We also consider the unemployment rate datacollected from Datastream. This variable is also available as series V2064894 from the CANSIMdatabase from January 1975.

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Some money variable are considered such as the Money Supply Growth which is the monthlygrowth of the Money Supply Index obtained as series V37173 from the CANSIM database. Themoney supply variable represents the unadjusted currency outside banks. The data set alsoincludes the Gross Domestic Product Growth which is obtained as the monthly growth in theseasonally adjusted GDP for all industries. We construct the GDP variable with two series(V329529 and V41881478 ) from the CANSIM database. The first one, the GDP at factor costin 1992 constant prices, allows going back before March 1981, but is now discontinued. Thesecond one, the GDP at basic prices in 2002 constant prices, is used as soon as possible sothat it is behind the GDP variable from March 1981. Although they differ slightly in theirmethodology, the series produce growths correlated at 0.95 in their common time span.

We consider some macro-financial variables related to the Bank of Canada Prime Rates.These series are from the CANSIM database. As these series are highly persistent, we usetheir lagged values, their lagged variation and their lagged values relative to their twelve-monthmoving average. We also include the composite leading indicator (CLI) growth. This informationvariable is available as series V7687 from the CANSIM database. According to Statistics Canada,the CLI is comprised of ten components which lead cyclical activity and together represent allmajor categories of GDP. It thus reflects a variety of mechanisms that can cause business cycles.The components are an housing index, the business and personal services employment, the TSE300 Index, the money supply M1, the U.S. Composite Leading Indicator, the average workweek hours, the new orders in durable goods, the shipments/inventories of finished goods, thefurniture and appliance sales and other durable goods sales.

4.2.2 Financial market characteristic variables

As in Chretien and Coggins (2012), we consider several financial information variables. Thesevariables are related to equity valuation ratios and the market-related variables. Among thesefinancial variables, there are two annual dividend yield variables for the S&P/TSX CompositeIndex. The realized dividend yield is computed from the difference between the one-year totalreturn of the index and its one-year price return. The data come from the CFMRC databaseand start in January 1976. We also obtain a forward-looking dividend yield available as seriesV122628 in the CANSIM database from Statistics Canada. This series is described as taking theindicated dividend to be paid per share of stock over the next 12 months and dividing it by thecurrent price of the stock. We also consider the Price-Earnings Ratio. The price-earnings ratioof the S&P/TSX Composite Index is obtained from the CANSIM database as series V122629.It corresponds to the current market price divided by the earnings in the latest fiscal year. Wealso considered the Earnings-Price Ratio which is defined as one over the price-earnings ratio.The correlation between Price-Earnings Ratio and Earnings-Price Ratio is around -0.57.

The data set also includes the Previous Equity Premium which is simply the lagged EQP andattempts to capture the predictive information in the equity premium of the previous month. Wealso consider two Volume Growth variables for the TSX exchange. The volume of shares growthvariable which is the growth in the monthly number of shares transacted and the dollar volumegrowth which describes the growth in the monthly value of shares traded. The data come fromthe CANSIM database as series V37413 and V37412, respectively, and start in January 1976.We also include a Stock Variance variable obtained as the sum of squared daily returns on the

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Canadian stock market. The data are from the CFMRC database. This variable is computedfrom the daily returns of the value-weighted CFMRC equity market index from January 1976to January 1977 and the S&P/TSX Composite Index thereafter. The issuing activity variablewhich is the net equity expansion defined as the ratio of the twelve-month moving sums of netissues divided by the current market capitalization is included in the data set. This variablecome from the CFMRC database. The price and number of shares for each individual stock isused to obtain the total market capitalization.

As in Chretien and Coggins (2012), we have included the Cross-Sectional Beta Price ofRisk variable computed as the difference in betas between value and growth portfolios. Recentempirical evidence show that this information variable predicts the equity premium for the U.S.and a group of international countries (excluding Canada). Following this empirical research,we construct this variable from the Canadian value and growth portfolios available on KennethFrench’s website. Specifically starting in January 1980, the Cross-Sectional Beta Price of Riskvariable is equal to the 36-month average rolling beta of the four value portfolios minus the36-month average rolling beta of the four growth portfolios.

4.2.3 Interest rates variables

We also include some information variables that are related to the interest rates. For instance, weconsider the treasury bill yields, its lagged monthly variation and its lagged value relative to itstwelve-month moving average. We consider the annualized yield-to-maturity of the three-monthGovernment of Canada treasury bill. The treasury bill data is taken from the CFMRC databaseand is also available as series V122541 in the CANSIM database. We use its lagged value directlyas an information variable. Moreover, we consider the long-term government bond yields definedas the average yield-to-maturity of the Government of Canada treasury bonds with a maturity often years or more. It is available in the CFMRC database and in the CANSIM database as seriesV122487. As for treasury bill yields, we use its lagged value, its lagged variation and its laggedvalue relative to its twelve-month moving average. The correlations with their correspondingtreasury bill variable are 0.93 for the long-term government bond yields, 0.46 for the laggedvalue of the long-term government bond yields, and 0.65 for lagged variation of the long-termgovernment bond yields.

Also, we include the term premium defined as the difference between long-term governmentbond yields and the short-term treasury bill yields. Additionally, we consider three creditpremium (or default spread) variables. The first credit premium is the difference between theyield on long-term corporate bonds and long-term government bond yields. As in Chretienand Coggins (2012), we construct a long history of the corporate yields and we combine threedifferent series. From January 1976 to October 1977, we use the series V35752 from the CANSIMdatabase, the Scotia-McLeod Canada Long-Term All-Corporate Yield Index. From November1977 to June 2007, we take the Scotia Capital Canada All-Corporations Long-Term bond yieldseries from CFMRC, also available as series V122518 in the CANSIM database. From July 2007,we take the yield from the Merrill Lynch Canada Corporate Bond Index from Bloomberg. In aneffort to avoid mixing three different series, a second yield spread variable is computed as thedifference between the yield on the three-month prime corporate paper (series V122491) and thetreasury bill yields. This short-term credit premium variable goes back to January 1976 and has

11

a correlation of 0.17 with the credit premium. Then, we form a return-based credit premiumvariable as the difference between long-term corporate bond and long-term government bondreturns. For the corporate bond returns, we use series V35754 (the Scotia-McLeod CanadaLong-Term All-Corporate Total Return Index) from January 1976 to October 2002 and thereturns from the Merrill Lynch Canada Corporate Bond Index (F9C0) thereafter. We obtainthe government bond returns from the CFMRC database.

4.3 Descriptive statistics

Panel A of figure 1 depicts the Canadian equity premium over the period 1976m1-2011m12. Ashighlighted by the vertical lines on the figure, we can split the dynamic of the Canadian equitypremium into five sub-periods corresponding to five important periods in the Canadian stocksmarkets. The first sub-period covers 1976m1 to 1980m1 and is associated with the increasinginterest rates and inflation concerns of the early 1980s. The second sub-period covers 1980m2to 1987m9 and ends with the October 1987 crash. The third period covers 1987m10 to 1998m6and stops before the Russian debt default and the Long Term Capital Management bankruptcyof August 1998. The fourth period covers 1998m7 to 2007m1 and is associated with the USsubprime crisis and last period covers 2007m2 to 2011m12.

Figure 1: Canadian equity premium and extracted factorsPanel A shows the monthly return of the Canadian equity premium defined as the excess return of the Canadian

stock market index over the risk-free interest rate. Panel B shows the extracted factor using principal components

analysis. The sample period is 1976m2-2011m12.

Panel A: Canadian equity premium

-.2

-.2

-.2-.1

-.1

-.10

0

0.1

.1

.1.2

.2

.2Canadian Equity Premium (EQP)

Canad

ian Eq

uity P

remium

(EQP

)

Canadian Equity Premium (EQP)1975m1

1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1

Panel B of figure 1 plots the extracted factors and figure 2 displays the Kernel densityestimate of the monthly returns of the Canadian equity premium. The sample period is 1976m1-2011m12. Figure 2 shows that the Canadian equity premium density is close to a standardnormal density, although with thicker tails. Additionally, panels A and B of table 1 presentthe mean, standard deviation, minimum and maximum of the equity premium as well as theestimated factors respectively for the full period, the first sub-period (1976m1-1980m1), the

12

Panel B: Extracted factors

-.1

-.1

-.10

0

0.1

.1

.1.2

.2

.21975m1

1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1f1

f1

f1f2

f2

f2f3

f3

f3f4

f4

f4f5

f5

f5f6

f6

f6

second sub-period (1980m2-1987m9), the third sub-period (1988m10-1998m6), the fourth sub-period (1998m7-2007m1) and the fifth sub-period (2007m2-2011m12). For the full sample, themean of the Canadian equity premium is 0.4% and the standard deviation is 4.7%. For thesubsamples, the mean and the standard deviation of the Canadian equity premium vary from0.1% to 1.2% and from 4% to 5.2%, respectively. Figures 5 to 10 depict the estimates of the sixfactors for the full sample using PCA.

5 Empirical results

5.1 In-Sample (IS) forecasting

After the estimation of the common factors by the principal components analysis method, weassess the In-Sample (IS) predictability of the Canadian equity premium by examining thet-statistics of the parameters α, β and γ in the following forecasting equation. Our targetregression coefficients are estimated using Ordinary Least Squared (OLS) and the statisticalsignificance thresholds are from Welch and Goyal (2008) and are computed from bootstrappedF -statistics. The general univariate regression model is given by

EQPt+1|t = α+ β′(L)Ft−q + γ′(L)EQPt−p+1 + εt+1, (5.1)

where EQPt|t+1 is the forecasting of the Canadian equity premium and α and γ are estimatedby using linear regression method. The results of the principal components analysis lead toretain six common factors F1, F2, F3, F4, F5 and F6 and zero lag (q = 0 and p = 1). Thus, the

13

Table 1: Summary statistics for full and sub-sample period. Factors 1 to 6 are estimated usingPCA.Panel A: Full-sample and sub-sample descriptive statistics.

Variables Mean Standard Dev. Minimum Maximum Obs.

full sample period (1976m2-2011m12)

eqp 0.004 0.047 -0.235 0.132 431

F1 0.167 0.006 0.153 0.193 431

F2 -0.001 0.032 -0.023 0.163 431

F3 0.000 0.025 -0.056 0.083 431

F4 0.000 0.021 -0.103 0.053 431

F5 0.000 0.014 -0.026 0.037 431

F6 0.000 0.012 -0.041 0.026 431

first sample period (1976m2-1980m1)

eqp 0.012 0.045 -0.103 0.108 48

F1 0.158 0.003 0.153 0.164 48

F2 -0.017 0.001 -0.020 -0.015 48

F3 0.001 0.025 -0.041 0.051 48

F4 -0.002 0.022 -0.055 0.043 48

F5 0.008 0.006 -0.002 0.027 48

F6 0.002 0.014 -0.031 0.027 48

second sample period (1980m2-1987m9)

eqp 0.002 0.052 -0.185 0.132 92

F1 0.166 0.002 0.160 0.172 92

F2 -0.014 0.005 -0.023 -0.003 92

F3 0.002 0.028 -0.056 0.083 92

F4 -0.002 0.026 -0.103 0.046 92

F5 0.014 0.011 -0.004 0.037 92

F6 0.004 0.011 -0.020 0.026 92

14

Panel B: Sub-sample descriptive statistics

Variables Mean Standard Dev. Minimum Maximum Obs.

third sample period (1987m10-1998m6)

eqp 0.002 0.040 -0.235 0.073 129

F1 0.168 0.005 0.160 0.185 129

F2 0.003 0.026 -0.018 0.163 129

F3 0.000 0.024 -0.049 0.059 129

F4 0.001 0.020 -0.047 0.053 129

F5 0.002 0.009 -0.017 0.024 129

F6 0.000 0.010 -0.023 0.020 129

fourth sample period (1998m7-2007m1)

eqp 0.005 0.048 -0.205 0.116 103

F1 0.170 0.007 0.163 0.193 103

F2 0.017 0.053 -0.010 0.163 103

F3 -0.001 0.024 -0.052 0.064 103

F4 0.000 0.019 -0.083 0.043 103

F5 -0.009 0.009 -0.022 0.018 103

F6 -0.002 0.011 -0.028 0.018 103

fifth sample period (2007m2-2011m12)

eqp 0.001 0.050 -0.168 0.114 59

F1 0.168 0.002 0.164 0.172 59

F2 -0.007 0.003 -0.013 0.004 59

F3 -0.001 0.022 -0.037 0.063 59

F4 0.002 0.017 -0.032 0.044 59

F5 -0.017 0.005 -0.027 -0.005 59

F6 -0.005 0.012 -0.041 0.014 59

15

Figure 2: Density of Canadian equity premium

0

0

02

2

24

4

46

6

68

8

810

10

10Density of Canadian equity premium

Dens

ity o

f Can

adian

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ity p

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um

Density of Canadian equity premium-.3

-.3

-.3-.2

-.2

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-.1

-.10

0

0.1

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.2

.2EQP

EQP

EQPkernel = epanechnikov, bandwidth = .01

kernel = epanechnikov, bandwidth = .01

kernel = epanechnikov, bandwidth = .01Kernel density estimate of EQPKernel density estimate of EQP

Kernel density estimate of EQP

Note: The figure shows the Kernel density estimate of the monthly return of the Canadian equity premium definedas the excess return of the Canadian stock market index over the risk-free interest rate. The sample period is1976m01-2011m12

forecasting equation is given by

EQPt+1|t = α+ β1F1t + β2F2t + β3F3t + β4F4t + β5F5t + β6F6t + γEQPt + εt+1, (5.2)

As the graph of the Canadian equity premium is split in five sub-samples, we conduct theIn-Sample exercise for each sub-sample as well as for the full sample. Panels A, B and C oftable 2 evaluate the IS forecasting results for the full sample and for the sub-samples. In eachpanel, we present the coefficient estimates, the standard errors, the p-value and the adjustedR2 of the regression. It is worth mentioning that we present results from model 1 and model2, where model 1 is called the factor-augmented vector autoregression model (FAVAR) and themodel 2 is called the pure factor model, which does not include any AR components (i.e. thelag of EQP).

The tables show that the constant term and some of estimated factors are statisticallysignificant in predicting of the Canadian equity premium over the full sample and the sub-samples considered. The most successful factors for full sample IS forecasting results are F1, F4,F5 and F6 and the constant α. The adjusted R2 of the full sample estimation is 4.4%, which ishigher than the results reported in Welch and Goyal (2008) and Chretien and Coggins (2012).Overall, our IS results for the full sample suggest that the estimated factors have significantpredictive power in the forecasting of the Canadian equity premium.

The IS forecasting results for the first sub-sample show that factors F1, F2, F3, F4, F5 andF6 are statistically significant with an adjusted R2 around to 48%. The best model remainsthe factor augmented vector-autoregressive (FAVAR) model for this sub-sample results. The

16

Table 2: Regression of monthly Canadian equity premium on estimated factors

Panel A: IS periods (1976m2-2011m12 and 1976m2-1980m1)

The table presents the OLS beta parameter estimates for multivariate regressions of both models 1 and2 using the full sample and the subsample 1976m2-1980m1. The Canadian equity premium is measured atmonthly frequency. The t-statistics using Newey and West (1987) corrected standard errors are reported inparentheses. Bootstrapped p-values are reported in the table as well as the adjusted R2 of the regression.Statistical significance of the estimated coefficients are based on of the t-statistics thresholds, significance isbased on bootstrapped confidence intervals. ∗ ∗ ∗ indicates significance at 1%, ∗∗ indicates significance at 5%and ∗ at 10%.

1976m2-2011m12Model 1 Model 2

Coef. Std. err. p-value Coef. Std. err. p-value

β1 -1.224∗∗ (0.534) 0.022 -1.287∗∗ (0.527) 0.015

β2 0.096 (0.097) 0.322 0.102 (0.096) 0.287

β3 -0.152∗ (0.090) 0.092 -0.144 (0.089) 0.108

β4 -0.227∗∗ (0.113) 0.045 -0.266∗∗ (0.106) 0.013

β5 -0.285∗ (0.164) 0.083 -0.302∗ (0.163) 0.064

β6 -0.495∗∗∗ (0.189) 0.009 -0.517∗∗∗ (0.188) 0.006γ 0.054 (0.051) 0.288α 0.208∗∗ (0.089) 0.020 0.219∗∗ (0.088) 0.013

Adj. R2 4.43%∗∗∗ Prob > F 4.46%∗∗∗ Prob > FObs. 430 = 0.0005 431 = 0.0003



β1 -10.309∗∗ (4.430) 0.025 -12.428∗∗∗ (4.067) 0.004

β2 -78.048∗∗∗ (15.001) 0.000 -82.313∗∗∗ (14.713) 0.000

β3 1.447∗∗∗ (0.372) 0.000 1.483∗∗∗ (0.361) 0.000

β4 -0.666∗∗ (0.292) 0.028 -0.654∗∗ (0.257) 0.015

β5 -8.685∗∗∗ (2.335) 0.001 -8.636∗∗∗ (2.340) 0.001

β6 -2.908∗∗∗ (0.779) 0.001 -2.758∗∗∗ (0.774) 0.001γ -0.197 (0.123) 0.117α 0.418 (0.524) 0.430 0.679 (0.475) 0.160


17

IS results for the second sub-sample show that F1, F5, F6 and the constant are statisticallysignificant. The adjusted R2 for this sub-sample is 14.6% for the FAVAR model and 15% for thepure factor model. Overall, the IS results with the second sub-sample corroborate the empiricalpower of the extracted factors.

The less significant results are those found using the third and the fifth sub-samples. Forethese sub-samples, most of the factors and the variables considered are not statistically sig-nificant. These results may be caused by a structural break and suggest an instability of theestimated parameters5. In this paper, we do not consider structural breaks and we suppose thatthe factor structure remains stable over the target period.

The IS forecasting results with the fourth sub-sample show that factors F5 and F6 arestatistically significant. The adjusted R2 is 4% for the FAVAR model and 4.6% for the purefactor model.

In summary, our in-sample (IS) results indicate that the extracted factors significantly con-tribute to forecasting the Canadian equity premium. Therefore, the dynamic factor model ap-pears to be an interesting method to capture the relation between the Canadian equity premiumand the informations variables.

5An overwhelming idea to capture this structural break is to consider a dynamic factor model with time-varyingparameter model (TVP) and structural breaks.

18

Panel B: IS periods (1980m2-1989m9 and 1987m10-1998m6)

The table presents the OLS beta parameter estimates for multivariate regressions of both models 1 and2 using the subsample 1980m2-1989m9 and the subsample 1987m10-1998m6. The Canadian equity premiumis measured at monthly frequency. The t-statistics using Newey and West (1987) corrected standard errorsare reported in parentheses. Bootstrapped p-values are reported in the table as well as the adjusted R2 ofthe regression. Statistical significance of the estimated coefficients are based on of the t-statistics thresholds,significance is based on bootstrapped confidence intervals. ∗ ∗ ∗ indicates significance at 1%, ∗∗ indicatessignificance at 5% and ∗ at 10%.



β1 -4.890∗ (2.569) 0.060 -4.511∗ (2.521) 0.077

β2 -2.140 (1.518) 0.162 -2.136 (1.515) 0.162

β3 -0.279 (0.195) 0.156 -0.310 (0.191) 0.108

β4 -0.382 (0.247) 0.126 -0.285 (0.216) 0.191

β5 -2.756∗∗∗ (0.809) 0.001 -2.591∗∗∗ (0.781) 0.001

β6 -1.218∗∗ (0.562) 0.033 -1.198∗∗ (0.560) 0.035γ -0.100 (0.123) 0.420α 0.829∗ (0.467) 0.001 0.763∗ (0.446) 0.001




β1 -0.955 (1.910) 0.618 -1.019 (1.910) 0.595

β2 0.064 (0.180) 0.720 0.055 (0.180) 0.759

β3 -0.156 (0.148) 0.295 -0.156 (0.148) 0.296

β4 -0.227 (0.202) 0.264 -0.187 (0.199) 0.348

β5 -0.642 (0.965) 0.507 -0.535 (0.959) 0.578

β6 -0.807∗ (0.481) 0.096 -0.734 (0.476) 0.126γ -0.095 (0.091) 0.302α 0.164 (0.318) 0.607 0.174 (0.318) 0.584

Adj. R2 5.61%∗∗ Prob > F 5.55%∗∗ Prob > FObs. 129 = 0.0499 129 = 0.0425

19

Panel C: IS periods (1998m7-2007m1 and 2007m2-2011m12)

The table presents the OLS beta parameter estimates for multivariate regressions of both models 1 and2 using the subsample 1998m7-2007m1 and the subsample 2007m2-2011m12. The Canadian equity premiumis measured at monthly frequency. The t-statistics using Newey and West (1987) corrected standard errorsare reported in parentheses. Bootstrapped p-values are reported in the table as well as the adjusted R2 ofthe regression. Statistical significance of the estimated coefficients are based on of the t-statistics thresholds,significance is based on bootstrapped confidence intervals. ∗ ∗ ∗ indicates significance at 1%, ∗∗ indicatessignificance at 5% and ∗ at 10%.



β1 -1.856 (2.965) 0.533 -1.870 (2.956) 0.528

β2 0.558 (0.566) 0.326 0.574 (.564) 0.311

β3 0.054 (0.218) 0.805 0.062 (.217) 0.777

β4 -0.316 (0.273) 0.249 -0.360 (.263) 0.174

β5 -2.881∗ (1.651) 0.084 -3.020∗ (1.632) 0.067

β6 -1.347∗∗ (0.616) 0.031 -1.480∗∗ (.580) 0.012γ 0.071 (0.108) 0.513α 0.282 (0.486) 0.563 0.283 (0.485) 0.560

Adj. R2 4.04% Prob > F 4.61% Prob > FObs. 103 = 0.1409 103 = 0.1029



β1 -8.032 (8.296) 0.338 -9.646 (8.283) 0.249

β2 -0.546 (2.369) 0.819 0.030 (2.351) 0.990

β3 -0.497 (0.323) 0.130 -0.620∗ (.313) 0.053

β4 -0.219 (0.486) 0.654 -0.308 (.486) 0.530

β5 0.761 (4.698) 0.872 1.239 (4.725) 0.794

β6 0.484 (1.726) 0.780 0.806 (1.725) 0.642γ 0.193 (0.141) 0.175α 1.360 (1.475) 0.361 1.645 (1.473) 0.269


20

5.2 Out-of-sample (OOS) forecasting

As our IS results highlight the predictive power of the dynamic factor model, an interesting andfrequently used way to quantify the out-of-sample (OOS) forecasting performance based on realdata is to compute the mean squared error (MSE) of the forecasts. The MSE statistics thatwe use in our approach is close to the lost function based on the mean squared forecast errorproposed by Diebold and Mariano (1995). Since our goal is to assess the predictive power ofthe macro-financial information variables in the forecasting of the Canadian equity premium,we use three benchmark models. The first benchmark model is an autoregressive model withone lag, AR(1). The historical average and the historical median models are used as the secondand the third benchmark model, respectively. Thus, the predictive power of the dynamic factormodel is evaluated by comparing the forecast errors of the estimated model relative to those ofthe estimated benchmark models.

The forecast performance comparison is performed in a simulated out-of-sample frameworkwhere the MSE calculation are done using a fully recursive methodology. In the first step,the dynamic factor model is estimated using a predetermined sub-sample and a simulated real-time forecasting is done from the rest of the sample. In practice, the out-of-sample (OOS)forecast uses only the data available up to the time at which the forecast is made. Let h bethe number of observations in the OOS evaluation window. T − h then corresponds to thenumber of observations in the initial estimation window. Likewise, let ε(BM) and ε(DFM) denotethe forecasting errors associated with the benchmark models and the dynamic factor model,respectively.

As in Welch and Goyal (2008), the mean squared errors (MSE) and the OOS statistics usedin our framework are computed as:

OOS-R2 = 1−MSE(DFM)

MSE(BM), and MSEι = (1/h)

h∑t=1

ε2ι .

OOS-R2

= 1− (1−R2)×(h− k2

h− k1

),

OOS-∆RMSE =√MSE(BM) −

√MSE(DFM),

OOS-MSE-T =

√h+ 1− k1

h+ 1− k2

[d√h

se(d)

],

OOS-MSE-F = (h− (k1 − k2))×(MSE(BM) −MSE(DFM)

MSE(DFM)

),

(5.3)

where k1 and k2 denote the number of parameters in the FAVAR model and the benchmark

model, respectively. d is defined as MSE(BM) −MSE(DFM)6. The adjusted out-of-sample R

2

statistics that we use in this paper is close to the standard in-sample R2

statistics. Although wereport both MSE-F and MSE-T statistics, we chose to use the MSE-F for significance level inthis framework because Clark and McCracken (2001) find that MSE-F has higher power thanMSE-T . Our MSE-F thresholds are based on the bootstrap results from McCracken (2007).

6For more details about d, see McCracken (2007) and Welch and Goyal (2008)

21

Table 3: OOS results of models 1 and 2 with an evaluation window of 160 observationsThis table presents the out-of-sample (OOS) statistics for the Canadian equity premium forecasts at monthly

frequency with model 1 (EQPt+1|t = α+ β1F1t + β2F2t + β3F3t + β4F4t + β5F5t + β6F6t + γEQPt + ε1,t+1) and

model 2 (EQPt+1|t = α+β1F1t+β2F2t+β3F3t+β4F4t+β5F5t+β6F6t+ε2,t+1 ) using an evaluation window of 160observations. Estimated factors are explained in the previous section. Stock returns are price changes, includingdividends, of the S&P/TSX. All numbers are in percent except the t-stat and OOS-∆MSE-F. The OOS-R2,OOS-∆RMSE, OOS-∆MSE-F and MSE-T are defined above. A star next to OOS-R2 is based on significancelevel of MSE-F statistic in McCracken (2004), which tests for equal MSE of the targeted models forecast and thebenchmark models forecast. One-sided critical values of MSE statistics are from bootstrapped distributions inMcCracken (2007).

Benchmark AR Model Average Mean Model Median Model

Model 1: EQPT+1|T = α+∑6

j=1 βjFjT + γEQPT + ε1,T+1

OOS-R2(%) 0.951∗ 3.428∗∗ 3.617∗∗

MSE-T (1.348) (2.098) (2.099)OOS-∆RMSE(%) 0.114 0.190 0.195OOS-∆MSE-F 7.615 12.728 13.138


j=1 βjFjT + ε2,T+1

OOS-R2(%) −0.185 2.959∗∗ 3.149∗∗

MSE-T (0.986) ( 1.895) (1.928)OOS-∆RMSE(%) 0.087 0.163 0.168OOS-∆MSE-F 5.820 10.919 11.242

22

Based on Massimiliano et al. (2002), the general forecast model is given by equation (5.2),and all forecasting models are specified and estimated as a linear projection of the one stepahead variable, EQP , onto extracted factors including the lagged value of the Canadian equitypremium. In contrast to Welch and Goyal (2008) that argues that the historical mean hasdone as well at forecasting the equity premium as any of the more complex empirical modelsthat have been brought to bear on this issue, our contribution is to show that the factor-augmented vector autoregressive (FAVAR) model as well as the pure factor model performbetter than the benchmark models. A positive value of ∆RMSE shows that the dynamic factormodel outperforms the benchmark model. As the R2 increases with the numbers of informationvariables in the forecasting equation, we also use the R2 statistics to analyze the performanceof the dynamic factor model. Again, a positive value of the out-of-sample statistics R2 alsohighlights the prediction power of macroeconomic and financial information variables in theforecasting of the Canadian equity premium. As argued in Campbell and Thompson (2008),very small R2 statistics are relevant for investors because they can generate large utility gains.∆RMSE and ∆MSE-F are also used to capture the predictive power of the factor-augmentedautoregressive (FAVAR) model and the pure factor model. Indeed a positive value of ∆MSE-Fminus its critical value confirms the good performance of the dynamic factor model.

In this section and in table 3, we present results an OOS valuation window of 160 obser-vations. Thus, using an initial estimation window of T − h = 270 observations. This choiceis in accordance with the minimum window size of 240 observations advocated statistically byMcCracken (2007). Results with other initial estimation windows and OOS evaluation windowsare presented in the section on robustness checks. According to the sign of our OOS statis-tics, results in table 3 show that our forecasting models outperform the AR model, the averagemean and the median models. The OOS statistics for the FAVAR model against the AR model(R2(0.951%), ∆RMSE(0.114%) and ∆MSE-F (7.65)) are positive and statistically significant.Likewise, the OOS statistics when using the median and the average mean as benchmarks arepositive and statistically significant. Except compared to the AR benchmark, the OOS statis-tics for the pure factor model are also positive and statically significant. Thus, our forecastingmodels outperform the benchmark models when the size of the evaluation window is set to 160.Overall, our results confirms the findings in Ludvigson and Ng (2010a) and Chretien and Cog-gins (2012) concerning the predictive power of macroeconomic information variables. We finda signficant out-of-sample performance for the forecasting models considered in contrast to theresults in Welch and Goyal (2008) and Campbell and Thompson (2008).

5.3 Robustness checks

It is not clear how to choose the windows over which a regression model is estimated andsubsequently evaluated. This is even more important for OOS tests. Although any choice isnecessarily ad-hoc in the end, the criteria are clear. As discussed in Chretien and Coggins (2012),it is important to have enough initial data to get a reliable regression estimate at the start ofevaluation period, and it is important to have an evaluation period that is long enough to berepresentative. Also, Welch and Goyal (2008) emphasize that out-of-sample performance of theequity premium has been particularly poor for predictive regressions in the last few decades. Toinvestigate the robustness of the OOS results to the choice of OOS evaluation windows, we redo

23

the OOS exercise by exploring four other evaluation window sizes: 240 months, 200 months, 120months and 80 months.

Panels A and B of table 4 present results from these four evaluation windows. For h = 240,our OOS-∆RMSE statistics are 0.050%, 0.105% and 0.110% for the model 1 when using theAR model, historical mean model and median model as benchmark, respectively. The last two

results are statistically significant, with corresponding R2

of 1.8% and 2.0%. This supportsthe fact that our dynamic factor model outperforms the benchmark models, and these resultsare robust to the OOS-∆MSE-F statistics. For h = 200 and h = 120, our results are similarand confirm the outperformance of the FAVAR model. It is worth mentioning that the OOSperformance of model 2 is also better in the predicting of the Canadian equity premium thanthe benchmark models. The OOS results become insignificant only when the evaluation windowsize falls to 80 observations (the last 80 months of our sample).

Concerning the benchmarks, our OOS results show that the AR(1) model is a better predictorfor the Canadian equity premium than the average mean and the median models. Interestingly,the OOS performance of our forecasting models depicts an inverted U-shape function of the OOSevaluation window size. Indeed, the OOS results weaken when h is larger than 200 periods andsmaller than 120 periods. This may be created by unreliable estimation when barh is too largeor forecast evaluaten when barh is too small. This robustness analysis shows that the dynamicfactor model can yield useful out-of-sample forecasts. Our results confirm those in Ludvigsonand Ng (2010a) that the use of macroeconomic and financial variables provide considerablein-sample and out-sample predictive ability for the extracted factors.

6 Economic interpretation

A fair criticism of dynamic factor models is that the estimated factors are difficult to interpret.This section sheds light on the economic interpretation of the extracted factors by pooling ourdataset into blocks, and conducting correlation analysis as well as an IS analysis with factorsextracted within each block. The first subsection presents a correlation analysis between theextracted factors and some information variables. In the second subsection, we consider thedata in blocks and conduct both correlation and IS investigations.

6.1 Correlation with common information variables

In this subsection, we compute the cross-correlation matrix between extracted factors and ninewell-known macro-financial variables from the predictability literature. The selected macro-financial variables are: The previous equity premium (zeqp), the lagged value of the dividend-price ratio (dp lag), the lagged monthly variation of of the three-month Government of CanadaTreasury bill (tbillv lag), the lagged value of the term premium (term lag), the lagged valueof the short-term credit premium (credit lag), the lagged value of the monthly growth in theConsumer Price Index (inflation), the lagged value of the gross domestic product (gdp lag), thelagged value of the unemployment rate (unemp) and the lagged value of the composite leadingindicator growth (lead lag).

Table 5 lists the cross-correlation statistics between extracted factors and selected macro-financial variables. Bold figures emphasize the highest algebraic correlation for each factor. Ou

24

Table 4: OOS robustness checks of models 1 and 2 against benchmarks

Panel A: h = 240 and h = 200This table presents the out-of-sample (OOS) statistics for the Canadian equity premium forecasts at monthly

frequency with model 1 (EQPt+1|t = α+ β1F1t + β2F2t + β3F3t + β4F4t + β5F5t + β6F6t + γEQPt + ε1,t+1) and

model 2 (EQPt+1|t = α + β1F1t + β2F2t + β3F3t + β4F4t + β5F5t + β6F6t + ε2,t+1 ) using an evaluation windowof 240 and 200 observations. Estimated factors are explained in the previous section. Stock returns are pricechanges, including dividends, of the S&P/TSX. All numbers are in percent except the t-stat and OOS-∆MSE-F.The OOS-R2, OOS-∆RMSE, OOS-∆MSE-F and MSE-T are defined above. A star next to OOS-R2 is basedon significance level of MSE-F statistic in McCracken (2004), which tests for equal MSE of the targeted modelsforecast and the benchmark models forecast. One-sided critical values of MSE statistics are from bootstrappeddistributions in McCracken (2007).

OOS periods OOS Statistics AR Model Average Mean Model Median Model

h = 240



OOS-R2(%) -0.268 1.804∗ 2.041∗




OOS-R2(%) -0.986 1.525∗ 1.763∗


h = 200



OOS-R2(%) 0.915∗ 3.189∗∗ 3.379∗∗




OOS-R2(%) -0.149 2.656∗∗ 2.847∗∗


25

Panel B: h = 120 and h = 80

This table presents the out-of-sample (OOS) statistics for the Canadian equity premium forecasts at

monthly frequency with model 1 (EQPt+1|t = α+β1F1t +β2F2t +β3F3t +β4F4t +β5F5t +β6F6t +γEQPt +ε1,t+1)

and model 2 (EQPt+1|T = α + β1F1t + β2F2t + β3F3t + β4F4t + β5F5t + β6F6t + ε2,t+1 ) using an evaluationwindow of 120 and 80 observations. Estimated factors are explained in the previous section. Stock returnsare price changes, including dividends, of the S&P/TSX. All numbers are in percent except the t-stat andOOS-∆MSE-F. The OOS-R2, OOS-∆RMSE, OOS-∆MSE-F and MSE-T are defined above. A star next toOOS-R2 is based on significance level of MSE-F statistic in McCracken (2004), which tests for equal MSE of thetargeted models forecast and the benchmark models forecast. One-sided critical values of MSE statistics arefrom bootstrapped distributions in McCracken (2007).

OOS periods OOS Statistics AR Model Average Mean Model Median Model

h = 120



OOS-R2(%) -0.249 3.093∗∗ 3.159∗




OOS-R2(%) -2.256 2.028∗ 2.094∗


h = 80



OOS-R2(%) -5.967 -2.739 -2.332MSE-T (0.383) (1.011) (1.050)OOS-∆RMSE(%) 0.049 0.148 0.157OOS-∆MSE-F 1.652 4.961 5.343



OOS-R2(%) -8.479 -3.734 -3.323MSE-T (-0.021) (0.716) (0.786)OOS-∆RMSE(%) -0.003 0.096 0.105OOS-∆MSE-F -0.100 3.199 3.506

26

Table 5: Cross-correlation statistics between factors and information variables.

eqp F1 F2 F3 F4 F5 F6 zeqpzeqp 0.109 -0.066 -0.015 0.078 -0.299 -0.096 -0.112 1.000dp lag 0.036 -0.415 -0.324 0.055 -0.036 0.625 0.134 0.033tbillv lag -0.085 -0.018 -0.048 -0.096 0.127 0.035 0.094 -0.144term lag 0.079 0.057 0.370 -0.027 -0.015 -0.546 -0.102 0.079credit lag -0.048 0.121 0.050 0.010 0.074 -0.297 -0.271 -0.118inflation 0.006 -0.153 -0.211 -0.122 -0.171 0.423 0.176 0.009gdp lag 0.023 -0.170 -0.090 0.046 -0.076 0.198 0.095 -0.023unemp 0.075 -0.122 0.103 -0.028 -0.123 0.321 0.101 0.092lead lag 0.127 -0.009 0.112 0.062 -0.189 -0.172 0.021 0.394

results show F1 is negatively correlated to the dividend-price ratio (-0.415). F2 has a correlationof 0.37 with the term premium. F3 is negatively correlated to the consumer price index (-0.122).F4 captures predominantly the previous equity premium inversely and F5 has a correlation of0.625 with the dividend-price ratio. F6 is negatively correlated to the short-term credit premium.Regarding the size of the correlations, our results support the idea that extracted factors arelinear combination of information variables. In a sense, the factors capture more informationthan the one in commonly used predictors.

6.2 Factors by categories of information variables

One reason that the estimated factors are difficult to interpret is that the factors are estimatedfrom a large panel of data without taking full advantage of the data structure. This subsectionproposes an economic interpretation of the factors by pooling our information variables intothree categories: Pure financial sector, which includes only financial information variables, pureeconomic information variables and prices information variables. We extract two factors fromthe financial set, three factors from the economic set and one factor from the prices set. Thenumber of factors is chosen in accordance to the number of variables in each block.

Once we have the extracted factors by blocks, we conduct a correlation analysis to assesstheir correlation with factors extracted without categorizing our dataset. We also perform an ISanalysis using the factors extracted by blocks. This IS exercise aims to test the contribution ofeach category in the forecasting of the Canadian equity premium. In the stock market, cyclicalmovements are driven by co-movements in financial and real variables. Thus, this analysisis useful for the interpretation of the factors and allows us to assess whether the extractedfactors by category have unconditional predictive power for the Canadian equity premium. Asin Ludvigson and Ng (2007), ffin1 and ffin2 are financial factors. Correspondingly, feco1 , feco2 and

feco3 are economic factors and fpri is a price factor. Tables 6 reports results from the correlationanalysis.

ffin1 is strongly correlated to F2 with a correlation of 0.996 and has a correlation of 0.711

with F1. ffin2 is negatively correlated with F4. F1 has a positive correlation of 0.69 with the first

economic factor. F5 is strongly correlated with the third economic faction with a correlation

27

Table 6: Cross-correlation statistics between factors and factors by blocks

ffin1 ffin2 feco1 feco2 feco3 fpri

F1 0.711 -0.086 0.690 0.083 0.220 -0.017

F2 0.996 -0.132 -0.021 -0.021 0.282 -0.036

F3 0.024 0.640 -0.161 -0.860 -0.023 -0.029

F4 -0.049 -0.757 -0.000 -0.511 0.024 0.017

F5 0.072 -0.005 -0.127 0.003 -0.947 0.008

F6 0.020 -0.005 0.045 -0.000 0.104 0.196

of −0.947. Figures 3 and 4 display the correlation between ffin1 and F2, and between feco3 and

F5. Table 7 lists the cross-correlation statistics between the factors extracted by blocks andthe aforementioned well-known macro-financial variables. Although the factors are extractedby categories, our results show that the factors reflect a combination of numerous informationvariables.

Table 7: Cross-correlation statistics between factors by blocks and information variables.

eqp ffin1 ffin2 feco1 feco2 feco3 fpri zeqp

zeqp 0.109 -0.007 0.270 -0.098 0.080 0.056 -0.033 1.000dp lag 0.036 -0.274 0.100 -0.349 -0.022 -0.684 -0.005 0.033tbillv lag -0.085 -0.052 -0.151 0.034 0.019 -0.033 0.358 -0.144term lag 0.079 0.321 -0.054 -0.217 0.021 0.595 -0.020 0.079credit lag -0.048 0.022 -0.049 0.171 -0.045 0.285 -0.226 -0.118inflation 0.006 -0.171 0.074 -0.075 0.194 -0.436 0.082 0.009gdp lag 0.023 -0.071 0.095 -0.190 -0.002 -0.205 0.165 -0.023unemp 0.075 0.128 0.055 -0.327 0.082 -0.305 -0.048 0.092lead lag 0.127 0.109 0.163 -0.126 0.037 0.177 0.117 0.394

Panels A, B and C of table 8 report the results of the predictive regression of the Canadianequity premium on the extracted financial, economic and price factors. Additionally, to inves-tigate whether the predictive power varies over the sample, we consider regressions using theprevious five sub-periods. The relevancy of macroeconomic variables in explaining Canadianequity premium is reinforced with these IS results, which show that economic and price factorsadds substantial predictive power beyond the financial factors. The full-sample IS results showthat economic factors are statistically significant with an adjusted R2 around 3.9%. However,financial factors appear to be not statistically significant. Furthermore, when considering thesub-samples regressions, our results show that the economic and price factors often have a strongpredictive power in the forecasting of the Canadian equity premium. Only in the 1976-1980 sub-periods do the financial factors appear relevant. When comparing the adjusted R2 of IS resultswith and without categorization of the variables, our results show that predictability withoutcategorization is better than it is with a categorization of the variables.

28

Figure 3: Correlation between factors: ffin1 and F2

0

0

0.1

.1

.1.2

.2

.2.3

.3

.3.4

.4

.41975m1

1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1F_fin1

F_fin1

F_fin1F_2

F_2

F_2

Note: The figure depicts the monthly estimated factors F2 and Ffin1. F2 is the second factor obtained without acategorization of the data and Ffin1 is the first factor estimated with the pure financial block. The sample periodis 1976m2-2011m12.

Figure 4: Correlation between factors: feco3 and F5

-.04

-.04

-.04-.02

-.02

-.020

0

0.02

.02

.02.04

.04

.041975m1

1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1F_eco3

F_eco3

F_eco3F_5

F_5

F_5

Note: TThe figure depicts the monthly estimated factors F5 and Feco3. F5 is the fifth factor obtained withouta categorization of the data and Feco3 is the third factor estimated with the pure macroeconomic block. Thesample period is 1976m2-2011m12.

29

Table 8: Regression of monthly Canadian equity premium on estimated factors by blocks

Panel A: IS periods (1976m2-2011m12 and 1976m2-1980m1)

This table reports estimates from OLS regressions of Canadian equity premium on the estimated factorsby sector and the lagged value of the equity premium. ffin

1 , ffin2 , feco

1 , feco2 , feco

3 and fpri denote the factorsestimated by the method of principal components using a panel of data over the period 1976m2-2011m12 withindividual series from the financial, economic and price pool, respectively. The t-statistics using Newey and West(1987) corrected standard errors are reported in parentheses. Bootstrapped p-values are reported in the table aswell as the adjusted R2 of the regression. Statistical significance of the estimated coefficients are based on of thet-statistics thresholds, significance is based on bootstrapped confidence intervals. ∗ ∗ ∗ indicates significance at1%, ∗∗ indicates significance at 5% and ∗ at 10%.



ffin1 -0.041 (0.031) 0.181 -0.041 (0.031) 0.179

ffin2 0.018 (0.046) 0.685 0.038 (0.043) 0.379

feco1 -1.157∗∗ (0.466) 0.014 -1.237∗∗∗ (0.460) 0.007

feco2 0.210∗∗ (0.084) 0.013 0.228∗∗∗ (0.083) 0.007

feco3 0.235 (0.154) 0.127 0.256∗ (0.153) 0.096

fpri -0.106∗∗ (0.042) 0.012 -0.108∗∗ (0.042) 0.011zeqp 0.073 (0.050) 0.145

Const. 0.226∗∗ (0.089) -0.011 0.241∗∗∗ (0.088) 0.006




ffin1 -18.355∗∗∗ (6.664) 0.009 -19.375∗∗∗ (6.521) 0.005

ffin2 1.744∗∗∗ (0.614) 0.007 1.846∗∗∗ (0.606) 0.004

feco1 4.520∗∗ (1.683) 0.011 4.007∗∗ (1.617) 0.017

feco2 0.244 (0.199) 0.228 0.252 (0.198) 0.209

feco3 -1.617∗∗ (0.764) 0.041 -1.636∗∗ (0.754) 0.036

fpri -0.0258∗ (0.132) 0.058 -0.272∗∗ (0.130) 0.043zeqp -0.137 (0.143) 0.345

Const. -0.445 (0.325) 0.178 -0.332 (0.310) 0.291


30

Panel B: IS periods (1980m2-1987m9 and 1987m10-1998m6)

This table reports estimates from OLS regressions of Canadian equity premium on the estimated factorsby sector and the lagged value of the equity premium. ffin

1 , ffin2 , feco

1 , feco2 , feco




ffin1 -0.591 (0.720) 0.414 -0.609 (0.718) 0.398

ffin2 0.092 (0.108) 0.398 0.061 (0.098) 0.536

feco1 -4.584∗∗ (2.077) 0.030 -4.315∗∗ (2.033) 0.037

feco2 0.382∗ (0.195) 0.054 0.362∗ (0.192) 0.064

feco3 1.414∗∗ (0.556) 0.013 1.317∗∗ (0.536) 0.016

fpri -0.245∗∗∗ (0.090) 0.008 -0.239∗∗∗ (0.089) 0.009zeqp -0.083 (0.122) 0.497

Const. 0.918∗∗ (0.409) 0.027 0.866∗∗ (0.401) 0.033




ffin1 -0.060 (0.062) 0.339 -0.063 (0.062) 0.314

ffin2 0.029 (0.083) 0.726 0.016 (0.082) 0.841

feco1 -2.756∗∗ (1.105) 0.014 -2.659∗∗ (1.100) 0.017

feco2 0.240∗ (0.127) 0.061 0.220∗ (0.125) 0.081

feco3 -0.432 (0.566) 0.446 -0.450 (0.565) 0.428

fpri -0.099 (0.070) 0.158 -0.090 (0.069) 0.193zeqp -0.085 (0.091) 0.352

Const. 0.531∗∗ (0.211) 0.013 0.512∗∗ (0.210) 0.016

Adj. R2 5.03%∗ Prob > F 5.13%∗ Prob > FObs. 129 =0.0647 129 = 0.0520

31

Panel C: IS periods (1998m7-2007m1 and 2007m2-2011m12)

This table report estimates from OLS regressions of Canadian equity premium on the estimated factorsby sector and the lagged value of the equity premium. ffin

1 , ffin2 , feco

1 , feco2 , feco




ffin1 -0.025 (0.041) 0.536 -0.027 (0.041) 0.513

ffin2 0.118 (0.104) 0.258 0.154 (0.100) 0.127

feco1 1.328 (2.378) 0.578 1.885 (2.346) 0.424

feco2 -0.042 (0.221) 0.850 -0.037 (0.222) 0.867

feco3 -0.360 (1.007) 0.722 -0.683 (0.979) 0.487

fpri -0.009 (0.169) 0.957 -0.012 (0.170) 0.942zeqp 0.138 (0.107) 0.202

Const. -0.244 (0.452) 0.590 -0.347 (0.446) 0.439

Adj. R2 -0.92% Prob > F -1.60% Prob > FObs. 103 = 0.5350 103 = 0.6248



ffin1 -0.344 (1.071) 0.749 -0.059 (1.059) 0.956

ffin2 -0.099 (0.182) 0.588 -0.138 (0.181) 0.449

feco1 -5.404∗ (3.116) 0.089 -5.875∗ (3.122) 0.065

feco2 0.420 (0.320) 0.195 0.525∗ (0.313) 0.099

feco3 0.128 (1.267) 0.920 0.401 (1.261) 0.752

fpri -0.010 (0.103) 0.921 -0.023 (0.104) 0.825zeqp .191 (0.141) 0.180

Const. 1.055∗ (0.619) 0.094 1.132∗ (0.622) 0.074


32

7 Conclusion

This paper contributes to the literature on the ability of information variables to predict theequity premium by showing that macro-financial variables have important predictive power forthe Canadian equity premium. To do so, we use a dynamic factor analysis to summarize theinformation from a large number of macro-financial series. An advantage of such model relies onthe fact that they allow to summarize in the first few principal components a significant fractionof the overall covariation among the potential information variables in the panel.

Our main results are that, we find strong predictable variation in the Canadian equity pre-mium that is associated with macroeconomic activity. Specifically, the in-sample (IS) resultsindicate that the extracted factors significantly contribute to predict the Canadian equity pre-mium. Moreover, the out-of-sample (OOS) results confirm the predictive power of the extractedfactors. Furthermore, our results show that factors extracted only from the financial variablesare not statistically significant for the full sample. This result may be explained by the factthat specifications using pure financial variables omit pertinent information about future equityreturns associated with macroeconomic variables. Factors extracted only from the macroeco-nomic variables are statistically significant in the IS exercise. This suggests predictable businesscycle variation in the expected Canadian equity premium that is not captured by the yield curveor valuation ratios. Overall, the dynamic factor model appears to be an interesting method tocapture the relation between the Canadian equity premium and the informations variables.

33

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37

APPENDIX

Figure 5: Estimated factor 1

.15

.15

.15.16

.16

.16.17

.17

.17.18

.18

.18.19

.19

.19Estimated of Factor 1Es

timat

ed o

f Fac

tor 1

Estimated of Factor 11975m1

1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1

Note: The figure shows the monthly estimated factor 1 using principal components analysis (PCA). The sampleperiod is 1976m2-2011m12.


-.05

-.05

-.050

0

0.05

.05

.05.1

.1

.1.15

.15

.15Estimated of Factor 2

Estim

ated

of F

acto

r 2


1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1


38


-.05

-.05

-.050

0

0.05

.05

.05.1

.1


Estim

ated

of F

acto

r 3


1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1



-.1

-.1

-.1-.05

-.05

-.050

0

0.05

.05


Estim

ated

of F

acto

r 4


1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1

Note: The figure shows the monthly estimated factor 4 using principal components analysis (PCA). The sampleperiod is 1976m2-2011m12

39


-.04

-.04

-.04-.02

-.02

-.020

0

0.02

.02

.02.04

.04


Estim

ated

of F

acto

r 5


1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1



-.04

-.04

-.04-.02

-.02

-.020

0

0.02

.02

.02.04

.04


Estim

ated

of F

acto

r 6


1975m1

1975m11980m1

1980m1

1980m11985m1

1985m1

1985m11990m1

1990m1

1990m11995m1

1995m1

1995m12000m1

2000m1

2000m12005m1

2005m1

2005m12010m1

2010m1

2010m1


40

Data Appendix:

Table 9: Description of the dataThis table lists the short name of each series, its mnemonic (the series label used in the source database), thetransformation applied to the series, a brief data description and the category in which a serie is included. Allseries are from Cansim, the Global Insights Basic Economics Database and the Canadian Financial MarketsResearch Centre (CFMRC). In the transformation column, ln denotes logarithm, ∆ ln and ∆2 ln denote the firstand second difference of the logarithm.

No. Gr. Trans. Description

Panel A: Financial block1 1 No Equity Premium2 1 No DY Lag3 1 No DYf lag4 1 No DP lag5 1 No PE lag6 1 No EP lag7 1 No Previous Equity Premium8 1 No VOLG lag9 1 No VOLGd lag10 1 No SVAR lag11 1 No ISSUE lag12 1 No TBILL lag13 1 No TBILLv lag14 1 No TBILLr lag15 1 No LTGOV lag16 1 No LTGOVv lag17 1 No LTGOVr lag18 1 No TERM lag19 1 No CREDIT lag20 1 No CREDITs lag21 1 No CREDITr lag22 1 ∆ ln v122485: Government of Canada marketable bonds, average yield: 3-5 year23 1 ∆ ln v122486: Government of Canada marketable bonds, average yield: 5-10 year24 1 ∆ ln v122487: Government of Canada marketable bonds, average yield: over 10 y.25 1 ∆ ln v122491: Prime corporate paper rate: 3 month26 1 ∆ ln v122493: Chartered bank - non chequable savings deposits27 1 ∆ ln v122495: Chartered bank administered interest rates - prime business28 1 ∆ ln v122497: Average residential mortgage lending rate: 5 year29 1 ∆ ln v122504: Bankers’ acceptances: 1 month

41


30 1 ∆ ln v122509: Prime corporate paper rate: 1 month31 1 ∆ ln v122510: First coupon of Canada Savings Bonds32 1 ∆ ln v122514: Overnight money market financing, 7 day average33 1 ∆ ln v122515: Chartered bank - 5 year personal fixed term34 1 ∆ ln v122521: Chartered bank - conventional mortgage: 5 year35 1 ∆ ln v122530: Financial market statistics; Bank rate36 1 ∆ ln v122531: Treasury bills: 3 month37 1 ∆ ln v122541: Treasury bill auction - average yields: 3 month38 1 ∆ ln v122544: Selected Government of Canada benchmark bond yields: long term39 1 ∆ ln v122558: Government of Canada marketable bonds, average yield: 1-3 year40 1 ∆ ln v122620: S&P/TSX Composite Index, close

Panel B: Economic block41 2 ∆2 PRIMEr lag42 2 ∆2 LEAD lag43 2 ∆2 FX lag44 2 ∆2 FXv lag45 2 ∆2 FXr lag46 2 ∆ ENERG lag47 2 ∆ MMG lag48 2 No GDP lag49 2 No UNEMP lag50 2 No MONEYG lag51 2 No GDPG lag52 2 No PRIME lag53 2 ∆ ln v2062815: Unemployment rate (rate); Both sexes; 15 years and over54 2 ∆ ln v2062816: Participation rate (rate); Both sexes; 15 years and over55 2 ∆ ln v2062817: Employment rate (rate); Both sexes; 15 years and over56 2 ∆ ln v2062824: Unemployment rate (rate); Males; 15 years and over57 2 ∆ ln v2062833: Unemployment rate (rate); Females; 15 years and over58 2 ∆ ln v2062834: Participation rate (rate); Females; 15 years and over59 2 ∆ ln v2062842: Unemployment rate (rate); Both sexes; 15 to 24 years60 2 ∆ ln v2062843: Participation rate (rate); Both sexes; 15 to 24 years61 2 ∆ ln v2062844: Employment rate (rate); Both sexes; 15 to 24 years62 2 ∆ ln v2062897: Participation rate (rate); Both sexes; 20 to 24 years63 2 ∆ ln v2062952: Employment rate (rate); Both sexes; 25 to 54 years64 2 ∆ ln v2091597 - Canada; Unemployment rate (rate); Females; 15 y. and over

42


65 2 ∆ ln v2091177 - Canada; Unemployment rate (rate); Both sexes; 15 y. and over66 2 ∆ ln v2091178 - Canada; Unemployment rate (rate); Both sexes; 15 to 24 years67 2 ∆ ln v2091184 - Canada; Unemployment rate (rate); Both sexes; 15 to 64 years68 2 ∆ ln v2091198 - Canada; Participation rate (rate); Both sexes; 15 years and over69 2 ∆ ln v2091205 - Canada; Participation rate (rate); Both sexes; 15 to 64 years70 2 ∆ ln v2091219 - Canada; Employment rate (rate); Both sexes; 15 years and over71 2 ∆ ln v2091387 - Canada; Unemployment rate (rate); Males; 15 years and over72 2 ∆ ln v2091226 - Canada; Employment rate (rate); Both sexes; 15 to 64 years73 2 ∆ ln v2091226 - Canada; Employment rate (rate); Both sexes; 15 to 64 years74 2 ∆2 ln v122642: monthly average or average credit measure; Canada75 2 ∆2 ln v2062809: Population; Both sexes; 15 years and over76 2 ∆2 ln v2062810: Labour force; Both sexes; 15 years and over77 2 ∆2 ln v2062811: Employment; Both sexes; 15 years and over78 2 ∆2 ln v2062812: Employment full-time; Both sexes; 15 years and over79 2 ∆2 ln v2062813: Employment part-time; Both sexes; 15 years and over80 2 ∆2 ln v2062814: Unemployment; Both sexes; 15 years and over81 2 ∆2 ln v2062818: Population; Males; 15 years and over82 2 ∆2 ln v2062819: Labour force; Males; 15 years and over83 2 ∆2 ln v2062820: Employment; Males; 15 years and over84 2 ∆2 ln v2062821: Employment full-time; Males; 15 years and over85 2 ∆2 ln v2062822: Employment part-time; Males; 15 years and over86 2 ∆2 ln v2062823: Unemployment; Males; 15 years and over87 2 ∆2 ln v2062828: Labour force; Females; 15 years and over88 2 ∆2 ln v2062829: Employment; Females; 15 years and over89 2 ∆2 ln v2062830: Employment full-time; Females; 15 years and over90 2 ∆2 ln v2062831: Employment part-time; Females; 15 years and over91 2 ∆2 ln v2062832: Unemployment; Females; 15 years and over92 2 ∆2 ln v2062836: Population; Both sexes; 15 to 24 years93 2 ∆2 ln v2062837: Labour force; Both sexes; 15 to 24 years94 2 ∆2 ln v2062838: Employment; Both sexes; 15 to 24 years95 2 ∆2 ln v2062845: Population; Males; 15 to 24 years96 2 ∆2 ln v2062846: Labour force; Males; 15 to 24 years97 2 ∆2 ln v2062847: Employment; Males; 15 to 24 years98 2 ∆2 ln v2062854: Population; Females; 15 to 24 years99 2 ∆2 ln v2062855: Labour force; Females; 15 to 24 years100 2 ∆2 ln v2062856: Employment; Females; 15 to 24 years

43


101 2 ∆2 ln v2062865: Employment; Both sexes; 15 to 19 years102 2 ∆2 ln v2062892: Employment; Both sexes; 20 to 24 years103 2 ∆2 ln v2062919: Employment; Both sexes; 25 years and over104 2 ∆2 ln v2062946: Employment; Both sexes; 25 to 54 years105 2 ∆2 ln v2062953: Population; Males; 25 to 54 years106 2 ∆2 ln v2062954: Labour force; Males; 25 to 54 years107 2 ∆2 ln v2062955: Employment; Males; 25 to 54 years108 2 ∆2 ln v2062973: Employment; Both sexes; 55 years and over109 2 ∆2 ln v2062980: Population; Males; 55 years and over110 2 ∆2 ln v2062981: Labour force; Males; 55 years and over111 2 ∆2 ln v2062982: Employment; Males; 55 years and over112 2 ∆2 ln v2062989: Population; Females; 55 years and over113 2 ∆2 ln v2062990: Labour force; Females; 55 years and over114 2 ∆2 ln v2062991: Employment; Females; 55 years and over115 2 ∆2 ln v2064891: Employment full-time; Both sexes; 15 years and over116 2 ∆2 ln v2066969: Public sector employees; Both sexes117 2 ∆2 ln v2066970: Private sector employees; Both sexes118 2 ∆2 ln v2066971: Self-employed; Both sexes119 2 ∆2 ln v2067134: Public sector employees; Both sexes120 2 ∆2 ln v2067135: Private sector employees; Both sexes121 2 ∆2 ln v2067136: Self-employed; Both sexes122 2 ∆2 ln v2091030: Population; Both sexes; 15 years and over123 2 ∆2 ln v2091031: Population; Both sexes; 15 to 24 years124 2 ∆2 ln v2091033: Population; Both sexes; 25 to 44 years125 2 ∆2 ln v2091035: Population; Both sexes; 25 to 54 years126 2 ∆2 ln v2091037: Both sexes; 15 to 64 years127 2 ∆2 ln v2091048 :Population; Both sexes; 65 years and over128 2 ∆2 ln v2091051 :Labour force; Both sexes; 15 years and over129 2 ∆2 ln v2091052 :Labour force; Both sexes; 15 to 24 years130 2 ∆2 ln v2091054 :Labour force; Both sexes; 25 to 44 years131 2 ∆2 ln v2091058 :Labour force; Both sexes; 15 to 64 years132 2 ∆2 ln v2091072 :Employment; Both sexes; 15 years and over133 2 ∆2 ln v2091073 :Employment; Both sexes; 15 to 24 years134 2 ∆2 ln v2091075 :Employment; Both sexes; 25 to 44 years135 2 ∆2 ln v2091079 :Employment; Both sexes; 15 to 64 years

44

Description Economic Data


136 2 ∆2 ln v2091093 :Full-time employment; Both sexes; 15 years and over137 2 ∆2 ln v2091114 :Part-time employment; Both sexes; 15 years and over138 2 ∆2 ln v2091135 :Unemployment; Both sexes; 15 years and over139 2 ∆2 ln v2091240 :Population; Males; 15 years and over140 2 ∆2 ln v2091241 :Population; Males; 15 to 24 years141 2 ∆2 ln v2091242 :Population; Males; 25 years and over142 2 ∆2 ln v2091247 :Population; Males; 15 to 64 years143 2 ∆2 ln v2091261 :Labour force; Males; 15 years and over144 2 ∆2 ln v2091262 :Labour force; Males; 15 to 24 years145 2 ∆2 ln v2091263 :Labour force; Males; 25 years and over146 2 ∆2 ln v2091268 :Labour force; Males; 15 to 64 years147 2 ∆2 ln v2091270 :Labour force; Males; 20 to 24 years148 2 ∆2 ln v2091279 :Labour force; Males; 65 years and over149 2 ∆2 ln v2091282 :Employment; Males; 15 years and over150 2 ∆2 ln v2091283 :Employment; Males; 15 to 24 years151 2 ∆2 ln v2091284 :Employment; Males; 25 years and over152 2 ∆2 ln v2091289 :Employment; Males; 15 to 64 years153 2 ∆2 ln v2091291 :Employment; Males; 20 to 24 years154 2 ∆2 ln v2091300 :Employment; Males; 65 years and over155 2 ∆2 ln v2091303 :Full-time employment; Males; 15 years and over156 2 ∆2 ln v2091324 :Part-time employment; Males; 15 years and over157 2 ∆2 ln v2091345 :Unemployment; Males; 15 years and over158 2 ∆2 ln v2091450 :Population; Females; 15 years and over159 2 ∆2 ln v2091451 :Population; Females; 15 to 24 years160 2 ∆2 ln v2091452 :Population; Females; 25 years and over161 2 ∆2 ln v2091457 :Population; Females; 15 to 64 years162 2 ∆2 ln v2091471 :Labour force; Females; 15 years and over163 2 ∆2 ln v2091472 :Labour force; Females; 15 to 24 years164 2 ∆2 ln v2091473: Labour force; Females; 25 years and over (x 1,000)165 2 ∆2 ln v2091478: Labour force; Females; 15 to 64 years (x 1,000)166 2 ∆2 ln v2091480: Labour force; Females; 20 to 24 years (x 1,000)167 2 ∆2 ln v2091489: Labour force; Females; 65 years and over (x 1,000)168 2 ∆2 ln v2091492: Employment; Females; 15 years and over (x 1,000)169 2 ∆2 ln v2091493: Employment; Females; 15 to 24 years (x 1,000)

45


170 2 ∆2 ln v2091494: Employment; Females; 25 years and over (x 1,000)171 2 ∆2 ln v2091499: Employment; Females; 15 to 64 years (x 1,000)172 2 ∆2 ln v2091510: Employment; Females; 65 years and over (x 1,000)173 2 ∆2 ln v2091513: Full-time employment; Females; 15 years and over (x 1,000)174 2 ∆2 ln v2091534: Part-time employment; Females; 15 years and over (x 1,000)

Panel C: Price block175 3 No INF lag176 3 ∆2 ln Consumer Price Index All-items CPI177 3 ∆2 ln CPI Food including non-alcoholic beverages178 3 ∆2 ln CPI Transportation179 3 ∆2 ln CPI Gasoline180 3 ∆2 ln CPI Health and personal care181 3 ∆2 ln CPI Recreation, education and reading182 3 ∆2 ln CPI Alcoholic beverages and tobacco products CPI183 3 ∆2 ln Core Consumer Price Index (CPI) (Bank of Canada definition)184 3 ∆2 ln CPI All-items excluding food and energy185 3 ∆2 ln CPI All-items excluding energy186 3 ∆2 ln CPI Electricity, natural gas, fuel oil and other fuels187 3 ∆2 ln CPI Goods are physical or tangible commodities188 3 ∆2 ln CPI Valuable work performed by an individual or organization

46

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